Absolute magnitude

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<ul id="toc-Stars_and_galaxies-sublist" class="vector-toc-list"> <li id="toc-Apparent_magnitude" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Apparent_magnitude"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Apparent magnitude</span> </div> </a> <ul id="toc-Apparent_magnitude-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bolometric_magnitude" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bolometric_magnitude"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Bolometric magnitude</span> </div> </a> <ul id="toc-Bolometric_magnitude-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solar_System_bodies_(H)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solar_System_bodies_(H)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Solar System bodies (<span><i>H</i></span>)</span> </div> </a> <button aria-controls="toc-Solar_System_bodies_(H)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Solar System bodies (<span><i>H</i></span>) subsection</span> </button> <ul id="toc-Solar_System_bodies_(H)-sublist" class="vector-toc-list"> <li id="toc-Apparent_magnitude_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Apparent_magnitude_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Apparent magnitude</span> </div> </a> <ul id="toc-Apparent_magnitude_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximations_for_phase_integral_q(α)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approximations_for_phase_integral_q(α)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Approximations for phase integral <span><i>q</i>(<i>α</i>)</span></span> </div> </a> <ul id="toc-Approximations_for_phase_integral_q(α)-sublist" class="vector-toc-list"> <li id="toc-Planets_as_diffuse_spheres" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Planets_as_diffuse_spheres"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Planets as diffuse spheres</span> </div> </a> <ul id="toc-Planets_as_diffuse_spheres-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-More_advanced_models" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#More_advanced_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>More advanced models</span> </div> </a> <ul id="toc-More_advanced_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asteroids" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Asteroids"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Asteroids</span> </div> </a> <ul id="toc-Asteroids-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cometary_magnitudes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cometary_magnitudes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Cometary magnitudes</span> </div> </a> <ul id="toc-Cometary_magnitudes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Meteors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Meteors"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Meteors</span> </div> </a> <ul id="toc-Meteors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Absolute magnitude</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 71 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-71" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">71 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Absolute_magnitude" title="Absolute magnitude – Afrikaans" lang="af" hreflang="af" data-title="Absolute magnitude" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Absolute_Helligkeit" title="Absolute Helligkeit – Alemannic" lang="gsw" hreflang="gsw" data-title="Absolute Helligkeit" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%AF%D8%B1_%D9%85%D8%B7%D9%84%D9%82" title="قدر مطلق – Arabic" lang="ar" hreflang="ar" data-title="قدر مطلق" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Magnit%C3%BA_absoluta" title="Magnitú absoluta – Asturian" lang="ast" hreflang="ast" data-title="Magnitú absoluta" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Choa%CC%8Dt-t%C3%B9i_t%C3%A9ng-kip" title="Choa̍t-tùi téng-kip – Minnan" lang="nan" hreflang="nan" data-title="Choa̍t-tùi téng-kip" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%B0%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D0%B7%D0%BE%D1%80%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D1%96%D1%87%D1%8B%D0%BD%D1%8F" title="Абсалютная зорная велічыня – Belarusian" lang="be" hreflang="be" data-title="Абсалютная зорная велічыня" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%B0%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D0%B7%D0%BE%D1%80%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D1%96%D1%87%D1%8B%D0%BD%D1%8F" title="Абсалютная зорная велічыня – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Абсалютная зорная велічыня" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BD%D0%B0_%D0%B7%D0%B2%D0%B5%D0%B7%D0%B4%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Абсолютна звездна величина – Bulgarian" lang="bg" hreflang="bg" data-title="Абсолютна звездна величина" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%9A%E0%BD%B4%E0%BD%82%E0%BD%A6%E0%BC%8B%E0%BD%90%E0%BD%B4%E0%BD%96%E0%BC%8B%E0%BD%82%E0%BD%A6%E0%BD%A3%E0%BC%8B%E0%BD%86%E0%BC%8D" title="ཚུགས་ཐུབ་གསལ་ཆ། – Tibetan" lang="bo" hreflang="bo" data-title="ཚུགས་ཐུབ་གསལ་ཆ།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Apsolutna_magnituda" title="Apsolutna magnituda – Bosnian" lang="bs" hreflang="bs" data-title="Apsolutna magnituda" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Magnitud_absoluta" title="Magnitud absoluta – Catalan" lang="ca" hreflang="ca" data-title="Magnitud absoluta" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C3%87%C4%83%D0%BB%D1%82%C4%83%D1%80%D0%BB%D0%B0_%D0%B0%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BB%C4%83_%D0%BA%D0%B0%D0%BF" title="Çăлтăрла абсолютлă кап – Chuvash" lang="cv" hreflang="cv" data-title="Çăлтăрла абсолютлă кап" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Absolutn%C3%AD_hv%C4%9Bzdn%C3%A1_velikost" title="Absolutní hvězdná velikost – Czech" lang="cs" hreflang="cs" data-title="Absolutní hvězdná velikost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Absolut_st%C3%B8rrelsesklasse" title="Absolut størrelsesklasse – Danish" lang="da" hreflang="da" data-title="Absolut størrelsesklasse" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Absolute_Helligkeit" title="Absolute Helligkeit – German" lang="de" hreflang="de" data-title="Absolute Helligkeit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Absoluutne_t%C3%A4hesuurus" title="Absoluutne tähesuurus – Estonian" lang="et" hreflang="et" data-title="Absoluutne tähesuurus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%80%CF%8C%CE%BB%CF%85%CF%84%CE%BF_%CE%BC%CE%AD%CE%B3%CE%B5%CE%B8%CE%BF%CF%82" title="Απόλυτο μέγεθος – Greek" lang="el" hreflang="el" data-title="Απόλυτο μέγεθος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Magnitud_absoluta" title="Magnitud absoluta – Spanish" lang="es" hreflang="es" data-title="Magnitud absoluta" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Absoluta_magnitudo" title="Absoluta magnitudo – Esperanto" lang="eo" hreflang="eo" data-title="Absoluta magnitudo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Magnitude_absolutu" title="Magnitude absolutu – Basque" lang="eu" hreflang="eu" data-title="Magnitude absolutu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%AF%D8%B1_%D9%85%D8%B7%D9%84%D9%82_(%D8%A7%D8%AE%D8%AA%D8%B1%D8%B4%D9%86%D8%A7%D8%B3%DB%8C)" title="قدر مطلق (اخترشناسی) – Persian" lang="fa" hreflang="fa" data-title="قدر مطلق (اخترشناسی)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Magnitude_absolue" title="Magnitude absolue – French" lang="fr" hreflang="fr" data-title="Magnitude absolue" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Magnitude_absoluta" title="Magnitude absoluta – Galician" lang="gl" hreflang="gl" data-title="Magnitude absoluta" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%88%EB%8C%80%EB%93%B1%EA%B8%89" title="절대등급 – Korean" lang="ko" hreflang="ko" data-title="절대등급" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D6%81%D5%A1%D6%80%D5%B1%D5%A1%D5%AF_%D5%A1%D5%BD%D5%BF%D5%B2%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Բացարձակ աստղային մեծություն – Armenian" lang="hy" hreflang="hy" data-title="Բացարձակ աստղային մեծություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A4%AA%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%B7_%E0%A4%95%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A4%BF%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="निरपेक्ष कांतिमान – Hindi" lang="hi" hreflang="hi" data-title="निरपेक्ष कांतिमान" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Apsolutna_magnituda" title="Apsolutna magnituda – Croatian" lang="hr" hreflang="hr" data-title="Apsolutna magnituda" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Magnitudo_mutlak" title="Magnitudo mutlak – Indonesian" lang="id" hreflang="id" data-title="Magnitudo mutlak" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Magnitudine_assoluta" title="Magnitudine assoluta – Italian" lang="it" hreflang="it" data-title="Magnitudine assoluta" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%91%D7%94%D7%99%D7%A8%D7%95%D7%AA_%D7%9E%D7%95%D7%97%D7%9C%D7%98%D7%AA" title="בהירות מוחלטת – Hebrew" lang="he" hreflang="he" data-title="בהירות מוחלטת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mwangaza_halisi" title="Mwangaza halisi – Swahili" lang="sw" hreflang="sw" data-title="Mwangaza halisi" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Mayitid_absol" title="Mayitid absol – Haitian Creole" lang="ht" hreflang="ht" data-title="Mayitid absol" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Absol%C5%ABtais_spo%C5%BEums" title="Absolūtais spožums – Latvian" lang="lv" hreflang="lv" data-title="Absolūtais spožums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Absolut_Magnitude" title="Absolut Magnitude – Luxembourgish" lang="lb" hreflang="lb" data-title="Absolut Magnitude" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Absoliutinis_ry%C5%A1kis" title="Absoliutinis ryškis – Lithuanian" lang="lt" hreflang="lt" data-title="Absoliutinis ryškis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Abszol%C3%BAt_f%C3%A9nyess%C3%A9g" title="Abszolút fényesség – Hungarian" lang="hu" hreflang="hu" data-title="Abszolút fényesség" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D0%BF%D1%81%D0%BE%D0%BB%D1%83%D1%82%D0%BD%D0%B0_%D1%95%D0%B2%D0%B5%D0%B7%D0%B4%D0%B5%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Апсолутна ѕвездена величина – Macedonian" lang="mk" hreflang="mk" data-title="Апсолутна ѕвездена величина" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%87%E0%B4%B5%E0%B4%B2%E0%B4%95%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B4%AE%E0%B4%BE%E0%B4%A8%E0%B4%82" title="കേവലകാന്തിമാനം – Malayalam" lang="ml" hreflang="ml" data-title="കേവലകാന്തിമാനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A4%AA%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%B7_%E0%A4%A6%E0%A5%83%E0%A4%B6%E0%A5%8D%E0%A4%AF%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4" title="निरपेक्ष दृश्यप्रत – Marathi" lang="mr" hreflang="mr" data-title="निरपेक्ष दृश्यप्रत" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Magnitud_mutlak" title="Magnitud mutlak – Malay" lang="ms" hreflang="ms" data-title="Magnitud mutlak" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Absolute_helderheid" title="Absolute helderheid – Dutch" lang="nl" hreflang="nl" data-title="Absolute helderheid" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B5%B6%E5%AF%BE%E7%AD%89%E7%B4%9A" title="絶対等級 – Japanese" lang="ja" hreflang="ja" data-title="絶対等級" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Absolutt_st%C3%B8rrelsesklasse" title="Absolutt størrelsesklasse – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Absolutt størrelsesklasse" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Absolutt_storleiksklasse" title="Absolutt storleiksklasse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Absolutt storleiksklasse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Magnitud_absoluda" title="Magnitud absoluda – Occitan" lang="oc" hreflang="oc" data-title="Magnitud absoluda" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mutlaq_yulduz_kattaligi" title="Mutlaq yulduz kattaligi – Uzbek" lang="uz" hreflang="uz" data-title="Mutlaq yulduz kattaligi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%82%E0%A8%B0%E0%A8%A8_%E0%A8%9A%E0%A8%AE%E0%A8%95" title="ਪੂਰਨ ਚਮਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਪੂਰਨ ਚਮਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%A8%D8%B4%D9%BE%DA%93%D9%87_%DA%81%D9%84%D8%A7" title="بشپړه ځلا – Pashto" lang="ps" hreflang="ps" data-title="بشپړه ځلا" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Absolutna_wielko%C5%9B%C4%87_gwiazdowa" title="Absolutna wielkość gwiazdowa – Polish" lang="pl" hreflang="pl" data-title="Absolutna wielkość gwiazdowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Magnitude_absoluta" title="Magnitude absoluta – Portuguese" lang="pt" hreflang="pt" data-title="Magnitude absoluta" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Magnitudine_absolut%C4%83" title="Magnitudine absolută – Romanian" lang="ro" hreflang="ro" data-title="Magnitudine absolută" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D0%B7%D0%B2%D1%91%D0%B7%D0%B4%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Абсолютная звёздная величина – Russian" lang="ru" hreflang="ru" data-title="Абсолютная звёздная величина" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Absolute_magnitude" title="Absolute magnitude – Scots" lang="sco" hreflang="sco" data-title="Absolute magnitude" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Magnit%C3%B9dini_assuluta" title="Magnitùdini assuluta – Sicilian" lang="scn" hreflang="scn" data-title="Magnitùdini assuluta" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple badge-Q70893996 mw-list-item" title=""><a href="https://simple.wikipedia.org/wiki/Absolute_magnitude" title="Absolute magnitude – Simple English" lang="en-simple" hreflang="en-simple" data-title="Absolute magnitude" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Absol%C3%BAtna_hviezdna_ve%C4%BEkos%C5%A5" title="Absolútna hviezdna veľkosť – Slovak" lang="sk" hreflang="sk" data-title="Absolútna hviezdna veľkosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Absolutni_izsev" title="Absolutni izsev – Slovenian" lang="sl" hreflang="sl" data-title="Absolutni izsev" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%BF%D1%81%D0%BE%D0%BB%D1%83%D1%82%D0%BD%D0%B0_%D0%B7%D0%B2%D0%B5%D0%B7%D0%B4%D0%B0%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Апсолутна звездана величина – Serbian" lang="sr" hreflang="sr" data-title="Апсолутна звездана величина" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Apsolutna_magnituda" title="Apsolutna magnituda – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Apsolutna magnituda" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Magnitudo_mutlak" title="Magnitudo mutlak – Sundanese" lang="su" hreflang="su" data-title="Magnitudo mutlak" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Absolut_magnitud" title="Absolut magnitud – Swedish" lang="sv" hreflang="sv" data-title="Absolut magnitud" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Lubusang_kalakhan" title="Lubusang kalakhan – Tagalog" lang="tl" hreflang="tl" data-title="Lubusang kalakhan" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Azal_amagdez" title="Azal amagdez – Kabyle" lang="kab" hreflang="kab" data-title="Azal amagdez" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82_%D0%B9%D0%BE%D0%BB%D0%B4%D1%8B%D0%B7%D1%87%D0%B0_%D0%B7%D1%83%D1%80%D0%BB%D1%8B%D0%BA" title="Абсолют йолдызча зурлык – Tatar" lang="tt" hreflang="tt" data-title="Абсолют йолдызча зурлык" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%82%E0%B8%8A%E0%B8%95%E0%B8%B4%E0%B8%A1%E0%B8%B2%E0%B8%95%E0%B8%A3%E0%B8%AA%E0%B8%B1%E0%B8%A1%E0%B8%9A%E0%B8%B9%E0%B8%A3%E0%B8%93%E0%B9%8C" title="โชติมาตรสัมบูรณ์ – Thai" lang="th" hreflang="th" data-title="โชติมาตรสัมบูรณ์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mutlak_parlakl%C4%B1k" title="Mutlak parlaklık – Turkish" lang="tr" hreflang="tr" data-title="Mutlak parlaklık" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BD%D0%B0_%D0%B7%D0%BE%D1%80%D1%8F%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Абсолютна зоряна величина – Ukrainian" lang="uk" hreflang="uk" data-title="Абсолютна зоряна величина" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%E1%BA%A5p_sao_tuy%E1%BB%87t_%C4%91%E1%BB%91i" title="Cấp sao tuyệt đối – Vietnamese" lang="vi" hreflang="vi" data-title="Cấp sao tuyệt đối" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BB%9D%E5%AF%B9%E6%98%9F%E7%AD%89" title="绝对星等 – Wu" lang="wuu" hreflang="wuu" data-title="绝对星等" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B5%95%E5%B0%8D%E6%98%9F%E7%AD%89" title="絕對星等 – Cantonese" lang="yue" hreflang="yue" data-title="絕對星等" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Measure of the luminosity of celestial objects</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the brightness of stars. For the science fiction magazine, see <a href="/wiki/Absolute_Magnitude_(magazine)" title="Absolute Magnitude (magazine)">Absolute Magnitude (magazine)</a>.</div> <p class="mw-empty-elt"> </p><p>In <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <b>absolute magnitude</b> (<b><span class="texhtml mvar" style="font-style:italic;">M</span></b>) is a measure of the <a href="/wiki/Luminosity" title="Luminosity">luminosity</a> of a <a href="/wiki/Celestial_object" class="mw-redirect" title="Celestial object">celestial object</a> on an inverse <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">logarithmic</a> <a href="/wiki/Magnitude_(astronomy)" title="Magnitude (astronomy)">astronomical magnitude</a> scale; the more luminous (intrinsically bright) an object, the lower its magnitude number. An object's absolute magnitude is defined to be equal to the <a href="/wiki/Apparent_magnitude" title="Apparent magnitude">apparent magnitude</a> that the object would have if it were viewed from a distance of exactly 10 <a href="/wiki/Parsec" title="Parsec">parsecs</a> (32.6 <a href="/wiki/Light-year" title="Light-year">light-years</a>), without <a href="/wiki/Extinction_(astronomy)" title="Extinction (astronomy)">extinction</a> (or dimming) of its light due to absorption by <a href="/wiki/Interstellar_medium" title="Interstellar medium">interstellar matter</a> and <a href="/wiki/Cosmic_dust" title="Cosmic dust">cosmic dust</a>. By hypothetically placing all objects at a standard reference distance from the observer, their luminosities can be directly compared among each other on a magnitude scale. For <a href="/wiki/Solar_System" title="Solar System">Solar System</a> bodies that shine in reflected light, a different definition of <a class="mw-selflink-fragment" href="#Solar_System_bodies_(H)">absolute magnitude (H)</a> is used, based on a standard reference distance of one <a href="/wiki/Astronomical_unit" title="Astronomical unit">astronomical unit</a>. </p><p>Absolute magnitudes of stars generally range from approximately −10 to +20. The absolute magnitudes of galaxies can be much lower (brighter). </p><p>The more luminous an object, the smaller the numerical value of its absolute magnitude. A difference of 5 magnitudes between the absolute magnitudes of two objects corresponds to a ratio of 100 in their luminosities, and a difference of n magnitudes in absolute magnitude corresponds to a luminosity ratio of 100<sup>n/5</sup>. For example, a star of absolute magnitude M<sub>V</sub> = 3.0 would be 100 times as luminous as a star of absolute magnitude M<sub>V</sub> = 8.0 as measured in the V filter band. The <a href="/wiki/Sun" title="Sun">Sun</a> has absolute magnitude M<sub>V</sub> = +4.83.<sup id="cite_ref-SunAbs_1-0" class="reference"><a href="#cite_note-SunAbs-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Highly luminous objects can have negative absolute magnitudes: for example, the <a href="/wiki/Milky_Way" title="Milky Way">Milky Way</a> galaxy has an absolute <a href="/wiki/UBV_photometric_system" title="UBV photometric system">B magnitude</a> of about −20.8.<sup id="cite_ref-Karachentsev_2-0" class="reference"><a href="#cite_note-Karachentsev-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>As with all astronomical <a href="/wiki/Magnitude_(astronomy)" title="Magnitude (astronomy)">magnitudes</a>, the absolute magnitude can be specified for different <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> ranges corresponding to specified <a href="/wiki/Filter_(optics)" class="mw-redirect" title="Filter (optics)">filter</a> bands or <a href="/wiki/Passband" title="Passband">passbands</a>; for stars a commonly quoted absolute magnitude is the <b>absolute visual magnitude</b>, which uses the visual (V) band of the spectrum (in the <a href="/wiki/UBV_photometric_system" title="UBV photometric system">UBV photometric system</a>). Absolute magnitudes are denoted by a capital M, with a subscript representing the filter band used for measurement, such as M<sub>V</sub> for absolute magnitude in the V band. </p><p>An object's absolute <i>bolometric</i> magnitude (M<sub>bol</sub>) represents its total <a href="/wiki/Luminosity" title="Luminosity">luminosity</a> over all <a href="/wiki/Wavelengths" class="mw-redirect" title="Wavelengths">wavelengths</a>, rather than in a single filter band, as expressed on a logarithmic magnitude scale. To convert from an absolute magnitude in a specific filter band to absolute bolometric magnitude, a <a href="/wiki/Bolometric_correction" title="Bolometric correction">bolometric correction</a> (BC) is applied.<sup id="cite_ref-Flower1996_3-0" class="reference"><a href="#cite_note-Flower1996-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Stars_and_galaxies">Stars and galaxies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=1" title="Edit section: Stars and galaxies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light-years, 308.57 petameters or 308.57 <a href="/wiki/Orders_of_magnitude_(numbers)#1012" title="Orders of magnitude (numbers)">trillion</a> kilometres). A star at 10 parsecs has a <a href="/wiki/Parallax" title="Parallax">parallax</a> of 0.1″ (100 <a href="/wiki/Minute_of_arc" class="mw-redirect" title="Minute of arc">milliarcseconds</a>). Galaxies (and other <a href="/wiki/Nebula" title="Nebula">extended objects</a>) are much larger than 10 parsecs; their light is radiated over an extended patch of sky, and their overall brightness cannot be directly observed from relatively short distances, but the same convention is used. A galaxy's magnitude is defined by measuring all the light radiated over the entire object, treating that integrated brightness as the brightness of a single point-like or star-like source, and computing the magnitude of that point-like source as it would appear if observed at the standard 10 parsecs distance. Consequently, the absolute magnitude of any object <i>equals</i> the apparent magnitude it <i>would have</i> if it were 10 parsecs away. </p><p>Some stars visible to the naked eye have such a low absolute magnitude that they would appear bright enough to outshine the <a href="/wiki/Planet" title="Planet">planets</a> and cast shadows if they were at 10 parsecs from the Earth. Examples include <a href="/wiki/Rigel" title="Rigel">Rigel</a> (−7.8), <a href="/wiki/Deneb" title="Deneb">Deneb</a> (−8.4), <a href="/wiki/Zeta_Puppis" title="Zeta Puppis">Naos</a> (−6.2), and <a href="/wiki/Betelgeuse" title="Betelgeuse">Betelgeuse</a> (−5.8). For comparison, <a href="/wiki/Sirius" title="Sirius">Sirius</a> has an absolute magnitude of only 1.4, which is still brighter than the <a href="/wiki/Sun" title="Sun">Sun</a>, whose absolute visual magnitude is 4.83. The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75.<sup id="cite_ref-Strobel_4-0" class="reference"><a href="#cite_note-Strobel-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Casagrande_5-0" class="reference"><a href="#cite_note-Casagrande-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Absolute magnitudes of stars generally range from approximately −10 to +20. The absolute magnitudes of galaxies can be much lower (brighter). For example, the giant <a href="/wiki/Elliptical_galaxy_M87" class="mw-redirect" title="Elliptical galaxy M87">elliptical galaxy M87</a> has an absolute magnitude of −22 (i.e. as bright as about 60,000 stars of magnitude −10). Some <a href="/wiki/Active_galactic_nuclei" class="mw-redirect" title="Active galactic nuclei">active galactic nuclei</a> (<a href="/wiki/Quasars" class="mw-redirect" title="Quasars">quasars</a> like <a href="/wiki/CTA-102" title="CTA-102">CTA-102</a>) can reach absolute magnitudes in excess of −32, making them the most luminous persistent objects in the observable universe, although these objects can vary in brightness over astronomically short timescales. At the extreme end, the optical afterglow of the gamma ray burst <a href="/wiki/GRB_080319B" title="GRB 080319B">GRB 080319B</a> reached, according to one paper, an absolute <a href="/wiki/Photometric_system" title="Photometric system">r magnitude</a> brighter than −38 for a few tens of seconds.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Apparent_magnitude">Apparent magnitude</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=2" title="Edit section: Apparent magnitude"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Apparent_magnitude" title="Apparent magnitude">Apparent magnitude</a></div> <p>The Greek astronomer <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a> established a numerical scale to describe the brightness of each star appearing in the sky. The brightest stars in the sky were assigned an apparent magnitude <span class="texhtml"><i>m</i> = 1</span>, and the dimmest stars visible to the naked eye are assigned <span class="texhtml"><i>m</i> = 6</span>.<sup id="cite_ref-Carroll_7-0" class="reference"><a href="#cite_note-Carroll-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitude <span class="texhtml mvar" style="font-style:italic;">M</span> and apparent magnitude <span class="texhtml mvar" style="font-style:italic;">m</span> from any distance <span class="texhtml mvar" style="font-style:italic;">d</span> (in <a href="/wiki/Parsec" title="Parsec">parsecs</a>, with 1 pc = 3.2616 <a href="/wiki/Light-year" title="Light-year">light-years</a>) are related by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100^{\frac {m-M}{5}}={\frac {F_{10}}{F}}=\left({\frac {d}{10\;\mathrm {pc} }}\right)^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>100</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>−</mo> <mi>M</mi> </mrow> <mn>5</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mi>F</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mn>10</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100^{\frac {m-M}{5}}={\frac {F_{10}}{F}}=\left({\frac {d}{10\;\mathrm {pc} }}\right)^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01232643d188516aafe583f1d0757bd7706bf4e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.824ex; height:6.509ex;" alt="{\displaystyle 100^{\frac {m-M}{5}}={\frac {F_{10}}{F}}=\left({\frac {d}{10\;\mathrm {pc} }}\right)^{2},}"></span> where <span class="texhtml mvar" style="font-style:italic;">F</span> is the radiant flux measured at distance <span class="texhtml mvar" style="font-style:italic;">d</span> (in parsecs), <span class="texhtml"><i>F</i><sub>10</sub></span> the radiant flux measured at distance <span class="texhtml">10 pc</span>. Using the <a href="/wiki/Common_logarithm" title="Common logarithm">common logarithm</a>, the equation can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=m-5\log _{10}(d_{\text{pc}})+5=m-5\left(\log _{10}d_{\text{pc}}-1\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>m</mi> <mo>−</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>pc</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>5</mn> <mo>=</mo> <mi>m</mi> <mo>−</mo> <mn>5</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>pc</mtext> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=m-5\log _{10}(d_{\text{pc}})+5=m-5\left(\log _{10}d_{\text{pc}}-1\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51a777c8dbca81cccbfccd6cb2dec1f5bdc63789" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.411ex; height:3.009ex;" alt="{\displaystyle M=m-5\log _{10}(d_{\text{pc}})+5=m-5\left(\log _{10}d_{\text{pc}}-1\right),}"></span> where it is assumed that <a href="/wiki/Extinction_(astronomy)" title="Extinction (astronomy)">extinction from gas and dust</a> is negligible. Typical extinction rates within the <a href="/wiki/Milky_Way" title="Milky Way">Milky Way</a> galaxy are 1 to 2 magnitudes per kiloparsec, when <a href="/wiki/Dark_nebula" title="Dark nebula">dark clouds</a> are taken into account.<sup id="cite_ref-Unsoeld2013_8-0" class="reference"><a href="#cite_note-Unsoeld2013-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>For objects at very large distances (outside the Milky Way) the luminosity distance <span class="texhtml"><i>d</i><sub>L</sub></span> (distance defined using luminosity measurements) must be used instead of <span class="texhtml mvar" style="font-style:italic;">d</span>, because the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean</a> approximation is invalid for distant objects. Instead, <a href="/wiki/General_relativity" title="General relativity">general relativity</a> must be taken into account. Moreover, the <a href="/wiki/Cosmological_redshift" class="mw-redirect" title="Cosmological redshift">cosmological redshift</a> complicates the relationship between absolute and apparent magnitude, because the radiation observed was shifted into the red range of the spectrum. To compare the magnitudes of very distant objects with those of local objects, a <a href="/wiki/K_correction" title="K correction">K correction</a> might have to be applied to the magnitudes of the distant objects. </p><p>The absolute magnitude <span class="texhtml mvar" style="font-style:italic;">M</span> can also be written in terms of the apparent magnitude <span class="texhtml mvar" style="font-style:italic;">m</span> and <a href="/wiki/Stellar_parallax" title="Stellar parallax">stellar parallax</a> <span class="texhtml mvar" style="font-style:italic;">p</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=m+5\left(\log _{10}p+1\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mn>5</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=m+5\left(\log _{10}p+1\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e43be16876666d00464ce3dcbe3cd3132e7f3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.222ex; height:2.843ex;" alt="{\displaystyle M=m+5\left(\log _{10}p+1\right),}"></span> or using apparent magnitude <span class="texhtml mvar" style="font-style:italic;">m</span> and <a href="/wiki/Distance_modulus" title="Distance modulus">distance modulus</a> <span class="texhtml mvar" style="font-style:italic;">μ</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=m-\mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>m</mi> <mo>−</mo> <mi>μ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=m-\mu .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9587f91377e5c0483c4903768bd8df83f244fe6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.47ex; height:2.676ex;" alt="{\displaystyle M=m-\mu .}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Rigel" title="Rigel">Rigel</a> has a visual magnitude <span class="texhtml"><i>m</i><sub>V</sub></span> of 0.12 and distance of about 860 light-years: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {V} }=0.12-5\left(\log _{10}{\frac {860}{3.2616}}-1\right)=-7.0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.12</mn> <mo>−</mo> <mn>5</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>860</mn> <mn>3.2616</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>7.0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {V} }=0.12-5\left(\log _{10}{\frac {860}{3.2616}}-1\right)=-7.0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49c95c9f4f90a7077ab6bbccb6d100cf9ce2c7b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.821ex; height:6.176ex;" alt="{\displaystyle M_{\mathrm {V} }=0.12-5\left(\log _{10}{\frac {860}{3.2616}}-1\right)=-7.0.}"></span> </p><p><a href="/wiki/Vega" title="Vega">Vega</a> has a parallax <span class="texhtml mvar" style="font-style:italic;">p</span> of 0.129″, and an apparent magnitude <span class="texhtml"><i>m</i><sub>V</sub></span> of 0.03: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {V} }=0.03+5\left(\log _{10}{0.129}+1\right)=+0.6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.03</mn> <mo>+</mo> <mn>5</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0.129</mn> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>+</mo> <mn>0.6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {V} }=0.03+5\left(\log _{10}{0.129}+1\right)=+0.6.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598756ff51524af96197f3574baa7857037ace98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.211ex; height:2.843ex;" alt="{\displaystyle M_{\mathrm {V} }=0.03+5\left(\log _{10}{0.129}+1\right)=+0.6.}"></span> </p><p>The <a href="/wiki/Black_Eye_Galaxy" title="Black Eye Galaxy">Black Eye Galaxy</a> has a visual magnitude <span class="texhtml"><i>m</i><sub>V</sub></span> of 9.36 and a distance modulus <span class="texhtml mvar" style="font-style:italic;">μ</span> of 31.06: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {V} }=9.36-31.06=-21.7.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>9.36</mn> <mo>−</mo> <mn>31.06</mn> <mo>=</mo> <mo>−</mo> <mn>21.7.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {V} }=9.36-31.06=-21.7.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cff0c6166ba5f86ae0b52c1928d4ac41e7150a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.776ex; height:2.509ex;" alt="{\displaystyle M_{\mathrm {V} }=9.36-31.06=-21.7.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Bolometric_magnitude">Bolometric magnitude</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=4" title="Edit section: Bolometric magnitude"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Apparent_bolometric_magnitude" class="mw-redirect" title="Apparent bolometric magnitude">Apparent bolometric magnitude</a></div> <p>The absolute <a href="/wiki/Bolometer" title="Bolometer">bolometric</a> magnitude (<span class="texhtml"><i>M</i><sub>bol</sub></span>) takes into account <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> at all <a href="/wiki/Wavelengths" class="mw-redirect" title="Wavelengths">wavelengths</a>. It includes those unobserved due to instrumental <a href="/wiki/Passband" title="Passband">passband</a>, the Earth's atmospheric absorption, and <a href="/wiki/Extinction_(astronomy)" title="Extinction (astronomy)">extinction by interstellar dust</a>. It is defined based on the <a href="/wiki/Luminosity" title="Luminosity">luminosity</a> of the stars. In the case of stars with few observations, it must be computed assuming an <a href="/wiki/Effective_temperature" title="Effective temperature">effective temperature</a>. </p><p>Classically, the difference in bolometric magnitude is related to the luminosity ratio according to:<sup id="cite_ref-Carroll_7-1" class="reference"><a href="#cite_note-Carroll-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {bol,\star } }-M_{\mathrm {bol,\odot } }=-2.5\log _{10}\left({\frac {L_{\star }}{L_{\odot }}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mo>,</mo> <mo>⋆</mo> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mo>,</mo> <mo>⊙</mo> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {bol,\star } }-M_{\mathrm {bol,\odot } }=-2.5\log _{10}\left({\frac {L_{\star }}{L_{\odot }}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667fb13447c5d2e8e5d498a51dd6f56fc6082518" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.679ex; height:6.176ex;" alt="{\displaystyle M_{\mathrm {bol,\star } }-M_{\mathrm {bol,\odot } }=-2.5\log _{10}\left({\frac {L_{\star }}{L_{\odot }}}\right)}"></span> which makes by inversion: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L_{\star }}{L_{\odot }}}=10^{0.4\left(M_{\mathrm {bol,\odot } }-M_{\mathrm {bol,\star } }\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0.4</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mo>,</mo> <mo>⊙</mo> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mo>,</mo> <mo>⋆</mo> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L_{\star }}{L_{\odot }}}=10^{0.4\left(M_{\mathrm {bol,\odot } }-M_{\mathrm {bol,\star } }\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c769f506d233ca8cb31b628cf5a505c92405a63d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.997ex; height:5.676ex;" alt="{\displaystyle {\frac {L_{\star }}{L_{\odot }}}=10^{0.4\left(M_{\mathrm {bol,\odot } }-M_{\mathrm {bol,\star } }\right)}}"></span> where </p> <ul><li><span class="texhtml"><i>L</i><sub>⊙</sub></span> is the Sun's luminosity (bolometric luminosity)</li> <li><span class="texhtml"><i>L</i><sub>★</sub></span> is the star's luminosity (bolometric luminosity)</li> <li><span class="texhtml"><i>M</i><sub>bol,⊙</sub></span> is the bolometric magnitude of the Sun</li> <li><span class="texhtml"><i>M</i><sub>bol,★</sub></span> is the bolometric magnitude of the star.</li></ul> <p>In August 2015, the <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">International Astronomical Union</a> passed Resolution B2<sup id="cite_ref-IAU_XXIX_9-0" class="reference"><a href="#cite_note-IAU_XXIX-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> defining the <a href="/wiki/Zero_Point_(photometry)" class="mw-redirect" title="Zero Point (photometry)">zero points</a> of the absolute and apparent <a href="/wiki/Bolometric_magnitude" class="mw-redirect" title="Bolometric magnitude">bolometric magnitude</a> scales in SI units for power (<a href="/wiki/Watt" title="Watt">watts</a>) and irradiance (W/m<sup>2</sup>), respectively. Although bolometric magnitudes had been used by astronomers for many decades, there had been systematic differences in the absolute magnitude-luminosity scales presented in various astronomical references, and no international standardization. This led to systematic differences in bolometric corrections scales.<sup id="cite_ref-IAU2015B2_10-0" class="reference"><a href="#cite_note-IAU2015B2-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Combined with incorrect assumed absolute bolometric magnitudes for the Sun, this could lead to systematic errors in estimated stellar luminosities (and other stellar properties, such as radii or ages, which rely on stellar luminosity to be calculated). </p><p>Resolution B2 defines an absolute bolometric magnitude scale where <span class="texhtml"><i>M</i><sub>bol</sub> = 0</span> corresponds to luminosity <span class="texhtml"><i>L</i><sub>0</sub> = <span class="nowrap"><span data-sort-value="7028301280000000000♠"></span>3.0128<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>28</sup> W</span></span>, with the zero point <a href="/wiki/Luminosity" title="Luminosity">luminosity</a> <span class="texhtml"><i>L</i><sub>0</sub></span> set such that the Sun (with nominal luminosity <span class="nowrap"><span data-sort-value="7026382800000000000♠"></span>3.828<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>26</sup> W</span>) corresponds to absolute <a href="/wiki/Bolometric_magnitude" class="mw-redirect" title="Bolometric magnitude">bolometric magnitude</a> <span class="texhtml"><i>M</i><sub>bol,⊙</sub> = 4.74</span>. Placing a <a href="/wiki/Radiation" title="Radiation">radiation</a> source (e.g. star) at the standard distance of 10 <a href="/wiki/Parsecs" class="mw-redirect" title="Parsecs">parsecs</a>, it follows that the zero point of the apparent bolometric magnitude scale <span class="texhtml"><i>m</i><sub>bol</sub> = 0</span> corresponds to <a href="/wiki/Irradiance" title="Irradiance">irradiance</a> <span class="texhtml"><i>f</i><sub>0</sub> = <span class="nowrap"><span data-sort-value="6992251802100199999♠"></span>2.518<span style="margin-left:.25em;">021</span><span style="margin-left:.25em;">002</span><span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−8</sup> W/m<sup>2</sup></span></span>. Using the IAU 2015 scale, the nominal total <a href="/wiki/Solar_irradiance" title="Solar irradiance">solar irradiance</a> ("<a href="/wiki/Solar_constant" title="Solar constant">solar constant</a>") measured at 1 <a href="/wiki/Astronomical_unit" title="Astronomical unit">astronomical unit</a> (<span class="nowrap"><span data-sort-value="7003136100000000000♠"></span>1361 W/m<sup>2</sup></span>) corresponds to an apparent bolometric magnitude of the <a href="/wiki/Sun" title="Sun">Sun</a> of <span class="texhtml"><i>m</i><sub>bol,⊙</sub> = −26.832</span>.<sup id="cite_ref-IAU2015B2_10-1" class="reference"><a href="#cite_note-IAU2015B2-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Following Resolution B2, the relation between a star's absolute bolometric magnitude and its luminosity is no longer directly tied to the Sun's (variable) luminosity: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {bol} }=-2.5\log _{10}{\frac {L_{\star }}{L_{0}}}\approx -2.5\log _{10}L_{\star }+71.197425}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>≈</mo> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msub> <mo>+</mo> <mn>71.197425</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {bol} }=-2.5\log _{10}{\frac {L_{\star }}{L_{0}}}\approx -2.5\log _{10}L_{\star }+71.197425}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90135c68fb08570a24124901ee38456ad483ac67" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:50.578ex; height:5.676ex;" alt="{\displaystyle M_{\mathrm {bol} }=-2.5\log _{10}{\frac {L_{\star }}{L_{0}}}\approx -2.5\log _{10}L_{\star }+71.197425}"></span> where </p> <ul><li><span class="texhtml"><i>L</i><sub>★</sub></span> is the star's luminosity (bolometric luminosity) in <a href="/wiki/Watt" title="Watt">watts</a></li> <li><span class="texhtml"><i>L</i><sub>0</sub></span> is the zero point luminosity <span class="nowrap"><span data-sort-value="7028301280000000000♠"></span>3.0128<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>28</sup> W</span></li> <li><span class="texhtml"><i>M</i><sub>bol</sub></span> is the bolometric magnitude of the star</li></ul> <p>The new IAU absolute magnitude scale permanently disconnects the scale from the variable Sun. However, on this SI power scale, the nominal solar luminosity corresponds closely to <span class="texhtml"><i>M</i><sub>bol</sub> = 4.74</span>, a value that was commonly adopted by astronomers before the 2015 IAU resolution.<sup id="cite_ref-IAU2015B2_10-2" class="reference"><a href="#cite_note-IAU2015B2-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude <span class="texhtml"><i>M</i><sub>bol</sub></span> as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\star }=L_{0}10^{-0.4M_{\mathrm {bol} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.4</mn> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\star }=L_{0}10^{-0.4M_{\mathrm {bol} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45782bc6cd74c40686421c5c962eec7d17f473bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.848ex; height:3.009ex;" alt="{\displaystyle L_{\star }=L_{0}10^{-0.4M_{\mathrm {bol} }}}"></span> using the variables as defined previously. </p><p><span class="anchor" id="Solar_System"></span><span class="anchor" id="Solar_system"></span><span class="anchor" id="Solar_System_bodies"></span><span class="anchor" id="Solar_system_bodies"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Solar_System_bodies_(H)"><span id="Solar_System_bodies_.28H.29"></span>Solar System bodies (<span class="texhtml mvar" style="font-style:italic;"><i>H</i></span>)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=5" title="Edit section: Solar System bodies (H)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Solar_System_bodies_(H)"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For an introduction, see <a href="/wiki/Magnitude_(astronomy)" title="Magnitude (astronomy)">Magnitude (astronomy)</a>.</div> <table class="wikitable floatright" style="text-align:center; font-size:0.9em;"> <caption>Abs Mag (H)<br />and Diameter<br />for asteroids<br />(<a href="/wiki/Albedo#Astronomical_albedo" title="Albedo">albedo</a>=0.14)<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th>H</th> <th>Diameter </th></tr> <tr> <td>10</td> <td>36 km </td></tr> <tr> <td>12.7</td> <td>10 km </td></tr> <tr> <td>15</td> <td>3.6 km </td></tr> <tr id="1km"> <td>17.7</td> <td>1 km </td></tr> <tr> <td>19.2</td> <td>510 m </td></tr> <tr> <td>20</td> <td>360 m </td></tr> <tr id="PHA"> <td>22</td> <td>140 m </td></tr> <tr> <td>22.7</td> <td>100 m </td></tr> <tr> <td>24.2</td> <td>51 m </td></tr> <tr> <td>25</td> <td>36 m </td></tr> <tr id="Chelyabinsk"> <td>26.6</td> <td>17 m </td></tr> <tr> <td>27.7</td> <td>10 m </td></tr> <tr> <td>30</td> <td>3.6 m </td></tr> <tr id="1m"> <td>32.7</td> <td>1 m </td></tr></tbody></table> <p>For <a href="/wiki/Planet" title="Planet">planets</a> and <a href="/wiki/Asteroid" title="Asteroid">asteroids</a>, a definition of absolute magnitude that is more meaningful for non-stellar objects is used. The absolute magnitude, commonly called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, is defined as the <a href="/wiki/Apparent_magnitude" title="Apparent magnitude">apparent magnitude</a> that the object would have if it were one <a href="/wiki/Astronomical_unit" title="Astronomical unit">astronomical unit</a> (AU) from both the <a href="/wiki/Sun" title="Sun">Sun</a> and the observer, and in conditions of ideal solar opposition (an arrangement that is impossible in practice).<sup id="cite_ref-Luciuk_12-0" class="reference"><a href="#cite_note-Luciuk-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Because Solar System bodies are illuminated by the Sun, their brightness varies as a function of illumination conditions, described by the <a href="/wiki/Phase_angle_(astronomy)" title="Phase angle (astronomy)">phase angle</a>. This relationship is referred to as the <a href="/wiki/Phase_curve_(astronomy)" title="Phase curve (astronomy)">phase curve</a>. The absolute magnitude is the brightness at phase angle zero, an arrangement known as <a href="/wiki/Opposition_(astronomy)" title="Opposition (astronomy)">opposition</a>, from a distance of one AU. </p> <div class="mw-heading mw-heading3"><h3 id="Apparent_magnitude_2">Apparent magnitude</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=6" title="Edit section: Apparent magnitude"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Phase_angle_explanation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Phase_angle_explanation.png/250px-Phase_angle_explanation.png" decoding="async" width="250" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Phase_angle_explanation.png/375px-Phase_angle_explanation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Phase_angle_explanation.png/500px-Phase_angle_explanation.png 2x" data-file-width="756" data-file-height="842" /></a><figcaption>The phase angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> can be calculated from the distances body-sun, observer-sun and observer-body, using the <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a>.</figcaption></figure> <p>The absolute magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> can be used to calculate the apparent magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> of a body. For an object <a href="/wiki/Reflection_(physics)" title="Reflection (physics)">reflecting</a> sunlight, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> are connected by the relation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>H</mi> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>S</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>O</mi> </mrow> </msub> </mrow> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdbb07677b36ea91a1e4cd9312ea29d75658389" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.484ex; height:7.509ex;" alt="{\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is the <a href="/wiki/Phase_angle_(astronomy)" title="Phase angle (astronomy)">phase angle</a>, the angle between the body-Sun and body–observer lines. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a31da5be7f430c1b77531ab6f470bef393b7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.366ex; height:2.843ex;" alt="{\displaystyle q(\alpha )}"></span> is the <a href="/wiki/Bond_albedo#Phase_integral" title="Bond albedo">phase integral</a> (the <a href="/wiki/Integral" title="Integral">integration</a> of reflected light; a number in the 0 to 1 range).<sup id="cite_ref-Karttunen2016_13-0" class="reference"><a href="#cite_note-Karttunen2016-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>By the <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a>, we have: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\alpha }={\frac {d_{\mathrm {BO} }^{2}+d_{\mathrm {BS} }^{2}-d_{\mathrm {OS} }^{2}}{2d_{\mathrm {BO} }d_{\mathrm {BS} }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\alpha }={\frac {d_{\mathrm {BO} }^{2}+d_{\mathrm {BS} }^{2}-d_{\mathrm {OS} }^{2}}{2d_{\mathrm {BO} }d_{\mathrm {BS} }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c31e7dd1e7df3505cd964cc3573fcedeb238ae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.284ex; height:6.509ex;" alt="{\displaystyle \cos {\alpha }={\frac {d_{\mathrm {BO} }^{2}+d_{\mathrm {BS} }^{2}-d_{\mathrm {OS} }^{2}}{2d_{\mathrm {BO} }d_{\mathrm {BS} }}}.}"></span> </p><p>Distances: </p> <ul><li><span class="texhtml"><i>d</i><sub>BO</sub></span> is the distance between the body and the observer</li> <li><span class="texhtml"><i>d</i><sub>BS</sub></span> is the distance between the body and the Sun</li> <li><span class="texhtml"><i>d</i><sub>OS</sub></span> is the distance between the observer and the Sun</li> <li><span class="texhtml"><i>d</i><sub>0</sub></span>, a <a href="/wiki/Unit_conversion" class="mw-redirect" title="Unit conversion">unit conversion</a> factor, is the constant 1 <a href="/wiki/Astronomical_Unit" class="mw-redirect" title="Astronomical Unit">AU</a>, the average distance between the Earth and the Sun</li></ul> <div class="mw-heading mw-heading3"><h3 id="Approximations_for_phase_integral_q(α)"><span id="Approximations_for_phase_integral_q.28.CE.B1.29"></span>Approximations for phase integral <span class="serif-fonts" style="font-family: 'Georgia Pro', Georgia, 'DejaVu Serif', Times, 'Times New Roman', FreeSerif, 'DejaVu Math TeX', 'URW Bookman L', serif;"><i>q</i>(<i>α</i>)</span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=7" title="Edit section: Approximations for phase integral q(α)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a31da5be7f430c1b77531ab6f470bef393b7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.366ex; height:2.843ex;" alt="{\displaystyle q(\alpha )}"></span> depends on the properties of the reflecting surface, in particular on its <a href="/wiki/Surface_roughness" title="Surface roughness">roughness</a>. In practice, different approximations are used based on the known or assumed properties of the surface. The surfaces of terrestrial planets are generally more difficult to model than those of gaseous planets, the latter of which have smoother visible surfaces.<sup id="cite_ref-Karttunen2016_13-1" class="reference"><a href="#cite_note-Karttunen2016-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Planets_as_diffuse_spheres">Planets as diffuse spheres</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=8" title="Edit section: Planets as diffuse spheres"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Diffuse_reflector_sphere_disk.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Diffuse_reflector_sphere_disk.png/240px-Diffuse_reflector_sphere_disk.png" decoding="async" width="240" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Diffuse_reflector_sphere_disk.png/360px-Diffuse_reflector_sphere_disk.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Diffuse_reflector_sphere_disk.png/480px-Diffuse_reflector_sphere_disk.png 2x" data-file-width="1877" data-file-height="1032" /></a><figcaption>Diffuse reflection on sphere and flat disk</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Diffuse_reflection_model_phase_functions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Diffuse_reflection_model_phase_functions.svg/240px-Diffuse_reflection_model_phase_functions.svg.png" decoding="async" width="240" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Diffuse_reflection_model_phase_functions.svg/360px-Diffuse_reflection_model_phase_functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Diffuse_reflection_model_phase_functions.svg/480px-Diffuse_reflection_model_phase_functions.svg.png 2x" data-file-width="990" data-file-height="765" /></a><figcaption>Brightness with phase for diffuse reflection models. The sphere is 2/3 as bright at zero phase, while the disk can't be seen beyond 90 degrees.</figcaption></figure> <p>Planetary bodies can be approximated reasonably well as <a href="/wiki/Lambertian_diffuse_lighting_model" class="mw-redirect" title="Lambertian diffuse lighting model">ideal diffuse reflecting</a> <a href="/wiki/Sphere" title="Sphere">spheres</a>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> be the phase angle in <a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>, then<sup id="cite_ref-Whitmell1907_14-0" class="reference"><a href="#cite_note-Whitmell1907-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(\alpha )={\frac {2}{3}}\left(\left(1-{\frac {\alpha }{180^{\circ }}}\right)\cos {\alpha }+{\frac {1}{\pi }}\sin {\alpha }\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(\alpha )={\frac {2}{3}}\left(\left(1-{\frac {\alpha }{180^{\circ }}}\right)\cos {\alpha }+{\frac {1}{\pi }}\sin {\alpha }\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbb231280b2657cdbe2e96b15cafcc8b17f7290" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.961ex; height:6.176ex;" alt="{\displaystyle q(\alpha )={\frac {2}{3}}\left(\left(1-{\frac {\alpha }{180^{\circ }}}\right)\cos {\alpha }+{\frac {1}{\pi }}\sin {\alpha }\right).}"></span> A full-phase diffuse sphere reflects two-thirds as much light as a diffuse flat disk of the same diameter. A quarter phase (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f0fb8d6c2247b522d8bcb4a62b5c0a9f9ee7e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.965ex; height:2.343ex;" alt="{\displaystyle \alpha =90^{\circ }}"></span>) has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f17c9906369545ffcc4da833960f8be6e29a9cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.778ex; height:3.343ex;" alt="{\textstyle {\frac {1}{\pi }}}"></span> as much light as full phase (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ecf24b900c7f4cd88b7d8eab67ca3f039f12a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.803ex; height:2.343ex;" alt="{\displaystyle \alpha =0^{\circ }}"></span>). </p><p>By contrast, a <i>diffuse disk reflector model</i> is simply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(\alpha )=\cos {\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(\alpha )=\cos {\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/761d89fe5ad605f8a5c25354fbb725ef6175f166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.451ex; height:2.843ex;" alt="{\displaystyle q(\alpha )=\cos {\alpha }}"></span>, which isn't realistic, but it does represent the <a href="/wiki/Opposition_surge" title="Opposition surge">opposition surge</a> for rough surfaces that reflect more uniform light back at low phase angles. </p><p>The definition of the <a href="/wiki/Geometric_albedo" title="Geometric albedo">geometric albedo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, a measure for the reflectivity of planetary surfaces, is based on the diffuse disk reflector model. The absolute magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, diameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> (in <a href="/wiki/Kilometer" class="mw-redirect" title="Kilometer">kilometers</a>) and geometric albedo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> of a body are related by<sup id="cite_ref-sizemagnitude_15-0" class="reference"><a href="#cite_note-sizemagnitude-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Mag_formula_16-0" class="reference"><a href="#cite_note-Mag_formula-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-H_derivation_17-0" class="reference"><a href="#cite_note-H_derivation-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\frac {1329}{\sqrt {p}}}\times 10^{-0.2H}\mathrm {km} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1329</mn> <msqrt> <mi>p</mi> </msqrt> </mfrac> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.2</mn> <mi>H</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">m</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\frac {1329}{\sqrt {p}}}\times 10^{-0.2H}\mathrm {km} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68a03e5a58b4b4d48398edc5f9ff6d9928c8f077" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.556ex; height:6.176ex;" alt="{\displaystyle D={\frac {1329}{\sqrt {p}}}\times 10^{-0.2H}\mathrm {km} ,}"></span> or equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=5\log _{10}{\frac {1329}{D{\sqrt {p}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1329</mn> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>p</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=5\log _{10}{\frac {1329}{D{\sqrt {p}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97284b6248d20ec077221ad1f662a6f688e6fcb2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.459ex; height:6.176ex;" alt="{\displaystyle H=5\log _{10}{\frac {1329}{D{\sqrt {p}}}}.}"></span> </p><p>Example: The <a href="/wiki/Moon" title="Moon">Moon's</a> absolute magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> can be calculated from its diameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=3474{\text{ km}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mn>3474</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> km</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=3474{\text{ km}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4352c55ef1eb9506fe166d35a8f08a524f4d92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.417ex; height:2.176ex;" alt="{\displaystyle D=3474{\text{ km}}}"></span> and <a href="/wiki/Geometric_albedo" title="Geometric albedo">geometric albedo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0.113}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>0.113</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=0.113}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f61104622149fd7b39760e6f4eebe672051988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.654ex; height:2.509ex;" alt="{\displaystyle p=0.113}"></span>:<sup id="cite_ref-Albedo_18-0" class="reference"><a href="#cite_note-Albedo-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=5\log _{10}{\frac {1329}{3474{\sqrt {0.113}}}}=+0.28.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1329</mn> <mrow> <mn>3474</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>0.113</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>+</mo> <mn>0.28.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=5\log _{10}{\frac {1329}{3474{\sqrt {0.113}}}}=+0.28.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7654e2798e81c2c855a8a74db025ccf234621b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.353ex; height:6.176ex;" alt="{\displaystyle H=5\log _{10}{\frac {1329}{3474{\sqrt {0.113}}}}=+0.28.}"></span> We have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{BS}=1{\text{ AU}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> AU</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{BS}=1{\text{ AU}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff04f6b4a76ea1f8654e7815a1061b69e4addb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.076ex; height:2.509ex;" alt="{\displaystyle d_{BS}=1{\text{ AU}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{BO}=384400{\text{ km}}=0.00257{\text{ AU}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>O</mi> </mrow> </msub> <mo>=</mo> <mn>384400</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> km</mtext> </mrow> <mo>=</mo> <mn>0.00257</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> AU</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{BO}=384400{\text{ km}}=0.00257{\text{ AU}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5adcac3e0e5c0814e56131d6da0971a0d6a9ab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.193ex; height:2.509ex;" alt="{\displaystyle d_{BO}=384400{\text{ km}}=0.00257{\text{ AU}}.}"></span> At <a href="/wiki/Lunar_phases" class="mw-redirect" title="Lunar phases">quarter phase</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle q(\alpha )\approx {\frac {2}{3\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle q(\alpha )\approx {\frac {2}{3\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0326e10504f453869044f6ac9f6e42aca8c0d537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.065ex; height:3.676ex;" alt="{\textstyle q(\alpha )\approx {\frac {2}{3\pi }}}"></span> (according to the diffuse reflector model), this yields an apparent magnitude of <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=+0.28+5\log _{10}{\left(1\cdot 0.00257\right)}-2.5\log _{10}{\left({\frac {2}{3\pi }}\right)}=-10.99.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>+</mo> <mn>0.28</mn> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>0.00257</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>10.99.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=+0.28+5\log _{10}{\left(1\cdot 0.00257\right)}-2.5\log _{10}{\left({\frac {2}{3\pi }}\right)}=-10.99.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ace54773785a158bd417552a6e79ea29756b506" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.015ex; height:6.176ex;" alt="{\displaystyle m=+0.28+5\log _{10}{\left(1\cdot 0.00257\right)}-2.5\log _{10}{\left({\frac {2}{3\pi }}\right)}=-10.99.}"></span> The actual value is somewhat lower than that, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=-10.0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>10.0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=-10.0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4c9e9119c2ac0ffbb14bcb195208c6dc848f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.728ex; height:2.343ex;" alt="{\displaystyle m=-10.0.}"></span> This is not a good approximation, because the phase curve of the Moon is too complicated for the diffuse reflector model.<sup id="cite_ref-Luciuk2_19-0" class="reference"><a href="#cite_note-Luciuk2-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> A more accurate formula is given in the following section. </p> <div class="mw-heading mw-heading4"><h4 id="More_advanced_models">More advanced models</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=9" title="Edit section: More advanced models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because Solar System bodies are never perfect diffuse reflectors, astronomers use different models to predict apparent magnitudes based on known or assumed properties of the body.<sup id="cite_ref-Karttunen2016_13-2" class="reference"><a href="#cite_note-Karttunen2016-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> For planets, approximations for the correction term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2.5\log _{10}{q(\alpha )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2.5\log _{10}{q(\alpha )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be77c8c8e1a0735fccd5bbd45740c1abd4d4d498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.769ex; height:2.843ex;" alt="{\displaystyle -2.5\log _{10}{q(\alpha )}}"></span> in the formula for <span class="texhtml mvar" style="font-style:italic;">m</span> have been derived empirically, to match <a href="/wiki/Phase_curve_(astronomy)" title="Phase curve (astronomy)">observations at different phase angles</a>. The approximations recommended by the <a href="/wiki/Astronomical_Almanac" title="Astronomical Almanac">Astronomical Almanac</a><sup id="cite_ref-Mallama_and_Hilton_20-0" class="reference"><a href="#cite_note-Mallama_and_Hilton-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> are (with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> in degrees): </p> <table class="wikitable"> <tbody><tr> <th>Planet </th> <th>Referenced calculation<sup id="cite_ref-IMCCE_21-0" class="reference"><a href="#cite_note-IMCCE-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> </th> <th>Approximation for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2.5\log _{10}{q(\alpha )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2.5\log _{10}{q(\alpha )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be77c8c8e1a0735fccd5bbd45740c1abd4d4d498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.769ex; height:2.843ex;" alt="{\displaystyle -2.5\log _{10}{q(\alpha )}}"></span> </th></tr> <tr> <td><a href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</a> </td> <td>−0.4 </td> <td>−0.613 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +6.328\times 10^{-2}\alpha -1.6336\times 10^{-3}\alpha ^{2}+3.3644\times 10^{-5}\alpha ^{3}-3.4265\times 10^{-7}\alpha ^{4}+1.6893\times 10^{-9}\alpha ^{5}-3.0334\times 10^{-12}\alpha ^{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>6.328</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>α</mi> <mo>−</mo> <mn>1.6336</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3.3644</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3.4265</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>7</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1.6893</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>3.0334</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>12</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +6.328\times 10^{-2}\alpha -1.6336\times 10^{-3}\alpha ^{2}+3.3644\times 10^{-5}\alpha ^{3}-3.4265\times 10^{-7}\alpha ^{4}+1.6893\times 10^{-9}\alpha ^{5}-3.0334\times 10^{-12}\alpha ^{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fcd7ceb903a3b6ad4c0ed5b42f99949f9ead25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:113.61ex; height:2.843ex;" alt="{\displaystyle +6.328\times 10^{-2}\alpha -1.6336\times 10^{-3}\alpha ^{2}+3.3644\times 10^{-5}\alpha ^{3}-3.4265\times 10^{-7}\alpha ^{4}+1.6893\times 10^{-9}\alpha ^{5}-3.0334\times 10^{-12}\alpha ^{6}}"></span> </td></tr> <tr> <td><a href="/wiki/Venus_(planet)" class="mw-redirect" title="Venus (planet)">Venus</a> </td> <td>−4.4 </td> <td>−4.384 </td> <td> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1.044\times 10^{-3}\alpha +3.687\times 10^{-4}\alpha ^{2}-2.814\times 10^{-6}\alpha ^{3}+8.938\times 10^{-9}\alpha ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1.044</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mi>α</mi> <mo>+</mo> <mn>3.687</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2.814</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>8.938</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1.044\times 10^{-3}\alpha +3.687\times 10^{-4}\alpha ^{2}-2.814\times 10^{-6}\alpha ^{3}+8.938\times 10^{-9}\alpha ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f2716368de2f00dc31d0d9c8f446267c0b7c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:70.622ex; height:2.843ex;" alt="{\displaystyle -1.044\times 10^{-3}\alpha +3.687\times 10^{-4}\alpha ^{2}-2.814\times 10^{-6}\alpha ^{3}+8.938\times 10^{-9}\alpha ^{4}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }<\alpha \leq 163.7^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>α</mi> <mo>≤</mo> <msup> <mn>163.7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }<\alpha \leq 163.7^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca47621fd783f91d62f42d3ffb1573abf7128be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.252ex; height:2.509ex;" alt="{\displaystyle 0^{\circ }<\alpha \leq 163.7^{\circ }}"></span>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +240.44228-2.81914\alpha +8.39034\times 10^{-3}\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>240.44228</mn> <mo>−</mo> <mn>2.81914</mn> <mi>α</mi> <mo>+</mo> <mn>8.39034</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +240.44228-2.81914\alpha +8.39034\times 10^{-3}\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e1c5c22246bb5fa31b92f58890c7848cfc4170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:44.206ex; height:2.843ex;" alt="{\displaystyle +240.44228-2.81914\alpha +8.39034\times 10^{-3}\alpha ^{2}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 163.7^{\circ }<\alpha <179^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>163.7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>α</mi> <mo><</mo> <msup> <mn>179</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 163.7^{\circ }<\alpha <179^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ea216936d9f7a15ec98db07a93a2b421629aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.577ex; height:2.343ex;" alt="{\displaystyle 163.7^{\circ }<\alpha <179^{\circ }}"></span>)</li></ul> </td></tr> <tr> <td><a href="/wiki/Earth" title="Earth">Earth</a> </td> <td>− </td> <td>−3.99 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1.060\times 10^{-3}\alpha +2.054\times 10^{-4}\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1.060</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mi>α</mi> <mo>+</mo> <mn>2.054</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1.060\times 10^{-3}\alpha +2.054\times 10^{-4}\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64825dc4064500bf18639089a5c92f49cda4d9f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.268ex; height:2.843ex;" alt="{\displaystyle -1.060\times 10^{-3}\alpha +2.054\times 10^{-4}\alpha ^{2}}"></span> </td></tr> <tr> <td><a href="/wiki/Moon" title="Moon">Moon</a><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </td> <td>0.2 </td> <td>+0.28 </td> <td> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +2.9994\times 10^{-2}\alpha -1.6057\times 10^{-4}\alpha ^{2}+3.1543\times 10^{-6}\alpha ^{3}-2.0667\times 10^{-8}\alpha ^{4}+6.2553\times 10^{-11}\alpha ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2.9994</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>α</mi> <mo>−</mo> <mn>1.6057</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3.1543</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>2.0667</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>8</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>6.2553</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>11</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2.9994\times 10^{-2}\alpha -1.6057\times 10^{-4}\alpha ^{2}+3.1543\times 10^{-6}\alpha ^{3}-2.0667\times 10^{-8}\alpha ^{4}+6.2553\times 10^{-11}\alpha ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c81854324401a712750eb87d9a25cefdd67781c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:95.433ex; height:2.843ex;" alt="{\displaystyle +2.9994\times 10^{-2}\alpha -1.6057\times 10^{-4}\alpha ^{2}+3.1543\times 10^{-6}\alpha ^{3}-2.0667\times 10^{-8}\alpha ^{4}+6.2553\times 10^{-11}\alpha ^{5}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \leq 150^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≤</mo> <msup> <mn>150</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \leq 150^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5206c3be20523a88db1ad70091ca097fd447bbee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.128ex; height:2.509ex;" alt="{\displaystyle \alpha \leq 150^{\circ }}"></span>, before full Moon)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +3.3234\times 10^{-2}\alpha -3.0725\times 10^{-4}\alpha ^{2}+6.1575\times 10^{-6}\alpha ^{3}-4.7723\times 10^{-8}\alpha ^{4}+1.4681\times 10^{-10}\alpha ^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>3.3234</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>α</mi> <mo>−</mo> <mn>3.0725</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6.1575</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>4.7723</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>8</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1.4681</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>10</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +3.3234\times 10^{-2}\alpha -3.0725\times 10^{-4}\alpha ^{2}+6.1575\times 10^{-6}\alpha ^{3}-4.7723\times 10^{-8}\alpha ^{4}+1.4681\times 10^{-10}\alpha ^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f922abb1a6ec69e2b5e2a97dd42afb606c88ba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:95.433ex; height:2.843ex;" alt="{\displaystyle +3.3234\times 10^{-2}\alpha -3.0725\times 10^{-4}\alpha ^{2}+6.1575\times 10^{-6}\alpha ^{3}-4.7723\times 10^{-8}\alpha ^{4}+1.4681\times 10^{-10}\alpha ^{5}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \leq 150^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≤</mo> <msup> <mn>150</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \leq 150^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5206c3be20523a88db1ad70091ca097fd447bbee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.128ex; height:2.509ex;" alt="{\displaystyle \alpha \leq 150^{\circ }}"></span>, after full Moon)</li></ul> </td></tr> <tr> <td><a href="/wiki/Mars_(planet)" class="mw-redirect" title="Mars (planet)">Mars</a> </td> <td>−1.5 </td> <td>−1.601 </td> <td> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +2.267\times 10^{-2}\alpha -1.302\times 10^{-4}\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2.267</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>α</mi> <mo>−</mo> <mn>1.302</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2.267\times 10^{-2}\alpha -1.302\times 10^{-4}\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b1bf24517a1e34ff836e5af2dfae22ed45e16b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.268ex; height:2.843ex;" alt="{\displaystyle +2.267\times 10^{-2}\alpha -1.302\times 10^{-4}\alpha ^{2}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }<\alpha \leq 50^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>α</mi> <mo>≤</mo> <msup> <mn>50</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }<\alpha \leq 50^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef535b5bd77829c3b6bc748f886df3186ca67c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.28ex; height:2.509ex;" alt="{\displaystyle 0^{\circ }<\alpha \leq 50^{\circ }}"></span>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1.234-2.573\times 10^{-2}\alpha +3.445\times 10^{-4}\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1.234</mn> <mo>−</mo> <mn>2.573</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>α</mi> <mo>+</mo> <mn>3.445</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1.234-2.573\times 10^{-2}\alpha +3.445\times 10^{-4}\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/351619052c107704fbef84d29b9566942eb7c1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:42.405ex; height:2.843ex;" alt="{\displaystyle +1.234-2.573\times 10^{-2}\alpha +3.445\times 10^{-4}\alpha ^{2}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 50^{\circ }<\alpha \leq 120^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>50</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>α</mi> <mo>≤</mo> <msup> <mn>120</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 50^{\circ }<\alpha \leq 120^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c41775cc9987d96dafb29950118d07a39af5a694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.605ex; height:2.509ex;" alt="{\displaystyle 50^{\circ }<\alpha \leq 120^{\circ }}"></span>)</li></ul> </td></tr> <tr> <td><a href="/wiki/Jupiter_(planet)" class="mw-redirect" title="Jupiter (planet)">Jupiter</a> </td> <td>−9.4 </td> <td>−9.395 </td> <td> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3.7\times 10^{-4}\alpha +6.16\times 10^{-4}\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3.7</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mi>α</mi> <mo>+</mo> <mn>6.16</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3.7\times 10^{-4}\alpha +6.16\times 10^{-4}\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b590f75e6c57230affec23388a7e58edc3dfbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.78ex; height:2.843ex;" alt="{\displaystyle -3.7\times 10^{-4}\alpha +6.16\times 10^{-4}\alpha ^{2}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \leq 12^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≤</mo> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \leq 12^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d78b3851afd0c5e247e7ef4ae853adce4a5373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.965ex; height:2.509ex;" alt="{\displaystyle \alpha \leq 12^{\circ }}"></span>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -0.033-2.5\log _{10}{\left(1-1.507\left({\frac {\alpha }{180^{\circ }}}\right)-0.363\left({\frac {\alpha }{180^{\circ }}}\right)^{2}-0.062\left({\frac {\alpha }{180^{\circ }}}\right)^{3}+2.809\left({\frac {\alpha }{180^{\circ }}}\right)^{4}-1.876\left({\frac {\alpha }{180^{\circ }}}\right)^{5}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>0.033</mn> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1.507</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>0.363</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>0.062</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2.809</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>1.876</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -0.033-2.5\log _{10}{\left(1-1.507\left({\frac {\alpha }{180^{\circ }}}\right)-0.363\left({\frac {\alpha }{180^{\circ }}}\right)^{2}-0.062\left({\frac {\alpha }{180^{\circ }}}\right)^{3}+2.809\left({\frac {\alpha }{180^{\circ }}}\right)^{4}-1.876\left({\frac {\alpha }{180^{\circ }}}\right)^{5}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e97f6a10867b5bb780799057d3b706c86a24f6f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:109.178ex; height:6.176ex;" alt="{\displaystyle -0.033-2.5\log _{10}{\left(1-1.507\left({\frac {\alpha }{180^{\circ }}}\right)-0.363\left({\frac {\alpha }{180^{\circ }}}\right)^{2}-0.062\left({\frac {\alpha }{180^{\circ }}}\right)^{3}+2.809\left({\frac {\alpha }{180^{\circ }}}\right)^{4}-1.876\left({\frac {\alpha }{180^{\circ }}}\right)^{5}\right)}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >12^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <msup> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >12^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e057673768c07c68a13b784b5ee085bccc736f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.965ex; height:2.343ex;" alt="{\displaystyle \alpha >12^{\circ }}"></span>)</li></ul> </td></tr> <tr> <td><a href="/wiki/Saturn_(planet)" class="mw-redirect" title="Saturn (planet)">Saturn</a> </td> <td>−9.7 </td> <td>−8.914 </td> <td> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1.825\sin {\left(\beta \right)}+2.6\times 10^{-2}\alpha -0.378\sin {\left(\beta \right)}e^{-2.25\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1.825</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mn>2.6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>α</mi> <mo>−</mo> <mn>0.378</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2.25</mn> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1.825\sin {\left(\beta \right)}+2.6\times 10^{-2}\alpha -0.378\sin {\left(\beta \right)}e^{-2.25\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbea44e31f4c1fa98e92429c44ff23ff04b45e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.151ex; height:3.176ex;" alt="{\displaystyle -1.825\sin {\left(\beta \right)}+2.6\times 10^{-2}\alpha -0.378\sin {\left(\beta \right)}e^{-2.25\alpha }}"></span> (for planet and rings, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <6.5^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <msup> <mn>6.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <6.5^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86233453473ce8bb70f85304e4855f83cebb14a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.612ex; height:2.343ex;" alt="{\displaystyle \alpha <6.5^{\circ }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta <27^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo><</mo> <msup> <mn>27</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta <27^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5c62deb7c02f70a39905aef8d3b2dedcccf7d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.81ex; height:2.676ex;" alt="{\displaystyle \beta <27^{\circ }}"></span>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -0.036-3.7\times 10^{-4}\alpha +6.16\times 10^{-4}\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>0.036</mn> <mo>−</mo> <mn>3.7</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mi>α</mi> <mo>+</mo> <mn>6.16</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -0.036-3.7\times 10^{-4}\alpha +6.16\times 10^{-4}\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac80b216f1e5d3b43cc31f5a1840e0f08b7f326b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:38.917ex; height:2.843ex;" alt="{\displaystyle -0.036-3.7\times 10^{-4}\alpha +6.16\times 10^{-4}\alpha ^{2}}"></span> (for the globe alone, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \leq 6^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≤</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \leq 6^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d840cb637a5c664b0ae34819415ef86069c30f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.803ex; height:2.509ex;" alt="{\displaystyle \alpha \leq 6^{\circ }}"></span>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +0.026+2.446\times 10^{-4}\alpha +2.672\times 10^{-4}\alpha ^{2}-1.505\times 10^{-6}\alpha ^{3}+4.767\times 10^{-9}\alpha ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>0.026</mn> <mo>+</mo> <mn>2.446</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mi>α</mi> <mo>+</mo> <mn>2.672</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1.505</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4.767</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +0.026+2.446\times 10^{-4}\alpha +2.672\times 10^{-4}\alpha ^{2}-1.505\times 10^{-6}\alpha ^{3}+4.767\times 10^{-9}\alpha ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12501d0f2dfad65d7cbf5717558114ed32404021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:78.759ex; height:2.843ex;" alt="{\displaystyle +0.026+2.446\times 10^{-4}\alpha +2.672\times 10^{-4}\alpha ^{2}-1.505\times 10^{-6}\alpha ^{3}+4.767\times 10^{-9}\alpha ^{4}}"></span> (for the globe alone, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6^{\circ }<\alpha <150^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>α</mi> <mo><</mo> <msup> <mn>150</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6^{\circ }<\alpha <150^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466ee90b00b231fe3934fae019200ab025b4156c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.443ex; height:2.343ex;" alt="{\displaystyle 6^{\circ }<\alpha <150^{\circ }}"></span>)</li></ul> </td></tr> <tr> <td><a href="/wiki/Uranus_(planet)" class="mw-redirect" title="Uranus (planet)">Uranus</a> </td> <td>−7.2 </td> <td>−7.110 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -8.4\times 10^{-4}\phi '+6.587\times 10^{-3}\alpha +1.045\times 10^{-4}\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>8.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo>+</mo> <mn>6.587</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mi>α</mi> <mo>+</mo> <mn>1.045</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -8.4\times 10^{-4}\phi '+6.587\times 10^{-3}\alpha +1.045\times 10^{-4}\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caf194d6903be99f7fe2d1ec0437beedba7958e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:49.648ex; height:3.009ex;" alt="{\displaystyle -8.4\times 10^{-4}\phi '+6.587\times 10^{-3}\alpha +1.045\times 10^{-4}\alpha ^{2}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <3.1^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <msup> <mn>3.1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <3.1^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47d1d187c3e74653dce6310b57e33ca56fd48e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.612ex; height:2.343ex;" alt="{\displaystyle \alpha <3.1^{\circ }}"></span>) </td></tr> <tr> <td><a href="/wiki/Neptune_(planet)" class="mw-redirect" title="Neptune (planet)">Neptune</a> </td> <td>−6.9 </td> <td>−7.00 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +7.944\times 10^{-3}\alpha +9.617\times 10^{-5}\alpha ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>7.944</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mi>α</mi> <mo>+</mo> <mn>9.617</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +7.944\times 10^{-3}\alpha +9.617\times 10^{-5}\alpha ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc98b740334d1f05c95253d4cdc53f9504bf4e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.268ex; height:2.843ex;" alt="{\displaystyle +7.944\times 10^{-3}\alpha +9.617\times 10^{-5}\alpha ^{2}}"></span> (for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <133^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <msup> <mn>133</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <133^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4763bd715e388e2072b6265d01c5194aa02ac25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.128ex; height:2.343ex;" alt="{\displaystyle \alpha <133^{\circ }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>2000.0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>></mo> <mn>2000.0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t>2000.0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c88b73a26963516110c9f7ef7029a222105bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.397ex; height:2.176ex;" alt="{\displaystyle t>2000.0}"></span>) </td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="theader">The different halves of the Moon, as seen from Earth</div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Daniel_Hershman_-_march_moon_(by).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Daniel_Hershman_-_march_moon_%28by%29.jpg/200px-Daniel_Hershman_-_march_moon_%28by%29.jpg" decoding="async" width="200" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Daniel_Hershman_-_march_moon_%28by%29.jpg/300px-Daniel_Hershman_-_march_moon_%28by%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Daniel_Hershman_-_march_moon_%28by%29.jpg/400px-Daniel_Hershman_-_march_moon_%28by%29.jpg 2x" data-file-width="514" data-file-height="600" /></a></span></div><div class="thumbcaption">Moon at first quarter</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Waning_gibbous_moon_near_last_quarter_-_23_Sept._2016.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Waning_gibbous_moon_near_last_quarter_-_23_Sept._2016.png/200px-Waning_gibbous_moon_near_last_quarter_-_23_Sept._2016.png" decoding="async" width="200" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Waning_gibbous_moon_near_last_quarter_-_23_Sept._2016.png/300px-Waning_gibbous_moon_near_last_quarter_-_23_Sept._2016.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Waning_gibbous_moon_near_last_quarter_-_23_Sept._2016.png/400px-Waning_gibbous_moon_near_last_quarter_-_23_Sept._2016.png 2x" data-file-width="2472" data-file-height="2820" /></a></span></div><div class="thumbcaption">Moon at last quarter</div></div></div></div></div> <p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> is the effective inclination of <a href="/wiki/Saturn%27s_rings" class="mw-redirect" title="Saturn's rings">Saturn's rings</a> (their tilt relative to the observer), which as seen from Earth varies between 0° and 27° over the course of one Saturn orbit, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5304c739c4deec1d259e3235a419e6177fe77a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.07ex; height:2.843ex;" alt="{\displaystyle \phi '}"></span> is a small correction term depending on Uranus' sub-Earth and sub-solar latitudes. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is the <a href="/wiki/Common_Era" title="Common Era">Common Era</a> year. Neptune's absolute magnitude is changing slowly due to seasonal effects as the planet moves along its 165-year orbit around the Sun, and the approximation above is only valid after the year 2000. For some circumstances, like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \geq 179^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≥</mo> <msup> <mn>179</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \geq 179^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36173ad3a4c232ceb6bc80c021e1a7b12b71cf12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.128ex; height:2.509ex;" alt="{\displaystyle \alpha \geq 179^{\circ }}"></span> for Venus, no observations are available, and the phase curve is unknown in those cases. The formula for the Moon is only applicable to the <a href="/wiki/Near_side_of_the_Moon" title="Near side of the Moon">near side of the Moon</a>, the portion that is visible from the Earth. </p><p>Example 1: On 1 January 2019, <a href="/wiki/Venus_(planet)" class="mw-redirect" title="Venus (planet)">Venus</a> was <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{BS}=0.719{\text{ AU}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>0.719</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> AU</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{BS}=0.719{\text{ AU}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9698b6464f436962c5011b19a2a34b016f62b7dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.211ex; height:2.509ex;" alt="{\displaystyle d_{BS}=0.719{\text{ AU}}}"></span> from the Sun, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{BO}=0.645{\text{ AU}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>O</mi> </mrow> </msub> <mo>=</mo> <mn>0.645</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> AU</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{BO}=0.645{\text{ AU}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/158348d23db426a655bf8799409cb5481de6feff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.404ex; height:2.509ex;" alt="{\displaystyle d_{BO}=0.645{\text{ AU}}}"></span> from Earth, at a phase angle of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =93.0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mn>93.0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =93.0^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e9570a5ab2dd282c01a5bc6a8c195bb8e4c0c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.775ex; height:2.343ex;" alt="{\displaystyle \alpha =93.0^{\circ }}"></span> (near quarter phase). Under full-phase conditions, Venus would have been visible at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=-4.384+5\log _{10}{\left(0.719\cdot 0.645\right)}=-6.09.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>4.384</mn> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mn>0.719</mn> <mo>⋅</mo> <mn>0.645</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>6.09.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=-4.384+5\log _{10}{\left(0.719\cdot 0.645\right)}=-6.09.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982a84e9ad8f40086858eca5b89d79700c2c8fa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.638ex; height:2.843ex;" alt="{\displaystyle m=-4.384+5\log _{10}{\left(0.719\cdot 0.645\right)}=-6.09.}"></span> Accounting for the high phase angle, the correction term above yields an actual apparent magnitude of <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=-6.09+\left(-1.044\times 10^{-3}\cdot 93.0+3.687\times 10^{-4}\cdot 93.0^{2}-2.814\times 10^{-6}\cdot 93.0^{3}+8.938\times 10^{-9}\cdot 93.0^{4}\right)=-4.59.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>6.09</mn> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1.044</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>93.0</mn> <mo>+</mo> <mn>3.687</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>93.0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2.814</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>93.0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>8.938</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>93.0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>4.59.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=-6.09+\left(-1.044\times 10^{-3}\cdot 93.0+3.687\times 10^{-4}\cdot 93.0^{2}-2.814\times 10^{-6}\cdot 93.0^{3}+8.938\times 10^{-9}\cdot 93.0^{4}\right)=-4.59.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80db823b02d36b306983978a88d27362bafdf620" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:113.663ex; height:3.343ex;" alt="{\displaystyle m=-6.09+\left(-1.044\times 10^{-3}\cdot 93.0+3.687\times 10^{-4}\cdot 93.0^{2}-2.814\times 10^{-6}\cdot 93.0^{3}+8.938\times 10^{-9}\cdot 93.0^{4}\right)=-4.59.}"></span> This is close to the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=-4.62}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>4.62</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=-4.62}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9d1e2a98d2c46816be82ad903652a4f1c9d3c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.081ex; height:2.343ex;" alt="{\displaystyle m=-4.62}"></span> predicted by the Jet Propulsion Laboratory.<sup id="cite_ref-JPLHorizonsVenus_23-0" class="reference"><a href="#cite_note-JPLHorizonsVenus-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Example 2: At <a href="/wiki/First_quarter" class="mw-redirect" title="First quarter">first quarter phase</a>, the approximation for the Moon gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -2.5\log _{10}{q(90^{\circ })}=2.71.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">(</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>2.71.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -2.5\log _{10}{q(90^{\circ })}=2.71.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0299d9616dee29332c7090c235d97474e43ee15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.54ex; height:2.843ex;" alt="{\textstyle -2.5\log _{10}{q(90^{\circ })}=2.71.}"></span> With that, the apparent magnitude of the Moon is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m=+0.28+5\log _{10}{\left(1\cdot 0.00257\right)}+2.71=-9.96,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>+</mo> <mn>0.28</mn> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>0.00257</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mn>2.71</mn> <mo>=</mo> <mo>−</mo> <mn>9.96</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m=+0.28+5\log _{10}{\left(1\cdot 0.00257\right)}+2.71=-9.96,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45b0b77cdf7cfbe8fa5b8a6a9f6f3891f6b3ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.641ex; height:2.843ex;" alt="{\textstyle m=+0.28+5\log _{10}{\left(1\cdot 0.00257\right)}+2.71=-9.96,}"></span> close to the expected value of about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -10.0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>10.0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -10.0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe2ebd76b5fdda50f114d19cda95bb42ac4c120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.942ex; height:2.343ex;" alt="{\displaystyle -10.0}"></span>. At <a href="/wiki/Last_quarter" class="mw-redirect" title="Last quarter">last quarter</a>, the Moon is about 0.06 mag fainter than at first quarter, because that part of its surface has a lower albedo. </p><p>Earth's <a href="/wiki/Albedo" title="Albedo">albedo</a> varies by a factor of 6, from 0.12 in the cloud-free case to 0.76 in the case of <a href="/wiki/Altostratus_clouds" class="mw-redirect" title="Altostratus clouds">altostratus cloud</a>. The absolute magnitude in the table corresponds to an albedo of 0.434. Due to the variability of the <a href="/wiki/Weather" title="Weather">weather</a>, Earth's apparent magnitude cannot be predicted as accurately as that of most other planets.<sup id="cite_ref-Mallama_and_Hilton_20-1" class="reference"><a href="#cite_note-Mallama_and_Hilton-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Asteroids">Asteroids</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=10" title="Edit section: Asteroids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ceres_opposition_effect.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Ceres_opposition_effect.png/240px-Ceres_opposition_effect.png" decoding="async" width="240" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Ceres_opposition_effect.png/360px-Ceres_opposition_effect.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Ceres_opposition_effect.png/480px-Ceres_opposition_effect.png 2x" data-file-width="1247" data-file-height="392" /></a><figcaption>Asteroid <a href="/wiki/1_Ceres" class="mw-redirect" title="1 Ceres">1 Ceres</a>, imaged by the <a href="/wiki/Dawn_(spacecraft)" title="Dawn (spacecraft)">Dawn</a> spacecraft at phase angles of 0°, 7° and 33°. The strong difference in brightness between the three is real. The left image at 0° phase angle shows the brightness surge due to the <a href="/wiki/Opposition_effect" class="mw-redirect" title="Opposition effect">opposition effect</a>.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Asteroid_HG_phase_integrals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Asteroid_HG_phase_integrals.svg/240px-Asteroid_HG_phase_integrals.svg.png" decoding="async" width="240" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Asteroid_HG_phase_integrals.svg/360px-Asteroid_HG_phase_integrals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Asteroid_HG_phase_integrals.svg/480px-Asteroid_HG_phase_integrals.svg.png 2x" data-file-width="990" data-file-height="765" /></a><figcaption>Phase integrals for various values of G</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Slope_parameter_G.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Slope_parameter_G.png/240px-Slope_parameter_G.png" decoding="async" width="240" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Slope_parameter_G.png/360px-Slope_parameter_G.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Slope_parameter_G.png/480px-Slope_parameter_G.png 2x" data-file-width="1057" data-file-height="603" /></a><figcaption>Relationship between the slope parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> and the opposition surge. Larger values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> correspond to a less pronounced opposition effect. For most asteroids, a value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=0.15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>0.15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=0.15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c7c69a064c04892198229dc1ecd1735595b5a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.059ex; height:2.176ex;" alt="{\displaystyle G=0.15}"></span> is assumed, corresponding to an opposition surge of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.3{\text{ mag}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> mag</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.3{\text{ mag}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ec188ac5d50e0428fc4d0c8367251783a49e38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.813ex; height:2.509ex;" alt="{\displaystyle 0.3{\text{ mag}}}"></span>.</figcaption></figure> <p>If an object has an atmosphere, it reflects light more or less isotropically in all directions, and its brightness can be modelled as a diffuse reflector. Bodies with no atmosphere, like asteroids or moons, tend to reflect light more strongly to the direction of the incident light, and their brightness increases rapidly as the phase angle approaches <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd0e1e92cf5770c2bfbb1de8b4b7bf904c9deef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle 0^{\circ }}"></span>. This rapid brightening near opposition is called the <a href="/wiki/Opposition_effect" class="mw-redirect" title="Opposition effect">opposition effect</a>. Its strength depends on the physical properties of the body's surface, and hence it differs from asteroid to asteroid.<sup id="cite_ref-Karttunen2016_13-3" class="reference"><a href="#cite_note-Karttunen2016-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>In 1985, the <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">IAU</a> adopted the <a href="/wiki/Semi-empirical" class="mw-redirect" title="Semi-empirical">semi-empirical</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle HG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle HG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fb62307b508eca86c308e23e26eaa6404e5d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.89ex; height:2.176ex;" alt="{\displaystyle HG}"></span>-system, based on two parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> called <i>absolute magnitude</i> and <i>slope</i>, to model the opposition effect for the <a href="/wiki/Ephemeris" title="Ephemeris">ephemerides</a> published by the <a href="/wiki/Minor_Planet_Center" title="Minor Planet Center">Minor Planet Center</a>.<sup id="cite_ref-MPC1985_24-0" class="reference"><a href="#cite_note-MPC1985-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>H</mi> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>S</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>O</mi> </mrow> </msub> </mrow> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>−</mo> <mn>2.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdbb07677b36ea91a1e4cd9312ea29d75658389" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.484ex; height:7.509ex;" alt="{\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )},}"></span> </p><p>where </p> <ul><li>the phase integral is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(\alpha )=\left(1-G\right)\phi _{1}\left(\alpha \right)+G\phi _{2}\left(\alpha \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>G</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>G</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(\alpha )=\left(1-G\right)\phi _{1}\left(\alpha \right)+G\phi _{2}\left(\alpha \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8483cfc52688f586ab8f127f51a174af438af92a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.405ex; height:2.843ex;" alt="{\displaystyle q(\alpha )=\left(1-G\right)\phi _{1}\left(\alpha \right)+G\phi _{2}\left(\alpha \right)}"></span> and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi _{i}\left(\alpha \right)=\exp {\left(-A_{i}\left(\tan {\frac {\alpha }{2}}\right)^{B_{i}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi _{i}\left(\alpha \right)=\exp {\left(-A_{i}\left(\tan {\frac {\alpha }{2}}\right)^{B_{i}}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b4f2e32af14f166519e7421066187c04758dc1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.903ex; height:4.843ex;" alt="{\textstyle \phi _{i}\left(\alpha \right)=\exp {\left(-A_{i}\left(\tan {\frac {\alpha }{2}}\right)^{B_{i}}\right)}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb857369fbc28dab4b5d76bb835e6f196b1d1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=1}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}=3.332}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>3.332</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}=3.332}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef16a6ee9b1305ab176d5c607477778dc564113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.192ex; height:2.509ex;" alt="{\displaystyle A_{1}=3.332}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}=1.862}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1.862</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2}=1.862}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c176a91d10256319243388d6d49a8c45e01b98ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.192ex; height:2.509ex;" alt="{\displaystyle A_{2}=1.862}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}=0.631}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.631</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}=0.631}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab41d9722f5e1cb662c539dd717023fb557aa8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.213ex; height:2.509ex;" alt="{\displaystyle B_{1}=0.631}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2}=1.218}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1.218</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2}=1.218}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0efb57e8ade576d9c5b88df64b6e519a6013aca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.213ex; height:2.509ex;" alt="{\displaystyle B_{2}=1.218}"></span>.<sup id="cite_ref-Lagerkvist_25-0" class="reference"><a href="#cite_note-Lagerkvist-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li></ul> <p>This relation is valid for phase angles <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <120^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <msup> <mn>120</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <120^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189adceeae0a768cf0219a188225b48c767b564d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.128ex; height:2.343ex;" alt="{\displaystyle \alpha <120^{\circ }}"></span>, and works best when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <20^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <msup> <mn>20</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <20^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e465d27ab58c0a02c702b4068d6e31a606971d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.965ex; height:2.343ex;" alt="{\displaystyle \alpha <20^{\circ }}"></span>.<sup id="cite_ref-dymock_26-0" class="reference"><a href="#cite_note-dymock-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>The slope parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> relates to the surge in brightness, typically <span class="nowrap"><span data-sort-value="6999300000000000000♠"></span>0.3 mag</span>, when the object is near opposition. It is known accurately only for a small number of asteroids, hence for most asteroids a value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=0.15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>0.15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=0.15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c7c69a064c04892198229dc1ecd1735595b5a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.059ex; height:2.176ex;" alt="{\displaystyle G=0.15}"></span> is assumed.<sup id="cite_ref-dymock_26-1" class="reference"><a href="#cite_note-dymock-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> In rare cases, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> can be negative.<sup id="cite_ref-Lagerkvist_25-1" class="reference"><a href="#cite_note-Lagerkvist-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-JPLdoc_27-0" class="reference"><a href="#cite_note-JPLdoc-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> An example is <a href="/wiki/101955_Bennu" title="101955 Bennu">101955 Bennu</a>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=-0.08}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo>−</mo> <mn>0.08</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=-0.08}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e170e92eb57ace935d2a7084c07fda5b985a70b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.868ex; height:2.343ex;" alt="{\displaystyle G=-0.08}"></span>.<sup id="cite_ref-Bennu_28-0" class="reference"><a href="#cite_note-Bennu-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>In 2012, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle HG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle HG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fb62307b508eca86c308e23e26eaa6404e5d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.89ex; height:2.176ex;" alt="{\displaystyle HG}"></span>-system was officially replaced by an improved system with three parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ea4f4668b8334c8a7d3d284b0fd22131ef5f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645011b0c6933a02f5f7d84624f78220d747427e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}"></span>, which produces more satisfactory results if the opposition effect is very small or restricted to very small phase angles. However, as of 2022, this <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle HG_{1}G_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle HG_{1}G_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/429bc42e469d4ad1fb12f03e9f09ae7c06053139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.826ex; height:2.509ex;" alt="{\displaystyle HG_{1}G_{2}}"></span>-system has not been adopted by either the Minor Planet Center nor <a href="/wiki/Jet_Propulsion_Laboratory" title="Jet Propulsion Laboratory">Jet Propulsion Laboratory</a>.<sup id="cite_ref-Karttunen2016_13-4" class="reference"><a href="#cite_note-Karttunen2016-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Shevchenko2016_29-0" class="reference"><a href="#cite_note-Shevchenko2016-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>The apparent magnitude of asteroids <a href="/wiki/Light_curve" title="Light curve">varies as they rotate</a>, on time scales of seconds to weeks depending on their <a href="/wiki/Rotation_period" class="mw-redirect" title="Rotation period">rotation period</a>, by up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\text{ mag}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> mag</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\text{ mag}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/846b93a5a10a3f08e34e9ac760600cd6b4b0029d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.004ex; height:2.509ex;" alt="{\displaystyle 2{\text{ mag}}}"></span> or more.<sup id="cite_ref-lc_30-0" class="reference"><a href="#cite_note-lc-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> In addition, their absolute magnitude can vary with the viewing direction, depending on their <a href="/wiki/Axial_tilt" title="Axial tilt">axial tilt</a>. In many cases, neither the rotation period nor the axial tilt are known, limiting the predictability. The models presented here do not capture those effects.<sup id="cite_ref-dymock_26-2" class="reference"><a href="#cite_note-dymock-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Karttunen2016_13-5" class="reference"><a href="#cite_note-Karttunen2016-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Cometary_magnitudes">Cometary magnitudes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=11" title="Edit section: Cometary magnitudes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The brightness of <a href="/wiki/Comet" title="Comet">comets</a> is given separately as <i>total magnitude</i> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></span>, the brightness integrated over the entire visible extend of the <a href="/wiki/Coma_(cometary)" class="mw-redirect" title="Coma (cometary)">coma</a>) and <i>nuclear magnitude</i> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecebe334d5cadc3ffcf245eb02919034d7a2ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{2}}"></span>, the brightness of the core region alone).<sup id="cite_ref-MPES_31-0" class="reference"><a href="#cite_note-MPES-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Both are different scales than the magnitude scale used for planets and asteroids, and can not be used for a size comparison with an asteroid's absolute magnitude <span class="texhtml mvar" style="font-style:italic;">H</span>. </p><p>The activity of comets varies with their distance from the Sun. Their brightness can be approximated as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}=M_{1}+2.5\cdot K_{1}\log _{10}{\left({\frac {d_{BS}}{d_{0}}}\right)}+5\log _{10}{\left({\frac {d_{BO}}{d_{0}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2.5</mn> <mo>⋅</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>S</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>O</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}=M_{1}+2.5\cdot K_{1}\log _{10}{\left({\frac {d_{BS}}{d_{0}}}\right)}+5\log _{10}{\left({\frac {d_{BO}}{d_{0}}}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efdac1e9a2f6a3c1637ec9ed93fff34b36cdeaae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.473ex; height:6.176ex;" alt="{\displaystyle m_{1}=M_{1}+2.5\cdot K_{1}\log _{10}{\left({\frac {d_{BS}}{d_{0}}}\right)}+5\log _{10}{\left({\frac {d_{BO}}{d_{0}}}\right)}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{2}=M_{2}+2.5\cdot K_{2}\log _{10}{\left({\frac {d_{BS}}{d_{0}}}\right)}+5\log _{10}{\left({\frac {d_{BO}}{d_{0}}}\right)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>2.5</mn> <mo>⋅</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>S</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>O</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{2}=M_{2}+2.5\cdot K_{2}\log _{10}{\left({\frac {d_{BS}}{d_{0}}}\right)}+5\log _{10}{\left({\frac {d_{BO}}{d_{0}}}\right)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf20622ae4cd476504f9524e3eb7e4e0f134ab22" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.12ex; height:6.176ex;" alt="{\displaystyle m_{2}=M_{2}+2.5\cdot K_{2}\log _{10}{\left({\frac {d_{BS}}{d_{0}}}\right)}+5\log _{10}{\left({\frac {d_{BO}}{d_{0}}}\right)},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1,2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1,2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/012a22ac20ac8f8d831f2eed78ee4a59f58e728a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.374ex; height:2.343ex;" alt="{\displaystyle m_{1,2}}"></span> are the total and nuclear apparent magnitudes of the comet, respectively, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1,2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1,2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1636773c5b9d9a3e946e75a65bd1fc0603c1da18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.588ex; height:2.843ex;" alt="{\displaystyle M_{1,2}}"></span> are its "absolute" total and nuclear magnitudes, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{BS}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{BS}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a375b1faf52b9a9bf9e296165f03591bb094b683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.749ex; height:2.509ex;" alt="{\displaystyle d_{BS}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{BO}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>O</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{BO}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/285427d64187c90f0b720e65b0753cf7b48c8ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.942ex; height:2.509ex;" alt="{\displaystyle d_{BO}}"></span> are the body-sun and body-observer distances, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4740381c16ea98c4132510daa642e93c1e42c049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{0}}"></span> is the <a href="/wiki/Astronomical_Unit" class="mw-redirect" title="Astronomical Unit">Astronomical Unit</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1,2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1,2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a95ac5eb20dcefa5d7bb7c2ee7ed3b2001b17a5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.307ex; height:2.843ex;" alt="{\displaystyle K_{1,2}}"></span> are the slope parameters characterising the comet's activity. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63b09257ec17c60892f31beed55b0f15608ebcb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.327ex; height:2.176ex;" alt="{\displaystyle K=2}"></span>, this reduces to the formula for a purely reflecting body (showing no cometary activity).<sup id="cite_ref-Meisel1976_32-0" class="reference"><a href="#cite_note-Meisel1976-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>For example, the lightcurve of comet <a href="/wiki/C/2011_L4" class="mw-redirect" title="C/2011 L4">C/2011 L4 (PANSTARRS)</a> can be approximated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}=5.41{\text{, }}K_{1}=3.69.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5.41</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>, </mtext> </mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>3.69.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}=5.41{\text{, }}K_{1}=3.69.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22dad0a725b0082b939d0f54cd32a146040f6a5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.675ex; height:2.509ex;" alt="{\displaystyle M_{1}=5.41{\text{, }}K_{1}=3.69.}"></span><sup id="cite_ref-COBS_2011L4_33-0" class="reference"><a href="#cite_note-COBS_2011L4-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> On the day of its perihelion passage, 10 March 2013, comet PANSTARRS was <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.302{\text{ AU}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.302</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> AU</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.302{\text{ AU}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e3758b42a9c2d033fb0c23fa1bff96186d5db94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.364ex; height:2.176ex;" alt="{\displaystyle 0.302{\text{ AU}}}"></span> from the Sun and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.109{\text{ AU}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.109</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> AU</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.109{\text{ AU}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98df893b46907b19ecd5dbf01b82cbee90ee9dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.364ex; height:2.176ex;" alt="{\displaystyle 1.109{\text{ AU}}}"></span> from Earth. The total apparent magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></span> is predicted to have been <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}=5.41+2.5\cdot 3.69\cdot \log _{10}{\left(0.302\right)}+5\log _{10}{\left(1.109\right)}=+0.8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5.41</mn> <mo>+</mo> <mn>2.5</mn> <mo>⋅</mo> <mn>3.69</mn> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mn>0.302</mn> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mn>1.109</mn> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mo>+</mo> <mn>0.8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}=5.41+2.5\cdot 3.69\cdot \log _{10}{\left(0.302\right)}+5\log _{10}{\left(1.109\right)}=+0.8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7883f4199b80abadcb89938040b5a87cc4ef3a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.582ex; height:2.843ex;" alt="{\displaystyle m_{1}=5.41+2.5\cdot 3.69\cdot \log _{10}{\left(0.302\right)}+5\log _{10}{\left(1.109\right)}=+0.8}"></span> at that time. The Minor Planet Center gives a value close to that, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1}=+0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>+</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}=+0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995f29e19fdd10ccc0dadc99fcbd3d418a56cc4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.973ex; height:2.509ex;" alt="{\displaystyle m_{1}=+0.5}"></span>.<sup id="cite_ref-MPC2011L4_34-0" class="reference"><a href="#cite_note-MPC2011L4-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable sortable" style="font-size: 0.9em;"> <caption>Absolute magnitudes and sizes of comet nuclei </caption> <tbody><tr> <th>Comet </th> <th>Absolute<br />magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d686fc81d1d1eb3ae54e78aeee8957baf6718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}"></span><sup id="cite_ref-kidger_35-0" class="reference"><a href="#cite_note-kidger-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </th> <th>Nucleus<br />diameter </th></tr> <tr> <td><a href="/wiki/Comet_Sarabat" class="mw-redirect" title="Comet Sarabat">Comet Sarabat</a></td> <td>−3.0</td> <td>≈100 km? </td></tr> <tr> <td><a href="/wiki/Comet_Hale-Bopp" class="mw-redirect" title="Comet Hale-Bopp">Comet Hale-Bopp</a></td> <td>−1.3</td> <td>60 ± 20 km </td></tr> <tr> <td><a href="/wiki/Comet_Halley" class="mw-redirect" title="Comet Halley">Comet Halley</a></td> <td>4.0</td> <td>14.9 x 8.2 km </td></tr> <tr> <td>average new comet</td> <td>6.5</td> <td>≈2 km<sup id="cite_ref-Hughes_36-0" class="reference"><a href="#cite_note-Hughes-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/C/2014_UN271_(Bernardinelli-Bernstein)" class="mw-redirect" title="C/2014 UN271 (Bernardinelli-Bernstein)">C/2014 UN<sub>271</sub> (Bernardinelli-Bernstein)</a></td> <td>6.7<sup id="cite_ref-Bernardinelli_37-0" class="reference"><a href="#cite_note-Bernardinelli-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup></td> <td>60–200 km?<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ATel14759_39-0" class="reference"><a href="#cite_note-ATel14759-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/289P/Blanpain" title="289P/Blanpain">289P/Blanpain</a> (during 1819 outburst)</td> <td>8.5<sup id="cite_ref-Yoshida_40-0" class="reference"><a href="#cite_note-Yoshida-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup></td> <td>320 m<sup id="cite_ref-Jewitt_41-0" class="reference"><a href="#cite_note-Jewitt-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>289P/Blanpain (normal activity)</td> <td>22.9<sup id="cite_ref-JPL_289_42-0" class="reference"><a href="#cite_note-JPL_289-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></td> <td>320 m </td></tr></tbody></table> <p>The absolute magnitude of any given comet can vary dramatically. It can change as the comet becomes more or less active over time or if it undergoes an outburst. This makes it difficult to use the absolute magnitude for a size estimate. When comet <a href="/wiki/289P/Blanpain" title="289P/Blanpain">289P/Blanpain</a> was discovered in 1819, its absolute magnitude was estimated as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}=8.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>8.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}=8.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18189572c6b42038ebddc4f64f48363696b83816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.379ex; height:2.509ex;" alt="{\displaystyle M_{1}=8.5}"></span>.<sup id="cite_ref-Yoshida_40-1" class="reference"><a href="#cite_note-Yoshida-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> It was subsequently lost and was only rediscovered in 2003. At that time, its absolute magnitude had decreased to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}=22.9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>22.9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}=22.9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71737dbd3ecf8ad0f226b8629e295561bd593ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.541ex; height:2.509ex;" alt="{\displaystyle M_{1}=22.9}"></span>,<sup id="cite_ref-JPL_289_42-1" class="reference"><a href="#cite_note-JPL_289-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> and it was realised that the 1819 apparition coincided with an outburst. 289P/Blanpain reached naked eye brightness (5–8 mag) in 1819, even though it is the comet with the smallest nucleus that has ever been physically characterised, and usually doesn't become brighter than 18 mag.<sup id="cite_ref-Yoshida_40-2" class="reference"><a href="#cite_note-Yoshida-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Jewitt_41-1" class="reference"><a href="#cite_note-Jewitt-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>For some comets that have been observed at heliocentric distances large enough to distinguish between light reflected from the coma, and light from the nucleus itself, an absolute magnitude analogous to that used for asteroids has been calculated, allowing to estimate the sizes of their nuclei.<sup id="cite_ref-Lamy2004_43-0" class="reference"><a href="#cite_note-Lamy2004-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Meteors">Meteors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=12" title="Edit section: Meteors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Meteor" title="Meteor">meteor</a>, the standard distance for measurement of magnitudes is at an altitude of 100 km (62 mi) at the observer's <a href="/wiki/Zenith" title="Zenith">zenith</a>.<sup id="cite_ref-IMO_44-0" class="reference"><a href="#cite_note-IMO-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SSD_Glossary_45-0" class="reference"><a href="#cite_note-SSD_Glossary-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Araucaria_Project" title="Araucaria Project">Araucaria Project</a></li> <li><a href="/wiki/Hertzsprung%E2%80%93Russell_diagram" title="Hertzsprung–Russell diagram">Hertzsprung–Russell diagram</a> – relates absolute magnitude or <a href="/wiki/Luminosity" title="Luminosity">luminosity</a> versus spectral color or surface <a href="/wiki/Temperature" title="Temperature">temperature</a>.</li> <li><a href="/wiki/Jansky" title="Jansky">Jansky</a> - the preferred unit for radio astronomy – linear in power/unit area</li> <li><a href="/wiki/List_of_most_luminous_stars" title="List of most luminous stars">List of most luminous stars</a></li> <li><a href="/wiki/Photographic_magnitude" title="Photographic magnitude">Photographic magnitude</a></li> <li><a href="/wiki/Surface_brightness" title="Surface brightness">Surface brightness</a> – the <i>magnitude</i> for extended objects</li> <li><a href="/wiki/Zero_point_(photometry)" title="Zero point (photometry)">Zero point (photometry)</a> – the typical calibration point for star flux</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-SunAbs-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-SunAbs_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html">"Sun Fact Sheet"</a>. <a href="/wiki/NASA_Goddard_Space_Flight_Center" class="mw-redirect" title="NASA Goddard Space Flight Center">NASA Goddard Space Flight Center</a><span class="reference-accessdate">. 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Pearson. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontomo00bwca_613/page/n59">60</a>–62. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-44284-0" title="Special:BookSources/978-0-321-44284-0"><bdi>978-0-321-44284-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Modern+Astrophysics&rft.pages=60-62&rft.edition=2nd&rft.pub=Pearson&rft.date=2007&rft.isbn=978-0-321-44284-0&rft.au=Carroll%2C+B.+W.&rft.au=Ostlie%2C+D.+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontomo00bwca_613&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> <li id="cite_note-Unsoeld2013-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Unsoeld2013_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUnsöld,_A.Baschek,_B.2013" class="citation cs2">Unsöld, A.; Baschek, B. (2013), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PrzoCAAAQBAJ&pg=PA331"><i>The New Cosmos: An Introduction to Astronomy and Astrophysics</i></a> (5th ed.), <a href="/wiki/Springer_Science_%26_Business_Media" class="mw-redirect" title="Springer Science & Business Media">Springer Science & Business Media</a>, p. 331, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3662043561" title="Special:BookSources/978-3662043561"><bdi>978-3662043561</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+Cosmos%3A+An+Introduction+to+Astronomy+and+Astrophysics&rft.pages=331&rft.edition=5th&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-3662043561&rft.au=Uns%C3%B6ld%2C+A.&rft.au=Baschek%2C+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPrzoCAAAQBAJ%26pg%3DPA331&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> <li id="cite_note-IAU_XXIX-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-IAU_XXIX_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.iau.org/news/announcements/detail/ann15023/">"IAU XXIX General Assembly Draft Resolutions Announced"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">8 July</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=IAU+XXIX+General+Assembly+Draft+Resolutions+Announced&rft_id=http%3A%2F%2Fwww.iau.org%2Fnews%2Fannouncements%2Fdetail%2Fann15023%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> <li id="cite_note-IAU2015B2-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-IAU2015B2_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IAU2015B2_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-IAU2015B2_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMamajekTorresPrsaHarmanec2015" class="citation cs2">Mamajek, E. E.; Torres, G.; Prsa, A.; Harmanec, P.; Asplund, M.; Bennett, P. D.; Capitaine, N.; Christensen-Dalsgaard, J.; Depagne, E.; Folkner, W. M.; Haberreiter, M.; Hekker, S.; Hilton, J. L.; Kostov, V.; Kurtz, D. W.; Laskar, J.; Mason, B. D.; Milone, E. F.; Montgomery, M. M.; Richards, M. T.; Schou, J.; Stewart, S. G. (13 August 2015), <a rel="nofollow" class="external text" href="https://www.iau.org/static/resolutions/IAU2015_English.pdf">"IAU 2015 Resolution B2 on Recommended Zero Points for the Absolute and Apparent Bolometric Magnitude Scales"</a> <span class="cs1-format">(PDF)</span>, <i>Resolutions Adopted at the General Assemblies</i>, IAU Inter-Division A-G Working Group on Nominal Units for Stellar & Planetary Astronomy, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1510.06262">1510.06262</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015arXiv151006262M">2015arXiv151006262M</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Resolutions+Adopted+at+the+General+Assemblies&rft.atitle=IAU+2015+Resolution+B2+on+Recommended+Zero+Points+for+the+Absolute+and+Apparent+Bolometric+Magnitude+Scales&rft.date=2015-08-13&rft_id=info%3Aarxiv%2F1510.06262&rft_id=info%3Abibcode%2F2015arXiv151006262M&rft.aulast=Mamajek&rft.aufirst=E.+E.&rft.au=Torres%2C+G.&rft.au=Prsa%2C+A.&rft.au=Harmanec%2C+P.&rft.au=Asplund%2C+M.&rft.au=Bennett%2C+P.+D.&rft.au=Capitaine%2C+N.&rft.au=Christensen-Dalsgaard%2C+J.&rft.au=Depagne%2C+E.&rft.au=Folkner%2C+W.+M.&rft.au=Haberreiter%2C+M.&rft.au=Hekker%2C+S.&rft.au=Hilton%2C+J.+L.&rft.au=Kostov%2C+V.&rft.au=Kurtz%2C+D.+W.&rft.au=Laskar%2C+J.&rft.au=Mason%2C+B.+D.&rft.au=Milone%2C+E.+F.&rft.au=Montgomery%2C+M.+M.&rft.au=Richards%2C+M.+T.&rft.au=Schou%2C+J.&rft.au=Stewart%2C+S.+G.&rft_id=https%3A%2F%2Fwww.iau.org%2Fstatic%2Fresolutions%2FIAU2015_English.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://cneos.jpl.nasa.gov/tools/ast_size_est.html">CNEOS Asteroid Size Estimator</a></span> </li> <li id="cite_note-Luciuk-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Luciuk_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuciuk,_M." class="citation cs2">Luciuk, M., <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180920225551/http://www.asterism.org/tutorials/tut35%20Magnitudes.pdf"><i>Astronomical Magnitudes</i></a> <span class="cs1-format">(PDF)</span>, p. 8, archived from <a rel="nofollow" class="external text" href="http://www.asterism.org/tutorials/tut35%20Magnitudes.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 20 September 2018<span class="reference-accessdate">, retrieved <span class="nowrap">11 January</span> 2019</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astronomical+Magnitudes&rft.pages=8&rft.au=Luciuk%2C+M.&rft_id=http%3A%2F%2Fwww.asterism.org%2Ftutorials%2Ftut35%2520Magnitudes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> <li id="cite_note-Karttunen2016-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Karttunen2016_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Karttunen2016_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Karttunen2016_13-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Karttunen2016_13-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Karttunen2016_13-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Karttunen2016_13-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarttunen,_H.Kröger,_P.Oja,_H.Poutanen,_M.2016" class="citation book cs1">Karttunen, H.; Kröger, P.; Oja, H.; Poutanen, M.; Donner, K. J. (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ndd2DQAAQBAJ&pg=PA163"><i>Fundamental Astronomy</i></a>. Springer. p. 163. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783662530450" title="Special:BookSources/9783662530450"><bdi>9783662530450</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Astronomy&rft.pages=163&rft.pub=Springer&rft.date=2016&rft.isbn=9783662530450&rft.au=Karttunen%2C+H.&rft.au=Kr%C3%B6ger%2C+P.&rft.au=Oja%2C+H.&rft.au=Poutanen%2C+M.&rft.au=Donner%2C+K.+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dndd2DQAAQBAJ%26pg%3DPA163&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> <li id="cite_note-Whitmell1907-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whitmell1907_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitmell,_C._T.1907" class="citation cs2">Whitmell, C. T. (1907), <a rel="nofollow" class="external text" href="http://adsbit.harvard.edu/full/seri/Obs../0030//0000097.000.html">"Brightness of a planet"</a>, <i>The Observatory</i>, <b>30</b>: 97, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1907Obs....30...96W">1907Obs....30...96W</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Observatory&rft.atitle=Brightness+of+a+planet&rft.volume=30&rft.pages=97&rft.date=1907&rft_id=info%3Abibcode%2F1907Obs....30...96W&rft.au=Whitmell%2C+C.+T.&rft_id=http%3A%2F%2Fadsbit.harvard.edu%2Ffull%2Fseri%2FObs..%2F0030%2F%2F0000097.000.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> <li id="cite_note-sizemagnitude-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-sizemagnitude_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruton,_D." class="citation cs2">Bruton, D., <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110723191750/http://www.physics.sfasu.edu/astro/asteroids/sizemagnitude.html"><i>Conversion of Absolute Magnitude to Diameter for Minor Planets</i></a>, Stephen F. Austin State University, archived from <a rel="nofollow" class="external text" href="http://www.physics.sfasu.edu/astro/asteroids/sizemagnitude.html">the original</a> on 23 July 2011<span class="reference-accessdate">, retrieved <span class="nowrap">12 January</span> 2019</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conversion+of+Absolute+Magnitude+to+Diameter+for+Minor+Planets&rft.pub=Stephen+F.+Austin+State+University&rft.au=Bruton%2C+D.&rft_id=http%3A%2F%2Fwww.physics.sfasu.edu%2Fastro%2Fasteroids%2Fsizemagnitude.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> <li id="cite_note-Mag_formula-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mag_formula_16-0">^</a></b></span> <span class="reference-text">The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1329{\text{ km}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1329</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> km</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1329{\text{ km}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94acf8fe546609ef534113c860da936089145f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.394ex; height:2.176ex;" alt="{\displaystyle 1329{\text{ km}}}"></span> factor can be computed as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\text{ AU}}\cdot 10^{H_{\text{Sun}}/5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> AU</mtext> </mrow> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sun</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\text{ AU}}\cdot 10^{H_{\text{Sun}}/5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/685714b9c0a1048475492fbd8e57235976967d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.865ex; height:2.843ex;" alt="{\displaystyle 2{\text{ AU}}\cdot 10^{H_{\text{Sun}}/5}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\text{Sun}}=-26.76}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sun</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>26.76</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\text{Sun}}=-26.76}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae78deec0ded2e8e1859785d586128712c40a28e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.109ex; height:2.509ex;" alt="{\displaystyle H_{\text{Sun}}=-26.76}"></span>, the absolute magnitude of the Sun, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1{\text{ AU}}=1.4959787\times 10^{8}{\text{ km}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> AU</mtext> </mrow> <mo>=</mo> <mn>1.4959787</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> km</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1{\text{ AU}}=1.4959787\times 10^{8}{\text{ km}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88b9c0b0586aebd9132a7a625481e30df17f1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; 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Retrieved <span class="nowrap">16 May</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Solar+System+Dynamics+Glossary+%E2%80%93+Absolute+magnitude+of+Solar+System+bodies&rft.pub=NASA+Jet+Propulsion+Laboratory&rft_id=http%3A%2F%2Fssd.jpl.nasa.gov%2F%3Fglossary%26term%3DH&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+magnitude" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_magnitude&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://astro.pas.rochester.edu/~aquillen/ast142/costanti.html">Reference zero-magnitude fluxes</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20030222000548/http://astro.pas.rochester.edu/~aquillen/ast142/costanti.html">Archived</a> 22 February 2003 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.iau.org/">International Astronomical Union</a></li> <li><a rel="nofollow" class="external text" href="http://www.fxsolver.com/solve/share/knc8CRLY7WMX324G2I7GXg==/">Absolute Magnitude of a Star calculator</a></li> <li><a rel="nofollow" class="external text" href="http://www.astronomynotes.com/starprop/s4.htm">The Magnitude system</a></li> <li><a rel="nofollow" class="external text" href="http://webcitation.org/query.php">About stellar magnitudes</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211027163807/https://webcitation.org/query.php">Archived</a> 27 October 2021 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://simbad.u-strasbg.fr/sim-fid.pl">Obtain the magnitude of any star</a> – <a href="/wiki/SIMBAD" title="SIMBAD">SIMBAD</a></li> <li><a rel="nofollow" class="external text" href="http://www.minorplanetcenter.org/iau/lists/Sizes.html">Converting magnitude of minor planets to diameter</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20010302182040/http://neo.jpl.nasa.gov/glossary/h.html">Another table for converting asteroid magnitude to estimated diameter</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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globule</a></li> <li><a href="/wiki/Young_stellar_object" title="Young stellar object">Young stellar object</a> <ul><li><a href="/wiki/Protostar" title="Protostar">Protostar</a></li> <li><a href="/wiki/Pre-main-sequence_star" title="Pre-main-sequence star">Pre-main-sequence</a></li> <li><a href="/wiki/Herbig_Ae/Be_star" title="Herbig Ae/Be star">Herbig Ae/Be</a></li> <li><a href="/wiki/T_Tauri_star" title="T Tauri star">T Tauri</a></li></ul></li> <li><a href="/wiki/Herbig%E2%80%93Haro_object" title="Herbig–Haro object">Herbig–Haro object</a></li> <li><a href="/wiki/Hayashi_track" title="Hayashi track">Hayashi track</a></li> <li><a href="/wiki/Henyey_track" title="Henyey track">Henyey track</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Stellar_evolution" title="Stellar evolution">Evolution</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Main_sequence" title="Main sequence">Main sequence</a></li> <li><a href="/wiki/Red-giant_branch" title="Red-giant branch">Red-giant branch</a></li> <li><a href="/wiki/Horizontal_branch" title="Horizontal branch">Horizontal branch</a> <ul><li><a href="/wiki/Red_clump" title="Red clump">Red clump</a></li></ul></li> <li><a href="/wiki/Asymptotic_giant_branch" title="Asymptotic giant branch">Asymptotic giant branch</a> <ul><li><a href="/wiki/Post-AGB_star" title="Post-AGB star">post-AGB</a></li> <li><a href="/wiki/Super-AGB_star" title="Super-AGB star">super-AGB</a></li></ul></li> <li><a href="/wiki/Blue_loop" title="Blue loop">Blue loop</a></li> <li><a href="/wiki/Planetary_nebula" title="Planetary nebula">Planetary nebula</a> <ul><li><a href="/wiki/Protoplanetary_nebula" title="Protoplanetary nebula">Protoplanetary</a></li></ul></li> <li><a href="/wiki/Wolf%E2%80%93Rayet_nebula" title="Wolf–Rayet nebula">Wolf–Rayet nebula</a></li> <li><a href="/wiki/PG_1159_star" title="PG 1159 star">PG1159</a></li> <li><a href="/wiki/Dredge-up" title="Dredge-up">Dredge-up</a></li> <li><a href="/wiki/OH/IR_star" title="OH/IR star">OH/IR</a></li> <li><a href="/wiki/Instability_strip" title="Instability strip">Instability strip</a></li> <li><a href="/wiki/Luminous_blue_variable" title="Luminous blue variable">Luminous blue variable</a></li> <li><a href="/wiki/Stellar_population" title="Stellar population">Stellar population</a></li> <li><a href="/wiki/Supernova" title="Supernova">Supernova</a> <ul><li><a href="/wiki/Superluminous_supernova" title="Superluminous supernova">Superluminous</a></li> <li><a href="/wiki/Hypernova" title="Hypernova">Hypernova</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Stellar_classification" title="Stellar classification">Classification</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Early-type_star" class="mw-redirect" title="Early-type star">Early</a></li> <li><a href="/wiki/Late-type_star" class="mw-redirect" title="Late-type star">Late</a></li> <li>Main sequence <ul><li><a href="/wiki/O-type_main-sequence_star" title="O-type main-sequence star">O</a></li> <li><a href="/wiki/B-type_main-sequence_star" title="B-type main-sequence star">B</a></li> <li><a href="/wiki/A-type_main-sequence_star" title="A-type main-sequence star">A</a></li> <li><a href="/wiki/F-type_main-sequence_star" title="F-type main-sequence star">F</a></li> <li><a href="/wiki/G-type_main-sequence_star" title="G-type main-sequence star">G</a></li> <li><a href="/wiki/K-type_main-sequence_star" title="K-type main-sequence star">K</a></li> <li><a href="/wiki/Red_dwarf" title="Red dwarf">M</a></li></ul></li> <li><a href="/wiki/Subdwarf" title="Subdwarf">Subdwarf</a> <ul><li><a href="/wiki/Subdwarf_O_star" title="Subdwarf O star">O</a></li> <li><a href="/wiki/Subdwarf_B_star" title="Subdwarf B star">B</a></li></ul></li> <li><a href="/wiki/Wolf%E2%80%93Rayet_star" title="Wolf–Rayet star">WR</a></li> <li><a href="/wiki/OB_star" title="OB star">OB</a></li> <li><a href="/wiki/Subgiant" title="Subgiant">Subgiant</a></li> <li><a href="/wiki/Giant_star" title="Giant star">Giant</a> <ul><li><a href="/wiki/Blue_giant" title="Blue giant">Blue</a></li> <li><a href="/wiki/Red_giant" title="Red giant">Red</a></li> <li><a href="/wiki/Yellow_giant" class="mw-redirect" title="Yellow giant">Yellow</a></li></ul></li> <li><a href="/wiki/Bright_giant" class="mw-redirect" title="Bright giant">Bright giant</a></li> <li><a href="/wiki/Supergiant" title="Supergiant">Supergiant</a> <ul><li><a href="/wiki/Blue_supergiant" title="Blue supergiant">Blue</a></li> <li><a href="/wiki/Red_supergiant" title="Red supergiant">Red</a></li> <li><a href="/wiki/Yellow_supergiant" title="Yellow supergiant">Yellow</a></li></ul></li> <li><a href="/wiki/Hypergiant" title="Hypergiant">Hypergiant</a> <ul><li><a href="/wiki/Yellow_hypergiant" title="Yellow hypergiant">Yellow</a></li></ul></li> <li><a href="/wiki/Carbon_star" title="Carbon star">Carbon</a> <ul><li><a href="/wiki/S-type_star" title="S-type star">S</a></li> <li><a href="/wiki/CN_star" title="CN star">CN</a></li> <li><a href="/wiki/CH_star" title="CH star">CH</a></li></ul></li> <li><a href="/wiki/White_dwarf" title="White dwarf">White dwarf</a></li> <li><a href="/wiki/Chemically_peculiar_star" title="Chemically peculiar star">Chemically peculiar</a> <ul><li><a href="/wiki/Am_star" title="Am star">Am</a></li> <li><a href="/wiki/Ap_and_Bp_stars" title="Ap and Bp stars">Ap/Bp</a></li> <li><a href="/wiki/CEMP_star" title="CEMP star">CEMP</a></li> <li><a href="/wiki/Mercury-manganese_star" title="Mercury-manganese star">HgMn</a></li> <li><a href="/wiki/Helium-weak_star" title="Helium-weak star">He-weak</a></li> <li><a href="/wiki/Barium_star" title="Barium star">Barium</a></li> <li><a href="/wiki/Lambda_Bo%C3%B6tis_star" title="Lambda Boötis star">Lambda Boötis</a></li> <li><a href="/wiki/Lead_star" title="Lead star">Lead</a></li> <li><a href="/wiki/Technetium_star" title="Technetium star">Technetium</a></li></ul></li> <li><a href="/wiki/Be_star" title="Be star">Be</a> <ul><li><a href="/wiki/Shell_star" title="Shell star">Shell</a></li></ul></li> <li><a href="/wiki/B(e)_star" title="B(e) star">B[e]</a></li> <li><a href="/wiki/Helium_star" title="Helium star">Helium</a> <ul><li><a href="/wiki/Extreme_helium_star" title="Extreme helium star">Extreme</a></li></ul></li> <li><a href="/wiki/Blue_straggler" title="Blue straggler">Blue straggler</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Compact_star" class="mw-redirect" title="Compact star">Remnants</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Compact_star" class="mw-redirect" title="Compact star">Compact star</a></li> <li><a href="/wiki/IRAS_00500%2B6713" title="IRAS 00500+6713">Parker's star</a></li> <li><a href="/wiki/White_dwarf" title="White dwarf">White dwarf</a> <ul><li><a href="/wiki/Helium_planet" title="Helium planet">Helium planet</a></li></ul></li> <li><a href="/wiki/Neutron_star" title="Neutron star">Neutron</a> <ul><li><a href="/wiki/Radio-quiet_neutron_star" title="Radio-quiet neutron star">Radio-quiet</a></li> <li><a href="/wiki/Pulsar" title="Pulsar">Pulsar</a> <ul><li><a href="/wiki/Binary_pulsar" title="Binary pulsar">Binary</a></li> <li><a href="/wiki/X-ray_pulsar" title="X-ray pulsar">X-ray</a></li></ul></li> <li><a href="/wiki/Magnetar" title="Magnetar">Magnetar</a></li></ul></li> <li><a href="/wiki/Stellar_black_hole" title="Stellar black hole">Stellar black hole</a></li> <li><a href="/wiki/X-ray_binary" title="X-ray binary">X-ray binary</a> <ul><li><a href="/wiki/X-ray_burster" title="X-ray burster">Burster</a></li></ul></li> <li><a href="/wiki/Soft_gamma_repeater" title="Soft gamma repeater">SGR</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hypothetical_star" title="Hypothetical star">Hypothetical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blue_dwarf_(red-dwarf_stage)" title="Blue dwarf (red-dwarf stage)">Blue dwarf</a></li> <li><a href="/wiki/Black_dwarf" title="Black dwarf">Black dwarf</a></li> <li><a href="/wiki/Exotic_star" title="Exotic star">Exotic</a> <ul><li><a href="/wiki/Boson_star" class="mw-redirect" title="Boson star">Boson</a></li> <li><a href="/wiki/Electroweak_star" class="mw-redirect" title="Electroweak star">Electroweak</a></li> <li><a href="/wiki/Strange_star" title="Strange star">Strange</a></li> <li><a href="/wiki/Preon_star" class="mw-redirect" title="Preon star">Preon</a></li> <li><a href="/wiki/Planck_star" title="Planck star">Planck</a></li> <li><a href="/wiki/Dark_star_(dark_matter)" title="Dark star (dark matter)">Dark</a></li> <li><a href="/wiki/Dark-energy_star" title="Dark-energy star">Dark-energy</a></li> <li><a href="/wiki/Quark_star" title="Quark star">Quark</a></li> <li><a href="/wiki/Q_star" title="Q star">Q</a></li></ul></li> <li>Black hole star <ul><li><a href="/wiki/Black_star_(semiclassical_gravity)" title="Black star (semiclassical gravity)">Black</a></li> <li>Hawking</li> <li><a href="/wiki/Quasi-star" title="Quasi-star">Quasi-star</a></li></ul></li> <li><a href="/wiki/Gravastar" title="Gravastar">Gravastar</a></li> <li><a href="/wiki/Thorne%E2%80%93%C5%BBytkow_object" title="Thorne–Żytkow object">Thorne–Żytkow object</a></li> <li><a href="/wiki/Iron_star" title="Iron star">Iron</a></li> <li><a href="/wiki/Blitzar" title="Blitzar">Blitzar</a></li> <li><a href="/wiki/White_hole" title="White hole">White hole</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Stellar_nucleosynthesis" title="Stellar nucleosynthesis">Nucleosynthesis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deuterium_fusion" title="Deuterium fusion">Deuterium burning</a></li> <li><a href="/wiki/Lithium_burning" title="Lithium burning">Lithium burning</a></li> <li><a href="/wiki/Proton%E2%80%93proton_chain" title="Proton–proton chain">Proton–proton chain</a></li> <li><a href="/wiki/CNO_cycle" title="CNO cycle">CNO cycle</a></li> <li><a href="/wiki/Helium_flash" title="Helium flash">Helium flash</a></li> <li><a href="/wiki/Triple-alpha_process" title="Triple-alpha process">Triple-alpha process</a></li> <li><a href="/wiki/Alpha_process" title="Alpha process">Alpha process</a></li> <li><a href="/wiki/Carbon-burning_process" title="Carbon-burning process">C burning</a></li> <li><a href="/wiki/Neon-burning_process" title="Neon-burning process">Ne burning</a></li> <li><a href="/wiki/Oxygen-burning_process" title="Oxygen-burning process">O burning</a></li> <li><a href="/wiki/Silicon-burning_process" title="Silicon-burning process">Si burning</a></li> <li><a href="/wiki/S-process" title="S-process">s-process</a></li> <li><a href="/wiki/R-process" title="R-process">r-process</a></li> <li><a href="/wiki/P-process" title="P-process">p-process</a></li> <li><a href="/wiki/Nova" title="Nova">Nova</a> <ul><li><a href="/wiki/Symbiotic_nova" title="Symbiotic nova">Symbiotic</a></li> <li><a href="/wiki/Nova_remnant" title="Nova remnant">Remnant</a></li> <li><a href="/wiki/Luminous_red_nova" title="Luminous red nova">Luminous red nova</a></li> <li><a href="/wiki/Nova#Recurrent_novae" title="Nova">Recurrent</a></li> <li><a href="/wiki/Micronova" title="Micronova">Micronova</a></li></ul></li> <li><a href="/wiki/Supernova_nucleosynthesis" title="Supernova nucleosynthesis">Supernova</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Stellar_structure" title="Stellar structure">Structure</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Stellar_core" title="Stellar core">Core</a></li> <li><a href="/wiki/Convection_zone" title="Convection zone">Convection zone</a> <ul><li><a href="/wiki/Microturbulence" title="Microturbulence">Microturbulence</a></li> <li><a href="/wiki/Solar-like_oscillations" title="Solar-like oscillations">Oscillations</a></li></ul></li> <li><a href="/wiki/Radiation_zone" class="mw-redirect" title="Radiation zone">Radiation zone</a></li> <li><a href="/wiki/Stellar_atmosphere" title="Stellar atmosphere">Atmosphere</a> <ul><li><a href="/wiki/Photosphere" title="Photosphere">Photosphere</a></li> <li><a href="/wiki/Starspot" title="Starspot">Starspot</a></li> <li><a href="/wiki/Chromosphere" title="Chromosphere">Chromosphere</a></li> <li><a href="/wiki/Stellar_corona" title="Stellar corona">Stellar corona</a></li> <li><a href="/wiki/Alfv%C3%A9n_surface" title="Alfvén surface">Alfvén surface</a></li></ul></li> <li><a href="/wiki/Stellar_wind" title="Stellar wind">Stellar wind</a> <ul><li><a href="/wiki/Stellar-wind_bubble" title="Stellar-wind bubble">Bubble</a></li> <li><a href="/wiki/Bipolar_outflow" title="Bipolar outflow">Bipolar outflow</a></li></ul></li> <li><a href="/wiki/Accretion_disk" title="Accretion disk">Accretion disk</a> <ul><li><a href="/wiki/Protoplanetary_disk" title="Protoplanetary disk">Protoplanetary disk</a></li> <li><a href="/wiki/Proplyd" title="Proplyd">Proplyd</a></li></ul></li> <li><a href="/wiki/Asteroseismology" title="Asteroseismology">Asteroseismology</a> <ul><li><a href="/wiki/Helioseismology" title="Helioseismology">Helioseismology</a></li></ul></li> <li><a href="/wiki/Circumstellar_dust" title="Circumstellar dust">Circumstellar dust</a></li> <li><a href="/wiki/Cosmic_dust" title="Cosmic dust">Cosmic dust</a></li> <li><a href="/wiki/Circumstellar_envelope" title="Circumstellar envelope">Circumstellar envelope</a></li> <li><a href="/wiki/Eddington_luminosity" title="Eddington luminosity">Eddington luminosity</a></li> <li><a href="/wiki/Kelvin%E2%80%93Helmholtz_mechanism" title="Kelvin–Helmholtz mechanism">Kelvin–Helmholtz mechanism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Stellar_designations_and_names" title="Stellar designations and names">Designation</a></li> <li><a href="/wiki/Stellar_dynamics" title="Stellar dynamics">Dynamics</a></li> <li><a href="/wiki/Effective_temperature" title="Effective temperature">Effective temperature</a></li> <li><a href="/wiki/Luminosity" title="Luminosity">Luminosity</a></li> <li><a href="/wiki/Stellar_kinematics" title="Stellar kinematics">Kinematics</a></li> <li><a href="/wiki/Stellar_magnetic_field" title="Stellar magnetic field">Magnetic field</a></li> <li><a class="mw-selflink selflink">Absolute magnitude</a></li> <li><a href="/wiki/Stellar_mass" title="Stellar mass">Mass</a></li> <li><a href="/wiki/Metallicity" title="Metallicity">Metallicity</a></li> <li><a href="/wiki/Stellar_rotation" title="Stellar rotation">Rotation</a></li> <li><a href="/wiki/Starlight" title="Starlight">Starlight</a></li> <li><a href="/wiki/Variable_star" title="Variable star">Variable</a></li> <li><a href="/wiki/Photometric_system" title="Photometric system">Photometric system</a></li> <li><a href="/wiki/Color_index" title="Color index">Color index</a></li> <li><a href="/wiki/Hertzsprung%E2%80%93Russell_diagram" title="Hertzsprung–Russell diagram">Hertzsprung–Russell diagram</a></li> <li><a href="/wiki/Color%E2%80%93color_diagram" title="Color–color diagram">Color–color diagram</a></li> <li><a href="/wiki/Str%C3%B6mgren_sphere" title="Strömgren sphere">Strömgren sphere</a></li> <li><a href="/wiki/Kraft_break" title="Kraft break">Kraft break</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Star_system" title="Star system">Star systems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_star" title="Binary star">Binary</a> <ul><li><a href="/wiki/Contact_binary" title="Contact binary">Contact</a></li> <li><a href="/wiki/Common_envelope" title="Common envelope">Common envelope</a></li> <li><a href="/wiki/Eclipsing_binary" class="mw-redirect" title="Eclipsing binary">Eclipsing</a></li> <li><a href="/wiki/Symbiotic_binary" title="Symbiotic binary">Symbiotic</a></li></ul></li> <li><a href="/wiki/Star_system#Multiple_star_systems" title="Star system">Multiple</a></li> <li><a href="/wiki/Star_cluster" title="Star cluster">Cluster</a> <ul><li><a href="/wiki/Open_cluster" title="Open cluster">Open</a></li> <li><a href="/wiki/Globular_cluster" title="Globular cluster">Globular</a></li> <li><a href="/wiki/Super_star_cluster" title="Super star cluster">Super</a></li></ul></li> <li><a href="/wiki/Planetary_system" title="Planetary system">Planetary system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Earth-centric<br />observations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sun" title="Sun">Sun</a> <ul><li><a href="/wiki/Solar_eclipse" title="Solar eclipse">Solar eclipse</a></li> <li><a href="/wiki/Solar_radio_emission" title="Solar radio emission">Solar radio emission</a></li> <li><a href="/wiki/Solar_System" title="Solar System">Solar System</a></li> <li><a href="/wiki/Sunlight" title="Sunlight">Sunlight</a></li></ul></li> <li><a href="/wiki/Pole_star" title="Pole star">Pole star</a></li> <li><a href="/wiki/Circumpolar_star" title="Circumpolar star">Circumpolar</a></li> <li><a href="/wiki/Constellation" title="Constellation">Constellation</a></li> <li><a href="/wiki/Asterism_(astronomy)" title="Asterism (astronomy)">Asterism</a></li> <li><a href="/wiki/Magnitude_(astronomy)" title="Magnitude (astronomy)">Magnitude</a> <ul><li><a href="/wiki/Apparent_magnitude" title="Apparent magnitude">Apparent</a></li> <li><a href="/wiki/Extinction_(astronomy)" title="Extinction (astronomy)">Extinction</a></li> <li><a href="/wiki/Photographic_magnitude" title="Photographic magnitude">Photographic</a></li></ul></li> <li><a href="/wiki/Radial_velocity" title="Radial velocity">Radial velocity</a></li> <li><a href="/wiki/Proper_motion" title="Proper motion">Proper motion</a></li> <li><a href="/wiki/Stellar_parallax" title="Stellar parallax">Parallax</a></li> <li><a href="/wiki/Photometric-standard_star" title="Photometric-standard star">Photometric-standard</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_stars" title="Lists of stars">Lists</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_proper_names_of_stars" title="List of proper names of stars">Proper names</a> <ul><li><a href="/wiki/List_of_Arabic_star_names" title="List of Arabic star names">Arabic</a></li> <li><a href="/wiki/Chinese_star_names" class="mw-redirect" title="Chinese star names">Chinese</a></li></ul></li> <li><a href="/wiki/List_of_star_extremes" title="List of star extremes">Extremes</a> <ul><li><a href="/wiki/List_of_most_massive_stars" title="List of most massive stars">Most massive</a></li> <li><a href="/wiki/List_of_hottest_stars" title="List of hottest stars">Highest temperature</a></li> <li><a href="/wiki/List_of_coolest_stars" title="List of coolest stars">Lowest temperature</a></li> <li><a href="/wiki/List_of_largest_known_stars" class="mw-redirect" title="List of largest known stars">Largest volume</a></li> <li><a href="/wiki/List_of_smallest_known_stars" title="List of smallest known stars">Smallest volume</a></li> <li><a href="/wiki/List_of_brightest_stars" title="List of brightest stars">Brightest</a></li> <li><a href="/wiki/Historical_brightest_stars" title="Historical brightest stars">Historical brightest</a></li> <li><a href="/wiki/List_of_most_luminous_stars" title="List of most luminous stars">Most luminous</a></li> <li><a href="/wiki/List_of_nearest_stars" title="List of nearest stars">Nearest</a> <ul><li><a href="/wiki/List_of_nearest_bright_stars" title="List of nearest bright stars">bright</a></li></ul></li> <li><a href="/wiki/List_of_the_most_distant_astronomical_objects#List_of_most_distant_objects_by_type" title="List of the most distant astronomical objects">Most distant</a></li></ul></li> <li><a href="/wiki/List_of_stars_with_resolved_images" title="List of stars with resolved images">With resolved images</a></li> <li><a href="/wiki/List_of_multiplanetary_systems" title="List of multiplanetary systems">With multiple exoplanets</a></li> <li><a href="/wiki/List_of_brown_dwarfs" title="List of brown dwarfs">Brown dwarfs</a></li> <li><a href="/wiki/List_of_red_dwarfs" title="List of red dwarfs">Red dwarfs</a></li> <li><a href="/wiki/List_of_white_dwarfs" title="List of white dwarfs">White dwarfs</a></li> <li><a href="/wiki/List_of_novae_in_the_Milky_Way_galaxy" title="List of novae in the Milky Way galaxy">Milky Way novae</a></li> <li><a href="/wiki/List_of_supernovae" title="List of supernovae">Supernovae</a> <ul><li><a href="/wiki/List_of_supernova_candidates" title="List of supernova candidates">Candidates</a></li> <li><a href="/wiki/List_of_supernova_remnants" title="List of supernova remnants">Remnants</a></li></ul></li> <li><a href="/wiki/List_of_planetary_nebulae" title="List of planetary nebulae">Planetary nebulae</a></li> <li><a href="/wiki/Timeline_of_stellar_astronomy" title="Timeline of stellar astronomy">Timeline of stellar astronomy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Substellar_object" title="Substellar object">Substellar object</a> <ul><li><a href="/wiki/Brown_dwarf" title="Brown dwarf">Brown dwarf</a> <ul><li><a href="/wiki/Brown-dwarf_desert" title="Brown-dwarf desert">Desert</a></li> <li><a href="/wiki/Sub-brown_dwarf" title="Sub-brown dwarf">Sub</a></li></ul></li> <li><a href="/wiki/Planet" title="Planet">Planet</a></li></ul></li> <li><a href="/wiki/Galactic_year" title="Galactic year">Galactic year</a></li> <li><a href="/wiki/Galaxy" title="Galaxy">Galaxy</a></li> <li><a href="/wiki/Guest_star_(astronomy)" title="Guest star (astronomy)">Guest</a></li> <li><a href="/wiki/Gravity" title="Gravity">Gravity</a></li> <li><a href="/wiki/Intergalactic_star" title="Intergalactic star">Intergalactic</a></li> <li><a href="/wiki/Planet-hosting_stars" class="mw-redirect" title="Planet-hosting stars">Planet-hosting stars</a></li> <li><a href="/wiki/Tidal_disruption_event" title="Tidal disruption event">Tidal disruption event</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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Absolute magnitude - Wikipedia