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id="toc-Planck's_law_of_blackbody_radiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Planck's_law_of_blackbody_radiation"> <div class="vector-toc-text"> 2.1 Planck's law of blackbody radiation </div> </a> <ul id="toc-Planck's_law_of_blackbody_radiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wien's_displacement_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wien's_displacement_law"> <div class="vector-toc-text"> 2.2 Wien's displacement law </div> </a> <ul id="toc-Wien's_displacement_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stefan–Boltzmann_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stefan–Boltzmann_law"> <div class="vector-toc-text"> 2.3 Stefan–Boltzmann law </div> </a> <ul id="toc-Stefan–Boltzmann_law-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 3 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Human-body_emission" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Human-body_emission"> <div class="vector-toc-text"> 3.1 Human-body emission </div> </a> <ul id="toc-Human-body_emission-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Temperature_relation_between_a_planet_and_its_star" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Temperature_relation_between_a_planet_and_its_star"> <div class="vector-toc-text"> 3.2 Temperature relation between a planet and its star </div> </a> <ul id="toc-Temperature_relation_between_a_planet_and_its_star-sublist" class="vector-toc-list"> <li id="toc-Effective_temperature_of_Earth" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Effective_temperature_of_Earth"> <div class="vector-toc-text"> 3.2.1 Effective temperature of Earth </div> </a> <ul id="toc-Effective_temperature_of_Earth-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cosmology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cosmology"> <div class="vector-toc-text"> 3.3 Cosmology </div> </a> <ul id="toc-Cosmology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Light_bulb" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Light_bulb"> <div class="vector-toc-text"> 3.4 Light bulb </div> </a> <ul id="toc-Light_bulb-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 4 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Balfour_Stewart" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Balfour_Stewart"> <div class="vector-toc-text"> 4.1 Balfour Stewart </div> </a> <ul id="toc-Balfour_Stewart-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gustav_Kirchhoff" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gustav_Kirchhoff"> <div class="vector-toc-text"> 4.2 Gustav Kirchhoff </div> </a> <ul id="toc-Gustav_Kirchhoff-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Doppler_effect" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Doppler_effect"> <div class="vector-toc-text"> 5 Doppler effect </div> </a> <ul id="toc-Doppler_effect-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"> 6 See also </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"> 7 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 7.1 Bibliography </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 8 Further reading </div> </a> <ul id="toc-Further_reading-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"> 9 External links </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" > <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" > Toggle the table of contents </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">Black-body radiation</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 37 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-37" aria-hidden="true" > 37 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A5%D8%B4%D8%B9%D8%A7%D8%B9_%D8%A7%D9%84%D8%AC%D8%B3%D9%85_%D8%A7%D9%84%D8%A3%D8%B3%D9%88%D8%AF" title="إشعاع الجسم الأسود – Arabic" lang="ar" hreflang="ar" data-title="إشعاع الجسم الأسود" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%95%E0%A7%83%E0%A6%B7%E0%A7%8D%E0%A6%A3%E0%A6%95%E0%A6%BE%E0%A6%AF%E0%A6%BC_%E0%A6%AC%E0%A6%BF%E0%A6%95%E0%A6%BF%E0%A7%B0%E0%A6%A3" title="কৃষ্ণকায় বিকিৰণ – Assamese" lang="as" hreflang="as" data-title="কৃষ্ণকায় বিকিৰণ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%83%E0%A6%B7%E0%A7%8D%E0%A6%A3%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A4%E0%A7%81_%E0%A6%AC%E0%A6%BF%E0%A6%95%E0%A6%BF%E0%A6%B0%E0%A6%A3" title="কৃষ্ণবস্তু বিকিরণ – Bangla" lang="bn" hreflang="bn" data-title="কৃষ্ণবস্তু বিকিরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Pelydriad_corff_du" title="Pelydriad corff du – Welsh" lang="cy" hreflang="cy" data-title="Pelydriad corff du" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Hohlraumstrahlung" title="Hohlraumstrahlung – German" lang="de" hreflang="de" data-title="Hohlraumstrahlung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Musta_keha_kiirgus" title="Musta keha kiirgus – Estonian" lang="et" hreflang="et" data-title="Musta keha kiirgus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Radiaci%C3%B3n_de_cuerpo_negro" title="Radiación de cuerpo negro – Spanish" lang="es" hreflang="es" data-title="Radiación de cuerpo negro" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Nigra-korpa_radiado" title="Nigra-korpa radiado – Esperanto" lang="eo" hreflang="eo" data-title="Nigra-korpa radiado" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gorputz_beltzaren_erradiazio" title="Gorputz beltzaren erradiazio – Basque" lang="eu" hreflang="eu" data-title="Gorputz beltzaren erradiazio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B4_%D8%AC%D8%B3%D9%85_%D8%B3%DB%8C%D8%A7%D9%87" title="تابش جسم سیاه – Persian" lang="fa" hreflang="fa" data-title="تابش جسم سیاه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Rayonnement_du_corps_noir" title="Rayonnement du corps noir – French" lang="fr" hreflang="fr" data-title="Rayonnement du corps noir" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Rada%C3%ADocht_d%C3%BAchoirp" title="Radaíocht dúchoirp – Irish" lang="ga" hreflang="ga" data-title="Radaíocht dúchoirp" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%9D%91%EC%B2%B4_%EB%B0%A9%EC%82%AC" title="흑체 방사 – Korean" lang="ko" hreflang="ko" data-title="흑체 방사" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%83%E0%A4%B7%E0%A5%8D%E0%A4%A3%E0%A4%BF%E0%A4%95%E0%A4%BE_%E0%A4%B5%E0%A4%BF%E0%A4%95%E0%A4%BF%E0%A4%B0%E0%A4%A3" title="कृष्णिका विकिरण – Hindi" lang="hi" hreflang="hi" data-title="कृष्णिका विकिरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Zra%C4%8Denje_crnog_tijela" title="Zračenje crnog tijela – Croatian" lang="hr" hreflang="hr" data-title="Zračenje crnog tijela" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Radiasi_benda-hitam" title="Radiasi benda-hitam – Indonesian" lang="id" hreflang="id" data-title="Radiasi benda-hitam" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%A8%D7%99%D7%A0%D7%AA_%D7%92%D7%95%D7%A3_%D7%A9%D7%97%D7%95%D7%A8" title="קרינת גוף שחור – Hebrew" lang="he" hreflang="he" data-title="קרינת גוף שחור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Feketetest-sug%C3%A1rz%C3%A1s" title="Feketetest-sugárzás – Hungarian" lang="hu" hreflang="hu" data-title="Feketetest-sugárzás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%8D%E0%B4%B2%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%95%E0%B5%8D_%E0%B4%AC%E0%B5%8B%E0%B4%A1%E0%B4%BF_%E0%B4%B1%E0%B5%87%E0%B4%A1%E0%B4%BF%E0%B4%AF%E0%B5%87%E0%B4%B7%E0%B5%BB" title="ബ്ലാക്ക് ബോഡി റേഡിയേഷൻ – Malayalam" lang="ml" hreflang="ml" data-title="ബ്ലാക്ക് ബോഡി റേഡിയേഷൻ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%94%E1%80%80%E1%80%BA%E1%80%91%E1%80%8A%E1%80%BA_%E1%80%9B%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%81%E1%80%BC%E1%80%8A%E1%80%BA%E1%80%96%E1%80%BC%E1%80%AC%E1%80%91%E1%80%BD%E1%80%80%E1%80%BA%E1%80%81%E1%80%BC%E1%80%84%E1%80%BA%E1%80%B8" title="အနက်ထည် ရောင်ခြည်ဖြာထွက်ခြင်း – Burmese" lang="my" hreflang="my" data-title="အနက်ထည် ရောင်ခြည်ဖြာထွက်ခြင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%BB%92%E4%BD%93%E6%94%BE%E5%B0%84" title="黒体放射 – Japanese" lang="ja" hreflang="ja" data-title="黒体放射" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no badge-Q70894304 mw-list-item" title=""><a href="https://no.wikipedia.org/wiki/Svart_str%C3%A5ling" title="Svart stråling – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Svart stråling" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Svart_str%C3%A5ling" title="Svart stråling – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Svart stråling" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AC%E0%A8%B2%E0%A9%88%E0%A8%95-%E0%A8%AC%E0%A9%8C%E0%A8%A1%E0%A9%80_%E0%A8%B0%E0%A9%87%E0%A8%A1%E0%A9%80%E0%A8%8F%E0%A8%B8%E0%A8%BC%E0%A8%A8" title="ਬਲੈਕ-ਬੌਡੀ ਰੇਡੀਏਸ਼ਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਬਲੈਕ-ਬੌਡੀ ਰੇਡੀਏਸ਼ਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A8%D9%84%DB%8C%DA%A9_%D8%A8%D8%A7%DA%88%DB%8C_%D8%B1%DB%8C%DA%88%DB%8C_%D8%A7%DB%8C%D8%B4%D9%86" title="بلیک باڈی ریڈی ایشن – Western Punjabi" lang="pnb" hreflang="pnb" data-title="بلیک باڈی ریڈی ایشن" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Radia%C3%A7%C3%A3o_de_corpo_negro" title="Radiação de corpo negro – Portuguese" lang="pt" hreflang="pt" data-title="Radiação de corpo negro" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Blackbody_radiation" title="Blackbody radiation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Blackbody radiation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Zra%C4%8Denje_crnog_tela" title="Zračenje crnog tela – Serbian" lang="sr" hreflang="sr" data-title="Zračenje crnog tela" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Zra%C4%8Denje_crnog_tijela" title="Zračenje crnog tijela – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Zračenje crnog tijela" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9C%E0%B9%88%E0%B8%A3%E0%B8%B1%E0%B8%87%E0%B8%AA%E0%B8%B5%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A7%E0%B8%B1%E0%B8%95%E0%B8%96%E0%B8%B8%E0%B8%94%E0%B8%B3" title="การแผ่รังสีของวัตถุดำ – Thai" lang="th" hreflang="th" data-title="การแผ่รังสีของวัตถุดำ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kara_cisim_%C4%B1%C5%9F%C4%B1n%C4%B1m%C4%B1" title="Kara cisim ışınımı – Turkish" lang="tr" hreflang="tr" data-title="Kara cisim ışınımı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B8%D0%BF%D1%80%D0%BE%D0%BC%D1%96%D0%BD%D1%8E%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F_%D1%87%D0%BE%D1%80%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D1%96%D0%BB%D0%B0" title="Випромінювання чорного тіла – Ukrainian" lang="uk" hreflang="uk" data-title="Випромінювання чорного тіла" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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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">Thermal electromagnetic radiation</div> Black-body radiation is the <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal</a> <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> within, or surrounding, a body in <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> with its environment, emitted by a <a href="/wiki/Black_body" title="Black body">black body</a> (an idealized opaque, non-reflective body). It has a specific, <a href="/wiki/Continuous_spectrum" class="mw-redirect" title="Continuous spectrum">continuous spectrum</a> of <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a>, inversely related to intensity, that depend only on the body's <a href="/wiki/Temperature" title="Temperature">temperature</a>, which is assumed, for the sake of calculations and theory, to be uniform and constant.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a><a href="#cite_note-Landsberg-4">[4]</a><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Black_body.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Black_body.svg/310px-Black_body.svg.png" decoding="async" width="310" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Black_body.svg/465px-Black_body.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Black_body.svg/620px-Black_body.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>As the temperature of a black body decreases, the emitted thermal radiation decreases in intensity and its maximum moves to longer wavelengths. Shown for comparison is the classical <a href="/wiki/Rayleigh%E2%80%93Jeans_law" title="Rayleigh–Jeans law">Rayleigh–Jeans law</a> and its <a href="/wiki/Ultraviolet_catastrophe" title="Ultraviolet catastrophe">ultraviolet catastrophe</a>.</figcaption></figure> A perfectly insulated enclosure which is in thermal equilibrium internally contains blackbody radiation, and will emit it through a hole made in its wall, provided the hole is small enough to have a negligible effect upon the equilibrium. The thermal radiation spontaneously emitted by many ordinary objects can be approximated as blackbody radiation. Of particular importance, although planets and stars (including the <a href="/wiki/Earth" title="Earth">Earth</a> and <a href="/wiki/Sun" title="Sun">Sun</a>) are neither in thermal equilibrium with their surroundings nor perfect black bodies, blackbody radiation is still a good first approximation for the energy they emit. <a href="#cite_note-Morison-5">[5]</a> The term black body was introduced by <a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Gustav Kirchhoff</a> in 1860.<a href="#cite_note-6">[6]</a> Blackbody radiation is also called <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal radiation</a>, cavity radiation, complete radiation or temperature radiation. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=1" title="Edit section: Theory">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Spectrum">Spectrum</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=2" title="Edit section: Spectrum">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Blacksmith_at_work.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Blacksmith_at_work.jpg/330px-Blacksmith_at_work.jpg" decoding="async" width="330" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Blacksmith_at_work.jpg/495px-Blacksmith_at_work.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Blacksmith_at_work.jpg/660px-Blacksmith_at_work.jpg 2x" data-file-width="1600" data-file-height="1242" /></a><figcaption>Blacksmiths judge workpiece temperatures by the colour of the glow.<a href="#cite_note-7">[7]</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gluehfarben_no_language_horizontal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Gluehfarben_no_language_horizontal.svg/660px-Gluehfarben_no_language_horizontal.svg.png" decoding="async" width="660" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Gluehfarben_no_language_horizontal.svg/990px-Gluehfarben_no_language_horizontal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Gluehfarben_no_language_horizontal.svg/1320px-Gluehfarben_no_language_horizontal.svg.png 2x" data-file-width="1002" data-file-height="102" /></a><figcaption>This blacksmith's colourchart stops at the melting temperature of steel</figcaption></figure> Black-body radiation has a characteristic, continuous <a href="/wiki/Spectral_energy_distribution" title="Spectral energy distribution">frequency spectrum</a> that depends only on the body's temperature,<a href="#cite_note-Kogure-8">[8]</a> called the Planck spectrum or <a href="/wiki/Planck%27s_law" title="Planck's law">Planck's law</a>. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at <a href="/wiki/Room_temperature" title="Room temperature">room temperature</a> most of the emission is in the <a href="/wiki/Infrared" title="Infrared">infrared</a> region of the <a href="/wiki/Electromagnetic_spectrum" title="Electromagnetic spectrum">electromagnetic spectrum</a>.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a><a href="#cite_note-Planck_1914-11">[11]</a> As the temperature increases past about 500 degrees <a href="/wiki/Celsius" title="Celsius">Celsius</a>, black bodies start to emit significant amounts of visible light. Viewed in the dark by the human eye, the first faint glow appears as a "ghostly" grey (the visible light is actually red, but low intensity light activates only the eye's grey-level sensors). With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises.<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a> When the body appears white, it is emitting a substantial fraction of its energy as <a href="/wiki/Ultraviolet_radiation" class="mw-redirect" title="Ultraviolet radiation">ultraviolet radiation</a>. The <a href="/wiki/Sun" title="Sun">Sun</a>, with an <a href="/wiki/Effective_temperature" title="Effective temperature">effective temperature</a> of approximately 5800 K,<a href="#cite_note-14">[14]</a> is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the <a href="/wiki/Visible_spectrum" title="Visible spectrum">visible spectrum</a>, but with significant power in the ultraviolet as well. Blackbody radiation provides insight into the <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> state of cavity radiation. <div class="mw-heading mw-heading3"><h3 id="Black_body">Black body</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=3" title="Edit section: Black body">edit</a>]</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">Main article: <a href="/wiki/Black_body" title="Black body">Black body</a></div> All normal (<a href="/wiki/Baryon" title="Baryon">baryonic</a>) matter emits electromagnetic radiation when it has a temperature above <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a>. The radiation represents a conversion of a body's internal energy into electromagnetic energy, and is therefore called <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal radiation</a>. It is a <a href="/wiki/Spontaneous_process" title="Spontaneous process">spontaneous process</a> of radiative distribution of <a href="/wiki/Entropy" title="Entropy">entropy</a>. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Color_temperature_black_body_800-12200K.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Color_temperature_black_body_800-12200K.svg/512px-Color_temperature_black_body_800-12200K.svg.png" decoding="async" width="512" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Color_temperature_black_body_800-12200K.svg/768px-Color_temperature_black_body_800-12200K.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Color_temperature_black_body_800-12200K.svg/1024px-Color_temperature_black_body_800-12200K.svg.png 2x" data-file-width="512" data-file-height="131" /></a><figcaption>Color of a black body from 800 K to 12200 K. This range of colors approximates the range of colors of stars of different temperatures, as seen or photographed in the night sky.</figcaption></figure> Conversely, all normal matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a>, is called a black body. When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. Its emission is called blackbody radiation. The concept of the black body is an idealization, as perfect black bodies do not exist in nature.<a href="#cite_note-Planck_1914_42-15">[15]</a> However, <a href="/wiki/Graphite" title="Graphite">graphite</a> and <a href="/wiki/Carbon_black" title="Carbon black">lamp black</a>, with emissivities greater than 0.95, are good approximations to a black material. Experimentally, blackbody radiation may be established best as the ultimately stable steady state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective.<a href="#cite_note-Planck_1914_42-15">[15]</a> A closed box with walls of graphite at a constant temperature with a small hole on one side produces a good approximation to ideal blackbody radiation emanating from the opening.<a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a> Blackbody radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.<a href="#cite_note-Planck_1914_42-15">[15]</a> In equilibrium, for each frequency, the intensity of radiation which is emitted and reflected from a body relative to other frequencies (that is, the net amount of radiation leaving its surface, called the spectral radiance) is determined solely by the equilibrium temperature and does not depend upon the shape, material or structure of the body.<a href="#cite_note-Caniou-18">[18]</a> For a black body (a perfect absorber) there is no reflected radiation, and so the spectral radiance is entirely due to emission. In addition, a black body is a diffuse emitter (its emission is independent of direction). Blackbody radiation becomes a visible glow of light if the temperature of the object is high enough.<a href="#cite_note-19">[19]</a> The <a href="/wiki/Draper_point" title="Draper point">Draper point</a> is the temperature at which all solids glow a dim red, about 798 K.<a href="#cite_note-20">[20]</a> At 1000 K, a small opening in the wall of a large uniformly heated opaque-walled cavity (such as an oven), viewed from outside, looks red; at 6000 K, it looks white. No matter how the oven is constructed, or of what material, as long as it is built so that almost all light entering is absorbed by its walls, it will contain a good approximation to blackbody radiation. The spectrum, and therefore color, of the light that comes out will be a function of the cavity temperature alone. A graph of the spectral radiation intensity plotted versus frequency(or wavelength) is called the blackbody curve. Different curves are obtained by varying the temperature. <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Pahoehoe_toe.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Pahoehoe_toe.jpg/250px-Pahoehoe_toe.jpg" decoding="async" width="250" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Pahoehoe_toe.jpg/375px-Pahoehoe_toe.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Pahoehoe_toe.jpg/500px-Pahoehoe_toe.jpg 2x" data-file-width="800" data-file-height="500" /></a><figcaption>The temperature of a <a href="/wiki/Lava#Pāhoehoe" title="Lava">Pāhoehoe</a> lava flow can be estimated by observing its color. The result agrees well with other measurements of temperatures of lava flows at about 1,000 to 1,200 °C (1,830 to 2,190 °F).</figcaption></figure> When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the blackbody curve. This means that the blackbody curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of <a href="/wiki/Kirchhoff%27s_law_of_thermal_radiation" title="Kirchhoff's law of thermal radiation">Kirchhoff's law of thermal radiation</a>: the blackbody curve is characteristic of thermal light, which depends only on the <a href="/wiki/Temperature" title="Temperature">temperature</a> of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>.<a href="#cite_note-Huang-21">[21]</a> When the black body is small, so that its size is comparable to the wavelength of light, the absorption is modified, because a small object is not an efficient absorber of light of long wavelength, but the principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium. In the laboratory, blackbody radiation is approximated by the radiation from a small hole in a large cavity, a <a href="/wiki/Hohlraum" title="Hohlraum">hohlraum</a>, in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature. (This technique leads to the alternative term cavity radiation.) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. Absorption occurs regardless of the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the <a href="/wiki/Power_spectral_density" class="mw-redirect" title="Power spectral density">spectrum</a> of the hole's radiation (that is, the amount of light emitted from the hole at each wavelength) will be continuous, and will depend only on the temperature and the fact that the walls are opaque and at least partly absorptive, but not on the particular material of which they are built nor on the material in the cavity (compare with <a href="/wiki/Emission_spectrum" title="Emission spectrum">emission spectrum</a>). The <a href="/wiki/Radiance" title="Radiance">radiance</a> or observed intensity is not a function of direction. Therefore, a black body is a perfect <a href="/wiki/Lambert%27s_cosine_law" title="Lambert's cosine law">Lambertian</a> radiator. Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The <a href="/wiki/Emissivity" title="Emissivity">emissivity</a> of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, it is typical in engineering to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength so that the emissivity is a constant. This is known as the gray body assumption. <figure typeof="mw:File/Thumb"><a href="/wiki/File:WMAP_2012.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/300px-WMAP_2012.png" decoding="async" width="300" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/450px-WMAP_2012.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/600px-WMAP_2012.png 2x" data-file-width="4096" data-file-height="2048" /></a><figcaption>Nine-year <a href="/wiki/WMAP" class="mw-redirect" title="WMAP">WMAP</a> image (2012) of the <a href="/wiki/Cosmic_microwave_background_radiation" class="mw-redirect" title="Cosmic microwave background radiation">cosmic microwave background radiation</a> across the universe.<a href="#cite_note-Space-20121221-22">[22]</a><a href="#cite_note-arXiv-20121220-23">[23]</a></figcaption></figure> With non-black surfaces, the deviations from ideal blackbody behavior are determined by both the surface structure, such as roughness or granularity, and the chemical composition. On a "per wavelength" basis, real objects in states of <a href="/wiki/Thermodynamic_equilibrium#Local_and_global_equilibrium" title="Thermodynamic equilibrium">local thermodynamic equilibrium</a> still follow <a href="/wiki/Kirchhoff%27s_law_(thermodynamics)" class="mw-redirect" title="Kirchhoff's law (thermodynamics)">Kirchhoff's Law</a>: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less radiation than an ideal black body; the incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body. In <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, objects such as <a href="/wiki/Star" title="Star">stars</a> are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect blackbody spectrum is exhibited by the <a href="/wiki/Cosmic_microwave_background_radiation" class="mw-redirect" title="Cosmic microwave background radiation">cosmic microwave background radiation</a>. <a href="/wiki/Hawking_radiation" title="Hawking radiation">Hawking radiation</a> is the hypothetical blackbody radiation emitted by <a href="/wiki/Black_hole" title="Black hole">black holes</a>, at a temperature that depends on the mass, charge, and spin of the hole. If this prediction is correct, black holes will very gradually shrink and evaporate over time as they lose mass by the emission of photons and other particles. A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that, for most practical purposes, such a black body does not emit in the visible range.<a href="#cite_note-24">[24]</a> The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of <a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">quantum mechanics</a>. <div class="mw-heading mw-heading3"><h3 id="Further_explanation">Further explanation</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=4" title="Edit section: Further explanation">edit</a>]</div> According to the Classical Theory of Radiation, if each <a href="/wiki/Fourier_mode" class="mw-redirect" title="Fourier mode">Fourier mode</a> of the equilibrium radiation (in an otherwise empty cavity with perfectly reflective walls) is considered as a degree of freedom capable of exchanging energy, then, according to the <a href="/wiki/Equipartition_theorem" title="Equipartition theorem">equipartition theorem</a> of classical physics, there would be an equal amount of energy in each mode. Since there are an infinite number of modes, this would imply infinite <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a>, as well as a nonphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the <a href="/wiki/Ultraviolet_catastrophe" title="Ultraviolet catastrophe">ultraviolet catastrophe</a>. In the longer <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a> this deviation is not so noticeable, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc23768f43b26085f80e1882a94b31d24abd653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.571ex; height:2.176ex;" alt="{\displaystyle h\nu }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nh\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>h</mi> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nh\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ac490bbd0db1a5a12dabf9f177412ddd96d257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.966ex; height:2.176ex;" alt="{\displaystyle nh\nu }"> are very small. In the shorter wavelengths of the ultraviolet range, however, classical theory predicts the energy emitted tends to infinity, hence the ultraviolet catastrophe. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures (see also <a href="/wiki/Wien%27s_displacement_law" title="Wien's displacement law">Wien's law</a>). Instead, in the quantum treatment of this problem, the numbers of the energy modes are <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantized</a>, attenuating the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The modes that had more energy than the thermal energy of the substance itself were not considered, and because of quantization modes having infinitesimally little energy were excluded. Thus for shorter wavelengths very few modes (having energy more than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc23768f43b26085f80e1882a94b31d24abd653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.571ex; height:2.176ex;" alt="{\displaystyle h\nu }">) were allowed, supporting the data that the energy emitted is reduced for wavelengths less than the wavelength of the observed peak of emission. Notice that there are two factors responsible for the shape of the graph, which can be seen as working opposite to one another. Firstly, shorter wavelengths have a larger number of modes associated with them. This accounts for the increase in spectral radiance as one moves from the longest wavelengths towards the peak at relatively shorter wavelengths. Secondly, though, at shorter wavelengths more energy is needed to reach the threshold level to occupy each mode: the more energy needed to excite the mode, the lower the probability that this mode will be occupied. As the wavelength decreases, the probability of exciting the mode becomes exceedingly small, leading to fewer of these modes being occupied: this accounts for the decrease in spectral radiance at very short wavelengths, left of the peak. Combined, they give the characteristic graph.<a href="#cite_note-25">[25]</a> Calculating the blackbody curve was a major challenge in <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> during the late nineteenth century. The problem was solved in 1901 by <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a> in the formalism now known as <a href="/wiki/Planck%27s_law" title="Planck's law">Planck's law</a> of blackbody radiation.<a href="#cite_note-26">[26]</a> By making changes to <a href="/wiki/Wien%27s_radiation_law" class="mw-redirect" title="Wien's radiation law">Wien's radiation law</a> (not to be confused with Wien's displacement law) consistent with <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> and <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, he found a mathematical expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, which is to say that it existed in integer multiples of some quantity. <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a> built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a>. These theoretical advances eventually resulted in the superseding of classical electromagnetism by <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. These quanta were called <a href="/wiki/Photon" title="Photon">photons</a> and the blackbody cavity was thought of as containing a <a href="/wiki/Photon_gas" title="Photon gas">gas of photons</a>. In addition, it led to the development of quantum probability distributions, called <a href="/wiki/Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac statistics</a> and <a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein statistics</a>, each applicable to a different class of particles, <a href="/wiki/Fermion" title="Fermion">fermions</a> and <a href="/wiki/Boson" title="Boson">bosons</a>. The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the <a href="/wiki/Stefan%E2%80%93Boltzmann_law" title="Stefan–Boltzmann law">Stefan–Boltzmann law</a>. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible—indeed, the radiation of visible light increases <a href="/wiki/Monotonic_function" title="Monotonic function">monotonically</a> with temperature.<a href="#cite_note-Landau-27">[27]</a> The Stefan–Boltzmann law also says that the total radiant heat energy emitted from a surface is proportional to the fourth power of its <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a>. The law was formulated by Josef Stefan in 1879 and later derived by Ludwig Boltzmann. The formula E = σT4 is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and σ = 5.670367×10−8 W·m−2⋅K−4 is the <a href="/wiki/Stefan%E2%80%93Boltzmann_constant" class="mw-redirect" title="Stefan–Boltzmann constant">Stefan–Boltzmann constant</a>.<a href="#cite_note-28">[28]</a> <div class="mw-heading mw-heading2"><h2 id="Equations">Equations</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=5" title="Edit section: Equations">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Planck's_law_of_blackbody_radiation">Planck's law of blackbody radiation</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=6" title="Edit section: Planck's law of blackbody radiation">edit</a>]</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/Planck%27s_law" title="Planck's law">Planck's law</a></div> Planck's law states that<a href="#cite_note-Rybicki_1979_22-29">[29]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /kT}-1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>h</mi> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /kT}-1}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2da0af42d06c76a3a5fd5435ca083134202b39" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.516ex; height:6.176ex;" alt="{\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /kT}-1}},}"> where <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist" style="padding-left: 1.5em"><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\nu }(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\nu }(T)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c94ca2424e3bc90a5acceef97eb5918bf11672d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.313ex; height:2.843ex;" alt="{\displaystyle B_{\nu }(T)}"> is the spectral radiance (the <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> per unit <a href="/wiki/Solid_angle" title="Solid angle">solid angle</a> and per unit of area normal to the propagation) density of frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"> radiation per unit <a href="/wiki/Frequency" title="Frequency">frequency</a> at thermal equilibrium at temperature <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">. Units: power / [area × solid angle × frequency].</li><li><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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>;</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum;</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>;</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"> is the <a href="/wiki/Frequency" title="Frequency">frequency</a> of the electromagnetic radiation;</li><li><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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is the absolute <a href="/wiki/Temperature" title="Temperature">temperature</a> of the body (in cgs units).</li></ul></div> For a black body surface, the spectral radiance density (defined per unit of area normal to the propagation) is independent of the angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> of emission with respect to the normal. However, this means that, following <a href="/wiki/Lambert%27s_cosine_law" title="Lambert's cosine law">Lambert's cosine law</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\nu }(T)\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\nu }(T)\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6f4b86c4669da1bbb776c6a8a044d5c9561604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.289ex; height:2.843ex;" alt="{\displaystyle B_{\nu }(T)\cos \theta }"> is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a71d74eaa07a7412594b9fa1363018156966d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.301ex; height:2.843ex;" alt="{\displaystyle 1/\cos \theta }"> with respect to an area normal to the propagation direction. At oblique angles, the solid angle spans involved do get smaller, resulting in lower aggregate intensities. The emitted energy flux density or irradiance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\nu }(T,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\nu }(T,E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea33830684ef559a40cb85381d75cc9786fb0996" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.123ex; height:2.843ex;" alt="{\displaystyle B_{\nu }(T,E)}">, is related to the photon flux density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\nu }(T,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{\nu }(T,E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd45b52cb5a5e79bc3e1c3bfb2719419ec8d7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.356ex; height:2.843ex;" alt="{\displaystyle b_{\nu }(T,E)}"> through<a href="#cite_note-30">[30]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\nu }(T,E)=Eb_{\nu }(T,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\nu }(T,E)=Eb_{\nu }(T,E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b111d8492a14fa7954d373625408678bdc40bf35" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.353ex; height:2.843ex;" alt="{\displaystyle B_{\nu }(T,E)=Eb_{\nu }(T,E)}"> <div class="mw-heading mw-heading3"><h3 id="Wien's_displacement_law">Wien's displacement law</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=7" title="Edit section: Wien's displacement law">edit</a>]</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/Wien%27s_displacement_law" title="Wien's displacement law">Wien's displacement law</a></div> Wien's displacement law shows how the spectrum of blackbody radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of frequency. A consequence of Wien's displacement law is that the wavelength at which the intensity per unit wavelength of the radiation produced by a black body has a local maximum or peak, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\text{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\text{peak}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e275c1654d1730ba49f8aa8f5593f8e82dafb588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.921ex; height:2.843ex;" alt="{\displaystyle \lambda _{\text{peak}}}">, is a function only of the temperature: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\text{peak}}={\frac {b}{T}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>T</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\text{peak}}={\frac {b}{T}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d42bc250d93be21fca4d01872971b38d2d69d5d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.139ex; height:5.343ex;" alt="{\displaystyle \lambda _{\text{peak}}={\frac {b}{T}},}"> where the constant b, known as Wien's displacement constant, is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {hc}{k}}{\frac {1}{5+W_{0}(-5e^{-5})}}\approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>c</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>5</mn> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {hc}{k}}{\frac {1}{5+W_{0}(-5e^{-5})}}\approx }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e8d2419ceba8e76cda245b84cc48a94876db90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.918ex; height:6.176ex;" alt="{\displaystyle {\frac {hc}{k}}{\frac {1}{5+W_{0}(-5e^{-5})}}\approx }"> 2.897771955×10−3 m K.<a href="#cite_note-31">[31]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f541f57fd799ba5137a2e50a1a728dde4306c06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{0}}"> is the <a href="/wiki/Lambert_W_function" title="Lambert W function">Lambert W function</a>. So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\text{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\text{peak}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e275c1654d1730ba49f8aa8f5593f8e82dafb588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.921ex; height:2.843ex;" alt="{\displaystyle \lambda _{\text{peak}}}"> is approximately 2898 μm/T, with the temperature given in kelvins. At a typical room temperature of 293 K (20 °C), the maximum intensity is at 9.9 μm. Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by<a href="#cite_note-32">[32]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{\text{peak}}=T\times 5.879...\times 10^{10}\ \mathrm {Hz} /\mathrm {K} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mo>=</mo> <mi>T</mi> <mo>×</mo> <mn>5.879...</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{\text{peak}}=T\times 5.879...\times 10^{10}\ \mathrm {Hz} /\mathrm {K} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122e29240370cb6c949730e723d53b7c206e75e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.542ex; height:3.343ex;" alt="{\displaystyle \nu _{\text{peak}}=T\times 5.879...\times 10^{10}\ \mathrm {Hz} /\mathrm {K} .}"> In unitless form, the maximum occurs when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}(1-x/3)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}(1-x/3)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336656828d960d0a8b7e6fb33e1dc2732a339318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.984ex; height:2.843ex;" alt="{\displaystyle e^{x}(1-x/3)=1}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=h\nu /kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>h</mi> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=h\nu /kT}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c1f591ad76b28a09e789f144f143ffe830ebbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.009ex; height:2.843ex;" alt="{\displaystyle x=h\nu /kT}">. The approximate numerical solution is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\approx 2.82}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≈</mo> <mn>2.82</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\approx 2.82}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3a30808f5db560164711b5bf4c74fa4255a1ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.562ex; height:2.176ex;" alt="{\displaystyle x\approx 2.82}">. At a typical room temperature of 293 K (20 °C), the maximum intensity is for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"> = 17 THz. <div class="mw-heading mw-heading3"><h3 id="Stefan–Boltzmann_law">Stefan–Boltzmann law</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=8" title="Edit section: Stefan–Boltzmann law">edit</a>]</div> By integrating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\nu }(T)\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\nu }(T)\cos(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17718bf3031b3421a4b011285156fd688ff2b23b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.711ex; height:2.843ex;" alt="{\displaystyle B_{\nu }(T)\cos(\theta )}"> over the frequency the radiance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> (units: power / [area × solid angle] ) is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {2\pi ^{5}}{15}}{\frac {k^{4}T^{4}}{c^{2}h^{3}}}{\frac {1}{\pi }}=\sigma T^{4}{\frac {\cos(\theta )}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>15</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo>=</mo> <mi>σ</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {2\pi ^{5}}{15}}{\frac {k^{4}T^{4}}{c^{2}h^{3}}}{\frac {1}{\pi }}=\sigma T^{4}{\frac {\cos(\theta )}{\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1797e762b59d358162aba6ea16ad0d64481359ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.162ex; height:6.009ex;" alt="{\displaystyle L={\frac {2\pi ^{5}}{15}}{\frac {k^{4}T^{4}}{c^{2}h^{3}}}{\frac {1}{\pi }}=\sigma T^{4}{\frac {\cos(\theta )}{\pi }}}"> by using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }dx\,{\frac {x^{3}}{e^{x}-1}}={\frac {\pi ^{4}}{15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>15</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }dx\,{\frac {x^{3}}{e^{x}-1}}={\frac {\pi ^{4}}{15}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d42a77077f0b59c6a585f9327ecb9fb470d6993f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.077ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{\infty }dx\,{\frac {x^{3}}{e^{x}-1}}={\frac {\pi ^{4}}{15}}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\equiv {\frac {h\nu }{kT}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\equiv {\frac {h\nu }{kT}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86745d0cafda8ef18afae72c22386c55ea5bdfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.112ex; height:5.509ex;" alt="{\displaystyle x\equiv {\frac {h\nu }{kT}}}"> and with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \equiv {\frac {2\pi ^{5}}{15}}{\frac {k^{4}}{c^{2}h^{3}}}=5.670373\times 10^{-8}\mathrm {\frac {W}{m^{2}K^{4}}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>15</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>5.670373</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">W</mi> <mrow> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi mathvariant="normal">K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \equiv {\frac {2\pi ^{5}}{15}}{\frac {k^{4}}{c^{2}h^{3}}}=5.670373\times 10^{-8}\mathrm {\frac {W}{m^{2}K^{4}}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794d61764371c301c213bc00042645c468304d8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.175ex; height:6.009ex;" alt="{\displaystyle \sigma \equiv {\frac {2\pi ^{5}}{15}}{\frac {k^{4}}{c^{2}h^{3}}}=5.670373\times 10^{-8}\mathrm {\frac {W}{m^{2}K^{4}}} }"> being the <a href="/wiki/Stefan%E2%80%93Boltzmann_constant" class="mw-redirect" title="Stefan–Boltzmann constant">Stefan–Boltzmann constant</a>. On a side note, at a distance d, the intensity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dI}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dI}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771d4c30bf007af363211fcaf091380ec667cb0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle dI}"> per area <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dA}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1d35422f3d3b3cfcfd9b9aebe9176599fed24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.959ex; height:2.176ex;" alt="{\displaystyle dA}"> of radiating surface is the useful expression <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dI=\sigma T^{4}{\frac {\cos \theta }{\pi d^{2}}}dA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>I</mi> <mo>=</mo> <mi>σ</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>π</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dI=\sigma T^{4}{\frac {\cos \theta }{\pi d^{2}}}dA}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34dbdd066895ee2970838252a249d3a05eb5b6d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.974ex; height:5.676ex;" alt="{\displaystyle dI=\sigma T^{4}{\frac {\cos \theta }{\pi d^{2}}}dA}"> when the receiving surface is perpendicular to the radiation. By subsequently integrating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> over the solid angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> for all azimuthal angle (0 to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }">) and polar angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> from 0 to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}">, we arrive at the <a href="/wiki/Stefan%E2%80%93Boltzmann_law" title="Stefan–Boltzmann law">Stefan–Boltzmann law</a>: the power j* emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{\star }=\sigma T^{4},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>=</mo> <mi>σ</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j^{\star }=\sigma T^{4},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5783cbdef0bd591124c7cd03ec0bc4923116e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.888ex; height:3.009ex;" alt="{\displaystyle j^{\star }=\sigma T^{4},}"> We used <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \cos \theta \,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi /2}\cos \theta \sin \theta \,d\theta \,d\phi =\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \cos \theta \,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi /2}\cos \theta \sin \theta \,d\theta \,d\phi =\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e9f22283ed9530282200c5dd2dc70d99141ee5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.134ex; height:6.343ex;" alt="{\displaystyle \int \cos \theta \,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi /2}\cos \theta \sin \theta \,d\theta \,d\phi =\pi .}"> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=9" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Human-body_emission">Human-body emission</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=10" title="Edit section: Human-body emission">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/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 img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:288px;max-width:288px"><div class="trow"><div class="tsingle" style="width:231px;max-width:231px"><div class="thumbimage"><a href="/wiki/File:Human-Visible.jpg" class="mw-file-description"><img alt="Photo of person in the visible spectrum" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Human-Visible.jpg/229px-Human-Visible.jpg" decoding="async" width="229" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/9b/Human-Visible.jpg 1.5x" data-file-width="289" data-file-height="211" /></a></div></div></div><div class="trow"><div class="tsingle" style="width:286px;max-width:286px"><div class="thumbimage"><a href="/wiki/File:Human-Infrared.jpg" class="mw-file-description"><img alt="Photo of a person in the infrared spectrum, shifted to the visible" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Human-Infrared.jpg/284px-Human-Infrared.jpg" decoding="async" width="284" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/44/Human-Infrared.jpg 1.5x" data-file-width="355" data-file-height="214" /></a></div><div class="thumbcaption">Much of a person's energy is radiated away in the form of long-wave <a href="/wiki/Infrared" title="Infrared">infrared</a> (LWIR) light. Some materials are transparent in the infrared, but opaque to visible light, as is the plastic bag in this thermal (LWIR) camera image (bottom). Other materials are transparent to visible light, but opaque or reflective in the infrared, noticeable by the darkness of the man's glasses.</div></div></div></div></div> The human body radiates energy as <a href="/wiki/Infrared" title="Infrared">infrared</a> light. The net power radiated is the difference between the power emitted and the power absorbed: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>net</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>emit</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>absorb</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/932fa80dc1bd1948f3eca5f02e6c4999002a7f5f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.003ex; height:2.509ex;" alt="{\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}.}"> Applying the Stefan–Boltzmann law, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{net}}=A\sigma \varepsilon \left(T^{4}-T_{0}^{4}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>net</mtext> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mi>σ</mi> <mi>ε</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{net}}=A\sigma \varepsilon \left(T^{4}-T_{0}^{4}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0006b8edea9645f752dd1a0afee2b15127932137" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.203ex; height:3.343ex;" alt="{\displaystyle P_{\text{net}}=A\sigma \varepsilon \left(T^{4}-T_{0}^{4}\right),}"> where A and T are the body surface area and temperature, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> is the <a href="/wiki/Emissivity" title="Emissivity">emissivity</a>, and T0 is the ambient temperature. The total surface area of an adult is about 2 m2, and the mid- and far-infrared <a href="/wiki/Emissivity" title="Emissivity">emissivity</a> of skin and most clothing is near unity, as it is for most nonmetallic surfaces.<a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a> Skin temperature is about 33 °C,<a href="#cite_note-35">[35]</a> but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C.<a href="#cite_note-36">[36]</a> Hence, the net radiative heat loss is about <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}=\mathrm {100~W} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>net</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>emit</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>absorb</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>100</mn> <mtext> </mtext> <mi mathvariant="normal">W</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}=\mathrm {100~W} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8e12eb53deb346bb6d2ce37241fbd3eb32460cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.558ex; height:2.509ex;" alt="{\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}=\mathrm {100~W} .}"> The total energy radiated in one day is about 8 <a href="/wiki/Megajoule" class="mw-redirect" title="Megajoule">MJ</a>, or 2000 kcal (food <a href="/wiki/Calorie" title="Calorie">calories</a>). <a href="/wiki/Basal_metabolic_rate" title="Basal metabolic rate">Basal metabolic rate</a> for a 40-year-old male is about 35 kcal/(m2·h),<a href="#cite_note-Harris1918-37">[37]</a> which is equivalent to 1700 kcal per day, assuming the same 2 m2 area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.<a href="#cite_note-38">[38]</a> There are other important thermal loss mechanisms, including <a href="/wiki/Convection" title="Convection">convection</a> and <a href="/wiki/Evaporation" title="Evaporation">evaporation</a>. Conduction is negligible – the <a href="/wiki/Nusselt_number" title="Nusselt number">Nusselt number</a> is much greater than unity. Evaporation by <a href="/wiki/Perspiration" title="Perspiration">perspiration</a> is only required if radiation and convection are insufficient to maintain a steady-state temperature (but evaporation from the lungs occurs regardless). Free-convection rates are comparable, albeit somewhat lower, than radiative rates.<a href="#cite_note-39">[39]</a> Thus, radiation accounts for about two-thirds of thermal energy loss in cool, still air. Given the approximate nature of many of the assumptions, this can only be taken as a crude estimate. Ambient air motion, causing forced convection, or evaporation reduces the relative importance of radiation as a thermal-loss mechanism. Application of <a href="/wiki/Wien%27s_displacement_law" title="Wien's displacement law">Wien's law</a> to human-body emission results in a peak wavelength of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\text{peak}}=\mathrm {\frac {2.898\times 10^{-3}~K\cdot m}{305~K}} =\mathrm {9.50~\mu m} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2.898</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mtext> </mtext> <mi mathvariant="normal">K</mi> <mo>⋅</mo> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mn>305</mn> <mtext> </mtext> <mi mathvariant="normal">K</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>9.50</mn> <mtext> </mtext> <mi>μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\text{peak}}=\mathrm {\frac {2.898\times 10^{-3}~K\cdot m}{305~K}} =\mathrm {9.50~\mu m} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83862c9aded974ebcc73b90cf56b9f95509050a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.452ex; height:5.843ex;" alt="{\displaystyle \lambda _{\text{peak}}=\mathrm {\frac {2.898\times 10^{-3}~K\cdot m}{305~K}} =\mathrm {9.50~\mu m} .}"> For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micrometer range. <div class="mw-heading mw-heading3"><h3 id="Temperature_relation_between_a_planet_and_its_star">Temperature relation between a planet and its star</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=11" title="Edit section: Temperature relation between a planet and its star">edit</a>]</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/Planetary_equilibrium_temperature" title="Planetary equilibrium temperature">Planetary equilibrium temperature</a></div> The blackbody law may be used to estimate the temperature of a planet orbiting the Sun. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Erbe.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Erbe.gif/300px-Erbe.gif" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Erbe.gif/450px-Erbe.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8f/Erbe.gif 2x" data-file-width="545" data-file-height="401" /></a><figcaption>Earth's longwave thermal <a href="/wiki/Earth%27s_energy_budget#Outgoing_energy" title="Earth's energy budget">radiation</a> intensity, from clouds, atmosphere and ground</figcaption></figure> The temperature of a planet depends on several factors: <ul><li>Incident radiation from its star</li> <li>Emitted radiation of the planet (for example, <a href="/wiki/Earth%27s_energy_budget#Outgoing_energy" title="Earth's energy budget">Earth's infrared glow</a>)</li> <li>The <a href="/wiki/Albedo" title="Albedo">albedo</a> effect causing a fraction of light to be reflected by the planet</li> <li>The <a href="/wiki/Greenhouse_effect" title="Greenhouse effect">greenhouse effect</a> for planets with an atmosphere</li> <li>Energy generated internally by a planet itself due to <a href="/wiki/Radioactive_decay" title="Radioactive decay">radioactive decay</a>, <a href="/wiki/Tidal_heating" title="Tidal heating">tidal heating</a>, and <a href="/wiki/Kelvin%E2%80%93Helmholtz_mechanism" title="Kelvin–Helmholtz mechanism">adiabatic contraction due to cooling</a>.</li></ul> The analysis only considers the Sun's heat for a planet in a Solar System. The <a href="/wiki/Stefan%E2%80%93Boltzmann_law" title="Stefan–Boltzmann law">Stefan–Boltzmann law</a> gives the total <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> (energy/second) that the Sun emits: <figure typeof="mw:File/Frame"><a href="/wiki/File:Sun-Earth-Radiation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/1e/Sun-Earth-Radiation.png" decoding="async" width="256" height="185" class="mw-file-element" data-file-width="256" data-file-height="185" /></a><figcaption>The Earth only has an absorbing area equal to a two dimensional disk, rather than the surface of a sphere.</figcaption></figure> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {S\ emt}}=4\pi R_{\rm {S}}^{2}\sigma T_{\rm {S}}^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mtext> </mtext> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>σ</mi> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {S\ emt}}=4\pi R_{\rm {S}}^{2}\sigma T_{\rm {S}}^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe7cdef2cc3d51c493eb4f313930da577a05f15" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.565ex; height:3.176ex;" alt="{\displaystyle P_{\rm {S\ emt}}=4\pi R_{\rm {S}}^{2}\sigma T_{\rm {S}}^{4}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(1)</td></tr></tbody></table> where <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="padding-left: 1.5em"><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7f4b8e7c7fc824dd8c82008b7cceac27f60bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle \sigma \,}"> is the <a href="/wiki/Stefan%E2%80%93Boltzmann_law" title="Stefan–Boltzmann law">Stefan–Boltzmann constant</a>,</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\rm {S}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\rm {S}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f46b98284037d279b9d06d55d8996dd5ad55ac9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.891ex; height:2.509ex;" alt="{\displaystyle T_{\rm {S}}\,}"> is the effective temperature of the Sun, and</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\rm {S}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\rm {S}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa9d8ca03ae16b778d9e28b38a7e0b92b525b09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.297ex; height:2.509ex;" alt="{\displaystyle R_{\rm {S}}\,}"> is the radius of the Sun.</li></ul></div> The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {SE}}=P_{\rm {S\ emt}}\left({\frac {\pi R_{\rm {E}}^{2}}{4\pi D^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mtext> </mtext> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {SE}}=P_{\rm {S\ emt}}\left({\frac {\pi R_{\rm {E}}^{2}}{4\pi D^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e11e2087cbe48d3fb64b81dc758007cc7a2f47" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.191ex; height:7.509ex;" alt="{\displaystyle P_{\rm {SE}}=P_{\rm {S\ emt}}\left({\frac {\pi R_{\rm {E}}^{2}}{4\pi D^{2}}}\right)}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(2)</td></tr></tbody></table> where <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="padding-left: 1.5em"><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\rm {E}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\rm {E}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee28b583a733b8405e704b80c8fb7263fbbbe401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.503ex; height:2.509ex;" alt="{\displaystyle R_{\rm {E}}\,}"> is the radius of the planet, and</li><li><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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1393ae1816376a473758d4b30cd5f36b8823eecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.311ex; height:2.176ex;" alt="{\displaystyle D\,}"> is the distance between the <a href="/wiki/Sun" title="Sun">Sun</a> and the planet.</li></ul></div> Because of its high temperature, the Sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction <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><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 }"> of this energy where <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><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 }"> is the <a href="/wiki/Albedo" title="Albedo">albedo</a> or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9afa7876fb8b4fb8c4d8039ebed6cd1cbc4781cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.49ex; height:2.343ex;" alt="{\displaystyle 1-\alpha }"> of the Sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {abs}}=(1-\alpha )\,P_{\rm {SE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {abs}}=(1-\alpha )\,P_{\rm {SE}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c649654b10959582c88881a7e59cc2fe0475653" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.651ex; height:2.843ex;" alt="{\displaystyle P_{\rm {abs}}=(1-\alpha )\,P_{\rm {SE}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(3)</td></tr></tbody></table> Even though the planet only absorbs as a circular area <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi R^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi R^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4bdad3187c4001ed7079d18509f3b52d18479c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.15ex; height:2.676ex;" alt="{\displaystyle \pi R^{2}}">, it emits in all directions; the spherical surface area being <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi R^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi R^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ebf047836476dc57dd8c3758e0fb2aeb950ef8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.313ex; height:2.676ex;" alt="{\displaystyle 4\pi R^{2}}">. If the planet were a perfect black body, it would emit according to the <a href="/wiki/Stefan%E2%80%93Boltzmann_law" title="Stefan–Boltzmann law">Stefan–Boltzmann law</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {emt\,bb}}=4\pi R_{\rm {E}}^{2}\sigma T_{\rm {E}}^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>σ</mi> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {emt\,bb}}=4\pi R_{\rm {E}}^{2}\sigma T_{\rm {E}}^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/503f7f44fa8d862aad196d0b9b807cc162925339" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.49ex; height:3.176ex;" alt="{\displaystyle P_{\rm {emt\,bb}}=4\pi R_{\rm {E}}^{2}\sigma T_{\rm {E}}^{4}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(4)</td></tr></tbody></table> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\rm {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\rm {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a189c1d77b098b176532fcdad1b5320cc16119ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.709ex; height:2.509ex;" alt="{\displaystyle T_{\rm {E}}}"> is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {abs}}=P_{\rm {emt\,bb}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {abs}}=P_{\rm {emt\,bb}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/587d504595d26ee0e936e89be05400b75a95c8fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.885ex; height:2.509ex;" alt="{\displaystyle P_{\rm {abs}}=P_{\rm {emt\,bb}}}">, is known as the <a href="/wiki/Effective_temperature" title="Effective temperature">effective temperature</a>. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the Sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\epsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\epsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17caf7fa86c6bc3e815fc9621cc42620cab8556d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.059ex; height:2.343ex;" alt="{\displaystyle {\overline {\epsilon }}}"> of the radiation that a black body would emit where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\epsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\epsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17caf7fa86c6bc3e815fc9621cc42620cab8556d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.059ex; height:2.343ex;" alt="{\displaystyle {\overline {\epsilon }}}"> is the average emissivity in the IR range. The power emitted by the planet is then: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {emt}}={\overline {\epsilon }}\,P_{\rm {emt\,bb}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {emt}}={\overline {\epsilon }}\,P_{\rm {emt\,bb}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c949e8259dbb786d626a635219f21e97bbf55f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.686ex; height:2.676ex;" alt="{\displaystyle P_{\rm {emt}}={\overline {\epsilon }}\,P_{\rm {emt\,bb}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(5)</td></tr></tbody></table> For a body in <a href="/wiki/Radiative_equilibrium#Definitions_of_radiative_equilibrium#radiative_exchange_equilibrium" title="Radiative equilibrium">radiative exchange equilibrium</a> with its surroundings, the rate at which it emits <a href="/wiki/Radiant_energy" title="Radiant energy">radiant energy</a> is equal to the rate at which it absorbs it:<a href="#cite_note-Prevost_1791-40">[40]</a><a href="#cite_note-41">[41]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {abs}}=P_{\rm {emt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {abs}}=P_{\rm {emt}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed9fb7f5cb3821ca848724d6d3385aae7eb85a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.67ex; height:2.509ex;" alt="{\displaystyle P_{\rm {abs}}=P_{\rm {emt}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(6)</td></tr></tbody></table> Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, TP: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{P}=T_{S}{\sqrt {\frac {R_{S}{\sqrt {\frac {1-\alpha }{\overline {\varepsilon }}}}}{2D}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mover> <mi>ε</mi> <mo accent="false">¯</mo> </mover> </mfrac> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>D</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{P}=T_{S}{\sqrt {\frac {R_{S}{\sqrt {\frac {1-\alpha }{\overline {\varepsilon }}}}}{2D}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e15190e9e45f01119f11e3f2d0c8b88de2fe20" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.231ex; height:8.343ex;" alt="{\displaystyle T_{P}=T_{S}{\sqrt {\frac {R_{S}{\sqrt {\frac {1-\alpha }{\overline {\varepsilon }}}}}{2D}}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(7)</td></tr></tbody></table> In other words, given the assumptions made, the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet. Notice that a gray (flat spectrum) ball where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\alpha )={\overline {\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\alpha )={\overline {\varepsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1da37f80e96ed95258cbe7f991094b0abb706ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.597ex; height:2.843ex;" alt="{\displaystyle (1-\alpha )={\overline {\varepsilon }}}"> comes to the same temperature as a black body no matter how dark or light gray. <div class="mw-heading mw-heading4"><h4 id="Effective_temperature_of_Earth">Effective temperature of Earth</h4>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=12" title="Edit section: Effective temperature of Earth">edit</a>]</div> Substituting the measured values for the Sun and Earth yields: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\rm {S}}=5772\ \mathrm {K} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>5772</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\rm {S}}=5772\ \mathrm {K} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7b0e05697b0bb187e8ac25aaa70d00e4e922ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.288ex; height:2.509ex;" alt="{\displaystyle T_{\rm {S}}=5772\ \mathrm {K} ,}"><a href="#cite_note-NASA-42">[42]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\rm {S}}=6.957\times 10^{8}\ \mathrm {m} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>6.957</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\rm {S}}=6.957\times 10^{8}\ \mathrm {m} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc497e85102da91aec0fe886dfc275d8262e7a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.688ex; height:3.009ex;" alt="{\displaystyle R_{\rm {S}}=6.957\times 10^{8}\ \mathrm {m} ,}"><a href="#cite_note-NASA-42">[42]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=1.496\times 10^{11}\ \mathrm {m} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mn>1.496</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=1.496\times 10^{11}\ \mathrm {m} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc70f9a0861835bdfdc18e0dd2c38d71ae50bea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.524ex; height:3.009ex;" alt="{\displaystyle D=1.496\times 10^{11}\ \mathrm {m} ,}"><a href="#cite_note-NASA-42">[42]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0.309\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>0.309</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0.309\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf05360dca670f5d6726630f957a37f9fca856e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.463ex; height:2.176ex;" alt="{\displaystyle \alpha =0.309\ }"><a href="#cite_note-Cole-43">[43]</a></li></ul> With the average emissivity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\varepsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2218c42327a03e8cd7483b392523f02e6e20cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.198ex; height:2.343ex;" alt="{\displaystyle {\overline {\varepsilon }}}"> set to unity, the <a href="/wiki/Effective_temperature" title="Effective temperature">effective temperature</a> of the Earth is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\rm {E}}=254.356\ \mathrm {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>254.356</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\rm {E}}=254.356\ \mathrm {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55f784c10f131c114e41127837097b89e4f324af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.818ex; height:2.509ex;" alt="{\displaystyle T_{\rm {E}}=254.356\ \mathrm {K} }"> or −18.8 °C. This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring <a href="/wiki/Greenhouse_effect" title="Greenhouse effect">greenhouse effects</a> (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrums<a href="#cite_note-44">[44]</a>). The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the Moon as a good estimate. The albedo and emissivity of the Moon are about 0.1054<a href="#cite_note-Saari-45">[45]</a> and 0.95<a href="#cite_note-46">[46]</a> respectively, yielding an estimated temperature of about 1.36 °C. Estimates of the Earth's average albedo vary in the range 0.3–0.4, resulting in different estimated effective temperatures. Estimates are often based on the <a href="/wiki/Solar_constant" title="Solar constant">solar constant</a> (total insolation power density) rather than the temperature, size, and distance of the Sun. For example, using 0.4 for albedo, and an insolation of 1400 W m−2, one obtains an effective temperature of about 245 K.<a href="#cite_note-47">[47]</a> Similarly using albedo 0.3 and solar constant of 1372 W m−2, one obtains an effective temperature of 255 K.<a href="#cite_note-48">[48]</a><a href="#cite_note-49">[49]</a><a href="#cite_note-50">[50]</a> <div class="mw-heading mw-heading3"><h3 id="Cosmology">Cosmology</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=13" title="Edit section: Cosmology">edit</a>]</div> The <a href="/wiki/Cosmic_microwave_background" title="Cosmic microwave background">cosmic microwave background</a> radiation observed today is the most perfect blackbody radiation ever observed in nature, with a temperature of about 2.7 K.<a href="#cite_note-White-51">[51]</a> It is a "snapshot" of the radiation at the time of <a href="/wiki/Decoupling_(cosmology)" title="Decoupling (cosmology)">decoupling</a> between matter and radiation in the early universe. Prior to this time, most matter in the universe was in the form of an ionized plasma in thermal, though not full thermodynamic, equilibrium with radiation. According to Kondepudi and Prigogine, at very high temperatures (above 1010 K; such temperatures existed in the very early universe), where the thermal motion separates protons and neutrons in spite of the strong nuclear forces, electron-positron pairs appear and disappear spontaneously and are in thermal equilibrium with electromagnetic radiation. These particles form a part of the black body spectrum, in addition to the electromagnetic radiation.<a href="#cite_note-52">[52]</a> A black body at room temperature (23 °C (296 K; 73 °F)) radiates mostly in the <a href="/wiki/Infrared" title="Infrared">infrared</a> spectrum, which cannot be perceived by the human eye,<a href="#cite_note-53">[53]</a> but can be sensed by some reptiles. As the object increases in temperature to about 500 °C (773 K; 932 °F), the emission spectrum gets stronger and extends into the human visual range, and the object appears dull red. As its temperature increases further, it emits more and more orange, yellow, green, and then blue light (and ultimately beyond violet, <a href="/wiki/Ultraviolet" title="Ultraviolet">ultraviolet</a>). <div class="mw-heading mw-heading3"><h3 id="Light_bulb">Light bulb</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=14" title="Edit section: Light bulb">edit</a>]</div> <a href="/wiki/Tungsten_filament" class="mw-redirect" title="Tungsten filament">Tungsten filament</a> lights have a continuous black body spectrum with a cooler colour temperature, around 2,700 K (2,430 °C; 4,400 °F), which also emits considerable energy in the infrared range. Modern-day <a href="/wiki/Fluorescent" class="mw-redirect" title="Fluorescent">fluorescent</a> and <a href="/wiki/LED" class="mw-redirect" title="LED">LED</a> lights, which are more efficient, do not have a continuous black body emission spectrum, rather emitting directly, or using combinations of phosphors that emit multiple narrow spectrums. <figure typeof="mw:File/Thumb"><a href="/wiki/File:PlanckianLocus.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/PlanckianLocus.png/303px-PlanckianLocus.png" decoding="async" width="303" height="341" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/PlanckianLocus.png/455px-PlanckianLocus.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/PlanckianLocus.png/606px-PlanckianLocus.png 2x" data-file-width="1300" data-file-height="1462" /></a><figcaption>The color (<a href="/wiki/Chromaticity" title="Chromaticity">chromaticity</a>) of blackbody radiation scales inversely with the temperature of the black body; the <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of such colors, shown here in <a href="/wiki/CIE_1931_color_space" title="CIE 1931 color space">CIE 1931 x,y space</a>, is known as the <a href="/wiki/Planckian_locus" title="Planckian locus">Planckian locus</a>.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=15" title="Edit section: History">edit</a>]</div> In his first memoir, <a href="/wiki/Augustin-Jean_Fresnel" title="Augustin-Jean Fresnel">Augustin-Jean Fresnel</a> (1788–1827) responded to a view he extracted from a French translation of <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>'s <a href="/wiki/Optics" title="Optics">Optics</a>. He says that Newton imagined particles of light traversing space uninhibited by the <a href="/wiki/Caloric_theory" title="Caloric theory">caloric</a> medium filling it, and refutes this view (never actually held by Newton) by saying that a black body under illumination would increase indefinitely in heat.<a href="#cite_note-54">[54]</a> <div class="mw-heading mw-heading3"><h3 id="Balfour_Stewart">Balfour Stewart</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=16" title="Edit section: Balfour Stewart">edit</a>]</div> In 1858, <a href="/wiki/Balfour_Stewart" title="Balfour Stewart">Balfour Stewart</a> described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature.<a href="#cite_note-Stewart_1858-55">[55]</a> Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of <a href="/wiki/Pierre_Prevost_(physicist)" title="Pierre Prevost (physicist)">Pierre Prevost</a> and of <a href="/wiki/John_Leslie_(physicist)" title="John Leslie (physicist)">John Leslie</a>. He wrote, "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power." Stewart's statement assumed a general principle: that there exists a body or surface that has the greatest possible absorbing and radiative power for every wavelength and equilibrium temperature. Stewart was concerned with selective thermal radiation, which he investigated using plates which selectively radiated and absorbed different wavelengths. He discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Stokes-<a href="/wiki/Helmholtz_reciprocity" title="Helmholtz reciprocity">Helmholtz reciprocity</a> principle. His research did not consider that properties of rays are dependent on wavelength, and he did not use tools such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared. <div class="mw-heading mw-heading3"><h3 id="Gustav_Kirchhoff">Gustav Kirchhoff</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=17" title="Edit section: Gustav Kirchhoff">edit</a>]</div> In 1859, <a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Gustav Robert Kirchhoff</a> reported the coincidence of the wavelengths of spectrally resolved lines of absorption and emission of visible light. Importantly for thermal physics, he also observed that bright lines or dark lines were apparent depending on the temperature difference between emitter and absorber.<a href="#cite_note-56">[56]</a> Kirchhoff then went on to consider some bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at a temperature T. Here is used a notation different from Kirchhoff's. Here, the emitting power E(T, i) denotes a dimensioned quantity, the total radiation emitted by a body labeled by index i at temperature T. The total absorption ratio a(T, i) of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature T . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio E(T, i) / a(T, i) of emitting power to absorptivity is a dimensioned quantity, with the dimensions of emitting power, because a(T, i) is dimensionless. Also here the wavelength-specific emitting power of the body at temperature T is denoted by E(λ, T, i) and the wavelength-specific absorption ratio by a(λ, T, i) . Again, the ratio E(λ, T, i) / a(λ, T, i) of emitting power to absorptivity is a dimensioned quantity, with the dimensions of emitting power. In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers.<a href="#cite_note-57">[57]</a> His theoretical proof was and still is considered by some writers to be invalid.<a href="#cite_note-Siegel-58">[58]</a><a href="#cite_note-Schirrmacher_2001-59">[59]</a> His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorptivity has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) has one and the same value for all bodies. In this report there was no mention of black bodies. In 1860, still not knowing of Stewart's measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation emitted and absorbed by a body in equilibrium, the dimensioned total radiation ratio E(T, i) / a(T, i) has one and the same value common to all bodies.<a href="#cite_note-Kirchhoff_1860c-60">[60]</a> Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid.<a href="#cite_note-Siegel-58">[58]</a><a href="#cite_note-Schirrmacher_2001-59">[59]</a> But more importantly, it relied on a new theoretical postulate of "perfectly black bodies," which is the reason why one speaks of Kirchhoff's law. Such black bodies showed complete absorption in their infinitely thin most superficial surface. They correspond to Balfour Stewart's reference bodies, with internal radiation, coated with lamp-black. They were not the more realistic perfectly black bodies later considered by Planck. Planck's black bodies radiated and absorbed only by the material in their interiors; their interfaces with contiguous media were only mathematical surfaces, capable neither of absorption nor emission, but only of reflecting and transmitting with refraction.<a href="#cite_note-61">[61]</a> Kirchhoff's proof considered an arbitrary non-ideal body labeled i as well as various perfect black bodies labeled BB. It required that the bodies be kept in a cavity in thermal equilibrium at temperature T. His proof intended to show that the ratio E(λ, T, i) / a(λ, T, i) was independent of the nature i of the non-ideal body, however partly transparent or partly reflective it was. His proof first argued that for wavelength λ and at temperature T, at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power E(λ, T, BB), with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorptivity a(λ, T, BB) of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorptivity E(λ, T, BB) / a(λ, T, BB) is again just E(λ, T, BB), with the dimensions of power. Kirchhoff considered thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature T. He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio E(λ, T, i) / a(λ, T, i) was equal to E(λ, T, BB), which may now be denoted Bλ (λ, T). Bλ (λ, T) is a continuous function, dependent only on λ at fixed temperature T, and an increasing function of T at fixed wavelength λ. It vanishes at low temperatures for visible wavelengths, which does not depend on the nature i of the arbitrary non-ideal body (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing). Thus <a href="/wiki/Kirchhoff%27s_law_of_thermal_radiation" title="Kirchhoff's law of thermal radiation">Kirchhoff's law of thermal radiation</a> can be stated: For any material at all, radiating and absorbing in thermodynamic equilibrium at any given temperature T, for every wavelength λ, the ratio of emissive power to absorptivity has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by Bλ (λ, T). (For our notation Bλ (λ, T), Kirchhoff's original notation was simply e.)<a href="#cite_note-Kirchhoff_1860c-60">[60]</a><a href="#cite_note-62">[62]</a><a href="#cite_note-63">[63]</a><a href="#cite_note-64">[64]</a><a href="#cite_note-65">[65]</a><a href="#cite_note-66">[66]</a> Kirchhoff announced that the determination of the function Bλ (λ, T) was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. Occasionally by historians that function Bλ (λ, T) has been called "Kirchhoff's (emission, universal) function,"<a href="#cite_note-67">[67]</a><a href="#cite_note-68">[68]</a><a href="#cite_note-69">[69]</a><a href="#cite_note-70">[70]</a> though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoff's universality principle was worked on and debated by various physicists over the same time, and later.<a href="#cite_note-Schirrmacher_2001-59">[59]</a> Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewart's, and in some respects it was so.<a href="#cite_note-Siegel-58">[58]</a> Kirchhoff's 1860 paper did not mention the second law of thermodynamics, and of course did not mention the concept of entropy which had not at that time been established. In a more considered account in a book in 1862, Kirchhoff mentioned the connection of his law with <a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's principle</a>, which is a form of the second law.<a href="#cite_note-71">[71]</a> According to Helge Kragh, "Quantum theory owes its origin to the study of thermal radiation, in particular to the "blackbody" radiation that Robert Kirchhoff had first defined in 1859–1860."<a href="#cite_note-72">[72]</a> <div class="mw-heading mw-heading2"><h2 id="Doppler_effect">Doppler effect</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=18" title="Edit section: Doppler effect">edit</a>]</div> The <a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">relativistic Doppler effect</a> causes a shift in the frequency f of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency f': <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'=f{\frac {1-{\frac {v}{c}}\cos \theta }{\sqrt {1-v^{2}/c^{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'=f{\frac {1-{\frac {v}{c}}\cos \theta }{\sqrt {1-v^{2}/c^{2}}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f672b14d597605305a7baafc747e3578d4b181e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.597ex; height:7.176ex;" alt="{\displaystyle f'=f{\frac {1-{\frac {v}{c}}\cos \theta }{\sqrt {1-v^{2}/c^{2}}}},}"> where v is the velocity of the source in the observer's rest frame, θ is the angle between the velocity vector and the observer-source direction measured in the reference frame of the source, and c is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>.<a href="#cite_note-73">[73]</a> This can be simplified for the special cases of objects moving directly towards (θ = π) or away (θ = 0) from the observer, and for speeds much less than c. Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation. For the case of a source moving directly towards or away from the observer, this reduces to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'=T{\sqrt {\frac {c-v}{c+v}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>c</mi> <mo>−</mo> <mi>v</mi> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>v</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'=T{\sqrt {\frac {c-v}{c+v}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf827a220ae6349aae879af5a81110d682ee02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.921ex; height:6.176ex;" alt="{\displaystyle T'=T{\sqrt {\frac {c-v}{c+v}}}.}"> Here v > 0 indicates a receding source, and v < 0 indicates an approaching source. This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the <a href="/wiki/Cosmic_microwave_background_radiation" class="mw-redirect" title="Cosmic microwave background radiation">cosmic microwave background radiation</a>, which exhibits a dipole anisotropy from the Earth's motion relative to this blackbody radiation field. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=19" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Bolometer" title="Bolometer">Bolometer</a></li> <li><a href="/wiki/Color_temperature" title="Color temperature">Color temperature</a></li> <li><a href="/wiki/Draper_point" title="Draper point">Draper point</a></li> <li><a href="/wiki/Infrared_thermometer" title="Infrared thermometer">Infrared thermometer</a></li> <li><a href="/wiki/Photon_polarization" title="Photon polarization">Photon polarization</a></li> <li><a href="/wiki/Planck%27s_law" title="Planck's law">Planck's law</a></li> <li><a href="/wiki/Pyrometer" title="Pyrometer">Pyrometer</a></li> <li><a href="/wiki/Rayleigh%E2%80%93Jeans_law" title="Rayleigh–Jeans law">Rayleigh–Jeans law</a></li> <li><a href="/wiki/Sakuma%E2%80%93Hattori_equation" title="Sakuma–Hattori equation">Sakuma–Hattori equation</a></li> <li><a href="/wiki/Terahertz_radiation" title="Terahertz radiation">Terahertz radiation</a></li> <li><a href="/wiki/Thermography" title="Thermography">Thermography</a></li> <li><a href="/wiki/Wien_approximation" title="Wien approximation">Wien approximation</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=20" title="Edit section: References">edit</a>]</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-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFLoudon2000">Loudon 2000</a>, Chapter 1. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFMandelWolf1995">Mandel & Wolf 1995</a>, Chapter 13. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFKondepudiPrigogine1998">Kondepudi & Prigogine 1998</a>, Chapter 11. </li> <li id="cite_note-Landsberg-4"><a href="#cite_ref-Landsberg_4-0">^</a> <a href="#CITEREFLandsberg1990">Landsberg 1990</a>, Chapter 13. </li> <li id="cite_note-Morison-5"><a href="#cite_ref-Morison_5-0">^</a> <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 id="CITEREFIan_Morison2008" class="citation book cs1">Ian Morison (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yrV8vvJzgWkC&pg=PA48">Introduction to Astronomy and Cosmology</a>. J Wiley & Sons. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-03333-3" title="Special:BookSources/978-0-470-03333-3"><bdi>978-0-470-03333-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Astronomy+and+Cosmology&rft.pages=48&rft.pub=J+Wiley+%26+Sons&rft.date=2008&rft.isbn=978-0-470-03333-3&rft.au=Ian+Morison&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyrV8vvJzgWkC%26pg%3DPA48&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> From (Kirchhoff, 1860) (Annalen der Physik und Chemie), p. 277: "Der Beweis, welcher für die ausgesprochene Behauptung hier gegeben werden soll, … vollkommen schwarze, oder kürzer schwarze, nennen." (The proof, which shall be given here for the proposition stated [above], rests on the assumption that bodies are conceivable which in the case of infinitely small thicknesses, completely absorb all rays that fall on them, thus [they] neither reflect nor transmit rays. I will call such bodies "completely black [bodies]" or more briefly "black [bodies]".) See also (Kirchhoff, 1860) (Philosophical Magazine), p. 2. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDustin2018" class="citation web cs1">Dustin (18 December 2018). <a rel="nofollow" class="external text" href="https://blacksmithu.com/how-blacksmiths-measure-temperature/">"How Do Blacksmiths Measure The Temperature Of Their Forge And Steel?"</a>. Blacksmith U.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Blacksmith+U&rft.atitle=How+Do+Blacksmiths+Measure+The+Temperature+Of+Their+Forge+And+Steel%3F&rft.date=2018-12-18&rft.au=Dustin&rft_id=https%3A%2F%2Fblacksmithu.com%2Fhow-blacksmiths-measure-temperature%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-Kogure-8"><a href="#cite_ref-Kogure_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTomokazu_KogureKam-Ching_Leung2007" class="citation book cs1">Tomokazu Kogure; Kam-Ching Leung (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qt5sueHmtR4C&pg=PA41">"§2.3: Thermodynamic equilibrium and blackbody radiation"</a>. The astrophysics of emission-line stars. Springer. p. 41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-34500-0" title="Special:BookSources/978-0-387-34500-0"><bdi>978-0-387-34500-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A72.3%3A+Thermodynamic+equilibrium+and+blackbody+radiation&rft.btitle=The+astrophysics+of+emission-line+stars&rft.pages=41&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-387-34500-0&rft.au=Tomokazu+Kogure&rft.au=Kam-Ching+Leung&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dqt5sueHmtR4C%26pg%3DPA41&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> Wien, W. (1893). Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie, Sitzungberichte der Königlich-Preußischen Akademie der Wissenschaften (Berlin), 1893, 1: 55–62. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Lummer, O., Pringsheim, E. (1899). Die Vertheilung der Energie im Spectrum des schwarzen Körpers, Verhandlungen der Deutschen Physikalischen Gessellschaft (Leipzig), 1899, 1: 23–41. </li> <li id="cite_note-Planck_1914-11"><a href="#cite_ref-Planck_1914_11-0">^</a> <a href="#CITEREFPlanck1914">Planck 1914</a> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="/wiki/John_William_Draper" title="John William Draper">Draper, J.W.</a> (1847). On the production of light by heat, London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, series 3, 30: 345–360. <a rel="nofollow" class="external autonumber" href="https://archive.org/stream/londonedinburghp30lond#page/344/mode/2up">[1]</a> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFPartington1949">Partington 1949</a>, pp. 466–467, 478 </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFGoodyYung1989">Goody & Yung 1989</a>, pp. 482, 484 </li> <li id="cite_note-Planck_1914_42-15">^ <a href="#cite_ref-Planck_1914_42_15-0">a</a> <a href="#cite_ref-Planck_1914_42_15-1">b</a> <a href="#cite_ref-Planck_1914_42_15-2">c</a> <a href="#CITEREFPlanck1914">Planck 1914</a>, p. 42 </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFWien1894">Wien 1894</a> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFPlanck1914">Planck 1914</a>, p. 43 </li> <li id="cite_note-Caniou-18"><a href="#cite_ref-Caniou_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_Caniou1999" class="citation book cs1">Joseph Caniou (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X-aFGcf6pOEC&pg=PA107">"§4.2.2: Calculation of Planck's law"</a>. Passive infrared detection: theory and applications. Springer. p. 107. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-8532-2" title="Special:BookSources/0-7923-8532-2"><bdi>0-7923-8532-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A74.2.2%3A+Calculation+of+Planck%27s+law&rft.btitle=Passive+infrared+detection%3A+theory+and+applications&rft.pages=107&rft.pub=Springer&rft.date=1999&rft.isbn=0-7923-8532-2&rft.au=Joseph+Caniou&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DX-aFGcf6pOEC%26pg%3DPA107&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMekhrenginMeshkovskiiTashkinovGuryev2019" class="citation journal cs1">Mekhrengin, M.V.; Meshkovskii, I.K.; Tashkinov, V.A.; Guryev, V.I.; Sukhinets, A.V.; Smirnov, D.S. (June 2019). "Multispectral pyrometer for high temperature measurements inside combustion chamber of gas turbine engines". Measurement. 139: 355–360. <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/2019Meas..139..355M">2019Meas..139..355M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.measurement.2019.02.084">10.1016/j.measurement.2019.02.084</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:116260472">116260472</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Measurement&rft.atitle=Multispectral+pyrometer+for+high+temperature+measurements+inside+combustion+chamber+of+gas+turbine+engines&rft.volume=139&rft.pages=355-360&rft.date=2019-06&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A116260472%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.measurement.2019.02.084&rft_id=info%3Abibcode%2F2019Meas..139..355M&rft.aulast=Mekhrengin&rft.aufirst=M.V.&rft.au=Meshkovskii%2C+I.K.&rft.au=Tashkinov%2C+V.A.&rft.au=Guryev%2C+V.I.&rft.au=Sukhinets%2C+A.V.&rft.au=Smirnov%2C+D.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ._R._Mahan2002" class="citation book cs1">J. 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D.; E. M. Lifshitz (1996). Statistical Physics (3rd Edition Part 1 ed.). Oxford: Butterworth–Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-65314-2" title="Special:BookSources/0-521-65314-2"><bdi>0-521-65314-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Physics&rft.place=Oxford&rft.edition=3rd+Edition+Part+1&rft.pub=Butterworth%E2%80%93Heinemann&rft.date=1996&rft.isbn=0-521-65314-2&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=E.+M.+Lifshitz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/Stefan-Boltzmann-law">"Stefan-Boltzmann law"</a>. <a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a>. 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Stefan-Boltzmann+law&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.date=2019&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2FStefan-Boltzmann-law&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-Rybicki_1979_22-29"><a href="#cite_ref-Rybicki_1979_22_29-0">^</a> <a href="#CITEREFRybickiLightman1979">Rybicki & Lightman 1979</a>, p. 22 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJenny_Nelson2002" class="citation book cs1">Jenny Nelson (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1142/p276">The Physics of Solar Cells</a>. Imperial College Press. p. 19. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2Fp276">10.1142/p276</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-86094-340-9" title="Special:BookSources/978-1-86094-340-9"><bdi>978-1-86094-340-9</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+Physics+of+Solar+Cells&rft.pages=19&rft.pub=Imperial+College+Press&rft.date=2002&rft_id=info%3Adoi%2F10.1142%2Fp276&rft.isbn=978-1-86094-340-9&rft.au=Jenny+Nelson&rft_id=https%3A%2F%2Fdoi.org%2F10.1142%2Fp276&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://physics.nist.gov/cgi-bin/cuu/Value?bwien">"Wien wavelength displacement law constant"</a>. 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Proc Natl Acad Sci USA. 4 (12): 370–3. <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/1918PNAS....4..370H">1918PNAS....4..370H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.4.12.370">10.1073/pnas.4.12.370</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1091498">1091498</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16576330">16576330</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc+Natl+Acad+Sci+USA&rft.atitle=A+Biometric+Study+of+Human+Basal+Metabolism&rft.volume=4&rft.issue=12&rft.pages=370-3&rft.date=1918&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1091498%23id-name%3DPMC&rft_id=info%3Apmid%2F16576330&rft_id=info%3Adoi%2F10.1073%2Fpnas.4.12.370&rft_id=info%3Abibcode%2F1918PNAS....4..370H&rft.au=Harris+J%2C+Benedict+F&rft.au=Benedict&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1091498&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevine,_J2004" class="citation journal cs1">Levine, J (2004). <a rel="nofollow" class="external text" href="http://ajpendo.physiology.org/cgi/content/full/286/5/E675">"Nonexercise activity thermogenesis (NEAT): environment and biology"</a>. 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Retrieved 2007-06-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Heat+Transfer+and+the+Human+Body&rft.au=DrPhysics.com&rft_id=http%3A%2F%2Fwww.drphysics.com%2Fconvection%2Fconvection.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-Prevost_1791-40"><a href="#cite_ref-Prevost_1791_40-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrevost1791" class="citation journal cs1">Prevost, P. (1791). "Mémoire sur l'équilibre du feu". 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(2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Bgsy66mJ5mYC&q=blackbody+emissivity+greenhouse+intitle:Planetary-Science+inauthor:cole&pg=RA3-PA382">Planetary Science: The Science of Planets Around Stars</a> (1st ed.). IOP Publishing. pp. 36–37, 380–382. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7503-0815-X" title="Special:BookSources/0-7503-0815-X"><bdi>0-7503-0815-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Planetary+Science%3A+The+Science+of+Planets+Around+Stars&rft.pages=36-37%2C+380-382&rft.edition=1st&rft.pub=IOP+Publishing&rft.date=2002&rft.isbn=0-7503-0815-X&rft.au=Cole%2C+George+H.+A.&rft.au=Woolfson%2C+Michael+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBgsy66mJ5mYC%26q%3Dblackbody%2Bemissivity%2Bgreenhouse%2Bintitle%3APlanetary-Science%2Binauthor%3Acole%26pg%3DRA3-PA382&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> Principles of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. From Chapter 3 which is available online <a rel="nofollow" class="external text" href="http://www-das.uwyo.edu/~deshler/Atsc4400_5400_Climate/PierreHumbert_Climate_Ch3.pdf">here</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120328102328/http://www-das.uwyo.edu/~deshler/Atsc4400_5400_Climate/PierreHumbert_Climate_Ch3.pdf">Archived</a> March 28, 2012, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, p. 12 mentions that Venus' blackbody temperature would be 330 K "in the zero albedo case", but that due to atmospheric warming, its actual surface temperature is 740 K. </li> <li id="cite_note-Saari-45"><a href="#cite_ref-Saari_45-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaariShorthill1972" class="citation journal cs1">Saari, J. M.; Shorthill, R. W. (1972). "The Sunlit Lunar Surface. I. Albedo Studies and Full Moon". The Moon. 5 (1–2): 161–178. <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/1972Moon....5..161S">1972Moon....5..161S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00562111">10.1007/BF00562111</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119892155">119892155</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Moon&rft.atitle=The+Sunlit+Lunar+Surface.+I.+Albedo+Studies+and+Full+Moon&rft.volume=5&rft.issue=1%E2%80%932&rft.pages=161-178&rft.date=1972&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119892155%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00562111&rft_id=info%3Abibcode%2F1972Moon....5..161S&rft.aulast=Saari&rft.aufirst=J.+M.&rft.au=Shorthill%2C+R.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> Lunar and Planetary Science XXXVII (2006) 2406 </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichael_D._Papagiannis1972" class="citation book cs1">Michael D. 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Taylor & Francis. pp. 10–11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-677-04000-4" title="Special:BookSources/978-0-677-04000-4"><bdi>978-0-677-04000-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Space+physics+and+space+astronomy&rft.pages=10-11&rft.pub=Taylor+%26+Francis&rft.date=1972&rft.isbn=978-0-677-04000-4&rft.au=Michael+D.+Papagiannis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSpgOAAAAQAAJ%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillem_Jozef_Meine_MartensJan_Rotmans1999" class="citation book cs1">Willem Jozef Meine Martens & Jan Rotmans (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=o1SELkgK6PcC&q=Earth+effective-temperature+albedo+blackbody+0.3&pg=RA1-PA53">Climate Change an Integrated Perspective</a>. 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Selsis (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bA_uR3iwzQUC&q=Earth+effective-temperature+albedo+blackbody+0.3&pg=PA279">"The Prebiotic Atmosphere of the Earth"</a>. In Pascale Ehrenfreund; et al. (eds.). Astrobiology: Future Perspectives. Springer. pp. 279–280. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-2587-7" title="Special:BookSources/978-1-4020-2587-7"><bdi>978-1-4020-2587-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Prebiotic+Atmosphere+of+the+Earth&rft.btitle=Astrobiology%3A+Future+Perspectives&rft.pages=279-280&rft.pub=Springer&rft.date=2004&rft.isbn=978-1-4020-2587-7&rft.au=F.+Selsis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbA_uR3iwzQUC%26q%3DEarth%2Beffective-temperature%2Balbedo%2Bblackbody%2B0.3%26pg%3DPA279&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> Wallace, J.M., Hobbs, P.V. (2006). Atmospheric Science. 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Partington">Partington, J.R.</a> (1949), p. 466. </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGillispie1960" class="citation book cs1"><a href="/wiki/Charles_Coulston_Gillispie" title="Charles Coulston Gillispie">Gillispie, Charles Coulston</a> (1960). <a rel="nofollow" class="external text" href="https://archive.org/details/edgeofobjectivit00char/page/408">The Edge of Objectivity: An Essay in the History of Scientific Ideas</a>. Princeton University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/edgeofobjectivit00char/page/408">408–9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-02350-6" title="Special:BookSources/0-691-02350-6"><bdi>0-691-02350-6</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+Edge+of+Objectivity%3A+An+Essay+in+the+History+of+Scientific+Ideas&rft.pages=408-9&rft.pub=Princeton+University+Press&rft.date=1960&rft.isbn=0-691-02350-6&rft.aulast=Gillispie&rft.aufirst=Charles+Coulston&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fedgeofobjectivit00char%2Fpage%2F408&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> </li> <li id="cite_note-Stewart_1858-55"><a href="#cite_ref-Stewart_1858_55-0">^</a> <a href="#CITEREFStewart1858">Stewart 1858</a> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <a href="#CITEREFKirchhoff1860a">Kirchhoff 1860a</a> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <a href="#CITEREFKirchhoff1860b">Kirchhoff 1860b</a> </li> <li id="cite_note-Siegel-58">^ <a href="#cite_ref-Siegel_58-0">a</a> <a href="#cite_ref-Siegel_58-1">b</a> <a href="#cite_ref-Siegel_58-2">c</a> <a href="#CITEREFSiegel1976">Siegel 1976</a> </li> <li id="cite_note-Schirrmacher_2001-59">^ <a href="#cite_ref-Schirrmacher_2001_59-0">a</a> <a href="#cite_ref-Schirrmacher_2001_59-1">b</a> <a href="#cite_ref-Schirrmacher_2001_59-2">c</a> <a href="#CITEREFSchirrmacher2001">Schirrmacher 2001</a> </li> <li id="cite_note-Kirchhoff_1860c-60">^ <a href="#cite_ref-Kirchhoff_1860c_60-0">a</a> <a href="#cite_ref-Kirchhoff_1860c_60-1">b</a> <a href="#CITEREFKirchhoff1860c">Kirchhoff 1860c</a> </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <a href="#CITEREFPlanck1914">Planck 1914</a>, p. 11 </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <a href="#CITEREFChandrasekhar1950">Chandrasekhar 1950</a>, p. 8 </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <a href="#CITEREFMilne1930">Milne 1930</a>, p. 80 </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <a href="#CITEREFRybickiLightman1979">Rybicki & Lightman 1979</a>, pp. 16–17 </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <a href="#CITEREFMihalasWeibel-Mihalas1984">Mihalas & Weibel-Mihalas 1984</a>, p. 328 </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <a href="#CITEREFGoodyYung1989">Goody & Yung 1989</a>, pp. 27–28 </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <a href="/wiki/Friedrich_Paschen" title="Friedrich Paschen">Paschen, F.</a> (1896), personal letter cited by <a href="#CITEREFHermann1971">Hermann 1971</a>, p. 6 </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <a href="#CITEREFHermann1971">Hermann 1971</a>, p. 7 </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <a href="#CITEREFKuhn1978">Kuhn 1978</a>, pp. 8, 29 </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <a href="#CITEREFMehraRechenberg1982">Mehra & Rechenberg 1982</a>, pp. 26, 28, 31, 39 </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> <a href="#CITEREFKirchhoff1862/1882">Kirchhoff & 1862/1882</a>, p. 573 </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <a href="#CITEREFKragh1999">Kragh 1999</a>, p. 58 </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> The Doppler Effect, T. P. Gill, Logos Press, 1965 </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=21" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChandrasekhar1950" class="citation book cs1"><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar, S.</a> (1950). Radiative Transfer. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Radiative+Transfer&rft.pub=Oxford+University+Press&rft.date=1950&rft.aulast=Chandrasekhar&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodyYung1989" class="citation book cs1">Goody, R. M.; <a href="/wiki/Yuk_L._Yung" title="Yuk L. Yung">Yung, Y. L.</a> (1989). Atmospheric Radiation: Theoretical Basis (2nd ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-510291-8" title="Special:BookSources/978-0-19-510291-8"><bdi>978-0-19-510291-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atmospheric+Radiation%3A+Theoretical+Basis&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=1989&rft.isbn=978-0-19-510291-8&rft.aulast=Goody&rft.aufirst=R.+M.&rft.au=Yung%2C+Y.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHermann1971" class="citation book cs1"><a href="/w/index.php?title=Armin_Hermann&action=edit&redlink=1" class="new" title="Armin Hermann (page does not exist)">Hermann, A.</a> (1971). <a rel="nofollow" class="external text" href="https://archive.org/details/genesisofquantum00herm">The Genesis of Quantum Theory</a>. Nash, C.W. (transl.). <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-08047-8" title="Special:BookSources/0-262-08047-8"><bdi>0-262-08047-8</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+Genesis+of+Quantum+Theory&rft.pub=MIT+Press&rft.date=1971&rft.isbn=0-262-08047-8&rft.aulast=Hermann&rft.aufirst=A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgenesisofquantum00herm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> a translation of Frühgeschichte der Quantentheorie (1899–1913), Physik Verlag, Mosbach/Baden.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirchhoff1860a" class="citation journal cs1"><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff, G.</a>; [27 October 1859] (1860a). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015049219333;view=1up;seq=684">"Über die Fraunhofer'schen Linien"</a> [On Fraunhofer's lines]. <a href="/wiki/Monatsberichte_der_K%C3%B6niglich_Preussischen_Akademie_der_Wissenschaften_zu_Berlin" class="mw-redirect" title="Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin">Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin</a>: 662–665.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monatsberichte+der+K%C3%B6niglich+Preussischen+Akademie+der+Wissenschaften+zu+Berlin&rft.atitle=%C3%9Cber+die+Fraunhofer%27schen+Linien&rft.pages=662-665&rft.date=1860&rft.aulast=Kirchhoff&rft.aufirst=G.&rft.au=%5B27+October+1859%5D&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dmdp.39015049219333%3Bview%3D1up%3Bseq%3D684&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirchhoff1860b" class="citation journal cs1"><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff, G.</a>; [11 December 1859] (1860b). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015049219333;view=1up;seq=811">"Über den Zusammenhang zwischen Emission und Absorption von Licht und Wärme"</a> [On the relation between emission and absorption of light and heat]. <a href="/wiki/Monatsberichte_der_K%C3%B6niglich_Preussischen_Akademie_der_Wissenschaften_zu_Berlin" class="mw-redirect" title="Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin">Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin</a>: 783–787.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monatsberichte+der+K%C3%B6niglich+Preussischen+Akademie+der+Wissenschaften+zu+Berlin&rft.atitle=%C3%9Cber+den+Zusammenhang+zwischen+Emission+und+Absorption+von+Licht+und+W%C3%A4rme&rft.pages=783-787&rft.date=1860&rft.aulast=Kirchhoff&rft.aufirst=G.&rft.au=%5B11+December+1859%5D&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dmdp.39015049219333%3Bview%3D1up%3Bseq%3D811&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirchhoff1860c" class="citation journal cs1"><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff, G.</a> (1860c). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=umn.31951d00326548g;view=1up;seq=291">"Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht"</a> [On the relation between bodies' emission capacity and absorption capacity for heat and light]. <a href="/wiki/Annalen_der_Physik_und_Chemie" class="mw-redirect" title="Annalen der Physik und Chemie">Annalen der Physik und Chemie</a>. 109 (2): 275–301. <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/1860AnP...185..275K">1860AnP...185..275K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.18601850205">10.1002/andp.18601850205</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik+und+Chemie&rft.atitle=Ueber+das+Verh%C3%A4ltniss+zwischen+dem+Emissionsverm%C3%B6gen+und+dem+Absorptionsverm%C3%B6gen+der+K%C3%B6rper+f%C3%BCr+W%C3%A4rme+and+Licht&rft.volume=109&rft.issue=2&rft.pages=275-301&rft.date=1860&rft_id=info%3Adoi%2F10.1002%2Fandp.18601850205&rft_id=info%3Abibcode%2F1860AnP...185..275K&rft.aulast=Kirchhoff&rft.aufirst=G.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dumn.31951d00326548g%3Bview%3D1up%3Bseq%3D291&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"> Translated by Guthrie, F. as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirchhoff1860" class="citation journal cs1 cs1-prop-long-vol">Kirchhoff, G. (1860). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=pst.000068485634;view=1up;seq=19">"On the relation between the radiating and absorbing powers of different bodies for light and heat"</a>. <a href="/wiki/Philosophical_Magazine" title="Philosophical Magazine">Philosophical Magazine</a>. Series 4, volume 20: 1–21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+the+relation+between+the+radiating+and+absorbing+powers+of+different+bodies+for+light+and+heat&rft.volume=Series+4%2C+volume+20&rft.pages=1-21&rft.date=1860&rft.aulast=Kirchhoff&rft.aufirst=G.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dpst.000068485634%3Bview%3D1up%3Bseq%3D19&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirchhoff1862/1882" class="citation cs2"><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff, G.</a> (1882) [1862], "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht", Gessamelte Abhandlungen, Leipzig: Johann Ambrosius Barth, pp. 571–598</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ueber+das+Verh%C3%A4ltniss+zwischen+dem+Emissionsverm%C3%B6gen+und+dem+Absorptionsverm%C3%B6gen+der+K%C3%B6rper+f%C3%BCr+W%C3%A4rme+und+Licht&rft.btitle=Gessamelte+Abhandlungen&rft.place=Leipzig&rft.pages=571-598&rft.pub=Johann+Ambrosius+Barth&rft.date=1882&rft.aulast=Kirchhoff&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKondepudiPrigogine1998" class="citation book cs1"><a href="/wiki/Dilip_Kondepudi" title="Dilip Kondepudi">Kondepudi, D.</a>; <a href="/wiki/Ilya_Prigogine" title="Ilya Prigogine">Prigogine, I.</a> (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-97393-9" title="Special:BookSources/0-471-97393-9"><bdi>0-471-97393-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Thermodynamics.+From+Heat+Engines+to+Dissipative+Structures&rft.pub=John+Wiley+%26+Sons&rft.date=1998&rft.isbn=0-471-97393-9&rft.aulast=Kondepudi&rft.aufirst=D.&rft.au=Prigogine%2C+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKragh1999" class="citation book cs1"><a href="/wiki/Helge_Kragh" title="Helge Kragh">Kragh, H.</a> (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumgeneratio0000krag">Quantum Generations: a History of Physics in the Twentieth Century</a>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-01206-7" title="Special:BookSources/0-691-01206-7"><bdi>0-691-01206-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Generations%3A+a+History+of+Physics+in+the+Twentieth+Century&rft.pub=Princeton+University+Press&rft.date=1999&rft.isbn=0-691-01206-7&rft.aulast=Kragh&rft.aufirst=H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumgeneratio0000krag&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuhn1978" class="citation book cs1"><a href="/wiki/Thomas_Kuhn" title="Thomas Kuhn">Kuhn, T. S.</a> (1978). Black–Body Theory and the Quantum Discontinuity. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-502383-8" title="Special:BookSources/0-19-502383-8"><bdi>0-19-502383-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Black%E2%80%93Body+Theory+and+the+Quantum+Discontinuity&rft.pub=Oxford+University+Press&rft.date=1978&rft.isbn=0-19-502383-8&rft.aulast=Kuhn&rft.aufirst=T.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandsberg1990" class="citation book cs1">Landsberg, P. T. (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0gnWL7tmxm0C&pg=PA208">Thermodynamics and statistical mechanics</a> (Reprint ed.). <a href="/wiki/Courier_Dover_Publications" class="mw-redirect" title="Courier Dover Publications">Courier Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66493-7" title="Special:BookSources/0-486-66493-7"><bdi>0-486-66493-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+and+statistical+mechanics&rft.edition=Reprint&rft.pub=Courier+Dover+Publications&rft.date=1990&rft.isbn=0-486-66493-7&rft.aulast=Landsberg&rft.aufirst=P.+T.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0gnWL7tmxm0C%26pg%3DPA208&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLavenda1991" class="citation book cs1"><a href="/wiki/Bernard_H._Lavenda" title="Bernard H. Lavenda">Lavenda, Bernard Howard</a> (1991). Statistical Physics: A Probabilistic Approach. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. pp. 41–42. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54607-8" title="Special:BookSources/978-0-471-54607-8"><bdi>978-0-471-54607-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Physics%3A+A+Probabilistic+Approach&rft.pages=41-42&rft.pub=John+Wiley+%26+Sons&rft.date=1991&rft.isbn=978-0-471-54607-8&rft.aulast=Lavenda&rft.aufirst=Bernard+Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoudon2000" class="citation book cs1"><a href="/wiki/Rodney_Loudon" title="Rodney Loudon">Loudon, R.</a> (2000) [1973]. The Quantum Theory of Light (third ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850177-3" title="Special:BookSources/0-19-850177-3"><bdi>0-19-850177-3</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+Quantum+Theory+of+Light&rft.edition=third&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=0-19-850177-3&rft.aulast=Loudon&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMandelWolf1995" class="citation book cs1"><a href="/wiki/Leonard_Mandel" title="Leonard Mandel">Mandel, L.</a>; <a href="/wiki/Emil_Wolf" title="Emil Wolf">Wolf, E.</a> (1995). Optical Coherence and Quantum Optics. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-41711-2" title="Special:BookSources/0-521-41711-2"><bdi>0-521-41711-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optical+Coherence+and+Quantum+Optics&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=0-521-41711-2&rft.aulast=Mandel&rft.aufirst=L.&rft.au=Wolf%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMehraRechenberg1982" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Jagdish_Mehra" title="Jagdish Mehra">Mehra, J.</a>; <a href="/wiki/Helmut_Rechenberg" title="Helmut Rechenberg">Rechenberg, H.</a> (1982). The Historical Development of Quantum Theory. Vol. 1, part 1. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90642-8" title="Special:BookSources/0-387-90642-8"><bdi>0-387-90642-8</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+Historical+Development+of+Quantum+Theory&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=0-387-90642-8&rft.aulast=Mehra&rft.aufirst=J.&rft.au=Rechenberg%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMihalasWeibel-Mihalas1984" class="citation book cs1"><a href="/wiki/Dimitri_Mihalas" title="Dimitri Mihalas">Mihalas, D.</a>; Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-503437-6" title="Special:BookSources/0-19-503437-6"><bdi>0-19-503437-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Radiation+Hydrodynamics&rft.pub=Oxford+University+Press&rft.date=1984&rft.isbn=0-19-503437-6&rft.aulast=Mihalas&rft.aufirst=D.&rft.au=Weibel-Mihalas%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilne1930" class="citation journal cs1 cs1-prop-long-vol"><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne, E.A.</a> (1930). "Thermodynamics of the Stars". <a href="/w/index.php?title=Handbuch_der_Astrophysik&action=edit&redlink=1" class="new" title="Handbuch der Astrophysik (page does not exist)">Handbuch der Astrophysik</a>. 3, part 1: 63–255.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Handbuch+der+Astrophysik&rft.atitle=Thermodynamics+of+the+Stars&rft.volume=3%2C+part+1&rft.pages=63-255&rft.date=1930&rft.aulast=Milne&rft.aufirst=E.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMüller-Kirsten2013" class="citation book cs1">Müller-Kirsten, Harald J.W. (2013). Basics of Statistical Physics (2nd ed.). <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4449-53-3" title="Special:BookSources/978-981-4449-53-3"><bdi>978-981-4449-53-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basics+of+Statistical+Physics&rft.edition=2nd&rft.pub=World+Scientific&rft.date=2013&rft.isbn=978-981-4449-53-3&rft.aulast=M%C3%BCller-Kirsten&rft.aufirst=Harald+J.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPartington1949" class="citation book cs1"><a href="/wiki/J.R._Partington" class="mw-redirect" title="J.R. Partington">Partington, J.R.</a> (1949). An Advanced Treatise on Physical Chemistry. Volume 1. Fundamental Principles. The Properties of Gases. <a href="/wiki/Longman" title="Longman">Longmans, Green and Co.</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Advanced+Treatise+on+Physical+Chemistry.+Volume+1.+Fundamental+Principles.+The+Properties+of+Gases&rft.pub=Longmans%2C+Green+and+Co.&rft.date=1949&rft.aulast=Partington&rft.aufirst=J.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlanck1914" class="citation book cs1"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1914) [1912]. <a rel="nofollow" class="external text" href="https://archive.org/details/theoryofheatradi00planrich">The Theory of Heat Radiation</a>. translated by Masius, M. <a href="/w/index.php?title=P._Blakiston%27s_Sons_%26_Co.&action=edit&redlink=1" class="new" title="P. Blakiston's Sons & Co. (page does not exist)">P. Blakiston's Sons & Co.</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Heat+Radiation&rft.pub=P.+Blakiston%27s+Sons+%26+Co.&rft.date=1914&rft.aulast=Planck&rft.aufirst=M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryofheatradi00planrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRybickiLightman1979" class="citation book cs1">Rybicki, G. B.; <a href="/wiki/Alan_Lightman" title="Alan Lightman">Lightman, A. P.</a> (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LtdEjNABMlsC">Radiative Processes in Astrophysics</a>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-82759-2" title="Special:BookSources/0-471-82759-2"><bdi>0-471-82759-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Radiative+Processes+in+Astrophysics&rft.pub=John+Wiley+%26+Sons&rft.date=1979&rft.isbn=0-471-82759-2&rft.aulast=Rybicki&rft.aufirst=G.+B.&rft.au=Lightman%2C+A.+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLtdEjNABMlsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchirrmacher2001" class="citation book cs1"><a href="/w/index.php?title=Arne_Schirrmacher&action=edit&redlink=1" class="new" title="Arne Schirrmacher (page does not exist)">Schirrmacher, A.</a> (2001). Experimenting theory: the proofs of Kirchhoff's radiation law before and after Planck. <a href="/w/index.php?title=M%C3%BCnchner_Zentrum_f%C3%BCr_Wissenschafts_und_Technikgeschichte&action=edit&redlink=1" class="new" title="Münchner Zentrum für Wissenschafts und Technikgeschichte (page does not exist)">Münchner Zentrum für Wissenschafts und Technikgeschichte</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experimenting+theory%3A+the+proofs+of+Kirchhoff%27s+radiation+law+before+and+after+Planck&rft.pub=M%C3%BCnchner+Zentrum+f%C3%BCr+Wissenschafts+und+Technikgeschichte&rft.date=2001&rft.aulast=Schirrmacher&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiegel1976" class="citation journal cs1">Siegel, D.M. (1976). "Balfour Stewart and Gustav Robert Kirchhoff: two independent approaches to "Kirchhoff's radiation law"". <a href="/wiki/Isis_(journal)" title="Isis (journal)">Isis</a>. 67 (4): 565–600. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F351669">10.1086/351669</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/794025">794025</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:37368520">37368520</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Isis&rft.atitle=Balfour+Stewart+and+Gustav+Robert+Kirchhoff%3A+two+independent+approaches+to+%22Kirchhoff%27s+radiation+law%22&rft.volume=67&rft.issue=4&rft.pages=565-600&rft.date=1976&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A37368520%23id-name%3DS2CID&rft_id=info%3Apmid%2F794025&rft_id=info%3Adoi%2F10.1086%2F351669&rft.aulast=Siegel&rft.aufirst=D.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart1858" class="citation journal cs1"><a href="/wiki/Balfour_Stewart" title="Balfour Stewart">Stewart, B.</a> (1858). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015005551075;view=1up;seq=165">"An account of some experiments on radiant heat"</a>. <a href="/wiki/Transactions_of_the_Royal_Society_of_Edinburgh" class="mw-redirect" title="Transactions of the Royal Society of Edinburgh">Transactions of the Royal Society of Edinburgh</a>. 22: 1–20. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0080456800031288">10.1017/S0080456800031288</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122316368">122316368</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Royal+Society+of+Edinburgh&rft.atitle=An+account+of+some+experiments+on+radiant+heat&rft.volume=22&rft.pages=1-20&rft.date=1858&rft_id=info%3Adoi%2F10.1017%2FS0080456800031288&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122316368%23id-name%3DS2CID&rft.aulast=Stewart&rft.aufirst=B.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dmdp.39015005551075%3Bview%3D1up%3Bseq%3D165&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWien1894" class="citation journal cs1"><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien, W.</a> (1894). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=wu.89048352843;view=1up;seq=148">"Temperatur und Entropie der Strahlung"</a> [Temperature and entropy of radiation]. <a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a>. 288 (5): 132–165. <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/1894AnP...288..132W">1894AnP...288..132W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.18942880511">10.1002/andp.18942880511</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=Temperatur+und+Entropie+der+Strahlung&rft.volume=288&rft.issue=5&rft.pages=132-165&rft.date=1894&rft_id=info%3Adoi%2F10.1002%2Fandp.18942880511&rft_id=info%3Abibcode%2F1894AnP...288..132W&rft.aulast=Wien&rft.aufirst=W.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dwu.89048352843%3Bview%3D1up%3Bseq%3D148&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=22" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKroemer,_HerbertKittel,_Charles1980" class="citation book cs1">Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-1088-9" title="Special:BookSources/0-7167-1088-9"><bdi>0-7167-1088-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermal+Physics&rft.edition=2nd&rft.pub=W.+H.+Freeman+Company&rft.date=1980&rft.isbn=0-7167-1088-9&rft.au=Kroemer%2C+Herbert&rft.au=Kittel%2C+Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTipler,_PaulLlewellyn,_Ralph2002" class="citation book cs1">Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-4345-0" title="Special:BookSources/0-7167-4345-0"><bdi>0-7167-4345-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Physics&rft.edition=4th&rft.pub=W.+H.+Freeman&rft.date=2002&rft.isbn=0-7167-4345-0&rft.au=Tipler%2C+Paul&rft.au=Llewellyn%2C+Ralph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABlack-body+radiation" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Black-body_radiation&action=edit&section=23" title="Edit section: External links">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://iwant2study.org/ospsg/index.php//605">Blackbody radiation JavaScript Interactives</a> Blackbody radiation by Fu-Kwun Hwang and Loo Kang Wee</li> <li><a rel="nofollow" class="external text" href="http://www.spectralcalc.com/blackbody/blackbody.html">Calculating Blackbody Radiation</a> Interactive calculator with Doppler Effect. Includes most systems of units.</li> <li><a rel="nofollow" class="external text" href="http://academo.org/demos/colour-temperature-relationship/">Color-to-Temperature demonstration</a> at Academo.org</li> <li><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/coobod.html#c1">Cooling Mechanisms for Human Body</a> – From Hyperphysics</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120103021555/http://www.x20.org/library/thermal/blackbody.htm">Descriptions of radiation emitted by many different objects</a></li> <li><a rel="nofollow" class="external text" href="http://webphysics.davidson.edu/Applets/java11_Archive.html">Blackbody Emission Applet</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100609050537/http://webphysics.davidson.edu/Applets/java11_Archive.html">Archived</a> 2010-06-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/BlackbodySpectrum/">"Blackbody Spectrum"</a> by Jeff Bryant, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>, 2007.</li></ul> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 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[\"CITEREFPartington1949\"] = 1,\n [\"CITEREFPlanck1901\"] = 1,\n [\"CITEREFPlanck1914\"] = 1,\n [\"CITEREFPrevost1791\"] = 1,\n [\"CITEREFRybickiLightman1979\"] = 1,\n [\"CITEREFSaariShorthill1972\"] = 1,\n [\"CITEREFSchirrmacher2001\"] = 1,\n [\"CITEREFSiegel1976\"] = 1,\n [\"CITEREFStewart1858\"] = 1,\n [\"CITEREFTipler,_PaulLlewellyn,_Ralph2002\"] = 1,\n [\"CITEREFTomokazu_KogureKam-Ching_Leung2007\"] = 1,\n [\"CITEREFWhite1999\"] = 1,\n [\"CITEREFWien1894\"] = 1,\n [\"CITEREFWillem_Jozef_Meine_MartensJan_Rotmans1999\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Authority control\"] = 1,\n [\"Citation\"] = 1,\n [\"Cite book\"] = 32,\n [\"Cite encyclopedia\"] = 1,\n [\"Cite journal\"] = 15,\n [\"Cite news\"] = 1,\n [\"Cite web\"] = 10,\n [\"Convert\"] = 4,\n [\"DEFAULTSORT:Black Body\"] = 1,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"EquationRef\"] = 7,\n [\"Harvid\"] = 3,\n [\"Harvnb\"] = 30,\n [\"ISBN\"] = 2,\n [\"Main\"] = 4,\n [\"Math\"] = 46,\n [\"Multiple image\"] = 1,\n [\"Mvar\"] = 2,\n [\"Nowrap\"] = 5,\n [\"NumBlk\"] = 7,\n [\"Portal bar\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n [\"Short description\"] = 1,\n [\"Unbulleted list\"] = 3,\n [\"Val\"] = 8,\n [\"Webarchive\"] = 2,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-gznpt","timestamp":"20241125133435","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Black-body radiation","url":"https:\/\/en.wikipedia.org\/wiki\/Black-body_radiation","sameAs":"http:\/\/www.wikidata.org\/entity\/Q900097","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q900097","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-06-15T04:29:56Z","dateModified":"2024-11-13T14:11:22Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/1\/19\/Black_body.svg","headline":"thermal electromagnetic radiation"}</script> </body> </html>