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.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">Not to be confused with <a href="/wiki/Atmospheric_diffraction" title="Atmospheric diffraction">Atmospheric diffraction</a>.</div> Atmospheric refraction is the deviation of <a href="/wiki/Light" title="Light">light</a> or other <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic wave</a> from a straight line as it passes through the <a href="/wiki/Atmosphere" title="Atmosphere">atmosphere</a> due to the variation in <a href="/wiki/Air_density" class="mw-redirect" title="Air density">air density</a> as a function of <a href="/wiki/Height" title="Height">height</a>.<a href="#cite_note-1">[1]</a> This refraction is due to the velocity of light through <a href="/wiki/Air" class="mw-redirect" title="Air">air</a> decreasing (the <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a> increases) with increased density. Atmospheric <a href="/wiki/Refraction" title="Refraction">refraction</a> near the ground produces <a href="/wiki/Mirage" title="Mirage">mirages</a>. Such refraction can also <a href="/wiki/Looming_and_similar_refraction_phenomena" title="Looming and similar refraction phenomena">raise or lower</a>, or stretch or shorten, the images of distant objects without involving mirages. <a href="/wiki/Turbulence" title="Turbulence">Turbulent</a> air can make distant objects appear to <a href="/wiki/Twinkling" title="Twinkling">twinkle or shimmer</a>. The term also applies to the <a href="/wiki/Refraction_of_sound" class="mw-redirect" title="Refraction of sound">refraction of sound</a>. Atmospheric refraction is considered in measuring the position of both celestial and terrestrial objects. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Atmospheric_refraction_-_sunset_and_sunrise.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Atmospheric_refraction_-_sunset_and_sunrise.png/220px-Atmospheric_refraction_-_sunset_and_sunrise.png" decoding="async" width="220" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Atmospheric_refraction_-_sunset_and_sunrise.png/330px-Atmospheric_refraction_-_sunset_and_sunrise.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Atmospheric_refraction_-_sunset_and_sunrise.png/440px-Atmospheric_refraction_-_sunset_and_sunrise.png 2x" data-file-width="1760" data-file-height="1126"></a><figcaption>Diagram showing displacement of the <a href="/wiki/Sun" title="Sun">Sun</a>'s image at <a href="/wiki/False_sunrise" title="False sunrise">sunrise</a> and <a href="/wiki/False_sunset" title="False sunset">sunset</a></figcaption></figure> Astronomical or celestial refraction causes <a href="/wiki/Astronomical_object" title="Astronomical object">astronomical objects</a> to appear higher above the horizon than they actually are. Terrestrial refraction usually causes terrestrial objects to <a href="/wiki/Mirage#Superior_mirage" title="Mirage">appear higher</a> than they actually are, although in the afternoon when the air near the ground is heated, the rays can curve upward making objects <a href="/wiki/Mirage#Inferior_mirage" title="Mirage">appear lower</a> than they actually are. Refraction not only affects visible light rays, but all <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>, although in varying degrees. For example, in the <a href="/wiki/Visible_spectrum" title="Visible spectrum">visible spectrum</a>, blue is more affected than red. This may cause astronomical objects to appear <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersed</a> into a spectrum in high-resolution images. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:The_Swimming_Moon.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/The_Swimming_Moon.jpg/220px-The_Swimming_Moon.jpg" decoding="async" width="220" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/The_Swimming_Moon.jpg/330px-The_Swimming_Moon.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/The_Swimming_Moon.jpg/440px-The_Swimming_Moon.jpg 2x" data-file-width="1280" data-file-height="1039"></a><figcaption>The atmosphere refracts the image of a <a href="/wiki/Lunar_phase" title="Lunar phase">waxing</a> crescent <a href="/wiki/Moon" title="Moon">Moon</a> as it sets into the horizon.<a href="#cite_note-2">[2]</a></figcaption></figure> Whenever possible, <a href="/wiki/Astronomer" title="Astronomer">astronomers</a> will schedule their observations around the times of <a href="/wiki/Culmination" title="Culmination">culmination</a>, when celestial objects are highest in the sky. Likewise, sailors will not <a href="/wiki/Celestial_navigation#Angular_measurement" title="Celestial navigation">shoot</a> a star below 20° above the horizon. If observations of objects near the horizon cannot be avoided, it is possible to equip an <a href="/wiki/Optical_telescope" title="Optical telescope">optical telescope</a> with control systems to compensate for the shift caused by the refraction. If the dispersion is also a problem (in case of broadband high-resolution observations), atmospheric refraction correctors (made from pairs of rotating glass <a href="/wiki/Prism_(optics)" title="Prism (optics)">prisms</a>) can be employed as well. Since the amount of atmospheric refraction is a function of the <a href="/wiki/Lapse_rate" title="Lapse rate">temperature gradient</a>, <a href="/wiki/Temperature" title="Temperature">temperature</a>, <a href="/wiki/Pressure" title="Pressure">pressure</a>, and <a href="/wiki/Humidity" title="Humidity">humidity</a> (the amount of <a href="/wiki/Water_vapor" title="Water vapor">water vapor</a>, which is especially important at mid-<a href="/wiki/Infrared" title="Infrared">infrared</a> wavelengths), the amount of effort needed for a successful compensation can be prohibitive. Surveyors, on the other hand, will often schedule their observations in the afternoon, when the magnitude of refraction is minimum. Atmospheric refraction becomes more severe when temperature gradients are strong, and refraction is not uniform when the atmosphere is heterogeneous, as when <a href="/wiki/Turbulence" title="Turbulence">turbulence</a> occurs in the air. This causes suboptimal <a href="/wiki/Astronomical_seeing" title="Astronomical seeing">seeing</a> conditions, such as the <a href="/wiki/Twinkling" title="Twinkling">twinkling</a> of <a href="/wiki/Star" title="Star">stars</a> and various deformations of the <a href="/wiki/Sun" title="Sun">Sun</a>'s apparent shape soon before <a href="/wiki/Sunset" title="Sunset">sunset</a> or after <a href="/wiki/Sunrise" title="Sunrise">sunrise</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Astronomical_refraction">1 Astronomical refraction</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Calculating_refraction">1.1 Calculating refraction</a></li> <li class="toclevel-2 tocsection-3"><a href="#Random_refraction_effects">1.2 Random refraction effects</a></li> </ul> </li> <li class="toclevel-1 tocsection-4"><a href="#Terrestrial_refraction">2 Terrestrial refraction</a></li> <li class="toclevel-1 tocsection-5"><a href="#See_also">3 See also</a></li> <li class="toclevel-1 tocsection-6"><a href="#Notes">4 Notes</a></li> <li class="toclevel-1 tocsection-7"><a href="#References">5 References</a></li> <li class="toclevel-1 tocsection-8"><a href="#Further_reading">6 Further reading</a></li> <li class="toclevel-1 tocsection-9"><a href="#External_links">7 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Astronomical_refraction">Astronomical refraction</h2> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=1" title="Edit section: Astronomical refraction" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg/220px-SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg" decoding="async" width="220" height="159" class="mw-file-element" data-file-width="2887" data-file-height="2089"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 159px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg/220px-SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg" data-width="220" data-height="159" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg/330px-SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg/440px-SB_DouglasPreserve_SunAtmosphericEffects_2017_cropped2.jpg 2x" data-class="mw-file-element"> </a><figcaption>Atmospheric refraction distorting the Sun’s disk into an uneven shape as it sets in the lower horizon.</figcaption></figure> Astronomical refraction deals with the angular position of celestial bodies, their appearance as a point source, and through differential refraction, the shape of extended bodies such as the Sun and Moon.<a href="#cite_note-3">[3]</a> Atmospheric refraction of the light from a star is zero in the <a href="/wiki/Zenith" title="Zenith">zenith</a>, less than 1′ (one <a href="/wiki/Arcminute" class="mw-redirect" title="Arcminute">arc-minute</a>) at 45° apparent <a href="/wiki/Horizontal_coordinate_system" title="Horizontal coordinate system">altitude</a>, and still only 5.3′ at 10° altitude; it quickly increases as altitude decreases, reaching 9.9′ at 5° altitude, 18.4′ at 2° altitude, and 35.4′ at the <a href="/wiki/Horizon" title="Horizon">horizon</a>;<a href="#cite_note-Allen1976-4">[4]</a> all values are for 10 °C and 1013.25 <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">hPa</a> in the visible part of the spectrum. On the horizon, refraction is slightly greater than the apparent diameter of the Sun, so when the bottom of the sun's disc appears to touch the horizon, the sun's true altitude is negative. If the atmosphere suddenly vanished at this moment, one couldn't see the sun, as it would be entirely below the horizon. By convention, <a href="/wiki/Sunrise" title="Sunrise">sunrise</a> and <a href="/wiki/Sunset" title="Sunset">sunset</a> refer to times at which the Sun's upper limb appears on or disappears from the horizon and the standard value for the Sun's true altitude is −50′: −34′ for the refraction and −16′ for the Sun's <a href="/wiki/Semidiameter" title="Semidiameter">semi-diameter</a>. The altitude of a celestial body is normally given for the center of the body's disc. In the case of the <a href="/wiki/Moon" title="Moon">Moon</a>, additional corrections are needed for the Moon's <a href="/wiki/Parallax#Lunar_parallax" title="Parallax">horizontal parallax</a> and its apparent semi-diameter; both vary with the Earth–Moon distance. Refraction near the horizon is highly variable, principally because of the variability of the <a href="/wiki/Lapse_rate" title="Lapse rate">temperature gradient</a> near the Earth's surface and the geometric sensitivity of the nearly horizontal rays to this variability. As early as 1830, <a href="/wiki/Friedrich_Bessel" class="mw-redirect" title="Friedrich Bessel">Friedrich Bessel</a> had found that even after applying all corrections for temperature and pressure (but not for the temperature gradient) at the observer, highly precise measurements of refraction varied by ±0.19′ at two degrees above the horizon and by ±0.50′ at a half degree above the horizon.<a href="#cite_note-5">[5]</a> At and below the horizon, values of refraction significantly higher than the nominal value of 35.4′ have been observed in a wide range of climates. Georg Constantin Bouris measured refraction of as much of 4° for stars on the horizon at the <a href="/wiki/National_Observatory_of_Athens" title="National Observatory of Athens">Athens Observatory</a><a href="#cite_note-Young2004-6">[6]</a> and, during his ill-fated <a href="/wiki/Imperial_Trans-Antarctic_Expedition" title="Imperial Trans-Antarctic Expedition">Endurance expedition</a>, <a href="/wiki/Sir_Ernest_Shackleton" class="mw-redirect" title="Sir Ernest Shackleton">Sir Ernest Shackleton</a> recorded refraction of 2°37′:<a href="#cite_note-Shackleton1919-7">[7]</a> “The sun which had made ‘positively his last appearance’ seven days earlier surprised us by lifting more than half its disk above the horizon on May 8. A glow on the northern horizon resolved itself into the sun at 11 am that day. A quarter of an hour later the unreasonable visitor disappeared again, only to rise again at 11:40 am, set at 1 pm, rise at 1:10 pm and set lingeringly at 1:20 pm. These curious phenomena were due to refraction which amounted to 2° 37′ at 1:20 pm. The temperature was 15° below 0° Fahr., and we calculated that the refraction was 2° above normal.” Day-to-day variations in the weather will affect the exact times of sunrise and sunset<a href="#cite_note-SchaeferLiller1990-8">[8]</a> as well as moon-rise and moon-set, and for that reason it generally is not meaningful to give rise and set times to greater precision than the nearest minute.<a href="#cite_note-Meeus1991-9">[9]</a> More precise calculations can be useful for determining day-to-day changes in rise and set times that would occur with the standard value for refraction<a href="#cite_note-11">[note 1]</a> if it is understood that actual changes may differ because of unpredictable variations in refraction. Because atmospheric refraction is nominally 34′ on the horizon, but only 29′ at 0.5° above it, the setting or rising sun seems to be flattened by about 5′ (about 1/6 of its apparent diameter). <div class="mw-heading mw-heading3"><h3 id="Calculating_refraction">Calculating refraction</h3> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=2" title="Edit section: Calculating refraction" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Young<a href="#cite_note-Young2004-6">[6]</a><a href="#cite_note-Young2006-12">[11]</a> distinguished several regions where different methods for calculating astronomical refraction were applicable. In the upper portion of the sky, with a <a href="/wiki/Zenith_distance" class="mw-redirect" title="Zenith distance">zenith distance</a> of less than 70° (or an altitude over 20°), various simple refraction formulas based on the index of refraction (and hence on the temperature, pressure, and humidity) at the observer are adequate. Between 20° and 5° of the horizon the temperature gradient becomes the dominant factor and numerical integration, using a method such as that of Auer and Standish<a href="#cite_note-AuerStandish2000-13">[12]</a> and employing the <a href="/wiki/Lapse_rate" title="Lapse rate">temperature gradient</a> of the <a href="/wiki/International_Standard_Atmosphere" title="International Standard Atmosphere">standard atmosphere</a> and the measured conditions at the observer, is required. Closer to the horizon, actual measurements of the changes with height of the local temperature gradient need to be employed in the numerical integration. Below the astronomical horizon, refraction is so variable that only crude estimates of astronomical refraction can be made; for example, the observed time of sunrise or sunset can vary by several minutes from day to day. As <a href="/wiki/The_Nautical_Almanac" title="The Nautical Almanac">The Nautical Almanac</a> notes, "the actual values of …the refraction at low altitudes may, in extreme atmospheric conditions, differ considerably from the mean values used in the tables."<a href="#cite_note-14">[13]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:BennettAtmRefractVsAlt.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/BennettAtmRefractVsAlt.png/220px-BennettAtmRefractVsAlt.png" decoding="async" width="220" height="147" class="mw-file-element" data-file-width="1000" data-file-height="668"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 147px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/BennettAtmRefractVsAlt.png/220px-BennettAtmRefractVsAlt.png" data-width="220" data-height="147" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/BennettAtmRefractVsAlt.png/330px-BennettAtmRefractVsAlt.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/BennettAtmRefractVsAlt.png/440px-BennettAtmRefractVsAlt.png 2x" data-class="mw-file-element"> </a><figcaption>Plot of refraction vs. altitude using Bennett's 1982 formula</figcaption></figure> Many different formulas have been developed for calculating astronomical refraction; they are reasonably consistent, differing among themselves by a few minutes of arc at the horizon and becoming increasingly consistent as they approach the zenith. The simpler formulations involved nothing more than the temperature and pressure at the observer, powers of the <a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent">cotangent</a> of the apparent altitude of the astronomical body and in the higher order terms, the height of a fictional homogeneous atmosphere.<a href="#cite_note-15">[14]</a><a href="#cite_note-16">[15]</a> The simplest version of this formula, which Smart held to be only accurate within 45° of the zenith, is:<a href="#cite_note-17">[16]</a><a href="#cite_note-W&C-18">[17]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=(n_{0}-1)\cot h_{\mathrm {a} }\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>cot</mi> <mo>⁡</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=(n_{0}-1)\cot h_{\mathrm {a} }\,,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/319cbd6e444e82388b0e5d2ce072a001d592dc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.424ex; height:2.843ex;" alt="{\displaystyle R=(n_{0}-1)\cot h_{\mathrm {a} }\,,}"></noscript><span class="lazy-image-placeholder" style="width: 20.424ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/319cbd6e444e82388b0e5d2ce072a001d592dc65" data-alt="{\displaystyle R=(n_{0}-1)\cot h_{\mathrm {a} }\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where R is the refraction in <a href="/wiki/Radians" class="mw-redirect" title="Radians">radians</a>, n0 is the <a href="/wiki/Index_of_refraction" class="mw-redirect" title="Index of refraction">index of refraction</a> at the observer (which depends on the temperature, pressure, and humidity), and ha is the apparent altitude angle of the astronomical body. An early simple approximation of this form, which directly incorporated the temperature and pressure at the observer, was developed by <a href="/wiki/George_Comstock_(astronomer)" title="George Comstock (astronomer)">George Comstock</a>:<a href="#cite_note-Comstock1890-19">[18]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\frac {21.5b}{273+t}}\cot h_{\mathrm {a} }\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>21.5</mn> <mi>b</mi> </mrow> <mrow> <mn>273</mn> <mo>+</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>cot</mi> <mo>⁡</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {21.5b}{273+t}}\cot h_{\mathrm {a} }\,,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfe629c7157572b3bcad56bb4a5e4122df52200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.167ex; height:5.509ex;" alt="{\displaystyle R={\frac {21.5b}{273+t}}\cot h_{\mathrm {a} }\,,}"></noscript><span class="lazy-image-placeholder" style="width: 20.167ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfe629c7157572b3bcad56bb4a5e4122df52200" data-alt="{\displaystyle R={\frac {21.5b}{273+t}}\cot h_{\mathrm {a} }\,,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where R is the refraction in seconds of arc, b is the <a href="/wiki/Atmospheric_pressure" title="Atmospheric pressure">atmospheric pressure</a> in <a href="/wiki/Millimeters_of_mercury" class="mw-redirect" title="Millimeters of mercury">millimeters of mercury</a>, and t is the temperature in <a href="/wiki/Celsius" title="Celsius">Celsius</a>. Comstock considered that this formula gave results within one arcsecond of <a href="/wiki/Friedrich_Bessel" class="mw-redirect" title="Friedrich Bessel">Bessel</a>'s values for refraction from 15° above the horizon to the zenith.<a href="#cite_note-Comstock1890-19">[18]</a> A further expansion in terms of the third power of the cotangent of the apparent altitude incorporates H0, the <a href="/wiki/Air_mass_(astronomy)#Homogeneous_atmosphere" title="Air mass (astronomy)">height of the homogeneous atmosphere</a>, in addition to the usual conditions at the observer:<a href="#cite_note-W&C-18">[17]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=(n_{0}-1)(1-H_{0})\cot h_{\mathrm {a} }-(n_{0}-1)[H_{0}-{\frac {1}{2}}(n_{0}-1)]\cot ^{3}h_{\mathrm {a} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cot</mi> <mo>⁡</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=(n_{0}-1)(1-H_{0})\cot h_{\mathrm {a} }-(n_{0}-1)[H_{0}-{\frac {1}{2}}(n_{0}-1)]\cot ^{3}h_{\mathrm {a} }.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/355d5b7d9b399cffc9a91350d35c7b27dbb2d219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:64.637ex; height:5.176ex;" alt="{\displaystyle R=(n_{0}-1)(1-H_{0})\cot h_{\mathrm {a} }-(n_{0}-1)[H_{0}-{\frac {1}{2}}(n_{0}-1)]\cot ^{3}h_{\mathrm {a} }.}"></noscript><span class="lazy-image-placeholder" style="width: 64.637ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/355d5b7d9b399cffc9a91350d35c7b27dbb2d219" data-alt="{\displaystyle R=(n_{0}-1)(1-H_{0})\cot h_{\mathrm {a} }-(n_{0}-1)[H_{0}-{\frac {1}{2}}(n_{0}-1)]\cot ^{3}h_{\mathrm {a} }.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> A version of this formula is used in the <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">International Astronomical Union</a>'s Standards of Fundamental Astronomy; a comparison of the IAU's algorithm with more rigorous ray-tracing procedures indicated an agreement within 60 <a href="/wiki/Minute_and_second_of_arc" title="Minute and second of arc">milliarcseconds</a> at altitudes above 15°.<a href="#cite_note-20">[19]</a> Bennett<a href="#cite_note-Bennett1982-21">[20]</a> developed another simple empirical formula for calculating refraction from the apparent altitude which gives the refraction R in arcminutes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\cot \left(h_{\mathrm {a} }+{\frac {7.31}{h_{\mathrm {a} }+4.4}}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7.31</mn> <mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>4.4</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\cot \left(h_{\mathrm {a} }+{\frac {7.31}{h_{\mathrm {a} }+4.4}}\right)\,.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1cf7ecacfa7ac2ac5e443481f36fe7c3d82fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.692ex; height:6.176ex;" alt="{\displaystyle R=\cot \left(h_{\mathrm {a} }+{\frac {7.31}{h_{\mathrm {a} }+4.4}}\right)\,.}"></noscript><span class="lazy-image-placeholder" style="width: 26.692ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1cf7ecacfa7ac2ac5e443481f36fe7c3d82fc0" data-alt="{\displaystyle R=\cot \left(h_{\mathrm {a} }+{\frac {7.31}{h_{\mathrm {a} }+4.4}}\right)\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> This formula is used in the <a href="/wiki/United_States_Naval_Observatory" title="United States Naval Observatory">U. S. Naval Observatory</a>'s Vector Astrometry Software,<a href="#cite_note-22">[21]</a> and is reported to be consistent with Garfinkel's<a href="#cite_note-23">[22]</a> more complex algorithm within 0.07′ over the entire range from the zenith to the horizon.<a href="#cite_note-Meeus1991-9">[9]</a><a href="#cite_note-Bennett1982-21">[20]</a> Sæmundsson<a href="#cite_note-Saemundsson1986-24">[23]</a> developed an inverse formula for determining refraction from true altitude; if h is the true altitude in degrees, refraction R in arcminutes is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=1.02\cot \left(h+{\frac {10.3}{h+5.11}}\right)\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mn>1.02</mn> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>h</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10.3</mn> <mrow> <mi>h</mi> <mo>+</mo> <mn>5.11</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=1.02\cot \left(h+{\frac {10.3}{h+5.11}}\right)\,;}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b734518f74fb6de8f6eb39779576c3c1f48711dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.267ex; height:6.176ex;" alt="{\displaystyle R=1.02\cot \left(h+{\frac {10.3}{h+5.11}}\right)\,;}"></noscript><span class="lazy-image-placeholder" style="width: 30.267ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b734518f74fb6de8f6eb39779576c3c1f48711dc" data-alt="{\displaystyle R=1.02\cot \left(h+{\frac {10.3}{h+5.11}}\right)\,;}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> the formula is consistent with Bennett's to within 0.1′. The formulas of Bennet and Sæmundsson assume an <a href="/wiki/Atmospheric_pressure" title="Atmospheric pressure">atmospheric pressure</a> of 101.0 kPa and a temperature of 10 °C; for different pressure P and temperature T, refraction calculated from these formulas is multiplied by<a href="#cite_note-Meeus1991-9">[9]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {P}{101}}\,{\frac {283}{273+T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mn>101</mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>283</mn> <mrow> <mn>273</mn> <mo>+</mo> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P}{101}}\,{\frac {283}{273+T}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38477140cd1c2b9e66c8b0520559752d371113e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.511ex; height:5.343ex;" alt="{\displaystyle {\frac {P}{101}}\,{\frac {283}{273+T}}}"></noscript><span class="lazy-image-placeholder" style="width: 13.511ex;height: 5.343ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38477140cd1c2b9e66c8b0520559752d371113e" data-alt="{\displaystyle {\frac {P}{101}}\,{\frac {283}{273+T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Refraction increases approximately 1% for every 0.9 kPa increase in pressure, and decreases approximately 1% for every 0.9 kPa decrease in pressure. Similarly, refraction increases approximately 1% for every 3 °C decrease in temperature, and decreases approximately 1% for every 3 °C increase in temperature. <div class="mw-heading mw-heading3"><h3 id="Random_refraction_effects">Random refraction effects</h3> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=3" title="Edit section: Random refraction effects" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Seeing_Moon.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/e/ef/Seeing_Moon.gif" decoding="async" width="150" height="150" class="mw-file-element" data-file-width="150" data-file-height="150"></noscript><span class="lazy-image-placeholder" style="width: 150px;height: 150px;" data-src="//upload.wikimedia.org/wikipedia/commons/e/ef/Seeing_Moon.gif" data-width="150" data-height="150" data-class="mw-file-element"> </a><figcaption>The animated image of the <a href="/wiki/Moon" title="Moon">Moon</a>'s surface shows the <a href="/wiki/Twinkling" title="Twinkling">effects</a> of atmospheric <a href="/wiki/Turbulence" title="Turbulence">turbulence</a> on the view.</figcaption></figure> <a href="/wiki/Turbulence" title="Turbulence">Turbulence</a> in <a href="/wiki/Earth%27s_atmosphere" class="mw-redirect" title="Earth's atmosphere">Earth's atmosphere</a> <a href="/wiki/Light_scattering" class="mw-redirect" title="Light scattering">scatters</a> the light from stars, making them appear brighter and fainter on a time-scale of <a href="/wiki/Millisecond" title="Millisecond">milliseconds</a>. The slowest components of these fluctuations are visible as <a href="/wiki/Twinkling" title="Twinkling">twinkling</a> (also called scintillation). Turbulence also causes small, sporadic motions of the star image, and produces rapid distortions in its structure. These effects are not visible to the <a href="/wiki/Naked_eye" title="Naked eye">naked eye</a>, but can be easily seen even in small telescopes. They perturb <a href="/wiki/Astronomical_seeing" title="Astronomical seeing">astronomical seeing</a> conditions. Some telescopes employ <a href="/wiki/Adaptive_optics" title="Adaptive optics">adaptive optics</a> to reduce this effect. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Terrestrial_refraction">Terrestrial refraction</h2> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=4" title="Edit section: Terrestrial refraction" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Levelling_refraction" title="Levelling refraction">Levelling refraction</a></div> Terrestrial refraction, sometimes called geodetic refraction, deals with the apparent angular position and measured distance of terrestrial bodies. It is of special concern for the production of precise <a href="/wiki/Cartography" title="Cartography">maps</a> and <a href="/wiki/Surveying" title="Surveying">surveys</a>.<a href="#cite_note-25">[24]</a><a href="#cite_note-Brunner_1984_p.-26">[25]</a> Since the line of sight in terrestrial refraction passes near the earth's surface, the magnitude of refraction depends chiefly on the temperature gradient near the ground, which varies widely at different times of day, seasons of the year, the nature of the terrain, the state of the weather, and other factors.<a href="#cite_note-27">[26]</a> As a common approximation, terrestrial refraction is considered as a constant bending of the ray of light or line of sight, in which the ray can be considered as describing a circular path. A common measure of refraction is the coefficient of refraction. Unfortunately there are two different definitions of this coefficient. One is the ratio of the radius of the Earth to the radius of the line of sight,<a href="#cite_note-Hirt2010-28">[27]</a> the other is the ratio of the angle that the line of sight subtends at the center of the Earth to the angle of refraction measured at the observer.<a href="#cite_note-Bomford1980_236-29">[28]</a> Since the latter definition only measures the bending of the ray at one end of the line of sight, it is one half the value of the former definition. The coefficient of refraction is directly related to the local vertical temperature gradient and the atmospheric temperature and pressure. The larger version of the coefficient k, measuring the ratio of the radius of the Earth to the radius of the line of sight, is given by:<a href="#cite_note-Hirt2010-28">[27]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=503{\frac {P}{T^{2}}}\left(0.0343+{\frac {dT}{dh}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>503</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>0.0343</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>T</mi> </mrow> <mrow> <mi>d</mi> <mi>h</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=503{\frac {P}{T^{2}}}\left(0.0343+{\frac {dT}{dh}}\right),}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c90d6e96eacccaf3740d74df75ed4d605fc14f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.237ex; height:6.176ex;" alt="{\displaystyle k=503{\frac {P}{T^{2}}}\left(0.0343+{\frac {dT}{dh}}\right),}"></noscript><span class="lazy-image-placeholder" style="width: 29.237ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c90d6e96eacccaf3740d74df75ed4d605fc14f" data-alt="{\displaystyle k=503{\frac {P}{T^{2}}}\left(0.0343+{\frac {dT}{dh}}\right),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where temperature T is given in <a href="/wiki/Kelvin" title="Kelvin">kelvins</a>, pressure P in <a href="/wiki/Bar_(unit)" title="Bar (unit)">millibars</a>, and height h in meters. The angle of refraction increases with the coefficient of refraction and with the length of the line of sight. Although the straight line from your eye to a distant mountain might be blocked by a closer hill, the ray may curve enough to make the distant peak visible. A convenient method to analyze the effect of refraction on visibility is to consider an increased effective radius of the Earth Reff, given by<a href="#cite_note-Young2006-12">[11]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\text{eff}}={\frac {R}{1-k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\text{eff}}={\frac {R}{1-k}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a13c6fb5428cd6492d17f8182635c0c4465185f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.529ex; height:5.509ex;" alt="{\displaystyle R_{\text{eff}}={\frac {R}{1-k}},}"></noscript><span class="lazy-image-placeholder" style="width: 13.529ex;height: 5.509ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a13c6fb5428cd6492d17f8182635c0c4465185f" data-alt="{\displaystyle R_{\text{eff}}={\frac {R}{1-k}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where R is the radius of the Earth and k is the coefficient of refraction. Under this model the ray can be considered a straight line on an Earth of increased radius. The curvature of the refracted ray in <a href="/wiki/Arc_seconds" class="mw-redirect" title="Arc seconds">arc seconds</a> per meter can be computed using the relationship<a href="#cite_note-30">[29]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sigma }}=16.3{\frac {P}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right)\cos \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>σ</mi> </mfrac> </mrow> <mo>=</mo> <mn>16.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>0.0342</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>T</mi> </mrow> <mrow> <mi>d</mi> <mi>h</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sigma }}=16.3{\frac {P}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right)\cos \beta }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b85d2ef48f9305c608bf5a4fa81fb6b5baceee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.022ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sigma }}=16.3{\frac {P}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right)\cos \beta }"></noscript><span class="lazy-image-placeholder" style="width: 35.022ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b85d2ef48f9305c608bf5a4fa81fb6b5baceee6" data-alt="{\displaystyle {\frac {1}{\sigma }}=16.3{\frac {P}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right)\cos \beta }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where 1/σ is the curvature of the ray in arcsec per meter, P is the pressure in millibars, T is the temperature in kelvins, and β is the angle of the ray to the horizontal. Multiplying half the curvature by the length of the ray path gives the angle of refraction at the observer. For a line of sight near the horizon cos β differs little from unity and can be ignored. This yields <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =8.15{\frac {LP}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>8.15</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>0.0342</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>T</mi> </mrow> <mrow> <mi>d</mi> <mi>h</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =8.15{\frac {LP}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right),}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348435a73eb7605a677dbdddf0c2bb4d447d55d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.905ex; height:6.176ex;" alt="{\displaystyle \Omega =8.15{\frac {LP}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right),}"></noscript><span class="lazy-image-placeholder" style="width: 30.905ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348435a73eb7605a677dbdddf0c2bb4d447d55d2" data-alt="{\displaystyle \Omega =8.15{\frac {LP}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where L is the length of the line of sight in meters and Ω is the refraction at the observer measured in arc seconds. A simple approximation is to consider that a mountain's apparent altitude at your eye (in degrees) will exceed its true altitude by its distance in kilometers divided by 1500. This assumes a fairly horizontal line of sight and ordinary air density; if the mountain is very high (so much of the sightline is in thinner air) divide by 1600 instead.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=5" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <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: 30em;"> <ul><li><a href="/wiki/Polar_circle#Effect_of_atmospheric_refraction" title="Polar circle">Polar circle#Effect of atmospheric refraction</a></li> <li><a href="/wiki/Sound_wave" class="mw-redirect" title="Sound wave">Sound waves</a> affected by air density variations lead to <a href="/wiki/Atmospheric_focusing" title="Atmospheric focusing">atmospheric focusing</a>.</li> <li><a href="/wiki/Air_mass_(astronomy)" title="Air mass (astronomy)">Air mass (astronomy)</a></li> <li><a href="/wiki/Atmospheric_optics" title="Atmospheric optics">Atmospheric optics</a></li> <li><a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a></li> <li><a href="/wiki/Fata_Morgana_(mirage)" title="Fata Morgana (mirage)">Fata Morgana (mirage)</a></li> <li><a href="/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a></li> <li><a href="/wiki/Looming_and_similar_refraction_phenomena" title="Looming and similar refraction phenomena">Looming and similar refraction phenomena</a></li> <li><a href="/wiki/Novaya_Zemlya_effect" title="Novaya Zemlya effect">Novaya Zemlya effect</a></li> <li><a href="/wiki/Radio_propagation" title="Radio propagation">Radio propagation</a></li> <li><a href="/wiki/Line-of-sight_propagation#Atmospheric_refraction" title="Line-of-sight propagation">Line-of-sight propagation#Atmospheric refraction</a></li> <li><a href="/wiki/Ray_tracing_(physics)" title="Ray tracing (physics)">Ray tracing (physics)</a></li> <li><a href="/wiki/Shen_Kuo" title="Shen Kuo">Shen Kuo</a></li> <li><a href="/wiki/Terrestrial_atmospheric_lens" title="Terrestrial atmospheric lens">Terrestrial atmospheric lens</a></li></ul></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Notes">Notes</h2> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=6" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-11"><a href="#cite_ref-11">^</a> For an example see Meeus 2002<a href="#cite_note-Meeus2002-10">[10]</a> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=7" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> It is common in studies of refraction to use the term height to express vertical distance above the ground, or <a href="/wiki/Geodetic_datum#Vertical_datum" title="Geodetic datum">vertical datum</a> and <a href="/wiki/Horizontal_coordinate_system" title="Horizontal coordinate system">altitude</a> to express angular height above the <a href="/wiki/Horizon" title="Horizon">horizon</a>. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</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) 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The Astronomical Journal. 127 (6): 3622–3637. <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/2004AJ....127.3622Y">2004AJ....127.3622Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F420806">10.1086/420806</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astronomical+Journal&rft.atitle=Sunset+Science.+IV.+Low-Altitude+Refraction&rft.volume=127&rft.issue=6&rft.pages=3622-3637&rft.date=2004&rft_id=info%3Adoi%2F10.1086%2F420806&rft_id=info%3Abibcode%2F2004AJ....127.3622Y&rft.aulast=Young&rft.aufirst=Andrew+T.&rft_id=https%3A%2F%2Fdoi.org%2F10.1086%252F420806&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-Shackleton1919-7"><a href="#cite_ref-Shackleton1919_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShackleton1919" class="citation book cs1">Shackleton, Sir Ernest (1919). South: the story of Shackleton's last expedition. London: Century Publishing. p. 49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7126-0111-5" title="Special:BookSources/978-0-7126-0111-5"><bdi>978-0-7126-0111-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=South%3A+the+story+of+Shackleton%27s+last+expedition&rft.place=London&rft.pages=49&rft.pub=Century+Publishing&rft.date=1919&rft.isbn=978-0-7126-0111-5&rft.aulast=Shackleton&rft.aufirst=Sir+Ernest&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-SchaeferLiller1990-8"><a href="#cite_ref-SchaeferLiller1990_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchaeferLiller1990" class="citation journal cs1"><a href="/wiki/Bradley_Schaefer" title="Bradley Schaefer">Schaefer, Bradley E.</a>; <a href="/wiki/William_Liller" title="William Liller">Liller, William</a> (1990). <a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F132705">"Refraction near the horizon"</a>. Publications of the Astronomical Society of the Pacific. 102: 796–805. <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/1990PASP..102..796S">1990PASP..102..796S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F132705">10.1086/132705</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publications+of+the+Astronomical+Society+of+the+Pacific&rft.atitle=Refraction+near+the+horizon&rft.volume=102&rft.pages=796-805&rft.date=1990&rft_id=info%3Adoi%2F10.1086%2F132705&rft_id=info%3Abibcode%2F1990PASP..102..796S&rft.aulast=Schaefer&rft.aufirst=Bradley+E.&rft.au=Liller%2C+William&rft_id=https%3A%2F%2Fdoi.org%2F10.1086%252F132705&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-Meeus1991-9">^ <a href="#cite_ref-Meeus1991_9-0">a</a> <a href="#cite_ref-Meeus1991_9-1">b</a> <a href="#cite_ref-Meeus1991_9-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeeus1991" class="citation book cs1">Meeus, Jean (1991). Astronomical algorithms (1st English ed.). Richmond, Va.: Willmann-Bell. pp. 102–103. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-943396-35-4" title="Special:BookSources/978-0-943396-35-4"><bdi>978-0-943396-35-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astronomical+algorithms&rft.place=Richmond%2C+Va.&rft.pages=102-103&rft.edition=1st+English&rft.pub=Willmann-Bell&rft.date=1991&rft.isbn=978-0-943396-35-4&rft.aulast=Meeus&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-Meeus2002-10"><a href="#cite_ref-Meeus2002_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeeus2002" class="citation book cs1">Meeus, Jean (2002). [Mathematical astronomy morsels] (1st English ed.). Richmond, Va.: Willmann-Bell. p. 315. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-943396-74-3" title="Special:BookSources/978-0-943396-74-3"><bdi>978-0-943396-74-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%5BMathematical+astronomy+morsels%5D&rft.place=Richmond%2C+Va.&rft.pages=315&rft.edition=1st+English&rft.pub=Willmann-Bell&rft.date=2002&rft.isbn=978-0-943396-74-3&rft.aulast=Meeus&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-Young2006-12">^ <a href="#cite_ref-Young2006_12-0">a</a> <a href="#cite_ref-Young2006_12-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoung2006" class="citation journal cs1">Young, Andrew T. (2006). "Understanding Astronomical Refraction". The Observatory. 126: 82–115. <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/2006Obs...126...82Y">2006Obs...126...82Y</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Observatory&rft.atitle=Understanding+Astronomical+Refraction&rft.volume=126&rft.pages=82-115&rft.date=2006&rft_id=info%3Abibcode%2F2006Obs...126...82Y&rft.aulast=Young&rft.aufirst=Andrew+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-AuerStandish2000-13"><a href="#cite_ref-AuerStandish2000_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAuerStandish2000" class="citation journal cs1">Auer, Lawrence H.; Standish, E. Myles (2000). <a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F301325">"Astronomical Refraction: Computation for All Zenith Angles"</a>. Astronomical Journal. 119 (5): 2472–2474. <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/2000AJ....119.2472A">2000AJ....119.2472A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F301325">10.1086/301325</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:121417663">121417663</a>. <q>This paper and the method presented in it were submitted for publication in 1970 July. Unfortunately, the referee did not understand the utility of our new approach, and for personal reasons we did not have the time to argue the point sufficiently. We did distribute preprints, and the method has become, with improved atmospheric models, the technique of choice for the computation of refraction (see, e.g., Seidelmann [Explanatory Supplement to the Astronomical Almanac,] 1992).</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astronomical+Journal&rft.atitle=Astronomical+Refraction%3A+Computation+for+All+Zenith+Angles&rft.volume=119&rft.issue=5&rft.pages=2472-2474&rft.date=2000&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121417663%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1086%2F301325&rft_id=info%3Abibcode%2F2000AJ....119.2472A&rft.aulast=Auer&rft.aufirst=Lawrence+H.&rft.au=Standish%2C+E.+Myles&rft_id=https%3A%2F%2Fdoi.org%2F10.1086%252F301325&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">The nautical almanac for the year 1988, Washington / London: United States Naval Observatory / Her Majesty's Stationery Office, 1986, p. 261, <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/1987nay..book......">1987nay..book......</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+nautical+almanac+for+the+year+1988&rft.place=Washington+%2F+London&rft.pages=261&rft.pub=United+States+Naval+Observatory+%2F+Her+Majesty%27s+Stationery+Office&rft.date=1986&rft_id=info%3Abibcode%2F1987nay..book......&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFletcher1952" class="citation cs2">Fletcher, A. (1952), "Astronomical Refraction at Low Altitudes in Marine Navigation", The Journal of Navigation, 5 (4), London: 307–330, <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%2FS0373463300045033">10.1017/S0373463300045033</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1469-7785">1469-7785</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:129233309">129233309</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Navigation&rft.atitle=Astronomical+Refraction+at+Low+Altitudes+in+Marine+Navigation&rft.volume=5&rft.issue=4&rft.pages=307-330&rft.date=1952&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A129233309%23id-name%3DS2CID&rft.issn=1469-7785&rft_id=info%3Adoi%2F10.1017%2FS0373463300045033&rft.aulast=Fletcher&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWittmann1997" class="citation cs2">Wittmann, A. D. (1997), "Astronomical refraction: formulas for all zenith distances", Astronomische Nachrichten, 318 (5): 305–312, <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/1997AN....318..305W">1997AN....318..305W</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%2Fasna.2113180507">10.1002/asna.2113180507</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astronomische+Nachrichten&rft.atitle=Astronomical+refraction%3A+formulas+for+all+zenith+distances&rft.volume=318&rft.issue=5&rft.pages=305-312&rft.date=1997&rft_id=info%3Adoi%2F10.1002%2Fasna.2113180507&rft_id=info%3Abibcode%2F1997AN....318..305W&rft.aulast=Wittmann&rft.aufirst=A.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmart1977" class="citation cs2"><a href="/wiki/William_Marshall_Smart" title="William Marshall Smart">Smart, W. M.</a> (1977), Text-Book on Spherical Astronomy (sixth ed.), Cambridge University Press, pp. 61–62, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-29180-4" title="Special:BookSources/978-0-521-29180-4"><bdi>978-0-521-29180-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Text-Book+on+Spherical+Astronomy&rft.pages=61-62&rft.edition=sixth&rft.pub=Cambridge+University+Press&rft.date=1977&rft.isbn=978-0-521-29180-4&rft.aulast=Smart&rft.aufirst=W.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-W&C-18">^ <a href="#cite_ref-W&C_18-0">a</a> <a href="#cite_ref-W&C_18-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoolardClemence1966" class="citation cs2"><a href="/wiki/Edgar_W._Woolard" title="Edgar W. Woolard">Woolard, Edgar W.</a>; <a href="/wiki/Gerald_M._Clemence" class="mw-redirect" title="Gerald M. Clemence">Clemence, Gerald M.</a> (1966), Spherical Astronomy, New York and London: Academic Press, pp. 82–83</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spherical+Astronomy&rft.place=New+York+and+London&rft.pages=82-83&rft.pub=Academic+Press&rft.date=1966&rft.aulast=Woolard&rft.aufirst=Edgar+W.&rft.au=Clemence%2C+Gerald+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-Comstock1890-19">^ <a href="#cite_ref-Comstock1890_19-0">a</a> <a href="#cite_ref-Comstock1890_19-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFComstock1890" class="citation cs2">Comstock, George C. (1890), "A Simple Approximate Formula for Refraction", Sidereal Messenger, 9: 186, <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/1890SidM....9..185.">1890SidM....9..185.</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sidereal+Messenger&rft.atitle=A+Simple+Approximate+Formula+for+Refraction&rft.volume=9&rft.pages=186&rft.date=1890&rft_id=info%3Abibcode%2F1890SidM....9..185.&rft.aulast=Comstock&rft.aufirst=George+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" 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 class="citation cs2"><a rel="nofollow" class="external text" href="http://www.iausofa.org/sofa_ast_f.pdf">Standards Of Fundamental Astronomy; SOFA Astrometry Tools</a> (PDF) (Software version 11; Document 1.6 ed.), International Astronomical Union, 2014, pp. 12, 71–73, retrieved 23 June 2016, <q>The accuracy of the result is limited by the corrections for refraction, which use a simple A tan ζ + B tan3 ζ model. Providing the meteorological parameters are known accurately and there are no gross local effects, the predicted observed coordinates should be within 0".05 (optical) 1"(radio) for ζ < 70°, better than 30" (optical or radio) at 85° and better than 0°.3 (optical) or 0°.5 (radio) at the horizon.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Standards+Of+Fundamental+Astronomy%3B+SOFA+Astrometry+Tools&rft.pages=12%2C+71-73&rft.edition=Software+version+11%3B+Document+1.6&rft.pub=International+Astronomical+Union&rft.date=2014&rft_id=http%3A%2F%2Fwww.iausofa.org%2Fsofa_ast_f.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-Bennett1982-21">^ <a href="#cite_ref-Bennett1982_21-0">a</a> <a href="#cite_ref-Bennett1982_21-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBennett1982" class="citation journal cs1">Bennett, G.G. (1982). "The Calculation of Astronomical Refraction in Marine Navigation". Journal of Navigation. 35 (2): 255–259. <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/1982JNav...35..255B">1982JNav...35..255B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0373463300022037">10.1017/S0373463300022037</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:140675736">140675736</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Navigation&rft.atitle=The+Calculation+of+Astronomical+Refraction+in+Marine+Navigation&rft.volume=35&rft.issue=2&rft.pages=255-259&rft.date=1982&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A140675736%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0373463300022037&rft_id=info%3Abibcode%2F1982JNav...35..255B&rft.aulast=Bennett&rft.aufirst=G.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaplan2011" class="citation cs2">Kaplan, G. H. (21 March 2011), <a rel="nofollow" class="external text" href="http://aa.usno.navy.mil/software/novas/novas_f/NOVAS_F3.1.f">"SUBROUTINE REFRAC"</a>, NOVAS Fortran source code, Vers. F3.1 (Computer Program), Washington, D.C.: U. S. 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Geophysical Research, 115 (D21): D21102, <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/2010JGRD..11521102H">2010JGRD..11521102H</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1029%2F2010JD014067">10.1029/2010JD014067</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/20.500.11937%2F2972">20.500.11937/2972</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geophysical+Research&rft.atitle=Monitoring+of+the+refraction+coefficient+in+the+lower+atmosphere+using+a+controlled+setup+of+simultaneous+reciprocal+vertical+angle+measurements&rft.volume=115&rft.issue=D21&rft.pages=D21102&rft.date=2010&rft_id=info%3Ahdl%2F20.500.11937%2F2972&rft_id=info%3Adoi%2F10.1029%2F2010JD014067&rft_id=info%3Abibcode%2F2010JGRD..11521102H&rft.aulast=Hirt&rft.aufirst=Christian&rft.au=Guillaume%2C+Sebastian&rft.au=Wisbar%2C+Annemarie&rft.au=B%C3%BCrki%2C+Beat&rft.au=Sternberg%2C+Harald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> <li id="cite_note-Bomford1980_236-29"><a href="#cite_ref-Bomford1980_236_29-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBomford1980" class="citation cs2"><a href="/wiki/Guy_Bomford" title="Guy Bomford">Bomford, Guy</a> (1980), <a href="/wiki/Geodesy_(book)" title="Geodesy (book)">Geodesy</a> (4 ed.), Oxford: Oxford University Press, p. 236, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851946-1" title="Special:BookSources/978-0-19-851946-1"><bdi>978-0-19-851946-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geodesy&rft.place=Oxford&rft.pages=236&rft.edition=4&rft.pub=Oxford+University+Press&rft.date=1980&rft.isbn=978-0-19-851946-1&rft.aulast=Bomford&rft.aufirst=Guy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </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="CITEREFBomford1980" class="citation cs2"><a href="/wiki/Guy_Bomford" title="Guy Bomford">Bomford, Guy</a> (1980), <a href="/wiki/Geodesy_(book)" title="Geodesy (book)">Geodesy</a> (4 ed.), Oxford: Oxford University Press, p. 235, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851946-1" title="Special:BookSources/978-0-19-851946-1"><bdi>978-0-19-851946-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geodesy&rft.place=Oxford&rft.pages=235&rft.edition=4&rft.pub=Oxford+University+Press&rft.date=1980&rft.isbn=978-0-19-851946-1&rft.aulast=Bomford&rft.aufirst=Guy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Further_reading">Further reading</h2> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=8" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLehnvan_der_Werf2005" class="citation journal cs1">Lehn, Waldemar H.; van der Werf, Siebren (2005). "Atmospheric refraction: a history". Applied Optics. 44 (27): 5624–5636. <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/2005ApOpt..44.5624L">2005ApOpt..44.5624L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FAO.44.005624">10.1364/AO.44.005624</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-6935">0003-6935</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/16201423">16201423</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Optics&rft.atitle=Atmospheric+refraction%3A+a+history&rft.volume=44&rft.issue=27&rft.pages=5624-5636&rft.date=2005&rft_id=info%3Adoi%2F10.1364%2FAO.44.005624&rft.issn=0003-6935&rft_id=info%3Apmid%2F16201423&rft_id=info%3Abibcode%2F2005ApOpt..44.5624L&rft.aulast=Lehn&rft.aufirst=Waldemar+H.&rft.au=van+der+Werf%2C+Siebren&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFilippenko1982" class="citation journal cs1">Filippenko, A. 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Telescopes and Instruments V, 3356, <a href="/wiki/Jet_Propulsion_Laboratory" title="Jet Propulsion Laboratory">Jet Propulsion Laboratory</a>: 665, <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/1998SPIE.3356..665W">1998SPIE.3356..665W</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1117%2F12.324434">10.1117/12.324434</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2014%2F19082">2014/19082</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:120030054">120030054</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Space+Telescopes+and+Instruments+V&rft.atitle=Very+High-Resolution+Space+Telescope+Using+the+Earth+Atmosphere+as+the+Objective+Lens&rft.volume=3356&rft.pages=665&rft.date=1998-03-20&rft_id=info%3Ahdl%2F2014%2F19082&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120030054%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1117%2F12.324434&rft_id=info%3Abibcode%2F1998SPIE.3356..665W&rft.aulast=Wang&rft.aufirst=Yu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKipping2019" class="citation cs2">Kipping, David (18 July 2019), "The "Terrascope": On the Possibility of Using the Earth as an Atmospheric Lens", Publications of the Astronomical Society of the Pacific, 131 (1005), <a href="/wiki/Columbia_University" title="Columbia University">Columbia University</a>: 114503, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1908.00490">1908.00490</a>, <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/2019PASP..131k4503K">2019PASP..131k4503K</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1538-3873%2Fab33c0">10.1088/1538-3873/ab33c0</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:199064594">199064594</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publications+of+the+Astronomical+Society+of+the+Pacific&rft.atitle=The+%22Terrascope%22%3A+On+the+Possibility+of+Using+the+Earth+as+an+Atmospheric+Lens&rft.volume=131&rft.issue=1005&rft.pages=114503&rft.date=2019-07-18&rft_id=info%3Aarxiv%2F1908.00490&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A199064594%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1538-3873%2Fab33c0&rft_id=info%3Abibcode%2F2019PASP..131k4503K&rft.aulast=Kipping&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtmospheric+refraction" class="Z3988"></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Atmospheric_refraction&action=edit&section=9" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button 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id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D8%A7%D9%86%D9%83%D8%B3%D8%A7%D8%B1_%D9%81%D9%8A_%D8%A7%D9%84%D8%BA%D9%84%D8%A7%D9%81_%D8%A7%D9%84%D8%AC%D9%88%D9%8A" title="الانكسار في الغلاف الجوي – Arabic" lang="ar" hreflang="ar" data-title="الانكسار في الغلاف الجوي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%80%D1%8D%D1%84%D1%80%D0%B0%D0%BA%D1%86%D1%8B%D1%8F" title="Астранамічная рэфракцыя – Belarusian" lang="be" hreflang="be" data-title="Астранамічная рэфракцыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Refracci%C3%B3_atmosf%C3%A8rica" title="Refracció atmosfèrica – Catalan" lang="ca" hreflang="ca" data-title="Refracció atmosfèrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs badge-Q70894304 mw-list-item" title=""><a href="https://cs.wikipedia.org/wiki/Atmosf%C3%A9rick%C3%A1_refrakce" title="Atmosférická refrakce – Czech" lang="cs" hreflang="cs" data-title="Atmosférická refrakce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Astronomische_Refraktion" title="Astronomische Refraktion – German" lang="de" hreflang="de" data-title="Astronomische Refraktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%AE%CE%B9%CE%BD%CE%B7_%CE%B4%CE%B9%CE%AC%CE%B8%CE%BB%CE%B1%CF%83%CE%B7" title="Γήινη διάθλαση – Greek" lang="el" hreflang="el" data-title="Γήινη διάθλαση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Refracci%C3%B3n_atmosf%C3%A9rica" title="Refracción atmosférica – Spanish" lang="es" hreflang="es" data-title="Refracción atmosférica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9fraction_atmosph%C3%A9rique" title="Réfraction atmosphérique – French" lang="fr" hreflang="fr" data-title="Réfraction atmosphérique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EA%B8%B0_%EA%B5%B4%EC%A0%88" title="대기 굴절 – Korean" lang="ko" hreflang="ko" data-title="대기 굴절" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Atmosferska_refrakcija" title="Atmosferska refrakcija – Croatian" lang="hr" hreflang="hr" data-title="Atmosferska refrakcija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Rifrazione_atmosferica" title="Rifrazione atmosferica – Italian" lang="it" hreflang="it" data-title="Rifrazione atmosferica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%91%D7%99%D7%A8%D7%94_%D7%90%D7%98%D7%9E%D7%95%D7%A1%D7%A4%D7%99%D7%A8%D7%99%D7%AA" title="שבירה אטמוספירית – Hebrew" lang="he" hreflang="he" data-title="שבירה אטמוספירית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Astronomesch_Refraktioun" title="Astronomesch Refraktioun – Luxembourgish" lang="lb" hreflang="lb" data-title="Astronomesch Refraktioun" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atmosferin%C4%97_refrakcija" title="Atmosferinė refrakcija – Lithuanian" lang="lt" hreflang="lt" data-title="Atmosferinė refrakcija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pembiasan_atmosfera" title="Pembiasan atmosfera – Malay" lang="ms" hreflang="ms" data-title="Pembiasan atmosfera" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Astronomische_refractie" title="Astronomische refractie – Dutch" lang="nl" hreflang="nl" data-title="Astronomische refractie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%A7%E6%B0%97%E5%B7%AE" title="大気差 – Japanese" lang="ja" hreflang="ja" data-title="大気差" 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 mw-list-item"><a href="https://no.wikipedia.org/wiki/Atmosf%C3%A6risk_refraksjon" title="Atmosfærisk refraksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Atmosfærisk refraksjon" 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/Atmosf%C3%A6risk_refraksjon" title="Atmosfærisk refraksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Atmosfærisk refraksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Refrakcja_atmosferyczna" title="Refrakcja atmosferyczna – Polish" lang="pl" hreflang="pl" data-title="Refrakcja atmosferyczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Refra%C3%A7%C3%A3o_atmosf%C3%A9rica" title="Refração atmosférica – Portuguese" lang="pt" hreflang="pt" data-title="Refração atmosférica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D1%81%D1%82%D1%80%D0%BE%D0%BD%D0%BE%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%80%D0%B5%D1%84%D1%80%D0%B0%D0%BA%D1%86%D0%B8%D1%8F" title="Астрономическая рефракция – Russian" lang="ru" hreflang="ru" data-title="Астрономическая рефракция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B4%DA%A9%D8%A7%D9%86%D8%AF%D9%86%DB%95%D9%88%DB%95%DB%8C_%D8%AA%DB%95%D9%88%D8%B4%DA%A9" title="شکاندنەوەی تەوشک – Central Kurdish" lang="ckb" hreflang="ckb" data-title="شکاندنەوەی تەوشک" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Atmosferska_refrakcija" title="Atmosferska refrakcija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Atmosferska refrakcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Atmosf%C3%A4risk_refraktion" title="Atmosfärisk refraktion – Swedish" lang="sv" hreflang="sv" data-title="Atmosfärisk refraktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B5%D1%84%D1%80%D0%B0%D0%BA%D1%86%D1%96%D1%8F_%D1%81%D0%B2%D1%96%D1%82%D0%BB%D0%B0_%D0%B2_%D0%B0%D1%82%D0%BC%D0%BE%D1%81%D1%84%D0%B5%D1%80%D1%96" title="Рефракція світла в атмосфері – Ukrainian" lang="uk" hreflang="uk" data-title="Рефракція світла в атмосфері" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%A7%E6%B0%A3%E6%8A%98%E5%B0%84" title="大氣折射 – Chinese" lang="zh" hreflang="zh" data-title="大氣折射" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 8 February 2024, at 06:38 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" 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<script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b7f745dd4-49wnh","wgBackendResponseTime":172,"wgPageParseReport":{"limitreport":{"cputime":"0.585","walltime":"0.768","ppvisitednodes":{"value":2600,"limit":1000000},"postexpandincludesize":{"value":79943,"limit":2097152},"templateargumentsize":{"value":3361,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":11,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":131114,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 608.162 1 -total"," 40.45% 245.974 2 Template:Reflist"," 20.05% 121.921 23 Template:Citation"," 13.05% 79.342 1 Template:Authority_control"," 12.51% 76.098 1 Template:Cite_web"," 11.53% 70.093 1 Template:Short_description"," 8.72% 53.049 1 Template:Portal_bar"," 6.10% 37.103 8 Template:Cite_journal"," 6.08% 36.994 2 Template:Pagetype"," 5.39% 32.809 1 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