Atmospheric refraction

Not to be confused withAtmospheric diffraction.

Diagram showing displacement of theSun's image atsunriseandsunset

Atmospheric refractionis the deviation oflightor otherelectromagnetic wavefrom a straight line as it passes through theatmospheredue to the variation inair densityas a function ofheight.This refraction is due to the velocity of light throughairdecreasing (therefractive indexincreases) with increased density. Atmosphericrefractionnear the ground producesmirages.Such refraction can alsoraise or lower,or stretch or shorten, the images of distant objects without involving mirages.Turbulentair can make distant objects appear totwinkle or shimmer.The term also applies to therefraction of sound.Atmospheric refraction is considered in measuring the position of both celestial and terrestrial objects.

Astronomical or celestial refraction causesastronomical objectsto appear higher above the horizon than they actually are. Terrestrial refraction usually causes terrestrial objects toappear higherthan they actually are, although in the afternoon when the air near the ground is heated, the rays can curve upward making objectsappear lowerthan they actually are.

Refraction not only affects visible light rays, but allelectromagnetic radiation,although in varying degrees. For example, in thevisible spectrum,blue is more affected than red. This may cause astronomical objects to appeardispersedinto a spectrum in high-resolution images.

The atmosphere refracts the image of awa xingcrescentMoonas it sets into the horizon.

Whenever possible,astronomerswill schedule their observations around the times ofculmination,when celestial objects are highest in the sky. Likewise, sailors will notshoota star below 20° above the horizon. If observations of objects near the horizon cannot be avoided, it is possible to equip anoptical telescopewith 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 glassprisms) can be employed as well.

Since the amount of atmospheric refraction is a function of thetemperature gradient,temperature,pressure,andhumidity(the amount ofwater vapor,which is especially important at mid-infraredwavelengths), 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 whenturbulenceoccurs in the air. This causes suboptimalseeingconditions, such as thetwinklingofstarsand various deformations of theSun's apparent shape soon beforesunsetor aftersunrise.

Astronomical refraction

Atmospheric refraction distorting the Sun’s disk into an uneven shape as it sets in the lower horizon.

Astronomical refractiondeals 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.

Atmospheric refraction of the light from a star is zero in thezenith,less than 1′ (onearc-minute) at 45° apparentaltitude,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 thehorizon;all values are for 10 °C and 1013.25hPa 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,sunriseandsunsetrefer 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'ssemi-diameter.The altitude of a celestial body is normally given for the center of the body's disc. In the case of theMoon,additional corrections are needed for the Moon'shorizontal parallaxand its apparent semi-diameter; both vary with the Earth–Moon distance.

Refraction near the horizon is highly variable, principally because of the variability of thetemperature gradientnear the Earth's surface and the geometric sensitivity of the nearly horizontal rays to this variability. As early as 1830,Friedrich Besselhad 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.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 theAthens Observatoryand, during his ill-fatedEndurance expedition,Sir Ernest Shackletonrecorded refraction of 2°37′:

“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 sunsetas 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.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 refractionif 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).

Calculating refraction

Youngdistinguished several regions where different methods for calculating astronomical refraction were applicable. In the upper portion of the sky, with azenith distanceof 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 Standishand employing thetemperature gradientof thestandard atmosphereand 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. AsThe Nautical Almanacnotes, "the actual values of…the refraction at low altitudes may, in extreme atmospheric conditions, differ considerably from the mean values used in the tables."

Plot of refraction vs. altitude using Bennett's 1982 formula

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 thecotangentof theapparentaltitude of the astronomical body and in the higher order terms, the height of a fictional homogeneous atmosphere.The simplest version of this formula, which Smart held to be only accurate within 45° of the zenith, is:

$R=(n_{0}-1)\cot h_{\mathrm {a} }\,,$

whereRis the refraction inradians,n₀is theindex of refractionat the observer (which depends on the temperature, pressure, and humidity), andh_ais theapparentaltitude angle of the astronomical body.

An early simple approximation of this form, which directly incorporated the temperature and pressure at the observer, was developed byGeorge Comstock:

$R={\frac {21.5b}{273+t}}\cot h_{\mathrm {a} }\,,$

whereRis the refraction in seconds of arc,bis theatmospheric pressureinmillimeters of mercury,andtis the temperature inCelsius.Comstock considered that this formula gave results within one arcsecond ofBessel's values for refraction from 15° above the horizon to the zenith.

A further expansion in terms of the third power of the cotangent of the apparent altitude incorporatesH₀,theheight of the homogeneous atmosphere,in addition to the usual conditions at the observer:

$R=(n_{0}-1)(1-H_{0})\cot h_{\mathrm {a} }-(n_{0}-1)[H_{0}-{\frac {1}{2}}(n_{0}-1)]\cot ^{3}h_{\mathrm {a} }.$

A version of this formula is used in theInternational Astronomical Union'sStandards of Fundamental Astronomy;a comparison of the IAU's algorithm with more rigorous ray-tracing procedures indicated an agreement within 60milliarcsecondsat altitudes above 15°.

Bennettdeveloped another simple empirical formula for calculating refraction from the apparent altitude which gives the refractionRin arcminutes:

$R=\cot \left(h_{\mathrm {a} }+{\frac {7.31}{h_{\mathrm {a} }+4.4}}\right)\,.$

This formula is used in theU. S. Naval Observatory'sVector Astrometry Software,and is reported to be consistent with Garfinkel'smore complex algorithm within 0.07′ over the entire range from the zenith to the horizon.Sæmundssondeveloped an inverse formula for determining refraction fromtruealtitude; ifhis the true altitude in degrees, refractionRin arcminutes is given by

$R=1.02\cot \left(h+{\frac {10.3}{h+5.11}}\right)\,;$

the formula is consistent with Bennett's to within 0.1′. The formulas of Bennet and Sæmundsson assume anatmospheric pressureof 101.0 kPa and a temperature of 10 °C; for different pressurePand temperatureT,refraction calculated from these formulas is multiplied by

${\frac {P}{101}}\,{\frac {283}{273+T}}$

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.

Random refraction effects

The animated image of theMoon's surface shows theeffectsof atmosphericturbulenceon the view.

TurbulenceinEarth's atmospherescattersthe light from stars, making them appear brighter and fainter on a time-scale ofmilliseconds.The slowest components of these fluctuations are visible astwinkling(also calledscintillation).

Turbulence also causes small, sporadic motions of the star image, and produces rapid distortions in its structure. These effects are not visible to thenaked eye,but can be easily seen even in small telescopes. They perturbastronomical seeingconditions. Some telescopes employadaptive opticsto reduce this effect.

Terrestrial refraction

Further information:Levelling refraction

Terrestrial refraction,sometimes calledgeodetic refraction,deals with the apparent angular position and measured distance of terrestrial bodies. It is of special concern for the production of precisemapsandsurveys.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.

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,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.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 coefficientk,measuring the ratio of the radius of the Earth to the radius of the line of sight, is given by:

$k=503{\frac {P}{T^{2}}}\left(0.0343+{\frac {dT}{dh}}\right),$

where temperatureTis given inkelvins,pressurePinmillibars,and heighthin 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 EarthR_eff,given by

$R_{\text{eff}}={\frac {R}{1-k}},$

whereRis the radius of the Earth andkis 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 inarc secondsper meter can be computed using the relationship

${\frac {1}{\sigma }}=16.3{\frac {P}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right)\cos \beta$

where 1/σ is the curvature of the ray in arcsec per meter,Pis the pressure in millibars,Tis 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

$\Omega =8.15{\frac {LP}{T^{2}}}\left(0.0342+{\frac {dT}{dh}}\right),$

whereLis 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.

See also

• Polar circle#Effect of atmospheric refraction
• Sound wavesaffected by air density variations lead toatmospheric focusing.
• Air mass (astronomy)
• Atmospheric optics
• Electromagnetic radiation
• Fata Morgana (mirage)
• Ibn al-Haytham
• Looming and similar refraction phenomena
• Novaya Zemlya effect
• Radio propagation
• Line-of-sight propagation#Atmospheric refraction
• Ray tracing (physics)
• Shen Kuo
• Terrestrial atmospheric lens

Further reading

• Lehn, Waldemar H.; van der Werf, Siebren (2005). "Atmospheric refraction: a history".Applied Optics.44(27): 5624–5636.Bibcode:2005ApOpt..44.5624L.doi:10.1364/AO.44.005624.ISSN0003-6935.PMID16201423.
• Filippenko, A. V. (1982)."The importance of atmospheric differential refraction in spectrophotometry".Publ. Astron. Soc. Pac.94:715–721.Bibcode:1982PASP...94..715F.doi:10.1086/131052.
• Hotine, Martin(1969),"Atmospheric Refraction",Mathematical Geodesy,ESSA Monograph, vol. 2, Washington, DC: U.S. Department of Commerce, Environmental Science Services Administration,Bibcode:1969mage.book.....H
• Nener, Brett D.; Fowkes, Neville; Borredon, Laurent (2003), "Analytical modesl of optical refraction in the troposphere",J. Opt. Soc. Am.,20(5): 867–875,Bibcode:2003JOSAA..20..867N,doi:10.1364/JOSAA.20.000867,PMID12747434,S2CID21222910
• Thomas, Michael E.; Joseph, Richard I. (1996),"Astronomical Refraction"(PDF),Johns Hopkins APL Technical Digest,17:279–284
• Wang, Yu (20 March 1998), Bely, Pierre Y; Breckinridge, James B (eds.), "Very High-Resolution Space Telescope Using the Earth Atmosphere as the Objective Lens",Space Telescopes and Instruments V,3356,Jet Propulsion Laboratory:665,Bibcode:1998SPIE.3356..665W,doi:10.1117/12.324434,hdl:2014/19082,S2CID120030054
• 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),Columbia University:114503,arXiv:1908.00490,Bibcode:2019PASP..131k4503K,doi:10.1088/1538-3873/ab33c0,S2CID199064594

External links

• Young, Andrew T.,Annotated bibliography of mirages, green flashes, atmospheric refraction, etc.,retrieved3 May2016
• Young, Andrew T.,Astronomical Refraction,retrieved3 May2016