# Atmospheric refraction

Diagram showing displacement of the Sun's image at sunrise and sunset

Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of altitude. Atmospheric refraction near the ground produces mirages and can make distant objects appear to shimmer or ripple, elevated or lowered, stretched or shortened with no mirage involved. The term also applies to the refraction of sound.

Atmospheric refraction causes astronomical objects to appear higher in the sky than they are in reality. It affects not only lightrays but all electromagnetic radiation, although in varying degrees (see dispersion in optics). For example in visible light, blue is more affected than red. This may cause astronomical objects to be spread out into a spectrum in high-resolution images.

Whenever possible, astronomers will schedule their observations around the time of culmination of an object when it is highest in the sky. Likewise sailors will never shoot a star which is not at least 20° or more above the horizon. If observations close to the horizon cannot be avoided, it is possible to equip a telescope with control systems to compensate for the shift caused by the refraction. If the dispersion is a problem too, (in case of broadband high-resolution observations) atmospheric refraction correctors can be employed as well (made from pairs of rotating glass prisms). But as the amount of atmospheric refraction is a function of temperature and pressure as well as humidity (the amount of water vapour is especially important at mid-infrared wavelengths) the amount of effort needed for a successful compensation can be prohibitive.

Atmospheric refraction becomes more severe when the atmospheric refraction is not homogenous, when there is turbulence in the air for example. This is the cause of twinkling of the stars and deformation of the shape of the sun at sunset and sunrise.

## Values

Atmospheric refraction is zero in the zenith, less than 1′ (one arcminute) at 45° apparent altitude, 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 horizon (Allen 1976, 125); all values are for 10 °C and 101.3 kPa in the visible part of the spectrum.

On the horizon, refraction is slightly greater than the apparent diameter of the Sun. Therefore when it appears that the full disc of the sun is just above the horizon, if it weren't for the atmosphere, no part of the sun's disc would be visible. By convention, sunrise and sunset refer to times at which the Sun’s upper limb appears on or disappears from the horizon; the standard value for the Sun’s true altitude is −50′: −34′ for the refraction and −16′ for the Sun’s semidiameter (the altitude of a celestial body is normally given for the centre of the body’s disc). In the case of the Moon, additional corrections are needed for the Moon’s horizontal parallax and its apparent semidiameter; both vary with the Earth–Moon distance.

Day-to-day variations in the weather will affect the exact times of sunrise and sunset (Schaefer and Liller 1990) as well as moonrise and moonset, and for that reason it generally is not meaningful to give rise and set times to greater precision than the nearest minute (Meeus 1991, 103). 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 (for example Meeus 2002, 315) if it is understood that actual changes may differ because of unpredictable variations in refraction.

Because atmospheric refraction is 34′ on the horizon itself, 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

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

Rigorous calculation of refraction requires numerical integration, using a method such as that of Auer and Standish (2000). Bennett (1982) developed a simple empirical formula for calculating refraction from the apparent altitude, using the algorithm of Garfinkel (1967) as the reference; if ha is the apparent altitude in degrees, refraction R in arcminutes is given by

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

the formula is accurate to within 0.07′ for the altitude range 0°–90° (Meeus 1991, 102). Sæmundsson (1986) developed a formula for determining refraction from true altitude; if h is the true altitude in degrees, refraction R in 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′. Both formulas assume an atmospheric pressure 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

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

(Meeus 1991, 103). 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

An animated image of the Moon's surface showing the effects of Earth's atmosphere on the view

Turbulence in the atmosphere magnifies and de-magnifies star images, making them appear brighter and fainter on a time-scale of milliseconds. The slowest components of these fluctuations are visible as twinkling (also called "scintillation").

Turbulence also causes small random motions of the star image, and produces rapid changes in its structure. These effects are not visible to the naked eye, but are easily seen even in small telescopes. They are called "seeing" by astronomers.

## References

• Allen, C. W., C. W. (1976), Astrophysical Quantities (3rd ed. ed.), London: Athlone, ISBN 0-485-11150-0
• Auer, Lawrence H.; Standish, E. Myles (2000), "Astronomical Refraction: Computation for All Zenith Angles", Astronomical Journal 119 (5): 2472–2474., Bibcode:2000AJ....119.2472A
• Bennett, G.G. (1982), "The Calculation of Astronomical Refraction in Marine Navigation", Journal of Navigation 35: 255–259, Bibcode:1982JNav...35..255B, doi:10.1017/S0373463300022037
• Filippenko, A. V. (1982). "The importance of atmospheric differential refraction in spectrophotometry". Publ. Astron. Soc. Pac: 715–721. Bibcode:1982PASP...94..715F.
• Garfinkel, B. (1967), "Astronomical Refraction in a Polytropic Atmosphere", Astronomical Journal 72: 235–254, Bibcode:1967AJ.....72..235G
• Meeus, Jean. 1991. Astronomical Algorithms. Richmond, Virginia: Willmann-Bell, Inc. ISBN 0-943396-35-2
• Meeus, Jean. 2002. More Mathematical Astronomy Morsels. Richmond, Virginia: Willmann-Bell, Inc. ISBN 0-943396-74-3
• 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
• Sæmundsson, Þorsteinn (1986), "Astronomical Refraction", Sky and Telescope 72: 70
• Schaefer, Bradley E.; Liller, William (1990), "Refraction near the horizon", Publications of the Astronomical Society of the Pacific 102: 796–805, Bibcode:1990PASP..102..796S
• Thomas, Michael E.; Joseph, Richard I. (1996), "Astronomical Refraction", Johns Hopkins Apl. Technical Digest 17: 279–284