page_title: Atmospheric refraction
section_title: Calculating refraction
hierarchical_section_title: Atmospheric refraction / Astronomical refraction / Calculating refraction
caption_attribution_description: English: Plot of atmospheric refraction vs. apparent altitude, using G.G. Bennett’s 1982 formula. Author: Jeff Conrad
context_page_description: 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 height. This refraction is due to the velocity of light through air, decreasing with increased density. Atmospheric refraction near the ground produces mirages. Such refraction can also raise or lower, or stretch or shorten, the images of distant objects without involving mirages. Turbulent air can make distant objects appear to twinkle or shimmer. The term also applies to the refraction of sound. Atmospheric refraction is considered in measuring the position of both celestial and terrestrial objects.
  Astronomical or celestial refraction causes astronomical objects to appear higher above the horizon than they actually are. Terrestrial refraction usually causes terrestrial objects to appear higher than they actually are, although in the afternoon when the air near the ground is heated, the rays can curve upward making objects appear lower than they actually are.
  Refraction not only affects visible light rays, but all electromagnetic radiation, although in varying degrees.
context_section_description: Young distinguished several regions where different methods for calculating astronomical refraction were applicable. In the upper portion of the sky, with a zenith distance 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 and employing the temperature gradient of the standard atmosphere 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 The Nautical Almanac 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."
  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 cotangent of the apparent altitude 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:
  where R is the refraction in radians, n₀ is the index of refraction at the observer (which depends on the temperature and pressure), and hₐ is the apparent altitude of the astronomical body.
  An early simple approximation of this form, which directly incorporated the temperature and pressure at the observer, was developed by George Comstock:
  where R is the refraction in seconds of arc, b is the barometric pressure in millimeters of mercury, and t is the Celsius temperature. Comstock considered that this formula gave results within one arcsecond of Bessel'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 incorporates H₀, the height of the homogeneous atmosphere, in addition to the usual conditions at the observer:
  A version of this formula is used in the International Astronomical Union's Standards of Fundamental Astronomy; a comparison of the IAU's algorithm with more rigorous ray-tracing procedures indicated an agreement within 60 milliarcseconds at altitudes above 15°.
  Bennett developed another simple empirical formula for calculating refraction from the apparent altitude which gives the refraction R in arcminutes:
  This formula is used in the U. S. Naval Observatory's Vector Astrometry Software, and is reported to be consistent with Garfinkel's more complex algorithm within 0.07′ over the entire range from the zenith to the horizon. Sæmundsson 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
  the formula is consistent with Bennett's to within 0.1′. The formulas of Bennet and Sæmundsson 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
  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.
caption: Plot of refraction vs. altitude using Bennett's 1982 formula
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