page_title: Air mass (astronomy)
section_title: Homogeneous spherical atmosphere with elevated observer
hierarchical_section_title: Air mass (astronomy) / Calculation / Atmospheric models / Homogeneous spherical atmosphere with elevated observer
caption_attribution_description: English: Air mass for elevated observer in homogeneous spherical atmosphere
context_page_description: In astronomy, air mass or airmass is the "amount of air that one is looking through" when seeing a star or other celestial source from below Earth's atmosphere. It is formulated as the integral of air density along the light ray.
  As it penetrates the atmosphere, light is attenuated by scattering and absorption; the thicker atmosphere through which it passes, the greater the attenuation. Consequently, celestial bodies when nearer the horizon appear less bright than when nearer the zenith. This attenuation, known as atmospheric extinction, is described quantitatively by the Beer–Lambert law.
  "Air mass" normally indicates relative air mass, the ratio of absolute air masses at oblique incidence relative to that at zenith. So, by definition, the relative air mass at the zenith is 1. Air mass increases as the angle between the source and the zenith increases, reaching a value of approximately 38 at the horizon. Air mass can be less than one at an elevation greater than sea level; however, most closed-form expressions for air mass do not include the effects of the observer's elevation, so adjustment must usually be accomplished by other means.
context_section_description: In the figure at right, an observer at O is at an elevation  above sea level in a uniform radially symmetrical atmosphere of height . The path length of a light ray at zenith angle  is ;  is the radius of the Earth. Applying the law of cosines to triangle OAC,
  expanding the left- and right-hand sides, eliminating the common terms, and rearranging gives
  Solving the quadratic for the path length s, factoring, and rearranging,
  The negative sign of the radical gives a negative result, which is not physically meaningful. Using the positive sign, dividing by , and cancelling common terms and rearranging gives the relative air mass:
  With the substitutions  and , this can be given as
  When the observer’s elevation is zero, the air mass equation simplifies to
  In the limit of grazing incidence, the absolute airmass equals the distance to the horizon.
  Furthermore, if the observer is elevated, the horizon zenith angle can be greater than 90°.
caption: Air mass for elevated observer in homogeneous spherical atmosphere
exif: {}