I'm having trouble obtaining this formula. Considering the curvature of the Earth (R is the Earth radius) and a non-vertical direction (zenith angle θ ), the relation between h and path length L in the atmosphere is: h = L cos ⁡ θ + 1 2 L 2 R sin 2 ⁡ θ h is the atmosphere's height.I understand the first term (which due to the inclination) but I can't find a way to get the second term (which is introduced by considering the "roundness" of the Earth)Any help would be greatly appreciated

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I&#039;m having trouble obtaining this formula. Considering the curvature of the Earth (R is the Earth radius) and a non-vertical direction (zenith angle       θ  ), the relation between h and path length L in the atmosphere is:  h  =  L  cos  ⁡      θ    +      1    2              L      2        R        sin    2    ⁡      θ  h is the atmosphere&#039;s height.I understand the first term (which due to the inclination) but I can&#039;t find a way to get the second term (which is introduced by considering the &quot;roundness&quot; of the Earth)Any help would be greatly appreciated

Miah Carlson · Accepted Answer

On the triangle formed by the point of entry of the light ray into the atmosphere (we assume the atmosphere is a sphere of finite radius), the point of observation and the Earth&#039;s center we can apply the law of cosines to find that  h  =            R      2        +          L      2        +    2    L    R    cos    ⁡    θ    −  RWe can Taylor expand this function in powers of   L      /    R assuming that   L  ≪  R (which means the Earth is very big compared to it&#039;s atmosphere so it&#039;s approximately flat). Keeping terms up to quadratic order we find  h  =  L  cos  ⁡  θ  +            L      2              2      R            sin    2    ⁡  θ  +      O        (                  L        3                    R        2              )

Considering the curvature of the Earth (R is the Earth radius) and a non-vertical direction (zenith angle theta), the relation between h and path length L in the atmosphere is: h=L cos theta+1/2 (L^2)/(R) sin^2 theta

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