Atmosphere as a function of Zenith angle?

[Ai52487963]

Homework Statement

Find x, the distance from your location to space, as a function of the zenith angle. The height at 0 degrees, H, is ~100km.

A picture helps more: http://img23.imageshack.us/img23/3630/picture1orr.jpg

Homework Equations

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The Attempt at a Solution

I've tried using the law of cosines to express the distance in terms of the Earth's radius and other known constants, but it seems like I'm over complicating the situation.

 

[mgb_phys]

You can probably assume a plane - 100km is small compared to the radius of the earth.

 

[Ai52487963]

You can probably assume a plane - 100km is small compared to the radius of the earth.

One http://www.astro.ufl.edu/~oliver/ast3722/lectures/EffectOfAtmosphere/EffectAtmos.pdf" I founddoes a taylor expansion, but why would I want to do that?

 

[Redbelly98]

There are 2 approaches to this problem that I can think of.

1. Using the "flat Earth" approximation, as mgb_phys suggested. In this case it's a straightforward trig problem.

2. Consider a spherical Earth. In this case you need to find the intersection of a line and a circle (the upper atmosphere's edge). More tedious than #1, but I think it's possible.

Note, for angles approaching horizontal, the flat-Earth approximation answer approaches infinity rather than a finite value.