Atmosphere as a function of Zenith angle?

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Homework Help Overview

The discussion revolves around determining the distance from a location to space as a function of the zenith angle, with a specific height of approximately 100 km at 0 degrees zenith angle. The problem involves geometric considerations related to the Earth's shape and atmosphere.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss using the law of cosines and consider the implications of approximating the Earth as flat versus spherical. Questions arise regarding the necessity of a Taylor expansion and the validity of different geometric approaches.

Discussion Status

Several approaches have been suggested, including a flat Earth approximation and a spherical Earth model. Participants are exploring the implications of these methods, particularly concerning the behavior of the solution as angles approach horizontal. No consensus has been reached, but various interpretations and methods are being actively discussed.

Contextual Notes

Participants note that the height of 100 km may be considered small relative to the Earth's radius, which influences the choice of approximation. There is also mention of potential complications in the calculations for angles approaching horizontal.

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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


?



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

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