- #1
tomwilliam2
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Homework Statement
Given the diagram below, showing the path of a geocentric satellite S flying over a ground station G, find an expression for the geocentric semi-angle ##\phi## in terms of ##\epsilon##, the radius of the Earth ##R_E##, and the height of the orbit ##h##.
Homework Equations
This should just involve basic trigonometry.
The Attempt at a Solution
I've extended the line from the centre of the Earth, through G, and out through the orbit path of the satellite. Now, I've drawn a vertical down from the satellite S parallel with the existing straight line until it intersects that extended line. I've called ##r## the height above the Earth of this point where my additional lines intersect.
$$\cos \phi = \frac{R_E + r}{R_E+h}$$
$$\cos \phi = \frac{R_E}{R_E+h}+\frac{r}{R_E+h}$$
I'm not sure if I've made any progress here, as I need to bring the angle ##\epsilon## into it. I imagine I could use some trig identities as well. I actually know the answer, but can't quite get there. The answer is:
$$\phi = -\epsilon + \cos^{-1} \left(\frac{R_E}{R_E+h}\cos \epsilon \right)$$
Any help greatly appreciated.