Look For An Equation?

Question

A surveillance satellite circles Earth at a height of h miles above the surface. Suppose that d is the distance, IN MILES, on the surface of the Earth that can be seen from the satellite.

(a) Find an equation that relates the central angle theta to the height h.

(b) Find an equation that relates the observable distance d and theta.

(c) Find an equation that relates d and h.


NOTE: The radius of Earth is 3960 miles.

Answer 1

I'll go for d being the straight-line distance from satellite to horizon, and the central angle theta being that subtended by d as viewed from the center of the earth. Then you have the right triangle described above with hypotenuse = r+h, and
1) theta = arcsin(d/(r+h)), d defined in (3)
2) d = (r+h)*sin(theta)
3) d = sqrt((r+h)^2 - r^2)

Answer 2

Draw cross-section. You would get Earth as a circle of radius R. Satellite is point at distance R+h. Draw tangential line to Earth. At tangential point radius makes straight angle with your observation line. You got straight triangle with hypothenuse R+h and one side R. The this side is visible from satellite is arcsin(R/(R+h)). Total central angle is twice this value because you can see the same angle the other side too. So answer for a is theta=2arcsin(R/(R+h)).

(b). If distance d is second side of the triangle then d=Rcos(theta/2). If distance d is measured on ground from the point where satellite is then d=R(Pi-theta)/2.

(c) From Pithagorean theorem second side of the triangle d = Sqrt((r+h)^2-R^2) = Sqrt(2Rh+h^2)

Answer 3

You do not clearly define the "central angle". I will proceed on the basis that theta is half the angle subtended at the satellite by the distance d. For brevity I write r = radius of earth (assumed spherical). Angles are in radians.

Then :

1. theta = arcsin r/(r+h)

2. d=2 x r x theta

3. eliminating theta from 1 and 2 gives:

d = 2 x r x arcsin r/(r+h)

Bramble