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Satellite Orbit Analysis (Posted on 2025-01-23)

Assume that Earth is a sphere with radius R. A satellite has an elliptical orbit with the center of Earth at one focus. The lowest point of the orbit is 5R above the surface of Earth, when the satellite is directly above the North Pole. The highest point of the orbit is 11R above the surface of Earth, when the satellite is directly above the South Pole. What is the height of the satellite above the surface of Earth, when the satellite is directly above the equator?

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soln

| Comment 1 of 2

Ans = 7R

Calling R=1 and drawing the system in the plane of the ellipse,

with 0 at the center, Earth center at +3, and giving the major axis

length 2a =18 puts apogee at 12 and perigee at 6 from Earth's center.

First we solve for the semi-minor axis b using x = +/-c = +/-3 as the

foci:

x^2/a^2 +y^2/b^2 = 1 [eq.1]

x^2/9^2 + y^2/b^2 =1

for all ellipses:

c = sqrt (a^2 + b^2)

3 = sqrt (9^2 + b^2)

gives b = 6 sqrt 2

The satellite is over the equator at the coordinate (x,y)=(3,s)

plugging into eq. 1:

3^2/81 + s^2/72 = 1

s=8

So the satellite is 8 units above the center,

or 7 R above the equator.

https://www.desmos.com/calculator/z7pptutmen

Edited on January 23, 2025, 11:53 am

Posted by Steven Lord on 2025-01-23 09:42:06

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