Question

1. compute the orbital radius and height above the earth for a GPS satellite based on its orbital period of 12 hours.

2. Estimate the ground footprint, in square miles, of a single satellite using simple trigonometry to determine what percentage of the earth’s surface can be “seen” from a single satellite.

3. Continue this “back of the envelope” analysis to estimate the total number of GPS satellites that are needed in order for a GPS receiver to have visibility of at least four satellite simultaneously for “almost” 100% of the time from “almost” anywhere on earth.
4. Compute the distance from a point on the earth to a GPS satellite that is located at the horizon. Express your answer in meters, and compare it to the distance in meters to a satellite that is directly overhead.
5. Compute the travel time, at the speed of light, for each of these two signals.

Answer 1

Answer :

Given details :

Time period T = 12 hours

According to Newton's form of Kepler's law,

Time period of satellite and Mass of object (around which satellite is rotating) are dependent as,

​​​​​​(equation 1)

where

T = Time period of satellite=12 hours =43,200 s (given)

R = Distance between centers of satellite and object.

G  

M = Mass of central object = 5.972 × 10²⁴ kg.

Putting in equation 1,

On solving we get

Distance between earth & satellite R = 26608244.8 m

R = 26608.2448 km or aproximately 26608.2 km

Radius of earth = 6378100 m = 6378.1 km

Height of satellite = 26608.2 - 6378.1 = 20230.1 km