For circular orbits around Earth, we found
T2 = (4p2/g RE2) r3
with T in seconds and r in meters. The distance of a satellite from the center of Earth in meters is an inconveniently big number, even before we raise it to the 3rd power. We may however multiply the expression on the right by (RE3/RE3) = 1 and then rearrange the terms:
T2 = (4p2/g RE2) (RE3/RE3) r3 = (4p2RE/g) (r/RE)3
The ratio r' = (r/RE) is the orbital distance measured in units of the Earth's radius. That number is usually between 1 (at the Earth's surface, r = RE) and 60 (at the Moon's orbit, r ~ 60 RE).
Also, that ratio is always the same, whether r and RE are in meters, yards or nautical miles, as long as both r and RE are measured in the same units. The other term is calculated below, with multiplication denoted by blank spaces between parentheses; you may check it with your calculator.
(4p2RE/g) = (4) (9.87) (6 371 000)/9.81 = 25 638 838
Using computer notation for the square root
SQRT (25 638 838) = 5063.5
T= 5063.5 seconds SQRT(r')3 = 5063 sec r'SQRT=(r')
This is the practical form of Kepler's 3rd law for Earth satellites. Our imagined satellite skimming the surface of the Earth (r' = 1) has a period
T = 5063.5 sec = (5063.5/60) minutes = 84.4 minutes
The space shuttle must clear the atmosphere and goes a bit higher. Say it orbits at r' = 1. 05, with SQRT(r') = 1. 0247. Then
T = (5063.5) (1.05) (1.0247) = 5448 seconds = 90.8 minutes
International communication satellites are in the equatorial plane of the Earth and have orbits with a 24-hour period. As the Earth's rotates, they keep pace with it and always stay above the same spot. What is their distance?
Here T is known and we need to find r':
T = 24 hours = 86 400 sec = 5063.5 SQRT(r')3
If all numbers on the last line are equal, their squares are equal, too
(r')3 = (17.0632)2 = 291.156
Now you need a calculator able to derive cubic roots (or else, the 1/3 = 0.333. . . power). This gives
r' = 6.628 earth radii
as the distance of "synchronous" satellites. The satellites of the global positioning system (GPS), by which a small, handheld instument can tell one's location on the globe with amazing accuracy, are in 12-hour orbits. Can you calculate their distance?
Next Regular Stop: #22 Frames of Reference: The Basics
Author and curator: David P. Stern