Chapter -2(Lecture- 4) Kepler’s laws of planetary motion, Mathematical Formulation, Apogee & Perigee.

Discussion

What are Kepler’s three laws of planetary motion? Give the mathematical formulation of

Kepler’s third law of planetary motion. What do the terms perigee and apogee mean

when used to describe the orbit of a satellite orbiting the earth?

A satellite in an elliptical orbit around the earth has an apogee of 39,152 km and a perigee

of 500 km. What is the orbital period of this satellite? Give your answer in hours.

 Note: assume the average radius of the earth is 6,378.137 km and Kepler’s constant has the value 3.986004418x10^5 km^3/s^2.

Solution

Kepler’s three laws of planetary motion are

1. The orbit of any smaller body about a larger body is always an ellipse, with the

center of mass of the larger body as one of the two foci.

2. The orbit of the smaller body sweeps out equal areas in time (see Fig).

3. The square of the period of revolution of the smaller body about the larger body

equals a constant multiplied by the third power of the semimajor axis of the

orbital ellipse.

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• Kepler's laws of  Planetary Motion (Click Here)

The mathematical formulation of the third law is T^2 = (4pi^2a^3)/m, where T is the orbital period, a is the semimajor axis of the orbital ellipse, and m is Kepler’s constant.

The perigee of a satellite is the closest distance in the orbit to the earth; the apogee of a

satellite is the furthest distance in the orbit from the earth.

The semimajor axis of the ellipse = (39,152 + (2x6378.137) + 500)/2 = 26,204.137 km

The orbital period is T^2 = (4pi^2a^3)/m = (4pi^2(26,204.137)^3)/ 3.986004418x10^5 = 1,782,097,845.0.

Therefore, T = 42,214.90075 seconds = 11 hours 43 minutes 34.9 seconds