# Kepler and Gravitation

### Problems for Kepler's Third Law

Problem : A geostationary satellite remains at the same position above the earth's surface at all times. What height above the surface must it be (assume the orbit is circular)? (Hint: What must the period be?) ($M_e = 5.98 \times 10^{24}$ kilograms and $r_e = 6.38 \times 10^6$ meters).

Problem : If an ICBM is launched at Washington DC from China, how long will it take to arrive? The ICMB only fires its rockets at take-off and re-entry and is allowed to coast in an elliptical orbit reaching a maximum height of 2500 kilometers above the earth. The mass of the earth is $M_e = 5.98 \times 10^{24}$ kg and it has radius $r_e = 6.38 \times 10^6$ m. The mass of the ICBM is $m = 1000$ kilograms.

Problem : A satellite has a polar orbit about the earth with a radius of $5.49 \times 10^6$ meters above the earth's surface. On one orbit it passes over London. Approximately where will it pass over exactly one orbit later? The radius of the earth is $r_e = 6.38 \times 10^{6}$ meters and its mass is $5.98 \times 10^{24}$ kilograms.

Problem : What is the period of a low-earth orbit (that is when $r \approx r_e$)? Extra hard part: If a hole was dug right through the center of the earth and a mass was dropped down it, what would occur? Compare this to the above situation. ($r_e = 6.38 \times 10^6$ meters and $M_e = 5.98 \times 10^{24}$ kilograms and assume that the mass of the earth is evenly distributed through its volume).

Problem : Consider a binary star system with two stars orbiting about their common center of mass. Each star completes its orbit in 10.2 days. One has mass $50 \times 10^{30}$ kilograms and the other mass $20 \times 10^{30}$ kilograms. Assuming circular orbit, what is the distance between the stars (the stars will be on opposite sides of their center of mass, so the distance between them will be given by the sum of the orbital radii)?

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