1. A satellite is in a circular orbit around the earth at a height of 1000 km. What is the period of the satellite's orbit and its speed?
R= 1000 km + 6378 km = 7378 km = 7.378 x 106 m
M = 5.98 x 1024 kg
P2 = 4 pi2 R3 / (G M)
so, P = 6.30 x 103 s = 1.75 hr
v = 2 pi R / P = 7.35 km s-1
2. Callisto has a period of 16.69 days and a mean orbital radius of 1.883 x 109 m. Estimate the mass of Jupiter.
P = 16.69 days = 1.442 x 106 s
R = 1.883 x 109 m
M = 4 pi2 R3 / (G P2)
M = 1.9 x 1027 kg
3. When the Apollo 11 spacecraft orbited the Moon, its mass was 9.98 x 103 kg, its period was 119 min, and its mean distance from the Moon's center was 1.849 x 106 m. Assuming a circular orbit, find the mass of the moon and the orbital speed of the space craft.
M= 9.979 x 103 kg
P= 119 min = 7140 s
R= 1.849 x 106 m
M = 4 pi2 R3 / (G P2)
M = 7.33 x 1022 kg
v = 2 pi R / P = 1.6 km s-1
4. The Hubble Space Telescope (HST) is a 2.4m reflecting telescope which was deployed in low-Earth orbit (600 km) by the space shuttle Discovery in 1990. Telescope time on this unique facility is highly competitive (typically, the telescope is over subscribed by at least 6 to 1). For ease of scheduling, telescope time is allocated in units of ``orbits.'' Assuming that the telescope has an ``overhead'' of 60%, calculate how much on-source observing time available in a 15 orbit allocation. [Overhead includes time for target acquisition, read-out of the CCDs, and time when the Sun or Moon are ``up''].
R= 6378 km + 600 km = 6978 km
M = 5.98 x 1024 kg
P2 = 4 pi2 R3 / (G M)
P = 5799 s = 1.61 hr
15 orbits = 15 * 1.61 = 24.16 hr
at 40% efficiency (100 - 60%), T = 24.16 x 0.4 = 9.7 hr