Physics: Planets and Satellites: Keepler's Laws?

Question

In 1993 the spacecraft Galileo sent home an image of asteroid 243 Ida and an orbiting tiny moon (now known as Dactyl), the first confirmed example of an asteroid-moon system. In the image, the moon, which is 1.5 km wide, is 100 km from the center of the asteroid, which is 55 km long. The shape of the moon's orbit is not well known; assume it is circular with a period of 27 h.

(a) What is the mass of the asteroid?
(b) The volume of the asteroid is 14,100 km^3. What is the density (mass per unit volume) of the asteroid?

The answers are:
(a) 6 x 10 ^ 16 kg
(b) 4 x 10 ^ 3 kg/m^3
but I don't know how to get these answers, please help me!

Answer 1

OK, first I will convert all quantities to meters, seconds and kilograms to avoid problems with misplaced decimals and using wrong units. The orbit is a circle of radius 100 km or 100,000 meters and it takes 27 hours or 27*3600 = 97,200 seconds to complete. The orbit circumference is 2*100,000*pi = 628,000 meters. Dactyl travels that far in 97,200 seconds so its speed is 628,000/97,200 = 6.46 meters per second. From the source, orbital speed is v=SQRT(u(2/r-1/a)), but since r=a in a circular orbit it reduces to v=SQRT(u/a) or v^2=u/a, where a is 100,000 meters. So 6.46^2=u/100,000, and u=4,173,160. Now u is GM, where M is the mass and G is the universal gravitational constant, so 0.000417=GM, or M=4,173,160/G. G is 6.6742x10^-11 in units of meters and seconds and kilograms. That makes the mass 4,173,160/6.6742x10^-11 or 6.25x10^16 kg. Round it to 6x10^16. The volume is 14,100 km^3. A cubic km is 1x10^9 cubic meters so the volume is 1x10^9*14,100, or 1.41x10^13 m^3. Density is mass/volume, or 6x10^16/1.41x10^13=4.2x10^3 kg/m^3, round it to 4x10^3.
Tadaa!