In 1993 the spacecraft Galileo sent home an image of asteroid 243 Ida and a tiny

Question

In 1993 the spacecraft Galileo sent home an image of
asteroid 243 Ida and a tiny orbiting moon (now
known as Dactly), 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, measured
from the Galileo images, is 14,100 km
3
. What is the
density (mass per unit volume) of the asteroid?

Explanation / Answer

The equation for Orbital Period T in seconds is: T = 2p v(a3 / µ) where a = length of the orbit's semi-major axis and µ = Mass of the Central Body (the asteroid) times G, the gravitation constant (from: http://en.wikipedia.org/wiki/Orbital_period) Since we're solving for M, we re-write the above equation and then plug-and-chug: (Note that just the a's are cubed, not the entire fraction.. haven't quite gotten the hang of the equation editor yet!) T = 20 Hours, which equals 72,000 Seconds a = 100 km. This is because for a circular orbit, the semi-major axis is equivalent to the radius. After entering our values and solving, we get: M = 1.17158588 × 1015 kilograms For part 2, Density is simply Mass divided by Volume, so Divide 1.17158588 × 1015 kilograms by 14,100 km3 and you will have your answer

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