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 Dactyl. This was the first example of an asteroid-moon system.
The moon is only about 1.5 km wide and Ida is about 55 km long. The orbit of Dactyl is not well known, however, it can be used to make an estimate of Ida’s mass.
(a) If we assume that Dactyl’s orbit is circular, with a radius of 87 km, and the orbit has a period of 25 h, calculate the mass of asteroid Ida (in kg).
(b) If the volume of Ida, as determined from the Galileo images, is 1.4 × 10^4 km^3, calculate the density (mass per unit volume) of Ida (in kg/m^3)

Explanation / Answer

r = 87 km
t = 25 hours
v = 87 *2*pi /25 hour
v = 21.85 km/hr
v = 6.07 m/sec
a = v^2/r
a = 0.00042 m/s^2

Now to the Universal law of gravitation

F= G Mm/r^2
a = GM /r^2 ... for a satellite orbiting a main body.
M = a r^2/G
M = 0.00042m/s^2 (87,000m)^2/ 6.67300 × 10^-11 m3/kg s^2
M = 4.8 x 10^16 kg

b.
Density = Mass/volume
= 4.8x10^16kg/14000km^3
= 3.43 x10^12 kg/km^3
= 3.43 x 10^3 kg/m^3

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