Question

In 1993 the spacecraft Galileo sent home an image (the figure) of asteroid 243 Ida and a tiny orbiting 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, measured from the Galileo images, is 14100 km3. What is the density (mass per unit volume) of the asteroid?

Answer 1

a)

T = time period = 27 h = 27 (3600) sec = 97200 sec

r = radius of the orbit = 100 km = 10⁵ m

M = mass of asteroid = ?

using the equation

T² = 4 $\pi$ ² r³/(GM)

(97200)² = 4 (3.14)²  (10⁵)³/((6.67 x 10^-11) (M))

M = 6.3 x 10¹⁶ kg

b)

V = Volume = 14100 km³ = 14100 (1000)³ m³ = 1.41 x 10¹³ m³

density is given as

$\rho$ = M/V

$\rho$ = (6.3 x 10¹⁶)/(1.41 x 10¹³)

$\rho$ = 4468.1 kg/m³