Question

A satellite orbits an asteroid whose average density is 3x the density of water (1 g/cm3). The orbital period of the satellite is 20 hours and the semi-major axis is 1,000 km. What is the radius of the asteroid?

Answer 1

let M is the mass and R is the radius of the astoroid.

orbital velocity of the satellite,

vo = 2*pi*r/T

= 2*pi*1000*10^3/(20*60*60)

= 87.27 m/s

orbital speed of a satellite, vo = sqrt(G*M/r)

vo^2 = G*M/r

M = vo^2*r/G

= 87.27^2*1000*10^3/(6.67*10^-11)

= 1.1418*10^20 kg

density of the astoroid, rho = 3*1 g/cm^3

= 3000 kg/m^3

now use,rho = M/V

V = M/rho

(4/3*pi*R^3) = M/rho

R^3 = 3*M/(4*pi*rho)

R = (3*M/(4*pi*rho))^(1/3)

= (3*1.1418*10^20/(4*pi*3000))^(1/3)

= 208670 m (or) 208.6 km <<<<<<<<------ Answer