Question

Suppose that humans have created a colony outside of our solar system on a planet called Cheops5. Cheops5 has a mass of 1.00×1025 kg and a day that lasts 27.0 h (which defines the rotational period of the planet). The colony is located on the equator of the planet. The colonists set up a communications satellite which orbits Cheops5. The satellite has a circular orbit that keeps it positioned directly above the colony. Calculate the radius ? of the satellite's orbit in kilometers. The gravitational constant ? is 6.67×10−11 m3 /(kg·s2 ).

Answer 1

M = mass of the planet = 1 x 10²⁵ kg

T = Time period = 27 h = 27 x 60 x 60 sec = 97200 sec

r = radius of orbit

Using kepler's third law

T² = 4² r³ /(GM)

(97200)² = 4 (3.14)² r³ /((6.67 x 10^-11) (1 x 10²⁵ ))

r = 5.43 x 10⁷ m