A large satellite of Pluto known as Charon has an orbital ra

A large satellite of Pluto, known as Charon, has an orbital radius of 19.600 km and an orbital period of 6.39 days. What are the orbital periods of two small satellites orbiting Pluto, one at a distance of48000 km and the other at 64.000 km from the center of Pluto?

Solution

Centripetal force equal to the gravitational force

F_c= F_G

mv²/r= GMm/r² ,where M is Pluto\'s mass and m is the satellite mass.

Period, T  = (2 pi / w) = 2 pi sqrt (r^{3 /} GM)

T³ ~ r²--------------------Kepler\'s third law

T ~r^2/3

Hence T₂= T₁*[r₂/r₁]^2/3, where T₁ and r₁ are the period and radius of Charon

Orbital period for small satellite of pluto with radius 48000km, T₂= 6.39 days*[ 48000 km/19600]^2/3= 11.61 days

Orbital period for small satellite of pluto with radius 64000km, T₂\'= 6.39 days*[ 64000 km/19600]^2/3= 14.06 days