Question

In March 2006, two small satellites were discovered orbiting Pluto, one at a distance of 48,000 km and the other at 64,000 km. Pluto already was known to have a large satellite Charon, orbiting at 19,600 km with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital periods of the two small satellites without using the mass of Pluto.

Answer 1

Answer

given,

distance of first satellite = 48,000 Km

distance of second satellite = 64,000 Km

orbital period = 6.39 day

Using equation of time period

  $T = \dfrac{2\pi r^{3/2}}{\sqrt{Gm_{pluto}}}$

now, from the above equation we can say that only variable is Time period and r is the radii of orbit.

from the first satellite

   $\dfrac{T_{charon}}{r^{3/2}_{charon}}=\dfrac{T_{sat1}}{r^{3/2}_{sat1}}$

   $T_{sat1}=\dfrac{T_{charon}\ r^{3/2}}{r^{3/2}_{charon}}$

   $T_{sat1}=\dfrac{6.39\times (48000)^{3/2}}{19600^{3/2}}$

   $T_{sat1}=24.5\ days$

for second satellite

   $T_{sat2}=\dfrac{T_{charon}\ r^{3/2}_{sat2}}{r^{3/2}_{charon}}$

   $T_{sat1}=\dfrac{6.39\times (64000)^{3/2}}{19600^{3/2}}$

   $T_{sat1}=37.7\ days$