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.Part AAssuming that the satellites do not affect each other, find the orbital periods T1 and T2 of the two small satellites without using the mass of Pluto.Enter your answers numerically separated by a comma.

Answer 1

24.48 days

37.7 days

Step-by-step explanation:

r = Radius

s denotes satellite

C denotes Charon

Time period is given by

`T=(2\pi r^(1.5))/(√(2GM))`

So,

`T\propto r^(1.5)`

`(T_C)/(T_(s1))=(r_c^(1.5))/(r_(s1)^(1.5))\\\Rightarrow T_(s1)=(T_Cr_(s1)^(1.5))/(r_C^(1.5))\\\Rightarrow T_(s1)=\frac{6.39* 86400* {48000000}^(1.5)}{19600000^(1.5)}\\\Rightarrow T_(s1)=2115886.41242\ s\\\Rightarrow T_(s1)=24.48\ days`

The time period of the first satellite is 24.48 days

`T_(s2)=(T_Cr_(s2)^(1.5))/(r_C^(1.5))\\\Rightarrow T_(s2)=\frac{6.39* 86400* {64000000}^(1.5)}{19600000^(1.5)}\\\Rightarrow T_(s2)=3257620.23942\ s\\\Rightarrow T_(s2)=37.7\ days`

The time period of the second satellite is 37.7 days