Question

A moon orbits a planet every 42 hours with a mean orbital radius of .002819 AU. The mass of the moon is 8.932 x 1022 kg. Using Newton’s modification to Kepler’s 3rd law, calculate the Mass of the Planet in kg.

Answer 1

The mass of the planet is
`1.9407*10^(27)\ kg`

Step-by-step explanation:

Given that,

Time period = 42 hours = 151200 sec

Orbital radius = 0.002819 AU = 421716397.5 m

Mass of moon
`m=8.932*10^(22)\ kg`

We need to calculate the mass of the planet

Using Kepler’s third law

`T^2\propto a^3`

`T^2=(4\pi^2)/(G(M+m))* a^3`

Where, a = orbital radius

T = time period

G = gravitational constant

M = mass of moon

m = mass of planet

Put the value into the formula

`(151200)^2=(4\pi^2)/(6.673*10^(-11)(8.932*10^(22)+m))*(421716397.5)^3`

`(8.932*10^(22)+m)=(4\pi^2)/(6.673*10^(-11))*((421716397.5)^3)/((151200)^2)`

`(8.932*10^(22)+m)=1.94087*10^(27)`

`m=1.94087*10^(27)-8.932*10^(22)`

`m=1.9407*10^(27)\ kg`

Hence, The mass of the planet is
`1.9407*10^(27)\ kg`