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

Answer:

The mass of the planet  is $1.9407\times10^{27}\ kg$

Explanation:

Given that,

Time period = 42 hours = 151200 sec

Orbital radius = 0.002819 AU = 421716397.5 m

Mass of moon $m=8.932\times10^{22}\ kg$

We need to calculate the mass of the planet

Using Kepler’s third law

$T^2\propto a^3$

$T^2=\dfrac{4\pi^2}{G(M+m)}\times a^3$

Where, a = orbital radius

T = time period

G = gravitational constant

M = mass of moon

m = mass of planet

Put the value into the formula

$(151200)^2=\dfrac{4\pi^2}{6.673\times10^{-11}(8.932\times10^{22}+m)}\times(421716397.5)^3$

$(8.932\times10^{22}+m)=\dfrac{4\pi^2}{6.673\times10^{-11}}\times\dfrac{(421716397.5)^3}{(151200)^2}$

$(8.932\times10^{22}+m)=1.94087\times10^{27}$

$m=1.94087\times10^{27}-8.932\times10^{22}$

$m=1.9407\times10^{27}\ kg$

Hence, The mass of the planet  is $1.9407\times10^{27}\ kg$