Chapter 14, Problem 47A

In 1610, Galileo discovered four moons of Jupiter. (Today we know that there are more than 60!) 10, the innermost of the moons observed by Galileo, is $1.5\times {10}^8\text{m}$ from Jupiter’s center and has a period of $1.5\times {10}^5\text{s}$. Calculate Jupiter’s mass.

To determine

Jupiter’s mass.

Answer to Problem 47A

The mass of Jupiter is $M=1.95\times {10}^{27}\text{kg}$ .

Explanation of Solution

Given:

Distance of Jupiter’s moon from its center, $4.2\times {10}^8\text{m}$

Period of Jupiter’s moon, $1.5\times {10}^5\text{sec}$

Formula Used:

The orbital velocity of satellite is given as,

  $v=\sqrt{\frac{GM}{R}}$

Where, $G$ is universal gravitation constant, $M$ is mass of planet and $R$ is distance of satellite from the planet.

Calculation:

The gravitational constant, $G=6.674\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$

The velocity can also be given as,

  $v=\frac{d}{t}=\frac{2\pi R}{T}$

Here, $R$ is distance satellite from the planet and $T$ is orbital period

So,

  $\begin{array}{l}v=\sqrt{\frac{GM}{R}}=\frac{2\pi R}{T}\\ \Rightarrow M=\frac{4{\pi }^2{R}^3}{G{T}^2}=\frac{4\times 3.14\times 3.14\times {\left(4.2\times {10}^8\right)}^3}{\left(6.674\times {10}^{-11}\right)\times {\left(1.5\times {10}^5\right)}^2}\text{kg}\\ \Rightarrow M=1.95\times {10}^{27}\text{kg}\end{array}$

Conclusion:

Therefore, the mass of Jupiter is $M=1.95\times {10}^{27}\text{kg}$ .