Question

In 1610, Galileo used his telescope to discover four prominent moons around Jupiter. Their mean orbital radiiaand periodsTare as follows:

(a) Plot log a (y-axis) against log T (x-axis) and show that you get a straight line.

(b) Measure the slope of the line and compare it with the value that you expect from Kepler’s third law.

(c) Find the mass of Jupiter from the intercept of this line with the y axis.

Short answer

1. The graph is drawn below.
2. The slope of the graph of log a against log T is $\frac{2}{3}.$
3. The mass of Jupiter is $1.9\times {10}^{27}\text{kg.}$

Step-by-step solution

Step 1: Given

The mean orbital radii ‘a’ and periods ‘T’ for four moons of Jupiter.

Determining the concept

Plot the straight line graph of log a vs log T from the given data. Then, find its slope. Using Kepler’s third law and the intercept of the line, find the mass of Jupiter. According to Kepler’s third law, the squares of the orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits.

The formula is as follows:

According to Kepler’s third law,${\mathrm{T}}^2=\frac{4{\mathrm{\pi }}^2}{\mathrm{GM}}{\mathrm{a}}^3$

where T is time, G is gravitational constant, M is mass and a is the radius.

(a) Determining the straight line graph of log a against log t

The graph of log a against log T is given below,

(b) Determining the slope of the graph of log a against log t and comparing with it the expected value from Kepler's third law 

The slope of the graph of log a against log T is,

$\frac{\text{AB}}{\text{BC}}=\frac{1.8}{2.7}$

$\frac{\text{AB}}{\text{BC}}=\frac{2}{3}$

Kepler’s third law gives,

${\mathrm{T}}^2=\frac{4{\mathrm{\pi }}^2}{\mathrm{GM}}{\mathrm{a}}^3$

${\mathrm{a}}^3=\frac{\mathrm{GM}}{4{\mathrm{\pi }}^2}{\mathrm{T}}^2$

Taking the log of both sides,

$3\mathrm{loga}=\log \left(\frac{\mathrm{GM}}{4{\mathrm{\pi }}^2}\right)+2\mathrm{logT}$

$\mathrm{loga}=\frac{1}{3}\log \left(\frac{\mathrm{GM}}{4{\mathrm{\pi }}^2}\right)+\frac{2}{3}\mathrm{logT}$

This equation suggests that the graph for log a against log T is a straight line and 2/3 is the slope of the graph.

Therefore, the value of the slope found in the graph is exactly the same as the value found in Kepler’s third law.

Hence, the slope of the graph of log a against log T is$\frac{2}{3}.$

(c) Determining the mass of Jupiter from the intercept of the graph with the y-axis

The intercept of the graph is,

$\frac{1}{3}\log \left(\frac{\mathrm{GM}}{4{\mathrm{\pi }}^2}\right)=5.17$

So,

$\log \left(\frac{\mathrm{GM}}{4{\mathrm{\pi }}^2}\right)=3(5.17)$

$\left(\frac{\mathrm{GM}}{4{\mathrm{\pi }}^2}\right)=\text{Antilog}\left(15.51\right)$

$\begin{array}{c}\mathrm{M}=\frac{3.23\times {10}^{15}\left(4\right){\left(3.142\right)}^2}{6.67\times {10}^{-11}}\mathrm{kg}\\ =1.9\times {10}^{27}\mathrm{kg}\end{array}$

Hence, the mass of Jupiter from the intercept of the graph with the y-axis is $1.9\times {10}^{27}\text{kg}$.

Therefore, using Kepler’s third law, the mass of the planet can be found.