Question

4 analyze: how does the period change as the orbital radius increases?
5 interpret: select the graph tab, and check that the graph of t vs. a is selected. use the zoom controls to adjust the graph so you can see all your data and it fills the graph.
a what does the graph indicate?
b. does the graph of t vs. a form a straight line?

1. investigate: choose other options in the dropdown menus until you find a graph that makes a perfectly straight line. (if you cannot see all of your data, click the - controls to zoom out.)

which relationship makes a straight line?
when you have a line, click the camera ( ) icon to take a screenshot. right-click the image, and click copy image. paste it here:

1. calculate: on the table on the previous page, label the third column \a³\ and the fourth column \t².\ use a calculator to cube each value of a and square each value of t.

what do you notice?
keplers third law states that the cube of a planets orbital radius is proportional to the square of a planets period: a³ = kt² for some constant k. if the radius is measured in astronomical units, the period is measured in earth years, and the mass of the star is equal to the mass of our sun, the value of k is equal to 1 au³/y².

1. challenge: the orbital radii of the planets are given in the table below. calculate the period of each planet (in earth years), and then check your values with your teacher.

Explanation:

Step1: Use Kepler's third law formula

Kepler's third law is \(a^{3}=kT^{2}\). Given \(k = 1AU^{3}/y^{2}\), we can solve for \(T\) as \(T=\sqrt{\frac{a^{3}}{k}}\). Since \(k = 1\), the formula simplifies to \(T=\sqrt{a^{3}}\).

Step2: Calculate \(T\) for each planet

• Mercury: \(a = 0.39AU\). \(T=\sqrt{0.39^{3}}=\sqrt{0.059319}\approx0.24\) years.
• Venus: \(a = 0.72AU\). \(T=\sqrt{0.72^{3}}=\sqrt{0.373248}\approx0.61\) years.
• Earth: \(a = 1.00AU\). \(T=\sqrt{1.00^{3}} = 1\) year.
• Mars: \(a = 1.52AU\). \(T=\sqrt{1.52^{3}}=\sqrt{3.511808}\approx1.87\) years.
• Jupiter: \(a = 5.20AU\). \(T=\sqrt{5.20^{3}}=\sqrt{140.608}\approx11.86\) years.
• Saturn: \(a = 9.54AU\). \(T=\sqrt{9.54^{3}}=\sqrt{868.336}\approx29.47\) years.
• Uranus: \(a = 19.18AU\). \(T=\sqrt{19.18^{3}}=\sqrt{7089.07}\approx84.2\) years.
• Neptune: \(a = 30.06AU\). \(T=\sqrt{30.06^{3}}=\sqrt{27162.18}\approx164.8\) years.

Answer:

Planet	\(T\) (years)
Venus	\(0.61\)
Earth	\(1\)
Mars	\(1.87\)
Jupiter	\(11.86\)
Saturn	\(29.47\)
Uranus	\(84.2\)
Neptune	\(164.8\)