Using Kepler's 3rd Law to find Period of Venus

[Rajveer97]

Homework Statement

Deduce, from the equations employed in Q4 and Q5, the exponent n in the equation: T = k rn where k is a constant and T is the period of a satellite which orbits at a radius r from a massive object in space. Hence, how long is the “year” on Venus if its distance from the Sun is 72.4% of the Earth’s?

Homework Equations

The equations I used in the previous equations were simply a = (4(pi^2)r)/T^2 and a = GM/r^2 and basically combinations of the two

The Attempt at a Solution

So I started this question by trying to find the distance of the Earth from the Sun using T^2 = (4pi^2)/Gm (m being mass of the Sun) to which I got the answer 2.75x10^11 km

Then I substituted the r in the Kepler's equation with 0.724Re (to find distance of Venus from the Sun) but my answers seemed to have all been wrong so either I'm making some silly arithmetical error or my entire approach is wrong. Any help and hints would be appreciated, thank you :)

 

[James R]

Rejverr97:

The question asks you to write an equation of the form $T=k r^n$, using the equations you had previously. That is, you need to combine those equations algebraically to express the period $T$ in terms of the distance $r$. In the process, you'll find the exponent $n$ as well as the constant $k$ in terms of the constants in your previous equations.

Once you have that expression, you can work with ratios of the $T$ and $r$ values. If you do that, you should not need to work out intermediate values like the actual distance of Earth from the Sun in metres.

Does that help?

 

[Rajveer97]

Rejverr97:

The question asks you to write an equation of the form $T=k r^n$, using the equations you had previously. That is, you need to combine those equations algebraically to express the period $T$ in terms of the distance $r$. In the process, you'll find the exponent $n$ as well as the constant $k$ in terms of the constants in your previous equations.

Once you have that expression, you can work with ratios of the $T$ and $r$ values. If you do that, you should not need to work out intermediate values like the actual distance of Earth from the Sun in metres.

Does that help?

I should have mentioned, I did do that first step which gave me T = (2pi/root Gm) x r^3/2 . So the value I got for n was 3/2. Could you elaborate a bit more on what you mean later by working out the ratios of the T and r values? Because currently I can't think in my head how to do this without needing the distance of Earth from the Sun.

 

[haruspex]

I should have mentioned, I did do that first step which gave me T = (2pi/root Gm) x r^3/2 . So the value I got for n was 3/2. Could you elaborate a bit more on what you mean later by working out the ratios of the T and r values? Because currently I can't think in my head how to do this without needing the distance of Earth from the Sun.

You don't need the absolute distance of either planet from the Sun. You are given the ratio of the two radii, and you are asked for Venus' "year" as measured in Earth years, i.e. you are asked for the ratio of the two years.
Combine that with the formula you obtained in the first part of the question, T=kr^3/2.

 

[James R]

I should have mentioned, I did do that first step which gave me T = (2pi/root Gm) x r^3/2 . So the value I got for n was 3/2. Could you elaborate a bit more on what you mean later by working out the ratios of the T and r values? Because currently I can't think in my head how to do this without needing the distance of Earth from the Sun.

Ok. So, you have
$$T_{Earth}=\frac{2\pi}{\sqrt{Gm}}r_{Earth}^{3/2},~~T_{Venus}=\frac{2\pi}{\sqrt{Gm}}r_{Venus}^{3/2}$$
Two equations. You know $r_{Venus}/r_{Earth}=0.724$ and $T_{Earth}=1$ year. Can you use the two equations to find $T_{Venus}$?