A-level Physics (Advancing Physics)/Orbits/Worked Solutions

'''1. The semi-major axis of an elliptical orbit can be approximated reasonably accurately by the mean distance of the planet for the Sun. How would you test, using the data in the table above, that the inner planets of the Solar System obey Kepler's Third Law?'''

Divide T2 by R3 and for each planet and see if this value is roughly constant.

'''2. Perform this test. Does Kepler's Third Law hold?'''

So, Kepler's Third Law does hold for the inner planets, using this rough approximation for the semi-major axis.

3. If T2 &alpha; R3, express a constant C in terms of T and R.

$$C = \frac{T^2}{R^3}$$

This is the constant of proportionality. It should be roughly the same for all the planets around the Sun. Alternatively, you can use:

$$C = \frac{R^3}{T^2}$$

We will be using the former to answer the next two questions, but you should be able to get the same answers using the latter.

'''4. Io, one of Jupiter's moons, has a mean orbital radius of 421600 km, and a year of 1.77 Earth days. What is the value of C for Jupiter's moons?'''

$$C = \frac{T^2}{R^3} = \frac{(1.77 \times 24 \times 60 \times 60)^2}{(421600000)^3} = 3.12 \times 10^{-16}\mbox{ s}^2\mbox{m}^{-3}$$

5. Ganymede, another of Jupiter's moons, has a mean orbital radius of 1070400 km. How long is its year?

$$3.12 \times 10^{-16} = \frac{T^2}{1070400000^3}$$

$$T = \sqrt{3.12 \times 10^{-16} \times 1070400000^3} = 618665\mbox{ s} = 7.16\mbox{ days}$$

Which isn't that accurate, due to the approximations that we used.