A question about the orbital period of an asteroid with respect to earth?

Question

Q2. An asteroid is orbiting the sun in a circular orbit, the radius of which is 4.0 times that of the Earth's orbit (which you may also assume to be circular). What is the orbital period of the asteroid? Choose the option that is closest to your answer and pencil across one cell in row 2.

KEY FOR Q2

A) 2.5 years

B) 4.0 years

C) 8.0 years

D) 16 years

E) 32 years

F) 64 years

Answer 1

For this question you need to appreciate Kepler's Third Law, which states that

r^3 is proportional to t^2

If you know the ratios of the radii, then you can calculate the ratios of the periods and hence get your answer.

Answer 2

The orbital period, T=2*pi*sqrt(a^3/mu), where mu=GM, G is the gravitational constant and M is the mass of the sun.  a is the size of the semi-major axis of the elipse of the orbit, but since the orbit here is circular, this is simply the radius.  You know that the period of the earth (T2) is 1 year, so you want to find the ratio:

T1...2*pi*sqrt(a1^3/mu)...sqrt(a1^3)

--- = -------------------------- =--------------=sqrt(4^3/1^3 )=4^(3/2)=8y

T2...2*pi*sqrt(a2^3/mu)...sqrt(a2^3)

This is because pi and mu are equal for both planets, and so cancel.