Question

If an asteroid has a perihelion distance of 2.0 A.U. and an aphelion distance of 6.0 A.U., what is its orbital semi-major axis, eccentricity, and period? Thanks!?

Answer 1

Semi major 4AU, eccentricity 0.5, period 8 years.

Explanation:

If the semi major axis distance is a and the eccentricity is e, then the perihelion distance is `a(1-e)` and the aphelion distance is `a(1+e)`.

Substituting the values gives `a(1-e)=2` and `a(1+e)=6`.
Adding gives `2a=8`. This gives `a=4`AU.
subtracting gives `2ae=4` which gives `ae=2`. Substituting for a gives `e=1/2` or 0.5.

Kepler's 3rd law states that given the period T, the `T^2=a^3`. Substituting for a gives `t^2=64`. Taking the square root gives `T=8` years.