# Find Aphelion of an elliptical orbit given Perihelion and Orbital Period

## Homework Statement

Comet Halley approaches the Sun to within 0.570 AU, and
its orbital period is 75.6 years. (AU is the symbol for astronomical unit, where
1 AU = 1.50 x 1011 m is the mean Earth‐Sun distance.) How far from the Sun will
Halleyʹs comet travel before it starts its return journey?

## Homework Equations

There are none given, though I believe some of these are probably relevant.

T=2π√(a^3/(GM))

which is the equation relation period to the semi-major axis and a gravitational constant

Ke(perihelion) + Kp(perihilion)=Ke(aphelion) + Kp(aphelion)

L= mvrsinθ

## The Attempt at a Solution

This question is legitimately driving me insane. I feel like I am on the brink of solving it, but I just keep going around in circles.

Now, I understand that if I had the semi-major axis, to solve this would be relatively easy...but I don't.

Given this, I attempted to rearrange T=2π√(a^3/(GM))
to solve for a.
My calculus can be a bit dodgy but I came up with,
a is equal to the cube root of t^2/2π(GM)

or

a=3√(T2/2π(GM)

Angular momentum and total energy for the system are conserved and thus are equitable at the perihelion and aphelion.

At the aphelion sinθ=1
So angular momentum is mvr at angles 0 and 180, which both lie on the major axis.

at the perihelion gravitational potential energy = 0 and kinetic energy is 100%
Inversely, at the aphelion, the energy in the comet is 100% gravitational potential.
Which I guess means acceleration = 0.

I am really lost on this one guys.
anything!