How Does Angular Momentum Vary with Orbital Distance in Physics?

[SignaturePF]

Homework Statement

https://aapt.org/physicsteam/2010/upload/2009_F-ma.pdf

Homework Equations

L = mrv
L = Iω

The Attempt at a Solution

For a circular orbit:
Fc = Fg
mv^2/r = Gmm/r^2
v = √(GM/R)
Thus:
l = mR√(GM/R)
l = m√(GMR)

This means that LA > LC, eliminating choices B, C, and E.

Now, to compare B, C
I'm interested in finding a more rigorous approach, but here goes.
The point of intersection between the Circlular path that C orbits on and the elliptical path that B orbits.
We know that the velocity at the perihelion is greater than the aphelion, that is, the velocity of the intersection is the maximum velocity that B ever achieves. I then made an intelligent guess and postulated that thus B > C,
leading to LA > LB > LC

Could you suggest more rigor/principles to do this question?

 

[Simon Bridge]

If choices B,C & E are eliminated - what is left are:

(A) LA > LB > LC
(E) The relationship between the magnitudes is different at various instants in time.

Look at E.
Consider: conservation of angular momentum.

 

[tms]

l = m√(GMR)
...
Could you suggest more rigor/principles to do this question?

You've already done that: for B, $r$ is never less than C's and never more than A's.

 

[SignaturePF]

Ya I see that but isn't it root(GM/a), where a is the semi major axis for object B. Doesn't that mean that the radius in the numerator won't cancel with the semi major axis on the denominator?
That's where I was worried.