Is Tidal Circularisation of Orbits a Tricky Concept?

[thespoonftw]

Homework Statement

A body on an orbit with semi-major axis a and eccentricity e undergoes tidal circularisation.

Show that the orbit will circularise at a semi-major axis, a_circ, given by

a_circ = 2r_peri = 2a (1 − e).

Homework Equations

No equations given, but I think the following could be useful

E = -GMm/2a
e² = 1 - b²/a²

The Attempt at a Solution

An earlier part of the question hints at L conservation

Equating centripetal force and grav force for the circular orbit gives:
L = m (GMR)^0.5

Finding the velocity at the closest point in orbit r = a(1-e)

E = -GMm/2a = 1/2 mv² - GMm/a(1-e)

simplifies to
v² = GM(1+e)/r_p

Equating L²
L² = GMm² r_p (1+e) = GMm² r_c

Finally:
r_c = r_p (1+e)

This is close to the final answer, but not quite!
Somethings gone wrong somewhere but I'm sure what.. I've checked my working several times.
Sorry a lot of my working lines are missing, it's quite tricky to type them all out.

 

[mfb]

You have to find out which quantities are conserved during the process. Energy, angular momentum, or something else?

a_circ = 2r_peri = 2a (1 − e).

That cannot be the final semi-major axis. Consider the trivial case of e=0, for example, where the semi-major axis will certainly not double.
It could be twice the semi-major axis.

 

[thespoonftw]

Oh, well spotted with the trivial case.
Yea it looks like there's something wrong the question, and i think my method was fine.
Thanks for your time.