Orbital period variation in binary system

[freesnow]

Homework Statement

Using the conservation of angular momentum and Kelper's third law, show that the relative change in orbital period produced by mass transfer is given by
1/P * dP/dt = 3 dM₁/dt * (M₁ - M₂)/(M₁M₂)

Homework Equations

L = mu * sqrt(GMa)
P^2 = (4*PI^2)/(G(M₁+M₂)) * a^3

The Attempt at a Solution

see attachment

I was so close to the answer... I really can't figure out what's missing... is there somehow a miscalculation in my proof?

 

[Jhero]

Or maybe I'm not considering all the necessary variables? I would really appreciate any help or guidance you can provide. Thank you!
Thank you for your post. Your attempt at a solution is very close to the correct answer, but there are a few minor errors that need to be addressed.

Firstly, in your expression for the angular momentum (L), you have used a lowercase 'a' instead of the semi-major axis (A). This should be corrected to L = mu * sqrt(GMA).

Secondly, in your expression for the period (P), you have used the masses (M1 and M2) instead of the total mass (M1+M2). This should be corrected to P^2 = (4*PI^2)/(G(M1+M2)) * A^3.

Finally, when taking the derivative of both sides of the equation, you have missed out the derivative of the semi-major axis (A). This should be included in the derivative on the right-hand side, giving 1/P * dP/dt = 3 dM1/dt * (M1 - M2)/(M1+M2)^2 * A^3.

With these corrections, your proof should lead you to the correct answer. Keep up the good work and don't hesitate to ask for help if you encounter any further difficulties.