Deriving orbital angular momentum relation with time derivatives

[Kurdt]

Just looking for some advice on where my maths is going wrong with this. I have the following equation.

$$L_{orb}=(\frac{GD}{M})^\frac{1}{2}M_sM_p$$

and information that the time derivatives of L and M are zero. Also M_s varies with time along with D. I am supposed to arrive at the following equation.

$$\frac{\dot{D}}{D}=-2(1-\frac{M_s}{M_p})\frac{\dot{M_s}}{M_s}$$

I first brought the M over to be on the same side as the L as when I take the time derivative they will be 0 and then after taking the time derivative of what is left on the left hand side and rearranging a little I can only get

$$\frac{\dot{D}}{D}=-2\frac{\dot{M_s}}{M_s^2}$$

Any pointers as to where my maths fails. I realize it could have something to do with a substitution of variables but I'm assuming not as it seems unlikely at the minute and I wouldn't like to type out all the possibilities . Any help is much appreciated.

 

[arildno]

What's r and b?
(This looks like some sort of spin equation in a gravitational field, but..)
And M? Do you mean $$M_{p}$$?

 

[Kurdt]

sorry r and b was supposed to be part of the subscript I will change it and M is the sum of M_p and M _p.

 

[Kurdt]

Its just basically the orbital angular momentum in the Roche model of binary stars which I am studying at the minute. Trying to extract useful information on the rate of orbital decay.

 

[arildno]

Remove the $$G^{\frac{1}{2}}$$ over as well.
Then we have:
$$0=\frac{\dot{D}}{2\sqrt{D}}M_{s}M_{p}+\sqrt{D}\frac{d}{dt}(M_{s}M_{p})\rightarrow\frac{\dot{D}}{D}=-2\frac{\frac{d}{dt}(M_{s}M_{p})}{M_{s}M_{p}}$$

The desired expression is now readily obtained

 

[Kurdt]

Thanks for your help. I knew it was something simple that I'd missed.