How do I expand powers to solve these equations?

[AT36]

I have attached two images. Could you explain me how the result was obtained by expanding the powers?

 

[BvU]

Hello AT36,

Not without some more context. Please oblige and tell us what this is about and come with a complete problem statement.

 

[AT36]

Hello AT36,

Not without some more context. Please oblige and tell us what this is about and come with a complete problem statement.

Hi,
I have attached a paper. I am trying to derive the slant range rate formula. Under the topic 'Calculation of the Observable', range 'p' equation is clear but the next step, they applied some expansion of powers which I am not able to understand.

 

[BvU]

With $$\dot \rho = {\partial \rho \over \partial t}+ {\partial \rho \over \partial \theta}\, {\partial \theta \over \partial t}$$ and $${d\rho^2 \over dt} = 2\rho\dot\rho$$ the expression $\rho^2 = a^2+b^2+c^2$ differentiates to$$2\rho\dot\rho = 2a\dot a+2b\dot b+2c\dot c$$ where e.g. $$\dot a = {d\over dt} (x-r_s\cos\theta) = \dot x - \left ( {d\over dt} r_s \right ) \cos\theta - r_s {d\over dt} \cos\theta = \dot x - \dot r_s \cos\theta + \omega r_s \sin\theta $$ Straightforward so far, right ?

Next step is to exploit $r_s /r << 1$ (from the accompanying text ) and collect terms of 0^th and first order in $r_s /r$.

I commend you for not taking the result for granted, but writing it out in full is too much work for me before breakfast, so I'll leave it to you -- but with the promise of further help if you indicate where you get stuck