Deriving equation 7.27 from the orbit equation

[Clara Chung]

Homework Statement

Homework Equations

1/r = a/b² * (1+e cosθ)

The Attempt at a Solution

How do you derive 7.27?
I tried to substitute r in the orbit equation by (acosψ - ae) / cosθ and got
(cos ψ - e)/cosθ*(1+e cosθ)=b^2/a^2 I don't know what to do next. Please help, thank you.

 

[TSny]

(cos ψ - e)/cosθ*(1+e cosθ)=b^2/a^2

I believe this is OK. You just need to manipulate it to get (7.27). I don't know the quickest way, but I found something that works. I don't want to give too much away.
Rearrange your equation as (cos ψ - e)*(1+e cosθ)=(b^2/a^2)*cosθ. Substitute for b^2/a^2 in terms of e. Multiply everything out and see if you can get to the result.

 

[Clara Chung]

I tried hard by substituting (b^2/a^2) but still can't figure out the equation... Then I changed my mind to substitute for e. Thank you so much.
(cos ψ - e)*(1+e cosθ)=(b^2/a^2)*cosθ

(ecos ψ - 1+b^2/a^2)*(1+e cosθ)=(b^2/a^2)*cosθ*e

-(1-ecosψ)(1+ecosθ)+(b^2/a^2)e cosθ+(b^2/a^2)=(b^2/a^2)*cosθ*e

(1-ecosψ)(1+ecosθ)=(b^2/a^2)

 

[TSny]

Nice.