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Question about the Kepler equation

  • #1
304
14

Homework Statement


59.png


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.
 

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Answers and Replies

  • #2
TSny
Homework Helper
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(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.
 
  • #3
304
14
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)
 
  • #4
TSny
Homework Helper
Gold Member
12,411
2,846
Nice.
 

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