Speed of orbiting planet given eccentricity of orbit

  • Thread starter ephedyn
  • Start date
  • #1
170
1

Homework Statement


If the eccentricity of a planet's orbit about the sun is 0.4, find (a) the ratio of the lengths of the major to minor axes of the planet's orbit, and (b) the ratio of the speeds of the planet when it is at the ends of the major axis of its elliptical orbit.


Homework Equations



[itex]E=\dfrac{1}{2} mv^2 + \dfrac{-GMm}{r} + \dfrac{L^2}{2mr^2}[/itex]

The Attempt at a Solution



The first part is rather short and sweet:

[itex]\dfrac{a}{b}=\dfrac{1}{\sqrt{1-\epsilon^{2}}}=\dfrac{1}{\sqrt{1-0.4^{2}}}\approx1.0911[/itex]

But I have no idea how to proceed on the second part. If I take the ratio of the kinetic energies I get a very messy expression and have problem eliminating the total energy E. Is there a more elegant way to do this?
 

Answers and Replies

  • #2
911
18
hi

orbital speed is given by

[tex]v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)}[/tex]

where [tex]\inline{\mu =G(M+m)}[/tex] is the standard gravitational parameter. and r is the distance from the sun(focus). at the farthest point in the orbit from the sun
[tex]\inline{r=(1+e)a}[/tex] and the closest point we have [tex]\inline{r=(1-e)a}[/tex]
where a is semi major axis length. use this info

newton
 
Last edited:
  • #3
170
1
Hey thanks for your help. I tried using your approach but I couldn't simplify it so I tried using conservation of angular momentum and got the correct answer. Simpler than I expected.
 
  • #4
911
18
well algebra is pretty straight forward I guess.
 

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