Speed of orbiting planet given eccentricity of orbit

[ephedyn]

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?

 

[issacnewton]

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

 

[ephedyn]

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.

 

[issacnewton]

well algebra is pretty straight forward I guess.

 

[rwisz]

To find the ratio of the speeds of the planet at the ends of the major axis, we can use the fact that the angular momentum (L) is conserved in an elliptical orbit. This means that at each point in the orbit, the product of the distance (r) and the speed (v) will be constant. Therefore, we can set up the following equation:

rv=constant

At the ends of the major axis, the distance (r) will be equal to the length of the major axis (a), and the speed (v) will be the fastest in the orbit. Therefore, at this point, we can say:

a_{max}v_{max}=constant

Similarly, at the ends of the minor axis, the distance (r) will be equal to the length of the minor axis (b), and the speed (v) will be the slowest in the orbit. Therefore, at this point, we can say:

b_{max}v_{min}=constant

Dividing these two equations, we get:

\dfrac{a_{max}}{b_{max}}\dfrac{v_{max}}{v_{min}}=1

Using the ratio of the major to minor axes found earlier, we can substitute and solve for the ratio of the speeds:

\dfrac{v_{max}}{v_{min}}=\dfrac{b_{max}}{a_{max}}=\dfrac{1}{1.0911}\approx0.917

Therefore, the ratio of the speeds at the ends of the major axis is approximately 0.917. This means that the planet will be moving at about 91.7% of its maximum speed when it is at the ends of the major axis of its orbit.