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forestmine

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## Homework Statement

I don't have any particular problems I need help with. I'm just trying to wrap my head around some of the equations and their meanings. Any clarification would be great!

We have from Kepler's 3rd Law the following:

The period squared is proportional to the semi-major axis cubed.

Or,

P^2 = ((a^3)4*pi^2)/GM

and this equation in particular is the one body problem, where the mass of the body rotating around the sun is negligible in comparison to the sun, so M is the mass of the sun? If so, what's negligible mass?

And the two body problem would be the same thing, but,

((a^3)4*pi^2)/G(M+m)

Would the two-body problem apply for a planet's elliptical orbit around the sun?

From there, there is the equation for velocity along an elliptical orbit:

v^2 = GM (2/r - 1/a)

M in this case is the mass of the particular body whose velocity we are calculating? Say, the mass of the Earth?

And then there are these equations, which I'm a bit confused about.

Velocity at perihelion: v^2 = (GM/a)(1+e/1-e)

Velocity at aphelion: v^2 = (GM/a)(1-e/1+e)

So, if I'm given the eccentricity of the ellipse, I ought to use one of the last two equations, depending on whether it's at perihelion or aphelion? If a specific eccentricity is not given, I'm assuming I can use the equation v^2 = GM (2/r - 1/a), where r is a specified distance of either perihelion or aphelion?

And one last question. If the general equation for escape velocity is given by:

v=(2MG/r)^1/2

can I apply it in this form for elliptical orbits? If so, is M in this case again the mass of the body from which a particle or object is attempting to escape, and r is the distance out to that object?

Hopefully this is somewhat clear!

Thanks in advance!