# Eliptical Orbit and Eccentricity

1. Mar 26, 2009

### fogel1497

1. Problem statement
The space shuttle is in circular orbit of radius R around the earth. The pilot triggers a brief burn that imparts a forward impulse 'p' to the shuttle. After the burn, the shuttle is in an elliptical orbit which passes back through the point where the burn took place once per orbit. The mass of the shuttle is m.

From this we learn the following:

Mass Shuttle = m
Impulse = p

In previous parts of the problem were asked to find Vinitial which is the velocity before the
burn, and Vfinal which is the velocity after the burn. So these are also both knowns.

The first part of this question im having trouble with is:

What is the formula r(theta) for the elliptical orbit in polar coordinates r, (theta) with the origin at the center of the earth? Hint: you may use the eccentricity " of the elliptical orbit as a known parameter in your formula.

A given formula on our equation sheet is:
r(theta) = a(1 + Ecos(theta))

So I know from the problem statement that the shuttle passes through the original point at which the burn was initiated. Knowing this I know that a, which is a perpendicular line to the major-axis to the edge of the elipse, is equal to R. Therefore the answer to this question is:

r(theta) = R(1+Ecos(theta))

Is this correct?

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Now the other part Im having trouble with:

What is the value of the eccentricity 'E' of the orbit in terms of R, m, p, the mass of the earth M, and known physical constants. (Known physical consants are mass of earth and G-the gravitational constant).

The only equation given to me with E in it is the one i said before:
r(theta) = R(1+Ecos(theta))

But I cant think of anything to set that equal to in order to solve for E in terms of those constants. I mean what exactly is r(theta)? Is that the radius at a given angle of theta? Do I have to convert it back to cartesian coordinates to use it in my equations? What equation do i set it equal to. Im real confused.