Eliptical Orbit and Eccentricity

[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:

Radius = R
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 I am 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 I am 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 can't 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. I am real confused.

 

[lippyka]

Answer:In order to solve for the eccentricity 'E' of the orbit, you need to use the equation for orbital energy. The equation is as follows:Energy = -G*M*m/(2a) Where G is the gravitational constant, M is the mass of the Earth, m is the mass of the shuttle, and a is the semi-major axis of the ellipse. Using this equation and the information given, you can solve for E by rearranging the equation to solve for a, then plugging in the value of R for a. This will give you an expression for E in terms of R, m, p, M, and G.

 

[nlightner]

Hello, the formulas you have mentioned are correct. The formula for r(theta) in polar coordinates is r(theta) = a(1 + Ecos(theta)), where E is the eccentricity of the orbit. In this case, a is equal to the radius of the circular orbit, which is R. So the formula is r(theta) = R(1+Ecos(theta)). This formula gives the distance of the shuttle from the center of the earth at any given angle theta.

To find the value of eccentricity E, we can use the fact that the total energy of the shuttle in orbit is conserved. This means that the sum of the kinetic and potential energies at any point in the orbit is constant. We can use this to set up an equation and solve for E.

The kinetic energy of the shuttle in orbit is given by K = (1/2)mv^2, where m is the mass of the shuttle and v is the velocity. We can find the velocity before and after the burn using the given information.

Before the burn, the shuttle is in a circular orbit, so the velocity is given by Vinitial = sqrt(GM/R), where G is the gravitational constant and M is the mass of the earth.

After the burn, the shuttle is in an elliptical orbit, so the velocity is given by Vfinal = sqrt(GM(2/R - 1/a)), where a is the semi-major axis of the elliptical orbit.

Now, using the conservation of energy, we can set up the following equation:

Kinitial + Uinitial = Kfinal + Ufinal

where U is the potential energy, given by U = -GMm/r. Substituting in the values for K and U, we get:

(1/2)m(Vinitial)^2 - (GMm)/R = (1/2)m(Vfinal)^2 - (GMm)/r(theta)

Substituting in the values for Vinitial and Vfinal, we get:

(1/2)m(GM/R) - (GMm)/R = (1/2)m(GM(2/R - 1/a)) - (GMm)/r(theta)

Simplifying, we get:

GMm/R - GMm/R = GMm/R - (GMm)/r(theta)

Cancelling out the common terms, we get:

0 = (GMm)/R - (GMm)/r(theta)

Solving for r