Satellite Hohmann transfer problem

1. Mar 10, 2012

Sekonda

Hey guys, here is the problem:

A spacecraft is initially in a circular orbit of the Sun at the Earth’s orbital radius. It uses
a single brief rocket thrust parallel to its velocity to put it in a new orbit with aphelion
distance equal to the radius of Jupiter’s orbit.

What is the ratio of the spacecraft’s speeds just after and just before the rocket thrust?

I am aware, or at least I think this problem is similar to a Hohmann transfer orbit but I'm fairly confused on how to start resolving the problem; I think I need to determine the perihelion distance though I'm not sure why.

Thanks,
S

2. Mar 10, 2012

Staff: Mentor

As you surmised, the problem is very much like a Hohmann transfer problem, at least the transfer orbit part.

Make a sketch of the scenario and I think you'll find that the perihelion distance is fairly obvious. You should be able to determine the initial speed (prior to the thrust) given where the spacecraft is starting from. Finding the speed after the thrust will require determining some information about the transfer orbit.

3. Mar 11, 2012

Sekonda

I'm still a bit confused, the aphelion distance was easy enough, though I'm unsure how the satellite is entering the orbit, is it entering the aphelion side of perihelion side of the Jupiter orbit.

Am I effectively working out how much kinetic energy would have to be used to overcome the gravitational potential difference between the earths orbit and Jupiter's?

Thanks,
S

4. Mar 11, 2012

Staff: Mentor

Just draw two concentric circles, one representing the orbit of Earth and the other representing the orbit of Jupiter. A dot at the center can represent the position of the Sun. You should be able to draw in an elliptical transfer orbit that touches both circles at the ends of its major axis. What then is the major axis length?

Can you determine the total energy of the orbit from the length of the major axis?

5. Mar 13, 2012

Sekonda

Is the semi major axis 0.5(Rj+Re) where Rj is the radius of Jupiter's orbit and Re is the radius of Earth's orbit?

Can I the use the fact the the total energy at these points is given by E=-k/(2a)

where 'a' is the semimajor axis length? and k=GMm

6. Mar 13, 2012

Staff: Mentor

Yes it is.
Yes you can. Actually it's true for the entire orbit -- the total specific mechanical energy for an orbit is a constant.
k (or more conventionally, μ) is in this case just G*M, M being the mass of the Sun.

7. Mar 13, 2012