Satellite Hohmann transfer problem

[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

 

[gneill]

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

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.

 

[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 Earth's orbit and Jupiter's?

Thanks,
S

 

[gneill]

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 Earth's orbit and Jupiter's?

Thanks,
S

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?

 

[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

 

[gneill]

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?

Yes it is.

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

Yes you can. Actually it's true for the entire orbit -- the total specific mechanical energy for an orbit is a constant.

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

k (or more conventionally, μ) is in this case just G*M, M being the mass of the Sun.

 

[Sekonda]

Thanks for the prompt reply, also for the clarification and advice.

I believe I have solved the problem now, thanks again gneill!

 

[gneill]

Thanks for the prompt reply, also for the clarification and advice.

I believe I have solved the problem now, thanks again gneill!

You're welcome