- #1

Raphael30

- 12

- 4

## Homework Statement

We want to send a satellite from a low Earth orbit of 320 km to mars. Calculate the change in velocity required to join the transfer ellipse.

## Homework Equations

Earth velocity: (μ

_{S}/R

_{EarthRev})

^{1/2}

Transfer velocity at perihelion: (2μ

_{S}R

_{MarsRev}/(R

_{EarthRev}(R

_{EarthRev}+R

_{MarsRev}))

^{1/2}

Orbital velocity at LEO: (μ

_{E}/R

_{LEO})

^{1/2}

Escape velocity at LEO: (2μ

_{S}/R

_{EarthRev})

^{1/2}

## The Attempt at a Solution

Ok so the Earth velocity is about 29780 m/s, the transfer velocity has to be 32730 m/s at perihelion (when it meets the Earth's revolution orbit). The orbital velocity is 7722 m/s and the minimal escape velocity is 10920 m/s. At minimal escape velocity, the maximal total departure velocity of the satellite from the sun's referential would be 40 700 m/s. However, this velocity drops as the satellite moves away from Earth, the escape velocity tending towards 0 and the total velocity tending towards 32780, all this because of the gain in potential energy. Counting the sun's attraction, this means the satellite would follow an orbit around the sun fairly similar to the Earth's. We could then suppose that the velocity needed for the satellite to escape the Earth's orbit has to be added somehow to the velocity needed to join the transfer ellipse. However, I have absolutely no idea how to combine the kinetic and potential energy of a system within another system and therefore I don't know when to sum kinetic energies, when to sum velocities directly, etc. I know that the final answer should be a gain in velocity of about 3600 m/s, far less than the 6138 obtained by summing the missing velocities directly and that this has something to do with some kind of Oberth effect, but I really can't seem to figure it out.