Oberth Effect Combined with Gravity Assist

[lotsorbits]

Hi all,

I need help with an idea that I think should be simple, but is proving difficult.

When a spaceship goes near a planet, can it take advantage of both the gravity assist AND oberth effect, or can it only pick one?

1) From a gravity assist, the spaceship could pick up as much as Vf = 2U + v for an increase in speed

2) From the Oberth effect, it can pick up as much Vf = sqrt((Δv + sqrt(V^2 + Vesc^2))^2 - Vesc^2).

(Δv is burn, Vesc is escape velocity at planet peri, V is initial velocity, U is planet's orbital speed, Vf is the ship's final velocity after leaving the planet due to either effect)

How do I calculate, roughly, the final velocity from combining both effects? I think it should be simple, but I'm not seeing it. Enabling me to do this back-of-envelope calculation before I do the simulation will at least give me a rough idea what my final speeds should be after I construct my orbit.

Also, if you know of any good books on orbital dynamics that treat Oberth and Gravity Assist very complete and well, from 1rst principles, that would be really great too.

 

[Quick_Nick]

I think one issue here is that 2U + v is a maximum, on a parabola coming straight at a planet. So your Oberth assisted burn may very well be negligible. (probably just a mid-course correction by modern usage) So you're going to have to get a much better approximation for the burn to matter.

 

[Tommy]

I think you have a slight misunderstanding of the Oberth effect. It does NOT give you velocity - rather it makes the velocity you gain provide more energy - leaving you with a higher velocity at escape.

You can calculate the vectors for "gravity assist" and the results of a burn separately - and simply add the vectors to get a rough idea of the total. Note that this doesn't give you a true vector - the calculations for the direction of the velocity gains are relative to when you escape the gravity source while the assist and burn will both change the direction of your vector before you escape.

However, most of the time we make these calculations based on a "point gravity source", so the difference between the calculation and the actual result will be relatively small - within the "margin of error" for the calculation and easily corrected during the normal MCC's.