# Given an initial orbit calculate the necessary maneuver impulse to obtain a required final eccentricity and semi-major axis

The question is simple. Starting from an eccentric orbit I want to obtain a final orbit of which the final semi-major axis and the final eccentricity are known. You would have to determine the boost and the point in the orbit where it should be done to get these final conditions. That is, we have an a0 and an e0 and an af and an ef and I would like an algorithm that gives me the impulse and the point of the orbit where to do it.

• What research have you already done ? Have you read this ? Are there no constraints on parameters other than semi major axis and eccentricity ?
– AJN
Feb 9, 2022 at 14:01
• The point of the orbit at which to do it is going to be the point where the two orbits intersect. Delta-V requirement will be the same at either one. If the two orbits do not intersect, it will be impossible to do so with a single impulse in a Keplerian/Newtonian two-body situation. Feb 10, 2022 at 0:38

First, given $$a$$, the velocity at any $$r$$ is calculated from the vis viva equation:
$$v^2 = \mu (\frac 2r - \frac 1a)$$
Then, calculate the parameter of the orbit (ok, if you are a math type, the semi-latus rectum): $$p = a(1-e^2)$$ and then the angular momentum $$h= \sqrt{\mu p}$$. This gives the transverse component of velocity by the definition $$h = rv_t$$. From the Pythagorean equation, the radial component of velocity can be found.
So, you can calculate $$v_t$$ $$v_r$$ for both orbits then the delta velocity follows from change in each component.