How calculate the minimum Delta v necessary to change one given orbit into another given orbit?

Kolodez

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The question is to be understood in the idealized model where I have only one point mass and a vessel with neglectable mass in orbit, Newton mechanics. Thanks for your help!
 

Urwumpe

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Just calculate the velocity vectors at the point where the two orbits intersect to get the DV of a single-burn strategy.

You can have much less DV with multiple burns or by exploiting non-spherical gravity or multi-body effects.
 

Kolodez

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If the planes of the two orbits are different and if the orbits intersect only at the nearer node, then it might be cheaper to do the plane alignment at the further node and then to match the orbits at the next intersection. So even if we are in the one-spherical-body model and if there is an orbit intersection, the single-burn strategy does not have to be optimal. Is there a general algorithm to find the optimal one?
 

Ajaja

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Lambert's problem + numerical optimization?
 

Ajaja

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Although If you need to get into any orbit with specific LAN/inclination/eccentricity/AoP (with any anomaly) it's not Lambert.
Maybe, the minimum delta-v here can be derived from laws of conservation energy and momentum. But I'm not sure.
I think you will be needed some sort of additional parameters (how many burns, time limits, etc.), some initial guesses, and it comes to an optimization problem again.

And you are right about the single-burn strategy. Sometimes it's not optimal even without the plane change maneuver (e.g. https://en.wikipedia.org/wiki/Bi-elliptic_transfer ).
 
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Kolodez

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Ok, so no simple solution? :( Only the energy-mass-ratio of an orbit is certainly not enough since we can have very different orbits with the same energy. But yes, maybe combining it with angular momentum will yield something meaningful, so thank you for that idea!

Actually I want to land on Mercury after some swing-bys at Mercury and since this is very Delta-v-intensive, I want to be able to measure the "distance" between my orbit before/after the Mercury swing-by and the Mercury orbit in order to compare the quality of swing-by manouvers.
 

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Ok, so no simple solution? :( Only the energy-mass-ratio of an orbit is certainly not enough since we can have very different orbits with the same energy. But yes, maybe combining it with angular momentum will yield something meaningful, so thank you for that idea!

Actually I want to land on Mercury after some swing-bys at Mercury and since this is very Delta-v-intensive, I want to be able to measure the "distance" between my orbit before/after the Mercury swing-by and the Mercury orbit in order to compare the quality of swing-by manouvers.

Well, I would then simply look at the problem from the point of view of our mission. In this case, entering Mercury Orbit might be even a waste of fuel compared to a direct landing and "suicide burn".

Next, you want to minimize the total impulse (DV) for the landing burn to be able to bring as much mass as possible down to the surface. So, you might want to minimize the excess velocity of your spacecraft when you enter the SOI of Mercury. Which means, you want to arrive with a velocity vector that is as close as possible to the velocity vector of Mercury. And much more interesting, you could describe this with only two key performance indicators for each possible trajectory: Semimajor axis and flight path angle.
 
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