How much delta-v does it take to go from one arbitrary orbit to another?

This is, in theory, very basic rocket science. I can't seem to find a solution on here though, so I'm asking in the hopes you'll enlighten me and my amateurish ways.

I've got a set of Keplerian orbital elements $$e_0$$, $$a_0$$, $$i_0$$, $$\omega_0$$, $$\Omega_0$$, and $$\theta_0$$, and I'd like to get to a different orbit with orbital elements $$e$$, $$a$$, $$i$$, $$\omega$$, $$\Omega$$, and $$\theta$$. How do I calculate (a) the amount of delta-v I'll need for this maneuver or set of maneuvers, and (b) which maneuver or set of maneuvers I should make to do this optimally?

P.S. I get the strong sense that my question (b) comes down to the kind of numerical integration that takes endless small steps to estimate properties of different options, and so doesn't have a simple answer. I'm not sure if that's practically the case when dealing with only two or three bodies, of which the one making the maneuvers has a far smaller mass than the others. If there is a practical way of answering question (b) given these assumptions, I'd love to hear it. If not, then question (a) alone is already very helpful, and I'm really thankful for your time.

• It may be helpful to take a look at the simplest case: the Hohmann transfer orbit between 2 coplanar circular orbits. Mar 30, 2021 at 9:37
• Might you be over-engineering the whole idea? What it takes to go from one orbit to another depends first on the orbits and then on your object's position in the starting orbit. How could it be otherwise? What could "arbitrary" contribute but confusion? Without first explaining that, what could all the other detail contribute? Mar 30, 2021 at 20:59
• @RobbieGoodwin What I'm asking is, given any starting orbit, and any desired orbit, how do I find delta-v needed to go from A to B, and how do I find the burns that get me there? Most such questions on here are physics homework style, asking for a single case with a starting and ending orbit, rather than asking for a general framework for how to solve this problem. The word arbitrary indicates that this question applies for any set of starting and ending orbits, and not just one specific case. Is there anything else I left out, as I'm happy to make sure I'm as clear as possible :) Mar 31, 2021 at 20:48
• @TheEnvironmentalist Sadly, that wasn't what you actually Asked. If you could swap out the too-technical detail in the exposition, life might become easier. Perhaps most obviously "arbitrary" the way you used it doesn't at all apply to any orbits… it simply creates the confusion I first warned you about. You would need "random", meaning any, not "arbitrary" meaning chosen without clear reason. Doesn't physics homework depend on the general framework, or are you suggesting the process changes according to a given pair of orbits? Mar 31, 2021 at 20:55
• @RobbieGoodwin I'm looking for a formula or set of formulas that take a starting orbit as orbital parameters, a desired orbit as orbital parameters, and gives me the burns that take me from the first to the second, and how much delta-v I'll need. I can't see how that can be made any less technical, as it's a mathematical question at its heart. My use of the term "arbitrary" was for its mathematical definition, indicating a process, formula, or formulas that apply to any possible set of starting and ending orbits, while using random would imply the subtly different randomly sampled orbit Mar 31, 2021 at 21:07

I've got a set of Keplerian orbital elements $$e_0$$, $$a_0$$, $$i_0$$, $$\omega_0$$, $$\Omega_0$$, and $$\theta_0$$, and I'd like to get to a different orbit with orbital elements $$e$$, $$a$$, $$i$$, $$\omega$$, $$\Omega$$, and $$\theta$$. How do I calculate (a) the amount of delta-v I'll need for this maneuver or set of maneuvers, and (b) which maneuver or set of maneuvers I should make to do this optimally?

Before I get to answering your two questions, there are four additional parameters you haven't thought of. There the two epoch times at which those orbital elements apply, which I'll call $$T_0$$ and $$T$$ to match up with your nomenclature, the time $$t_0$$ at which the vehicle departs from the initial orbit, and the time $$t$$ at which the vehicle arrives at the final orbit. It takes a certain amount of delta-v to accomplish this transfer.

How do I calculate (a) the amount of delta-v I'll need for this maneuver or set of maneuvers?

Fortunately, the calculation of this delta-v does not require integration, at least so long as the problem is kept simple. So, keeping it simple, I'll assume that orbit are Keplerian. Suppose you pick a random $$t_0$$ and $$t$$ subject only to the constraint that $$t>t_0$$, and suppose the vehicle performs only two instantaneous burns, once to start the transfer at time $$t_0$$ and another to end it at time $$t$$. This is a boundary value problem. In general, finding a solution to boundary value problem is much harder than finding a solution to an initial value problem. Fortunately, there is an elegant, integration-free algorithm to solve for the delta-v needed to at the start and end of the transfer. As Christopher James Huff mentioned in his answer, this is Lambert's problem.

How do I calculate (b) which maneuver or set of maneuvers I should make to do this optimally?

There is absolutely no guarantee that a random choice regarding $$t_0$$ and $$t$$ were optimal. To the contrary; the odds are very good that a random choice will be highly suboptimal. But now you are asking about optimality as well. There have been papers galore written in the past 350 years about optimization, and papers galore will continue to be published well into the future. The approach used by the Jet Propulsion Laboratory and other organizations to plan missions from Earth to Mars is to pick times $$t_0$$ and $$t$$ from values on a grid.

Each point will have a cost; you mentioned total delta-v as the cost function. There are other cost functions. Dollars, for example, is a the ultimate cost function. Translating delta-v to dollars is nontrivial. So once again, keeping things simple, I'll assume total delta-v as the cost function.

Every $$t_0, t$$ pair will result in multiple $$\Delta v_0, \Delta v$$ solutions to Lambert's problem. Pick the cheapest in terms of total delta-v. Then try another $$t_0, t$$ pair, and another, and another. Do this on a $$t_0, t$$ grid and you'll have a sampling of the cost surface. Projecting this onto the $$t_0, t$$ plane using contour plotting visualization techniques will result in a visual representation. The bulls eyes on that contour plot (the pork chop plot to which Christopher James Huff alluded in his answer) provides a nice representation regarding when it's best to depart and arrive.

What you're looking for is Lambert's problem, which is used both for trajectory design and orbit determination, and to produce porkchop plots. Your hunch that this is not a simple problem is correct. pykep has a solver for Lambert's problem that supports multiple revolutions as well as solvers for various related problems such as low-thrust trajectories.