Maximum Delta-v Between Two Orbits

[Three1415]

I have been wondering about this for a long time: What is the maximum "distance" (i.e, delta-v) that can separate two orbits around the same body? Intuition tells me that there should be some maximal value, almost certainly as a function of the orbital parent's mass/density, that can exist, but I have yet to discover anything definitive on this. Does anyone here happen to know?

[Rakaydos]

I would guess it would be between lowest stable orbit prograde and lowest stable orbit retrograde, or twice your orbital velocity in D/V. Or lowest stable orbit to highest stable orbit, retrograde for the extra few m/s of cost, if that's larger.

That's just a gut feeling though. I'm assuming you mean minimum DV transfers between orbits- A brachistone transfer can burn a LOT of DV in a hurry, on an otherwise easy transfer.

[K^2]

The least upper bound on velocity of an object in orbit with respect to parent body is escape velocity. So the least upper bound on the delta-V is twice the escape velocity. Note that if the delta-V actually is exactly twice the escape velocity, then at least one of the objects is on the escape trajectory, but you can have a delta-V that's less than twice escape velocity by any arbitrarily small amount.

Anyways, the short answer is twice the escape velocity.

[Three1415]

Anyways, the short answer is twice the escape velocity.

That's more or less what I thought, but I was unsure if weird inclinations or orbital planes could mess with it, though it makes sense that they do not (i.e, escape velocity from any orbit around one body will be the same no matter the inclination, anomaly, etc.).

Now for the practical applications: I know exactly how much delta-v to bring to GNBC battles! It also means that someone should tell zekes that his 11 km/s warships are useless...

[Bill Phil]

Between two orbits? None. If the final orbit the craft enters is the target orbit, then it could have gone into many previous orbits after the initial orbit.

[AngelLestat]

The least upper bound on velocity of an object in orbit with respect to parent body is escape velocity. So the least upper bound on the delta-V is twice the escape velocity. Note that if the delta-V actually is exactly twice the escape velocity, then at least one of the objects is on the escape trajectory, but you can have a delta-V that's less than twice escape velocity by any arbitrarily small amount.

Anyways, the short answer is twice the escape velocity.

Ok I dint understand it

Many technical words in English together. But it seems the others get it.

There is not other way to explain this? math, graphs?

internet link?

[Red Iron Crown]

To simplify what K^2 said:

1. You cannot go faster than escape velocity and remain in orbit.

2. The greatest delta-V required to go between two orbits is between two that are just under escape velocity but in opposite directions.

3. Thus the greatest delta-V requirement is 2 x escape velocity.

Escape velocity is a function of the planet's mass and radius of the lowest stable orbit, so you are right in thinking that it is a function of the body's mass and density.

[cantab]

To simplify what K^2 said:

1. You cannot go faster than escape velocity and remain in orbit.

2. The greatest delta-V required to go between two orbits is between two that are just under escape velocity but in opposite directions.

3. Thus the greatest delta-V requirement is 2 x escape velocity.

Escape velocity is a function of the planet's mass and radius of the lowest stable orbit, so you are right in thinking that it is a function of the body's mass and density.

I believe the flaw in this argument is that you are implicitly putting requirements on the transfer other than "most efficient". An orbit just under escape speed is necessarily highly elliptical with an apoapsis arbitrarily distant from the primary, and at an arbitrary distance from the primary it takes a negligible amount of delta-V to change from any closed orbit to any other closed orbit.

If we limit ourselves to transferring between closed stable orbits then intuitively I would expect the most demanding orbital change to then be a 180-degree plane change of the lowest stable orbit. I believe (but have not proven) the most efficient transfer would be a bi-elliptic. That would be a burn to nearly escape speed, an arbitrarily small burn at apoapsis arbitrarily far from the primary, and a circularisation from nearly escape speed. At any given height escape velocity is √2 times circular orbit speed.

Therefore the maximum required delta-V for a transfer between two orbits is 2x(√2-1) or 82.8% of the speed of the lowest stable orbit. Far less than twice the escape speed, which is 2x√2 2.83 times the speed of the lowest stable orbit.

Of course you can use more - this analysis assumes that one you are given your orbits you want to get between them as efficiently as possible. There is also the big caveat that I am assuming an unlimited sphere of influence.

If we do not limit ourselves to closed orbits then of course we can choose two trajectories shooting through the system as fast as we like and have any delta-V requirement to transfer we like. If we do not limit ourselves to stable orbits, and instead allow suborbital trajectories, I am unsure what impact this would have.

PS: By considering the bi-elliptic transfer, rather than the Hohmann transfer, I think you can prove that a 180 degree plane change of the lowest stable orbit is indeed the most demanding orbital change. If either the initial or final orbit is any higher it will have a lower orbital speed and correspondingly a lower escape speed at that altitude, and thus require less delta-V to escape from or enter from escape.

Edited July 28, 2015 by cantab

[Red Iron Crown]

If we limit ourselves to transferring between closed stable orbits then intuitively I would expect the most demanding orbital change to then be a 180-degree plane change of the lowest stable orbit.

It is more demanding to perform a 180 degree plane change at periapsis of an orbit with the minimum safe Pe and an Ap just inside the SoI.

I guess it depends on how "maximum dV" is defined, one can spend an arbitrarily large amount of dV between any two orbits by burning inefficiently.

[AngelLestat]

Ok thanks Red, now I understand it, I was not imagine the case of a retrograde orbit or any kind of inclination change.

[Bill Phil]

To simplify what K^2 said:

1. You cannot go faster than escape velocity and remain in orbit.

2. The greatest delta-V required to go between two orbits is between two that are just under escape velocity but in opposite directions.

3. Thus the greatest delta-V requirement is 2 x escape velocity.

Escape velocity is a function of the planet's mass and radius of the lowest stable orbit, so you are right in thinking that it is a function of the body's mass and density.

Number 1 is technically incorrect. A hyperbolic orbit is still an orbit.

Although, if we want to set up limits, 2* eV is honestly pretty good.

[Red Iron Crown]

You're right, Bill Phil. I should have said "closed orbit".