Understanding interplanetary fuel efficiency

[jBar]

I was wondering why it is more fuel efficient to plot a maneuver so that my periapsis is as close as possible to another planet or moon when doing interplanetary transfers. Shouldn't it be more efficient to burn at a farther away periapsis such as when I burn farther away from the planet to adjust my orbit? It seems to me that by making my periapsis closer, the increase of velocity due to gravity would require more retrograde burning to cancel out the velocity. Obviously a close periapsis is good for aerobraking, but what about planets without an atmosphere?

Edited June 11, 2013 by jBar

[Tank Buddy]

Well, that's something called the Oberth Effect, which states a rocket traveling at a higher velocity has more useful energy than a rocket traveling at a lower velocity. For example, it is easier to raise your apoapsis from the periapsis, where you are traveling faster. In your case, it is easier to change your trajectory farther away from your target. Think of it like this: there are two lines that form an angle, those lines form a 10 degree angle. The further away from the vertex, the further apart the lines get. This is similar in orbital mechanics; if you adjust you inclination by x degrees, the further away you get from that point the further off from the original trajectory you will be. But say you wait until you are about to reach the SOI of a planet, but you want to change your inclination. You will have to put in more energy to achieve a similar result because you have to change drastically and quickly, versus gradually and slowly. If you venture out to planets further from Kerbol that Kerbin, you will not be intercepting the planet at your periapsis, and vice versa for planets lower than kerbin. If you plan to aero brake in Duna, Eve, Laythe, or Jool, it is best to push you orbit very close to the atmosphere or just inside of it. This way you will have to use less energy to change your trajectory while traveling at interplanetary speeds. If you want to get captured in a world without an atmosphere, it would be better to be further away because the planet's gravity would be accelerating you less.

[RoboRay]

The Oberth effect is derived from the kinetic energy of your fuel. The lower you are, the faster you are going, and the more kinetic energy you have.

Energy can't be destroyed, so when you burn the fuel, what happens to the kinetic energy it possesses? It gets transferred to your craft, giving you a free speed boost.

[tavert]

Oberth Effect. Escape or capture burns are changing your orbital energy, and it's best to change your orbital energy when most of that energy is kinetic, not potential, meaning low periapsis. Kinetic energy is a function of speed squared, so the slope of orbital energy versus delta-V is larger for higher speeds.

It does however depend what you're planning on doing once you get there. If you aren't landing on the planet, just hanging out in orbit and you don't care what orbit, then you might be able to save some fuel by capturing into a circular orbit just barely inside the SoI. But if you plan on landing, dropping that orbit down to the surface will be expensive (or rather the landing itself will be expensive since you're falling so far), you're better off capturing into a very low periapsis. If you have a mothership or fuel tank or something that doesn't need to land, you can capture into a highly elliptical orbit, with very low periapsis and very high apoapsis (just inside the SoI), leave the mothership / fuel tank there and drop the rest of the way with the lander on its own. Your lander will need more fuel if you do this though, to catch back up the the high mothership orbit if you need to rendezvous after landing.

Edited June 11, 2013 by tavert

[Frederf]

I like to think of Oberth as the efficiency of accelerating while fast.

If kinetic energy is proportional to v^2 (and it is) then the energy by going from speed v to v+1 depends on the value of v. Going from speed 999 to 1000 gives double the energy as going from speed 499 to 500. Try the math. Compare 2^2 - 1^2 and 3^2 - 2^2 and 4^2 - 3^2 and so on. However both take one deltaV worth of fuel so if you're going to spend fuel you want to be going fast when you do. The best way to go fast is to jump off a tall building which in orbital terms means to fall as close as possible to a massive body, converting the potential energy of gravity into speed. Being high and slow does you no good since the potential energy of being up high doesn't affect your fuel efficiency.

The reverse is true. Scrubbing off kinetic energy is best done fast because every quantity of reduction in speed equates to more kinetic energy lost. It may seem like there's less speed to lose at a higher orbit but even at 0 m/s you still have all that potential energy which is going to turn into KE if you try to land. If come in with 1000m/s excess orbital energy and you don't care about landing then a high orbit is fine. You still have to blow off some steam though so it's better to flyby the target close, burn just so you are captured and then let yourself drift to outer orbit before circularizing. If you can burn while fast it's always better.

[Bunsen]

If you want to change the inclination of your orbit, lower velocity is always better. Less total v means less delta-v for a given change in angle.

If you want to change your total energy, then higher velocity is better, as Herr Oberth tells us. I think it's easier to see how it works by focusing on the energy of your rocket, though: Kinetic energy goes as the square of speed, so a small change in speed produces a change in energy that's proportional to your initial speed. If you're familiar with calculus this is obvious; if not then take a look at the graph on http://www.wolframalpha.com/input/?i=x^2 and notice that the slope (i.e. change in y per unit change in x) gets steeper when x gets bigger.

So if you're coming in hot as hell from an interplanetary transfer and trying to make a capture burn without wasting precious fuel, your goal is to reduce your total energy enough that you don't escape the planet's SOI again. You can do that most efficiently with a course that takes you as close as possible to the planet, so your limited delta-v buys you as much delta-E as possible.

Edit: Um, yeah, what Frederf said while I was typing.

Edited June 11, 2013 by Bunsen

[jBar]

That actually made sense. Thanks so much. This game makes me want to take physics again. I accidentally posted the same question twice since I haven't posted before sorry.

[hansman1982]

I believe, from my unscientific KSP observations, it depends on what you want to do.

Go faster - best done when you are going fast. Easier to raise your apoapsis at the periapsis.

Go slower - best done when you are going slow. Easier to lower you periapsis at the apoapsis.

Change direction - best done as far away as possible.

[tavert]

It looks like a mod merged my answer into this thread, thanks

Go slower - best done when you are going slow. Easier to lower you periapsis at the apoapsis.

There are several ways in which you might want to go slower though. If you're on a hyperbolic trajectory, you have no apoapsis (mathematically it's negative, but that's a bit confusing so ignore that), the best place to slow down from a hyperbolic trajectory into a captured elliptical orbit is at periapsis (for reasons described by several different people above).

[hansman1982]

... the best place to slow down from a hyperbolic trajectory into a captured elliptical orbit is at periapsis (for reasons described by several different people above).

I guess I shouldn't take for granted that is a given.