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If a rocket ship with limited fuel has fallen into a decades-long elliptical orbit around the Sun, where in the orbit should it fire its engines in order to achieve escape velocity with a minimum of fuel? Likewise, where would the most efficient place in the orbit be to fire the engines in order to stay in-system but with a shorter orbital period?

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    $\begingroup$ Ok funny story, a student of mine is doing something very close to this for a differential equations project due next week. $\endgroup$ – BSteinhurst Apr 5 '17 at 3:55
  • $\begingroup$ This would be better to ask over at Space Exploration, but I guess it is on topic here as well. $\endgroup$ – SE - stop firing the good guys Apr 5 '17 at 5:53
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    $\begingroup$ How are you building a fictional world with this information? Looks like it's just real world physics / space exploration question with no world being built. $\endgroup$ – Mołot Apr 5 '17 at 5:58
  • $\begingroup$ Try it on Kerbal Space Program ! $\endgroup$ – EngelOfChipolata Apr 5 '17 at 7:04
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    $\begingroup$ @MichaelKjörling and the consensus seems to be that these are on-topic if the purpose is to understand real world to built a fictional one - I see no world building here. It's not the real world part that makes it off topic. It's lack of any world building part. $\endgroup$ – Mołot Apr 5 '17 at 11:28
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Fire the engines closer to the Sun for maximum efficiency. This is due to the Oberth effect.

Assume the spacecraft undergoes a burn when it is farther away from the Sun. The expelled propellant will have a certain amount of kinetic and potential energy, $E_p$. If an energy of $\Delta E$ is released during the burn, the spacecraft gains an energy of $\Delta E-E_p$.

Now, if the burn is done when the spacecraft is close to the Sun, $E_p$ is lower because the propellant has a lower potential energy. Therefore, the quantity $\Delta E-E_p$ is higher, and the spacecraft has more energy. This makes it a lot more efficient maneuver for transferring to a variety of orbits - or, in this case, leaving the Solar System entirely.

As SF. pointed out, the Oberth effect holds for all massive bodies. It therefore makes sense that you can use other planets, not just the Sun, to reach a higher orbit or escape velocity.

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    $\begingroup$ Note: you can gain a lot of velocity through gravity assists against planets, and Oberth maneuver near them. So yes, perform the burn near the Sun, but if you get a good opportunity, don't just burn until all the fuel is out, but try to get a fly-by against some big planet, and save some fuel for a burn then. $\endgroup$ – SF. Apr 5 '17 at 9:16
  • $\begingroup$ I can't do the math right now, but wouldn't there be a way to optimize for Oberth effect? Use a short retrograde burn at apogee to lower the perigree and have a more efficient main burn? Just guessing based on bi-elliptic transfers. $\endgroup$ – Matej Lieskovsky Apr 5 '17 at 13:03
  • $\begingroup$ @SF. That's a good point. Edited. $\endgroup$ – HDE 226868 Apr 5 '17 at 13:23
  • $\begingroup$ @MatejLieskovsky Probably, yes, although I don't know the details. $\endgroup$ – HDE 226868 Apr 5 '17 at 13:23
  • $\begingroup$ @MatejLieskovsky: Yes, although Oberth maneuver against the Sun bears the natural inherent limitation on how low you dare to go. But yes, for strongly elliptical orbits, a burn near apoapsis that brings periapsis lower does help. I believe you need initial eccentricity of order of 1:12 for it to be "worth it" (close relation to bielliptic transfer). $\endgroup$ – SF. Apr 5 '17 at 13:34
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As a rule of thumb, in orbital mechanics, actions you take on one side of an orbit take effect on the other side of an orbit.

So, in order to create an infinitely eccentric orbit (i.e. to break free of the Sun), you burn along the velocity vector at the point of closest approach to the Sun (periapsis or perehelion).

Similarly, you can make your orbit smaller by burning against the velocity vector at the same point.

What I've described is the Hohmann Transfer, which is the simplistic two burn system to move from one circular orbit to another, by moving through an elliptic orbit. In this case, you're starting in an elliptic orbit, so you only do the second stage.

There are some other things to consider. There are other types of orbital transfer with different burn characteristics that use less propellant.

Also, if you want to really get fuel efficient, you use a gravitational boost. So, for example, missions to the outer planets like Jupiter and Saturn typically boost down to use Venus, Earth and Mars (if they can) to slingshot themselves to the right orbit. It's counter-intuitive, but it's true.

As a great example of it, someone posted a challenge in the Kerbal Space Program forums to get to the Jupiter Analogue and back using the minimum size of space-craft. Someone used this technique to do it with obscenely little material and their calculations are documented in a spreadsheet.

If that's not enough, there's also the Interplanetary Transport Network, a series of paths through our Solar System using Lagrangian and other such points to really minimise fuel burn, at the cost of travel time.

A famous example of this was a Japanese spacecraft that malfunctioned (I think it was called Hiten: https://en.wikipedia.org/wiki/Hiten), and it still managed to get to the Moon using this technique.

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  • $\begingroup$ About going toward the Sun to move outwards in the solar system: Not really any more counter-intuitive than that, if you are behind another spacecraft in an orbit identical to yours, and you want to get ahead of it, you start out by burning retrograde to decellerate. (This drops you into a lower orbit, resulting in a higher angular velocity, which means that the time to passing through the second spacecraft's nadir is reduced from infinity to some finite value.) Then burn prograde to get back into your original higher orbit, and finally circularize to the appropriate extent. $\endgroup$ – a CVn Apr 5 '17 at 11:17
  • $\begingroup$ If, on the other hand, you burn prograde to begin with, that will take you into a higher orbit, resulting in a lower angular velocity, which means that you will fall further behind the spacecraft that you wanted to overtake. $\endgroup$ – a CVn Apr 5 '17 at 11:19
  • $\begingroup$ @Michael and then there is a realm of "forced orbits" to get really close, and the fun begins. $\endgroup$ – Mołot Apr 5 '17 at 15:14

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