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Is there an upper limit to the speed you can be traveling to obtain any benefit from the Oberth effect? I'm writing an SF novel and I have two fleets chasing each other around a star at high speed. Since the leading fleet's only objective is to escape the solar system, is there any point in using the Oberth effect if they are already at extremely high speed (e.g. 5 % of light speed)? Remember, their only goal is to escape the solar system; trajectory is not important (obviously it will be a hyperbolic pass). Assume they have a maximum accel of 300g, no real limit on burn time, and can manage the heat. And yes, it's an unrealistic problem, folks. I just like to try and make my books as believable as possible.

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In most star systems, if you can burn forever at 300 gees, orbital mechanics are things that happen to other people.

Coasting into the Solar System with a hyperbolic excess velocity of about $15\,000 \mathrm{km/s}$, without burning any fuel, you will only gain about $11 \mathrm{km/s}$ before hitting the surface of the Sun.

And your spacecraft can burn at 300 gees the whole way.

With these features, there is basically no point in doing a powered gravity assist on anything that isn't a stellar remnant, and then you have to start dealing with everything that general relativity brings to the table.

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The Oberth effect is a property of the chemical rockets we use. When a spacecraft is on a highly elliptical orbit, the difference in speed between the highest point (apoapsis) and lowest point (periapsis) is extremely large, with the largest velocity occuring at the periapsis. It turns out there are significant gains in chemical rocket efficiency if you use the engines at this point rather than at the slower speeds. Long story short, when you are traveling at the faster speed the chemicals you are shooting out the back of your rocket also have a higher kinetic energy and you can steal some of that to increase the rockets delta v. It's roughly analogous to how you'd like to launch rockets near the equator because the planet is moving the fastest along that line.

What this means is that depending on the type of technology your spacecraft are using, they might not benefit from it at all. But besides that, also notice that the benefit comes from the speedup that occurs at the the periapsis of an orbit. When you are already traveling at extremely high speeds the speedup from falling closer to the gravity source is negligible as is the time spent near the periapsis.

TLDR: 0 to near 0 benefit for futuristic spacecraft darting around the galaxy.

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