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My co-workers and I had a debate if there is an "orbital speed limit" for a given altitude over a celestial body. According to Kepler's laws it appears you cannot.

My co-worker proposed the following thought experiment to prove you could if you were also thrusting towards the planet. If you had infinite fuel, could you orbit around an arbitrary point in deep space (No gravity effects) with any given orbital radius? For example orbit a point going the nearly speed of light with a 1cm orbital radius as long as you had, and kept adjusting your thrust vector and had enough thrust to arrest and reverse your velocity.

We understand and agree if you turned you engines off you would fly away in a straight line.

However, I am under the impression that at any given speed you have a minimum orbital radius which you could never go under without slowing your orbital velocity down regardless if you are actively changing your thrust vector to achieve impossible natural orbits.

Am I correct?

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    $\begingroup$ Please clean up this question, it is very difficult to understand. For example, " According to Kepler's laws it appears you cannot." - cannot what? $\endgroup$ – Nickolai Feb 25 '15 at 15:13
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    $\begingroup$ Kepler's laws refer to a satellite that is moving only under the influence of gravity. If you posit that you have as much thrust as you want, you can do anything you want. $\endgroup$ – Ross Millikan Feb 25 '15 at 15:31
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No, you are not correct. You could simply thrust towards the point to keep going in a circle at any speed (below c).

However I would recommend that instead of wasting all that fuel, you put enough of that fuel in a separate tank so that you split your mass in half, and put that tank on a rope in the opposite position around the point. The speed of rotation will determine the required strength and therefore mass of the rope, but it will be less than an infinite amount of fuel.

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  • $\begingroup$ Splitting your mass in 3 parts and having 3 ropes make up a triangle might work better for orbiting something solid. I do wonder though: would such a set-up be stable? $\endgroup$ – RomanSt Feb 25 '15 at 13:37
  • $\begingroup$ I've never analyzed that configuration. Just from comparing the excitation modes of diatomic vs. triatomic molecules, the latter are much more complicated. I'd probably go for the simpler system. $\endgroup$ – Mark Adler Feb 25 '15 at 16:49
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    $\begingroup$ @romkyns: Without a central mass, both configurations should be stable, since, with the ropes under tension, they'll both be rigid. With a non-negligible central mass, though, I suspect there'd be similar issues as the well-known gravitational instability of Niven's Ringworld, which could break the symmetry and eventually cause the central mass to hit one of the "orbiting" bodies. $\endgroup$ – Ilmari Karonen Feb 25 '15 at 17:03
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I am under the impression that at any given speed you have a minimum orbital radius which you could never go under without slowing your orbital velocity down.

Your impression is wrong. Since you seem to appreciate thought experiments, let's say it this way; Spin your craft along its axis. Any axis, at any speed you want. Assuming your craft can take the stress (being pulled apart by centrifugal force as it's not a point mass and the farther from its axis of rotation the higher the force in a vector away from it), it will orbit itself at essentially zero orbital altitude at the rotation rate it can still tolerate.

OK, that's cheating, right? But is it? It matches your requirements of infinitesimally small orbital radius at possibly ludicrous speeds at its extreme ends. As long as you can find that unobtainium to build your spacecraft of imagination out of, so it can tolerate shear stress and it won't rip it apart. And you don't even need all that much of propellants to achieve that. You just cant two engines at 90° angle to the craft (or one with two nozzles), each facing in the other direction and let it rip.

My co-worker proposed the following though experiment to prove you could if you were also thrusting towards the planet.

So here's another thought experiment. What in the spin I describe, that cheating self-orbit one, is different from what we'd usually consider an orbit? Can you now see what your co-worker proposed with continuous thrust towards the orbital focus? It's essentially keeping things together, replacing strength of atomic bonds of our spacecraft of imagination's materials at micro scale with macro scale propulsion of imagination, counteracting centrifugal force that is the product of your mass and acceleration, which in our case is v2/r. As long as the thrust towards focus negates centrifugal force, you can orbit at any speed below c. Your co-worker is correct.

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With sufficient thrust you can achieve an arbitrary speed at an arbitrary radius. The magnitude of the required thrust, in the direction of the centre point of the orbit, is given by: $$ F = \frac{mv^2}{r}, $$ $m$ being the mass of the spacecraft, $v$ being the speed of the spacecraft and $r$ being the radius of the orbit.

Note that the more reaction mass you have, the larger the thrust, the more reaction mass you have to use. Which is part of the fundamental problems with rockets. Most of the fuel is used to get the fuel in orbit (or in this case, to keep it in orbit).

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Yes, you're right. The other answers already noted that you may need a lot of fuel. That's an understatement. You've got the tyranny of the rocket equation working against you, with the exponential factor. Just for a single orbit at speed v, you need to a delta-v of 2*pi*v.

In particular, this means that you may need an initial weight (including fuel) just to complete one orbit that would cause your whole ship to turn into a black hole. I'd call that a failure.

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    $\begingroup$ OP specified infinite fuel. $\endgroup$ – Russell Borogove Feb 25 '15 at 16:30
  • $\begingroup$ @RussellBorogove: Yes? Regardless of fuel, you still need reaction mass. Obviously an infinite amount of fuel has an infinite mass, which clearly is more than sufficient to form a black hole. And if you fall for the "let's ignore all the reasons why X is impossible", then obviously everything is possible making the question a priori pointless. $\endgroup$ – MSalters Feb 26 '15 at 17:27
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    $\begingroup$ @MSalters No, a question that explicitly ignores one impossibility is not pointless. Saying "X is impossible" when that's obvious doesn't help anyone - the OP doesn't think infinite fuel is possible, but his question is not related to fuel. This really isn't an answer to the question. $\endgroup$ – cpast Feb 27 '15 at 2:09
  • $\begingroup$ Not trying to rub anything in but source of propulsion (not necessarily involving any reaction mass at all) could also be external, for example solar sails, beam-powered propulsion, Bussard ramjet, mass driver rings, magsails,... $\endgroup$ – TildalWave Mar 7 '15 at 19:49

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