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View submission: Ask Anything Wednesday - Physics, Astronomy, Earth and Planetary Science
For a rocket powered by a fuel with a given specific impulse, is there a maximum size (mass) planet from which the rocket could escape?
My question is a theoretical question, not a practical one. So ignore things like material strength limits for rocket housings, atmospheric drag, etc. Also, I understand that a black hole is inescapable. But do the equations for thrust, escape velocity, etc, impose a much lower limit for a given specific impulse fuel?
Comment by nivlark at 14/09/2022 at 22:54 UTC
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The relevant physics is the Tsiolkovsky rocket equation[1], which says that the fraction of the rocket's mass which must be burnt as fuel exponentially tends towards 1 as the desired change in velocity increases. So theoretically there is no limit, but practically it would become impossible to construct a rocket with a high enough propellant mass fraction while maintaining its structural integrity, let alone having leftover payload capacity.
1: https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation
This[2] article has some more details, and suggests that this practical limit would be reched for a planet with radius about 50% larger than the Earth's.
2: https://www.nasa.gov/mission_pages/station/expeditions/expedition30/tryanny.html
Comment by loki130 at 15/09/2022 at 12:56 UTC
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If your mass ratio ("wet" mass with all fuel loaded / "dry" mass with no fuel) is unlimited, then in principle you can achieve any arbitrary delta-v (capacity to change velocity) for any given specific impulse by increasing the mass ratio (piling on more fuel).
However, if your rocket engine as some nonzero mass and some limited thrust it can achieve per engine mass, then on any given planet's surface there is a maximum mass ratio you can achieve before the rocket doesn't have enough thrust to lift itself and that fuel (for chemical rockets launching from Earth's surface this is generally something like 100 to 200:1 carried mass to engine mass; adding another rocket engine adds more dry mass, so you'd need to add more fuel to get to the same mass ratio, and the result is that this limit--and therefore the maximum achievable delta-v--is independent of scale). And in any moderately realistic scenario, a rocket with only barely enough thrust to lift off would be less efficient at escaping than one with substantially more thrust (generally something like 1.5 times the rocket's weight is ideal) and so need more delta-v, further reducing the maximum mass of planet it can reasonably escape for a given specific impulse and engine thrust/mass ratio.
If, rather than on a planet's surface, you're already in a stable orbit, there is no such requirement; neglecting any other external forces or perturbations, a rocket with arbitrarily small thrust should be able to eventually spiral its way out to escape (much as with ground launch, rockets with lower thrust relative to mass would be less efficient and require more delta-v to escape, but I don't think this would ever prevent escape entirely in this scenario, merely increase the mass ratio necessary to do so). But this is sort of cheating, as being in stable orbit implies that you're already most of the way to escaping.
Finally, if we wanted to get a little practical, fuel must generally be stored in some container, and there are generally limits to the maximum ratio of fuel mass to container mass, and for something like chemical or nuclear thermal rockets, this will usually be less than the maximum ratio of carried mass to engine mass that can be launched on the surface, so you end up with a lower maximum mass ratio. There are some tricks like staging (throwing away fuel tanks once empty and spare rockets once no longer needed to lift the remaining fuel) to partially get around these limits, but they only go so far. For modern chemical rockets, the maximum achievable mass ratio tends to be something like 20 to 40:1