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Is there a self-rounding object in our solar system whose mass is insufficient to prevent the highest jumping human from escaping its gravity?

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    $\begingroup$ "Jump into space" and "escaping its gravity" are two completely different things. 12 humans have jumped into space from the surface of the moon, repeatedly. $\endgroup$ – Russell Borogove Jan 22 at 22:07
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    $\begingroup$ @RussellBorogove for degenerate definitions of "space." $\endgroup$ – Carl Witthoft Jan 23 at 15:20
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    $\begingroup$ Why limit it to self-rounded or even rounded bodies? As the xkcd commented below points out, there's a question both of mass and diameter. And as a nit: 99% of all Olympians are crappy jumpers. They excel in other athletic activities. $\endgroup$ – Carl Witthoft Jan 23 at 15:23
  • $\begingroup$ @CarlWitthoft Do you have an alternative definition for where space begins relative to the moon’s surface? $\endgroup$ – Russell Borogove Jan 23 at 20:38
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No. Saturn's moon Mimas is the smallest body in the solar system known to be rounded through self-gravitation, and it still has a surface escape velocity of 159 m/s, far above the speed achievable by the best human athletes.

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    $\begingroup$ Yep, though you could jump about 153x higher than you can on Earth, assuming you put the same energy into your jump. I wonder what that would be like to do, even if it doesn't get you to leave altogether. $\endgroup$ – The_Sympathizer Jan 23 at 4:13
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    $\begingroup$ Sadly the needed delta-v is so large (about 150m/s) that even making the Olympian into a shot putting rocket during the long jump wouldn't help. I guess our best bet is getting jet packing enrolled as a sport in Olympics. Or, what if we stacked a bunch of athletes on top of each other and each pushed up the next? $\endgroup$ – jpa Jan 23 at 8:17
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    $\begingroup$ @jpa The image of a multi-stage rocket made out of people is something I'd never had the opportunity to think about before, and now, cannot get out of my head. $\endgroup$ – notovny Jan 23 at 15:11
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    $\begingroup$ Obligatory xkcd $\endgroup$ – LordHieros Jan 23 at 15:13
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    $\begingroup$ Off-topic, but I'm intrigued by the similarity in escape velocity (around 200 m/s) and name with the KSP moon Minmus. $\endgroup$ – Skyler Jan 23 at 18:37
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Mimas is the smallest known self-rounding body, and we've already asked: Could a human jump off Mimas without return? The answer is no.

But we all want to answer to be yes, so what if we drop the "jumping" requirement, and just ask if a human could escape a self-rounding body with only human power?

The surface escape velocity of Mimas is around 159 m/s, and the surface velocity at the equator is about 15 m/s. Let's assume a human plus necessary life support equipment is 200 pounds: how much energy is required to accelerate 200 pounds to (159 - 15) meters per second?

$$ 1/2 \times 200\:\mathrm{lbs} \times ((159 - 15)\:\mathrm{m/s})^2 = 640.6\:\mathrm{kJ} $$

That's not too much! An olympic competitor can produce 200 watts on a bicycle for hours, so at that power how long would it take to generate 640.6 kJ?

$$ {640.6\:\mathrm{kJ} \over 200\:\mathrm W} = 4703\:\mathrm s $$

or, about 1 hour and 19 minutes. Totally feasible, even if it takes twice as long after inefficiencies!

So while a human may not be able to jump off a self-rounding body, it would totally be feasible for a human to escape Mimas given some device which could store human-generated power over a couple hours and then release it in a short burst, like a space-grade catapult.

Would the acceleration be survivable? A very detailed survey of the literature tells me humans can survive 40 g's of acceleration (through they won't stay conscious for very long at that). But fortunately at that acceleration, reaching escape velocity takes only 0.37 seconds. Unpleasant for sure, but feasible!

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    $\begingroup$ How about a fancy velodrome? With no air to slow them down the top speed for a bicycle could be a lot higher than on Earth. The athlete pedals around and around it building up speed until they reach 160 meters per second, then go for the exit. $\endgroup$ – Barry Haworth Jan 24 at 1:54
  • $\begingroup$ Does it have to be 40g? How about 4g over 3.7 seconds, would those 3.7 seconds really matter given the low 0.064 m/s² surface gravity? $\endgroup$ – anrieff Jan 24 at 12:28
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    $\begingroup$ @anrieff it doesn't have to be 40g, but think of the distance required to accelerate to 330 mph over 3.7 seconds. Sounds more like a drag strip than a catapult, and I think the catapult is much more fun. $\endgroup$ – Phil Frost Jan 24 at 16:05
  • $\begingroup$ @PhilFrost, indeed :) $\endgroup$ – anrieff Jan 24 at 17:51
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Phobos. If you agree to strech the "self-rounding" part enough to include it.

Because of its rather complex form and composition, there are points over its surface where the escape velocity is below the average human's running speed.

At Deimos (even less round, but still...) you even don't have to look for a special place.

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    $\begingroup$ Good point about the average human running speed.It is not about how high you can jump, but how fast you can run. $\endgroup$ – Maxter Jan 23 at 18:01
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    $\begingroup$ @Maxter Humans utilize gravity while running, I doubt that the running speed on Earth is comparable to running speed on Deimos. $\endgroup$ – JiK Jan 23 at 18:29
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    $\begingroup$ That would make for a great new question. I think it could even be higher on Deimos. $\endgroup$ – Maxter Jan 23 at 18:32
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    $\begingroup$ Re which specific points on Phobos the escape velocity is below the average human's running speed? Presume it has mountains or something. SpaceEx.SE: Would it be possible to “fall off” Phobos? $\endgroup$ – smci Jan 23 at 19:07
  • $\begingroup$ @Maxter running is a sequence of jumps anyway. The less gravity, the more prominent become jumps (see Apollo videos or just try to walk underwater) $\endgroup$ – fraxinus Jan 24 at 9:15
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Almost certainly, yes, although no such body has been identified.

Normally to self-round a body needs to be far too big for a human to jump off. However, there's another possibility--a body that melted. Consider a very dirty sun-grazing comet. The ices burn off, but suppose it goes so close that the rocks themselves experience surface melting. (The pass will be too fast to melt all the way through.) High points melt and flow down. After many passes you'll get something that is basically round. The smaller the body the faster it will be rounded this way.

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    $\begingroup$ The "melting" poing needs some thinking. You cannot have liquid without pressure (read: atmosphere). At low enough pressure (space vacuum is pretty much enough) solids skip the liquid phase and sublimate. $\endgroup$ – fraxinus Jan 24 at 9:18
  • $\begingroup$ In fact there are several issues with your proposition: as already written a small amount of ice close enough to the sun would have sublimated quite rapidly instead of melting. Furthermore, it is very difficult to keep a round shape if the body is not massive enough. You need to reach hydrostatic equilibrium which requires a certain mass. Even Mimas hasn't reached it. It looks round but it isn't actually. see en.wikipedia.org/wiki/Hydrostatic_equilibrium#Planetary_geology $\endgroup$ – C.Champagne Jan 24 at 16:40
  • $\begingroup$ @fraxinus The low temperature liquids have substantial vapor pressures, but I'm talking about the heavy stuff. $\endgroup$ – Loren Pechtel Jan 25 at 5:15

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