This is relevant for the definition of a dwarf planet.
I presume the answer will be, well, if we can tell the mass of the body and guess the material. I don't find this very satisfactory because (1) may be impossible and (2) will have large error.
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$\begingroup$ It's not uncommon for it to be inferred based upon size (or potentially mass) rather than being directly measured by it's adherence to being spherical. $\endgroup$– Mitch GoshornCommented Oct 9, 2014 at 22:24
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$\begingroup$ You don't need to guess the material. At planetary scale, all materials behave more or less like fluids, and the dominant force is gravity. There are no "solids" at that scale. Solid is a small-scale concept. $\endgroup$– Florin AndreiCommented Dec 5, 2014 at 20:02
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1$\begingroup$ @FlorinAndrei: Well, for a borderline drawf planet, it is obviously on a boundary between self-gravitating into a sphere and not. It must be the case that it matters what the material is at that point. $\endgroup$– ThePopMachineCommented Dec 5, 2014 at 20:25
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$\begingroup$ a planetary body does not have to be in hydrostatic equilibrium to be considered geologically differentiated. Near-hydrostatic bodies can be seen as spheroids like the protoplanet 4 Vesta $\endgroup$– Declan KonroydCommented Oct 30, 2015 at 12:14
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$\begingroup$ Nobody appears to understand what the question is about. $\endgroup$– ProfRobCommented Dec 29, 2016 at 8:01
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2 Answers
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I think you're asking, "If we know an object's shape, can we determine if it is in in hydrostatic equilibrium?" If so, one might wonder if astronomers classify basketballs or ball bearings as being in hydrostatic equilibrium, since they're so spherical.
Below about 100 km in radius, the answer is in general no. Given some population of randomly lumpy objects (like asteroids), some of them will be close to the shape of a sphere purely on accident. The composition matters too -- an object of this size made of hydrogen gas would assume a spherical shape from hydrostatic equilibrium, but an object made of rock might not (like Mathilde below). We could make better predictions given detailed knowledge of the materials and environment of the object, but this is not always possible, as you mentioned. For small objects, intermolecular and atomic forces dominate gravity.
Once you get to a certain size of object, it becomes much easier to make a prediction about hydrostatic equilibrium. This is still very much context-dependent, and you still get complications from material composition, temperature, etc. However, atomic binding forces have some fixed strength, but gravity scales like mass. Given ordinary astrophysical materials, we can be very sure that a body like Jupiter is under hydrostatic equilibrium.
You can make some order of magnitude estimates by assuming that the atomic interaction energy must be at least as large as the thermal energy (Hughes and Cole 1995). If you check out equation 5 of that paper, you'll see an explicit expression for a radius that divides spherical objects and nonspherical ones. At some mass, the binding atomic energy becomes dwarfed by the gravitational potential, and you always get a spherical object.
tl;dr -- small objects no, big objects yes, medium objects may require detailed modeling.
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For a celestial body to be a sphere in hydrostatic equilibrium, it needs to be a fluid. Hydrostatic equilibrium does not make sense for solid bodies.
So Earth and Mars are not in hydrostatic equilibrium. It is spherical for the good reason of being a massive body where its own gravity is enough to avoid big irregularities, but it is not supported by (fluid) pressure, but by (solid) incompressibility and material resistance.
On the other hand, Jupiter and the Sun are in hydrostatic equilibrium since the force that avoids them collapsing is actually (fluid) pressure.
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3$\begingroup$ I believe the downvote (not from me) is because at large enough masses and over long enough times spans, even solids act like fluids in this respect. $\endgroup$ Commented Dec 4, 2014 at 15:24
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3$\begingroup$ Also thermal history. A body that is now 'solid' over certain timescales was surely pretty fluid at the time of its formation. After cooling it stayed the way it formed - spherical. $\endgroup$ Commented Dec 4, 2014 at 16:20
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4$\begingroup$ Earth and Mars are very much in hydrostatic equilibruim. Please don't post speculation or opinions as if they are fact, this is not Yahoo Answers. If you don't know much about a topic, then let somebody else who does know come along and answer it. I would have expected better from a user with over 3000 reputation on the site. $\endgroup$ Commented Dec 4, 2014 at 16:40
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6$\begingroup$ According to the definition of a planet adopted by the International Astronomical Union in 2006, planets and dwarf planets are objects that have sufficient gravity to overcome their own rigidity and assume hydrostatic equilibrium. So quite a lot of astrophysicists think Earth and Mars are in hydrostatic equilibrium... The difference between what you term fluid pressure and the incompressibility [sic] of solids seems of no consequence, since the same Physics applies to them both - all that changes is that the equation of state for a solid is much "harder"; not incompressible. $\endgroup$– ProfRobCommented Dec 11, 2014 at 12:02
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3$\begingroup$ I'm not trusting Wikipedia even though I reference it. In university we actually modelled at what point does rock start behaving as a fluid given geological time scales. It is actually not as high as you might expect, just about mountain sized. That is, a large mountain will 'flow' over geological time scales. And this makes sense: this actually describes what we see in nature. The rock is not rigid enough to support its own weight. Becoming actually round under the influence of its own gravity takes on the order of 10^18 KG or so for rock I believe was what we came up with. $\endgroup$ Commented Dec 11, 2014 at 12:47