View Full Version : Hydrostatic Equilibrium Question

2009-Jun-25, 02:11 PM
Sorry, question packed.

I've been thinking (actually had a dream) about a BB. It's a ball of iron or nickel/iron rounded by hydrostatic equilibrium (http://en.wikipedia.org/wiki/Hydrostatic_equilibrium) without a rocky surface.

What’s the minimum diameter of BB? The math appears to be there (above link) but beyond my skill level.

The smallest found so far (including the candidates) is Mimas (http://en.wikipedia.org/wiki/Mimas_(moon)) at about 400km but its density is very low (1/3 of the moon)

I re-read the “rigidity being overcome by gravity” and wondered if iron's rigity would need more mass than crumbled rock. But I can’t swear that’s how it really works.

Could such a thing exists and be too small to have been seen yet? At the same time wouldn’t the gravity affects be the same as something larger and allow for discover even if we couldn’t see it?

If such a thing did exist, it seems most likely be in the Kuiper belt but did iron make it out the far after the last local area super nova? I always thought the heavy metals stayed close in the inner system but I don’t know that for fact.

What would make such a thing? An ejected planet core caused by a collision with another large body like the Earth/Moon event?

What could have smacked it so hard as to get the iron moving out to a Neptunian orbit or beyond?

Taking the math a little farther is it possilbe to compute the mass needed to maintain hydrostatic equilibrium even after a collison meaning the surface is always smooth or does the rounding only happen in a molten state?

Fiery Phoenix
2009-Jun-25, 02:15 PM
I'd want to know the minimum mass required for a celestial body to achieve hydrostatic equilibrium.

2009-Jun-25, 04:52 PM
Excuse me, but what do you mean by "BB?" Big bang? A BB from a BB gun? Ballistic Body? Basic Body? Basaltic Body? What?

It wasn't mentioned in the link you provided.

Speaking of which, did you read this section (http://en.wikipedia.org/wiki/Hydrostatic_equilibrium#Astrophysics)of your link?

Regardless, while Mimas is the smallest round moon in our solar system, with a mean radius of 198 km, it has the same escape velocity as Tethys (http://en.wikipedia.org/wiki/Tethys_(moon)), which has a mean radius of 533 km, more than double that of Mimas. And like Mimas, Tethys has it's own large impact crater.

grant hutchison
2009-Jun-25, 05:48 PM
I'm thinking "ball bearing". Sometimes the name and initialism are applied to a single ball, rather than to the whole bearing assembly.

Grant Hutchison

2009-Jun-25, 06:54 PM
BB as in BB gun - not relative to the question just an easy reference back to it. But ball bearing works too.

2009-Jun-25, 06:56 PM
Regardless, while Mimas is the smallest round moon in our solar system, with a mean radius of 298 km ...

Correction 198km at least per the link sent

2009-Jun-25, 06:59 PM
I'd want to know the minimum mass required for a celestial body to achieve hydrostatic equilibrium.

That sounds good. Thanks for reducing that down.

2009-Jun-25, 07:15 PM
The reason I asked about diameter is; in the dream BB was VERY much smaller than I would have thought with a gravity higher then I would have thought.

Dreams are funny that way.

2009-Jun-25, 10:10 PM
Well there is spherical and almost spherical. Depending on where you draw the line, we could say there are no spherical bodies, not even in ball bearing factories as parts per trillion errors always occur. In hydrostatic equilibrium, celestial bodies, one meter size spheres plus or minus one part per thousand, likely form under some conditions, somewhere in the universe. Perhaps we are missing something in your question?
Any body that is hot enough to be a low viscosity liquid, in freefall should become approximately spherical in hours, unless it is rotating rapidly. Even nanometer size.
For small bodies, surface tension is more important than the bodies own gravity. Perhaps there is an in between size where neither hydrostatic equilibrium nor surface tension are effective in forming the spherical shape.
Heavy metals usually collect in the inner solar system, but sling shot maneuvers and impacts can expel items from the solar system, so there should be a few in the Oort cloud and beyond. Likely rare, but some. Neil

2009-Jun-26, 12:31 AM
Correction 198km at least per the link sent

Corrected - thank you.

Ken G
2009-Jun-28, 04:01 PM
The question is actually very difficult. As asked, I'll bet you can go above any mass and still not have a balance between pressure and gravity. Just imagine a very long rod of iron, of arbitrary length. So it sounds like what you are really asking is, what is the largest ball of iron you can form naturally and have it not be a sphere. But the problem there is, the only way to form solid metal objects is to start with a much larger object, which forms hot, melts, and has the metal sink to the center. That's always going to be a sphere. But then how do you extract it? You have to smash the object open, and then pieces of iron are going to break off. So then the question is, what is the largest chunk of iron you can break off and have it not be a sphere? If the iron was already molten, it may stay molten, and that makes it much easier to make into a sphere. If it started solid, and stays solid, and the piece that breaks off is not spherical, how big can it be and still not be a sphere? That depends on a property of solid iron caled the "bulk modulus", which controls how much stress it can support. It's probably a theoretical calculation that can be done, but I thnk you are on the right track of just looking at Mimas and what is out there. Solid iron would need to be a big larger to form into a sphere, so I'd say, a bit larger than Mimas.

Spaceman Spiff
2009-Jun-30, 06:56 PM
I'll add a few points to the last several in Ken G's post, above.

The behavior of materials under pressure is often plotted on a Stress vs. Strain diagram (http://en.wikipedia.org/wiki/Hooke%27s_law). Even "solid" materials will begin behaving "plastically" under sufficient pressure (the elastic limit (http://en.wikipedia.org/wiki/Plasticity_of_materials) and the material "yield (http://en.wikipedia.org/wiki/Elastic_limit)" are often used to describe this condition), and usually reach an ultimate strength (http://en.wikipedia.org/wiki/Tensile_strength#Typical_tensile_strengths). All of these are related to the Bulk Modulus (http://en.wikipedia.org/wiki/Bulk_modulus) in a manner that depends on the structure of material. The Bulk Modulus is another name for the inverse of a material's "compressibility". All of the above are all measured in units of pressure: N/m2 or Pascals (Pa), or millions of Pa: MPa, billions of Pa: GPa.

Since the OP asked about the concept of hydrostatic equilibrium (http://www.astronomynotes.com/starsun/s7.htm), let's look at what that can tell us about the "roundness" of an object. Solving the equation of hydrostatic equilibrium (http://burro.case.edu/Academics/Astr221/StarPhys/hydrostat.html) for a constant density (rho) sphere (not a terrible approximation for the smallest objects that become spherical) of mass M and radius R, one finds a central pressure of this configuration of:

Pc = (3/8pi) * GM2/R4,

where G is Newton's gravitational constant and pi = 3.1415926535897932...:)

Note that this central pressure required by hydrostatic equilibrium is also proportional to (R * rho)2. I have found that setting this pressure to exceed the ultimate strength (call it K for crushing, also a pressure) of the material (ice, rock, iron) yields an interesting back-of-the-envelope estimate of the minimum diameter D of an object (of a given composition and density rho) beyond which gravitational forces will likely produce a ~spheroid:

D > 170 km * (K/MPa)1/2 * (g cm-3/rho) .

So this minimum radius scales as the square root of the ultimate strength ("crushing" pressure K, in units of MPa), but scales inversely with density (in units of grams per cubic centimeter).

Rough estimates of K (here (http://en.wikipedia.org/wiki/Elastic_limit#Typical_yield_and_ultimate_strengths ) are some examples) and measures of density (rho):
ice: ~4 MPa and about 0.92 g/cc
rock: ~40 MPa and about 2.5 g/cc
iron: ~400 MPa and about 7.8 g/cc

Note: denser materials usually have larger values of K, but it's not a simple relation. So while icy compositions are less dense than rocky ones, they also have a much smaller value of K. Here then are the resulting minimum object diameter to achieve "spheredom" (to two significant figures):

Dmin(ice) ~ 370 km
Dmin(rock) ~ 430 km
Dmin(iron) ~ 440 km

Obviously, rock/ice or rock/iron mixtures will yield different minimum sizes. And as noted by Ken G this limit will depend on the melting history (which depends on mass and composition) and other histories of the object. Mimas is one of the smallest "round" objects and is composed largely of (H2O) ice with some rock, with a density of 1.15 g/cc, and its nearly round dimensions in diameter are: 415 x 394 x 381 km, right in line then with the above estimate. And Ken G's prediction of a slightly larger minimum diameter for iron compositions (than rocky) objects is illustrated here.

Several approximations went into the above estimation, but it gets one into the right ballpark.