My calculations were assuming that gravity is the only thing holding the planet together, and would work just fine for a completely liquid world, never mind the crumbs floating on top. The only caveat is that I assumed (for lack of any better assumption) that the shrinkage was such that the density increased by the same proportion everywhere.
Taking your suggestion about the only thing holding the planet together being gravity I did the following.
I wrote the equation for the new acceleration as a result of gravity at the new radius as
- gn = g*re[sup]2[/sup]/rn[sup]2[/sup].
gn is new acceleration, re is earth radius, rn is new earth radius, g is present acceleration from gravity at present earth size.
Then I wrote the equation for the angular velocity at the new radius assuming conservation of angular momentum.
- wn = we*re[sup]3[/sup]/rn[sup]3[/sup]
wn is new angular velocity and we is present angular velocity
The central acceleration at the new radius is
- a = wn[sup]2[/sup]*rn
Then I set the acceleration from gravity at the new radius equal to the central acceleration at the new radius.
- g*re[sup]2[/sup]/rn[sup]2[/sup] = wn[sup]2[/sup]*rn
That reduces to
rn = (we[sup]2[/sup]*re[sup]4[/sup]/g)[sup]1/3[/sup]
Then I plugged in the numbers for the constants and solved for rn.
I came up with rn = 597 miles.
wn is 0.2 rpm
gn and a are 1413 ft/sec[sup]2[/sup]
I think you want re[sup]2[/sup]/rn[sup]2[/sup], not cubed. I has dimensions of mass*radius[sup]2[/sup], and the ratio of the omegas will be the ratio of the Is.
Wait, what happens if the black hole has an angular momentum of greater than 1? Why isn’t this allowed, and why wasn’t I informed?
As the esteemed Stranger on a Train so promiscuously implied, a black hole with angular momentum parameter greater than 1 would be a naked singularity. Naked singularities are seriously bad news, and it might even be fair to say that nature abhors them (though you’d have to ask Stephen Hawking and Kip Thorne about that one). Fortunately, there’s no known way to produce a naked singularity in a universe that doesn’t already start with one, so there’s no risk of a black hole accidentally forming super-extreme.
OK, but say you’ve got your rotating massive star. Gravity contracts it…but our calculations show it will have an angular momentum >1. Uh-oh. Why can’t it just contract? Why are we positive that it would have to get rid of all that AM? What mechanism enforces this?
Basically, the interior mass will shrink into a black hole and the mass in the exterior will be spun out, 'cause it can’t contract faster than it gets thrown into an escape orbit, else it’ll be moving faster than light. So, Chronos is right; a naked singularity can’t form naturally under General Relativity (or at least not via gravitational collapse alone).
There’s all sorts of other wacky things going on at the same time, too, like superheavy atomic fusion/nucleosynthesis, frame dragging distortion, et cetera that complicate the simulation, but in the end, the dimensionless parameter Chronos mentions is the limit. Since this is directly in Chronos’ line of research, I’ll defer to him for the ugly details. Besides, naked singularities make me nervous.[sup]1[/sup]
Stranger
[sup]1[/sup]Let’s see if anyone gets that obscure Niven reference.
Makes perfect sense. You can’t spin faster than the speed of light.
According to my copy of the Math Tables from the *You are correct. My eyesight let me down when I looked in the Chemical Rubber Handbook. :smack:
Why won’t relativity kick in and cause an increase in mass rather than exceeding c? An increase in mass will allow conservation of momentum.
And that changes the formula for rn
rn = we[sup]2[/sup]*re[sup]2[/sup]/g
which gives
rn = 13.6 miles
wn = 58 rpm
gn and a = 2.7*10[sup]6[/sup]
We are now in agreement and the OP has his answer, at least for an earth without tensile strength.
However we disagree on your factor of 300 for the ratio of radii. I get only 291. 
I get a peripheral speed of 526 mps for the shrunken earth.
I’m not sure what you mean by your latter statement. The “increase in mass” from acceleration as the mass contracts to a singularity is, in a sense, actually what prevents the parameter from becoming greater than 1. As matter contracts, the rotation of the body speeds up; as that happens, more matter on the perimeter reaches escape velocity (some at a velocity approaching c) and is flung away by interia overcoming the centripetal force applied by gravity. This is a hard limit; gravitational attraction can’t overcome inertia to the point of accelerating the matter beyond c, or at least, not outside the ergosphere, and material inside the ergosphere (but outside the event horizon) will tend to slow the rotation of the black hole until it comes to equilibrium (i.e. the material either escapes or falls into the black hole; orbits within the ergosphere are inherently unstable).
Think of children on a merry-go-round; as they move inward, the rotational inertia decreases, and the rotation rate increases inversely proportional to the square of the radius. Now let’s change the behavior so that the rotation rate increases as an asymptotic function with decreasing radius. So, as more children move inward (thus increasing the rate of rotation) the ones on the outside are flung away amusingly, and the ones in the middle have to hang on tighter. They keep making their way inward, but the more they go in, the faster it rotates and most are flung out. So, only so many kids can be in the middle, and the ones who are the closest, or move the most quickly get their first, while the ones on the perimeter smash into the swing set, the jungle gym, and the fourth grade teacher who is blowing her whistle to stop all of this nonsense. (If it has not already been noted, I didn’t have a particularly happy childhood and frequently amused myself with images right in this line.)
Now, if you could somehow create exotic matter with a negative energy density (gravitationally repulsive) and figure out a way to focus it on a rotating mass so that it is forced into the event horizon, then maybe funky things could happen. Of course, we’re talking about a totally speculative form of matter and there would probably still be some finicky physical limitation that Chronos could then cite which would limit how much force you could apply. Nature’s a bitch, that way; she doesn’t like you to cause her to have a “wardrobe malfuction” with the fabric of her space-time continuum.
Stranger
I haven’t seen any studies of this which involve exotic negative matter, but I have seen some on trying to use normal matter to force a black hole super-extreme. In appropriate units, typical subatomic particles have angular momentum and charge much, much greater than their mass, so if you could just magically add enough without other consequences, you could increase the angular momentum or charge parameters far above 1. Except that there are other consequences. When a black hole is spinning enough or charged enough, it’ll actually repell particles with the same sign of angular momentum or charge, so you’d have to exert a force to push the particle in. In exerting this force, you’d be doing work, and all the work that you do ends up getting added to the mass of the black hole. Increasing the mass of the hole increases the amount of charge and angular momentum that’s allowed, so the resulting final state still isn’t extreme. I imagine that much the same process would occur with negative matter: Even though the matter itself might have negative mass, the energy you’d need to expend to force it in would more than compensate.
Well, if they make you nervous don’t use them to time travel and make duplicates of yourself; problem solved.
I got it. Whee !