Walking on a cube-shaped planet
straightdope.com
straightdope.com
"The Highest Possible Level of Development civilization. A gravely injured hermit comes to Trurl's house and tells Trurl of Klapaucius's adventure: Klapaucius wanders across an old robot, who tells him that he has logically deduced the existence of a civilization that reached the highest possible level of development (hence "HPLD"). He has inferred the existence of such a civilization by figuring that if there are different stages of development, there will be one that is the highest. He was then faced with a problem of identifying that one; as he noted, everyone claimed that theirs was the HPLD. Upon much research and thought, he decided that the only way to find it is by looking for a "wonder", i.e. something that has no rational explanation. Eventually Klapaucius discovers one such wonder: a star in the shape of a cube, orbited by a planet also shaped like a cube with the huge letters HPLD written on it."
Now I know how it is like, to be one of HPLD.
You probably know about this already but for the benefit of others I will post this:
http://en.wikipedia.org/wiki/List_of_ships_of_the_Culture_se...
Trurl, attempting to build a machine to write poetry:
I was born in a water moon. Some people, especially its inhabitants, called it a planet, but as it was only a little over two hundred kilometres in diameter, 'moon' seems the more accurate term. The moon was made entirely of water, by which I mean it was a globe that not only had no land, but no rock either, a sphere with no solid core at all, just liquid water, all the way down to the very centre of the globe.
If it had been much bigger the moon would have had a core of ice, for water, though supposedly incompressible, is not entirely so, and will change under extremes of pressure to become ice. (If you are used to living on a planet where ice floats on the surface of water, this seems odd and even wrong, but nevertheless it is the case.) The moon was not quite of a size for an ice core to form, and therefore one could, if one was sufficiently hardy, and adequately proof against the water pressure, make one's way down, through the increasing weight of water above, to the very centre of the moon.
Where a strange thing happened.
For here, at the very centre of this watery globe, there seemed to be no gravity. There was colossal pressure, certainly, pressing in from every side, but one was in effect weightless (on the outside of a planet, moon or other body, watery or not, one is always being pulled towards its centre; once at its centre one is being pulled equally in all directions), and indeed the pressure around one was, for the same reason, not quite as great as one might have expected it to be, given the mass of water that the moon was made up from.
This was, of course,—
http://upload.wikimedia.org/wikipedia/commons/0/08/Phase_dia...
g(r) = G * M(r) / r * r where M(r) is the mass inside a sphere of radius r. That's G (rho 4/3 pi r * r * r) / r * r or 4/3 rho G pi r.
Therefore, pressure = Integral[h=0:200km] rho * (4/3 rho G pi h) dh. This is 2/3 rho * rho G pi h * h, giving 5.6 mega pascals.
Looking at the phase diagram you pointed out, 6MPa is well within the liquid range. (Presumably the temperature would be around 4C). Even with a higher density it's at most about 7MPa, and you need to get to 1 GPa for water to always be a solid.
In other words, unless I did my math wrong or left something out of the calculations, this water sphere isn't close to one of the ice phases of water.
BTW, for more fun, since there's no heat source for the water (radioactive decay, heat from the phase change to ice, etc), then how does it stay liquid? If exposed to vacuum it would turn to vapor until it cooled down enough (about -60C) to freeze. There isn't enough mass to hold down much atmosphere, so I presume it's covered.
How also does it get enough energy to stay liquid? Most of the energy will be dumped into the top layer, so there will be a warm zone on top of a thermocline, like with Earth's oceans. If the energy comes from the sun, then do the poles freeze? If it doesn't rotate fast enough then the backside will freeze.
If ice does form, it reflects more heat than water so that region will stay ice.
Actually, I haven't read The Algebraist yet, but if I had to guess, it's a vaguely magical Culture explanation, like artificial suns surrounding the moon, or industrial processes within giving off enough waste heat to keep it warm, like the Puppeteer homeworld.
By the way, the pressure in the Mariana Trench is about 108 megapascals. The 7MPa I calculated is therefore about the same as 700 meters of depth.
Humans now, with the Atmospheric Diving System, have dived to 610 meters, and free divers have made it to 265m. We could probably make a (nuclear) sub that could cruise through the entire thing. The max depth for a steel sub is 250–400 meters and titanium to 1,000 meters.
A physician is a medical doctor. A physicist studies fundamental laws of our universe.
I think it would quickly lose it's tone of respect, and take on one of sarcasm.
Anyone here willing to take on the task of rendering an image of what this would look like?
So, if the Earth's mass, was all densely concentrated in a, say, 1 mile thick shell, you could drill a hole through the shell and experience total weightlessness when you popped out on the "inside" (assuming a total vacuum on the inside - if not - you'd experience a very small gravity toward the center based on the mass of the contained atmosphere).
Well, at the same rate as air at 1 atmosphere pressure will fill any vacuum container through a hole of that size.
You seem to suggest that fact that that speed somehow depends on the size of the hollow earth and the quantity of the atmosphere.
The pressure depends only on the height of the air column.
Edit: Actually never mind, I think the density of the shell is what threw me off! On the outside the net gravitational effect is the local shell (RHS) plus the rest of the shell (RHS), but just on the inside - the net effect is the remaining shell surface (RHS) minus the local shell (LHS). Because the effect of gravity is 1/r^2 everything works out!
So those giant mountains that comprise the vertexes of the cube would be pulling the oceans toward them. That works against your point. I don't know where the equilibrium between those forces sits, nor do I know how to calculate it.
Of course, this doesn't begin to account for the vectors, and it assumes that the density is uniform. But it should give us a sense of how much of the gravitational pull differs from our simplistic model.
An imagined sphere inside the cube holds 52% of the mass and so the remaining mass is divided among eight vertices, about 6% for each.
If you are off a ways from one of the points listed above, the sphere part (52%) will always pull towards the center. A lot of the force from the corners will be mitigated/cancelled by symmetry, and whatever is left over, again, it's only 6% maximum per corner.
The math is relatively easy with uniform density. Otherwise we have to start conjecturing on the density distribution, and without real-world feedback it degenerates to wondering if an infinite number of angels can dance on the head of a pin.
During the course of the experiment they had to measure the shape of the mountain quite carefully and they had the idea to join points of similar height together with a line on the map - inventing contour lines.
As soon as your test object is non-point you get tidal forces due to the field typically being nonuniform. As soon as your object is not spherically symmetric you get a field that looks nothing like that of a point mass.
I'd guess this is what the OP was recalling.
That doesn't apply to being on the face of a cube, though; that's not nearly far enough away, obviously. :)
I read (long time ago) that the horizon is 29km on a seashore. In other words if you see a ship disappear on a horizon, it was 29km away. And so his 3mi vs my 29km is a bit of a discrepancy. Can anyone set things straight here?
Wikipedia says 3.1 miles.
One family of orbits is parallel to a face of the cube, if I understand the paper even a little.
Correct me if I'm completely imagining things here, but is this how space is 'curved' by gravity? The disorted shape of the ocean and cube would reflect how the cube-planet felt to an observer within its gravitational pull.
Is it then too far-fetched to imagine that some freak process of nature or other might conceivably allow for the creation of a bizarre planet with an immensely dense spherical core and a lighter mantle and crust that take on the shape of a cube externally? Nature is not averse to giving us cubes, after all: http://www.gemstoneslist.com/pyrite.html
It's a torus of air in orbit. The trees look like integral signs because they align pointed toward the star, but have constant wind in opposite directions at either end, because the air there has different orbital speeds, being closer or further from the star.
EDIT: In a mashup of The Culture and Dilbert, a godlike nanotech Dogbert forces the hapless Homo Sapiens Dilbert to work on a "cubical".
http://books.google.com/books/about/Mathenauts.html?id=tHQsA...
Great book, BTW.
And the final moment of the film was merely playing with the idea that the intense gravity of the sun would alter time to the point where Capa would experience each nanosecond before his death, observing the nuclear reaction as it occured.
Pretty cool story, at least thats what I thought.
We walk on a sphere shaped planet, yet we do not "slide down" if we stand on the North Pole, yes? Same with a cubic one, as I imagine, the 90 degrees angle will be crossed without even realizing is there. I imagine we would perceive it as a flat planet.
As you can see, I did not take the whole thing too seriously. And that, of course, diminishes karma. So it goes. Cheers!
Yes, I can picture it as you describe it.
Well, if you throw out ALL the rules....
Not necessarily true. If the cube-shaped world has a cube-shaped moon orbiting at the same period as our moon, it will be sooner transformed into a ball by creepers.