Still, Hal Clement did a bang-up good job, and the error (or correction there of) doesn't negatively affect the story at all.
Also, setting up a geodynamo would be tough...
At least the discussion of gravitation is wrong. Somewhat unintuitively, the gravitational force on the inner part of the torus (the surface closest to the center, on the plane of the axes) is 0.
The pathological example is a hollow sphere of dense material. Outside the sphere, it "looks like" (if you just measure g) a solid planet. Inside the sphere, there is no gravitational field whatsoever, no matter how close you come to the surface.
Surprisingly, this holds no matter how large the sphere is. Suppose you're sitting on the inside of the surface of the sphere, and you decrease the radius a bit. Now the gravitational pull /away/ from this surface decreases like r^-2, so you would expect the gravity towards the surface (which is essentially unchanged) to increase. The issue is that the amount of material opposite you - the surface area of a sphere, really - also increases like r^2. (This is informal, but the best I can do for an intuitive explanation.)
Tough to find a good explanation of this online. http://physics.stackexchange.com/questions/364/gravity-on-a-...
Gravity is zero inside a hollow sphere or inside a hollow infinite cylinder. Here's the conceptual reason why. Imagine that you're near the left side of the sphere or cylinder. Then the mass to your left is closer to you, and therefore every kg of it exerts a strong gravitational pull (as force is proportional to 1/r^2). But although the mass to your right is farther away and thus exerts less gravitational force per kg, there's a lot more of it: roughly speaking, the amount of mass in a given direction is proportional to r^2 (times the solid angle it subtends). This factor of r^2 precisely cancels out the effects of the 1/r^2 weaker force per kg. (And yes, a careful derivation using calculus or Gauss's law would be more rigorous.)
None of that applies in the case of a torus. If I'm in the hole of the torus close to the left side, there's still a 1/r^2 difference in gravitational force per kg favoring the pull toward the left. But now, roughly speaking, the extra mass to my right is only greater by a factor of r (coming from the circumference of a circle rather than the surface area of a sphere). So the far side of the torus becomes less and less significant the father away it gets.
On top of all that, my impression from the article was that this sort of planet would require a really fast rate of rotation. I get the sense that the virtual "centrifugal" force plays a major role in the physics here. (Or if you prefer, that it's not reliable to ignore the effects of being in an accelerated reference frame.)
I find it difficult to believe that the author made such a glaring mistake, even after doing the simulations and all. Also, he says " the surface is an equipotential surface, so gravity (plus the centrifugal correction) is always perpendicular to it. " The spinning is the reason this ridiculous configuration is at least theoretically stable.
From my reading, the author's simulations presupposed what s/he believed the gravitational field is. (This is actually a reasonable thing to do, and I don't fault him for it.)
I agree with srl: gravitation must get gradually closer to zero in the inner part, where the molten lava, not being pulled down by any gravity, continually explodes due to the pressure of the internal gases, forming a fuzzy ring of (cooling?) debris, a gradually spreading chaotic mix of rocks, the greater part going towards the torus' real centre of gravity, leaving underlying molten rock exposed to follow their fate later on - in short a crumbling effect that cannot be arrested. This rocky chaos may later start collapsing in the middle of the hole, forming the seed of the future planet.
BTW any large enough asteroid impact would also spell doom to the whole torus planet and either wreak havoc, rip it apart (in interesting ways, depending on location and direction of the impact) or cause the central "hole" to collapse sooner, finally reducing the planet into a more spherical shape - enter the Torusmageddon :)
Just my two cents.
It's fun to speculate just how bizarre results you could get if you had a civilization with enough resources to do planetary scale arts projects just for the heck of it...
(And that's to say nothing of that area/volume ratio, what that implies for radiative heat rejection!)
It's all about the relative abilities. It's not completely unconcievable that some might transform planets the way we build funny houses, even if they are 'waste of precious building materials'.
NB I'm sure some Culture ships have probably built planets in idle moments - just for the hell of it.
> "there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one p such that f(p) = 0"
Which means they can all blow in a continuous path except for at least one point, so the wind at every point but 2 could be blowing east on a sphere.
Now, you may be thinking "but shouldn't the bits of planet-fluid under stronger gravity sink and push on the fluid around them and thus indirectly cause stuff to move around to the sides?" This is where equal potential really becomes important. For one bit of the planet to sink (losing potential energy), another bit of the planet would have to rise (gaining potential energy). If the shape of the planet currently conforms to an equipotential surface, any marginal redistribution of which bits are up and which bits are down will end up requiring at least as much total energy as the original configuration. Thus, the shape is a local optimum in the energy configuration space, and nothing will move.
that was what I was thinking.
So the reason we see gradients is because areas that are experiencing low force have lots of mass, and areas experiencing high force have little mass.
Looking at the diagrams, I guess I can see that is the case.
Thanks for clearing that up!
The Earth is highly plastic ... on geological timescales. The continents move, the Himalaya are rising at the rate of about 5 mm per year, and one theory of the post-Messinian flooding of the Mediterranean basin (https://en.wikipedia.org/wiki/Post-Messinian_flood_of_the_Me...) is that a fragment of one of the colliding plates in the region broke off and sank into the lithosphere.
The whole system is dynamic, and driven by both latent gravitational heat of formation and by radioactive decay (including postulated naturally occurring fission reactors within the outer core / inner mantel: http://www.nature.com/news/2008/080515/full/news.2008.822.ht... -- a fission reactor near the surface was found in Gabon, within Africa, active about 1.7 billion years ago: https://en.wikipedia.org/wiki/Natural_nuclear_fission_reacto...).
But it's an interesting rock we crawl about.
It's the very definition of a torus, at least when we're talking about fundamental polygons.
Suddenly I'm wondering how much of this applies to toroidal space stations.
For another take on mega-scale artificial worlds there is Alderson disk:
Such a small star would probably have to be kept burning artificially, as the smallest mass still allowing nuclear fusion is around 80 times the mass of Jupiter.
A space station that rotates for artificial gravity is completely different. In fact, its gravity is so little that it doesn't matter for any of the structural engineering. It can be completely neglected. The difference in mass between the two proposals is huge, by a factor of like 10^12.
Some idiots still believe it.