
Torus-Earth - Kutta
http://www.aleph.se/andart/archives/2014/02/torusearth.html
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dzuc
A few years ago I was wondering what this would look like and (less
accurately) modeled it in 3d:
[http://farm3.staticflickr.com/2630/4085959963_8dc8217283_o.j...](http://farm3.staticflickr.com/2630/4085959963_8dc8217283_o.jpg)

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roywiggins
That was fun. If you like this sort of thing, "Heavy Planet"[1] is a good hard
scifi book set on Mesklin, a fictional super-Earth which spins fast enough to
make gravity significantly less at the equator. It's also a good adventure
story, if you like 1950s scifi.

[1]
[http://en.wikipedia.org/wiki/Mesklin](http://en.wikipedia.org/wiki/Mesklin)

~~~
AlanSE
How does this compare to Mission of Gravity? I've had that on my shelf and
haven't got around to it yet. It's a very cool setting, and the author was
writing it before we had all the research on the subject we have today. And he
was correct thankfully!

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gliese1337
He actually wasn't 100% correct. If you get the omnibus "Heavy Planet" edition
(including a sequel novel "Starlight" and some short stories), it is explained
in an appendix that Hal Clement's estimation of the gravitational field was
somewhat off, resulting in erroneous values for the maximum field strength at
the poles.

Still, Hal Clement did a bang-up good job, and the error (or correction there
of) doesn't negatively affect the story at all.

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cossatot
This is cool! But the important question of what mantle convection, and
therefore tectonics, looks like on a toroidal earth is not addressed... for
the 'donut' earth where surface gravity varies by a factor of 3 on the
surface, one could envision subductions systems, possibly double (two-slab)
types, setting up at the northern and southern polar rings, while spreading
systems would set up at the inner and outer equators. I have no idea what
would happen to lighter, more differentiated arc rocks... would they gather in
the middle of a two-sided subduction system, even though that is a
gravitational high? Given the instability of these planets to perturbations,
it's possible that sufficient redistribution of mass due to tectonics and
crustal differentiation would be enough to rip them apart.

Also, setting up a geodynamo would be tough...

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gliese1337
It seems more likely that subduction would happen at the inner equator, and
spreading systems at the outer. This is addressed briefly in the article- when
plates move around the torus, they must compress to fit into the smaller
diameter of the inner equator, and stretch out when moving towards the larger
outer equator.

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srl
(Lest I come across as a complete ass: this is a cool piece, of course.)

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-...](http://physics.stackexchange.com/questions/364/gravity-on-a-
doughnut-shaped-mobius-planet)

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Steuard
I haven't done this calculation, so I'm not going to immediately say you're
wrong, but the example you use (a hollow sphere) doesn't generalize to other
shapes.

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.)

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srl
Dammit! You're right, I'm (very) wrong. (The stackexchange answer I used as a
"check" was wrong too. That'll show me.) I was lazily using Gauss's law and
ignoring the fact that a torus isn't spherically symmetric.

~~~
AlanSE
Could you please point out where a stackexchange answer was wrong on that
question? I'm a heavy user on this site and have written (and continue to
write) a lot on this subject there. Thanks.

~~~
sourkremlin
Item 3 on the answer from the user "Sklivvz♦". It was corrected in a comment.

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aruggirello
IMHO the surface of such a planet should be nothing more than molten lava, as
the outer parts of the torus are probably going to rotate much slower than the
inner parts (much like the rings of Saturn do). This would not allow for any
solid crust to form.

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.

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rcthompson
A torus is not a hollow sphere. The OP's article stated that the computed
surface of the torus was an equipotential surface, meaning that the apparent
gravitational force (a combination of actual gravity and centrifugal force) at
every point on that surface is straight down into the ground. OP even
calculates the strength of the apparent gravity at every point on and around
the torus. Assuming the planet started out in this toroidal shape, it would be
stable since the net force at every location is "straight down", so there
would not be anywhere where pieces would just fly off into space.

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_quasimodo
"perhaps due to an advanced civilization with more aesthetics than sanity" :)

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throwaway_yy2Di
Of course. An advanced civilization with engineering sense wouldn't waste
precious building materials on thousands of kilometers of planetary bedrock.

(And that's to say _nothing_ of that area/volume ratio, what that implies for
radiative heat rejection!)

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sharpneli
And a termite might think that an advanced civilization would never spend tons
of valuable vegetable matter for purely aesthetic reasons.

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'.

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debt
This reminds of something I've seen earlier: what if the Earth had rings like
Saturn? I'm more curious as to how this would affect us culturally; the effect
a toroidal earth would have on our collective consciousness, how we treat each
other, and our evolution.

[https://www.youtube.com/watch?v=UT2sQ7KIQ-E](https://www.youtube.com/watch?v=UT2sQ7KIQ-E)

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johnwatson11218
Not sure if it was in the article or not but I think the Hairy Ball theorem
from Topology would be important. Since earth is a sphere the wind can't blow
all in the same direction. Some currents must oppose others. In a torus the
wind could start blowing in one direction and stay that way forever.

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rtkwe
That's not what the Hairy Ball Theorem (HBT) says, the HBT says that there
must be a point on a sphere where a continuous vector field is zero. It
doesn't say they can't be pointing in the same direction except that point.

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johnwatson11218
this link
[http://uncyclopedia.wikia.com/wiki/Hairy_ball_theorem](http://uncyclopedia.wikia.com/wiki/Hairy_ball_theorem)
talks about the application to the torus. It states that it is possible to
completely comb the hair on a doughnut. That is the point I was trying to make
earlier. I think it does imply that the wind could start blowing in the same
direction if we lived on a torus. I think it would be the same if we lived
inside like in science fiction movies or if we lived on the surface with a
normal atmosphere.

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valtron
I can't tell if you're being sarcastic... but if not, this is the first
legitimate use of uncyclopedia as a source I've seen.

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GhotiFish
if planets are effectively amorphous liquids with no surface tensions at
planetary scales, then why would you see areas on the surface that experience
different amounts of gravitation. The rock of the planet experiences that
gravitation all the same as the surface of the rock, mass experiencing low
gravitation must equilibrate with mass experiencing high gravitation. So what
stops mass experiencing 0.65 units of gravity from displacing mass
experiencing 0.3 units? You would think this would result in somewhat
consistent gravity around the planet. This result is unexpected to me.

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gliese1337
Because "magnitude of the gravitational force" =/= "gravitational potential"
and potential is what matters- the shape is stable if the potential is equal
across the entire surface. This corresponds to the gravitational force vector
always being perpendicular to the surface (i.e., gravity always points
'down'). As long as the gravitational force vector is perpendicular to the
surface, nothing will get pulled sideways, therefore nothing will move,
therefore the shape is statically stable, and the magnitude of the
perpendicular force can vary as much as you like.

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.

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GhotiFish
> 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?"

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!

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moron4hire
I think it is funny to note that the old style of video game map, a square
area where walking over one edge makes you appear at the opposite edge, more
closely maps to a toroid than a sphere.

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quarterto
> more closely maps to a toroid than a sphere

It's the very _definition_ of a torus, at least when we're talking about
fundamental polygons[0].

[0]:
[http://en.wikipedia.org/wiki/Fundamental_polygon](http://en.wikipedia.org/wiki/Fundamental_polygon)

~~~
moron4hire
Yes, I just meant that, if you did wrap a square video game map into a torus,
it would look kind of funny as the middle would get stretched and the top and
bottom edges squashed. But it would certainly all be there, unlike a sphere
that would need parts of the map cut out.

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dognotdog
I do wonder if something like this exists, now. I'm familiar with the Dyson
ring and sphere, but I never thought of a torus. Even though after the sphere,
it seems the most plausible 'mode.'

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disputin
Useful for launches - a space cable car instead of a space elevator.

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dnautics
to be more specific, a funicular. A cable car would take energy to counteract
the lift on one side; a funicular would be very nicely counterbalanced.

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TehCorwiz
Larry Niven posited that a ring-shaped world would be roughly equivalent to a
partial Dyson sphere. Although I seem to recall that his were only habitable
on the inner surface.

Suddenly I'm wondering how much of this applies to toroidal space stations.

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utoku
Unfortunately they are not roughly equivalent. A solid shell or a ring around
a star is not a Dyson sphere. The correct Dyson sphere is composed of many
separate orbiting elements. A solid shell or semi-shell is unstable and is
bound to collide with the star or just wander away. Yes, the Ringworld is
unstable. This was one of Niven's mistakes and misunderstanding of Dyson
spheres.

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tutuca
I wouldn't say a "mistake" he explores that inestability in "Return to
Ringworld"...

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jessaustin
[http://tvtropes.org/pmwiki/pmwiki.php/Main/Retcon](http://tvtropes.org/pmwiki/pmwiki.php/Main/Retcon)

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NAFV_P
Sounds more plausible than this tripe:

[http://en.wikipedia.org/wiki/Hollow_Earth](http://en.wikipedia.org/wiki/Hollow_Earth)

Some idiots still believe it.

