2018-04-08
Heliocentric Helical Hoop World
(IMG) Image
How about putting a hoopworld around a sun? And another, while
we're at it. Having read about megastructures I haven't yet come
by anything like this, which probably means a) I haven't looked
deep enough, or b) it's so ridiculous that nobody has bothered
putting it down, or c) neither. Assuming c) and waiting anyone to
submit evidence for a) or b), here are some properties and problems
of this kind of a world based on a limited grasp on physics.
A Simpler case
A simpler case would be to put a single toroidal planet around the
Sun. Basically a mix of Niven_Ring and a hoopworld .If we wanted
to aim at Earth- like properties, we could set the distance and
rotational period to the same values as those of Earth's (1 au, ie.
1.496×10⁸ km, and a rotation per year), because the mass of the
orbiter is irrelevant.
(IMG) Image
How thick does it have to be to have Earth's gravity? Since the
major radius of our torus is much larger than the minor radius,
let's model it as an infinite cylinder, the gravity_of_which is
$$ g = {2 \lambda G \over x} = {2 \pi r_{minor}^2 \rho G \over
x} $$ where $\lambda$ is density per length, $r_{minor}$ the radius
of the cylinder, and $x$ the distance from the center. At the
surface $x=r_{minor}$, so
$$ g=2 \pi G \rho r_{minor}$$
where $\rho$ is the average density of the cylinder, and $G$ the
gravitational constant. We_get
$$r_{minor} = {g_{Terra} \over {2\pi G}} ≃ 4240 \mathrm{ km} $$
For comparison, the radius of Earth is about 6371 km.
*** Problems with rotation ***
What if we want to have day and night
on this world? How much of a strain would it put on the surface as
the shorter noon side rolls out to face outer space? In absolute
terms the difference_is $$ l_{outer}-l_{inner} \\
= 2\pi(r_{major}+r_{minor})- 2\pi(r_{major}-r_{minor}) \\
= 4\pi r_{minor} \\
= 4\pi 4240 \mathrm{ km}\\
≃ 53280 \mathrm{ km} $$
i...which sounds like a lot, but compared to the major
circumference of the torus, is_only $${\Delta l \over 2\pi
r_{major} }≃ 0.0057 \% $$
...which doesn't sound a lot, as does
5.7 cm per a kilometer, but when you put it in terms of the
Earth's circumference, you_get the equivalent of 2.27 kilometers of
stretching each day on the equator, which is definitely a lot.
Floor would be lava in that world.
I don't know how to wave that away. Given that our hypothetical
architects could put this kind of a world circling a sun in the
first place, they'd probably have a solution. Maybe the crust would
be of a substance that can take deformation in a stable way, maybe
they'd have slices of deformable material every x kilometers along
the torus etc, etc.
How about gyroscopic effects? The tube would want to continue to
roll around its axis while the year goes along, trying to chop the
whole ring into small independent pieces. Maybe. i*** Mass ***
Mass itself is another problem. The total mass of this single hoop
world would be $$\pi r_{minor}^2 \cdot 2\pi r_{major} \cdot \rho$$
which gives 2.93×10^2⁹ kg. There is little hope when looking at
what kind of masses_we_have_at_our_disposal (assuming that the case
would be about the same for other solar systems). You'd need a
star's worth of mass, and to convert it into heavier elements. Cue
magic. Maybe you'd implement the construction in an older universe
where heavier elements would be more abundant.
stevebowers of Orion's_Arm suggests building the ring at a young
star like T Tauri with the accretion disk still in place, and I
feel pretty stupid for not thinking about that on my own.
More Complex Configurations
Letting go of our already overstretched suspension of disbelief,
let's take the scenario further. Two torī orbiting each other like
twin planets, both in a helical arrangement, rotating daily.
Intuitively forming a helix would take some strain off the torus:
instead of having to alternate between the extremes of the daily
perihelion and aphelion (terms to be taken literally), the
circumference would remain nearer constant if a parts of the same
torus would be nearer while other parts would be farther from the
sun. In practice, however, it would be probably more correct to
think of the worlds as liquid and disregard any intuitions about
solid hoops, like having a variable local orbital speed in
different phases of the daily cycle.
Putting aside the problems with seismology, the arrangement itself
seems sound to me. Extremely delicate, but critically stable. Left
alone it would form lumps from the atmospheres, seas, and
ultimately solid masses gathering around even the most
infinitesimal of local mass anomalies. Maybe the torī wouldn't have
liquid cores and maybe seas would have a maximum width to reduce
downright flowing into a string of pearls. If the solid mass could
be kept in check, I assume the atmosphere and seas would play
along.
stevebowers suggested to make the whole thing out of water, which
would address (in part) both the problem of material and
stretching. It would be far easier to burn hydrogen and oxygen into
water than to transmute or haul heavier elements from god knows
where. You could cover the water ring with rafts where
needed. Exotic_phases_of_water or a 'regular' solid core could be
found in the center, as with any waterworld.
How to fight the subtle disturbances in the orbits? Carefully
operated active dampening masses rolling around the surface? This
is beyond me.
The failure of this kind of world, unraveling in time and space,
would provide interesting apocalyptic sci-fi scenarios. A ring of
uneven worlds, sections without air, sections with massive bulging
oceans. A looming chaos in the sky: a catastrophically failing
segment drawing a red line of glowing lava across the Milky Way,
getting closer and closer by the decade. You'd be in a hurry to
sail across that blob of an ocean or traverse the million kilometer
Desert of Very Little Air to buy yourself some time. Or develop a
space program and hop to the other hoop with a bit less destruction
going on. Or build a space station in the exact middle of the two
hoops.
There are some interesting choices to be made with the
configuration of these worlds. They could be asymmetrical (like
Earth and Moon) or Roche-like (with a different profile thanks to a
linear falloff of gravity). There could be 3 of them, or 3 in a
Rocheworld configuration, touching or not. 2 or 3, the individual
rings could rotate in either direction around the sun: the
direction would have no bearing on their mutual rotation around
each other.
Again, strange possibilities for sci-fi authors: imagine 2 tidally
locked hoops going the same direction, with space stations in the
middle, elevators to both worlds. Or different directions, with a
skyhook spinning in the middle on the ecliptic plane, touching down
twice a day along the torī, but returning to the same spot twice a
year. Or almost touching Roche torī with inner edges hurling by,
creating humongous storms and eventually touching, releasing a
fresh new hell on the tortured denizens.
I choose rather arbitrarily 2 locked worlds of identical mass,
orbiting the sun in opposing directions, and each other once per
day. Having a different sky each day makes it more interesting.
Something like this (not to scale):
(HTM) Video
Or this:
(HTM) Video
*** Distance ***
Masses and orbital period known, what
is the distance between them? I'm on dodgy ground here, but here
goes. Please comment if you can fix anything on this post.
$$a_{centripetal}=g_{infinite cylinder}\\ \omega^2 {x\over2} = 2
\pi r^2 \rho { G \over x}\\ \omega^2 x^2 = 4\pi r^2 \rho G\\ x =
\sqrt{4\pi r^2\rho G \over \omega^2} $$ where $x$ is the distance
between the 2 worlds and $\omega$ is the angular velocity (radians
per time). $x/2 $ because the body is rotating around the center of
the masses. This gives 125 300 km, about the third of the distance
between Earth and the Moon (380 995 km). In degrees, the other
torus would_be 3.878° wide (Moon: 0.5167°).
*** Magnetic Field ***
How about a magnetic field? If the torī would have a net charge
and swirl in opposite directions, they would be analogous to
parallel conductors with a current flowing opposite ways. They'd
generate a magnetic field with field lines roughly aligning with
their surfaces. If you took a compass and walked North, you'd keep
going round the world.
(IMG) Image
(IMG) Image
But would they have a net charge? I haven't been able to find a
good answer to whether planets accumulate a charge from the mainly
positive particles of the solar wind, or would any buildup of a
charge attract instantly an equal amount of opposite particles.
Surely there would be an imbalance, but I don't have the slightest
clue if it would be enough to create a magnetic field substantial
enough to shield the surface from radiation. If it did, there
probably wouldn't be any Aurora Borealis, since the field lines
wouldn't pass through the atmosphere, but would go through the
space between the torī. Would there be an analogue of Van Allen
belts, and where? Streaming between the worlds and their outer
rims? Lighting
If you live on the twin-side of your world, the twin will block the
sun at noon, but also make most of your night very bright, shining
at the sky. If you live on the dark side, your sky look pretty much
like a normal planet's sky, with the exception of the horizon
rising into thin shimmering lines in 2 directions.
(IMG) Image
The relative amount of light by latitude looks like this:
(IMG) Image
<-- spacewards twinwards -->
Thanks to the noon 'eclipse' at twin side, it gets slightly less
light in total. About 60° and 300° (60 degrees from the twin side)
gets most light from the additional nightly twin glow.
(Note: I don't have the mathematical muscle to simulate this with
much accuracy, so I just simulated the scene in Blender and
extracted these relative levels from baked texture maps.)
Since there isn't as big of an temperature gradient as with a
spherical planet with incidence angles from 0° to 90° and lighting
is quite homogenous too, there won't be dramatically different
climates. Slightly warmer night glow areas might create a low
pressure zones, creating upward wind that turns twin- and
spacewards and eventually after having cooled, return near the
ground. The differences are so slight that I doubt there would be
as distinct cells of circulation as on Earth.
There wouldn't be an orbital mechanical way to introduce seasons.
Axial tilt makes no sense for a torus world. Nor could you have its
orbit be elliptic to generate seasons: the torus would have to move
faster at perihelion than at aphelion, demanding the whole thing be
made from rubber.
But since we are talking megastructures, you could just have a
swarm of statites dimming the sun for half an year.
What kind of a sky would the hoops have, besides having each other
hanging overhead for those who live in the twinward side? Assuming
similar atmosphere as with Earth, the sky above and in the
direction of the tangent of $r_ {minor}$, but in the direction of
the $r_{major}$ tangent, ie. looking at the world extending to
space, the 'horizon' would seem redish, just like during sunrise
and sunset. This is because there will be more atmosphere for light
to travel through, and higher frequence (blue) light scatters away,
leaving reds. Something like this:
(IMG) Image
Summary
All in all as I looked into this, the option b) (being downright
ridiculous in practice) became more and more apparent. In any case
it's good food for thought and provides - as any megastructure -
interesting scenarios for science fiction, given a small star's
worth of unobtainium and some leeway with what is practically
possible or not.
I would really appreciate corrections, comments, and further ideas
and calculations (for example on weather) on this.