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(TFT) Re: TFT Digest V3 #721

DMG2 wrote:


doubling the radius = 1/40 grav      if you are still using center of mass.
(already posted)
There's an extra 0 running around in there. For the same mass planet, doubling the radius (either by inflating the planet somehow, or by moving up above the surface) -> 1/4 grav (not 1/40). Gravity goes like 1/r^2. 


the second post I had written....


But if we make all life 1/40th the size of earth normal....

we would have 160 x the surface area of earth

and...

everyone could walk on water.
??? If you mean surface tension would get a lot more important, this is true. It might be possible to manufacture something like snowshoes, whose users could walk on water by using the surface tension.
If you mean that the buoyancy of an object in water would become much greater than its weight, this is not true. The buoyancy is proportional to the *weight* of the displaced water, not the *mass*. The weight would go down with the local gravity. 


     The neatest by product of this is that objects near the center are
drawn outward.  Just as suggested people will be able to walk along the
inside of the hollow earth.  What a coincidence it is that I happen to be
working in the area planet creation and stable solar systems.  At this point
I can make no informed comment on tides though.


Yeah, cool ideas, physics-wise!

Nocstar wrote:


         If the physics of Cidri allows the surface inhabitants to be
Earth-like in all respects, how would the inhabitants of the core be
affected, assuming of course they could live in the hollow core.
Would their "gravity" be stronger or weaker? I would think it would
be stronger since the "surface of the inner core would be much less
than the outer surface. Should they decide to invade the outer world,
what would our intrepid characters be facing?


Well, that's an interesting question.

DMG2 said:


 Secondly as Newton invented
Calculus to prove, no matter the shape of the body gravity may be assumed to
originate from the center of mass.
Well, sort of. In the far-field approximation, meaning the point at which you want to know gravity is, say, 10 times farther away from the body than the largest dimension of the body, the body looks like a point mass at its center of mass. 

More interestingly, a spherical shell has two interesting properties:
1) For any point *outside* the spherical shell, the spherical shell looks like a point mass at its geometric center.
2) For any point *inside* the spherical shell, there isn't any apparent gravity. The parts of the shell on the "far" side cancel the parts on the "near" side exactly, for *any point* inside the shell.
So, for world-building, one could make a spherical shell world (which would take some *really strong* materials, but anyway...). Say diameter = 10 times that of Earth, mass = 100 times Earth mass. Surface area should also be 100 times that of Earth, but surface gravity would be the same.
Now *inside* the planet, assuming the rocks or whatever are the same density as Earth, 9/10 of the space will be empty, in the form of a big central bubble of zero-gravity. (Mass went up by *100, but volume went up by *1000.)
Ok, but zero gravity isn't that useful. So... we *spin* the planet, at say one rev per day (again, on the surface, that matches Earth). Centrifugal force will mean that stuff on the inside gets pulled away from the axis of spin. There'll be a big belt of ocean on the inside of the equator (if we throw in some water, and air, I mean). Acceleration due to spin will only be about 0.33 m/s, or about 1/30 normal Earth gravity, so characters that manage to get to this spot will be pretty strong compared to natives. Because of the weak acceleration, atmosphere scale height will be taller than on Earth, but at some point in the higher latitudes, the air will be too thin to go farther toward the poles (it'll all have flowed away to the inside of the equator, on top of the water). The poles, on the inside, would be in free-fall.
If those numbers don't work, you can keep scaling up. To keep same surface gravity as Earth, scale mass up like r^2. (Since interior volume scales like r^3, that means you keep getting a bigger and bigger central bubble.) Spin rate stays constant, so centrifugal force (-> "gravity" on the inside of the equator) scales like r. Don't let that get too big, though, because it will *also* apply on the outside of the equator, cancelling some of the real gravity.
Remember that centrifugal force points directly away from the axis of spin (a line), while gravity points directly toward the center of mass (a point). That means if your building materials are strong enough that the planet does not deform, all of the surfaces except the poles and the equator will be "tilted" compared to the local apparent (composite of gravity and centrifugal force) acceleration. If the planet *does* deform (or is built "deformed" to match the planned spin rate) then that effect can be cancelled on the outside surface, but never on the inside. However, then the math gets a *lot* more complicated and I'm not sure the whole thing works.
I guess I leave illumination of the interior as an exercise for the GM. Maybe the Mother of All Light Items is suspended right at the center? 


From: "Dan Tulloh" <dwtulloh61@cox.net>
Subject: Re: (TFT) hollow Cidri

...Even with the idea of 'reversing' gravity, other physics would just
be too wierd.  For example:  the curve of the earth is all wrong -
ships would no longer disappear beyond the horizon and in fact,
there is no horizon.
Right. Due to atmospheric haze, you couldn't see forever, but a long way. However you could probably see just *fine* straight across the planet. Map-makers would have it easy. Maybe there's some reason to have the oceans warm enough that there's always fog or clouds, to shut down infinite lines of sight? 

PvK wrote:


Just like in the middle of the Earth
(or, probably, off some distance because of the curvature of the shell, at
your zero point, with your anti-grav center balanced for 1g at the surface,
there would be zero gravity, and not more than 1g, as you wrote. Gravity
goes steadily down as you get closer to the zero point, because the mass
"over your head" is pulling you up, not down.
See above. For a qualitative proof, try this. For a spherical shell, we are trying to calculate acceleration due to gravitational attraction to the mass of the shell at at arbitrary point inside the shell.
Draw a line containing the arbitrary point and the center of the shell. Because everything is circularly symmetric about that line, any acceleration has to be right along that line, one direction or the other.
Now draw cones of the same half-angle, centered on the line you just drew, apexes at the arbitrary point, extending both ways to the shell. The "far" cone goes farther to reach the shell than the "near" cone.
Now consider the attraction on the arbitrary point due to the mass of the shell included in the bases of the two cones. The "far" cone contains more mass than the near cone, but that mass is farther away. In fact, the distance and the mass are exactly in the 1/r^2 ratio - which means the two cones' base areas attract the arbitrary point in equal and opposite directions - they cancel out exactly.
Making the cones bigger doesn't change this (you use the same argument on each additional increment of base area, so the curvature of the sphere doesn't matter). Eventually the cones are half-angle 90 degrees, meaning the include the whole shell, and they are still cancelling.
OK, now just consider the Earth (or whatever) as the superposition of a whole stack of spherical shells. As you go toward the center, the shells "above" (outside) of your radius from the center quit mattering to you, so the acceleration goes down. (Note, however, that the *pressure* does keep going up! Just not as fast as if all the mass were actually at the center.) At the center, you'd be in free fall (as you'd expect). 


Oh, the other question about living on the inside for me would be, what
would you see? Is the anti-grav also a light source? Does it turn on and
off every 24 hours, with little lights to resemble stars during the
night-time? Or some other optical (or magical) illusion to make it look
like a terrestrial sky with moving stars? Maybe there is a haze to prevent
anyone ever seeing the lights in the sky on the inside, and to prevent them
to seeing the rest of the interior terrain. I'd love to see someone do a 3D
computer render of the inside view in a hollow world that did not have a
haze, so you could see continents and stuff all over the sky. Sounds like
it would be time for a Lovecraftian sanity check for people from the
outside seeing that for the first time...
Very nice! I wrote the above before I read this, but great minds travel in similar gutters. 

Some of what you'd see:
Blue ocean band around the equator. Ocean is deepest at the equator - not many island chains there. 

Rivers run straight to the equator, like lines of longitude, in flat terrain.
Vegetation bands look like lines of latitude. Going toward the poles, weather is colder and air is thinner, like going up mountains.
Mountain ranges maybe reflect ranges on the "outside", like icebergs viewed from below? Dunno, depends on how it was built. Artificial mountains could also deflect rivers, possibly change weather patterns.
At the higher latitudes (and not very high, at that), the air gets thin, and then goes away, so there are stark rocks visible. Maybe some ice, maybe not. Depends on what drives weather. 

Ocean bubbling from the weather-driving heat source?

Central MoALI illuminating things?
I'm with Dan, this is eliciting strong flashbacks to when I read Rendezvous with Rama, not to Lord of the Rings. But whatever floats your boat.. 

PS: DMG2,


 >d := DistanceSqr(p1,p2);

d := d - (p2.corona_radius * p2.corona_radius);
d := 1/d;

 >d := d * p2.mass * GRAVITY_CONSTANT;
Doesn't this cause trouble when DistanceSqr(p1,p2) = (p2.corona_radius)^2? If they cancel exactly, 1/d gets infinite? 

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