Re: Insanely Silly Rationalization Time!

"Andrew Plotkin" <erkyrath@xxxxxxxxxx> wrote in message
news:e72pri$4j5$1@xxxxxxxxxxxxxxxxxxxx
Here, Logan Kearsley <chrono.surfer@xxxxxxxxxxx> wrote:
Alrighty, I'm trying to work out some basic laws of physics for a
sort-of-fantasy / sort-of-science-fiction universe I just came up with.
Among several strange and interesting differences from our universe,
this
one features flat planets with an edge that can be walked off of. I was
figuring on making this work by instituting a preferred direction for
gravity, with the force exerted between two particles being dependent on
the
angle between the line between the particles and the universal vertical,
so
if you're directly above or below something, gravity works you get a
strong
attraction, but if you're directly off to one side, there's no
attraction.

That ought to work out great for giving me large, slowly rotating flat
planets, but for two problems: first, could they hold an atmosphere, or
would there always be too much leakage around the edges?

I think it wouldn't be leakage; it would be a hurricane. You've got
air at (one hopes) 1 atmosphere, but nothing is holding that pressure
in around the edges. It would go whoosh and be gone.

If you play with the force equations some, you might get a result
where the leaked air is still bound to the overall plate-planet. My
first thought was that this would lead to horizontal double-tornadoes
around the plate's perimeter... but no, that's a perpetual motion
motion. (I *hope* you keep your silly-gravity force conservative!)

Try adding a trace of Newtonian gravity to your linear-gravity. That
should produce a plate-planet inside a very oblate spheroid of
atmosphere. Hm. Oblate? Now I'm not sure.

This would be a great adventure setting, in fact. The atmosphere
pancake would extend for thousands of miles, and would be nearly
zero-G -- you'd only need a trace of thrust (or lift-gas) to keep from
drifting in towards the planet. Sky-pirates ahoy! Of course, if you
cut in past the rim of the plate, you're suddenly plummeting...

I've played around a bit with a gravity equation of g=GMcos(theta)/r^2, and
here's what I've come up with:
Breaking it up into horizontal and vertical components,
gh=GMcos(theta)sin(theta)/r^2 & gv=GM(cos(theta)/r)^2

Letting z = universal vertical and x = universal horizontal, theta =
arctan(x/z) & r = sqrt(x^2+z^2):
g=GMz/(x^2+z^2)^(3/2)
gh=GMzx/(x^2+z^2)^2
gv=GMz^2/(x^2+z^2)^2

If we integrate gh across the surface of a disk to find the gravitational
field along the axis, we get
g = PiGz^2p[1/z^2-1/(z^2+r^2)], where p is the area density of the disk and
r is the radius, or
g = GM(z/r)^2[1/z^2-1/(z^2+r^2)]
The graph looks similar to that of the field on the axis of a disk in our
universe- nearly constant near the surface, then approaching an inverse
square farther away. Near the disk, it's less than the inverse square law,
but quickly becomes larger.
Off-axis, the field will be smaller, but not by much, except very near the
edge. The direction-dependence means that the influence of bits of mass in
the plate drops of very rapidly with horizontal distance, so you can
approximate the field at any point by just taking a circular slice out of
the disk around the point with r >> z and calculating the field from that.
So, the plate-planets in this universe ought to be able to hold an
atmosphere better than sphere-planets in our universe in the vertical
direction. But we already expected that.

If I wanted to be really rigorous about showing that a plate-planet could
hold atmosphere in the horizontal direction, I'd have to integrate gh over
the whole volume of the disk to get a function in terms of x for the field
extending in the horizontal plane of the disk, and then integrate that from
the edge of the disk to infinity to get binding energy / escape velocity,
and plug that into the Jeans escape formula... but that's a really nasty set
of integrals, so I'm not going to cheat.
Start by finding the horizontal field around the center of a vertical bar by
integrating gh over z. We get
g = Gpx[1/x^2-1/(x^2+r^2)], where p is the linear density of the bar and r
is 1/2 its length, or
g = GM(x/2r)[1/x^2-1/(x^2+r^2)]
Looks a lot like the equation for the disk's vertical gravity, except for
lacking a square. The graph, however, does not. It looks a lot like an
inverse square graph, but starts out lower and ends higher.
Now, we can use the bar to represent a vertical slice out of the
plate-planet. If we integrate that equation between the edges of the disk to
get the field for a thin slice across it, we get
g/w = (Gp/2)ln[(D+x)^2(x^2+z^2)/((D+x)^2+z^2)x^2], where D is the diameter
of the disk, z is half its thickness, and w is some small width.
If D is, say, 10,000km, z is 500km (1/10 as thick as it is wide), density is
5000kg/m^3, x is 100 km, and G is the same as in our universe, and w is just
1 meter, I get g ~= 1.2197E-8m/s^2.
Next, we would add up all of the slices across the disk, but that's where
the integration gets really nasty. So, I'm going to make a very rough,
hopefully under-estimate by setting w to the width of the disk and dividing
by 8 to account for the fact that it's probably roughly circular rather than
square and that we stuff off to the side has a reduced contribution due to
having to project the field vectors from 2 dimensions just into x. That
gives g ~= .0152m/s^2. Even really close to the disk, it doesn't get much
better- 1km from the edge, g ~= .0582m/s^2. Considering that bits of the
planet near the edge but off to one side or the other whose fields cancel
close to the edge will start contributing to the inward horizontal field as
one moves away from the disk, we can probably bump those estimates up a
little bit, and the field decreases rather slowly anyway, so if the
atmosphere were to thin out below, say, 200km from the edge, we'de be OK.
But since the horizontal field is so weak, and there's going to be nearly a
full bar of atmosphere pouring off the edge... it won't thin out that
quickly.

Dang.

I guess I'll have to come up with something really clever, or else come up
with a different angular dependency than cosine. Perhaps just strengthening
the point-source gravitational field in the vertical, rather than dropping
it off to zero in the horizontal. Or inventing some completely different
force to hold things together.

I wonder if some sort of gravitomagnetic force could be of use. I considered
trying to avoid breaking relativity by tying the preferred direction of
gravity to angular momentum, but I couldn't get the superposition to work
right. A gravitomagnetic field of some sort that would cause air flowing off
of the edges to spiral back over the center of the disk would be cool, but
as it is these disks can't rotate very quickly, so I would have to come up
with something very much different from our universe's gravitomagnetic
fields to make it work.

-l.
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