**Next message:**Nick Bostrom: "Re: Why just simulation?"**Previous message:**Nick Bostrom: "Re: Why just simulation?"**Maybe in reply to:**Alex Future Bokov: "Geometry of Dyson Clouds"**Next in thread:**Robert J. Bradbury: "Re: Geometry of Dyson Clouds"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

Alex Future Bokov writes:

*> Let's say you want to populate an orbital distance R around a planet
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*> as densely as possible with satellites, such that no two ever come
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*> closer than distance N from each other. Let's assume non-decaying
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*> orbits and that they are not constrained to any particular orbital
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*> angle. What kind of arrangement would these orbits have?
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*>
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*> Related question, maybe the same one-- what is the mathematical
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*> relationship between the number of intersecting orbital planes (all at
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*> distance R, but at different angles) and the number of satellites in
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*> each orbit? In other words, if you just have one orbit, then you can
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*> pack it with pi*2R/N satellites. If two such orbits intersect, then
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*> obviously each has to contain fewer satellites (they need to 'take
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*> turns' going through the two points of intersection) and so on for
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*> each additional orbit added to the mix, but what is the actual
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*> relationship?
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Let's imagine that you have a bunch of polar orbits. Start with one

satellite in each orbit. As they cross the north pole, arrange it so

that the first orbit's satellite crosses, then the next (in longitude)

orbit's satellite crosses, and so on. As each satellite crosses the north

pole, the next and previous satellites have to be about N distance away.

Therefore the number of polar orbits in use would be proportional to

R/N.

The satellite positions at any instant describe a spiral shape that

starts at the north pole at zero degrees of longitude, heads south and

passes through the equator at 90 degrees, then reaches the south pole at

180 degrees of longitude. It then heads back up again and crosses the

equator northbound at 90 degrees again, reaching the north pole at 180

degrees (these are the satellites which have just crossed the north pole).

This shape intersects itself but the satellites themselves are all in

polar orbits so only pass at the poles.

There are actually two satellites in each orbit in this model, one

crossing the north pole while the other crosses the south pole. But the

point is that the total number of satellites is still pi*2R/N, no more

than can be held in one orbit. The bottleneck is the point where the

orbits cross, and the satellites must be "out of phase" by at least N

miles for it to work, which is what you get with a single orbit as well.

So you haven't directly gained anything by using multiple orbits all

crossing at a single pair of points.

However with a single, packed-full orbit you can't add any more since

you can't cross it anywhere. With this orbit you can easily add an

equatorial orbit at right angles to the polar one. I think the effect

of the polar arrangement I described is that the spiral or figure-8

pattern would appear to rotate equatorially, around the poles, even

though the actual satellites are of course moving north-south. As each

new satellite crosses the poles the entire pattern has rotated by the

angle between polar orbits. I think this rotation speed should be the

same as the speed of an equatorial orbit. So you can add at least a

single equatorial orbit and pack it with satellites, leaving a gap where

the polar pattern crosses the equator. This basically gives us pi*4R/N

satellites (not counting the gap).

Then the question is, can you turn the equatorial orbit into the same

pattern as the polar orbits, a bunch of orbits crossing at a single point

on the equator? By itself this won't gain you anything, you can't put

more satellites into multiple orbits that cross at a point than into a

single orbit, as we saw before. But it might leave room for more orbits.

I don't know if it would work.

*> I have a hunch that when there are many orbits, it's best to angle
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*> them such that there are never more than two intersecting. I also have
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*> a hunch that for each R there is an optimal number of orbits beyond
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*> which any additional satellite capacity from more orbits would be
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*> offset by the decrease in the number of satellites each of the orbits
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*> would have to undergo in order to accomodate the new one. I don't know
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*> how to evaluate these hunches, but if they are correct, then I can
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*> sort of envision how a densely populated orbital distance would look--
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*> sort of like a hollowed out ball of yarn, with relatively large gaps.
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I think this is reasonable. It looks like the most favorable geometry

for two orbits to intersect is at a small angle, where both orbits are

moving in the same direction. Obviously any time orbits intersect we

must space the satellites within an orbit no closer than 2N, because

when a satellite crosses another orbit, the closest satellites in that

other orbit must be at +/- N from it. It might appear that the most

favorable geometry would be to intersect orbits at right angles, but

actually in the frame of reference of the moving satellites that makes

the intersection look like a 45 degree angle. So you need to increase

the spacing by a factor of sqrt(2). But with a grazing intersection,

the satellites fall neatly into place interleaved with each other.

Two orbits intersecting at small angles can each hold pi*R/N satellites,

so the pair of orbits can hold pi*2R/N, just as with a single orbit.

However given two orbits that intersect at relatively small angles, it

is not too difficult to position additional orbits that also intersect

at small angles (using a different intersection point, otherwise we've

got the polar orbits from above). We can't have the angles be *too*

small because then we can't fit additional orbits which cross the one

and then the other at the right points. The orbits have to separate by

enough that additional orbits can be added and fit into the gaps.

I haven't tried to work out the geometry in detail but it looks like it

is probably possible to get at least "several" such orbits into place

still crossing at angles less than 30 degrees or so. I doubt that you

can have dozens of them, though. Each orbit crosses every other orbit

and the angles are going to add up. But overall, this seems like the

most promising approach.

Sorry, I have no idea whether or where this problem might be more

formally studied.

Hal

**Next message:**Nick Bostrom: "Re: Why just simulation?"**Previous message:**Nick Bostrom: "Re: Why just simulation?"**Maybe in reply to:**Alex Future Bokov: "Geometry of Dyson Clouds"**Next in thread:**Robert J. Bradbury: "Re: Geometry of Dyson Clouds"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

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