Alex Future Bokov writes:
> Let's say you want to populate an orbital distance R around a planet
> as densely as possible with satellites, such that no two ever come
> closer than distance N from each other. Let's assume non-decaying
> orbits and that they are not constrained to any particular orbital
> angle. What kind of arrangement would these orbits have?
>
> Related question, maybe the same one-- what is the mathematical
> relationship between the number of intersecting orbital planes (all at
> distance R, but at different angles) and the number of satellites in
> each orbit? In other words, if you just have one orbit, then you can
> pack it with pi*2R/N satellites. If two such orbits intersect, then
> obviously each has to contain fewer satellites (they need to 'take
> turns' going through the two points of intersection) and so on for
> each additional orbit added to the mix, but what is the actual
> relationship?
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
> them such that there are never more than two intersecting. I also have
> a hunch that for each R there is an optimal number of orbits beyond
> which any additional satellite capacity from more orbits would be
> offset by the decrease in the number of satellites each of the orbits
> would have to undergo in order to accomodate the new one. I don't know
> how to evaluate these hunches, but if they are correct, then I can
> sort of envision how a densely populated orbital distance would look--
> sort of like a hollowed out ball of yarn, with relatively large gaps.
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
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