On 2008-01-17, Aztec50 <fasgard66@[EMAIL PROTECTED]
> wrote:
> Question #1: which will arrive first?
It's much easier if we can ignore the gravity of both Sun and Moon,
and just consider both ****ps and the Moon to be massless points
orbiting the Earth in initially circular orbit.
The ideal path for very low delta-V is to use half their delta-V to
thrust forward/backward as appropriate, changing the period of their
orbit, and just wait for the Moon to meet them after some number of
orbits. Then they use the other half of their delta-V to match
velocity again. However, this requires that their new orbital period
must have one of a specific set of integer ratios with the Moon's.
A slightly less ideal path involves thrusting at an angle to their
orbit, so that their new orbit crosses the Moon's in two locations,
with the second crossing point able to be tuned based on thrust angle.
This removes the need for an exact integer ratio, but at the cost of
some small amount of fuel.
However, it turns out that the amount of delta-V required to meet the
Moon within one orbit is very small by rocket standards. Multiple
orbits should never be required.
An orbit having perigee at the Moon's distance with 7/6 period differs
by 50m/s from the circular orbit. A rocket in this orbit will return
to its original position (relative to the Earth) after the Moon makes
1 full orbit plus 60 degrees. This would be suitable for the L4
craft, with a travel time of 7/6 of a month.
For the L5 (trailing) craft, an orbit with period 5/6 would be more
useful. This turns out to differ by 70 m/s and so requires more
delta-V than the L4 craft, but also reaches the moon faster (5/6
months instead of 7/6). If the L4 craft also has 70 m/s at each end
to spend, then it can get to the Moon in about 0.999 months.
So for very low delta-V, it depends greatly upon which craft is
closest to one of their "ideal" orbits.
For intermediate delta-V, I expect the L5 craft to win. As an example
of an intermediate delta-V: with 1 km/s (500 m/s each end), the L5 can
get there in 0.281 months by thrusting almost directly toward Earth
(85 degrees inward from orbital path). However the L4 can't get there
any sooner than 0.387 months, thrusting about 120 degrees outward from
the orbital path.
For very high delta-V, we can pretty much ignore gravity and they
should arrive at very similar times, but probably with the L5 winning
slightly.
> And then I reflected that Space****p A, in entering a retrograde
> orbit, will actually start to move in toward the Earth.
It depends upon delta-V. If it has less than twice orbital velocity
(about 2 km/s), then it can't possibly go into retrograde orbit at
all.
If it can go retrograde and heads inward, then the perigee will be
halfway between where it changes orbit and where it meets the Moon,
which in turn must be less than 60 degrees around the orbital path.
That means the path must be "flatter" than a circle and thus the
retrograde orbit must be higher-energy than the initial prograde
orbit. This can only happen with delta-V at least 4 times the orbital
velocity (i.e. more than about 4 km/s).
- Tim
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