Aztec50 wrote:
> Space****p A is at L4.
>
> Space****p B is at L5.
>
> They are racing to reach the Moon. Each has the same engines, and
> will go at the same speed.
Well, that's not how orbital mechanics works; you don't just point your
****p in a direction and go at a certain speed, you have to adjust your
orbit so that you get where you want. So the ****ps "travelling at the
same speed" really doesn't have much meaning.
How you'd get from L4 or L5 points to the secondary body -- or how you'd
accomplish any phase angle change in an orbit, for that matter --
depends on how much propulsive capability you have available, not only
in terms of deltavee, but also maximum thrust. Relatively high thrust
but low deltavee drives restricts to impulsive maneuvers (like basically
all of our chemical drive technology today). Very low thrust but
reasonable deltavee (e.g., ion drives or solar sails -- even though
they're not drives) means you can only adjust your orbit very gradually
and continuously. Very high deltavee (unreasonably high) and high
thrust allows you to perform ridiculous maneuvers like brachistochrones,
where you accelerate continuously to the midpoint and the decelerate to
the destination. (Note that even there, it's not the same as moving at
a constant speed, since that would get you there more slowly.)
> Question #1: which will arrive first? Question #2: if I didn't have
> much energy (and these ****ps weren't the supersleek dragsters they
> undoubtedly are), what would be the optimal/energy-efficient orbital
> "route" from each libration point to the Moon?
With a brachistochrone, the cruising speeds end up being so high that
for all intents and purposes you can ignore gravitational influences.
In this case, the two ****ps arrive at essentially the same time. But,
of course, this is really not practical.
There are really two broad approaches to make a phase angle change in
your orbit with realistic drives: a two-impulse transfer and a
four-impulse transfer. The two-impulse transfer puts you into a
different orbit which then swings around and intersects the original
orbit again, timed so that it will intersect when the desired position
p***** under it. To see that this is possible, consider that you're in
L4 -- ahead of the target -- and want to get back. Cancel your orbital
speed and apply a little push radially outward from the primary enough
so that the time it takes you to reach apoapsis and back is equally to
1/6 of the original orbital period. Then reapply your deltavee and get
back. (For L5, just make your bump outward longer so that it takes 5/6
of the orbital period.)
Needless to say, this takes a lot of deltavee, since you are in effect
canceling your orbital speed; these maneuvers require a huge change in
eccentricity. So they're not really idea, though depending on the
orbital system you're talking about, they may be reasonable. They also
take a bit of time, however.
A lower-deltavee route is to use orbital mechanics to let gravity do
your work for you: You make a transfer that puts you into a higher (or
lower) circular orbit, wait until your target orbits around, and then
make another transfer transfer back (each of these transfers requires
two burns). If you're in L4, then your destination is "behind" you in
orbit, so you want to rise up to a higher orbit, where your orbital
speed is lower, and let it catch up. If L5, then it's the opposite:
Your destination is "ahead" of you, so you want to descend to a lower
orbit and catch up to it.
These two broad categories blend into each other; it should be fairly
obvious that two-burn routes that exist that don't waste quite so much
fuel. Furthermore, you have options in how much fuel you burn to make
your required maneuvers, so you can trade fuel for time. This is one of
those essential cases in orbital mechanics where the details really
matter, as fuel expenditure and trip duration are inversely correlated
in a really crucial matter that makes it hard to overlook the details
and generalize.
> I.e., what I'm trying to figure out here is whether the fact that L4
> is "ahead" of the Moon's position in orbit is an advantage or
> disadvantage vis-a-vis L5's position "behind" the Moon's position in
> orbit. My first thought was that Space****p A will win easily (vis-a-
> vis Question #1), because while it moves toward the Moon, the Moon is
> moving toward it. But then I reflected that Space****p A, in going
> "backward", still has to compensate for the forces that are propelling
> it "forward." And then I reflected that Space****p A, in entering a
> retrograde orbit, will actually start to move in toward the Earth.
It's not totally clear to me which one wins out. It's certainly true
that a lower orbit (for some change in altitude) will orbit
differentially faster than a given outer orbit, it's also true that what
you're optimizing for is _deltavee_, not altitude, and orbital mechanics
being what it is, the same deltavee expenditure to get you into a lower
orbit will get you into a comparatively higher orbit, altitude-wise.
Whether that's enough to match the durations really depends on the
details, as they're highly sensitive to precisely what sort of orbit
changes you make, how quickly you do so, and at what deltavee. Neither
are going to take the same course.
All that said, I would guess that the lower orbit gets their first --
meaning the catch-up trajectory from L5 -- but I'm not absolutely sure.
--
Erik Max Francis && max@[EMAIL PROTECTED]
&& http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM, Y!M erikmaxfrancis
It's just another day / And nothing's any good
-- Sade
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