Re: Location, Location, Location!

From: h...@xxxxxxxxxxxxx (Henry Spencer)
Earth, approximating a HEEO trajectory, its speed is escape speed
*plus* that great-distance-speed.
No, not correct. What it gains as it falls into Earth's gravity well is
energy, not (directly) velocity.

Oops, thanks for the prompt correction.
It's change in potential energy that adds to knetic energy directly.
Thinking back in delta-vee, I should have thought deeper, that if
something is falling very slowly, it spends a lot time in each
range of altitude, so gravity has plenty of time accelerate it as
it passes through the range, whereas if it's already falling
quickly it spends only a little it of time in the same range so it
gains less acceleration while falling through the same range.
That would have told me just adding a fixed amount of velocity
wasn't correct, and maybe I would have then thought of the right
way adding a fixed amount of energy.

Here again -- although for different reasons -- you take not the
sum of velocities, but the square root of the sum of squares.

Yeah, funny how two different models, one for cartesian
coordinates, and one for converting speed to energy and back again,
work out per the same math.

That's sometimes called the Oberth effect.

But it's been known since Newton invented his gravity and mechanics
and the calculus to do all the math, so why would it be named after
somebody who wasn't even born until 1894. Or did nobody happen to
think of the particular situation until he did? But according to my
Google search, that effect is the *other* effect, the one of
maximizing effectiveness of thrust for delta-vee by firing rocket
while closest to the gravity-assist object. More on that later.

Hmm, in researching "Oberth effect" I found this sad gem posted to
sci.space.tech by Edward A Gedeon in 1994:
> Jupiter will be in the correct position for a fly-by in May of 2003.
> (heliographic longitude of Jupiter is 90 degrees less than Pluto)
Only trouble is, the idea is to get to Pluto before the year 2000,
while it still has an atmosphere.
Timing is everything...
Sad because we didn't even launch until years after 2000.

But there's a trick: If you can re-aim it just right, to pass near
Luna on the way in, Luna's pull on it can drain away most of that
excess energy...
Again, alas, not quite correct. A lunar flyby can take off
*some* of the excess energy, but there are limits. The extreme
case, not physically practical, would be a 180deg turn (as seen
from the Moon) around the Moon, and if oriented just right, that
would take off twice the Moon's orbital velocity... which is
unfortunately only about 1km/s. So about 2km/s is the most you can
lose (and this is at high altitude, so you don't have as much
leverage from the Oberth effect).

Indeed, relative to what delta-vee you'd use near Earth, you get a
lot less than 2 km/s reduced from escape speed. On the other hand,
you could fire a rocket while closest to Luna, but with a fullsized
rubble-pile asteroid that would just break it up, but with a
house-sized boulder that trick might work, but I guess it wouldn't
be of much use for achieving Earth orbit.

That's ample for a capture of something whose velocity at
infinity is quite small, but you'd have to choose just the right
asteroid -- most come past a lot faster than that.

That supports my main point, that because the asteroids arrive at
*faster* than escape speed (slower than I stupidly calculated by
adding velocities, but still much faster than escape speed, like
30-40% faster), you need to bleed off a lot of energy to capture
them. It kills my minor point, that you can bleed most of it off by
Lunar gravity-assist, in fact you can bleed off only about 5-10% of
escape, so if the excess is more than that you have a problem. But
we agree with another minor point that if you pick just the right
asteroid, with only a little more than escape speed, a Lunar
gravity assist might just make the difference between getting it
and not getting it.

Now it seems to me the only asteroids with very low extra-speed
would be those nearly in our own plane, in nearly circular orbit,
at nearly same distance from Sun as we are. The trouble with such
an asteroid is that it takes may years to gain or lose an orbit
compared to Earth and thereby get back near Earth again. So if we
plant a rocket on it when it passes by, to aim it for the next time
it passes by again, we gotta wait a long time from attach-rocket to
capture-asteroid.

Now suppose we start with an asteroid that is in 3:4 resonance with
Earth (3 of its years equal 4 of ours), and aim it so next time
around (4 years = 3 oribts later) it passes through a
Lunar-gravity-assist keyhole to change it to much better match
Earth's orbit, so *next* time around we can finally capture it.
Trouble is now instead of being in a 3:4 resonance, and coming back
in 4 years, it's in a 9:10 resonance and not coming back for 10
years (9 orbits), or even longer wait if we get it in an even
"better" (matching Earth's) orbit. The better the match to Earth's
orbit we achieve, the longer we have to wait until it drifts back
into phase with Earth.

I think the most practical thing to do is locate a house-size
boulder or somesuch and dislodge it from the asteroid, then attach
an ion rocket to that boulder, and deflect it in such a way that it
quickly (in just an orbit or two) returns to Earth-Luna vicinity
but at low excess speed (a tradeoff, but both criteria achievable
with a boulder instead of a whole asteroid), and then continue to
decellerate it as it approaches Earth-Luna system, and as it passes
Luna in a gravity assist, and as it gets really close to Earth, and
as it heads back out at near escape speed, to achieve just less
than escape speed before it gets away, so that we can capture it
the first time we get it back here, before the main asteroid it
came from is due back.

for an object of a finite size like the Moon, the physics doesn't
permit a full 180-degree turn around it, ...

I was thinking of a parabolic orbit. But I suppose if the incoming
speed is greater than Earth-escape, it must be *much* greater than
Luna-escape, so at best we can get a Luna-hyperbolic trajectory
that doesn't really change its angle of approch much, sigh. I guess
the best we can do is where the outgoing path of the hyperbolic
path (relative to Luna) is exactly backwards from Luna's orbital
motion, and incoming path nearly radial, so inward motion is
converted to backward motion, minus Luna's velocity, thereby
exactly subtracting Luna's orbital speed from incoming speed, which
however subtracts a much less amount from perigee speed (Oberth
effect again).

Given the assumptions of a 2km/s asteroid and a 180-degree turn,
your assessment is correct. But as noted above, the former is
uncommon and the latter unrealistic.

Yeah. Is there a table online listing the known Earth-passing
asteroids with least Earth-relative speed, showing also date when
they last passed Earth and when they'll next pass Earth?
.

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