Re: Deja Q and why I love Trek

Phillip Thorne <thorne@xxxxxxxxxxxxx> writes:

nebusj-@xxxxxxxxx (Joseph Nebus) wrote:
So they make several efforts at this, pulling the moon when
it's at perigee, the point closest to the planet's surface.
Now I've got to figure out why they're doing this already hard
task the hard way ...

On Tue, 15 Aug 2006, Steve Hall <shall1@xxxxxxxxxx> wrote:
A satellite is closest to the planet at perigee, but it is also
traveling fastest at perigee [...] Perhaps it's
easier to move the moon than to impart momentum to it? [...]

The usual sequence of maneuvers to change orbital altitude (while
staying in one plane; plane-change maneuvers are harder, according to
my copy of _Fundamentals of Astrodynamics_, Bate et al, 1971, which I
have never read all the way through because of the profusion of
radicals and matrices) is this:

1. You're in a low-altitude circular orbit.

2. Fire your engines to increase your velocity, thereby raising your
apoapse and making your orbit elliptical.

3. Once you're at apogee, fire your engines again to raise your
periapse, circularizing your new orbit.

You've got it right. Basically, any ballistic move (a short
in duration impulse imparted by whatever means) has its greatest effect
on altitude at the opposite end of the orbit. So, for instance, in
'Apollo 13' when we see the end of the lunar injection burn of the
third stage within sight of the Moon, we're seeing the filmmakers'
dramatic license. It was really done when the spacecraft was as far
from the Moon (or to be exact, where the Moon would be at the end of
theirtravel time) as it could get.

The formulas take some time to derive, but happily, plenty
of people derived them already;
http://www.braeunig.us/space/orbmech.htm
has a nice listing of the important ones.


(There's a sim-game of this at the Baltimore Science Museum, but it's
probably available in any realistic space-travel sim software.)

If the moon of Bre'el IV was in a highly elliptical orbit, with its
periapse dipping into the atmosphere and subjecting it to decay, then
the sensible thing to do would be to shove it [*] while at apoapse,
lifting that end of the orbit out of harm's way.

http://www.startrek.com/startrek/view/series/TNG/episode/68428.html

We don't get much orbital data about the moon, other than
that it needs 'four kilometers per second' to be safe, and that the
Enterprise in one attempt is able to impart a delta-v of '73 meters
per second'. Also, the elliptical orbit is such that the most likely
impact point can be calculated and Enterpise's first attempts don't
change the likely impact point much.

We're also told that it's a captured asteroid; these will
often have pretty eccentric orbits to start with, but will tend to
circularize their orbits given time. Also we know it's massive
enough that its close approaches cause large tides, but tides are
so dependent on surface conditions that I won't touch that source
of data.

With so little data it's hard to rule *much* out, but I'll
make a few assumptions and deductions:

1. Bre'el IV is pretty much like Earth in mass, rotation,
atmospheric height, and so on. This is because it's very easy to
get the mass of Earth and, more important, the gravitational constant
of the universe *times* the mass of Earth, accurately. Also, while
we can put in any number, that makes the problem too general to have
any hope of solving.

2. The moon naturally has an orbital period of less than a
day. They're facing a moon-surface impact event within a few days,
and we see the moon go through two perigees [1] in not that much
screen time.

3. Therefore, I will assume Bre'el has a 24-hour day, and
the moon's orbit is highly eccentric but has a period of 12 hours.
This lets it keep aiming at the same target despite minor changes
in the orbit.

4. So what is the perigee? I'll put it at 100 kilometers
above the surface of the planet, since that's a nice round number,
and around when something can be said to be entering the atmosphere.
Using Earth's physical data, then, that's a radius of perigee Rp of
6470 km.

5. To have an orbital period of 12 hours requires the
orbit's semimajor axis be 26,610 km. (Kepler's laws: for an orbit
period T, the semimajor axis r must satify the relationship
T^2 = (4*pi^2)/(G*M) * r^3, with G the gravitational constant and
M the mass of Earth.) This gives a radius of apogee Ra of 46,750 km.

6. To match these parameters, then, the velocity of the
moon at perigee must be Vp = sqrt( (2*G*M*Ra)/(Rp * (Ra + Rp)) ),
and at apogee must be Va = sqrt( (2*G*M*Rp)/ (Ra * (Ra + Rp)) ),
that is, Vp = 10.4 km/s and Va = 1.43 km/s.

Since escape velocity from the surface of the Earth, by the
way, is 11.2 km/s, this means Bre'el has to be must more massive than
the Earth, or have a much deeper atmosphere, since otherwise LaForge
can't need more than 1 km/s to get rid of the moon altogether. But
to carry on with the calculation on Earth's parameters:

7. In the first attempt LaForge, Data, and Q are able to
use the tractor beam and spindizzy field to give the moon an extra
73 meters per second of speed. Given a perigee radius Rp, and a
perigee velocity Vp, the apogee radius is
Ra = Rp / ( (2*G*M)/(Rp * Vp^2) - 1) -- and, neatly, given an apogee
radius Ra and apogee velocity Ra, the perigee radius is
Rp = Ra / ( (2*G*M)/(Ra * Va^2) - 1)

So, adding 73 m/s to the satellite's existing 10.4 km/s at
perigee makes the new radius of apogee 52,780 kilometers, or some
46,410 kilometers above the surface.

But, if they had done this from apogee, adding the same 73
m/s to the apogee's 1.43 km/s, then the new radius of perigee would
be 7,247 kilometers, or about 877 kilometers above the surface.
Call it 900 for the lack of significant digits.

Since `success' is keeping the moon from hitting the planet,
plainly, even their extremely weak impulse imparted on the moon would
have been better applied to the moon at apogee. The resistance from
the atmosphere falls quickly with altitude, and the extra 800 kilometers
would extend the moon's natural lifespan by quite a few orbits, plenty
for a graceful trimming.

A second application of the same 73 m/s at apogee would boost
the perigee to 1,700 kilometers above the surface, and a third
application would get them to 2,600 kilometers away from the ground at
its nearest.

Obviously these exact numbers aren't right, since Bre'el is
not identical to the Earth, but the qualitative argument is right.


To circularize the orbit at my assumed apogee requires a
velocity of 7.8 km/s; that'd need a hefty 6.37 km/s be added to the
moon's velocity at apogee. If they had that impulse available, they
could just as easily eject the moon from Bre'el's orbit altogether,
by applying about 1 km/s at perigee, but presumably the locals would
rather keep the moon if that's an option. In any case avoiding an
impact is easier still than ejecting the moon.


[1] Yeah, 'periapsis' and 'apoapsis' are the general terms,
but they solve a problem that doesn't exist, of making clear the
body your satellite orbits isn't Earth.


[*] Sorry, couldn't resist:
"You can take your moon and *shove it*!"
"I do not believe that is anatomically possible, by at least seven
orders of magnitude."

You'd be surprised ... a tiny bit can matter, at the right
moment.

--
Joseph Nebus
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Relevant Pages

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