starship-design: spinning up the Moon

[Steve VanDevender <stevev@efn.org>]

I did a little research into what it would really take to spin up the
Moon to a higher rotation rate.

The angular momentum of a body is in units of kg * m^2 / s, which you
can think of as kg (mass) * m (moment arm) * m/s (rotational velocity).

The Moon has a mass of some 7.35 * 10^22 kg and a radius of 1740 km.
Assuming it's a uniform-density sphere (not quite right, but sufficient
for approximation), its angular momentum for a given rotation rate w (in
1/s) will be m * 2/5 r^2 * w.  Its rotation rate is about 1 / 2.55 *
10^6 s.

The Moon's current angular momentum is therefore about 8.72 * 10^28
kg*m^2/s.  Spinning it up to have a rotation rate of 24 hours would
increase its angular momentum to 1.03 * 10^30 kg*m^2/s, for a total
change in angular momentum of 9.43 * 10^29 kg*m^2/s.

So let's suppose, just for the sake of argument, that we have a way to
smack comets or asteroids tangentially into the Moon's surface in such a
way as to perfectly transfer their linear momentum to the Moon's angular
momentum.  That would mean a mass M striking at a radius of 1740 km with
a velocity of v m/s, transferring a total angular momentum of M * v *
1.74 * 10^6 kg*m^2/s.

So to spin up the Moon, we need a quantity of momentum M * v = 5.42
kg*m/s.  So you can determine how much mass M would be required for a
given impact velocity v, or given mass M what velocity v it would have
to be traveling at to achieve the intended angular momentum change.

Let's say you can manage to smack those objects into the Moon at 50 km/s
(rather untenable, since that velocity's probably much too high to
expect that the momentum would be efficently transferred).  That means
you'd need a mass of comets/asteroids of 1.08 * 10^19 kg, or that you'd
need about 1/6780 the mass of the Moon in comets/asteroids impacting at
that velocity _and transferring ther momentum entirely_ to get the Moon
spun up.

With lower impact velocities the mass ratio goes up proportionally;
i.e. for impactors moving at 10 km/s you'd need five times as much mass
as for impactors moving at 50 km/s.  Even 1/10000 the mass of the Moon
might not seem like much, but that translates into a pretty huge
quantity of comets.