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Re:NIF:power company



Joel Rauber wrote:

A couple of questions for Bowman, I haven't read all he said carefully, but
here goes a couple of questions.

Can we not stipulate, for purposes of the thought experiment, that planet X
was so constructed out of materials of such rigidity that its shape would be
so close to spherical that plumb bobs would not hand perpendicular to the
normal to the surface of the planet, as Marlow suggested?

Maybe if planet X was really a mini-neutron star made of just nuclear matter
it would work. There is a real problem with conventional planet-forming
materials. Clearly the use of fluids such as gases and liquids is ruled out.
As far as ordinary solid materials (rock, ice and iron) is concerned I think
that we would still have some trouble. Recall that most asteroids are
irregularly shaped, but the very largest ones bigger than about 800 km in
diameter or so are spherical due to their gravity being sufficient to force
them into a round shape. We can use Newtonian gravity with a spherical
geometry and pretend the matter is incompressible to make some crude
estimates. Let _rho_= mass density of planet, R = planet's radius,
r = radial distance from planet's center (for some point inside or on the
planet), _omega_ = planet's spin angular velocity, P = pressure on the
material inside the planet, and g = (magnitude of) local acceleration of
gravity for the planet.

First of all it is straightforward to show that g(r) = (4_pi_/3)G_rho_r
for a point inside the planet (r < R). We can also show that
P(r) = (2_pi_/3)G(_rho_^2)(R^2 - r^2) when the planet is not rotating.
Now since for the planet to form in the first place it must have a net
inward directed effective gravity when the "centrifugal" rotational effects
are added to the local gravity. Also if this constraint was not satisfied
nobody living on the planet could stay on it easily if it had a negative
effective surface gravity. Since most solid rocky materials have a much
lower tensile strength than compressive strength it is probable that the
planet would fly apart if it had a negative effective surface gravity.
Since the centrifugal effect will be the strongest on the equator where
it will tend to cancel the inward gravitational acceleration we have the
constraint: (_omega_^2)r < (4_pi_/3)G_rho_r. Note that the factor of r
cancels on both sides of the inequality giving:
_omega_ < ((4_pi_/3)G_rho_)^(1/2). For typical rocky material we have
_rho_ = ~3300 kg/m^3. This estimate constrains _omega_ < 9.6 x 10^-4 rad/s
yielding an upper bound to the rotation rate. This bound is equivalent to
a minimum rotation period of 1.8 hr. This max. rotation rate will result
in a maximum coriolis effect 13 times stronger than that of Earth's.
Now we can use the fact that rocky objects of radius greater than 400 km tend
to be round to estimate the yield strength of rocky material. This is also
presumably typical of the largest size possible for our planet whose rocky
material's rigidity can maintain its shape against the gravitational and
centrifugal accelerations present in our our rotating planet X. The above
formulas for a round object of radius 400 km and density of 3300 kg/m^3
give a surface gravity g(R) = 0.37 m/s^2, a central pressure of
P(0) = 2.44 x 10^8 Pa = 2400 atm. On Earth a column of this rocky material
would have to be 7.5 km high to have a pressure this great at the bottom of
it. It seems interesting that this height tends to be typical of of Earth's
higher mountains. The summit of Mount Everest is about 8.8 km above sea
level but less than this high above the average Himalayan surface elevation
of the mountain bases. In addition, the Himalayas are not in isostatic
equilibrium as India is still pushing North into China and continuing to
lift the mountains higher than they would be in equilibrium; they would
subside to a lower elevation without this extra tectonic nudging. Notice that
a surface gravity of less than .4 m/s^2 with a comparable centrifugal
acceleration will not seem like a wild turntable or jet fighter ride. Also
the coriolis acceleration effect of just 13 times that of Earth will still
not be noticable enough to cause much of a curve ball when a fastball is
pitched in a baseball game.

We see that in order to create the planet X with the very rapid rotational
acceleration properties that the Rauber/Marlow discussion assumes while
maintaining a spherical shape, we need to find some material that can withstand
stresses *much* greater than normal rocky material is capable of withstanding.
Things would be much easier if a superdense rigid material could be found
since the max. _omega_ calculated above depends proportionally on the
sqrt(_rho_). I'm not sure which substance would be a good planet X building
material. I sure don't know the compressive and shear elasticity properties
of things like diamond at the stresses required for this problem, but even
diamond is *less dense* than rock. If more exotic matter is chosen (such as
white dwarf material strengthened by the degenerate electrons) I believe that
even this will behave more like an incompressible fluid rather than a fully
it would behave more like an incompressible fluid rather than a fully rigid
rigid solid. Maybe we would have go to nuclear matter? I sure don't know.[B

A second problem for this planet (besides finding a suitably rigid material
for the planet to maintain its spherical shape) is that in order to form the
planet in the first place as a sphere it would have to be built in a
nonrotating state. After it was assembled (presumably by some sort of
accretion or collapse process) angular momentum could be supplied to spin it
up. What kind of spin-up process could be used? The only one I can think of
is a glancing collision with a massive foreign body. Unfortunately, I suspect
that any collision sufficiently strong enough to properly spin the planet up
to the desired _omega_ would also be plenty strong to reshape (and probably
melt or destroy) the planet from its spherical shape.

In short I'm pessimistic about being able to build a spherical Planet X with
a rotation rate sufficient to see all the wonderful cool rotational fictitious
forces glaringly show up in everyday motions for simple objects manipulated by
the planet's inhabitants.

David Bowman
Georgetown College
dbowman@gtc.georgetown.ky.us