OC.Trei%CU20B@sri-unix.UUCP (04/25/84)
From: Peter G. Trei <OC.Trei@CU20B>
Well, it turns out that I am not the first to work out how
much energy is required to blow up a planet. Jef Poskanzker did it a
couple years ago, and much more correctly than my method. I assumed
that the velocity which each fragment must attain was the same. This
is not so; for a piece of planet at a given depth below the surface,
the effective escape velocity is that required to escape the gravity
generated only by the mass closer to the center than it. Thus, the
amount of energy required drops off towards zero for peices nearer and
nearer the center, and the calculation becomes non-trivial.
______________________________________________________________
Date: 20 Apr 84 15:48:38 PST (Friday)
From: Jef Poskanzer <Poskanzer.PA@XEROX.ARPA>
Subject: Death Star weapon.
To: oc.trei%cu20b@COLUMBIA-20.ARPA
Re: I wonder how many times in the past someone has actually done this
calculation!
At least once:
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Date: 21 Mar 1982 23:12:33-PST
From: jef at LBL-UNIX (Jef Poskanzer [rtsg])
To: SF-LOVERS at MIT-AI
Subject: Planetary Binding Energy
A while ago, I mentioned a back-of-the-envelope calculation I once did
to figure out the relative strengths of the gravitational and chemical
binding energies which hold our planet together. I claimed that the two
binding energies were, to within an order of magnitude, equal. Well, I
did the computation again, and it looks like I was just barely correct.
As I figure it, the chemical energy binding the Earth is one-tenth of
the gravitational energy. Thus the two are the same to within an order
of magnitude, but for all practical purposes the gravitational energy
dominates and our Earth behaves like a ball of liquid instead of a ball
of rock.
To compute the gravitatonal binding energy, I integrated the potential
energies of thin spherical shells of matter relative to the matter
they enclose. Assuming a constant-density, perfectly-spherical body,
the mass of a shell is (WARNING: fixed-width font required beyond this
point!)
2
ms = density * 4 pi r dr ,
and the mass inside a shell is
3
m = density * 4 / 3 pi r .
The potential energy of ms relative to m is
G m ms
Ums = - ------ .
r
Then the total potential energy is the integral
from r = 0 to r = R of Ums. The final result is
2 2 5
U = - G density 16/15 pi R .
This looks strange, but the dimensions are mass * length**2 / time**2,
which is energy, which is what we want. For the Earth, with a radius
of 6.37e8 cm and a density of 5.52 gm/cm**3, the result is 2.24e39
ergs.
The chemical binding energy looked a lot harder to compute to me, so I
settled for a really simple-minded method: I figured out how much
energy it would take to raise the temperture of the planet by 5000
degrees C. At that temperture very few things are chemically bound.
This probably is a gross over-estimate, but that's ok because it still
turns out smaller than the gravitational energy. For the Earth, with
a mass of 5.98e27 gm, and an arbitrarily chosen specific heat of 0.2
cal / gm degC, the energy required is 2e38 ergs - one tenth the
gravitational energy.
So, the total binding energy of the Earth is about 2.4e39 ergs, which
is quite an impressive ammount. If Alderaan was about the same size
as Earth, the Death Star would have had to use 100 billion tons of
antimatter fuel to destroy it! However, a cheaper method would be to
trigger a fusion chain reaction in the planet's oceans, as some
thought would happen here on Earth when we tested the H-bomb. Fusing
all the hydrogen in the Earth's oceans would release 1e41 ergs, more
than enough to disassemble the planet.
Cheerfully yours,
---
Jef
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Note: your number for the gravitational binding energy, 3.8307e33
Joules, or 3.8307e40 ergs, is ten times higher than mine. Your figure
for the mass needed to convert to energy is one hundred times higher,
because I slipped a decimal point. I should have said one TRILLION
tons of antimatter, and a like quantity of matter.
Ave Discordia!
---
Jef
_____________________________________________________________________
An interesting corollary of this is the observation that a
hollow sphere puts no net gravitational force on an object ANYWHERE
within it. This blows all the hollow-earth-civilisation stories out of
the water.
Whether it takes 4e33 or 2e32 Joules, thats still many many
cords of kindling. Here are one or two other methods:
1. A ray which depresses (or removes) either the negative charge on
the electron or the postive charge of the proton. The resulting electrostatic
repulsion would tear the planet apart VERY effectively. Larry Niven is also
responsible for this one, again as a handweapon. (He likes powerful guns!).
Back when the late-lamented Steady State Universe theory still had a few
diehard supporters, one of the mechanisms they proposed for the continuous
expansion was a slight disparity between the charges of protons and neutrons.
A VERY small difference is all that is required.
2. Asimov, in 'The Gods Themselves' had the Earth almost being
destroyed as a byproduct of an energy-production system which had only
one byproduct; the strength of the weak nuclear force was locally
decreased (increased?). This meant that nuclear fusion became easier
and easier as time went on. Indeed, had they left the generator
running too long, lighting a cigarette could have started a thermonuclear
chain reaction. A ray which could do this could cause a VERY BIG BANG.
Any more megalomaniac mechanisms out there?
Peter Trei
-------eder@ssc-vax.UUCP (05/02/84)
2 May 1984
The center of a planet is likely to contain a high concentration
of uranium and thorium. They are heavy and sink to the middle of planets
with fluid cores. Now Star Wars was set "A long, long time ago". U-235
has a shorter half-life than U-238, thus the concentration was higher earlier
in the life of your average planet. Far enough back, it would have been
a fissionable mix.
Now, what you do is fire a VERY high power laser at the planet. This
bores a hole to the center. Now fire a pulse of neutrons into the core,
which causes a small (relatively) detonation from enhanced fission. This
blows a hole in the middle of the planet, compressing the remainder of the
core. As the density increases, the rest of the material fissions.
KABLOOEY!
P.s. Hi, Peter. How do I send messages to you directly?
Dani Eder
Boeing Aerospace
ssc-vax!eder