MadSci Network: Astronomy Query:

### Re: Minimum size of asteroid or comet needed to break up the Earth?

Date: Wed Jul 7 23:27:14 1999
Posted By: Robert Macke, Grad student, Physics, Washington University
Area of science: Astronomy
ID: 930759053.As
Message:
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Dan,

The real issue here is not of the size of the object, but of the energy it
releases in the collision.  Even a very small asteroid could destroy the
earth if it traveled fast enough (though that's not very likely at all),
just as a cannon ball causes much more damage when fired from a cannon than
it causes if just thrown at its target by hand, because it releases much
more energy in the target if it travels faster.

So, how much energy do you need to destroy the earth?  Certainly, it is a
lot more than you need just to wipe out life.  Life exists in a very
fragile balance that can easily be disrupted worldwide by an object that
would otherwise just leave a crater the size of Ohio but leave the rest of
the planet untouched.  You need a lot more energy to split the Earth in
two.

Since we're talking hypothetical situations here, let's go to extremes.
(This simplifies the calculation a bit)  Let's say we wish to pull the
planet apart entirely, separating it dust grain from dust grain. The energy
it would take to do this is simply the opposite of the energy stored in the
gravitational collapse of the matter that formed the Earth.  I won't go
into the mathematics of it, since I am not sure of your calculus
background, but the concept is to calculate the energy difference for each
individual particle added to the Earth's surface as it collapses, and I
assume a planet of constant density (I presume that's a first-order
difference, and since we're just estimating anyway, that doesn't matter.)
Anyway, the energy is (3/5)*G*M*M/R, where G is the gravitational constant,
M is the mass of the Earth, and R is the radius of the Earth.  This gives
us 2.24*10^32 Joules, which is a huge amount.

Now, for the sake of argument, let's say that we have an object impacting
the Earth at escape velocity.  (That is, the speed it would have if it was
dropped from rest a very far distance away and permitted simply to fall
onto the Earth.)  That's something like 11,000 meters per second, which is
pretty fast.  How much mass would it need to have to produce the kind of
energy we need in the collision to destroy the planet?
The calculation isn't too difficult.  The energy of the rock is G*M*m/R,
where little m is the mass of the object, and the other variables are the
same as before.  This needs to be equal to (3/5)*G*M*M/R in order to
produce enough energy to destroy the Earth.  Solving the equation for m,
we see: m=(3/5)*M.
So, the mass of the asteroid is just 3/5 times the mass of the Earth.  In
other words, you need a planet or large moon to cause that kind of damage!

As for the Moon, the same calculation applies, so you would need something
about 3/5 times the mass of the Moon to destroy it.  However, since the
Moon itself is just over 1/85 the mass of the Earth (0.012 times the mass
of the Earth), it could not cause the destruction of the Earth if it fell
out of orbit. (It would certainly cause a lot of damage, though)

I should note that, in the early days of the Earth before it cooled,
scientists believe that it was hit by a large object roughly the size of
Mars (approx. 1/10 Earth's mass).  This removed a large part of the planet
and put it into permanent orbit, where it eventually coalesced into the
Moon.  Nevertheless, the rest of the planet fell back together to become
Earth as we know it, rather than flying apart completely.  Needless to say,
if there had been life at that time it would have been completely wiped
out. :)

---Bob Macke
S.B. Physics, MIT, 1996
M.A. Physics, Washington University in St. Louis, 1999

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