|MadSci Network: Astronomy|
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. :) I hope this answers your question, ---Bob Macke S.B. Physics, MIT, 1996 M.A. Physics, Washington University in St. Louis, 1999
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