Today we are going to do momentum.
Take a look at the diagram below (click on it to enlarge).
The impactor (as yesterday) of mass m hits the earth with velocity u m/s and immediately becomes a plasma. As it meets earth material (we assume molten rock) it shares its momentum with the new material. The result is a growing cone of tunneling material as shown in the diagram below. The semi-angle of the cone is α, its depth is d metres and its base radius is r = d * α.
Multiply the resulting volume by the density of molten rock ρ which is 5.5 tonnes per cubic metre and decide the final velocity v at which we believe further tunnelling will stop. Perhaps 10 metres per second?
Click to enlarge
Since γmu = ρα2d3v by conservation of momentum,
d = (γmu/ρα2v)1/3.
Now for some numbers. For the 1 kg impactor at 0.999c considered yesterday, with α = half a degree and with a final plasma velocity of 10 m/s, the crater-depth d = 1.2 km.
This 1.2 km deep crater is nothing like the journey to the centre of the earth we speculated about yesterday. If we increase the cone angle for a wider dissipation - say 2.5 degrees - then the depth decreases to only 400 metres.
Suppose we increase the mass of the impactor to a cubic metre of ice weighing one tonne, at the original half-degree spread angle? The penetration depth d is now around 12 km. Notice that the increase of mass of a thousandfold has only increased the penetration depth by a factor of ten. This is because of the cube law we see in the equation above.
What would get us to the centre of the earth? A million tons (109 kg) at 0.999999c would dig a crater 3,700 km deep. I reckon this would make a bit of a mess of the earth.
Here's the spreadsheet to play with (includes yesterday's).
A relativistic impactor is like a nuclear detonation
How does this compare with nuclear weapons? An approximate formula for the crater diameter d (km) of a nuclear explosion of M Megatons is simply:
d3 = M.
So a 27 Megaton detonation would produce a 3 km diameter crater. A rule of thumb states that the crater depth is around 1/5 of its diameter, so for the 27 Megaton device, the crater depth would be around 600 metres.
This is almost identical to the result given by the spreadsheet for a 1 kg impactor at 0.9c (kinetic energy = 27.84 Mt).
According to the Wikipedia article on the Orion spacecraft (powered by nuclear detonation) the initial plasma velocity of a 1 Megaton bomb is 10,000 km/sec. The velocity seems to scale linearly with the bomb yield suggesting that a 30 Megaton bomb might create a plasma shockfront expanding at a speed close to that of light. When this hits the ground, it might be indistinguishable from a relativistic impactor and therefore the cratering effects could be very similar.
