by WarDaft » Mon Feb 20, 2012 9:40 pm UTC
I guess we can approximate. I'm going to do my best to underestimate at each step, and so give a minimum radius for the impact with the ground.
Suppose it's 5 Mg, 1 meter squared cross section. At 50 km, it's going to encounter 1 gram of air (a bit more really, but close enough) every meter it falls. Lets get a lower bound by assuming the air particles slam into it so hard they get stuck, and just add to the mass (and so keeping as much energy as possible in the impactor.) So over 1 meter, its mass increases by 1 gram, and (using Newton to keep this simple) it's speed drops by just about 1.2 m/s, which represents the transfer of 3.6 TJ. This is, give or take, about a million times as much energy as it would take to vaporize 5 Mg of iron. So we know it's exploded by this point. (We could now make things totally circular and point out that a super high momentum plasma object would not have any elastic response to air and so simply soak it up as it went along, to justify our first assumption that the air molecules just get stuck in it.) Let's assume that it makes it this far before vaporizing however, so we have a 1 meter cross section high density plasma cloud, heading straight down at 60 Mm/s, 50 Km above the Earth's surface.
Let's assume a constant density of 1 gram/m^{3} for air for the next 10 Km. That means it will pick up 10 Kg of mass, and so slow to about 58,823 Km/s and have a total energy transfer of at least 3.495*10^{16} J from the original mass's kinetic energy. There's really no where else for that energy to go but to heat up the object even more, so should have a temperature of 1.56*10^{10} and correspondingly should have an internal pressure of 1.8*10^{11} ATMs. I'm simply not sure how to relate that to its rate of expansion, so here's our biggest approximation yet: I'm going to assume the density of the plasma object is uniform, then divide the pressure by 100 and assume the edge of the plasma cloud is accelerating as if acted on by that much force over 1 decimeter. Reversing the energy calculations suggests that it is now expanding outwards at 19 km/s, roughly 1/3000th of its current speed. A 1 meter cross section cylinder has a radius of 0.564 m. The next 3 km are therefore going to see an increase of at least 1 meter in radius, giving us a new cross section of at least 7.68 m^{2}. I cannot see any reason why the expansion of a uniform object under pressure in a relative vacuum (a ratio of 10^{11} makes the Earth's atmosphere effectively a vacuum by comparison) could not be done by integrating the internal pressure as an acceleration on the outermost material and maintained uniformity, but IANAP so I divided by 100 and applied it only over a small portion of the distance I was hoping for.
Ignoring all the energy picked up between 37 and 40 Km up, we can take our new cross section of 7.68 m^{2} and a new density of 6.3 g/m^{3}. Assuming constant size (to maintain underestimation) the next mere kilometer will result in soaking up another 48 Kg of air, or 144 kg over the next 3 km. The radius should have been at least 2.56 now with a volume of up to 70 m^{3}. Updating our cross section to 20.6 m^{2} and the density of air to 9.9 g/m^{3}, the next 3 Km of air promises to give us at least 600kg more mass. So the mass is now at least 5754 Kg with a downward velocity of 52 Mm/s and so the whole mess has transferred at least 1.18 * 10^{18} J from the initial objects kinetic energy into other forms of energy. That's more than 1/9th of it, and we're still up in the trace atmosphere. At this point the volume is at most 70 times our initial object's volume, and we've got at least 50 times as much energy in it, so it should still surely have at least 1/2 the pressure it had before, but with a larger radius, we can consider acceleration over a longer distance. Let's make it 6 times longer (since our radius is more than 5 times what we had before, this should be an even more accurate approximation) so we double the rate of expansion to 1 meter per 1 Km fallen. 31 Km up, the air has a density of 15.8 g/m^{3} and we have a cross section of 39.8 m^{2}.
The next 5 km promise to give us at least 3145 Kg of material. Our net pick up is 3899 Kg. Our current speed at 26 Km up is at most 33711 Km/s, so at least 3.94*10^{18} J has bled off from the initial kinetic energy. Our radius is now at least 8.56 m, our cross section is at least 230 m^{2}, and the density of air is 34.25 g/m^{3}. We are also no longer falling even close to 60 Mm/s, so we don't even need to consider acceleration due to pressure to increase the rate of expansion with respect to height, we're just not falling as fast. We'll assume the pressure compensates for the increase in material picked which would otherwise slow down the expansion (though it should do much more.) We'll round down a bit and say we expand 10m/6km. Why 6 km? Because we're going down to 20km next. The next 6 km should give us at least 33,465 kg of mass. Our speed is down to 7080 km/s, barely more than 10% of what we started with, and we're only a little more than half way, and still not into the actually dense portion of the atmosphere. At least 8/9ths of the kinetic energy we started with has been bled off. Our new radius is 18.56 meters, our new cross section is 1058 m^{2}, our new density of air is 88.9 g/m^{3}, and our new rate of expansion is almost 8 meters per km fallen.
The next 5 Km, we will pick up at least 480,949 Kg of mass. We're down to below 600 Km/s. Our rate of expansion is more than 10 times higher. At least 99% of the initial kinetic energy has been converted to other forms. Our radius is at least 58 m, and cross section at least 10,000 m^{2}. The density of air at this height has more than doubled to 194 g/m^{3}. We cannot assume the mass is a sphere at this point, but if it were, it would be no more than 3 times the density of the air at this atmosphere and half the density of air at sea level.
Given all that, it really looks like it's going to be an air burst, but if someone who was more confident of the mechanics could through a bit of math, that would be a lot better.
All Shadow priest spells that deal Fire damage now appear green.
Big freaky cereal boxes of death.