Relativistic Kill Vehicle What would happen if a significantly supra-molecular object (say ranking from grams to low kilotons) would be accelerated to relativistic speeds (>.10 c) such that its worldline would intersect with a planetary surface? 
Obviously, it would be a highly energetic event, more so with more mass and speed. But I'm not clear on whether any portion of the object would make it to the surface and generate a crater, or simply blow up in the stratosphere and generate a tremendous shockwave in the atmosphere? What combination of size and speed would be required to have long term planetary consequences for an Earth-type world (i.e. I'm thinking significantly more than a gentle hot wind)?
Most importantly, would a simple thin layer of gas say a few hundred million miles away be an effective defense? If not that, would anything we can feasibly imagine building?
For the purpose of I will ignore the consideration of how the impactor mass would reach this speed. Magical wand. 
 A: Assuming a direct hit - so traveling through about 50 km of atmosphere - at 0.1 c that would take about 2 ms if it didn't get slowed down too much by the atmosphere.
What about drag force? Let's assume a radius $r$, density $\rho$, mass $m = \frac43 \pi r^3 \rho$. If it is a sphere, it experiences a drag force $F=\frac12 \rho_a v^2 C_d A$. Putting $\rho_a=1 kg / m^3$ (a bit low at the surface of the earth, a bit high at 50 km), C_d = 0.47 (sphere), we find a force for a 10 cm radius object of
$$F = 0.5\cdot 1 \cdot (3\cdot 10^7)^2 \cdot 0.47 \cdot \pi \cdot 0.1^2=6.6\cdot 10^{12}N$$
That velocity squared term is a real killer...
The power dissipation at that velocity is
$$P = F v = 2\cdot 10^{20} W$$
While the kinetic energy is
$$KE = \frac12 m v^2 = \frac12 \frac43 \pi r^3 \rho v^2 = 15 TJ$$
This suggests you need to be more careful with the math - solve the equation of motion (which will have some mad deceleration). But if you just imagined that you used all that power to heat a column of air 50 km long and 0.5 m radius (that's about how far the shock wave travels in 2 ms) the temperature / pressure rise would be very substantial. Mass of air in that column roughly 8000 kg, with a heat capacity of about 1 kJ / kg / K, means that 15 TJ is enough to heat the air to 150,000 K which will give rise to quite a good pressure wave - but one that will dissipate quickly as you get further from the "column of fire".
Still - this sounds like something that earth might survive. But it would make quite a mess.
