Interstellar medium shock heating

How fast would a hypothetical spacecraft need to get to experience significant heating from interaction with the interstellar medium (ISM)? Significant, in this context, means a steady-state temperature (that at which radiative cooling balances the heat being generated) of a few hundred K or more. I am considering both the ISM and blueshifted photons here.

The means by which a spacecraft could reach such speeds are immaterial, or if it can be done at all, are irrelevant to this question.

An ideal answer will include a formula from which the temperature can be calculated for any velocity.

Interstellar space has about 100 ions, mostly protons and corresponding electrons, per cubic meter. So assume that we have a surface area of material one square meter in area with $$\gamma=2$$ or velocity $$v=.867c$$. The total energy of a proton impacting this surface is $$E=\gamma mc^2$$ and the kinetic energy is $$K=(\gamma-1)mc^2$$. In one second this square meter of surface sweeps out a volume of $$1.5\times 10^8m^3$$ which means it impacts around $$1.5\times 10^{10}$$ ions. Now the proton has mass $$1.7\times 10^{-27}kg$$ and if this energy is absorbed in this material it would absorb $$2.2\times 10^{-1}$$ joules of energy every second. The reader can make a table with higher $$\gamma$$.
• Not really, for even if one thinks there is a Lorentz contraction of the distance this $1m^2$ area sweeps out the number of ions in the volume swept out is the same. – Lawrence B. Crowell Jul 20 at 12:40