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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.

Demi
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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 $\gamma$.

Lawrence B. Crowell
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