If a human was moving fast enough how bright would the blue-shifted heat be? Would it be detectable by an average human eye on an average night?

Question

I guess another way to answer this question would be if our eyes had the capability to detect the peak black-body wavelength that is emitted by humans would it be enough radiation to detect if we were as sensitive to that radiation as we are to the visible spectrum.

A Wien's Law calculator I found online gives a value of 9,363 nm for a black-body at 309.5 K.

Other aspects are also interesting though (if calculable). On an average dark night how close would a person have to be to detect this radiation?

This problem is probably easier if we imagine humans could detect the peak wavelength but I'm also interested in how fast the blackbody would have to be traveling (and how close it would have to pass) for the light to be blue-shifted to a detectable wavelength using the range humans already have.

Answer 1

If you want to shift the blackbody spectrum into the visible range, then you need to give it an apparent temperature similar to the Sun at 5800 K (in practice, to get reasonable visible light you could manage with half that).

The temperature of a blackbody increases with redshift as $T_0(1+z)$, where $z$ is the redshift.

Thus you need $z = 5800/309.5 -1 = 17.7$. This corresponds to a speed of $0.994c$.

Surface brightness of a blackbody simply scales as $T^4$, so a human travelling towards you at this speed would be as bright as the surface of the Sun in the brief instant between them being so far away that they were unresolved by the eye to zooming past.

I'm struggling to understand the rest of your question. I guess the surface brightness at 309.5K is a factor $18.7^4=1.2\times 10^5$ fainter than the surface of the Sun. That sounds like a lot, but the eye has some sensitivity to light many orders of magnitude fainter than the Sun's surface. In fact, $10^5$ is of order the ratio of the brightness of the Sun to the full moon