Position of Neutron stars in H R diagram

• Why is that Neutron Stars are never depicted in an H R diagram? They can be placed in the bottom right corner but you will never find any diagram in literature showing neutron stars.

• The HR diagram is an observational diagram. Whilst neutron stars could be placed in the HR diagram in the same way as white dwarf stars are, it turns out to be impractical to do so because the photospheric luminosity and photospheric temperature of neutron stars is next to impossible to determine. The reason for this is two-fold: (i) Neutron stars start off very hot (interior temperatures of $\sim 10^{10}$K and photospheric temperatures of $\sim 10^{7}$K, but they cool very rapidly. Within 10,000 years after the originating supernova they will have cooled below a million degrees, then photon cooling takes over from neutrino losses and they cool to a few thousand degrees within 10 million years (e.g. http://adsabs.harvard.edu/abs/2004ARA%26A..42..169Y ). There are many uncertainties and unknowns in these processes. (ii) the photospheric emission is usually dwarfed by emission from the magnetosphere or luminosity due to accretion from a companion or the interstellar medium.

One can theoretically work out where neutron stars should be by assuming that the emission is like that from a blackbody and that the radius $R \sim 10$ km.

In that case neutron stars lie on a locus defined by
$$\frac{L}{L_{\odot}} = 1.9\times 10^{-9} \left(\frac{T}{10^{4}K}\right)^4$$
So, contrary to what you you say in your question, most neutron stars could be cool and very, very faint and spend the majority of their (cooling) lives at the bottom-left or even bottom-right of the HR diagram. There is actually a huge uncertainty over where neutron stars would appear on this locus. Their very low heat capacities means that any "reheating processes" could very effectively raise their temperatures. Such processes include Ohmic dissipation of the magnetic field, some kind of thermalisation of their rotational energy or accretion from the interstellar medium. For the latter, luminosities of $10^{20}-10^{21}$ W may be possible, implying effective temperatures of tens of thousands of Kelvin.

A neutron star at the same temperature as Sirius would have an absolute visual magnitude that was about 22 magnitudes fainter, $M_V \sim 23$. Another way of visualising this is that the neutron star cooling sequence is roughly parallel with the white dwarf cooling sequence but about 13 magnitudes fainter.

You never see this locus shown on an HR diagram because it is usually way off the bottom of the plot.