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Why is that neutron stars are never depicted in a Hertzsprung-Russell diagram? They can be placed in the bottom left corner but you will never find any diagram in literature showing neutron stars.

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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 $10^4 - 10^5$ years after the originating supernova they will have cooled below a million degrees, then photon cooling takes over from neutrino losses and they may cool to a few thousand degrees within 10 million years (e.g. Yakovlev & Pethick 2004). There are many uncertainties and unknowns in these processes - see below. (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 or even bottom-right of the HR diagram. There is actually a huge uncertainty over where neutron stars would appear on this locus. The only neutron stars with measured luminosities and temperatures are extremely young ($<10^{5}$ years) and these are still extremely hot $10^{6}K$ and quite luminous $\sim 0.1 L_{\odot}$. Old neutron stars are practically invisible, but 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 and there are no observed objects to populate it.

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  • $\begingroup$ This is fine that neutron star cools of with time but knowing the teperature and radius, luminous can be determined. SO why can't we define a region which includes both the young (hot) and old (cool) neutron stars? It evolves , not changes it properties abruptly. So there can be a continuous region for neutron stars in H R diangrams. $\endgroup$ – seeking_infinity Jan 2 '15 at 7:03
  • $\begingroup$ @seeking_infinity ? We can - that is the equation I have given you. Draw the line on your HR diagram. $\endgroup$ – ProfRob Jan 2 '15 at 9:47
  • $\begingroup$ Okay. This is resolved. Now I want to ask when we know we can, Why people never show it ? $\endgroup$ – seeking_infinity Jan 2 '15 at 11:03
  • $\begingroup$ @seeking_infinity Be the first! Seriously though - it's there in my answer. (a) Because there are no observed objects to put on it; (b) because you would have to extend your y-axis by about 15 magnitudes. And I guess (c) which is although you can draw a blackbody locus, neutron stars won't exactly be blackbodies; (d) You also need to correct the temperature for GR redshift (which depends on mass and radius), which is a minor complication. $\endgroup$ – ProfRob Jan 2 '15 at 11:29
  • $\begingroup$ Hi Rob, I wonder if you can help me with this question: physics.stackexchange.com/questions/457315/… The main thing I need to know is the minimum absolute magnitude of an ancient neutron star near the sun, heated only by infall of interstellar matter or whatever $\endgroup$ – Robert Walker Jan 28 '19 at 17:31
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The classic HR-diagram has the temperature increase to the right (so lower temperatures are on the right and higher on the left), as seen in the image below:

enter image description here
(source)

The neutron star has a surface temperature of about a million degrees, which would put it way off the chart towards the left, not towards the bottom right.

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  • $\begingroup$ Right. That was a mistake. They should be on bottom left. But why neutron stars are never shown in H R diagrams. $\endgroup$ – seeking_infinity Jan 1 '15 at 16:57
  • $\begingroup$ @seeking_infinity: the chart typically spans about 35k kelvin, you'd need to expand that to millions which would make the more typical stars (MS, giants, etc) too squished. $\endgroup$ – Kyle Kanos Jan 1 '15 at 16:58
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    $\begingroup$ Confusing myself now. Should have said - The photospheric temperature of a neutron star exceeds a million degrees for a tiny fraction of its lifetime. Most neutron stars are cold, dead cinders. Bottom-RIGHT once older than 10-100 million years I would say $\endgroup$ – ProfRob Jan 1 '15 at 19:55
  • $\begingroup$ Although accretion from the ISM could keep them hot, but I don't think a million degrees. $\endgroup$ – ProfRob Jan 1 '15 at 20:05
  • $\begingroup$ It would be fair to say that the only neutron stars that you could put on the HR diagram have temperatures of a million degrees, because these very young neutron stars are the only ones that have measurable temperatures and luminosities. $\endgroup$ – ProfRob Jan 1 '15 at 20:55

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