astrophysics - What is the temperature of the surface and core of a neutron star formed 12 billion years ago now equal to?

In what part of the spectrum is it radiating? In the infrared, in the microwave? Or is not radiating anymore at all?

In russian:

Чему сейчас равна температура поверхности и ядра нейтронной звезды, которая образовалась 12 миллиардов лет назад? В каком диапазоне она сейчас излучает? В инфракрасном, микроволновом? Или не излучает вообще?

Answer

This depends a lot on the heating and cooling mechanisms you have in your model. Certainly initially neutron stars cool when neutrinos created in nuclear reactions escape. However, this is highly sensitive to such effects as superconductivity of the nucleon species (which both suppresses nuclear reactions and reduces heat capacity) and the density profile (which is in turn determined by the endlessly debated equation of state for the matter). At some point, the star likely reaches a thermal equilibrium of sorts, cooling by blackbody radiation from the surface.

Of course cooling is not the only process. Just a few of the heating ideas that have been floated over the years include:

• Friction between superfluid and non-superfluid layers sapping energy out of the star's rotation;


• The latent heat of crystallization of the crust being released whenever the crust "cracks" due to its equilibrium obliquity being reduced as the star spins down;


• Conversion of magnetic field energy into heat via, e.g., pair production;


• Redistribution of species as the centrifugally-induced chemical potentials change with spindown;


• Interaction with the interstellar medium; and even


• Colliding and interacting with the occasional free-floating magnetic monopole, if you believe in such things.

A rather extensive review of all this and more can be found in this review by Tsuruta, if you have access. The paper is replete with cooling curves for all sorts of models - too many to summarize here. Extrapolating to $12~\mathrm{Gyr}$ leads to temperatures¹ of anywhere from a few Kelvin (obviously the lower limit based on the temperature of the CMB reservoir in which the system sits) to $10^4~\mathrm{K}$ (blackbody peaks in the UV) and higher.

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¹ Often the axes are labeled by luminosity rather than temperature, the conversion being simple enough given the model's radius. Also note that all observables in the paper are properly gravitationally redshifted; if you want to know about the radiation field just above the surface, you have to blueshift the photons back into potential well.