In our everyday experience termperature is due to the motion of atoms, molecules, etc. A neutron star, where protons and electrons are fused together to form neutrons, is nothing but a huge nucleus made up of neutrons. So, how does the concept of temperature arise?
Astrophysics – Determining the Temperature of a Neutron Star
astrophysicsneutron-starsstatistical mechanicstemperaturethermodynamics
Related Solutions
Conservation of energy and the electron-degenerate pressure.
For the neutron to decay you must have $$ n \to p + e^- + \bar{\nu}$$ or $$ n + \nu \to p + e^- \quad. $$
In either case that electron is going to stay around, but in addition to the neutrons being in a degenerate gas, the few remaining electrons are also degenerate, which means that adding a new one requires giving it momentum above the Fermi surface and the energy is not available.
The thermodynamic definition of temperature has been found to be emergent from the underlying particulate nature of matter. It is connected with an average over the kinetic energy of individual particles.
Here v is velocity of a molecule, m its mass, k_B the Bolzman constant and T the temperature
The kinetic energy requires to have a degree of freedom, which is fine in gases. In solids the degrees of freedom are the rotations and vibrations of the molecules, as the molecules themselves are bound and thus do not have degrees of freedom in space. The same for the internal constituents of molecules, atoms , etc. They exist in a bound state and a temperature cannot be defined for them. Their only contribution comes into contributing to the mass of the molecules.
One can stretch the definition by using the kinetic energy of a particle in the formula, and derive a temperature. All one is saying is that "this would be the temperature of an ensemble of particles that have this kinetic energy on average"
Another stretch of definitions is found here.
Thus at the subatomic level there does not exist a temperature for the bound quarks and gluons as no kinetic degree of freedom exists.
In the comment the quark matter subject has been broached. This is a hypothetical state of matter where the energies are such that the QCD asymptotic freedom behavior emerges. This can happen in two ways :
1) during the Big Bang ,
The earliest phases of the Big Bang are subject to much speculation. In the most common models the universe was filled homogeneously and isotropically with an incredibly high energy density and huge temperatures and pressures and was very rapidly expanding and cooling. Approximately 10−37 seconds into the expansion, a phase transition caused a cosmic inflation, during which the universe grew exponentially.[18] After inflation stopped, the universe consisted of a quark–gluon plasma, as well as all other elementary particles.
The temperatures here are defined by the kinetic energies of the hypothesized particles and it is supplied by the energy of the universe as it evolves after the Big Bang
2) and is searched for in ion ion collisions at LHC.
In these heavy-ion collisions the hundreds of protons and neutrons in two such nuclei smash into one another at energies of upwards of a few trillion electronvolts each. This forms a miniscule fireball in which everything “melts” into a quark-gluon plasma.
The temperature here is defined by the kinetic energy of quarks and gluons in the plasma that have degrees of freedom as for a while they are asymptotically free. The energy is supplied by the accelerator.
Best Answer
First, strictly speaking a neutron star is not a nucleus since it is bound together by gravity rather than the strong force.
Measuring a surface temperature for any star is deceptively simple. All that is needed is a spectrum, which gives the luminous flux (or similar quantity) as a function of photon wavelength. There will be a broad thermal peak somewhere in the spectrum, whose peak wavelength can be converted to a temperature using Wien's displacement law:
$$T=\frac{b}{\lambda_{\rm max}}$$
with $b\sim2.9\times10^{-3}\rm mK^{-1}$. Neutron stars peak in the x-ray, and picking a wavelength of $1\;\rm nm$ (roughly in the middle of the logarithmic x-ray spectrum) gives a temperature of about $3$ million $\rm K$, which is in the ballpark of what is typically quoted for a neutron star.
More broadly than the motion of atoms or molecules, you can think of temperature as a measurement of the internal (not bulk) kinetic energy of a collection of particles, and energy is trivially related to temperature via Boltzmann's constant (though to get a more carefully defined concept of temperature requires a bit more work, see e.g. any derivation of Wien's displacement law).