We find a new mechanism of neutron star radiation wherein radiation is produced by the stellar interior .
The main finding is that neutron star interior is transparent for collisionless electron sound , the same way as it is transparent for neutrinos .
In the presence of magnetic field the electron sound is coupled with electromagnetic radiation , such collective excitation is known as a fast magnetosonic wave .
At high densities the wave reduces to the zero sound in electron liquid , while near the stellar surface it is similar to electromagnetic wave in a medium .
We find that zero sound is generated by superfluid vortices in the stellar core .
Thermally excited helical vortex waves produce fast magnetosonic waves in the stellar crust which propagate toward the surface and transform into outgoing electromagnetic radiation .
The magnetosonic waves are partially absorbed in a thin layer below the surface .
The absorption is highly anisotropic , it is smaller for waves propagating closer to the magnetic field direction .
As a result , the vortex radiation is pulsed with the period of star rotation .
The vortex radiation has the spectral index \alpha \approx - 0.45 and can explain nonthermal radiation of middle-aged pulsars observed in the infrared , optical and hard X-ray bands .
The radiation is produced in the star interior , rather then in magnetosphere , which allows direct determination of the core temperature .
Comparing the theory with available spectra observations we find that the core temperature of the Vela pulsar is T \approx 8 \times 10 ^ { 8 } K , while the core temperature of PSR B0656+14 and Geminga exceeds 2 \times 10 ^ { 8 } K. This is the first measurement of the temperature of a neutron star core .
The temperature estimate rules out equation of states incorporating Bose condensations of pions or kaons and quark matter in these objects .
The estimate also allows us to determine the critical temperature of triplet neutron superfluidity in the Vela core T _ { c } = ( 7.5 \pm 1.5 ) \times 10 ^ { 9 } K which agrees well with recent data on behavior of nucleon interactions at high energies .
We also find that in the middle aged neutron stars the vortex radiation , rather then thermal conductivity , is the main mechanism of heat transfer from the stellar core to the surface .
The core radiation opens a possibility to study composition of neutron star crust by detection absorption lines corresponding to the low energy excitations of crust nuclei .
Bottom layers of the crust may contain exotic nuclei with the mass number up to 600 and the core radiation creates a perspective to study their properties .
In principle , zero sound can also be emitted by other mechanisms , rather than vortices .
In this case the spectrum of stellar radiation would contain features corresponding to such processes .
As a result , zero sound opens a perspective of direct spectroscopic study of superdense matter in the neutron star interiors .