  We find 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 and
form the fast magnetosonic wave. We find that electron 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 vortex
radiation has the spectral index -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 stellar interior 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=8*10^8K,
while the core temperature of PSR B0656+14 and Geminga exceeds 2*10^8K. 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. Based on the
temperature estimate and cooling models we determine the critical temperature
of triplet neutron superfluidity in the Vela core Tc=(7.5\pm 1.5)*10^9K which
agrees well with recent data on behavior of nucleon interactions at high
energies. Another finding is 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. Electron sound opens a
perspective of direct spectroscopic study of superdense matter in the neutron
star interiors.
