## 4. Models of cooling neutron starsTo illustrate the results of Sects. 2 and 3 we have performed
simulations of neutron-star cooling. We have used the same cooling
code as described by Levenfish & Yakovlev (1996). The general
relativistic effects are included explicitly. The stellar cores are
assumed to have the same equation of state (Prakash et al., 1988) as
has been used in Sect. 3. The maximum neutron-star mass, for this
equation of state, is . We consider
the stellar models with two masses. In the first case, the mass is
, the radius
km, and the central density
g cm For simplicity, the nucleons are assumed to be superfluid everywhere in the stellar core. We suppose that the proton superfluidity is of type (A), while the neutron superfluidity is of type (B). We make the simplified assumption that and are constant over the stellar core and can be treated as free parameters (see Sect. 3). Our cooling code includes the main traditional neutrino reactions
in the neutron star core (1)-(3), suppressed properly by neutron and
proton superfluids (Levenfish & Yakovlev 1996), supplemented by
the Cooper neutrino emission by neutrons and protons (Sects. 2 and 3).
In addition we include the neutrino emission due to electron-ion
bremsstrahlung in the neutron star crust using an approximate formula
by Maxwell (1979). The neutron-star heat capacity is assumed to be the
sum of the capacities of Figs. 6a and b show typical cooling curves (dependence of the
effective surface temperature
versus stellar age
Fig. 7 compares the standard cooling curves () with the available observations of thermal radiation from isolated neutron stars. The observational data are summarized in Table 2. We include the data on four pulsars (Vela, Geminga, PSR 0656+14, PSR 1055-52) and three radioquiet objects (RX J0822-43, 1E 1207-52, RX J0002+62) in supernova remnants. The pulsar ages are the dynamical ages except for Vela, where new timing results by Lyne et al. (1996) are used. Ages of radioquiet objects are associated to ages of their supernova remnants. The error bars give the confidence intervals of the redshifted effective surface temperatures obtained by two different methods. The first method consists in fitting the observed spectra by neutron-star hydrogen and/or helium atmosphere models (open circles), and the second one consists in fitting by the blackbody spectrum (filled circles). Dashed region encloses all standard cooling curves for a star with different and (from K to K). Notice that the Cooper-pair neutrinos introduce into the cooling theory some new "degree of freedom" which helps one to fit the observational data. For instance, we present a solid curve which hits five error bars for the atmosphere models at once. We would not be able to find a similar cooling curve if we neglected the Cooper-pair neutrino emission. The dashed curve in Fig. 7 shows that Cooper-pair neutrinos make the standard cooling very sensitive to the superfluid parameters. A minor change even of the one superfluid parameter yields quite a different cooling curve, which hits two error bars only. Such a sensitivity is important for constraining and from observations.
Thus, the majority of observations of thermal radiation from isolated neutron stars can be interpreted by the standard cooling with quite a moderate nucleon superfluidity in the stellar core. These moderate critical temperatures do not contradict to a wealth of microscopic calculations of and . Notice that it is easier for us to explain the "atmospheric" surface temperatures than the blackbody ones. This statement can be considered as an indirect argument in favour of the atmospheric interpretation of the thermal radiation. Although the theory of neutron-star atmospheres is not yet complete (e.g., Pavlov & Zavlin 1998) the atmospheric fits give more reasonable neutron-star parameters (radii, magnetic fields, distances, etc.) which are in better agreement with the parameters obtained by other independent methods. Even in our simplified model (one equation of state, two fixed neutron-star masses, constant and over the stellar core) it is possible to choose quite definite superfluid parameters to explain most of observations. However, one needs more elaborated models of cooling neutron stars to obtain reliable information on superfluid state of the neutron star cores. We expect to develop such models in the future making use of the results obtained in the present article. © European Southern Observatory (ESO) 1999 Online publication: March 1, 1999 |