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Astron. Astrophys. 343, 650-660 (1999)

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4. Models of cooling neutron stars

To 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 [FORMULA]. We consider the stellar models with two masses. In the first case, the mass is [FORMULA], the radius [FORMULA] km, and the central density [FORMULA] g cm-3 is below the threshold ([FORMULA]) of the direct Urca process; this is an example of the standard cooling. In the second case [FORMULA], [FORMULA] km, and [FORMULA] g cm-3. The powerful direct Urca process is open in a small central stellar kernel of radius 2.32 km and mass [FORMULA], producing enhanced cooling. Notice, that in calculations of the equation of state of the stellar core in our earlier articles (Levenfish & Yakovlev 1996, and references therein) the parameter [FORMULA] (standard saturation number density of baryons) was set equal to 0.165 fm-3. In the present article, we adopt a more natural choice [FORMULA] fm-3 and use the model of the rapidly cooling star with somewhat higher mass than before ([FORMULA]). The nucleon effective masses are set equal to 0.7 of their bare masses.

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 [FORMULA] and [FORMULA] 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 n, p, and e in the stellar core affected by n and p superfluids (Levenfish & Yakovlev 1994b). We have neglected the heat capacity of the crust due to small crustal masses for the chosen stellar models. Our cooling code is based on approximation of isothermal stellar interior valid for a star of age [FORMULA]-[FORMULA] yr, inside which the thermal relaxation is over. We use the relationship between the surface and interior stellar temperature obtained by Potekhin et al. (1997). We assume that the stellar atmosphere may contain light elements. Then we can compare our results with observations of thermal radiation interpreted using the hydrogen or helium atmosphere models. However the mass of light elements is postulated to be insufficient ([FORMULA]) to affect the cooling.

Figs. 6a and b show typical cooling curves (dependence of the effective surface temperature [FORMULA] versus stellar age t) for the neutron stars with superfluid cores. The effective temperature is redshifted (as detected by a distant observer). Fig. 6a displays the standard cooling, while 6b shows the enhanced cooling. It is seen that Cooper-pair neutrino emission can be either important or unimportant for the cooling of both types depending on [FORMULA], [FORMULA] and t. Its effect is to accelerate the cooling, and the stronger effect takes place if the neutron or proton superfluidity is switched on in the neutrino cooling era ([FORMULA] yr). As mentioned in Sect. 3, the neutrinos provided by neutron pairing can dominate the neutrino emission from the modified Urca process at temperatures [FORMULA] K. Therefore, if the neutron superfluidity with [FORMULA] K is switched on in the stellar core, the standard cooling is accelerated. The acceleration is especially dramatic if the modified Urca was suppressed by the proton superfluidity before the onset of the neutron superfluidity ([FORMULA]). Then the slow cooling looks like the enhanced one (cf Figs. 6a and b). On the other hand, Cooper-pair neutrinos can be unimportant. The example is given by the curve ([FORMULA], [FORMULA]) in Fig. 6a. In this case, the star enters the photon cooling era (with the photon surface luminosity much larger than the neutrino one) with the internal temperature [FORMULA] K. The superfluidity appears later, and has naturally almost no effect on the cooling. If [FORMULA] the Cooper neutrino emission by neutrons can dominate even the powerful direct Urca process and accelerate the enhanced cooling (Fig. 6b). In other cases (e.g., the curves ([FORMULA], [FORMULA]) and ([FORMULA], [FORMULA])) the effect of Cooper-pair neutrinos on the enhanced cooling can be unimportant.

[FIGURE] Fig. 6a and b. Redshifted surface temperature [FORMULA] versus age t for the standard a and enhanced b cooling of the 1.3 [FORMULA] and 1.48 [FORMULA] neutron stars, respectively. Dotted curves are for non-superfluid stars. Solid and dashed curves are for superfluid stars (marked with ([FORMULA], [FORMULA])) including and neglecting Cooper-pair neutrinos, respectively. Solid and dashed curves (7.8,7.8) in a , and (8.3,8.6) and (10,8.4) in b coincide.

Fig. 7 compares the standard cooling curves ([FORMULA]) 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 [FORMULA] star with different [FORMULA] and [FORMULA] (from [FORMULA] K to [FORMULA] 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 [FORMULA] and [FORMULA] from observations.

[FIGURE] Fig. 7. Standard cooling curves (1.3 [FORMULA]) compared with observations (Table 2). Error bars are 90%-95% estimates of [FORMULA] obtained by fitting the observed radiation spectra with black-body spectrum (filled circles) and atmosphere models (open circles). Dotted curve corresponds to a non-superfluid star. Solid and dashed curves labelled as in Fig. 6 are for superlfuid stars. Dashed region is formed by standard the curves for different [FORMULA] and [FORMULA].


Table 2. Observational data.
a) Confidence level of [FORMULA] (90% and 95.5% correspond to [FORMULA] and [FORMULA], respectively); dash means that the level is not indicated in cited references.
b) Method for interpretation of observation.
c) The mean age taken according to Craig et al. (1997).
d) According to Lyne et al. (1996).

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 [FORMULA] and [FORMULA]. 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 [FORMULA] and [FORMULA] 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.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999