Go to previous page Go up Go to next page

12.2 Cooling of isolated neutron stars

Neutron stars are born in the core collapse supernova explosions of massive stars, as briefly reviewed in Section 12.1. During the first tens of seconds, the newly formed proto-neutron star with a radius of ∼ 50 km stays very hot with temperatures on the order of 1011 – 1012 K. In the following stage, the star becomes transparent to neutrinos generated in its interior via various processes (see Section 11). Within ∼ 10 – 20 s the proto-neutron star thus rapidly cools down by powerful neutrino emission and shrinks into an ordinary neutron star. The last cooling stage, after about 104 – 105 years, is governed by the emission of thermal photons due to the diffusion of heat from the interior to the surface (for a recent review of neutron star cooling, see, for instance, [431318] and references therein). Neutron stars in X-ray binaries may be heated as a result of the accretion of matter from the companion star. Observational data and references have been collected on the UNAM webpage [212].

The cooling of a young neutron star is very sensitive to its crust physics including, for example, neutron superfluidity, as shown in Figure 68View Image. Superfluidity of free neutrons in the inner crust suppresses heat capacity. Moreover, superfluidity opens a new channel for neutrino emission. Indeed, the formation of a bound neutron pair liberates energy, which can be converted into a neutrino-antineutrino pair, as discussed in Section 11.6.

View Image

Figure 68: Redshifted surface temperatures (as seen by an observer at infinity) vs. age of neutron stars with different masses as compared with observation. Dot-dashed curves are calculated with only proton superfluidity in the core. Solid curves also include neutron superfluidity in the crust and outer core [428Jump To The Next Citation Point].

12.2.1 Thermal relaxation of the crust

Due to its relatively low neutrino emissivity, the crust of a newly-born neutron star cools less rapidly than the core and thus stays hotter. As a result, the surface temperature decreases slowly during the first ten to hundred years and then drops sharply when the cooling wave from the core reaches the surface as illustrated in Figure 68View Image. After time tw, the star becomes isothermal except for the very outer layers. The relaxation time tw of reaching a quasi-isothermal state depends, in particular, on the specific heat Cv and on the thermal conductivity κ of the inner crust (see Section 9.3) and is approximately given by [256167Jump To The Next Citation Point]

( rg)− 3∕2 Cv tw ∼ (ΔR )2 1 − -- ---, (277 ) R κ
where ΔR is the thickness of the crust, R the circumferential radius of the star and rg = 2GM ∕c2, the Schwarzschild radius. The ratio of specific heat Cv to thermal conductivity κ has to be taken at half nuclear saturation density, slightly lower than the crust bottom density ρ cc (see [428]; in general, the relaxation time is the most sensitive to κ and Cv in the density range 0.1ρcc < ρ < ρcc). The thermal conductivity of the crust comes mainly from electrons scattering off atomic nuclei and electrically charged impurities. It is crucially dependent on the structure and composition of the crust (see Section 9). The crustal specific heat is dominated by free neutrons if they are not superfluid. Otherwise the neutron specific heat is strongly suppressed and its contribution to the total heat capacity is negligible as can be seen in Figure 69View Image. However, the density range and the critical temperatures for neutron superfluidity in the crust are still not very well known. The presence of nuclear inhomogeneities can have a significant effect on the specific heat by reducing the neutron pairing correlations inside the nuclei especially in the shallow layers of the inner crust at densities 11 12 −3 ρ ∼ 10 –10 g cm [332361237294]. The cooling curves of a 1.5M ⊙ neutron star for different crust models are shown in Figure 70View Image. Observations of young neutron stars could thus put constraints on the thermal properties of the crust, which in turn depend on its structure and composition. Such young neutron stars have not been observed yet. One reason might be that neutron stars born in type II supernova explosions remain hidden by the expanding supernova envelopes for many years.
View Image

Figure 69: Neutron star specific heat at T = 109 K. Solid lines: partial heat capacities of ions (i), electrons (e) and free neutrons (n) in nonsuperfluid crusts, as well as of neutrons, protons (p) and electrons in nonsuperfluid cores. Dashed lines: heat capacities of free neutrons in the crust modified by superfluidity. Two particular models of weak and strong superfluidity are considered. The effects of the nuclear inhomogeneities on the free neutrons are neglected. From [167Jump To The Next Citation Point].
View Image

Figure 70: Effective surface temperature (as seen by an observer at infinity) of a 1.5 M ⊙ neutron star during the first hundred years for different crust models. Dotted lines: cooling without neutrino emission from the crust (upper line), infinite κ at ρ > 1010 g cm −3. Solid line: cooling curve for the best values of κ, C v, and Q ν. Dashed line C = 0 n: dripped neutrons heat capacity removed. Dashed curve κ: thermal conductivity calculated assuming point-like nuclei. Two other dashed lines: neutrino emission processes removed except for plasmon decay (pl) or electron-nucleus Bremsstrahlung (eZ). See also line 1.5M ⊙ in Table 2 of [167].

12.2.2 Observational constraints from thermal X-ray emission

In cooling simulations, the neutron star is usually decomposed into the stellar interior, which becomes isothermal after a few tens to hundreds of years and the outer heat blanketing (insulating) envelope, where temperature gradients persist due to low thermal conductivity. The boundary between the interior and the envelope is conventionally set at ρ = 1010 g cm −3. The relationship between the surface temperature Ts and the temperature Tb at the bottom of the heat blanketing envelope is very sensitive to the structure and the composition of the crust and to the presence of a magnetic field. The outermost envelope of a neutron star, composed mainly of iron (Section 3.1), may be covered by a thin layer of light elements due to accretion, which strongly enhances heat transport and increases the surface temperature for a given Tb (let us remember that the electron thermal conductivity in a Coulomb plasma of ions with charge Z varies as ∼ 1∕Z). Strong magnetic fields also affect heat transport, leading to a nonuniform surface-temperature distribution and, in particular, hot caps near the magnetic poles, as illustrated in Figure 71View Image.

View Image

Figure 71: Magnetic field lines and temperature distribution in a neutron star crust for an axisymmetric dipolar magnetic field B = 3 × 1012 G and an isothermal core with temperature Tcore = 106 K. The temperature is measured in units of Tcore. The magnetic field is confined to the crust. From [158].

The effects of different crust models on the relationship between the surface temperature Ts and the temperature Tb at the bottom of the heat-blanketing envelope, are illustrated in Figure 72View Image. Grigorian [176] recently argued that cooling models predicting neutron stars with an age between about 103 – 104 years to be hotter than those already observed, should be rejected since if such stars existed in our galaxy, they would have already been detected. This brightness constraint puts restrictions on the Ts –Tb relationship hence on crust models.

View Image

Figure 72: Relationship between the effective surface temperature Teff, as measured by an observer at infinity, and the local temperature at the bottom of the heat-blanketing envelope, Tb (at ρb = 1011 g cm −3). Calculations performed for the ground-state (Fe) and partly-accreted envelopes of mass ΔM. Numbers − 16,...,− 7 indicate log (ΔM ∕M ) 10 for a 1.4 M ⊙ star with surface gravity g14 = 2.43 (in units of 1014 cm s–2). Symbols in parentheses indicate chemical composition of accreted envelope.

  Go to previous page Go up Go to next page