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, [431, 318] 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 68. 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.
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 68. After time
, the star becomes isothermal except for the very outer layers. The relaxation
time
of reaching a quasi-isothermal state depends, in particular, on the specific heat
and on the
thermal conductivity
of the inner crust (see Section 9.3) and is approximately given by [256, 167
]
where
is the thickness of the crust,
the circumferential radius of the star and
,
the Schwarzschild radius. The ratio of specific heat
to thermal conductivity
has to be taken at half
nuclear saturation density, slightly lower than the crust bottom density
(see [428]; in general, the
relaxation time is the most sensitive to
and
in the density range
). 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 69. 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
[332, 361, 237, 294]. The cooling curves of a
neutron star for different
crust models are shown in Figure 70. 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.
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
. The relationship
between the surface temperature
and the temperature
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
(let us
remember that the electron thermal conductivity in a Coulomb plasma of ions with charge
varies as
). 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 71.
The effects of different crust models on the relationship between the surface temperature
and the
temperature
at the bottom of the heat-blanketing envelope, are illustrated in Figure 72.
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
relationship hence on crust models.