1 Astrophysical Constraints on Dense Matter in Neutron Stars
31
There are also neutrino emission processes in the crust of the star, such as
plasmon decay and neutrino pair bremsstrahlung from electron-ion and electronelectron interactions. These could contribute for middle-aged stars, when the core
processes are suppressed by the superfluidity gap and the crust is warm enough to
produce neutrinos.
1.5.3 Photon Luminosity and the Minimal Cooling Model
When neutrino emission has tapered off, further cooling of the star is controlled by
conductive transport in the dense layers of the star (where electrons are degenerate
and thus have large mean free paths) and by radiative transport nearer the surface.
There is a minimum in the overall efficiency of energy transport at the “sensitivity
layer” where conduction hands off to radiation (where electrons are only partially
degenerate). This thus acts as a bottleneck and largely determines the overall cooling
rate; the density at the sensitivity layer increases with increasing temperature. For
temperatures in the observed range ∼10 5−6 K the sensitivity layer is at a density
<10 9 g cm −3 , where pycnonuclear fusion is not guaranteed to convert light elements
to heavy ones. The electron thermal conductivity in liquid ions depends on their
charge Z as ∼1/Z [172], so differences in the composition of the upper layers (due,
e.g., to different fall-back after core collapse) could have an influence on the thermal
evolution of neutron stars.
Conduction dominates at high enough densities that magnetic fields are likely
to be unimportant: at typical densities ρ > ∼ 10 6 g cm −3 the Fermi energy is E F > ∼
m e c 2 , implying that the magnetic field needs to be B > ∼ B c = m e c 3 /( ¯
he) = 4.414×
10 13 G to have a significant influence. However, more moderate magnetic fields can
affect radiative transport near the surface. More specifically, suppression of electron
motion across field lines yields anisotropic conduction. This can produce strong
anisotropies in the emergent radiation, but the overall effect on cooling is relatively
small [172]. Ultimately, the radiation emerges with some spectrum and an effective
temperature T eff that can be defined as
T
4
eff ≡
1
4π
T
4
s (θ, φ) sin θdθ dφ .
(1.15)
Here, the local effective temperature T s (θ, φ) at each location (θ, φ) on the surface
is defined via σ SB T 4
s (θ, φ) = F (θ, φ) where F is the emergent photon flux and
these quantities are appropriately redshifted. If the emergent spectrum is close to
a blackbody then one can estimate the temperature without knowing the distance
to the star (see Sect. 1.4), but as we discussed earlier atmospheric effects cause the
spectrum to deviate from a blackbody form, which complicates inferences.
With this physics in place one can construct a surface temperature versus age
curve by including a specific choice for the uncertain core processes as well as
choosing the composition of the surface layers. A particularly useful choice is the
31
There are also neutrino emission processes in the crust of the star, such as
plasmon decay and neutrino pair bremsstrahlung from electron-ion and electronelectron interactions. These could contribute for middle-aged stars, when the core
processes are suppressed by the superfluidity gap and the crust is warm enough to
produce neutrinos.
1.5.3 Photon Luminosity and the Minimal Cooling Model
When neutrino emission has tapered off, further cooling of the star is controlled by
conductive transport in the dense layers of the star (where electrons are degenerate
and thus have large mean free paths) and by radiative transport nearer the surface.
There is a minimum in the overall efficiency of energy transport at the “sensitivity
layer” where conduction hands off to radiation (where electrons are only partially
degenerate). This thus acts as a bottleneck and largely determines the overall cooling
rate; the density at the sensitivity layer increases with increasing temperature. For
temperatures in the observed range ∼10 5−6 K the sensitivity layer is at a density
<10 9 g cm −3 , where pycnonuclear fusion is not guaranteed to convert light elements
to heavy ones. The electron thermal conductivity in liquid ions depends on their
charge Z as ∼1/Z [172], so differences in the composition of the upper layers (due,
e.g., to different fall-back after core collapse) could have an influence on the thermal
evolution of neutron stars.
Conduction dominates at high enough densities that magnetic fields are likely
to be unimportant: at typical densities ρ > ∼ 10 6 g cm −3 the Fermi energy is E F > ∼
m e c 2 , implying that the magnetic field needs to be B > ∼ B c = m e c 3 /( ¯
he) = 4.414×
10 13 G to have a significant influence. However, more moderate magnetic fields can
affect radiative transport near the surface. More specifically, suppression of electron
motion across field lines yields anisotropic conduction. This can produce strong
anisotropies in the emergent radiation, but the overall effect on cooling is relatively
small [172]. Ultimately, the radiation emerges with some spectrum and an effective
temperature T eff that can be defined as
T
4
eff ≡
1
4π
T
4
s (θ, φ) sin θdθ dφ .
(1.15)
Here, the local effective temperature T s (θ, φ) at each location (θ, φ) on the surface
is defined via σ SB T 4
s (θ, φ) = F (θ, φ) where F is the emergent photon flux and
these quantities are appropriately redshifted. If the emergent spectrum is close to
a blackbody then one can estimate the temperature without knowing the distance
to the star (see Sect. 1.4), but as we discussed earlier atmospheric effects cause the
spectrum to deviate from a blackbody form, which complicates inferences.
With this physics in place one can construct a surface temperature versus age
curve by including a specific choice for the uncertain core processes as well as
choosing the composition of the surface layers. A particularly useful choice is the