32
M. C. Miller
“minimal cooling model” (see [173] for a recent treatment), in which one assumes
no exotic components or direct URCA but does include Cooper pairing, crustal
bremsstrahlung, and all relevant photon processes. This represents the smallest
amount of cooling that is realistic. As a result, if there is evidence that at least some
neutron stars are significantly cooler than they are predicted to be in this model, that
might suggest exotic components (although it does not work the other way around;
conformity with minimal cooling might mean the suppression effects are important).
To see how this cooling model fares we now turn to observations.
1.5.4 Observations and Systematic Errors
Before discussing the observations we must issue a series of caveats. There are
many reasons why it is difficult to generate reliable points on a temperature-age
curve, including the interpretation of the spectrum, other possible heating sources,
and challenges with age estimation. We now discuss these in order before finally
evaluating the best current data on cooling neutron stars.
A neutron star with a surface effective temperature of 10 6 K and a radius of
10 km has a luminosity of L ∼ 10 32 erg s −1 , which at a distance of 3 kpc gives
a detector flux of F ≈ 7 × 10 −13 erg cm −2 s −1 and corresponds to ∼2 counts per
second for a 1000 cm 2 detector. It is therefore possible to get a reasonable number of
photons over a long observation, although the thermal peak of ∼ 0.2 keV(T /10 6 K)
is strongly susceptible to interstellar absorption.
There are also complications with the atmospheric model, as we discussed when
we examined radius estimates for cooling neutron stars. For example, compared
to a blackbody with the same effective temperature, an unmagnetized hydrogen
atmosphere has a strong excess at higher energies when absorption dominates the
opacity, because these opacities scale as ν −3 (e.g., [196]). Magnetized hydrogen has
less of an excess because strong fields increase the binding energy of atoms (e.g.,
Problem 3 in §112 of [128]), and heavier elements also have more bound electrons
and thus less of an opacity deficit at high energies compared to lighter elements
[152, 155, 184, 187]. Observational support for the diffusive burning of hydrogen
or helium [56, 58] may have been obtained from the evidence that the atmosphere
of Cas A is dominated by carbon [104]. As a result, unless there is a clear statistical
preference for one atmospheric model versus another (which is not currently the
case for any star), the temperature will be uncertain.
We must also be cautious because in addition to simple cooling there are various
heat sources that could contribute. These include magnetic dissipation (likely only
important for highly magnetic neutron stars) and magnetospheric emission. The
latter is likely to produce a nonthermal spectrum, which is indeed seen in some
stars. One could argue with some justice that if we are looking for cases where the
temperature is less than predicted by the minimal cooling model, other heat sources
will only mean that such evidence is even stronger. The tricky part comes when one
subtracts off nonthermal components to estimate the underlying thermal emission;
M. C. Miller
“minimal cooling model” (see [173] for a recent treatment), in which one assumes
no exotic components or direct URCA but does include Cooper pairing, crustal
bremsstrahlung, and all relevant photon processes. This represents the smallest
amount of cooling that is realistic. As a result, if there is evidence that at least some
neutron stars are significantly cooler than they are predicted to be in this model, that
might suggest exotic components (although it does not work the other way around;
conformity with minimal cooling might mean the suppression effects are important).
To see how this cooling model fares we now turn to observations.
1.5.4 Observations and Systematic Errors
Before discussing the observations we must issue a series of caveats. There are
many reasons why it is difficult to generate reliable points on a temperature-age
curve, including the interpretation of the spectrum, other possible heating sources,
and challenges with age estimation. We now discuss these in order before finally
evaluating the best current data on cooling neutron stars.
A neutron star with a surface effective temperature of 10 6 K and a radius of
10 km has a luminosity of L ∼ 10 32 erg s −1 , which at a distance of 3 kpc gives
a detector flux of F ≈ 7 × 10 −13 erg cm −2 s −1 and corresponds to ∼2 counts per
second for a 1000 cm 2 detector. It is therefore possible to get a reasonable number of
photons over a long observation, although the thermal peak of ∼ 0.2 keV(T /10 6 K)
is strongly susceptible to interstellar absorption.
There are also complications with the atmospheric model, as we discussed when
we examined radius estimates for cooling neutron stars. For example, compared
to a blackbody with the same effective temperature, an unmagnetized hydrogen
atmosphere has a strong excess at higher energies when absorption dominates the
opacity, because these opacities scale as ν −3 (e.g., [196]). Magnetized hydrogen has
less of an excess because strong fields increase the binding energy of atoms (e.g.,
Problem 3 in §112 of [128]), and heavier elements also have more bound electrons
and thus less of an opacity deficit at high energies compared to lighter elements
[152, 155, 184, 187]. Observational support for the diffusive burning of hydrogen
or helium [56, 58] may have been obtained from the evidence that the atmosphere
of Cas A is dominated by carbon [104]. As a result, unless there is a clear statistical
preference for one atmospheric model versus another (which is not currently the
case for any star), the temperature will be uncertain.
We must also be cautious because in addition to simple cooling there are various
heat sources that could contribute. These include magnetic dissipation (likely only
important for highly magnetic neutron stars) and magnetospheric emission. The
latter is likely to produce a nonthermal spectrum, which is indeed seen in some
stars. One could argue with some justice that if we are looking for cases where the
temperature is less than predicted by the minimal cooling model, other heat sources
will only mean that such evidence is even stronger. The tricky part comes when one
subtracts off nonthermal components to estimate the underlying thermal emission;