1 Astrophysical Constraints on Dense Matter in Neutron Stars
19
The key questions are therefore (1) how well can the distance be determined, (2) how
well can T eff,∞ be established, given that it depends on a spectral fit, and (3) how
well is the flux known, since only the thermal component is relevant?
There are two categories of sources that have been studied carefully in this
manner to estimate neutron star radii. The first is the so-called quiescent low-mass
X-ray binaries (qLMXBs). These are transiently accreting neutron stars that, in
the ideal case, do not accrete at all when they are not accreting actively. Some
qLMXBs are in globular clusters, so for those sources the distance to the qLMXBs
can be determined by measuring the distance to the cluster. If a qLMXB has
a phase of active accretion, then it has a steady supply of hydrogen or helium,
which should rise to the top and dominate the surface composition after a few
seconds [5]. In addition, the dipolar magnetic field strengths of LMXBs are weak,
typically ∼10 8 − 10 9 G averaged over the surface [7, 42]. Thus magnetic effects
will be minimal (note that magnetic fields can affect energy spectra significantly at
photon energies comparable to or less than the electron cyclotron energy ¯
hω B =
¯
heB/m e c = 11.6 keV(B/10 12 G)). Therefore, the spectra might be well-modeled
by nonmagnetic atmospheres. These characteristics are all good for estimating radii.
Guillot et al. [94] recently applied this method to five qLMXBs in globular clusters.
They assumed that all the stars have the same radius, that the atmospheres are
nonmagnetic and composed purely of hydrogen (an assumption that they supported
on the basis of the companion type in two cases), and that the surface emission was
uniform. With these assumptions, they found R = 9.1
+1.3
−1.5 km.
However, the atmospheric composition matters greatly. For example, [200] find
that whereas fits of hydrogen atmospheres to a qLMXB in the globular cluster
M28 yield a radius of R = 9 ± 3 km at 90% significance, a helium atmosphere
gives R = 14
+3
−8 km with an equally good fit (χ ν /dof = 0.87/141 for the H
atmosphere, 0.88/142 for the He atmosphere). Similarly, [52] fit data from a qLMXB
in the globular cluster M13 and find R = 9.0
+3.0
−1.5 km for an H atmosphere and
R = 14.6
+3.5
−3.1 km for a He atmosphere, again with a comparable quality of fit.
Although one might argue [94] that if the companion is hydrogen rich the neutron
star atmosphere will be as well, it has been proposed that after ∼10 2−4 years
diffusive burning of hydrogen will leave helium as the main surface composition
([56, 58, 193]; see [26] for the first step in a re-evaluation of this work in the light of
a better treatment of Coulomb separation of ions). Given that ∼15 years of RXTE
observations have not revealed any accretion-powered outbursts from any of the
qLMXBs included in the analysis of [94], the recurrence time could be significantly
greater than a century and thus the surface composition might be helium rather than
hydrogen (see [134] for a discussion of what the larger implied radii would mean
for dense matter).
Moreover, it is not necessarily valid to assume that the entire surface radiates
uniformly. Nonaxisymmetric emission could theoretically show up as detectable
pulses, but if the magnetic pole is close to the rotational pole (as was suggested to
explain phenomena including the lack of pulsations in most LMXBs; see [126, 127])
the pulsations could be undetectable [181, 182, 226] while leading to inferred radii
that are too low.
19
The key questions are therefore (1) how well can the distance be determined, (2) how
well can T eff,∞ be established, given that it depends on a spectral fit, and (3) how
well is the flux known, since only the thermal component is relevant?
There are two categories of sources that have been studied carefully in this
manner to estimate neutron star radii. The first is the so-called quiescent low-mass
X-ray binaries (qLMXBs). These are transiently accreting neutron stars that, in
the ideal case, do not accrete at all when they are not accreting actively. Some
qLMXBs are in globular clusters, so for those sources the distance to the qLMXBs
can be determined by measuring the distance to the cluster. If a qLMXB has
a phase of active accretion, then it has a steady supply of hydrogen or helium,
which should rise to the top and dominate the surface composition after a few
seconds [5]. In addition, the dipolar magnetic field strengths of LMXBs are weak,
typically ∼10 8 − 10 9 G averaged over the surface [7, 42]. Thus magnetic effects
will be minimal (note that magnetic fields can affect energy spectra significantly at
photon energies comparable to or less than the electron cyclotron energy ¯
hω B =
¯
heB/m e c = 11.6 keV(B/10 12 G)). Therefore, the spectra might be well-modeled
by nonmagnetic atmospheres. These characteristics are all good for estimating radii.
Guillot et al. [94] recently applied this method to five qLMXBs in globular clusters.
They assumed that all the stars have the same radius, that the atmospheres are
nonmagnetic and composed purely of hydrogen (an assumption that they supported
on the basis of the companion type in two cases), and that the surface emission was
uniform. With these assumptions, they found R = 9.1
+1.3
−1.5 km.
However, the atmospheric composition matters greatly. For example, [200] find
that whereas fits of hydrogen atmospheres to a qLMXB in the globular cluster
M28 yield a radius of R = 9 ± 3 km at 90% significance, a helium atmosphere
gives R = 14
+3
−8 km with an equally good fit (χ ν /dof = 0.87/141 for the H
atmosphere, 0.88/142 for the He atmosphere). Similarly, [52] fit data from a qLMXB
in the globular cluster M13 and find R = 9.0
+3.0
−1.5 km for an H atmosphere and
R = 14.6
+3.5
−3.1 km for a He atmosphere, again with a comparable quality of fit.
Although one might argue [94] that if the companion is hydrogen rich the neutron
star atmosphere will be as well, it has been proposed that after ∼10 2−4 years
diffusive burning of hydrogen will leave helium as the main surface composition
([56, 58, 193]; see [26] for the first step in a re-evaluation of this work in the light of
a better treatment of Coulomb separation of ions). Given that ∼15 years of RXTE
observations have not revealed any accretion-powered outbursts from any of the
qLMXBs included in the analysis of [94], the recurrence time could be significantly
greater than a century and thus the surface composition might be helium rather than
hydrogen (see [134] for a discussion of what the larger implied radii would mean
for dense matter).
Moreover, it is not necessarily valid to assume that the entire surface radiates
uniformly. Nonaxisymmetric emission could theoretically show up as detectable
pulses, but if the magnetic pole is close to the rotational pole (as was suggested to
explain phenomena including the lack of pulsations in most LMXBs; see [126, 127])
the pulsations could be undetectable [181, 182, 226] while leading to inferred radii
that are too low.