1 Astrophysical Constraints on Dense Matter in Neutron Stars
23
stars have radii R < 15 km and spin frequencies ν < 600 Hz [44, 50]. The current
best analyses of isolated millisecond pulsars and bursting stars yield radii that are
consistent with expectations but not very constraining; [38] find that J 0030+0451
has a radius >10.4 km at 99.9% confidence, and [39] find that J2124–3358 has a
radius >7.8 km at 68% confidence (both of these assume M = 1.4 M ), and [29]
find Rc 2 /GM > 4.2 at 90% confidence for the burster XTE J1814–338. Analyses
using the “oblate Schwarzschild” approximation (in which the star is allowed to be
oblate due to rotation but the spacetime is still assumed to be Schwarzschild) for
SAX J1808.4–3658 [159] and XTE J1807–294 [135] are similarly unconstraining.
The strongest current constraints from this method come from a recent analysis
of PSR J0437–4715 assuming a hydrogen atmosphere, for which the result is
R > 11.1 km at 3σ confidence [37].
A key assumption in the analysis of [37] is that the angular distribution of
radiation from a point on the surface can be described by the pattern that emerges
when the energy is deposited deep and propagates through a pure nonmagnetic
hydrogen atmosphere. This could be a correct assumption, but in addition to our
previous comments that hydrogen might not be the dominant surface composition,
we note that the assumption of deep deposition of energy (which for this source
comes from the return current of relativistic pairs from the magnetosphere) is based
on the idea that the current gives up its energy via Coulomb collisions and nothing
else. Given that plasma instabilities can shorten by orders of magnitude the column
depth of energy deposition (see [45] for a recent example in the context of how
AGN jet energy is injected in the intergalactic medium), this might not be a safe
assumption.
Similar caution is appropriate for analyses of the waveforms from accretionpowered pulsars. For example, the oblate Schwarzschild analyses of the accretionpowered pulsars SAX J1808.4–3658 [159] and XTE J1807–294 [135] use a
model with a blackbody component (assumed to be isotropic) and a Comptonized
component (assumed to have an angular variation ∝ 1 − a cos α, where α is the
angle from the surface normal and a is a free parameter). These authors also
included a scattered light component and assumed an infinitesimal spot. None of
these assumptions or models can be perfect, and it is not known how serious an
effect they have on the derived radii of the stars.
There are two reasons for the large credible regions that currently arise from
analyses of waveforms: (1) the total number of counts is small, and (2) there are
significant degeneracies between the parameters that produce the waveform. The
effects of both factors are expected to be addressed using the next generation of
large-area X-ray timing satellites. As discussed by Lo et al. [138], if a million counts
are received from the spot (comparable to the total number expected from combining
several bursts observed using LOFT, or to the integrated counts from non-accreting
neutron stars or bursting sources using NICER) and the center of the hot spot and
the observer are both within 10 ◦ of the rotational equator, then M and R can both
be obtained to 10% precision.
As with the other methods we discuss, a key question is the role of systematic
errors: if some aspect of the real system differs from what we assume in our model
23
stars have radii R < 15 km and spin frequencies ν < 600 Hz [44, 50]. The current
best analyses of isolated millisecond pulsars and bursting stars yield radii that are
consistent with expectations but not very constraining; [38] find that J 0030+0451
has a radius >10.4 km at 99.9% confidence, and [39] find that J2124–3358 has a
radius >7.8 km at 68% confidence (both of these assume M = 1.4 M ), and [29]
find Rc 2 /GM > 4.2 at 90% confidence for the burster XTE J1814–338. Analyses
using the “oblate Schwarzschild” approximation (in which the star is allowed to be
oblate due to rotation but the spacetime is still assumed to be Schwarzschild) for
SAX J1808.4–3658 [159] and XTE J1807–294 [135] are similarly unconstraining.
The strongest current constraints from this method come from a recent analysis
of PSR J0437–4715 assuming a hydrogen atmosphere, for which the result is
R > 11.1 km at 3σ confidence [37].
A key assumption in the analysis of [37] is that the angular distribution of
radiation from a point on the surface can be described by the pattern that emerges
when the energy is deposited deep and propagates through a pure nonmagnetic
hydrogen atmosphere. This could be a correct assumption, but in addition to our
previous comments that hydrogen might not be the dominant surface composition,
we note that the assumption of deep deposition of energy (which for this source
comes from the return current of relativistic pairs from the magnetosphere) is based
on the idea that the current gives up its energy via Coulomb collisions and nothing
else. Given that plasma instabilities can shorten by orders of magnitude the column
depth of energy deposition (see [45] for a recent example in the context of how
AGN jet energy is injected in the intergalactic medium), this might not be a safe
assumption.
Similar caution is appropriate for analyses of the waveforms from accretionpowered pulsars. For example, the oblate Schwarzschild analyses of the accretionpowered pulsars SAX J1808.4–3658 [159] and XTE J1807–294 [135] use a
model with a blackbody component (assumed to be isotropic) and a Comptonized
component (assumed to have an angular variation ∝ 1 − a cos α, where α is the
angle from the surface normal and a is a free parameter). These authors also
included a scattered light component and assumed an infinitesimal spot. None of
these assumptions or models can be perfect, and it is not known how serious an
effect they have on the derived radii of the stars.
There are two reasons for the large credible regions that currently arise from
analyses of waveforms: (1) the total number of counts is small, and (2) there are
significant degeneracies between the parameters that produce the waveform. The
effects of both factors are expected to be addressed using the next generation of
large-area X-ray timing satellites. As discussed by Lo et al. [138], if a million counts
are received from the spot (comparable to the total number expected from combining
several bursts observed using LOFT, or to the integrated counts from non-accreting
neutron stars or bursting sources using NICER) and the center of the hot spot and
the observer are both within 10 ◦ of the rotational equator, then M and R can both
be obtained to 10% precision.
As with the other methods we discuss, a key question is the role of systematic
errors: if some aspect of the real system differs from what we assume in our model