24
M. C. Miller
fits, how badly will our inferred mass and radius be skewed? There are clearly
an unlimited number of possible sources of systematic error, but an encouraging
conclusion from the work done by Lo et al. [138] with synthetic data is that even
if the assumed surface beaming pattern, spot shape, or spectrum differ significantly
from the actual ones, fits using the standard model do not simultaneously produce
(1) a statistically good fit, (2) apparently strong constraints on M and R, and
(3) significant bias in M and R. Thus, at least for the systematic differences from the
model explored by Lo et al. [138], if the fit is good and the constraints are strong,
the inferred values of the mass and radius are reliable.
Valuable extra information could be obtained from the identification of atomic
lines from the surfaces of rotating neutron stars. No such line has been confirmed,
and indeed even if a line-like feature is seen in a spectrum it is not trivial to identify
the z = 0 atomic transition corresponding to the line. One such identification
was claimed from an analysis of stacked bursts from EXO 0748−676 [65], but an
additional long look at the star found it in another state that had no lines at all,
whether zero redshift or from the surface, and thus was unable to confirm the lines
[66]. The spin frequency of this star is 552 Hz [88] rather than the originally claimed
45 Hz [227], and hence one might expect that Doppler smearing would make a sharp
line undetectable (although note that [20] suggest that sharp lines would still be
visible; if this result is confirmed, it means that there are better prospects for sharp
lines than previously thought). If future large-area instruments are able to not only
detect such features but also measure them precisely, then both the redshift from the
surface and the linear speed of the surface at the spot, as well as possibly even framedragging effects, could be inferred [30]. This would allow many degeneracies to be
broken and would lead to much more precise constraints on neutron star masses and
radii (and moments of inertia from frame-dragging). Note that such measurements
will only be possible from actively accreting stars, because heavy elements sink
in the atmospheres of isolated neutron stars within seconds [5]. It has also been
proposed that the equivalent width of the line will allow a measurement of the
surface gravity, and hence that M/R (from the redshift) and M/R 2 (from the surface
gravity) can be measured independently (e.g., [57]). In principle this is also possible
using a non-thermal continuum spectrum, but this would require exceptional data.
1.4.4 Maximum Spin Rate
Another method that has been suggested to constrain the radius (or more properly,
the average density) is measurements of spin frequencies: a high enough spin
frequency from any star would rule out the hardest equations of state. Unfortunately,
no confirmed spin frequency is high enough to place significant limits on dense
matter. The maximum spin frequency for a star of mass M and radius R is
roughly [63]
ν max = 1250 Hz (M/M )
1/2 (R/10 km)
−3/2 .
(1.14)
M. C. Miller
fits, how badly will our inferred mass and radius be skewed? There are clearly
an unlimited number of possible sources of systematic error, but an encouraging
conclusion from the work done by Lo et al. [138] with synthetic data is that even
if the assumed surface beaming pattern, spot shape, or spectrum differ significantly
from the actual ones, fits using the standard model do not simultaneously produce
(1) a statistically good fit, (2) apparently strong constraints on M and R, and
(3) significant bias in M and R. Thus, at least for the systematic differences from the
model explored by Lo et al. [138], if the fit is good and the constraints are strong,
the inferred values of the mass and radius are reliable.
Valuable extra information could be obtained from the identification of atomic
lines from the surfaces of rotating neutron stars. No such line has been confirmed,
and indeed even if a line-like feature is seen in a spectrum it is not trivial to identify
the z = 0 atomic transition corresponding to the line. One such identification
was claimed from an analysis of stacked bursts from EXO 0748−676 [65], but an
additional long look at the star found it in another state that had no lines at all,
whether zero redshift or from the surface, and thus was unable to confirm the lines
[66]. The spin frequency of this star is 552 Hz [88] rather than the originally claimed
45 Hz [227], and hence one might expect that Doppler smearing would make a sharp
line undetectable (although note that [20] suggest that sharp lines would still be
visible; if this result is confirmed, it means that there are better prospects for sharp
lines than previously thought). If future large-area instruments are able to not only
detect such features but also measure them precisely, then both the redshift from the
surface and the linear speed of the surface at the spot, as well as possibly even framedragging effects, could be inferred [30]. This would allow many degeneracies to be
broken and would lead to much more precise constraints on neutron star masses and
radii (and moments of inertia from frame-dragging). Note that such measurements
will only be possible from actively accreting stars, because heavy elements sink
in the atmospheres of isolated neutron stars within seconds [5]. It has also been
proposed that the equivalent width of the line will allow a measurement of the
surface gravity, and hence that M/R (from the redshift) and M/R 2 (from the surface
gravity) can be measured independently (e.g., [57]). In principle this is also possible
using a non-thermal continuum spectrum, but this would require exceptional data.
1.4.4 Maximum Spin Rate
Another method that has been suggested to constrain the radius (or more properly,
the average density) is measurements of spin frequencies: a high enough spin
frequency from any star would rule out the hardest equations of state. Unfortunately,
no confirmed spin frequency is high enough to place significant limits on dense
matter. The maximum spin frequency for a star of mass M and radius R is
roughly [63]
ν max = 1250 Hz (M/M )
1/2 (R/10 km)
−3/2 .
(1.14)