26
M. C. Miller
frequency less than that of the upper peak by a characteristic but not exactly constant
amount that may be related to the spin frequency of the star (but see [149] for a
dissenting opinion).
Most, but not all, modelers identify the upper peak frequency with an orbital
frequency at some characteristic radius around the star. If this is true, it means that
the star must fit inside that radius, as must the radius of the innermost stable circular
orbit (ISCO) predicted by general relativity; the latter condition follows because
matter inside the ISCO will fall rapidly towards the star and thus prevent it from
forming high quality factor oscillations. As derived by [157], for a neutron star with
a dimensionless rotation parameter j ≡ cJ /GM 2 these conditions limit the mass
and radius to M max = 2.2 M (1+0.75j )(1000 Hz/ν QPO ) and R max = 19.5 km(1+
0.2j )(1000 Hz/ν QPO ) for an upper peak frequency ν QPO . The highest confirmed
QPO frequencies are all less than 1300 Hz (see, e.g., [41] for a discussion of the
1330 Hz QPO once suggested for 4U 0614+09), so at this stage the constraints are
not restrictive. If broad iron lines from the inner disk are discovered simultaneously
with kilohertz QPOs, this will provide another measure of the mass because the
line breadth gives
√
M/r whereas the QPO frequency gives
M/r 3 at the orbital
radius r (see [49] for current data and [28] for future prospects; note that the Kerr
spacetime is not an adequate approximation for sufficiently rapidly rotating neutron
stars [156]). Similarly, if reverberation mapping can establish an absolute time scale
for the system, this might yield masses and radii [15].
If the orbital frequency of the ISCO is established for a star, then the mass of the
star is known to within a small uncertainty related to the star’s dimensionless angular
momentum parameter. After doubt was cast on initial claims of ISCO signatures
[240] because of the complex relation between count rate and QPO frequency in
these stars [150], recent analysis of the RXTE database has suggested that in many
stars the predicted sharp drop in quality factor and gradual drop in amplitude [157]
are seen at a frequency that is consistent across a wide range in count rate and X-ray
colors [16–19]. The independence of this behavior from proxies of mass accretion
rate such as count rate and colors led these authors to suggest that a spacetime
marker such as the ISCO was the most likely reason for the observed phenomena.
If so, this represents a confirmation of a key prediction of strong-gravity general
relativity (the ISCO), and implies masses greater than 2.0 M for some neutron
stars, which would be highly constraining on equations of state.
Such important implications demand careful examination. For example, [148]
notes that the maximum quality factor achieved by neutron star LMXBs, versus
their average luminosity, has a shape similar to the quality factor versus radius
in individual stars, and uses this to conclude that other factors operate and that
the drop in quality factor might not be caused by approach to the ISCO. Indeed,
other factors do influence the quality factor; for example, the high-luminosity stars
plotted by Méndez [148] have geometrically thick disks that cannot produce highQ oscillations. This is thus not directly relevant to the arguments made by Barret et
al. [16–19] because the stars they examined are all low-luminosity, and hence the
results of [148] do not address whether the ISCO causes the behavior observed in
M. C. Miller
frequency less than that of the upper peak by a characteristic but not exactly constant
amount that may be related to the spin frequency of the star (but see [149] for a
dissenting opinion).
Most, but not all, modelers identify the upper peak frequency with an orbital
frequency at some characteristic radius around the star. If this is true, it means that
the star must fit inside that radius, as must the radius of the innermost stable circular
orbit (ISCO) predicted by general relativity; the latter condition follows because
matter inside the ISCO will fall rapidly towards the star and thus prevent it from
forming high quality factor oscillations. As derived by [157], for a neutron star with
a dimensionless rotation parameter j ≡ cJ /GM 2 these conditions limit the mass
and radius to M max = 2.2 M (1+0.75j )(1000 Hz/ν QPO ) and R max = 19.5 km(1+
0.2j )(1000 Hz/ν QPO ) for an upper peak frequency ν QPO . The highest confirmed
QPO frequencies are all less than 1300 Hz (see, e.g., [41] for a discussion of the
1330 Hz QPO once suggested for 4U 0614+09), so at this stage the constraints are
not restrictive. If broad iron lines from the inner disk are discovered simultaneously
with kilohertz QPOs, this will provide another measure of the mass because the
line breadth gives
√
M/r whereas the QPO frequency gives
M/r 3 at the orbital
radius r (see [49] for current data and [28] for future prospects; note that the Kerr
spacetime is not an adequate approximation for sufficiently rapidly rotating neutron
stars [156]). Similarly, if reverberation mapping can establish an absolute time scale
for the system, this might yield masses and radii [15].
If the orbital frequency of the ISCO is established for a star, then the mass of the
star is known to within a small uncertainty related to the star’s dimensionless angular
momentum parameter. After doubt was cast on initial claims of ISCO signatures
[240] because of the complex relation between count rate and QPO frequency in
these stars [150], recent analysis of the RXTE database has suggested that in many
stars the predicted sharp drop in quality factor and gradual drop in amplitude [157]
are seen at a frequency that is consistent across a wide range in count rate and X-ray
colors [16–19]. The independence of this behavior from proxies of mass accretion
rate such as count rate and colors led these authors to suggest that a spacetime
marker such as the ISCO was the most likely reason for the observed phenomena.
If so, this represents a confirmation of a key prediction of strong-gravity general
relativity (the ISCO), and implies masses greater than 2.0 M for some neutron
stars, which would be highly constraining on equations of state.
Such important implications demand careful examination. For example, [148]
notes that the maximum quality factor achieved by neutron star LMXBs, versus
their average luminosity, has a shape similar to the quality factor versus radius
in individual stars, and uses this to conclude that other factors operate and that
the drop in quality factor might not be caused by approach to the ISCO. Indeed,
other factors do influence the quality factor; for example, the high-luminosity stars
plotted by Méndez [148] have geometrically thick disks that cannot produce highQ oscillations. This is thus not directly relevant to the arguments made by Barret et
al. [16–19] because the stars they examined are all low-luminosity, and hence the
results of [148] do not address whether the ISCO causes the behavior observed in