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M. Méndez and T. M. Belloni
6.6 QPO Frequency Correlations
Although the beat frequency model is unable to explain all observations, it is
reasonable to think that the kHz QPOs could in some way be connected to the
rotation of the neutron star. When the first source for which both burst oscillations
and kHz QPOs were discovered, 4U 1728−34, it was realised that Δν was around
the same value as the burst oscillation frequency [150]. However, the next source
with a double detection was 4U 1636−63, where Δν ∼ 270 Hz and the burst
oscillation frequency was 581 Hz, close to twice that value. After then, every time
a new source showed kHz QPOs and had an estimate of the spin period either
through burst oscillations or through a direct detection in the case of accreting
millisecond pulsars, it turned out that the latter were close to Δν or half of it. More
specifically, if the ν spin was slower than ∼400 Hz, Δν ∼ ν spin , if it was faster
Δν ∼ ν spin /2. The symbol ∼ here is to be intended as “close to”, since Δν is not
constant for any particular source, but varies over a range. However, it was later
realised that the data are also compatible with Δν being essentially constant around
305 Hz [104], especially after multiplying the kHz QPO frequencies of accreting
millisecond pulsars by 1.5, as suggested by an offset in the correlation with the lowfrequency QPO frequencies [96, 172]. The situation can be seen in Fig. 6.10. Notice
that the spin period of 4U 0614+09 was discovered after the original version of this
plot was published and its Δν values fall on the constant-Δν track rather than the
Δν ∼ ν spin /2 one.
It is interesting to compare the distribution of all Δν values available in the
literature and the distribution of detected (or derived from burst oscillations) spin
periods (see chapter by Patruno and Watts, this book) as shown in Fig. 6.11. The
distribution of Δν values, coming from a large number of sources, peaks around
300 Hz and is well approximated by a Gaussian with centroid 305 Hz. The
distribution of pulse periods, obviously less populated, is rather flat between 200
and 600 Hz. From these data, it appears that the kHz QPOs are not related to the
spin period of the neutron star, although in a number of sources Δν does increase
towards ν spin with decreasing ν upp (e.g. [105, 107, 108], but see [70]).
Going back to the correlations between kHz QPO frequencies and theoretical
models, it is interesting to produce an updated version of the plot shown in Fig. 6.6
(originally shown for a few sources in [148]), which gives Δν vs. ν upp , where all
published values from RXTE are shown (the same values used for the top panel of
Fig. 6.11). They can be seen in Fig. 6.12. Notice that a prediction of the relativisticprecession model is that, for these masses, Δν should not exceed ∼400 Hz, which
indeed is what is observed.
However, when dealing with pairs of values, in this case ν low and ν upp , it is best
to plot them one versus the other. This was done in [104]; in Fig. 6.13 we show a
new version of that plot with all published values included (the same values used
for Fig. 6.12). The predictions of the relativistic-precession model for a neutron-star
mass of 1.8, 2.0 and 2.2 solar masses (dashed lines) fit rather well the distribution
of points at low frequencies, but diverge slightly at high frequencies, as can also be
M. Méndez and T. M. Belloni
6.6 QPO Frequency Correlations
Although the beat frequency model is unable to explain all observations, it is
reasonable to think that the kHz QPOs could in some way be connected to the
rotation of the neutron star. When the first source for which both burst oscillations
and kHz QPOs were discovered, 4U 1728−34, it was realised that Δν was around
the same value as the burst oscillation frequency [150]. However, the next source
with a double detection was 4U 1636−63, where Δν ∼ 270 Hz and the burst
oscillation frequency was 581 Hz, close to twice that value. After then, every time
a new source showed kHz QPOs and had an estimate of the spin period either
through burst oscillations or through a direct detection in the case of accreting
millisecond pulsars, it turned out that the latter were close to Δν or half of it. More
specifically, if the ν spin was slower than ∼400 Hz, Δν ∼ ν spin , if it was faster
Δν ∼ ν spin /2. The symbol ∼ here is to be intended as “close to”, since Δν is not
constant for any particular source, but varies over a range. However, it was later
realised that the data are also compatible with Δν being essentially constant around
305 Hz [104], especially after multiplying the kHz QPO frequencies of accreting
millisecond pulsars by 1.5, as suggested by an offset in the correlation with the lowfrequency QPO frequencies [96, 172]. The situation can be seen in Fig. 6.10. Notice
that the spin period of 4U 0614+09 was discovered after the original version of this
plot was published and its Δν values fall on the constant-Δν track rather than the
Δν ∼ ν spin /2 one.
It is interesting to compare the distribution of all Δν values available in the
literature and the distribution of detected (or derived from burst oscillations) spin
periods (see chapter by Patruno and Watts, this book) as shown in Fig. 6.11. The
distribution of Δν values, coming from a large number of sources, peaks around
300 Hz and is well approximated by a Gaussian with centroid 305 Hz. The
distribution of pulse periods, obviously less populated, is rather flat between 200
and 600 Hz. From these data, it appears that the kHz QPOs are not related to the
spin period of the neutron star, although in a number of sources Δν does increase
towards ν spin with decreasing ν upp (e.g. [105, 107, 108], but see [70]).
Going back to the correlations between kHz QPO frequencies and theoretical
models, it is interesting to produce an updated version of the plot shown in Fig. 6.6
(originally shown for a few sources in [148]), which gives Δν vs. ν upp , where all
published values from RXTE are shown (the same values used for the top panel of
Fig. 6.11). They can be seen in Fig. 6.12. Notice that a prediction of the relativisticprecession model is that, for these masses, Δν should not exceed ∼400 Hz, which
indeed is what is observed.
However, when dealing with pairs of values, in this case ν low and ν upp , it is best
to plot them one versus the other. This was done in [104]; in Fig. 6.13 we show a
new version of that plot with all published values included (the same values used
for Fig. 6.12). The predictions of the relativistic-precession model for a neutron-star
mass of 1.8, 2.0 and 2.2 solar masses (dashed lines) fit rather well the distribution
of points at low frequencies, but diverge slightly at high frequencies, as can also be