4 Accreting Millisecond X-ray Pulsars
177
derivatives may in some cases not be so dissimilar. We have marked each AMXP in
Table 4.3 with a “w” for weak timing noise and “s” for strong. This distinction is
somehow arbitrary, but is useful to understand the probability of the pulse frequency
derivatives differing from the true spin frequency derivatives.
To verify whether the pulse frequency derivative is a robust indicator of the spin
frequency derivative ˙
ν s , one has to investigate the strength of φ N on the timescales
over which ˙
ν is measured. To do this, one can use Monte Carlo (MC) simulations
to estimate the true uncertainty on the pulse frequency derivative [123]. The MC
method works as follows: one fits a second-order polynomial to measure ν and ˙
ν
with standard χ 2 minimization techniques after having removed all other effects
(such as orbital variations) and obtains an estimate of ν ± σ ν and ˙
ν ± σ ˙
ν . If in the
phase residuals there are still substantial variations of the pulse phase, in excess
of that expected from measurement errors alone (i.e., φ M )—which can be verified
by checking whether χ 2 gives a statistically unacceptable fit—then the statistical
errors σ ν and σ ˙
ν are not good representations of the true uncertainties of the spin
parameters. Instead one can take the phase residuals, calculate a power spectrum
and simulate several thousand time series with nearly identical noise content as the
original phase residual time series [123]. At this point ν and ˙
ν can be measured
for each simulated time series, and a distribution of parameters constructed. The
standard deviation of the distribution of ν and ˙
ν provides a good representation of
the true uncertainties on the spin parameters. In this way one can immediately check
whether the noise content φ N affects the measured value of ν and ˙
ν. Applying this
technique has revealed discrepancies between pulse and spin parameters in several
AMXPs (Table 4.3). The most striking finding has been the non-detection of spin
frequency derivatives in several AMXPs, with upper limits below the expected ˙
ν s .
The use of MC simulations is not the final word on this problem, since the
origin of timing noise remains unexplained. It also has its limitations: if the lowest
Fourier frequency component of timing noise is comparable to the length of the
data segment over which ν and ˙
ν are measured, MC simulations cannot distinguish
pulse and spin parameters. It has been noted that the phase residuals obtained after
removing ν and ˙
ν from the pulse phases of two AMXPs (XTE J1814-338 and XTE
1807-294) are anti-correlated with variations in X-ray flux [239, 287]. The anticorrelation improves substantially if one fits a simple ν = const model, suggesting
that timing noise, which is related to the X-ray flux variations, is almost entirely
responsible for the measured ˙
ν [258]. Correlations (or anti-correlations) between
pulse phase residuals (with respect to a ν = const model) and X-ray flux have now
been found in at least six AMXPs where such studies have been carried out [258].
In some cases the correlation was striking (e.g., in XTE J1814-338 [127]) leaving
little doubt that the pulse frequency derivative ˙
ν is not the spin frequency derivative
˙
ν s . This discovery also suggests that it is the pulse phase φ and not its second
derivative ( ˙
ν) that correlates with the flux (and thus the mass accretion rate),
as would instead be expected from Eq. (4.7). One way to test these findings is to
measure “instantaneous” short timescale spin frequencies and check whether these
scale with the bolometric flux as ˙
ν s ∝ F
γ
bol , where γ is a scale factor that depends
on accretion disk structure. The bolometric flux is, however, almost never available,
177
derivatives may in some cases not be so dissimilar. We have marked each AMXP in
Table 4.3 with a “w” for weak timing noise and “s” for strong. This distinction is
somehow arbitrary, but is useful to understand the probability of the pulse frequency
derivatives differing from the true spin frequency derivatives.
To verify whether the pulse frequency derivative is a robust indicator of the spin
frequency derivative ˙
ν s , one has to investigate the strength of φ N on the timescales
over which ˙
ν is measured. To do this, one can use Monte Carlo (MC) simulations
to estimate the true uncertainty on the pulse frequency derivative [123]. The MC
method works as follows: one fits a second-order polynomial to measure ν and ˙
ν
with standard χ 2 minimization techniques after having removed all other effects
(such as orbital variations) and obtains an estimate of ν ± σ ν and ˙
ν ± σ ˙
ν . If in the
phase residuals there are still substantial variations of the pulse phase, in excess
of that expected from measurement errors alone (i.e., φ M )—which can be verified
by checking whether χ 2 gives a statistically unacceptable fit—then the statistical
errors σ ν and σ ˙
ν are not good representations of the true uncertainties of the spin
parameters. Instead one can take the phase residuals, calculate a power spectrum
and simulate several thousand time series with nearly identical noise content as the
original phase residual time series [123]. At this point ν and ˙
ν can be measured
for each simulated time series, and a distribution of parameters constructed. The
standard deviation of the distribution of ν and ˙
ν provides a good representation of
the true uncertainties on the spin parameters. In this way one can immediately check
whether the noise content φ N affects the measured value of ν and ˙
ν. Applying this
technique has revealed discrepancies between pulse and spin parameters in several
AMXPs (Table 4.3). The most striking finding has been the non-detection of spin
frequency derivatives in several AMXPs, with upper limits below the expected ˙
ν s .
The use of MC simulations is not the final word on this problem, since the
origin of timing noise remains unexplained. It also has its limitations: if the lowest
Fourier frequency component of timing noise is comparable to the length of the
data segment over which ν and ˙
ν are measured, MC simulations cannot distinguish
pulse and spin parameters. It has been noted that the phase residuals obtained after
removing ν and ˙
ν from the pulse phases of two AMXPs (XTE J1814-338 and XTE
1807-294) are anti-correlated with variations in X-ray flux [239, 287]. The anticorrelation improves substantially if one fits a simple ν = const model, suggesting
that timing noise, which is related to the X-ray flux variations, is almost entirely
responsible for the measured ˙
ν [258]. Correlations (or anti-correlations) between
pulse phase residuals (with respect to a ν = const model) and X-ray flux have now
been found in at least six AMXPs where such studies have been carried out [258].
In some cases the correlation was striking (e.g., in XTE J1814-338 [127]) leaving
little doubt that the pulse frequency derivative ˙
ν is not the spin frequency derivative
˙
ν s . This discovery also suggests that it is the pulse phase φ and not its second
derivative ( ˙
ν) that correlates with the flux (and thus the mass accretion rate),
as would instead be expected from Eq. (4.7). One way to test these findings is to
measure “instantaneous” short timescale spin frequencies and check whether these
scale with the bolometric flux as ˙
ν s ∝ F
γ
bol , where γ is a scale factor that depends
on accretion disk structure. The bolometric flux is, however, almost never available,