X-ray spectral timing: methods and interpretation Phil Uttley Why - - PDF document
X-ray spectral timing: methods and interpretation Phil Uttley Why spectral-timing? Spectra are complex, hard to disentangle and GX 339-4 hard state interpretations are often degenerate model-rich (we have physical
Phil Uttley
Spectra are complex, hard to
disentangle and interpretations are often degenerate – model-rich (we have physical interpretations!) but data-limited
We know these sources can
vary like crazy: time-averaged spectra may not be very meaningful
But, PSDs are hard to
interpret physically. Data- rich (lots of phenomenology!) but model-poor
2004 2009
GX 339-4 hard state
Various approaches: Fourier and energy combinations
Fourier space Energy space
0.5-1 keV 3-10 keV
0.02-0.03 Hz 2-6 Hz
3-10 vs 0.5-1 keV lags
References: Vaughan et al., 2003, MNRAS, 345, 1271 (rms spectrum) Nowak et al., 1999, ApJ, 517, 355 (coherence and lags)
Simply compare the PSDs for light curves made for different energy bands
6-15 keV 2-6 keV 7.5-15 keV 2-7.5 keV
Cyg X-1 softest state Cyg X-1 intermediate state
Difference only in normalisation Difference in shape and normalisation
How can we explain these behaviours? Can understand better when we understand rms spectrum
r(E,1,2) = P(E,) P
noise(E)
( )d
1 2
Fourier-frequency-resolved rms spectrum, by using property that the integral of the PSD (P()) equals the variance (Parseval’s theorem): If PSD in units of fractional rms2 Hz-1, the resulting rms is the fractional rms produced by variations in that frequency range. If PSD is not normalised by mean2 (it is in units of (counts s-1)2 Hz-1), the rms is in ‘detector units’ of counts s-1
(Pnoise is the observational noise level in the PSD)
Absolute rms spectra (detector units) are easier to fit with models, while fractional rms allows a direct comparison with the PSD energy dependence When PSD normalisation is lower at softer energies we might get:
r(E) x (E) log(E) r(E) x (E) log(E)
For change in PSD normalisation only, we expect this shape to stay the same with frequency (but rms-spectrum normalisation will change with frequency to match PSD shape)
r(E) x (E) log(E)
Absolute rms vs energy (don’t divide by mean) can be fitted just like a time-averaged spectrum:
Fractional rms looks something like this:
But absolute rms much easier to interpret! In the soft state, E-dependence of PSD normalisation is due to presence of a constant disc component – dilutes fractional rms (hence PSD) at low energies across entire frequency range
Cyg X-1 soft state mean rms
diskbb power- law
Fourier-resolved
rms-spectrum can pick out kHz QPO
Shows that drop
in fractional rms at low frequencies is due to constant disk blackbody contribution
(Gilfanov et al. 2003)
Constant components are easy to detect with absolute rms spectra. But we could imagine more complex patterns, e.g. power-law pivoting about some energy:
log(E)
log[F(E,t)]
r(E) log(E)
Pivoting produces rms-spectra that are softer or harder than the underlying power-law depending on whether observed E < Epivot or E > Epivot
Epivot Epivot
Differences in PSD shape with energy can be explained if different spectral components produce variations
E.g. in NS GX 340+0, the QPO and higher- frequency noise show rms spectra that look like Comptonised bb (prob. from boundary layer). But LF noise shows soft excess – low-frequency disc variations?
(Gilfanov et al. 2003)
Comptonised bb Soft variability due to diskbb?
Cross-spectrum and lags
Imagine 2 light curves, s(t) and h(t) which contain a signal which is correlated between 2 bands (denoted A) and a signal which is uncorrelated between the 2 bands (denoted N). The FTs at a given frequency can be written as:
S() = AS()exp(iAS )+ NS()exp(iNS) H() = AH()exp(iAH )+ NH()exp(iNH )
The cross-spectrum is defined as:
C() = ASAH exp i( AH AS )
[ ]+ ASNH exp i(NH AS ) [ ]
+AHNS exp i( AH NS )
[ ]+ NSNH exp i(NH NS ) [ ] C() = S*()H()
So that (dropping some of the ’s to save space) :
Cross-spectrum continued
C() = ASAH exp i( AH AS )
[ ]+ ASNH exp i(NH AS ) [ ]
+AHNS exp i( AH NS )
[ ]+ NSNH exp i(NH NS ) [ ]
Therefore averaging over many samples gives:
C() = Ccor() + Cuncor()
For different samples: argument is constant argument drawn from U(0,2)
C() Ccor() = ASAH exp i
[ ]
phase lag between h and s
Common convention: hard lags are positive, soft lags negative
The signal-to-noise of spectral timing measurements depend
AS NS , AH NH , ASAH NSNH
The first two scale with sqrt(count-rate) and apply when A>>N common for AGN, the third scales linearly with count rate and applies when A<<N, most commonly found in X-ray binaries.
Lags vs Fourier frequency: broadband noise
6-15 keV 2-6 keV Cyg X-1 hard state Cyg X-1 ‘soft’ state (Phase lag)/2 Time lag (phase lag)/(2)
Hard photons lag soft photons
The simplest expectation is that each Lorentzian
radius
Roughly log-linear energy dependence (Comptonisation – prob. not), but wiggles around 6-10 keV suggests role for reflection?
Nowak et al. 1999 Kotov et al. 2001
Lags vs Fourier frequency: QPOs
GRS 1915+105 (Muno et al. 2001)
‘Intrinsic’ coherence: what fraction of variability (after correcting for observational noise) in two bands is correlated between the bands?
Nowak & Vaughan 1997 (read for great overview of method)
I
2() =
C()
2 (noise term)
P
S() P noise,S
( ) P
H() P noise,H
( )
rms per channel:
can have low S/N
energies
Noise subtraction
can lead to negative variance – bias creeps in at high energies
GRO J1655-40
Cross-spectral equivalent of
rms spectrum
Measures shape of spectrum
that is correlated with chosen ‘reference band’
High S/N reference band
gives high S/N spectrum (much better than conventional technique!)
But caveats apply….
rms covariance
See Wilkinson & Uttley (2009) for time-domain version of method
Covariance spectra: pros and cons
Has high S/N and does
not suffer problem of –ve variances
But measures only
correlated component (can be complicated if spectral coherence < unity …but could also be useful to pick out physically connected components) rms covariance
Pushing below 3 keV: the hard state of GX 339-4
XMM-Newton 2 orbits in March 2004, we use pn data
in timing mode (160 ksec useful exposure)
Spectrum shows a typical hard state with evidence
for power-law, diskbb and reflection components
PSD also consistent with hard state
GX 339-4 2004 GX 339-4 2009
At low frequencies, variations in mdot are produced at larger radius in disc, modulating disc emission before propagating in to the corona on the disc viscous time-scale At high frequencies, variations in mdot are produced at small radius in disc or in corona itself. Only a fraction of disc emission can respond, but all of corona does, and coronal heating dominates variability disc reverberation
Key to success: modelling spectral- timing behaviour
We can understand spectral-timing behaviour in
terms of transfer function which is convolved with the input signal to produce the output light curve:
Time Time
Delay Input signal Transfer function Output signal
Differences in the transfer function for different
energies lead to energy-dependent PSD shape, lags etc.
Mapping the transfer function
f (t,E) = x(t)r(t,E)
Consider signal x(t), transfer function r(t,E) and resulting output light curve f(t,E). The convolution theorem of FTs gives:
F*(,E)F(,Eref ) = X()
2 R*(,E)R(,Eref )
So that: I.e. the cross-spectrum encodes information about the input signal and the transfer function Moreover, the cross-spectrum of the transfer function can be used to infer the observed cross-spectrum, hence covariance, lags etc, i.e. we can fit model transfer functions to the data
How lag-vs-energy spectra map the disk
Transfer function depends on emissivity vs light-travel
delay (size-scale). Selecting on Fourier-frequency can pick out different parts of the emissivity profile at that energy.
We can map the reflection and disc thermal emission
Iron line+reflection Reprocessed disk thermal emission
6.4 keV 5.5 keV 4 keV 1 keV 0.7 keV 0.3 keV
Using IXO-HTRS to measure the disk inner radius of Cyg X-1 in the hard state
(100 ksec exposure, select variations > 10 Hz)