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 interpretations!) but data-limited � � We know these sources can vary like crazy: time-averaged spectra may not be very 2004 meaningful 2009 � � But, PSDs are hard to interpret physically. Data- rich (lots of phenomenology!) but model-poor

Various approaches: Fourier and energy combinations Fourier space Energy space 0.02-0.03 Hz 2-6 Hz 0.5-1 keV 3-10 keV 3-10 vs 0.5-1 keV lags Spectral-timing techniques (the RXTE era) References: Vaughan et al., 2003, MNRAS, 345, 1271 (rms spectrum) Nowak et al., 1999, ApJ, 517, 355 (coherence and lags)

Energy-dependent PSDs Simply compare the PSDs for light curves made for different energy bands Cyg X-1 softest state Cyg X-1 intermediate state 7.5-15 keV 2-7.5 keV 6-15 keV 2-6 keV Difference only in normalisation Difference in shape and normalisation How can we explain these behaviours? Can understand better when we understand rms spectrum The rms spectrum With good timing data, best approach is to make a Fourier-frequency-resolved rms spectrum , by using property that the integral of the PSD ( P ( � )) equals the variance (Parseval’s theorem): ( P noise is the observational � 2 � r ( E , � 1 , � 2 ) = ( P ( E , � ) � P noise ( E ) ) d � noise level in the PSD) � 1 If PSD in units of fractional rms 2 Hz -1 , the resulting rms is the fractional rms produced by variations in that frequency range. If PSD is not normalised by mean 2 (it is in units of (counts s -1 ) 2 Hz -1 ), the rms is in ‘detector units’ of counts s -1

The fractional rms spectrum 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 ) r ( E ) or, maybe x ( E ) x ( E ) log( 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) Interpreting the rms spectrum 1 Absolute rms vs energy (don’t divide by mean) can be fitted just like a time-averaged spectrum: power- Cyg X-1 law r ( E ) soft state � Fractional x ( E ) rms looks mean something like this: log( E ) diskbb rms 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

rms spectrum of kHz QPOs � � 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) Interpreting the rms spectrum 2 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: E pivot log[ F ( E,t )] r ( E ) E pivot log( E ) log( E ) Pivoting produces rms-spectra that are softer or harder than the underlying power-law depending on whether observed E < E pivot or E > E pivot

Energy-dependent PSD shape changes Differences in PSD shape with energy can be explained if different spectral components produce variations on different time-scales E.g. in NS GX 340+0, the QPO and higher- frequency noise show rms spectra that look Comptonised bb like Comptonised bb (prob. from boundary layer). But LF noise shows soft excess – Soft variability due low-frequency disc to diskbb? variations? (Gilfanov et al. 2003) 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 ( � ) = A S ( � )exp( i � AS ) + N S ( � )exp( i � NS ) H ( � ) = A H ( � )exp( i � AH ) + N H ( � )exp( i � NH ) The cross-spectrum is defined as: C ( � ) = S * ( � ) H ( � ) So that (dropping some of the � ’s to save space) : [ ] + A S N H exp i ( � NH � � AS ) [ ] C ( � ) = A S A H exp i ( � AH � � AS ) + A H N S exp i ( � AH � � NS ) [ ] + N S N H exp i ( � NH � � NS ) [ ]

Cross-spectrum continued C ( � ) = A S A H exp i ( � AH � � AS ) [ ] + A S N H exp i ( � NH � � AS ) [ ] + A H N S exp i ( � AH � � NS ) [ ] + N S N H exp i ( � NH � � NS ) [ ] C( � ) = C cor ( � ) + C uncor ( � ) For different samples: argument is constant argument drawn from U(0,2 � ) ... Therefore averaging over many samples gives: phase lag C ( � ) � C cor ( � ) = A S A H exp i � � [ ] between h and s Common convention: hard lags are positive, soft lags negative A note on signal-to-noise The signal-to-noise of spectral timing measurements depend on which of the noise terms dominates, S/N can then be: A S A H A S A H , , N S N H N S N H 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 Hard photons lag soft photons Time lag (phase lag)/(2 �� ) (Phase lag)/2 � 6-15 keV 2-6 keV Cyg X-1 ‘soft’ state Cyg X-1 hard state Propagation model for lags The simplest expectation is that each Lorentzian originates at a different radius

Lags vs energy (Cyg X-1 hard state) Nowak et al. 1999 Kotov et al. 2001 Roughly log-linear energy dependence (Comptonisation – prob. not), but wiggles around 6-10 keV suggests role for reflection? Lags vs Fourier frequency: QPOs GRS 1915+105 (Muno et al. 2001)

Spectral coherence ‘Intrinsic’ coherence: what fraction of variability (after correcting for observational noise) in two bands is correlated between the bands? 2 � ( noise term ) C ( � ) 2 ( � ) = � I ( ) P ( ) P S ( � ) � P H ( � ) � P noise , S noise , H Nowak & Vaughan 1997 (read for great overview of method) New approaches, and pushing below 3 keV (the XMM-Newton era)

Problems with rms spectra � � rms per channel: can have low S/N esp. at high energies � � Noise subtraction can lead to negative variance – bias creeps in at high energies GRO J1655-40 Something new: covariance spectra � � 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 rms (much better than covariance conventional technique!) � � But caveats apply…. 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 rms � � But measures only covariance correlated component (can be complicated if spectral coherence < unity …but could also be useful to pick out physically connected components) BHXRB vs AGN S/N

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 Energy-dependence of PSD vs covariance spectra

Lag-energy spectra Interpretation 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

The future of spectral-timing 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: Input signal Transfer function Output signal = * Time Delay Time � � Differences in the transfer function for different energies lead to energy-dependent PSD shape, lags etc.

Mapping the transfer function Consider signal x ( t ), transfer function r ( t , E ) and resulting output light curve f ( t , E ). The convolution theorem of FTs gives: f ( t , E ) = x ( t ) � r ( t , E ) F ( � , E ) = X ( � ) R ( � , E ) � So that: 2 R * ( � , E ) R ( � , E ref ) F * ( � , E ) F ( � , E ref ) = X ( � ) 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 Iron line+reflection Reprocessed disk thermal emission 6.4 keV 1 keV 5.5 keV 0.7 keV 4 keV 0.3 keV � � 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