3.1.1 Lightcurve, colour-colour and hardness intensity diagram

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1 Chapter 3 X ray data analysis methods 3.1 Data Analysis Procedure The analysis and reduction procedure of astronomical data can be broadly classified into two categories - (1) count rate variations as a function of time/frequency, i.e., timing variability studies (2) count rate variability as a function of photon energy, i.e., energy spectral studies. These methods and few techniques which have been used for the purpose of data analysis in this work have been discussed below. Some special techniques like flux resolved spectroscopy have also been used and they have been described in relevant chapters Lightcurve, colour-colour and hardness intensity diagram Variations in X ray intensity as a function of time at a particular energy is known as lightcurve. Nature of astrophysical objects can sometimes be identified using X ray lightcurve, e.g., a rapidly rotating neutron star or pulsar produces highly regular, structured pulsations in the light curve whereas a black hole X ray binary mostly shows random fluctuations at variable intensity level. Another important aspect of lightcurve is the measure of the timescale in the range of few seconds to few hundreds of seconds at which different physical processes occur in an accretion disk or inflow and outflow of accreting material into compact object. In order to study the X ray flux contribution at different energies to the total X ray intensity variations, ratio of X ray intensities at different energy band are useful. Colour-colour diagram (CD) is defined as the plot 26

2 of hard colour vs soft colour. Although the choice of energy band varies depending upon the nature of the compact object, energy bands below 8 kev are used for soft colour while energy bands above 8 kev are used for hard color. Hardness intensity diagram (HID) is defined as the plot of hard colour as a function of X ray intensity. Both CD and HID provide a rough spectral evolution of an source which is importantly, a model independent approach. For example, the X ray outburst evolution of all transient black hole X ray binaries (BHXBs) produce a q shaped diagram in the HID. Neutron star X ray binaries, which have been broadly categorized into two classes depending on distinct nature of tracks in the CD and the HID, are known as Z sources and atoll sources. Physical parameters like mass accretion rate is found to evolve across the q diagram or the Z track which clearly brings out the importance of the CD and HID in understanding accretion into low mass X ray binaries. The role of Z track in NSXBs in understanding accretion mechanism has been discussed in Chapter 1. The importance of q diagram as an analysis tool is discussed below Importance of q - diagram in black hole X ray Binaries The X ray outbursts evolution in BHXBs using a hardness intensity diagram (HID) tracks a q shaped diagram where the hardness ratio is the ratio of detector counts in two energy bands, preferably a hard band ( kev) and a soft band ( kev) [Belloni, 2004; Homan et al., 2001; van der Klis, 2005]. Figure 3.1 shows a q diagram typically traced by the transient BHXB GX (also see, Dunn et al. [2008]) where the source evolution is shown by arrows from A to F. Along with the evolution of Radio jets along the outbursts, it is also known as unified X ray-radio jet model or disk-jet coupling model (see Chapter 1), first proposed by Fender, Belloni & Gallo [2004]. Although exact interpretation of emission process is difficult due to mix of soft and hard X ray emission in different bands, this model shows the evolution of X ray states and their transitions in transient BHXBs, described in Chapter 1. The vertical source track on the right side of Figure 3.1 (i.e., from A to B) corresponds to the low/hard or hard state with the highest hardness value during the rising phase of the outburst. This track start from the quiescent state where the source is below the detectable limit of RXTE/PCA. The spectra is mostly powerlaw dominated which is usually due to the inverse comptonization process. One interesting aspect of 27

3 Figure 3.1: A q diagram observed with the RXTE/PCA in GX (taken from Fender & Belloni [2012]). this part of the HID is that at the end of the track (i.e., near the point B), the source reaches the near Eddington luminosity while the spectra is dominated by hard powerlaw. This has been interpreted in terms of luminous hot accretion flow (LHAF) in the BHXBs [Yuan & Zdziarski, 2004]. LHAF also successfully explain the possible origin of strong Radio jets during this period [Liu & Wu, 2013]. Vertical track on the far left of the q diagram at low X ray colour corresponds to the high/soft or thermal state. Radio jets are suppressed during this period and the emission is dominated by thermal blackbody radiation. Remillard & McClintock [2006] classifications agree very well in these two regimes. However, for the region in the q diagram which is intermediate between extreme right and extreme left track, the states are described differently. The states in this region are further divided by Belloni et al. [2005] into soft intermediate and hard intermediate states, based on the X ray colours and the properties of the power continuum as well as quasi periodic oscillations (QPOs) in the PDS. Transition from one state to another has also signature in both X ray PDS as well as radio jets. For example, the transition from the soft intermediate to the soft state has been sharply marked by a superluminal radio jet in GX [Belloni et al., 2005]. For a long time, study of the outburst evolution in accretion disk remains 28

4 a matter of immense interests since it traces different physical processes in the disk at different luminosity and connects different physical parameters using an approach which is model-independent. There are two important limitations of lightcurve analysis: (1) Any pulsations with the time scales of less than a second can not be observed using lightcurve. Pulse frequency of the order of khz may originate from the inner accretion disk where gravitational force is strongest and plays an important role in deciding accretion geometry. (2) Depending upon the source spectral state, random fluctuations may have variable power which can not be decided from the lightcurve, CD or HID analysis. Fourier power spectral analysis is an important tool which overcomes both limitations Power spectra analysis Fourier power spectrum of a time series is a technique in which count rate is transformed from time to frequency space. This tool is useful in X ray variability studies, particularly when counting noise dominates the time series. In this method, the continuous time series is split up into blocks of equal time intervals and for each interval Fourier spectrum is calculated. These power spectra are then averaged to produce an average variability of the source for a specific time interval. Two normalization methods are generally used to plot power spectra: (1) Leahy normalization [Leahy et al., 1983] or (2) rms normalization. The inverse of the length of each segment of the time series provide the frequency resolution of the spectra whereas half of the inverse of the time resolution of the data provide the maximum frequency up to which the power spectra is computed. This is known as Nyquist frequency. All sorts of periodic and aperiodic variabilities appears either as a peak in the spectra or as a broad structure, usually known as noise. Among peak structures two are important (1) Highly coherent signals from pulsars which appear like a single frequency bin in the PDS. (2) Quasi-periodic oscillations (QPOs) which are narrow peaked features appear at different frequencies ranging from mhz (observed only from BHXBs) to khz (usually observed from NSXBs). Power spectra from different astrophysical sources are usually consists of superposition of different components, the origin of which is not clearly understood. Noise components and QPOs are usually fitted with Lorentzian functions which represent 29

5 RMS nomalized power (rms/mean) 2 Hz χ Frequency (Hz) Figure 3.2: An example of a power density spectrum of GRS where noise components as well as quasi-periodic oscillations are observed, is fitted with different model components and the residual is also shown. the Fourier transform of damped oscillations. The continuum is usually fitted with the power-law model while a broken power-law model is preferred if a low frequency break in power-law is observed. QPOs and highly coherent pulsations are usually characterized by three parameters : (1) characteristic frequency of the pulsation, f max, where the variance per logarithmic frequency interval is maximum. (2) Quality factor (q-factor) of the QPO which is defined as ν peak /FWHM. FWHM is the full width at half maximum of the Lorentzian function (3) rms amplitude (%) of the QPOs which is measured as the square root of the normalization, obtained by fitting a rms normalized power spectrum. It is observed that different power spectral features are associated with different energy spectral states which will be explored in details at Chapter 5. An example of a rms normalized PDS is shown in Figure 3.2 where the PDS continuum and QPOs are fitted with broken powerlaw and Lorentzians Cross spectra analysis : coherence function and time/phase lags A coherence function is the frequency dependent measure of the degree of linear correlation between two time series at different energies. For completely incoherent process, the ideal value of coherence function is 0 and its value is 1 for perfectly coherent time series. In practice, presence of Poisson counting noise in a signal, however, always reduces the coherence below 1 even for a perfectly synchronized time series. A random 30

6 part to each phase measurement has been contributed by Poisson noise, causing the misalignment of the phase in complex spectra. However, this noise always dominates at sufficiently high frequencies. Coherence is also difficult to measure at very low frequencies because of the low statistics due to the number of independent measurements is smaller. Although less explored than power density spectral (PDS) study, measurement of coherence is particularly useful when a finer study at a specific Fourier frequency is required like observations of Quasi Periodic Oscillations and its harmonics, break in the continuum of the PDS. To calculate coherence function, cross spectra techniques have been used. Complex cross spectra or complex spectral density is a complex quantity whose phase is a measure of the shift in phase between two time series at a particular frequency. This shift in frequency between two light curves are known as the phase lag which is used to calculate time lag. Study of lag properties and coherence function also has one more important aspect. Lag timescale in the order of micro-sec is usually observed from Neutron Star X ray binaries (NSXBs). Now, the light travel timescale from a standard accretion geometry involving Comptonizing corona is also of the order of micro-sec. Thus, it is believed that the lag measurement is the most powerful tool to probe accretion and radiation geometry which, till date, is yet to understand completely. A Fortran code is developed to compute coherence function, phase lag and time lag spectra between two time series in the following way. To calculate phase lag for each observation, the data is divided into two energy bands and produce a cross spectrum for every 16 s intervals. The cross spectrum is defined as the complex cross product X k ( f)=s k ( f)h k( f)= S k ( f) H k ( f) e i(φ H φ S ), (3.1) where S k (f) and H k (f) are the kth component of the Fourier Transform of time series s(t) and h(t), the soft and the hard light curves, at a frequency f k, and φ S & φ H are their phases, respectively. The phase lag between the signals in the two bands at Fourier frequency f k is φ k (f) = arg[x k (f)] =φ H - φ S (the position angle of X k (f) in the complex plane) and the corresponding Fourier time lag is φ k (f)/2πf. We calculated an average cross vector X by averaging complex values over multiple adjacent 16 s spectra and then finding the final value of φ k (f) versus Fourier frequency. The phase lag analysis 31

7 is limited between 0.1 Hz and 7 Hz as all QPOs and their harmonics lie within this region. Below 0.1 Hz, systematic errors become significant [Nowak et al., 1999a] while above 10 Hz errors due to binning effects are important [Crary et al., 1998]. In all cases, positive phase lag means that hard photons are lagging soft photons. As the effect due to dead-time is negligible, no dead-time correction is considered into our calculation. For all observations we have calculated the coherence function in different energy bands given by [Nowak et al., 1999a], γ 2 ( f)= S ( f)h( f) 2 S( f) 2 H( f) 2, (3.2) between FFTs for soft energy band (S( f)) and the FFTs for hard energy bands (H( f)), angle brackets indicating ensemble averages over Fourier frequencies and individual data segments [Vaughan & Nowak, 1997]. In case of Gaussian statistics, the error using one sigma uncertainty in γ 2 (f) is [Nowak et al., 1999a] δγ 2 ( f)= 2γ 2 ( f)[1 γ 2 ( f)] γ( f) n, (3.3) where n is the number of measurements considered in computing coherence function [Bendat & Piersol, 2010; Vaughan & Nowak, 1997]. A detailed discussion on the implication of coherence function can be found in Nowak et al. [1999a,b]. The uncertainty in phase φ k (f) (computed including taking account of the coherence function) [Nowak et al., 1999a] is - φ k ( f)=n 1/2 1 γ 2 ( f) 2γ 2 ( f) (3.4) where N is the number of lightcurves averaged over and γ 2 (f) is the coherence of two light curves without the correction for counting noise. Thus, the error on time lag is φ k (f)/2πf. Errors measured in this way provide more rigorous estimations than simple propagation errors [Nowak et al., 1999a]. Using above equations, we divide the total X ray counts (C) of each time series by a specific number of energy bands (N) in such a way that each energy band has 32

8 Figure 3.3: An example of a time lag spectrum (left panel) and the corresponding coherence function (right panel) are shown with 1σ error-bars as a function of energy. In both panels, the reference energy band is shown in hollow circles. sufficiently large number of counts so that the probability of running out of counting statistics for any energy band is sufficiently low. If the Higher limit (E h )- lower limit (E l ) of an energy band exceeds the value 10.0 and the integrated counts are less than 10% of actual counts, then it is discarded from our calculations. The highest energy band for which the integrated count is exactly equal to C/N, is chosen as the reference band. For this reason, reference band in different energy bands are different. First report on the measurement of lag is 27 µs time delays in 830 Hz QPOs between 4 6 and kev in the NSXB 4U , with high-energy photons lagging low-energy photons, and found upper limits to the time delays of 45 µs between and kev in 730 Hz QPOs in 4U and 30 µs between and kev in 870 Hz QPOs in 4U [Vaughan et al., 1997]. 3.2 X ray spectral analysis X ray spectra are used to understand the distribution of normalized counts of photons having different energies. X ray spectra from astrophysical objects are basically complex resultant of different physical mechanisms. Each of them contribute as a characteristic spectrum either in the form of X ray emission lines or in the shape of a continuum emission. However, the source spectra usually convolved with the background spectra. Hence to obtain exact scientific informations from the observed spectra, it 33

9 is important to understand physical conditions giving rise to radiations and it is also equally important to understand the detector response as well as intrinsic background spectra Detector response and spectral fitting In general, a detector actually produces pulse height distribution at different channels of the spectrometer at different photon energies. Hence the probability that an incident photon of a particular energy will be observed in a particular instrument channel is provided by a response matrix which has been used to determine the detector response. A good response matrix usually consider (1) energy to channel relationship (2) quantum efficiency as a function of photon energy and (3) the spectral redistribution within the detector. Depending on the nature of interaction, different peaks in the spectrum may appear and continuum shape may also change accordingly. For example, in case of photo electric interactions, if the photon, originated from the outer to the inner shell electron transition, is not absorbed by the detector then an additional peak appears in the spectrum at the energy equal to the difference between incident photon energy and escape photon energy. This is known as escape peak. In case of Compton interaction, the fraction of energy deposited on the detector appears as the Compton continuum. Thus for each energy of the incident photon, a broad spectra appears across all pulse height channels. Hence the detector response is important to understand and construct properly to obtain actual source spectra. Since the response matrix is non invertible, obtaining incident spectra from above equation is not possible. Hence we convolve the model spectra with detector response to obtain observed spectra. A fit statistic (usually χ 2 minimization method is used) is usually engaged to check whether model spectra correctly describe observed spectra. If the model spectra agrees well with the observed spectra with high confidence, then the observed spectra is regarded as true incident spectra. A routine software, known as XSpec, is available which reads pulse height spectra, response matrix and background spectra for similar channels, performs fitting of several model spectra and provides a set of parameters for which the χ 2 minimization is achieved. Among source model spectrum, standard blackbody, disk blackbody, thermal and non-thermal comptonization, powerlaw and Gaussian/Lorentzian line profiles 34

10 are important one. Parameters of a well-fitted spectrum usually provide physical condition and geometry of radiating material. For example, fitting with disk blackbody model provide inner disk temperature and inner disk radius of a radiating accretion disk. The source spectrum is usually modelled using additive combination of few physical/phenomenological models spectra modified by the photon absorption model due to the presence of Galactic neutral Hydrogen column. 35