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Anuj Nandi

Publications and source records attributed to Anuj Nandi.

At least 55 records · Page 3Linked to original sources

AstroSat and MAXI view of Cygnus X-1: Signature of an `extreme' soft nature

We present a detailed spectral and timing analysis of Cygnus X-1 with multi-epoch observations, during $2016$ to $2019$, by SXT and LAXPC on-board AstroSat. We model the spectra in broad energy range of $0.5\!-\!70.0\,\rm{keV}$ to study the evolution of spectral properties while Cygnus X-1 transited from hard state to an extreme soft state via intermediate states in 2017. Simultaneous timing features are also examined by modelling the power density spectra in $3.0\!-\!50.0\,\rm{keV}$ . We find that during high-soft state observations, made by AstroSat on Oct $24,\,2017$ (MJD $58050$), the energy spectrum of the source exhibits an inner disk temperature (kT$\rm_{in}$) of $0.46\!\pm\!0.01\,\rm{keV}$ , a very steep photon index ($Γ$) of $3.15\!\pm\!0.03$ along with a fractional disk flux contribution of $\sim\!45\%$. The power density spectrum in the range of $0.006\!-\!50.0\,\rm{Hz}$ is also very steep with a power-law index of $1.12\!\pm\!0.04$ along with a high RMS value of $\sim\!25\%$. Comparing the spectral softness of high-soft state with those of previously reported, we confirm that {\it AstroSat} observed Cygnus X-1 in the `softest' state. The lowest MAXI spectral hardness ratio of $\sim\!0.229$ corroborates the softest nature of the source. Moreover, we estimate the spin of the black hole by continuum-fitting method, which indicates that Cygnus X-1 is a maximally rotating `hole'. Further, Monte Carlo (MC) simulations are performed to estimate the uncertainty in spin parameter, which is constrained as a$_{\ast}>0.9981$ with $3σ$ confidence interval. Finally, we discuss the implications of our findings.

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Long-term XMM-Newton view of magnetar CXOU J010043.1$-$721134: Comprehensive spectral and temporal results

We present an in-depth analysis and results of eleven XMM-Newton datasets, spanning 2000 to 2016, of the anomalous X-ray Pulsar CXOU J010043.1$-$721134 which has been classified as a magnetar. We find a spin-period of 8.0275(1) s as of December 2016 and calculate the period derivative to be $(1.76\pm 0.02) \times 10^{-11}$ s s$^{-1}$, which translate to a dipolar magnetic field strength of $3.8\times 10^{14}$ G and characteristic age of $\sim 7200$ yr for the magnetar. It has a double-peaked pulse profile, with one broad and one narrow peak, in both soft ($0.3-1.3$ keV) and hard ($1.3-8$ keV) energy bands. The pulse fractions in the two energy bands are found to be consistent with constant values. These results are in agreement with previously published results for this source. Although two-component models produce acceptable fits to its energy spectra, single component models are much simpler and are able to explain the similarity of the pulse profiles in the low and high energy bands. We attempt fitting with four different single-component models and find that the best fit to the spectra is obtained by fitting a thermal Comptonization model with the photon index $(Γ)$ between $2.0-2.7$ and the electron temperature $(kT_e)$ between $0.5-0.9$ keV, for a seed blackbody photon distribution of 0.2 keV. Finally, we conclude by discussing our results briefly.

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A Machine Learning approach for classification of accretion states of Black hole binaries

In this paper, we employ Machine Learning algorithms on multi-mission observations for the classification of accretion states of outbursting Black hole X-ray binaries for the first time. Archival data from RXTE, Swift, MAXI and AstroSat observatories are used to generate the hardness intensity diagrams (HIDs) for outbursts of the sources XTE J1859+226 (1999 outburst), GX 339-4 (2002, 2004, 2007 and 2010 outbursts), IGR J17091-3624 (2016 outburst), and MAXI J1535-571 (2017 outburst). Based on variation of X-ray flux, hardness ratios, presence of various types of Quasi-periodic Oscillations (QPOs), photon indices and disk temperature, we apply clustering algorithms like K-Means clustering and Hierarchical clustering to classify the accretion states (clusters) of each outburst. As multiple parameters are involved in the classification process, we show that clustering algorithms club together the observations of similar characteristics more efficiently than the `standard' method of classification. We also infer that K-Means clustering provides more reliable results than Hierarchical clustering. We demonstrate the importance of the classification based on machine learning by comparing it with results from `standard' classification.

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Broadband 'spectro-temporal' features of extragalactic black hole binaries LMC X-1 and LMC X-3: An AstroSat perspective

We present the first results of extragalactic black hole X-ray binaries LMC X-1 and LMC X-3 using all the archival and legacy observations by AstroSat during the period of $2016-2020$. Broadband energy spectra ($0.5-20$ keV) of both sources obtained from the SXT and LAXPC on-board AstroSat are characterized by strong thermal disc blackbody component ($kT_{in}\sim1$keV, $f_{disc}>79\%$) along with a steep power-law ($Γ\sim2.4-3.2$). Bolometric luminosity of LMC X-1 varies from $7-10\%$ of Eddington luminosity ($L_{Edd}$) and for LMC X-3 is in the range $7-13\%$ of $L_{Edd}$. We study the long-term variation of the light curve using MAXI data and find the fractional variance to be $\sim25\%$ for LMC X-1 and $\sim53\%$ for LMC X-3. We examine the temporal properties of both sources and obtain fractional rms variability of PDS in the frequency range $0.002-10$ Hz to be $\sim9\%-17\%$ for LMC X-1, and $\sim7\%-11\%$ for LMC X-3. The `spectro-temporal' properties indicate both sources are in thermally dominated soft state. By modelling the spectra with relativistic accretion disc model, we determine the mass of LMC X-1 and LMC X-3 in the range $7.64-10.00$ $M_{\odot}$ and $5.35-6.22$ $M_{\odot}$ respectively. We also constrain the spin of LMC X-1 to be in the range $0.82-0.92$ and that of LMC X-3 in $0.22-0.41$ with 90\% confidence. We discuss the implications of our results in the context of accretion dynamics around the black hole binaries and compare it with the previous findings of both sources.

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AstroSat view of IGR J17091-3624 and GRS 1915+105: decoding the 'pulse' in the 'Heartbeat State'

IGR J17091--3624 is a transient galactic black hole which has a distinct quasi-periodic variability known as `heartbeat', similar to the one observed in GRS 1915+105. In this paper, we report the results of $\sim 125$ ks \textit{AstroSat} observations of this source during the 2016 outburst. For the first time a double peaked QPO (DPQ) is detected in a few time segments of this source with a difference of $δf ~\sim12$ mHz between the two peaks. The nature of the DPQ was studied based on hardness ratios and using the static as well as the dynamic power spectrum. Additionally, a low frequency (25--48 mHz) `heartbeat' single peak QPO (SPQ) was observed at different intervals of time along with harmonics ($50-95$ mHz). Broadband spectra in the range $0.7-23$ keV, obtained with \textit{SXT} and \textit{LAXPC}, could be fitted well with combination of a thermal Comptonisation and a multicolour disc component model. During \textit{AstroSat} observation, the source was in the Soft-Intermediate State (SIMS) as observed with \textit{Swift/XRT}. We present a comparative study of the `heartbeat' state variability in IGR J17091--3624 with GRS 1915+105. Significant difference in the timing properties is observed although spectral parameters ($Γ\sim2.1-2.4$ and $T_\mathrm{max}\sim0.6-0.8$ keV) in the broad energy band remain similar. Spectral properties of segments exhibiting SPQ and DPQ are further studied using simple phase resolved spectroscopy which does not show a significant difference. Based on the model parameters, we obtain the maximum ratio of mass accretion rate in GRS 1915+105 to that in IGR J17091--3624 as $\sim25:1$. We discuss the implications of our findings and comment on the physical origin of these exotic variabilities.

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AstroSat view of GRS 1915+105 during the Soft State: Detection of HFQPOs and estimation of Mass and Spin

We report the results of AstroSat observations of GRS 1915$+$105 obtained using 100 ks guaranteed-time (GT) during the soft state. The Color-Color Diagram (CCD) indicates a variability class of $δ$ with the detection of High Frequency QPO (HFQPO) in the power density spectra (PDS). The HFQPO is seen to vary in the frequency range of $67.96 - 70.62$ Hz with percentage rms $\sim 0.83 - 1.90$ % and significance varying from $1.63 - 7.75$. The energy dependent power spectra show that the HFQPO features are dominant only in $6 - 25$ keV energy band. The broadband energy spectra ($0.7 - 50$ keV) of SXT (Soft X-ray Telescope) and LAXPC (Large Area X-ray Proportional Counter) modelled with nthComp and powerlaw imply that the source has an extended corona in addition to a compact 'Comptonizing corona' that produces high energy emission and exhibits HFQPOs. The broadband spectral modelling indicates that the source spectra are well described by thermal Comptonization with electron temperature (kT$_{\rm e}$) of $2.07 - 2.43$ keV and photon-index ($Γ_{\rm nth}$) between $1.73-2.45$ with an additional powerlaw component of photon-index ($Γ_{\rm PL}$) between $2.94 - 3.28$. The norm of nthComp component is high ($\sim 8$) during the presence of strong HFQPO and low ($\sim 3$) during the absence of HFQPO. Further, we model the energy spectra with the kerrbb model to estimate the accretion rate, mass and spin of the source. Our findings indicate that the source accretes at super-Eddington rate of $1.17-1.31~ \dot{M}_{\rm Edd}$. Moreover, we find the mass and spin of the source as $12.44 - 13.09~M_{\odot}$ and $0.990-0.997$ with $90\%$ confidence suggesting that GRS 1915$+$105 is a maximally rotating stellar mass X-ray binary black hole source.

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AstroSat view of LMC X-2: Evolution of broadband X-ray spectral properties along a complete Z-track

In this paper, we report the first results of the extragalactic Z-source LMC X-2 obtained using the $\sim$ 140 ks observations with {\it Large Area X-ray Proportional Counter (LAXPC)} and {\it Soft X-ray Telescope (SXT)} onboard {\it AstroSat}. The HID created with the {\it LAXPC} data revealed a complete Z-pattern of the source, showing all the three branches. We studied the evolution of the broadband X-ray spectra in the energy range of $0.5-20.0$ keV along the Z-track, a first such study of this source. The X-ray spectra of the different parts of the Z-pattern were well described by an absorbed Comptonized component. An absence of the accretion disc component suggests that the disc is most probably obscured by a Comptonized region. The best fit electron temperature ($kT_e$) was found to be in the range of $1.7-2.1$ keV and optical depth ($τ$) was found to be in the range of $13.2-17.5$. The optical depth ($τ$) increased as the source moved from the normal/flaring branch (NB/FB) vertex to the upper part of the FB, suggesting a possible outflow triggered by a strong radiation pressure. The power density spectra (PDS) of HB and NB could be fitted with a pure power-law of index $α$ $\sim$ 1.68 and 0.83 respectively. We also found a weak evidence of QPO (2.8~$σ$) in the FB. The intrinsic luminosity of the source varied between $(1.03-1.79)$ $\times$ 10$^{38}$ ergs/s. We discuss our results by comparing with other Z-sources and the previous observations of LMC X-2.

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AstroSat and MAXI view of the Black Hole binary 4U 1630-472 during 2016 and 2018 Outbursts

We present an in-depth spectral and timing analysis of the Black Hole binary 4U 1630-472 during 2016 and 2018 outbursts as observed by \textit{AstroSat} and \textit{MAXI}. The extensive coverage of the outbursts with \textit{MAXI} is used to obtain the Hardness Intensity Diagram (HID). The source follows a `c'-shaped profile in agreement with earlier findings. Based on the HIDs of previous outbursts, we attempt to track the evolution of the source during a `super'-outburst and `mini'-outbursts. We model the broadband energy spectra ($0.7-20.0$ keV) of \textit{AstroSat} observations of both outbursts using phenomenological and physical models. No Keplerian disc signature is observed at the beginning of 2016 outburst. However, the disc appears within a few hours after which it remains prominent with temperature ($T_{in}$) $\sim$ 1.3 keV and increase in photon index ($Γ$) from 1.8 to 2.0, whereas the source was at a disc dominant state throughout the \textit{AstroSat} campaign of 2018 outburst. Based on the HIDs and spectral properties, we classify the outbursts into three different states - the `canonical' hard and soft states along with an intermediate state. Evolution of rms along different states is seen although no Quasi-periodic Oscillations (QPOs) are detected. We fit the observed spectra using a dynamical accretion model and estimate the accretion parameters. Mass of the black hole is estimated using inner disc radius, bolometric luminosity and two component flow model to be $3-9$ $M_{\odot}$. Finally, we discuss the possible implications of our findings.

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AstroSat/LAXPC view of GX 17+2: Spectral Evolution along the Z-track

In this paper, we present the first results obtained using $\sim$ 50 ks observations of the bright low-mass X-ray binary (LMXB) GX 17+2 with Large Area X-ray Proportional Counter (LAXPC) onboard {\it AstroSat}. The source traced out a complete Z-track in the hardness intensity diagram (HID). The spectra at different sections of the Z-diagram are well described by either a combination of a thermal Comptonization component, a power-law and a relativistic iron line or a model consisting of a thermal disk component, a single temperature blackbody, a power-law and a relativistic iron line. Fitting the spectra with both phenomenological models suggests that the power-law component is strong in the horizontal branch (HB), becomes weaker as the source moves down the normal branch (NB) and then again becomes stronger as the sources moves up the flaring branch (FB). However, we find that the strength of the power-law component is model dependent, although the trend in the variation of the power-law strength along the Z-track is similar. A simple model composed by a Comptonized emission and power-law component, convolved with the ionized reflection, also describes the spectra very well. A normal branch oscillation (NBO) with a centroid frequency 7.42$\pm$0.23 Hz, quality factor (Q) $\sim$ 4.88, rms 1.41$\pm$0.29\% and significance 5.1$σ$ is detected at the middle of the NB. The parameters of the Comptonized emission show a systematic evolution along the Z-diagram. The optical depth of the corona increases as the source moves up along the FB, suggesting possible trigger of an outflow or dumping of the disc material in to the corona by radiation pressure.

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Shocks in relativistic viscous accretion flows around Kerr black holes

We study the relativistic viscous accretion flows around the Kerr black holes. We present the governing equations that describe the steady state flow motion in full general relativity and solve them in 1.5D to obtain the complete set of global transonic solutions in terms of the flow parameters, namely specific energy (${\cal E}$), specific angular momentum (${\cal L}$) and viscosity ($α$). We obtain a new type of accretion solution which was not reported earlier. Further, we show for the first time to the best of our knowledge that viscous accretion solutions may contain shock waves particularly when flow simultaneously passes through both inner critical point ($r_{\rm in}$) and outer critical point ($r_{\rm out}$) before entering into the Kerr black holes. We examine the shock properties, namely shock location ($r_s$) and compression ratio ($R$, the measure of density compression across the shock front) and show that shock can form for a large region of parameter space in ${\cal L}-{\cal E}$ plane. We study the effect of viscous dissipation on the shock parameter space and find that parameter space shrinks as $α$ is increased. We also calculate the critical viscosity parameter ($α^{\rm cri}$) beyond which standing shock solutions disappear and examine the correlation between the black hole spin ($a_k$) and $α^{\rm cri}$. Finally, the relevance of our work is conferred where, using $r_s$ and $R$, we empirically estimate the oscillation frequency of the shock front ($ν_{QPO}$) when it exhibits Quasi-periodic (QP) variations. The obtained results indicate that the present formalism seems to be potentially viable to account for the QPO frequency in the range starting from milli-Hz to kilo-Hz as $0.386~{\rm Hz}\le ν_{QPO} \left(\frac{10M_\odot}{M_{BH}} \right) \le 1312$ Hz for $a_k=0.99$, where $M_{BH}$ stands for the black hole mass.

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AstroSat view of MAXI J1535-571: broadband spectro-temporal features

We present the results of Target of Opportunity (ToO) observations made with AstroSat of the newly discovered black hole binary MAXI J1535-571. We detect prominent C-type Quasi-periodic Oscillations (QPOs) of frequencies varying from 1.85 Hz to 2.88 Hz, along with distinct harmonics in all the AstroSat observations. We note that while the fundamental QPO is seen in the 3 - 50 keV energy band, the harmonic is not significant above ~ 35 keV. The AstroSat observations were made in the hard intermediate state, as seen from state transitions observed by MAXI and Swift. We attempt spectral modelling of the broadband data (0.7-80 keV) provided by AstroSat using phenomenological and physical models. The spectral modelling using nthComp gives a photon index in the range between 2.18-2.37 and electron temperature ranging from 21 to 63 keV. The seed photon temperature is within 0.19 to 0.29 keV. The high flux in 0.3 - 80 keV band corresponds to a luminosity varying from 0.7 to 1.07 L_Edd assuming the source to be at a distance of 8 kpc and hosting a black hole with a mass of 6 M$_{\odot}$. The physical model based on the two-component accretion flow gives disc accretion rates as high as ~ 1 $\dot{m}_{Edd}$ and halo rate ~ 0.2 $\dot{m}_{Edd}$ respectively. The near Eddington accretion rate seems to be the main reason for the unprecedented high flux observed from this source. The two-component spectral fitting of AstroSat data also provides an estimate of a black hole mass between 5.14 to 7.83 M$_{\odot}$.

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A Possible Model for the Long-Term Flares of Sgr A*

We examine the effects of magnetic field on low angular momentum flows with standing shock around black holes in two dimensions. The magnetic field brings change in behavior and location of the shock which results in regularly or chaotically oscillating phenomena of the flow. Adopting fiducial parameters like specific angular momentum, specific energy and magnetic field strength for the flow around Sgr A*, we find that the shock moves back and forth in the range 60--170R_g, irregularly recurring with a time-scale of ~5 days with an accompanying more rapid small modulation with a period of 25 hrs without fading, where R_g is the Schwarzschild radius. The time variability associated with two different periods is attributed to the oscillating outer strong shock, together with another rapidly oscillating inner weak shock. As a consequence of the variable shock location, the luminosities vary roughly by more than a factor of 3. The time-dependent behaviors of the flow are well compatible with luminous flares with a frequency of ~one per day and bright flares occurring every ~5 -- 10 days in the latest observations by Chandra, Swift and XMM-Newton monitoring of Sgr A*.

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Accretion-ejection in rotating black holes: a model for `outliers' track of radio-X-ray correlation in X-ray binaries

We study the global accretion-ejection solutions around a rotating black hole considering three widely accepted pseudo-Kerr potentials that satisfactorily mimic the space-time geometry of rotating black holes. We find that all the pseudo potentials provide standing shock solutions for large range of flow parameters. We identify the effective region of the shock parameter space spanned by energy ($\mathcal{E}_{\text{in}}$) and angular momentum ($λ_{\text{in}}$) measured at the inner critical point ($x_{\text{in}}$) and find that the possibility of shock formation becomes feeble when the viscosity parameter ($α$) is increased. In addition, we find that shock parameter space also depends on the adiabatic index ($γ$) of the flow and the shock formation continues to take place for a wide range of $γ$ as $1.5 \le γ\le 4/3$. For all the pseudo potentials, we calculate the critical viscosity parameter ($α_{\text{shock}}^{\text{cri}}$) beyond which standing shock ceases to exist and compare them as function of black hole spin ($a_k$). We observe that all the pseudo potentials under consideration are qualitatively similar as far as the standing shocks are concerned, however, they differ both qualitatively and quantitatively from each other for rapidly rotating black holes. Further, we compute the mass loss from the disc using all three pseudo potentials and find that the maximum mass outflow rate ($R^{\rm max}_{\dot m}$) weakly depends on the black hole spin. To validate our model, we calculate the maximum jet kinetic power using the accretion-ejection formalism and compare it with the radio jet power of low-hard state of the black hole X-ray binaries (hereafter XRBs). The outcome of our results indicate that XRBs along the `outliers' track might be rapidly rotating.

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Low angular momentum relativistic hot accretion flow around Kerr black holes with variable adiabatic index

We study the relativistic, time-independent, low angular momentum, inviscid, advective accretion flow around Kerr black hole. Considering the relativistic equation of state (REoS), we examine the transonic properties of the flow and find that there exists an upper bound of the location of the physically accepted critical point ($r^{\rm max}_{\rm out}$). However, no such limit exists when an ideal gas equation of state (IEoS) is assumed to describe the flow. Further, we calculate the global accretion solutions that contain shock waves and separate the domain of parameter space in angular momentum ($λ$) and energy (${\cal E}$) plane. We find ample disagreement between the shock parameter spaces obtained for REoS and IEoS, respectively. In general, post-shock flow (equivalently post-shock corona, hereafter PSC) is characterized by shock location ($r_s$) and compression ratio ($R$, measure of density compression across the shock front) which are uniquely determined for flow with given input parameters, namely $({\cal E}, λ)$. Using $r_s$ and $R$, we empirically compute the oscillation frequency ($ν_{QPO}$) of the shock front which is in general quasi-periodic (QP) in nature and retrace the domain of shock parameter space in the $r_s-R$ plane in terms of $ν_{QPO}$ for REoS around the weakly as well as rapidly rotating black holes. Finally, we indicate the relevance of the present work to explain the plausible origin of high frequency QPO (HFQPO) and its connection with the spin ($a_k$) of the Galactic black hole sources.

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Constraining the mass of the black hole GX 339-4 using spectro-temporal analysis of multiple outbursts

We carried out spectro-temporal analysis of the archived data from multiple outbursts spanning over the last two decades from the black hole X-ray binary GX 339-4. In this paper, the mass of the compact object in the X-ray binary system GX 339-4 is constrained based on three indirect methods. The first method uses broadband spectral modelling with a two component flow structure of the accretion around the black hole. The broadband data are obtained from {\it RXTE (Rossi X-ray Timing Explorer)} in the range 3.0 to 150.0 keV and from {\it Swift} and {\it NuSTAR (Nuclear Spectroscopic Telescope Array)} simultaneously in the range 0.5 to 79.0 keV. In the second method, we model the time evolution of Quasi-periodic Oscillation (QPO) frequencies, considering it to be the result of an oscillating shock that radially propagates towards or away from the compact object. The third method is based on scaling a mass dependent parameter from an empirical model of the photon index ($Γ$) - QPO ($ν$) correlation. We compare the results at 90 percent confidence from the three methods and summarize the mass estimate of the central object to be in the range $8.28 - 11.89~ M_{\odot}$.

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Spectral and Timing Properties of Atoll Source 4U 1705-44 : LAXPC/AstroSat Results

In this paper, we present the first results of spectral and timing properties of the atoll source 4U 1705-44 using $\sim$ 100 ks data obtained with Large Area X-ray Proportional Counter (LAXPC) onboard {\it AstroSat}. The source was in the high-soft state during our observations and traced out a {\it banana track} in the Hardness Intensity Diagram (HID). We study {\bf the} evolution of the Power Density Spectra (PDS) and the energy spectra along the HID. PDS show presence of a broad Lorentzian feature (Peaked Noise or PN) centered at $1-13$ Hz and a very low frequency noise (VLFN). The energy spectra can be described by sum of a thermal Comptonized component, a power-law and a broad iron line. The hard tail seen in the energy spectra is variable and contribute $4-30$\% of the total flux. The iron line seen in this source is broad (FWHM $\sim$ 2 keV) and strong (EW $\sim$ $369-512$ eV). Only relativistic smearing in the accretion disc can not explain the origin of this feature and requires other mechanism such as broadening by Comptonization process in the external part of the `Comptonized Corona'. A subtle and systematic evolution of the spectral parameters (optical depth, electron temperature etc.) is seen as the source moves along the HID. We study the correlation between frequency of the PN and the spectral parameters. PN frequency seems to be correlated with the strength of the corona. We discuss the implication of the results in the paper.

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Observational aspects of Outbursting Black Hole Sources - Evolution of Spectro-Temporal features and X-ray Variability

We report on our attempt to understand the outbursting profile of Galactic Black Hole (GBH) sources, keeping in mind the evolution of temporal and spectral features during the outburst. We present results of evolution of Quasi-periodic Oscillations (QPOs), spectral states and possible connection with Jet ejections during the outburst phase. Further, we attempt to connect the observed X-ray variabilities (i.e., `class' / `structured' variabilities, similar to GRS 1915+105) with spectral states of BH sources. Towards these studies, we consider three Black Hole sources that have undergone single (XTE J1859+226), a few (IGR J17091-3624) and many (GX 339-4) outbursts since the start of RXTE era. Finally, we model the broadband energy spectra (3 - 150 keV) of different spectral states using RXTE and NuSTAR observations. Results are discussed in the context of two component advective flow model, while constraining the mass of the three BH sources.

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Advective accretion flow properties around rotating black holes - application to GRO J1655-40

We examine the properties of the viscous dissipative accretion flow around rotating black holes in presence of mass loss. Considering thin disc approximation, we self-consistently calculate the inflow-outflow solutions and observe that the mass outflow rates decreases with the increase of viscosity parameter ($α$). Further, we carry out the model calculation of Quasi-periodic Oscillation frequency ($ν_{\rm QPO}$) that is frequently observed in black hole sources and observe that $ν^{\rm max}_{\rm QPO}$ increases with the increase of black hole spin ($a_k$). Then, we employ our model in order to explain the High Frequency Quasi-periodic Oscillations (HFQPOs) observed in black hole source GRO J1655-40. While doing this, we attempt to constrain the range of $a_k$ based on observed HFQPOs ($\sim$ 300 Hz and $\sim$ 450 Hz) for the black hole source GRO J1655-40.

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