Constraining the Coronal Heights and Readjustment Velocities based on the Detection of a Few hundred seconds Delays in the Z source GX 17+2

The geometry of accretion disk in low-mass X-ray binaries (LMXBs) is well developed viz. truncated accretion disk (Esin et al. 1997; Done et al. 2007) and accretion disk corona models (e.g. Church et al. 2006) but still debatable in terms of location, geometry and especially the variability caused in corona due to the inner accretion flow as well as jet. Though the picture is relatively clear in black hole X-ray binaries (BHBs) but the complexity remains in case of atoll and Z sources (Done et al. 2002; Di Salvo et al. 2002; Church et al. 2006; Church et al. 2014). At low luminosities $\sim$0.001–0.5 L_E (Done et al. 2007) atoll sources exhibit two prominent branches i.e. island and banana states in hardness intensity diagram (HID) or color-color diagram (CCD) (Hasinger & van der Klis 1989; van der Klis 2006). The island state is further divided into upper branch (UB), lower branch (LB) and a few sources exhibit an extreme island state (van Straaten et al. 2003) at the lowest detectable luminosity. A vivid X-ray spectral transition is noted as atoll’s moves across the various branches. The banana state is often found to be at high luminosity with a softer spectrum (Barret 2001; Church et al. 2014) whereas at lower luminosities, a hard tail is a key characteristic of the island state. These sources have two to five band-limited noise, peaked noise, 20-60 Hz quasi periodic oscillations (QPOs), typically show $\sim$ 200-1350 Hz QPOs and sometimes twin kilo Hertz (kHz) QPOs in their power density spectrum (PDS) (van Straaten et al. 2005; van der Klis 2006 and references therein).

On the other hand Z sources X-ray radiation close to Eddington luminosity (0.1-1 L_E) and trace a Z-like shape in the HID and CCD (Hasinger & van der Klis 1989; van der Klis 2006). The Z shape is broadly divided into three branches where the energy spectrum and power spectrum systematically changes viz. horizontal branch (HB), normal branch (NB) and flaring branch (FB) with two vertexes i.e. at HB to NB and NB to FB transitions. Further classification depicts two subgroups among Z sources, Cyg X-2 like (Cyg X-2, GX 340+0 and GX 5-1) and Sco X-1 like (Sco X-1, GX 17+2, and GX 349+2) based on the shape and orientation of their branches (Kuulkers et al. 1994, 1997). XTE J1701-462 is an arch-type source exhibiting characteristics of both Z sources (Cyg X-2 and Sco X-1 like) as well as atoll sources (Homan et al. 2007; Lin et al. 2007). At high luminosity, XTE J1701-462 displayed Sco X-1 like variability followed by Cyg X-2 like variability at relatively lower luminosity and atoll like behavioral at decay phase. In general the variability across the Z shape is often considered to increase in the mass accretion rate ($\dot{M}$) from HB to FB whereas an opposite scenario was proposed (Church et al. 2008; Jackson et al. 2009). On the contrary, it was suggested that $\dot{M}$ is constant along the branches and Z track could be due to instabilities either due to radiation pressure at inner radius/boundary layer or possibly caused by different solutions of accretion flows (Homan et al. 2002, 2010; Lin et al. 2007). High resolution spectral data unveil that inner disk radius appears to be close to the neutron star (NS) at different luminosities using NuSTAR, XMM-Newton and Suzaku (Cackett et al. 2012; Chiang et al. 2016; Ludlam et al. 2017). The PDS is often associated with a 10-60 Hz QPO at HB, 5-8 Hz QPO at NB (quite different from HB QPOs) along with kHz QPOs. It is found the QPO frequency increases from $\sim$ 10 Hz at the top of HB to 60 Hz at NB and a 5-8 Hz QPO is suddenly appears which is probably connected to a puffed-up disk (Titarchuck et al. 2001; Sriram et al. 2011), though a proper mechanism is still unknown.

Various models can explain the spectra of Z sources at different branches leading to a model degeneracy and because of this it is difficult to constrain the geometry of accretion disk. The soft (thermal) spectral component of Z source is modeled by a single temperature black body or a multicolor disk black body (Mitsuda et al. 1984) originating from the accretion disk while the non-thermal hard component is associated to a corona/Compton cloud, a jet (Di Salvo 2000, 2002; Agrawal & Sreekumar 2003; Falanga et al. 2006; Lin et al. 2009) or bulk motion Comptonization (Titarchuk & Zannias 1998; Farinelli et al. 2007). These models assume that hard X-ray emission is associated to a quasi spherical corona located in the inner region of the accretion around the NS surface. However in the accretion disk corona model wherein corona is extended over the Keplerian disk is often used to unfold the spectra of Z sources using a cut-off power-law model (Church & Balucinska-Church 2004; Church et al. 2008, 2014). Radio emission associated with jets were observed at HB and NB (Migliari & Fender 2006; Migliari et al. 2007; Fender et al. 2007) and in-fact a jet model can explain both spectral and timing variability of Z sources (Reig & Kylafis 2015, 2016) where soft photons from disk/boundary layer are comptonized up-scattered in the jet.

Cross correlation function (CCF) studies provide important evidences on the geometry and location of various soft and hard X-ray emitting regions in accretion disk. Low Fourier time scale of 16/32 s anti-correlated hard delays of the order of 10 –1000 s were reported in a few BHBs (Choudhury et al. 2005; Sriram et al. 2007, 2009, 2010) indicating towards a truncated accretion disk geometry. Similar delays both anti-correlated hard and soft X-rays were reported for Z and atoll sources (Cyg X-2, Lei et al. 2008, GX 5-1, Sriram et al. 2012; 4U 1735–44, Lei et al. 2013; 4U 1608–52, Wang et al. 2014). These studies especially in Z sources show that most of delays were detected when the sources were in HB and NB where a corona/jet is always present (Fender et al. 2009; Migliari et al. 2007, 2011). Based on a few 100 s delays, energy spectrum and PDS, Sriram et al. (2012) suggested that a truncated accretion disk geometry is needed to explain such longer delays. In case of the black hole binary GX 339-4, spectral studies associated with an anti-correlated delay indicated towards the condensation of corona in the inner region of accretion disk (Sriram et al. 2010; Meyer-Hofmeister et al. 2012). These detected delays were interpreted as the viscous/readjustment time scales in the inner region of accretion disk. At high frequencies (0.1-100 Hz), a few milli-second time lag (both soft and hard) were detected in Z sources, indicating toward the Comptonization process in the corona/jet (Vaughan et al. 1999; Kotov et al. 2001; Qu et al. 2004; Arevalo & Uttley 2006; Reig & Kylafis 2016) where soft photons are up-scattered causing the hard lags (van der Klis 1987; Qu et al. 2001; Li et al. 2013). Milli-seconds soft lags either can be explained by a shot model where soft photons lags to hard photons (Alpar & Shaham 1985) or in a two layer Comptonization model (Nobili et al. 2000).

GX 17+2 is a burster Z source, located at the distance of 13 kpc (Galloway et al. 2008) with a possible spin frequency of 584 Hz (Psaltis et al. 1999; Homan et al. 2002; Belloni et al. 2002) with no confirmed optical counterpart. Using the XMM-Newton data, an inclination $\leq$ 45^o was reported which is supported by the fact that no X-ray dips have been observed (Cackett et al. 2010). First a detailed spectral and temporal study for GX 17+2 was performed by Homan et al. (2002) and argued that $\dot{M}$ is not the unique parameter to explain the flux/luminosity evolution on the Z track and found to be constant in GX 17+2. A similar scenario was discussed based on the multi-component spectral model and illustrated the evolution of Z track with a constant $\dot{M}$ along with three different process viz. Comptonization in HB, existence of slim disk transition from a standard thin disk as the source approaches to NB and rapid decrease of thin disk radius on FB (Lin et al. 2012). A broad iron line and a low inclination of about 25^o–38^o was reported using NuSTAR data (Ludlam et al. 2017). Penninx et al. (1988) observed radio emission in GX 17+2, found to be strong in HB and weak in FB along the Z track. Migliari et al. (2007) performed simultaneous X-ray and radio observation and found similar results and strongly suggested that NB oscillation is connected to the jet. Based on the NuSTAR data, the inner disk radius was constrained at 10.3–13.0 km. In order to constrain the geometry of inner accretion disk in GX 17+2, we report the CCFs between soft and hard X-ray light curves along with temporal and spectral studies using the PCA & HEXTE data of RXTE satellite and NuSTAR satellite data.

The public archival data of Rossi-XTE satellite (Swank 1999) was analyzed for GX 17+2. The satellite consists of three primary instruments the Proportional Counter Array (PCA; Jahoda et al. 2006), the High Energy X-ray Timing Explorer (HEXTE; Rothschild et al. 1995; Gruber et al. 1996) and the All Sky Monitor (ASM; Levine et al. 1996). Among five proportional counter units PCU 2 data have been used for analysis as it was the longest observational and best calibrated unit. Energy dependent light curves were generated using the standard 2 mode data observations spanning more than 2000 s (e.g. Sriram et al. 2012). HEASOFT v6.8 software was used to reduce and analyse the data. Background light curves were generated using PCABACKEST V3.8 and appropriate background models were invoked (bright back ground $\geq$ 40 count s^-1 and faint background $\leq$ 40 count s^-1). Single bit, generic bit and event mode data were used to study the PDS. For spectral analysis, PCU 2 data were used and calibration uncertainties were ascertained by adding 0.5% systematic errors. Spectra from HEXTE cluster A data were extracted in 15-50 keV. For generating source and background spectra, hxtback command was utilized, with a rocking interval of 32 s and were corrected for dead-time corrections using hxtdead (e.g. Sriram et al. 2013). Various models were used to unfold the spectra made available in XSPEC V12.7.1 (Arnaud 1996). HID was generated using the data collected in the energy domain of 6.2-8.7 keV, 8.7-19.7 keV and 2.0-19.7 keV with bin size of 32 s after carefully selecting the channels using RXTE gain epochs and gain change effects.

The Nuclear Spectroscopic Telescope Array Mission (NuSTAR) data analysis was performed for both Focal Plane Modules A and B i.e. FPMA and FPMB instruments (Harrison et al. 2013) by using a 120” circular region (similar to Ludlam et al. 2017) for background subtracted energy dependent light curves and for source and background spectrum. Background subtraction was done by using the region of same radius away from the source. The NuSTAR Data Analysis Software (NuSTARDAS) which is integrated in the HEASoft package was used for data extraction. The NuSTAR pipeline nupipeline was used to run all the data processing tasks in sequence. Nupipeline was used to perform data processing until the first 2 stages which is data calibration and data screening in order to produce cleaned event files. Upon producing the cleaned event files, the nuproducts task was used for products extraction (stage 3). DS9 was used for region selection and the nuproducts task was run by providing appropriate source and background region files, along with appropriate bin sizes and energy ranges for background subtracted light curve extraction and the same nuproducts task was used to obtain source and background spectrum, with the corresponding response and ancilliary files. Upon obtaining both the FPMA and FPMB source and background spectrum, along with the ancilliary response matrices, they were combined using the ADDASCASPEC command. ADDRMF was used to combine and produce a single redistribution matrix file (RMF). The GRPPHA command was used to group the combined spectra to have a minimum of 25 counts per bin. As data span over a wide range, the observation span of RXTE and NuSTAR satellite data has been indicated on the ASM lightcurve in Figure 1.

CCF studies were performed using soft (2-5 keV, 3-5 keV) and hard X-ray (16-30 keV) light curves obtained from the PCU2 of RXTE and NuSTAR data. It was found from the RXTE data that whenever the source was in FB, the CCF does not show any delay with a positive CCF. In HB and NB (Figure 2, top), we found a positive correlation with no delays, correlation with soft and hard delays, anti-correlation with zero delays, anti-correlation with soft and hard delays (Figure 3 & Table 1). Hard/soft delay means that hard X-ray photons delayed to the soft/hard photons. It was found that delays were seen on a time scale of tens to a few hundred seconds and as such systemic trend is not observed from HB to NB. Similar behavior has been previously observed in Z type sources (Lei et al. 2008; Sriram et al. 2012; Ding et al. 2016). It is noted that whenever the anti-correlated delays occurred, CCF coefficients are often centered around $\sim$0.4–0.6 suggesting that hard X-ray photons are not linearly responding to soft photons. We found delays during which radio observation evidently point towards a jet (Migliari et al. 2007; see their Figure 1 & 2). In ObsID 700023-01-01-01, the first section clearly shows a correlated hard delay of 100$\pm$28 s (radio flux was high), followed by an anti-correlation and ending with an anti-correlated soft delay of –168$\pm$42 s (relatively low radio flux). If the NS surface/disk and hard photons are arriving from the base of the jet, then decrease in size of the base of jet would tend to decrease the hard flux too. The $\sim$100 s delay would be a time scale corresponding to change in the size of the base of jet.

NuSTAR data also revealed an anti-correlated hard X-ray delay of the order of $\sim$ 113$\pm$51 s with a CCF coefficient centered around $\sim$–0.6 at HB/NB branch (Figure 2, bottom) along with uncorrelated CCFs (Figure 4 & 5). The soft and hard energy band X-ray light curves clearly show that soft and hard flux is oppositely varying. In a truncated accretion disk scenario, the delay can be explained if initially the disk truncated away and due to radial movement of disk, it cools the inner corona resulting in relatively low hard flux. Or if the accretion disk is present close to the NS surface at about 1.03–1.30 ISCO as reported by Ludlam et al. (2017) then the observed hard delay could be a gradual decrease in the size of the region responsible for the hard flux, which could be the corona varying in size.

Power density spectra were obtained for ObsID 20053-03-03-01 and 70023-01-01-01 for three sections of the light curves (left and right panels of Figure 6). As discussed previously, ObsID 70023-01-01-01 shows relatively high degree of associated change in the X-ray and radio flux which is also associated with delay in CCF. It is observed that ObsID 20053-03-03-01 shows similar kind of light curve variation and hence we performed a comparative study. Both the PDSs are associated with HB with hardness ratios (HR) varying from 0.060-0.037 for 20053-03-03-01 and 0.046-0.064 for 70023-01-01-01.

In ObsID 200053-03-03-01, QPOs were found to vary from $\sim$ 32 Hz (section A) to $\sim$ 24 Hz (section B & C) and associated CCFs show soft X-ray delays of $\sim$- 207 s in section A, $\sim$ -668 s in section B and $\leq$-100 s in section C (Figure 3b). Whereas in the other observation, 70023-01-01-01, a $\sim$26 Hz QPO along with a harmonic was present in all the sections within error bars. From the CCF analysis, we observed a hard delay of $\sim$ 114 s in section A and a soft delay of $\sim$ –49 s in section C (Figure 3k and Table 1). Although in both observations a $\sim$26 Hz QPO was present, delays were found to drastically change. Assuming that observed delays are readjustment time scales associated with truncated accretion disk geometry, these results indicate that a QPO alone cannot constrain the accretion disk geometry.

To constrain and estimate spectral variation during delays, we investigated the ObsID 700023-01-01-01, for each section of the light curve (Figure 7) during which radio flux changed considerably along with X-ray flux. The source is in the upper horizontal branch during this observation (see Figure 2, Top). Section A, with a delay of 114 s, has a CCF of $\sim$ 0.6 and Section B shows a CCF of -0.3 with no delay, while section C shows a delay of -168 s having a CCF of -0.45 (Figure 3k). We unfolded spectra by two different models, first with a thermal Comptonization model viz. wabs(BB+Gaussian+ nthcomp) (Zdziarski et al. 1996), where we assumed that seed photons arriving from the NS and/or boundary layer are Comptonized in the Compton cloud. It was shown that such a thermal comptonization is needed for this source (Di Salvo et al. 2000), however we opted for nthcomp instead of CompTT model (Titarchuck 1994). We found that a Gaussian emission line is needed and its centroid energy was fixed at 6.5 keV and hydrogen equivalent column density N_H was fixed at 3 $\times$ 10²² cm^-2. In nthcomp model, there are five parameters viz. the asymptotic power-law photon index, electron temperature (kT_e), seed photon temperature, input type, kT_soft which decide which seed photon will Compton up-scatter either it could be from disk or black body, and the normalization. The black body seed photons were chosen to fit the continuum and kT_soft parameter was allowed to vary. It was noted that this model gave a reasonable fit without any addition of a power-law component in the energy range of 3-50 keV, however a power-law component was needed in the analysis by Migliari et al. (2007). Unfolded spectra along with their model components are shown in Figure 7. The asymptotic power-law index, $\Gamma_{nthcomp}$ = 2.34$\pm$ 0.24 and the electron temperature kT_e = 4.18 $\pm$ 0.85 keV in the first section were found to increase in the second and third sections ($\Gamma_{nthcomp}$ = 2.80$\pm$0.16, kT_e=7.59 $\pm$ 2.01 keV) (see Table 2). For ObsID 20053-03-03-01, we unfolded the spectra similarly using the model wabs(BB+Gaussian+nthcomp). A few of the parameters were fixed as discussed above. We found that kT_e increase from 2.15$\pm$0.85 keV (section A) to 5.82$\pm$2.20 keV (section C) and $\Gamma_{nthcomp}$ varies from 2.28$\pm$0.08 to 2.67$\pm$0.22.

In the next attempt we used accretion disk corona configuration (Church et al. 2012) and fitted the spectrum with a model wabs(BB+Gaussian+cutoffpl), where BB is black body emission from the NS and cutoffpl is the cut-off power-law associated with Comptonization in an extended corona over the disk. We found that all the parameters varied from the first to other sections. Church et al. (2012) fixed the power-law index $\Gamma$ of cutoffpl at 1.7, however we allowed it to vary. For the first section, $\Gamma$ = 1.13$\pm$0.24 changed to become a relatively steeper $\Gamma$ =1.76$\pm$0.17, $\Gamma$ =1.84$\pm$0.16 in the second and third sections and the cut-off energy E_c=5.31 $\pm$ 1.01 keV varied to higher values, E_c=8.32 $\pm$ 1.15 keV, E_c=8.93 $\pm$ 1.45 keV. Same model was used for unfolding the spectra of ObsID 20053-03-03-01. $\Gamma$ is found to be 0.93$\pm$0.16 in the first section, which varied to a value of $\Gamma$ =1.43$\pm$0.11 in the second section to a slightly lower value of $\Gamma$ =1.30$\pm$0.12 in the third section. The cut-off energy E_c went from 3.40 $\pm$ 0.55 keV in the first section to a relatively higher value of E_c=5.94 $\pm$ 0.65 keV in the second section and E_c=5.64 $\pm$ 0.73 keV in the third section.

Radio flux density is found to vary approximately from 4.5-1 mJy at 8 GHz and 4-2 mJy at 5 GHz (Migliari et al. 2007) during the observation duration of ObsID 70023-01-01-01 (see Figure 2 in Migliari et al. 2007). From both spectral fit attempts, it is clear that when the radio and X-ray fluxes were high (first section of the light curve) the kT_e was low with a relatively hard power-law component and the state of gradual fall of both radio and X-ray fluxes is associated with a corona of relatively high temperature electron distribution kT_e. Such a configuration of corona i.e. a compact low temperature optically thick corona is the characteristic of a steep power-law state/intermediate state in black hole X-ray binaries (Done & Kubota 2006; Done et al. 2007; Sriram et al. 2007, 2016) where often a transient jet is switched on/off (Fender et al. 2009).

Another observation (ObsID 80022-01-05-00) was analyzed, where soft and hard X-ray light curves are anti-correlated for almost 23 ks and a steep variation has occurred at $\sim$18 ks (soft X-ray is decreasing but hard X-ray is increasing, see Figure 3q). During this observation, the source was in the lower horizontal branch and upper normal branch. Same models as above were invoked to fit the 3-40 keV spectrum (as $>$ 40 keV S/N was low and hence was not considered) of each section (see Table 3 and Figure 7). The soft seed photon parameter kT_soft was tied to kT_BB. It was found that the electron temperature is kT_e $\sim$ 3 keV in all sections and no significant variations are observed in BB model parameters. A systematic residual is observed in the last section around $>$30 keV which can be modeled with a power-law component with an index, $\Gamma$=1.10 resulting in an improvement in the fit i.e.$\chi^{2}$/dof = 65/53 to $\chi^{2}$/dof = 36/53, F-test probability=4.4 $\times$ 10^-7 (see Table 3). Moreover it can be seen that the hard count rate increased in the final section of the light curve. This probably suggests a sudden appearance of jet in the inner region of accretion disk.

In GX 17+2 the CCFs analysis clearly show the correlated and anti-correlated soft and hard X-ray delays of a few tens to hundred seconds which are a characteristic feature often observed in Z and atoll sources (Lei et al. 2008 and Sriram et al. 2012). It is noted that the observed delays in the source was associated to the HB and NB during which a radio jet was observed to be present in a few observations (Migliari et al. 2007).

The spectral results suggest that the coronal properties have changed from a higher radio and X-ray flux to a low flux state. We found that disk normalization varied from section A to B & C (see Table 4). In order to properly constrain the inner disk radius, nthcomp model parameters were frozen. Inner disk radii were derived using a relation R_in (km) = 1.2 $\sqrt{(N/cosi)}$ $\times$ D / 10 kpc (Reynold & Miller 2013) and we noticed that R_in has changed from 9.3 km (section A) to 18.7 km (section B) (assuming i to be 25^∘ and 35^∘; Ludlam et al. 2017) but QPO frequency did not vary (see the right panel in Figure 5). Moreover, Migliari et al. (2007) showed that during this observation radio flux has varied consistently. In truncated accretion disk geometry, the QPO varies along with R_in (e.g. Done et al. 2007) however in this observation it is not so. This probably suggests that truncated disk front can be situated at any location. A similar spectral analysis for ObsID 20053-03-03-01 for section A, B and C indicated that R_in moved from 18 km (section A) to 31 km (section B & C) whereas QPO varied from 32 Hz to 24 Hz (Table 4 and Figure 6), which can be explained from the truncated accretion disk scenario. Instead of diskbb model, if BB model is used, we observe correlated changes of electron temperature of nthcomp model with QPO frequency i.e. as QPO frequency decreases the corona temperature increases (see Table 4). We noticed that in ObsID 20053-03-03-01 when QPO is at 24 Hz, R_in is around $\sim$ 30 km whereas in ObsID 70023-01-01-01, R_in varied from 9–20 km while QPO remained at $\sim$ 26-24 Hz.¹¹1Spectra were carefully corrected for gain variations during these epochs.

A few observations (Figure 3 & 6) exhibit delays in their CCFs although the QPO frequency was not found to vary. For example, the observation 20053-03-03-01, in section A shows a QPO at 32 Hz while a QPO at $\sim$24 Hz is observed in both section B & C (Figure 6). In respective CCFs, a soft X-ray delay varying from 207 s to 668 s from section A to B was noticed, while the delay is almost $<$100 s in section C. Assuming that the QPO is indicative parameter of the truncation radius, it can be concluded that the disk radius has changed from section A to B and the associated delays were found to be varying. Whereas in section B & C, significant variation of delays were noticed although QPO remained the same. This probably indicates that in section B there is an appreciable change in the disk–corona structure and this structure readjustment has settled down, hence relatively smaller delay in section C.

It was assumed that the readjustment time scale of disk in the inner region is the causative factor of the observed delays of a few tens to hundred seconds between 2-5 keV and 16-30 keV in GX 17+2 , but this was found to be only a few tens of seconds. Even the readjustment time scales of the vertical structure of disk due to the radiation pressure would also result in delays of a few seconds only. Hence we could conclude that the observed delays could not be just due to the readjustment time scales of the inner region of the disk but corona is also changing its size and structure. Thus we must consider the readjustment time scales of the coronal structure as well.

Assuming the observed delay is a combination of the readjustment time scale of the disk as well as corona, we can write,

where the first term is the readjustment timescale in the disk (assuming a Shakura and Sunyaev disk) and the second term is the readjustment timescale in corona.

We assume that the readjustment velocity in the coronal region is $\beta$ times that in the disk where $\beta$ is $\leq$ 1 (v_corona=$\beta$v_disk), since the coronal viscosity is less than the disk viscosity. We can rearrange the above equation to obtain the height of the coronal region,

where H_disk = 10⁸ $\alpha^{-1/10}$ $\dot{m}_{16}^{3/20}R_{10}^{9/8}f^{3/20}$ cm, $\rho$ = 7 $\times$ 10^-8 $\alpha^{-7/10}$ $\dot{m}^{11/20}$ $R^{-15/8}$ $f^{11/20}$ g cm^-3, f = (1-(R/R_s)^1/2)^1/4

Based on the above equations, we considered a delay of 207 s in the ObsID 20053-03-03-01 and considered R_disk to be 21 km in keeping with the observed QPO frequency of 32 Hz using the relation $\frac{R_{in}}{R_{g}}\leq 27\nu^{-0.35}\Bigg{[}{\frac{M}{2M_{\odot}}}\Bigg{]}^{-2/3}$ (Di Matteo & Psaltis 1999) and thus determined the coronal height to be $\sim$ 17 km ($\beta$=0.05) and 34 km ($\beta$=0.1) at that instant ($\dot{m}$ obtained from spectral fits; Table 4). Upon considering an observed soft delay of 668 s, based on the detected QPO frequency 24 Hz, we determine the coronal height to be $\sim$ 56 km ($\beta$=0.05) and 111 km ($\beta$=0.1). An $\alpha$ value of 0.1 was assumed for all the above calculations. We found that the inner disk radii R_in values are similar to that obtained from QPO frequency inner disk radii (see equation 1 and Table 4). We also evaluated the H_corona assuming the disk is at the last stable orbit i.e. R_disk = 12 km, which resulted in higher values.

In the truncated accretion disk scenario, the coronal quasi-spherical structure is assumed to be present inside the truncation radius, which suggests that the coronal radius should be of the order of truncated inner disk radius. Substituting R_disk = H_corona in equation 1 (R_disk being the inner truncation disk radius), one can constrain the $\beta$ factor which is of the order of 0.062–0.021 for a delay of 207 s & 668 s (ObsId 20053-03-03-01 section A & B) and $\beta$=0.125 & 0.085 for a delay of 114 s and 168 s in ObsID 70023-01-01-01 (section A and C). As shown above that coronal readjustment time scale is primarily responsible for the observed delays, we argue that $\beta$ is playing a key role. Based on NuSTAR spectrum study, we found that the disk is close to ISCO $\sim$ 14 km. Then proportionally the coronal height shall change based on the above discussed scenario and we found that $\beta$=0.08 for a delay of 100 s. Based on the above and this section we infer that the corona is varying independent of the location of the disk with different coronal readjustment time scales.

The disk truncation radius has been constrained using multiple models, which include the Relativistic Precession Model (RPM; Stella et al. 1999) , the Transition Layer Model (TLM; Titarchuk & Osherovich 1999) and the optically thin flow solution of the Advection Dominated Accretion Flow model (ADAF; Narayan & Yi 1995a, b). Apart from those, radius was constrained by applying the relations for the different inner structures in the truncated disk scenario, which include the quasi spherical region of the radiation pressure dominated disk, the magnetosphere radius and also the boundary layer radius (Popham & Sunyaev 2001). This section deals with each of the estimations separately.

The energy dependent cross correlation functions (CCFs) of GX 17+2 associated with soft and hard X-ray delays clearly show delay of the order of a few hundred seconds during HB & NB and are found to be highly correlated on FB with no delay. Highly correlated CCFs can be explained if both soft and hard X-rays are emitted from a confined zone, most probably arriving from the inner region of the accretion disk. The correlation coefficients of the anti-correlated and correlated CCFs of HB and NB are found to be lower than CCFs of FB indicating that the soft and hard X-ray emission are arriving from a broader region around the NS surface and hence the weaker correlation. It was observed that the CCF is highly positive with CC $>$ 0.8 with no delays in FB whereas this highly correlated feature of CCF is not seen in HB or NB where a lower CC has been observed with delays (e.g. Figure 3f). This is possibly due to the presence of an optically thick low electron temperature Compton cloud or a jet often associated with HB/NB with a characteristic of high variability. This scenario is closely matching with the results by Lin et al. (2012) where it was concluded that at constant mass accretion rate across the Z track, geometry of Comptonization region plays a pivotal role in forming the HB and a slim disk configuration up to the NB in the HID. Such a configuration of corona is often seen in black hole X-ray binaries (BHBs) where it is needed for launching a jet or an occurrence of jet line in HID (Done et al. 2007; Fender et al. 2009). During the episodic events of jet/ejection, it is often found that rapid transition occurs in BHBs where QPO types have rapidly varied or disappeared (e.g. Nespoli et al. 2003; Miller-Jone et al. 2012; Sriram et al. 2012, 2016). Z sources also display a rapid variability in QPO frequency during HB–NB or FB–NB transition (e.g. Casella et al. 2006).

From the CCF, PDS and spectral studies, we arrive at the following conclusions.

1. For the first time, the CCF between 3-5 keV and 16-30 keV showed a hard X-ray delay of 113 s using NuSTAR observation of GX 17+2 in one of the sections of the light curve where spectral studies indicate that the disk is close to the last stable orbit (see Table 6) and similar value was reported by Ludlam et al. (2017) . But in other sections of the light curve, CCF was found to be uncorrelated.

2. Based on the analysis of RXTE observation of GX 17+2, we found soft and hard X-ray delays of the order of a few hundred seconds which are indeed a characteristic feature associated with the HB and NB with a cross correlation coefficient arbitrarily spread in the domain $\sim$ 0.35 to 0.6. In the FB, the CCFs are strongly correlated with the coefficients $>$0.8. Though correlated and anti-correlated soft and hard delays were present across the HB and NB but such noticeable pattern of increase/decrease of delay values (i.e. higher/lower delays at the top of HB and lower/higher delays at the bottom of HB or NB) were not observed as the source traverses from the HB to NB or vice-versa.

3. In the case study performed for ObsID 70023-01-01-01 and 20053-03-03-01, we found that at similar QPO frequency $\sim$26 Hz the disk radii are different which indicates that disk truncation is not always correlated with QPO frequency. This result is evident from the section A and section C of ObsID 70023-01-01-01 where QPO has almost remained the same but disk radius varied from $\sim$10 km to $\sim$20 km.

4. From the observed delays in CCF, PDS and spectral studies, we conclude that corona is varying incoherently with the disk component, leading to different delays across the HB and NB. This is evident from ObsID 70023-01-01-01 and 20053-03-03-01 where for similar QPOs the observed delays were found to be different.

5. Coronal heights were constrained assuming that the delays are readjustment time scales of the inner truncated accretion disk region. Based on the delays, we found that on an average coronal heights are of the order of 20-35 km depending on the value of $\beta$ where $\beta$ is defined as the ratio of readjustment velocity of disk to the corona. Assuming that the coronal height is equal to the truncated radius we found that coronal readjustment velocities should be about $\beta$=0.06-0.12 times the readjustment velocities of disk. However it should be noted that the disk is close to the last stable orbit based on NuSTAR’s spectrum. Assuming that the disk is always at the last stable orbit the detection of delay in NuSTAR light curve suggests that coronal properties viz. size or geometry varies independent of the disk location. But NuSTAR has yet to observe the spectral transitions in GX 17+2 similar to GX 5-1 (Homan et al. 2018) where a clear change in the disk radius is observed.

6. We determined the disk truncation radius from various other methods as discussed above and found that it is of the order of 20-35 km.

It is evident from the above studies that both simultaneous timing and spectral observations are required in order to constrain the truncation radius of the accretion disk as well as to understand the corona geometry in Z sources and other X-ray binaries. Future long term observations from Astrosat and NuSTAR will be useful to constrain the timing and spectral variability and time-lag correlation at various spectral states in both Z and Atoll type neutron star sources during such delays, which could also constrain the size of corona and location of inner disk radius.