Inclination effects on the X-ray emission of Galactic black-hole binaries

TL;DR

This study investigates how the inclination angle of Galactic black-hole X-ray binaries affects the correlation between X-ray time lags and spectral properties, revealing that the correlation is stronger in systems viewed face-on.

Contribution

It demonstrates the inclination dependence of X-ray timing and spectral correlations and introduces a jet model explaining these observations.

Findings

Correlation is tight in low-inclination systems.

Amplitude of lags is larger at low and intermediate inclinations.

Jet model successfully reproduces observed behaviors.

Abstract

Galactic black-hole X-ray binaries (BHBs) emit a compact, optically thick, mildly relativistic radio jet when they are in the hard and hard-intermediate states. In these states, BHBs exhibit a correlation between the time lag of hard photons with respect to softer ones and the photon index of the power law component that characterizes the X-ray spectral continuum above $\sim$ 10 keV. The correlation, however, shows large scatter. Our objective is to investigate the role that the inclination of the system plays on the correlation between the time lag and the photon index. We find that the correlation between the time lag and the photon index is tight in low-inclination systems and becomes weaker in high-inclination systems. The amplitude of the lags is also larger at low and intermediate inclination angles than at high inclination. Our jet model that reproduces the process of Comptonization in an extended jet can account for the observations remarkably well.

Tables

Table 1. Table 1: List of outbursts and sources. The fourth column gives the total exposure time of all the observations analyzed. The fifth column gives the number of observations used in the spectral and timing analysis (blue empty circles in Fig.  1 ) over the total number (blue empty circles plus black dots in Fig.  1 ).

Object	Outburst	MJD interval	Total exposure	Number of	Inclination	$N_H$
	epoch		time (ks)	observations		$\times {10}^{22}$ cm-2
4U 1543–475	2002	52443–52565	275.4	20/112	Low	0.43 (1)
MAXI J1836–194	2011	55804–55896	116.9	69/74	Low	0.3 (7)
Cyg X–1	2003–2004	52693–53182	367.4	147/151	Low	0.54 (3)
GX 339–4	2002	52311–52884	495.6	48/267	Intermediate	0.4 (1)
	2004	53050–53498	565.2	152/328
	2007	53769–54678	572.6	191/347
	2010	54889–55618	326.0	107/317
XTE J1650–500	2002	52159–52447	327.9	47/182	Intermediate	0.7 (1)
Swift J1753.5-0127	2005	53553–53819	188.1	72/73	Intermediate	0.2 (5)
	2007–2011	54226–55894	607.2	275/278
XTE J1550–564	2000	51644–51741	128.7	36/236	High	0.65 (1)
	2001–2004	51938–53163	306.5	69/101
GRO J1655–40	2005	53423–53685	2238.3	68/501	High	0.8 (1)
H 1743–322	2003	52740–53055	668.0	42/194	High	2.4 (1)
	2004	53197–53329	79.1	19/49
	2005	53595–53668	46.4	11/23
	2008a	54746–54789	90.8	18/27
	2008b	54743–54849	68.6	36/38
	2009	54980–55039	93.4	15/51
	2010a	55220–55245	47.8	21/33
	2010b	55418–55470	106.8	25/58
	2011	55664–55735	48.2	19/39
XTE J1752–223	2009	55130–55414	408.2	113/207	Intermediate	0.6 (6)
XTE J1817–330	2006	53768–53950	382.3	20/140	High?	0.15 (1)
GS 1354–645	1997	50774–50840	50.5	7/9	High	3.72 (1)
MAXI J1543-564	2010	55465–55567	154.0	6/100	High	1.4 (2)
Swift J1842.5–1124	2008	54656–54803	83.2	31/49	High	0.4 (8)
XTE J1859+226	1999	51463–51749	356.5	25/127	High	0.34 (1)
XTE J1118+480	2000	51633–51764	138.7	47/50	High	0.01 (1)
	2005	53383–53428	71.7	27/39
MAXI J1659–152	2010	55465–55567	140.9	39/66	High?	0.27 (4)
(1) Dunn et al. (2010); (2) Stiele et al. (2012); (3) Hanke et al. (2009); (4) Jonker et al. (2012)
(5) Cadolle Bel et al. (2007); (6) Chun et al. (2013); (7) Russell et al. (2014)

Table 2. Table 2: Results of the linear regression and correlation analysis for Γ ≤ 2 Γ 2 \Gamma\leq 2 . Here ρ 𝜌 \rho is Pearson’s correlation coefficient, N 𝑁 N is the number of points, and p 𝑝 p the probability that the null hypothesis (no correlation) is true. ”Low” refers to systems with i ≤ 35 ∘ 𝑖 superscript 35 i\leq 35^{\circ} , ”Intermediate” to 35 ∘ < i ≤ 70 ∘ superscript 35 𝑖 superscript 70 35^{\circ}<i\leq 70^{\circ} , ”High” to i > 70 ∘ 𝑖 superscript 70 i>70^{\circ} .

	Slope	Intercept	$\rho$	N	$p$-value
Low	$0.0273\pm 0.0061$	$-0.0373\pm 0.0101$	0.88	7	$8.4\times {10}^{-3}$
Intermediate	$0.0279\pm 0.0024$	$-0.0338\pm 0.0041$	0.99	9	$2.4\times {10}^{-7}$
High	$0.0101\pm 0.0030$	$-0.0093\pm 0.0049$	0.75	8	$5.0\times {10}^{-2}$

Full text

*Aims. * Our objective is to investigate the role that the inclination plays on the correlation between the time lag and the photon index.

*Methods. * We have obtained RXTE energy spectra and light curves of a sample of black-hole binaries with different inclination angles. We have computed the photon index and the time lag between hard and soft photons and have performed a correlation and linear regression analysis of the two variables. We have also computed energy spectra and light curves of black-hole binaries using the Monte Carlo technique that reproduces the process of Comptonization in the jet. We account for the inclination effects by recording the photons that escape from the jet at different angles. From the simulated light curves and spectra we have obtained model-dependent photon index and time lags that we have compared with those obtained from the real data.

*Results. * We find that the correlation between the time lag and the photon index is tight in low-inclination systems and becomes weaker in high-inclination systems. The amplitude of the lags is also larger at low and intermediate inclination angles than at high inclination. We also find that the photon index and the time lag, obtained from the simulated spectra and light curves, also follow different relationships for different inclination angle ranges. Our jet model reproduces the observations remarkably well. The same set of models that reproduces the correlation for the low-inclination systems, also accounts for the correlation for intermediate- and high-inclination systems fairly well.

*Conclusions. * The large dispersion observed in the time lag - photon index correlation in BHBs can naturally be explained as an inclination effect. Comptonization in the jet explains the steeper dependence of the lags on the photon index in low/intermediate-inclination systems than in high-inclination ones.

1 Introduction

Observations over more than 30 years have revealed a common phenomenology in the X-ray properties of Galactic black-hole binaries (BHB). Most of them are transient sources that become active during a short period of time (several months) compared to the time they are in an off state (years). The realization that the source goes through different states during outbursts represents an important step forward in our understanding of the observational properties of BHBs. Each state is characterized by certain spectral (e.g., slope of the X-ray spectral continuum) and temporal (e.g., shape of the power spectral density, frequency of quasi-periodic oscillations) parameters, which vary smoothly within a given state, but may show sudden changes when the source changes state. An effective way to separate the observations into states is to plot the hardness - intensity diagram (HID). A BHB traces a $q$-shaped curve as the outburst progresses (Homan & Belloni, 2005; Remillard & McClintock, 2006; Belloni, 2010). Roughly speaking, the left and right branches correspond to the soft and the hard state, respectively, while the lines that connect these branches correspond to the intermediate state. The fact that the horizontal lines do not overlap, but are traced at different X-ray luminosity, indicates a strong hysteresis effect. An explanation of the hysteresis pattern and the direction in which the source moves in the HID (anticlockwise) has been given by Kylafis & Belloni (2015).

When BHBs are in the hard (HS) and hard-intermediate (HIMS) states, their X-ray spectrum in the 2-200 keV band is well represented by a power-law function that falls exponentially at high energies. Reflection components, such as iron line emission at 6.4-6.6 keV and an excess of emission around 20-30 keV, are commonly observed. BHBs also exhibit strong emission in the radio band, whose origin is attributed to a compact, partially optically thick, mildly relativistic jet.

There is growing evidence that the inclination of the binary orbit with respect to the observer’s line of sight plays an important role in the determination of the characteristics of the detected emission. Ponti et al. (2012) studied the winds emitted by the accretion disk and concluded that, because of the small opening angles, they are only observed in high-inclination systems. Muñoz-Darias et al. (2013) showed that inclination has a strong effect on the evolution of BHBs through the HID. The $q$-track in the HID of low-inclination systems displays more square shape, while that of high-inclination BHBs has more triangular shape. They also found that the accretion disks in high-inclination systems look hotter than in low-inclination systems.

The results of these two works are based on observations mainly in the soft state, when the radio jet is absent. Inclination-dependent differences have also been reported in the HS and the HIMS, when the jet is present. Heil et al. (2015) found that high-inclination BHBs display larger hardness ratios than low-inclination systems with similar power spectral shape. The rms variability, however, was not seen to be different in low- and high-inclination systems.

Motta et al. (2015) found that the amplitude of low-frequency type-C quasi-periodic oscillations (QPOs) is stronger for nearly edge-on systems (high inclination), while type-B QPOs are stronger when the accretion disk is closer to face-on (low inclination). In contrast, the broad-band noise associated with type-C QPOs is stronger in low-inclination sources. They concluded that these two types of QPOs and the broad-band noise associated with type-C QPOs correspond to different phenomena. While type-C QPOs are consistent with the truncated disk model and arise from relativistic precession of the inner hot flow, type-B QPOs are likely associated with the radio jet. The broad-band noise likely comes from fluctuations in the mass accretion rate.

van den Eijnden et al. (2017) performed a systematic analysis of the inclination dependence of phase lags associated with both Type-B and Type-C QPOs in Galactic black hole binaries. They found that the phase lag at the Type-C QPO frequency strongly depends on inclination, both in evolution with QPO frequency and sign. As the QPO frequency increases, the low-inclination systems tend to display larger positive (i.e. hard) lags, while high-inclination systems turn to negative (i.e. soft) lags.

Finally, Motta et al. (2018) investigated the correlation between the radio and the X-ray emission in BHBs and found that high-inclination objects tend to be radio-quiet, while low-inclination systems appear to be radio-loud.

Recently, we performed a detailed study of the HS and the HIMS of BHBs as a class and found a correlation between the photon index of the power-law component and the time lag of the hard photons with respect to the softer ones (Reig et al., 2018; Kylafis & Reig, 2018). We showed that up-scattering of low-energy photons (from the accretion disk) by highly-energetic electrons (in the jet) can explain the correlation. Although the correlation is statistically significant, it exhibits a large amount of scatter. The main goal of the present work is to investigate the effect of the orbital inclination on the correlation between the time lag and the photon index in BHBs. As in previous works that study the effects of inclination on the properties of BHBs, we assume that there is no intrinsic physical difference among BHBs, hence any systematic difference in the correlation at different inclinations must be attributed to this parameter. We conclude that the scatter in the correlation found by Reig et al. (2018) can be explained as an inclination effect. The scatter is due to the fact that we see the systems at different jet viewing angles.

2 Observations and data analysis

The data presented in this work has been downloaded from the Rossi X-ray Timing Explorer (RXTE) archive. We have followed the same analysis procedure as explained in Reig et al. (2018), with the only difference that here we have used the proportional counter array (PCA) only, to minimize the effect of reflection on the X-ray continuum.

Due to RXTE’s low-Earth orbit, the observations consist of a number of contiguous data intervals or pointings (typically 0.5–1 h long) interspersed with observational gaps produced by Earth occultations of the source and passages of the satellite through the South Atlantic Anomaly. For each observation, we have obtained the average energy spectrum over the energy range 2–25 keV and the light curves in the energy ranges $2-6$ keV, $9-15$ keV, and $2-15$ keV. The number of observations is given in Table 1. Due to the varying number of detectors (from 1 to 5) in the observations and to avoid calibration effects, for the spectral analysis, including the HID, we used only the proportional counter unit 2 (PCU2). For the timing analysis, we have used all PCUs that were on during the observations, because we have performed the timing analysis on segments of data 64 seconds long. In this way we increase the signal-to-noise with respect to the case of one detector only. To have a reasonable quality of the observations, we have considered only those with an average count rate111The reason that the rate of some data points (blue circles) goes below 20 c s*-1* in Fig. 1 is because that figure was made using rates from one PCU only, while the data selection was done with all the PCUs that were on. of at least 20 c s*-1* in the 2–15 keV energy range and at least 640 s of contiguous observations, i.e. without gaps (10 segments of 64 s each). To extract the observations that correspond to the HS and HIMS, we have selected observations with rms, in the 0.01–30 Hz frequency range and in the 2-15 keV energy range, larger than 10%. The time resolution of the light curves used to create the power spectra and to perform the time-lag analysis was $2^{-7}$ s.

The time lag has been computed for the $9-15$ keV photons with respect to $2-6$ keV ones and has resulted from the average of the time lag in the frequency range 0.05–5 Hz. To obtain the photon index, we have fitted an absorbed broken power-law model to the spectral continuum. The hydrogen column density has been fixed to the values given in Table 1. A narrow Gaussian component (line width \sigma\mathbin{\lower 3.0pt\hbox{\hbox to0.0pt{\raise 5.0pt\hbox{$\char 60\relax$}\hss}\mathchar 29208\relax}}0.9 keV) has been added to account for the iron line emission at around 6.4 keV. The photon index used in our analysis is the one that corresponds to the hard power law, that is, after the break. In support of the use of this phenomenological model is the fact that the reduced $\chi^{2}$ in 94% of the fits resulted in a value smaller or equal than 2 (85% smaller than 1.5).

2.1 Source selection

As a starting point, we have used the list of sources presented in Table 1 of Motta et al. (2018). However, because the inclination effects may be rather subtle, we wish to have the cleanest possible sample of sources. Thus, we have selected sources with i) well-sampled outbursts or a large number of observations in the RXTE archive, ii) densely populated HS, and iii) well-constrained values of the inclination (see Motta et al. 2015 and Motta et al. 2018 for an explanation of the method employed to infer the system inclination). We have chosen systems for which the range of the inclination values, given in Motta et al. (2018), clearly fall in one of the following categories: low-inclination BHBs (Li-BHBs), intermediate-inclination BHBs (IMi-BHBs), and high inclination BHBs (Hi-BHBs) depending on whether the angle between the observer’s line of sight and the perpendicular to the orbital plane is smaller than 35*∘, between 35∘* and 70*∘, and larger than 70∘*, respectively.

These constraints reduced the number of sources to nine, three for each group. These are (see Table 1): 4U 1543–475, MAXI J1836–194, and Cyg X–1 (Li-BHBs), GX 339–4, XTE J1650–500, and Swift J1753.5–0127 (IMi-BHBs) and XTE J1550–564, GRO J1655–40, and H 1743–322 (Hi-BHBs). We shall discuss other sources in Sect 3.2.

3 Results

We begin our analysis by generating the HID for each source as shown in Fig. 1. In this Figure, each data point represents the average count rate of one observation and was obtained using 16-s binned light curves. The black filled circles represent the complete data set, while the blue empty circles are the observations that we have selected to obtain time lag and photon index. They correspond to the HS and the HIMS. Star symbols separate approximately the HS from the HIMS and roughly corresponds to a photon index of $\Gamma\approx 2$ (see below).

Fig. 2 shows the correlation between the time lag and the photon index for Li-BHBs (top panel), IMi-BHBs (middle panel), and Hi-BHBs (bottom panel) for individual systems and individual outbursts. Li-BHBs show a very tight correlation. As the inclination increases, the scatter increases. To produce Fig. 2, the individual observations of each source were binned in $\Gamma$ bins of size $\Delta\Gamma=0.1$. The data points correspond to the weighted average of all the observations that fell in the corresponding bin.

Fig. 3 shows the average behavior for each one of the three groups in Fig. 2. Fig. 3 has been generated in the same way as Fig. 2, but this time all the observations of all the sources of the same group and of the same $\Gamma$ bin have been merged together and averaged.

3.1 Correlation analysis

We have performed three different, but related, analyses. First, we have examined how strong the correlation between the time lag and the photon index is. This has been done with the correlation coefficient $\rho$. The closer $\rho$ is to 1, the stronger the correlation is. Second, we have checked whether the correlation is statistically significant. This has been done by testing the null hypothesis that the two variables are uncorrelated, using the $t$-statistic $t=\rho\sqrt{(N-2)/(1-\rho^{2})}$, where $N$ is the number of data points. The smaller the probability $p$, the more significant the correlation is. Finally, we have performed a linear regression analysis to fit a straight line to the data. We have used the bisector BCES (bivariate correlated errors and intrinsic scatter) method following Akritas & Bershady (1996). This method takes into account both the individual errors and the intrinsic scatter. Moreover, this method is more appropriate than the traditional least squares estimator, because, since there is no a priory reason to choose one of the two variables as the independent variable, the BCES method uses the bisector of the two lines that correspond to the least-square fit of Y on X and X on Y.

The precise instant at which the source transits from the HS to the HIMS is difficult to identify with X-ray data only, because the properties of the HIMS are consistent with being the extension of those of the HS (Belloni, 2010). The $t_{\rm lag}-\Gamma$ correlation is positive at low $\Gamma$ and becomes negative at high $\Gamma$. Although the point at which the slope of the $t_{\rm lag}-\Gamma$ correlation changes may not coincide with the transition from the HS to the HIMS, the correlation becomes negative during most part of the HIMS (Kylafis & Reig, 2018). We have performed our analysis on the data points that contribute to the positive correlation, that is, for $\Gamma\leq 2$. In terms of spectral states, our analysis includes all the HS and perhaps the onset of the HIMS for some sources. To facilitate the visualization, we have drawn a vertical dashed-dotted line in Figs. 2 and 3 and marked the points with $\Gamma\approx 2$ with a magenta star in Fig. 1.

We have a pair $(\Gamma,t_{\rm lag})$ for each observation. Because of the large number of observations analyzed and for the sake of clarity, as we discussed above, the data presented in Figures 2 and 3 have been binned in $\Gamma$ with a bin size of $\Delta\Gamma=0.1$. The results of the analysis are given in Table 2 for the data sets shown in Fig. 3 and can be summarized as follows:

• •

Li-BHBs and IMi-BHBs display a distinct and strong correlation with correlation coefficients \rho\mathbin{\lower 3.0pt\hbox{\hbox to0.0pt{\raise 5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax}}0.9. In Hi-BHBs, the correlation is weaker.

• •

The correlation is significant above 99% confidence level for Li-BHB and IMi-BHBs and \mathbin{\lower 3.0pt\hbox{\hbox to0.0pt{\raise 5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax}}95% for Hi-BHBs. In other words, the probability that the two variables are uncorrelated is small.

• •

As the inclination increases, the scatter of the correlation also increases (Fig. 2).

• •

At \Gamma\mathbin{\lower 3.0pt\hbox{\hbox to0.0pt{\raise 5.0pt\hbox{$\char 60\relax$}\hss}\mathchar 29208\relax}}1.6, the amplitude of the lags is similar for all systems. Above this value, IMi-BHBs show, on average, longer time lags than the rest.

• •

The slope of the linear regression of the Li-BHBs is consistent with that of the IMi-BHBs within errors and significantly ($4.5\sigma$ and $11.5\sigma$, respectively) different from zero. In Hi-BHBs, the slope is different from zero at $\sim 3\sigma$.

3.2 Other sources

In this section, we investigate how the selection of sources affects the statistical analysis performed above. In Table 1 of Motta et al. (2018), there are two sources (XTE J1752-223 and XTE J1817–330) with an estimated inclination of $5-60^{\circ}$. Thus they belong to either the Li-BHB or the IMi-BHB group. In addition, another three sources have intermediate inclination, but we do not include these three sources in our analysis. A 0620–00 due to lack of $RXTE$ observations, GRS 1915+105 due to its peculiar behavior, and MAXI J1659–152 because its inclination is uncertain: while Motta et al. (2018) give an inclination angle in the rage $30-70^{\circ}$ based on the overall amplitude of the type-C QPO, Kuulkers et al. (2013) suggest $i\sim 65-80^{\circ}$ based on the presence of intensity drops, possibly attributed to absorption dips or partial eclipse.

Figure 4 shows the $\Gamma-t_{\rm lag}$ pairs for XTE J1752-223 and XTE J1817–330. While XTE J1752-223 shows a clear correlation, confirming it as a Li-BHB or IMi-BHB, the relationship between the lags and the photon index in XTE J1817–330 resembles that of Hi-BHBs. As Motta et al. (2015) pointed out, the inclination measurements mainly rely on the assumption that absorption dips and wind-related features are a strong indication of high-orbital inclination. No such features have been detected in XTE J1817–330. However, this non-detection does not guarantee that they may not appear in future observations. Thus, a Li-BHB, whose classification is based on these criteria, may turn into a Hi-BHB if new observations reveal dips in the X-ray light curve. The opposite, i.e., a Hi-BHB turning into a Li-BHB, cannot happen.

In addition to XTE J1550–564, GRO J1655–40, and H 1743–322, the list of sources of Motta et al. (2018) contains other Hi-BHBs with an apparently reliable estimate of the inclination: XTE J1908+094, XTE J1118+480, GS 1354–645, V404 Cygni, and XTE J1859+226. Of these five sources, we have not analyzed XTE J1908+094, because during most of the observations it was offset by $>25\arcmin$ with respect to the center of the field of view, since there was a nearby strong X-ray source 4U 1907+09 (that was actually the target of the observation). We remind that the PCA/RXTE had no imaging capabilities and that the field of view was $\sim 1^{\circ}$. The RXTE archive does not have data of V404 Cygni. We have also ignored XTE J1118+480 because the observations sample a very narrow part of the HS branch. In fact, during the 2005 outburst, the observation seems to cover the transition from the HS to the quiescent state at the very end of the decaying phase of the outburst. Another five sources, XTE J1748-288, Swift J1842.5–1124, IGR J17177–3656, 4U1630–47, and MAXI J1543-564 are classified as high-inclination objects, without specifying any value for the inclination angle. XTE J1748-288 and IGR J17177–3656 suffer from source confusion as there are other sources in the field of view and are offset with respect to its center. Also, 4U1630–47 was always in the soft state. The inclusion of GS 1354–645, XTE J1859+226, Swift J1842.5–1124, and MAXI J1543-564 to the initial group of Hi-BHBs does not alter significantly the slope of the correlation.

Figure 5 shows the slopes for different selections of sources. We do not see substantial differences. The largest difference is found when the source XTE J1752-223 is assumed to be a Li-BHB. However, no difference in the slope of the correlation is found if the source is included in the IM-BHB group. *Therefore, we conclude that the inclination angle of XTE J1752-223 must be in the range $35^{\circ}-70^{\circ}$. *

3.3 Jet model: Monte Carlo simulations

We have developed a model that simulates the process of inverse Compton scattering in a jet. Low-energy photons, presumably from the accretion disk, are up-scattered by energetic electrons moving outwards at mildly relativistic speeds in the jet. We assume the jet to be parabolic with a finite acceleration region.

The parameters of the model are: the optical depth along the jet’s axis $\tau_{\parallel}$, the width of the jet at its base $R_{0}$, the parallel, $v_{0}$, and perpendicular, $v_{\perp}$, components of the velocity, or equivalently the Lorenzt factor $\gamma=1/\sqrt{1-(v_{0}^{2}+v_{\perp}^{2})/c^{2}}$, the distance $z_{0}$ of the bottom of the jet from the black hole, the total height $H$ of the jet, the temperature $T_{bb}$ of the soft-photon input, the size $z_{1}$ and the exponent $p$ of the acceleration zone, where $v_{\parallel}(z)=(z/z_{1})^{p}\leavevmode\nobreak\ v_{0}$, for $z\leq z_{1}$, and $v_{\parallel}(z)=v_{0}$ for $z>z_{1}$.

The jet model used here is the same as the one used in Reig et al. (2018) and Kylafis & Reig (2018). The novelty of the version used in this work is that we now compute the dependence of the escaping photons as a function of the angle $\theta$ between the observer and the jet axis. We have used ten different bins in $w=\cos\theta$, each with size $\Delta w=0.1$, covering the range from $w=0$ to $w=1$. In terms of $w$, the different categories of sources are defined as: $0.8<w\leq 1$ for Li-BHBs, $0.3<w\leq 0.8$ for IMi-BHBs, and $0\leq w\leq 0.3$ for Hi-BHBs. We note that we have not found substantial differences by considering $0.9<w\leq 1$ for Li-BHBs and $0.3<w\leq 0.9$ for IMi-BHBs. Because the number of photons that scatter in the direction perpendicular to the jet axis ($0\leq w\leq 0.3$) is considerably smaller than at other directions, we increased the number of input photons by an order of magnitude with respect to our previous works. Here we have used $10^{8}$ photons. This increases the computing time but ensures good statistics in all the bins. Typically, the fraction of photons that escape from the jet distributes as 2–3% in $w=0.0-0.3$, 35–45% in $w=0.3-0.8$, and 40-45% in $w=0.8-1.0$. The rest, about $10-20$%, are emitted toward the accretion disk.

The various Monte Carlo models that we have run differ only in the value of the optical depth $\tau_{\parallel}$ (or equivalntly in the density) and the width of the jet at its base $R_{0}$, while the rest of the parameters are fixed at the following reference values: $z_{0}=5r_{g}$, $H=10^{5}r_{g}$, $v_{0}=0.8c$, $v_{\perp}=0.4c$, $z_{1}=50r_{g}$, $p=0.5$, and $T_{bb}=0.2$ keV, where $r_{g}=GM/c^{2}$ is the gravitational radius. We assume a black-hole mass of 10 $\leavevmode\nobreak\ {\rm M}_{\odot}$.

For each model and each angle bin, the code generates an energy spectrum and two light curves in the same energy bands as those chosen for the data analysis, namely $2-6$ keV and $9-15$ keV. The relevant parameters in this study are the photon index of the power-law spectral continuum and the time lag between the detection of hard photons ($9-15$ keV) with respect to softer photons ($2-6$ keV).

The shape of the simulated energy spectra is indeed well represented by a power-law with a roll over at high energies, as observed in real data. To determine the model photon index, we fit the spectra with a power-law and an exponential cutoff. The light curves are processed in the same way as the real light curves to derive the time lag. We compute the cross-spectrum, defined as $C(\nu_{j})=X_{1}^{*}(\nu_{j})X_{2}(\nu_{j})$, where $X_{i}(\nu_{j})$ is the Fourier transform of the time series and the asterisk denotes complex conjugate. The phase lag between the signals in the two bands at Fourier frequency $\nu_{j}$ is $\phi(\nu_{j})=\arg[C(\nu_{j})]$ [the position angle of $C(\nu_{j})$ in the complex plane] and the corresponding time lag $t_{\rm lag}(\nu_{j})=\phi(\nu_{j})/2\pi\,\nu_{j}$.

4 Comparison of the model with the observations

Our aim is to investigate whether Comptonization in an extended jet can explain the different correlations between the photon index and the time lag shown in Fig 2.

We assume that the jet axis is perpendicular to the orbital plane, hence the orbital inclination and the observation angle $\theta$ coincide. In other words, if $w\sim 1$, the observer sees the jet along its axis, whereas $w\sim 0$ corresponds to systems in which the observer sees the jet perpendicularly.

We have proceeded as follows: first we have chosen the models that reproduce the linear fit of Li-BHBs (top panel in Fig. 2 and black line and black squares in Fig. 3). That is, by changing $\tau_{\parallel}$ and $R_{0}$, we have built a series of models, whose $\Gamma_{\rm Li-BHB}$ and $t_{\rm lag,Li-BHB}$ match those of the observations. These two values have been obtained using only data that fell in the $w=0.8-1$ range, hence the subindex. Then, using the same models, i.e., the same pairs $(\tau_{\parallel},\leavevmode\nobreak\ R_{0})$, we have obtained ($\Gamma_{\rm IMi-BHB}$, $t_{\rm lag,IMi-BHB}$) and ($\Gamma_{\rm Hi-BHB}$, $t_{\rm lag,Hi-BHB}$), which now correspond to $w=0.3-0.8$ and $w=0-0.3$, respectively.

Our results are plotted in Fig. 6. The black symbols in this Figure are the same as the colored symbols in Fig. 2 (for \Gamma\mathbin{\lower 3.0pt\hbox{\hbox to0.0pt{\raise 5.0pt\hbox{$\char 60\relax$}\hss}\mathchar 29208\relax}}2.1), while the magenta stars correspond to the models. The line is simply the best linear fit to the model data points and is plotted for clarity. Given the simplicity of the model and the complexity and dissimilarity of the data, the agreement is remarkable.

To furher understand the effects of inclination, we have selected three models: one with $\tau_{\parallel}=10$ and $R_{0}=50r_{g}$ (big black empty square in Fig. 6), one with $\tau_{\parallel}=5$ and $R_{0}=140r_{g}$ (big green empty triangle in Fig. 6), and one with $\tau_{\parallel}=2.75$ and $R_{0}=250r_{g}$ (big red empty circle in Fig. 6). It is evident from the three panels of Fig. 6 that the same jet model produces different spectra and different time lags at different inclinations.

Heil et al. (2015) showed that high-inclination systems with a similar power spectral shape tend to show harder emission (measured as a hardness ratio) than lower inclination systems. Unfortunately, we cannot test this result with our model, because we cannot identify similar ”timing states”. In fact, our model predicts that the X-ray emission is softer at high inclination (note the shift toward the right of the three models in each panel of Fig. 6). That is, if we could see the same source from different viewing angles, we would find a softening of the spectrum (i.e., larger photon index) with increasing inclination. This does not necessarily mean that high inclination systems always display softer spectra than low-inclination systems. A high-inclination system may have a harder spectrum than a low-inclination system, but if we could see a high-inclination system at a lower angle, then our model predicts that the X-ray spectrum would be even harder. Unfortunately, we cannot see the same source from different viewing angles.

The detection of the 6.4 keV line implies that a reflection component should contribute to the continuum as well, possibly affecting the power-law photon index. However, we do not think that our results are significantly affected by reflection. First, Bagri et al. (2018) did not find significant differences in the photon index after adding reflection in the hard state of GX 339–4. They also performed their analysis using RXTE/PCA data over a similar energy range to the one we used. Second, we restrict the energy range of the spectral analysis to $E<25$ keV, i.e. leaving outside the energies at which the hump is most prominent. Finally, it appears that reflection dominates once the source moves well inside the HIMS (Plant et al., 2014), while we limited most of our study to the HS.

5 Discussion

In a recent work (Reig et al., 2018), we found that BHBs as a class exhibit a correlation between the power-law photon index and the time lag between hard and soft photons. When considered as a whole (many sources), the correlation shows a large amount of scatter. In this work, we show that the large scatter of the correlation can be explained as an inclination effect. Low- and intermediate-inclination systems show a steeper correlation than high-inclination systems.

In Reig et al. (2018), we demonstrated that Comptonization in the jet reproduces satisfactorily the observed relationship between the X-ray spectral continuum emission and the time lag of hard photons with respect to softer ones. However, due to the large dispersion in the data, concerns about how significantly the jet model can constrain the correlation remained. After all, fitting a model to a cloud of points is much less constraining that fitting a model to a tight correlation. These concerns vanished when we reproduced the very tight $t_{\rm lag}-\Gamma$ correlation in GX 339–4 (Kylafis & Reig, 2018). In the present work, we go one step further in demonstrating the potential of our jet model by reproducing the observed different $t_{\rm lag}-\Gamma$ correlations of BHBs as a function of inclination. We emphasize that the same set of models, that is, the same combination of $\tau_{\parallel}$ and $R_{0}$ that were selected to fit the Li-BHBs correlation were also used to reproduce the IMi-BHB and Hi-BHBs correlations and that the only difference was the angle at which the escaping photons were recorded.

Hi-BHBs exhibit a flatter relationship between photon index and lags. The largest difference is found at larger $\Gamma$, i.e., at low values of $\tau_{\parallel}$. We explain this as follows: the low$-\Gamma$ part of the correlation corresponds to a pure HS, in which presumably the jet is evolving. In Kylafis & Reig (2018), we showed that in this state the radius at the base of the jet increases and the optical depth along the jet decreases as $\Gamma$ increases. At low $\Gamma$, the jet is narrow and dense (large $\tau_{\parallel}$). Photons suffer a large number of scatterings in the lower part of the jet (acceleration region), but with a small mean free path, and escape the jet almost in all directions. As the jet radius increases and the optical depth decreases, photons are able to travel a larger distance along the jet and enter the region of large flow speed. Forward scattering of photons by the fast-flowing electrons results in their escape at small to moderate angles $\theta$. Toward the end of the HS, a fully developed jet is present. Its size is maximum and its optical depth is moderate. In this configuration, due to the bulk motion of the electrons in the outflow, most photons are scattered in the direction of the flow, have relatively large mean free paths and travel large distances, thus the hard photons have a long time lag. On the other hand, very few photons escape perpendicular to the jet axis (Hi-BHBs), and those that do have traveled short distances. Hence, high-inclination hard photons have short time lag.

5.1 Comptonization in the jet: a model that cannot be ignored

Transient BHBs are excellent laboratories to probe the Physics of accretion and relativistic ejection of matter. Over the past 20 years, and thanks to the state-of-the-art space detector technology on board X-ray missions, the phenomenological description of these states has reached an unprecedented degree of detail. We are now able to monitor changes in the broad (continuum) and discrete (lines) components of the energy spectrum and of a large number of timing parameters (rms, QPOs, lags). This extraordinary amount of information has not translated into a unified physical model and disagreement in even the most basic level exists. While there is general consensus that hard photons are produced by inverse Compton scattering, the geometry and properties of the comptonizing medium are still highly debated. Different models associate the comptonizing medium with different geometries. It could be an optically thin, very hot ”corona” in the vicinity of the compact object (Sunyaev & Titarchuk, 1980; Hua & Titarchuk, 1995; Zdziarski, 1998), an advection-dominated accretion flow (Narayan & Yi, 1994; Esin et al., 1997), a low angular momentum accretion flow (Ghosh et al., 2011; Garain et al., 2012), the base of a radio jet (Band & Grindlay, 1986; Georganopoulos et al., 2002; Markoff et al., 2005).

Since 2003, we have been demonstrating that Comptonization in an extended jet not only reproduces the general spectral and timing properties of transient BHBs, but also the more stringent constraints imposed by the correlation between spectral and timing parameters. In addition to quantitatively explaining the emerging spectrum from radio to hard X-rays (Giannios, 2005) and the evolution of the photon index and the time (phase)-lags as functions of Fourier frequency (Reig et al., 2003), improved versions of our original jet model have been able to explain the correlation between the time lag and the X-ray photon index in Cyg X–1 (Kylafis et al., 2008), in GX339–4 (Kylafis & Reig, 2018), and in the class of BHB as a whole (Reig et al., 2018). Similarly for the correlation between time lag and cut off energy in GX339–4 (Reig & Kylafis, 2015).

In this work, we have shown that the inclination of the system has a profound effect on the correlation between photon index and time lag. We have tested our model against this highly constraining result and passed it with success. Given the heterogeneity of the data, Fig. 6 represents a huge success of the model.

When one looks at the overall picture, BHBs as a whole show a similar pattern of variability, supporting the view that the process of accretion of matter and ejection of a jet is similar in all BHBs. After all, the definition of source states has been possible thanks to the repeatability of the behavioral patterns as the source goes through an outburst. The reality, however, is more complex than the overall picture. Separate outbursts even from the same source can look different. The HS may appear straight or slightly curved in the HID. The loop of the $q$ curve may have a rectangular or triangular shape rather than a rounded shape. A recent study has shown that a substantial fraction ($\sim$40%) of the outbursts in transient BHBs do not reach the soft state (Tetarenko et al., 2016). Differences between sources are most likely related to fundamental properties of black holes, such as mass and spin, while differences between outbursts of the same source can be attributed to different mass-transfer-accretion rates.

The mass and presumably the spin in transient BHBs vary significantly. We lack studies that link the actual values of these two parameters with the physics of accretion and ejection of matter. It is likely that different combinations of these parameters generate jets that differ in size, particle density, and velocity. Had we adjusted the jet parameters, such as the optical depth and the width of the jet separately in each group, the agreement in Fig. 6 would have been even better.

One very constraining relationship of our model is the fact that the parameters $\tau_{\parallel}$ and $R_{0}$ of the best-fit models follow a tight correlation. Although their actual absolute values may vary from source to source, the optical depth and the jet width vary in unison (Kylafis et al., 2008; Reig et al., 2018; Kylafis & Reig, 2018). As the source moves up along the HS branch (right branch in the HID), the Thomson optical depth along the jet decreases and the width of the jet increases. The variation of the width of the jet with luminosity is consistent with the idea that the accretion disk is truncated far away from the black hole during the HS and approaches it as the source transits to a softer state.

Since the jet is fed by the hot inner flow, the width $R_{0}$ of the base of the jet must be smaller than the extent of the hot inner flow, hence it must be smaller than the distance $R_{\rm tr}$ of the inner part of the accretion disk from the black hole. At low luminosity in the HS, $R_{0}<<R_{\rm tr}$. As the luminosity increases in the HS, $R_{0}$ also increases, but without exceeding $R_{\rm tr}$. At some luminosity, $R_{0}$ becomes comparable to $R_{\rm tr}$. This state would correspond to the transition between the HS and the HIMS. After that, in the HIMS, and as the source moves horizontally to the left in this branch (upper branch in the HID), the width decreases, while the optical depth remains fairly constant or decreases slightly. The relationship between $\tau_{\parallel}$ and $R_{0}$ for the models used in this work is given in Fig. 7. Because we have restricted our analysis to observations with (\Gamma\mathbin{\lower 3.0pt\hbox{\hbox to0.0pt{\raise 5.0pt\hbox{$\char 60\relax$}\hss}\mathchar 29208\relax}}2), the horizontal branch in the $\tau_{\parallel}$-$R_{0}$ plot (see Fig. 3 in Kylafis & Reig, 2018) is absent in Fig. 7.

Our jet model reproduces the expected trend between luminosity and truncation radius and provides a clear physical and independent prediction on the disk truncation radius. However, the values that we find at around the change of state from HS into HIMS ($\sim 300\,r_{g}$) are somehow larger than those obtained from spectral fits (Basak & Zdziarski, 2016; Jiang et al., 2019), X-ray reverberation (De Marco et al., 2015) and QPOs (Ingram et al., 2017), which typically give R_{\rm tr}\mathbin{\lower 3.0pt\hbox{\hbox to0.0pt{\raise 5.0pt\hbox{$\char 60\relax$}\hss}\mathchar 29208\relax}}100\,r_{g}.

In closing, we stress that the correlations between spectral (photon index, cutoff energy) and timing (time lag, rms variability, QPOs) parameters in BHBs imply that they are coupled and strongly suggest that these components appear to have a common underlying origin. The hysteresis of the HID adds another constraint (Kylafis & Belloni, 2015). So far, only Comptonization in an extended jet has been able to survive all these tests.

6 Conclusion

We have investigated the effect of the inclination on the correlation between the time lag and the photon index in BHBs. We have shown that although the correlation holds for all systems, there is a distinct inclination effect affecting the relationship between the two variables. High-inclination systems display, on average, a flatter correlation. This different behavior explains the large scatter in the $t_{\rm lag}-\Gamma$ correlation reported by Reig et al. (2018), where no distinction for inclination was made.

We have simulated the process of inverse Compton scattering in a jet and generated theoretical spectra and light curves. The most remarkable result of this work is the fact that we can reproduce the observed correlations between time lag and photon index for systems with different inclination angles, with the same set of models, by simply looking at the jet with different viewing angles.

Acknowledgements.

NDK thanks Asaf Pe’er for expressing his dislike for the large scatter exhibited in the time lag – photon index correlation reported in Reig et al. (2018). The authors also thank I. Papadakis for fruitful discussions that helped improve the final version of this paper.