The Radio/X-Ray Correlation in X-Ray Binaries—Insights from a Hard X-Ray Perspective

In Figure 1, we compare the two narrowband luminosities with the broadband luminosity, showing that the broadband luminosity can be estimated to be roughly 5–10 times the 3–9 keV luminosity and 10–20 times the 6–10 keV luminosity. However, this factor varies within the source evolution in the hard state. Most sources present a higher factor for lower luminosities that gradually decreases with increasing luminosity. The exceptions are XTE J1752–223, which show an increase of the luminosity ratio with increasing luminosity, and XTE J1118+480, which seem to present a constant factor between the narrow- and broadband luminosities. These differences might arise from XTE J1118+480 being at very low luminosities compared with other sources, and XTE J1752–223 having rather poor coverage. Because both narrowband luminosities show similar evolution with the broadband luminosity, the change in the luminosity ratio is not caused by an increased soft X-ray flux in the 3–6 keV band. Rather, the difference lies in the evolution of the hard X-ray spectra.

Zoom In Zoom Out Reset image size

In Figure 2, we have plotted the joint PCA and HEXTE spectra of GX 339–4 as an example, where the data and modeling show that the peak/cutoff temperature in the hard X-ray spectrum is anticorrelated with the luminosity. The spectra colored in red correspond to the pointings where the luminosity ratio starts to decrease at a broadband luminosity exceeding 5 × 10³⁷ erg s⁻¹. Thus, the increasing hard X-ray spectral curvature is causing the decrease in the luminosity ratio. This shows that in the bright, hard X-ray state the narrowband luminosity is not always a good proxy for the bolometric luminosity as it does not take into account the hard X-ray spectral evolution. In addition, using the broadband X-ray luminosity will necessarily affect to the radio/X-ray correlations measured for the sources which we will quantify in the next section.

Zoom In Zoom Out Reset image size

To construct the radio/X-ray correlations, we searched for quasi-simultaneous radio observations in the literature (see Section 2) and selected those which were observed within a day of the X-ray observations. On average, the radio observations were observed within 10 hr of the X-ray observations, with 50% within 6.6 hr and 75% within 16 hr. In Figure 3, we have plotted the 3–9 keV (left column) and 3–200 keV (right column) X-ray luminosities against the quasi-simultaneous, single-frequency radio luminosities in 8.4 GHz (top row) and 4.9 GHz (bottom row).

Zoom In Zoom Out Reset image size

For determining the radio/X-ray correlation coefficients, we calculated a linear regression of the logarithmic radio and X-ray luminosities from individual sources. Because both variables are measured quantities, we minimized the slope and the normalization of the regression in both directions. We used the hyper.fit package (Robotham & Obreschkow 2015) to fit a line to the data using the maximum likelihood method, including the intrinsic scatter of the model, and to estimate the errors on the slope and normalization. This method assumes that the data errors are Gaussian, the sample is drawn randomly from the model population and the intrinsic scatter is Gaussian. We also added corrections to the intrinsic scatter due to the small sample sizes as described in the Appendix A of Robotham & Obreschkow (2015). With the above assumptions, we show the best-fitting model lines in Figure 3, with the shaded areas representing 1σ errors on the slope and the intercept in addition to the intrinsic scatter. The 1σ error values on the slope are shown also in the panel legend for individual sources.

Using the broadband instead of the narrowband X-ray luminosity resulted in similar coefficients for GX 339–4 and XTE J1118+480 (i.e., the radio-loud group) for both single-frequency radio luminosities. For these sources, the slopes of the best-fit lines are consistent with being ξ_RX = 0.7–0.8. For the radio-quiet group, the slopes are slightly steeper. The 67% confidence limits (1σ) on the slope using broadband X-ray flux for H1743–322 are ξ_RX = 0.8–1.0 and ξ_RX = 1.1–1.5 for 8.4 and 4.9 GHz single-frequency radio luminosities, respectively. The correlation slope is not well-defined for Swift J1753.5–0127 when using the 8.4 GHz radio data, but for 4.9 GHz radio data the 67% confidence limits are ξ_RX = 1.0–1.2. For sources XTE J1752–233 and GRO 1655–40, which are found in between the radio-quiet and radio-loud groups, the slopes are very similar for both luminosity ranges but flatter for 4.9 GHz radio data.

In a broad sense, estimating the X-ray luminosity more accurately in taking into account the hard X-ray emission did not present a major effect on the radio/X-ray correlation. In the right panels of Figure 3, GX 339–4 and J1118+480 (the two radio-loud sources) form a well-defined correlation with a correlation index of 0.7–0.8, and all the radio-quiet sources lie below that correlation by about an order of magnitude. This factor of ∼10 difference is about the same when using the narrowband luminosity. Thus, the bolometric correction cannot explain the systematic radio-loud/radio-quiet dichotomy.

Overall, using the broadband X-ray luminosity presents a slight increase in the correlation coefficients (steepening of the slopes). In addition, it seems that the correlation slopes for the radio-quiet group are steeper than the radio-loud group. However, the radio/X-ray correlation slope for the X-ray luminous group (corresponds to an outburst rise) of the radio-loud source GX 339–4 is, in fact, more similar to the one for the radio-quiet group than the one calculated from the whole GX 339–4 sample when taking into account the broadband X-ray flux. Recently, Islam & Zdziarski (2018) came to the same conclusion. The more luminous group aligns (in both slope and normalization) with the radio-quiet sources (this can be best seen in the bottom-right panel of Figure 3, where the correlation coefficient is ∼1.1 when using only the X-ray brighter group, i.e., above 5 × 10³⁷ erg s⁻¹). This can imply that all sources could transit between the two different radio/X-ray tracks, depending on the X-ray luminosity.

The change to the steeper slope in GX 339–4 corresponds to the time when the source starts to shift from the luminosity ratio L_{3–200 keV}/L_{3–9 keV} = 10 to ${L}_{3-200\mathrm{keV}}/{L}_{3-9\mathrm{keV}}=5$ (Figure 1), and where the cutoff in the hard X-ray spectra appears (red group in Figure 2). In a similar fashion, we found that in the steeper track the hard X-ray spectra show similar morphology in all sources presenting a distinct hard X-ray cutoff below ∼300 keV, while elsewhere the cutoff is unconstrained meaning that the 90% confidence range for the cutoff energy pegs to 1000 keV in the spectral fitting and lies over the passband of HEXTE (Figure 4). Assuming that the spectrum arises from thermal Comptonization, this implies that the electrons producing the Comptonized emission are effectively cooler in the steeper track, which implies a morphological change in the accretion flow. The line fitted to the data points that present a measurable cutoff in the X-ray spectrum has a coefficient of 2.1 ± 0.1 (red line in Figure 4; the coefficient is 1.9 ± 0.1 when using 4.9 GHz radio data), that is steeper than the radio/X-ray correlation slope in any single source but could indicate the maximum amount of efficiency attainable by an XRB. Interestingly, one of the highest radio/X-ray correlation coefficient found among XRBs: ${L}_{{\rm{R}}}\,\propto {L}_{{\rm{X}}}^{1.8\pm 0.2}$ from MAXI J1836–194, could be then accommodated with varying accretion efficiency instead of the proposed scenario of varying jet Lorentz factor (Russell et al. 2015).

Zoom In Zoom Out Reset image size

In a recent paper, Espinasse & Fender (2018) studied the distribution of radio spectral indices in the radio/X-ray correlation. They found significant differences in the two populations of XRBs, such that the radio-quiet population presented steeper radio spectra as an average. The difference in the peaks of the distributions is ∼0.4, with radio-quiet and radio-loud sources presenting radio spectral indices ${\alpha }_{\mathrm{RQ}}\,\sim -0.2$ and ${\alpha }_{\mathrm{RL}}\sim 0.2$, respectively. In the framework of the fundamental plane , where the correlation coefficient can be estimated as ${\xi }_{\mathrm{RX}}\approx (1.42-0.67{\alpha }_{{\rm{R}}}){q}^{-1}$, where q is the accretion efficiency (see Heinz & Sunyaev 2003, their Equation 12(b)), this means a difference in ${\xi }_{\mathrm{RX}}$ of 1.3/q to 1.6/q. To relate this to the observed values of coefficients for the radio-loud group (${\xi }_{\mathrm{RX}}=0.7-0.8$) requires $q\sim 1.7$. Using this value for the radio-quiet group places the slope coefficient to ${\xi }_{\mathrm{RX}}=0.9$. Thus, having a lower spectral index in the radio, the radio-quiet sources should have higher slope coefficients, and ${\xi }_{\mathrm{RX}}=0.9$ is more or less consistent with the observed indices (collectively Gallo et al. 2012 found that the slope for the radio-quiet sources is 0.98 ± 0.08). It is thus possible, that both the radio-quiet and radio-loud groups differ only in normalization in the radio/X-ray plane. On the other hand, using the observables, i.e., the average radio/X-ray correlation indices for the individual radio-loud and radio-quiet sources used in this paper at 4.9 GHz (${\xi }_{\mathrm{RX},\mathrm{RL}}=0.7\pm 0.1$ and ${\xi }_{\mathrm{RX},\mathrm{RQ}}=1.2\pm 0.2$, respectively) and the corresponding radio spectral indices from Espinasse & Fender (2018); ${\alpha }_{\mathrm{RL}}\,=0.2\pm 0.2$ and ${\alpha }_{\mathrm{RQ}}=-0.2\pm 0.3$, we can place estimates to the value of q by rearranging the fundamental plane equation for ${\xi }_{\mathrm{RX}}$. Thus, the observed values can be reproduced with ${q}_{\mathrm{RL}}=1.9\pm 0.3$ and ${q}_{\mathrm{RQ}}=1.3\pm 0.3$, indicating that the different correlation slopes and radio spectral indices between the radio-loud and radio-quiet sources could be explained by assuming different X-ray radiative efficiencies (different values of q) as will be discussed in the next section.

In addition, no apparent connection between the radio flux and inclination or black hole spin were found in the study by Espinasse & Fender (2018). There can be a difference in the viscosity parameter from source to source, however, there seems to be no clear difference between sources that are radio-loud or radio-quiet (comparing sources in Espinasse & Fender 2018 to Tetarenko et al. 2018). The radiative efficiency in the accretion disk and/or in the jet can be different for the radio-loud and radio-quiet sources, however, as discussed below the former is most likely a function of the mass accretion rate and therefore contributes to the slope of the correlation. For the latter, Espinasse & Fender (2018) speculate that there can be distinct physical differences in the jets of the radio-loud and radio-quiet sources. The radio-loud sources could represent partially self-absorbed jets with continuous energy dissipation occurring via internal shocks (e.g., Jamil et al. 2010) and the radio-quiet sources could represent jets with more discrete ejecta that are "lit up" by a shock zone (similar to what has been observed from AGN, e.g., Marscher et al. 2008), which could then explain the differences in the radiative efficiency and radio spectral indices of the jet. However, this would require that the jet morphology should change when a source transits to/from the steeper track (e.g., H1743–322, XTE J1752–223, GRO 1655–40) and we would observe a shift in the radio spectral index. While double-frequency observations are scarce during the transitional phase for XRBs, GRO 1655–40 shows very little change at the radio spectral index or radio flux density during the transition with the radio spectral index staying at ${\alpha }_{{\rm{R}}}\gt 0.2$ (Shaposhnikov et al. 2007). H1743–322 shows a flat or inverted spectrum when the source is in the steep track; however, no information is available for the value of the radio spectral index elsewhere (Jonker et al. 2010). In addition, XTE 1752–223 shows both flat (during outburst rise) and optically thin (during outburst decay) spectral indices while remaining in the steep track (Brocksopp et al. 2013). These examples show that the radio spectral index and the single-frequency radio luminosity do not vary much in the source evolution along the fundamental plane during the transitional phase between the two tracks. Thus, the difference could be found in the amount of X-ray luminosity.

The association of the steep track with the evolution of the hard X-ray spectrum could be explained by a morphological change in the accretion flow. In this scenario, there is an increase in the accretion efficiency leading to an increased X-ray emission. The radio-quiet sources would be radiatively more efficient than the radio-loud sources. The emergence of the hard X-ray cutoff in the spectra when sources are in the radiatively efficient track implies effective cooling of the Compton upscattering electrons by a soft seed photon population.

For the radiatively inefficient accretion flows, it has been shown that the radiative efficiency changes with the mass accretion rate nonlinearly up to a critical luminosity where the accretion flow changes from ADAF or Type I luminous hot accretion flow (LHAF) to Type II LHAF or some cold disk/corona configuration (Xie & Yuan 2012). Close to the critical luminosity, ${L}_{\mathrm{cr},\mathrm{ADAF}}\approx 5{\theta }^{3/2}{\alpha }^{2}{\dot{M}}_{\mathrm{Edd}}$, where θ is the electron temperature in keV and α is the viscosity parameter, the efficiency exhibits an increase from 1% to 8% with very little change in the mass accretion rate, after which the efficiency is approximately constant. This framework has been proposed to explain the transitional track in the radio/X-ray plane, where the sources move horizontally, i.e., increasing their X-ray luminosity, while the radio luminosity stays constant (Xie & Yuan 2016).

After crossing the critical luminosity, the accretion flow would transfer into a two-component disk/corona configuration. The cold disk would form underneath the Comptonizing material gradually cooling it down as marked by the decreasing electron temperature (Figure 4), but remains not visible until the Comptonized layer has been cooled down exposing the underlying disk and shifting the source hardness fast to the softer part of the hardness-luminosity diagram (the "usual" state change). Recent work by Poutanen et al. (2018) showed that to reproduce similar hard power-law indices as observed in the hard X-ray state spectra of XRBs, more seed photons for Comptonization are needed in the form of cold clouds in the corona or cyclo-synchrotron radiation in the hot accretion flow. It has also been suggested that the accretion disk is already at the innermost stable circular orbit at the beginning of hard-to-soft state transition (Koljonen 2015), which indicates the formation of the cold accretion disk underneath the hot accretion flow during the hard state rise. The different state transition luminosities would be then moderated by the time the Comptonizing material has been cooled down.

Why different sources stay on the radiatively inefficient track and others transfer to the radiatively efficient track depend on the difference in their respective critical luminosities, that are a function of the temperature of the electron population and the viscosity parameter. If the electrons in the accretion flow can be kept at a hot temperature, the critical luminosity is larger, and the source can stay in the radiatively inefficient track longer. However, when the electrons start to cool, the critical luminosity drops below the current luminosity and the source shifts to the radiatively efficient track. As an example, In Figure 4, we show the critical luminosity with the viscosity parameter α = 0.2 (as estimated in Tetarenko et al. 2018 for GX 339–4) and electron temperature of 175 keV (corresponding to cutoff energies ∼350–500 keV). We find that it coincides well with the dividing luminosity of the two groups in GX 339–4.

In addition, Dinçer et al. (2014) showed that the fractional rms variability does not reach equal strength in radio-loud and radio-quiet sources despite similar spectral parameters (power-law index, luminosity) with the former presenting overall higher rms than the latter. In a similar fashion, Muñoz-Darias et al. (2011) found that the fractional rms variability decreased from 40% to 30% with luminosity in the rising hard state of GX 339–4. This reduction of the fractional rms in both cases could be associated with a second, less variable component diluting the observed rms variability. As discussed above, the formation of a cold disk underneath the hot flow would in addition to cooling down the electrons (and producing the spectral cutoff) reduce the observed rms variability. This is another observational evidence uniting the luminous hard state in GX 339–4 with the X-ray properties of the radio-quiet sources.

In many recent studies, evidence has been accumulated about inclination affecting to the X-ray observables in both spectral and timing domains. The spectral effects include harder spectra in the hard X-ray state (Heil et al. 2015), triangular hardness-intensity diagrams (HIDs) and hotter accretion disks (Muñoz-Darias et al. 2013) for high-inclination sources (e.g., H1743–322), while low inclination sources (e.g., GX 339–4) present more softer spectra, "boxy" HIDs and cooler disks. In the timing domain, low inclination sources have higher "hard line" slopes (Motta et al. 2018), i.e., the rate of decrease in the rms variability in the hard state discussed above, and weaker/stronger amplitudes of type-C/type-B quasi-periodic oscillations (Motta et al. 2015). Therefore, the apparent bolometric flux evolution in the radio/X-ray plane could be different for sources with different inclinations. In fact, for anisotropic models of X-ray emission (e.g., a slab corona) a difference in the received emission can be pronounced because the optical depth of the slab increases when viewed with higher inclinations. In addition, the reflected coronal emission from the accretion disk depends on the inclination of the source (Petrucci et al. 2001), and if the Comptonized emission comes deep from the gravitational field of a rapidly spinning black hole, gravitational redshift will deform the intrinsic spectrum depending on the viewing angle (Niedźwiecki 2005). Of course, evolution and/or the variability from source to source of the parameters of the electron population (temperature, optical depth) Compton upscattering the seed photon spectrum will also result in similar spectral effects, thus making determining the inclination effects a challenging task.

On the other hand, inclination is expected to affect to the radio flux received by the observer from the jet due to Doppler boosting. In Soleri & Fender (2011), a toy model for the jet assuming that ${L}_{{\rm{R}}}\propto {L}_{{\rm{X}}}{\delta }^{2}$, where δ is the Doppler boosting factor, was able to produce the spread of the radio/X-ray plane with variable inclination resulting to the radio emission from the high-inclination sources being Doppler de-boosted producing lower radio emission. There are also hints that the radio-loudness correlates with the orbital inclination (Motta et al. 2018). However, the observed evolution of the radio-quiet sources transferring to the ${L}_{{\rm{R}}}\propto {L}_{{\rm{X}}}^{\sim 0.7}$ track cannot be explained by Doppler boosting. Also, it is not certain whether the outer disk/orbital inclination corresponds to inner disk/jet inclination, as the spin axis of the black hole probably differs from the orbital inclination (Maccarone 2002; King & Nixon 2018), bringing further complications to the inclination dependence of the source properties.

All in all, it is certain that the inclination affects to both X-ray and radio luminosity in a nonlinear way due to the anisotropies most likely present in the geometries of the emitting components. The magnitude of this effect and its ability to explain both the increase in the X-ray emission (or the lack of increase in the radio emission) during the transitional phase between the two tracks and the higher radio/X-ray correlation index remains to be studied in the future.