Issue 
A&A
Volume 679, November 2023



Article Number  A81  
Number of page(s)  6  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/202346379  
Published online  16 November 2023 
A quantitative explanation of the radio – Xray correlation in blackhole Xray binaries
^{1}
University of Crete, Physics Department, University Campus,
70013
Heraklion, Greece
email: kylafis@physics.uoc.gr; pau@physics.uoc.gr; tsouros@physics.uoc.gr
^{2}
Institute of Astrophysics, Foundation for Research and TechnologyHellas,
71110
Heraklion, Greece
Received:
10
March
2023
Accepted:
18
August
2023
Context. The observed correlation between the radio and Xray fluxes in the hard state of blackhole Xray binaries (BHXRBs) has been in existence for over two decades. It is currently accepted that the hard Xrays in BHXRBs come from Comptonization in the corona and the radio emission from the relativistic jet (Lorentz γ ≫ 1), which is a narrow structure of a few R_{g} = GM/c^{2} at its base. The jet and the corona, however, are separate entities with hardly any communication between them, apart from the fact that both are fed from the accreting matter.
Aims. It is also widely accepted that the accretion flow around the black holes in BHXRBs consists of a thin outer disk and a hot inner flow. From this hot inner flow, which has a positive Bernoulli integral, an outflow must emanate in the hard and hardintermediate states of the source. By considering Compton upscattering of soft disk photons in the outflow (i.e., in the outflowing “corona”, which is a wider structure of tens to hundreds of R_{g} at its base) as the mechanism that produces the hard Xray spectrum, we have been able to quantitatively explain a number of observed correlations. Here, we investigate whether this outflowing corona can also explain the observed radio – Xray correlation. We remark that the outflowing corona (wide, with a low Lorentz γ) is completely separate from the relativistic jet (narrow, with a high Lorentz γ). The two may coexist, with the jet at the rotation axis and the corona around it.
Methods. We considered parabolic outflow models, which we have successfully used in the explanation of other correlations regarding GX 3394 in the hard and hardintermediate states, and computed the radio emission at 8.6 GHz coming from them, as well as the powerlaw photonnumber spectral index Γ of the Comptonized hard Xrays produced in them. Thus, we have a correlation between the computed radio flux F_{R} at 8.6 GHz and the computed spectral index Γ of the hard Xray spectrum. This correlation is a theoretical prediction, since both F_{R} and Γ are computed from the model and, to our knowledge, no such correlation has been constructed from observations. This prediction can be confirmed or proven wrong in future outbursts of GX 3394. Based upon observations, we also produced a correlation between the observed hard Xray flux F_{X} and the observed index Γ. Thus, for each value of Γ, observed or computed, we have the corresponding values of the observed F_{X} and the computed F_{R}, which we plotted against each other.
Results. For GX 3394, we found that our model calculations for F_{R} and Γ, with Γ as the link between the observed F_{X} and the computed F_{R}, successfully reproduce the observed correlation of F_{R} ∝ F_{X}^{0.6} in the hard state. In addition, in the hardintermediate state of GX 3394, this correlation breaks down and we predict that, in future outbursts of the source, the F_{R} will exhibit first a sudden increase and then a sharp drop within a very narrow range of values of F_{X}. Such a sharp drop of the F_{R} has been observed in other sources.
Conclusions. In our picture both the radio and the hard Xray emission come from the same region, namely the outflow, and it is therefore not surprising that they are correlated. Since in a parabolic outflow with constant outflow speed the density is largest at its bottom, the soft photons, coming from below, see something appearing similar to a “slab”, with a moderate optical depth (up to ten in the hard state) along the outflow and an order of magnitude larger in the perpendicular direction. We remark that it is a slab geometry that is invoked to explain the observed Xray polarization from BHXRBs. Because of this, we predict that the Xray polarization of GX 3394 will be parallel to the outflow in the hard state and perpendicular to it in the hardintermediate state.
Key words: radiation mechanisms: general / radiation mechanisms: nonthermal / Xrays: binaries / stars: black holes
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1 Introduction
Blackhole Xray binaries (BHXRBs) exhibit many observational correlations, the most prominent of which is the correlation seen in the hard state between the radio flux F_{R}, for example, at 8.6 GHz, and the Xray flux F_{X}, between 2 and 10 keV, for example (Hannikainen et al. 1998; Corbel et al. 2000, 2003; Gallo et al. 2003, Bright et al. 2020; Shaw et al. 2021). In the socalled standard track, the correlation is of the form ${F}_{\text{R}}\propto {F}_{\text{X}}^{\beta}$, where β ≈ 0.7 (Gallo et al. 2003; Corbel et al. 2003) or β ≈ 0.6 (Corbel et al. 2013). Radio and Xray emission also occurs in the hardintermediate state, but the relation between the two has not yet been studied.
Extensive theoretical work has been undertaken on both the radio and the Xray emissions from BHXRBs. It is well accepted that the hard Xrays originate by inverse Compton scattering in the corona around the black hole (see, e.g., Done et al. 2007; Kylafis & Belloni 2015) and the radio in the relativistic jet (e.g., Fender et al. 2004). Unlike windlike outflows, the jet is considered to be a narrow structure, on the order of 10 gravitational radii at its base (e.g., Zdziarski et al. 2022), with a Lorentz factor γ ≫ 1. The jet and the corona are disjoint, but the jet is fed from the accretion flow – corona system, and this led Heinz & Sunyaev (2003) to examine scale invariant jet models, where they showed analytically that F_{R} ∝ (Mṁ)^{17/12}, where M is the mass of the black hole and ṁ is the accretion rate in units of the Eddington rate. For radiatively inefficient accretion flows, such as the advection dominated accretion flows (ADAFs; Narayan & Yi 1994, 1995; Abramowicz et al. 1995), for which F_{X} ∝ Mṁ^{q} with q = 2.3, one has ${F}_{\text{R}}\propto {M}^{0.8}{F}_{\text{X}}^{0.6}$, which adeptly explains the observations (Merloni et al. 2003). If q is somewhat different than 2.3, such as q = 2, the observed F_{R}–F_{X} correlation cannot be explained.
A common picture of the accretion flow in BHXRBs in their hard and hardintermediate states consists of a geometrically thin, optically thick, outer disk (Shakura & Sunyaev 1973) and a hot inner flow, ADAF, which is geometrically thick and optically thin (Narayan & Yi 1994, 1995; Abramowicz et al. 1995). This hot inner flow is taken as the corona in many models, including Heinz & Sunyaev (2003) and Merloni et al. (2003). It is not widely recognized that, due to the positive Bernoulli integral of the hot inner flow, a mildly relativistic outflow must emanate from it (Blandford & Begelman 1999; Kazanas et al. 2015). In other words, the Comptonizing corona is not static, but is outflowing. This has tremendous implications.
We have been putting forward the idea that Comptonization in BHXRBs takes place mainly in the outflow (outflowing corona). As we discuss in Sect. 4, our picture explains a number of correlations, some of which have not been explained by any other model. Due to the magnetic field that is needed for the ejection of the outflow (e.g., Blandford & Payne 1982), the outflow also emits radio waves. Since it is the same electrons that perform the Compton upscattering of the soft photons and the radio emission by synchrotron, it is likely that the radio and the Xrays are correlated. This is what we demonstrate below, restricting our study to GX 3394, because it is well studied and we have previously modeled various correlations exhibited by this source (Reig & Kylafis 2015; Kylafis & Reig 2018; Kylafis et al. 2020).
Other studies have proposed an outflowing corona. The first such proposal was made by Beloborodov et al. (1999). Other works (e.g., Malzac et al. 2001; Markoff et al. 2005; Poutanen et al. 2023) have shown that the base of the outflow plays the role of the corona. We remark that the wellknown jet emitting disk – standard accretion disk (JEDSAD) model (Marcel et al. 2019; Barnier et al. 2022; see also previous work by this group), which is able to reproduce the spectral behavior (Xray and radio) of BHXRBs during their outbursts, relies on selfconsistent accretionejection solutions, unlike our simple model, which assumes the structure of the outflowing corona.
The structure of our paper is as follows. We describe our model in Sect. 2, present our results in Sect. 3, and discuss our findings in Sect. 4.
2 The model
We performed Monte Carlo simulations of Comptonization in an extended outflowing region. In our previous works, we referred to this outflowing corona as the jet, because in the past, anything outflowing was called a “jet”. However, this name now appears inappropriate. The outflow may or may not include a relativistic jet close to the blackhole axis. Our model consists of a parabolic outflowing corona with two symmetric lobes, where both Comptonization and radio emission occur. We subsequently describe the lobe along the positive z axis.
2.1 The outflowing corona
Guided by previous observations (Asada & Nakamura 2012; Kovalev et al. 2020), we consider a parabolic outflowing corona whose radius at height z > 0 above the black hole is given by $$R\left(z\right)={R}_{0}\text{\hspace{0.17em}\hspace{0.17em}}{\left(z/{z}_{0}\right)}^{1/2},$$(1)
where R_{0} is the radius of the outflow at its base, which is taken to be at a height z_{0} above the black hole. Here, R_{0} and z_{0} are parameters of the model.
At the bottom of the outflow, there is an acceleration region, beyond which the flow speed is constant and equal to v_{0}. Thus, the speed of the outflowing matter along the z axis is taken to be $${\upsilon}_{\Vert}\left(z\right)=\{\begin{array}{ll}{\upsilon}_{0}\text{\hspace{0.17em}}{\left(z/{z}_{1}\right)}^{a},\hfill & {z}_{0}\le z\le {z}_{1}\hfill \\ {\upsilon}_{0},\hfill & z>{z}_{1}\hfill \end{array},$$(2)
where v_{0}, z_{1}, and a are three of the parameters of our model.
If n_{e}(z) is the number density of electrons along the outflow, mass conservation requires that $$\dot{M}=2\pi {R}^{2}\left(z\right){m}_{\text{p}}{n}_{\text{e}}\text{\hspace{0.17em}}\left(z\right){\upsilon}_{\Vert}\left(z\right),$$(3)
where Ṁ is the massoutflow rate and m_{p} is the proton mass. The outflowing matter is considered to consist only of protons and electrons. Equation (3) implies that the number density of the electrons along the outflow is given by $${n}_{\text{e}}\left(z\right)=\{\begin{array}{ll}{n}_{1}\text{\hspace{0.17em}}{\left({z}_{1}/z\right)}^{a+1},\hfill & {z}_{0}\le z\le {z}_{1}\hfill \\ {n}_{1}\text{\hspace{0.17em}}\left({z}_{1}/z\right),\hfill & z>{z}_{1}\hfill \end{array},$$(4)
where n_{1} is the number density of electrons at z_{1}, while the density at the base of the outflow is n_{0} = n_{1}(z_{1}/z_{0})^{a+1}. The Thomson optical depth of the outflow along z is given by $${\tau}_{\Vert}={\displaystyle {\int}_{{z}_{0}}^{H}{n}_{\text{e}}\text{\hspace{0.17em}}\left(z\right){\sigma}_{\text{T}}\text{d}z},$$(5)
where σ_{T} is the Thomson cross section and H is the height of the outflow. Instead of n_{0}, we took τ_{‖} as a model parameter.
In the rest frame of the flow, the electrons are generally taken to have a powerlaw distribution of Lorentz γ, namely $${N}_{\text{e}}\left({\gamma}_{\text{co}}\right)={N}_{0}\text{\hspace{0.17em}}{\gamma}_{\text{co}}^{p},$$(6)
from γ_{min} to γ_{max}, where p, γ_{min}, and γ_{max} are parameters of the model. Readers can look below for details about N_{0}. Here, γ_{co} is the Lorentz factor of the electrons in the comoving frame.
In order to calculate the distribution of the electrons for an observer at rest, one would need to perform the transformation of the Lorentz factor from γ_{co} to γ. If one assumes that the velocity of the outflow is constant throughout, then it can be shown that γ is approximately proportional to γ_{co}, and thus Eq. (6) also holds for an observer at rest. Since the acceleration region is small, the contribution of the acceleration region to the radio emission of the outflow is small. Thus, we neglect the transformation from γ_{co} to γ and take the distribution of electrons for the observer at rest to be $${N}_{\text{e}}\text{\hspace{0.17em}}\left(\gamma \right)\simeq {N}_{0}{\gamma}^{p}.$$(6a)
The normalization N_{0}(z) can be calculated by integrating Eq. (6a) from γ_{min} to γ_{max}, and equating the expression with the comoving electron density. This yields $${N}_{0}\left(z\right)=n\left(z\right)\text{\hspace{0.17em}}\sqrt{1\frac{{\upsilon}_{\Vert}^{2}\left(z\right)}{{c}^{2}}}\left(p1\right){\gamma}_{\text{min}}^{p1}.$$(7)
Parameter values. As in our previous work, we fixed all of the parameters, except for R_{0} and τ_{‖}, which we varied. The rest of the parameters have the values z_{0} = 5R_{g}, where R_{g} = GM/c^{2} with M the blackhole mass, v_{0} = 0.8c, z_{1} = 50R_{g}, a = 1/2, p = 3, γ_{min} = 1.0, and γ_{max} = 500.
Our results and conclusions are insensitive to reasonable values of the above parameters. In particular, v_{0} can be significantly less than 0.8 and all of our previous work (see Sect. 4) is unaffected (Reig & Kylafis, in prep.). The crucial ingredient of our model is the parabolic shape of the outflow. Nonparabolic outflows might be necessary to explain the socalled outlier sources, but this will be the subject of a subsequent paper.
2.2 Radio emission
Ignoring synchrotron selfCompton (for a justification, see Giannios 2005), the equation for the transfer of radio photons in the outflow, in direction $\widehat{n}$, along which length is measured by s, is given by $$\frac{\text{d}I\left(v,s\right)}{\text{d}s}=j\left(v,s\right)a\left(v,s\right)I\left(v,s\right),$$(8)
where j(ν, s) and a(ν, s) are the emission and absorption coefficients, respectively, and I(ν, s) is the intensity at frequency ν at position s. The formal solution of this equation is $${I}_{v}={\displaystyle {\int}_{{s}_{1}}^{{s}_{2}}\text{d}s\text{\hspace{0.17em}}j\left(v,s\right)\text{\hspace{0.17em}exp\hspace{0.17em}}\left(\text{\hspace{0.17em}}{\displaystyle {\int}_{s}^{{s}_{2}}\text{d}s}\prime a\left(v,s\prime \right)\right)}.$$(9)
For $\widehat{n}$ perpendicular to the outflow axis, this is simplified to $${I}_{v}\left(z\right)=\frac{j\left(v,z\right)}{a\left(v,z\right)}\left[1\text{exp}\left\{a\left(v,z\right)R\left(z\right)\right\}\right],$$(10)
where R(z) is given by Eq. (1), since the emission and absorption coefficients only depend on z. The total power radiated per unit frequency, per unit solid angle is thus given by $$\frac{\text{d}{E}_{v}}{\text{d}tdv\text{d\Omega}}=2\pi \text{\hspace{0.17em}}{\displaystyle {\int}_{{z}_{0}}^{H}\text{d}z{I}_{v}\left(z\right)R\left(z\right)}.$$(11)
We next needed to specify the absorption and emission coefficients. Since the radio emission of the outflow is due to synchrotronemitting electrons, whose Lorentz factors follow a power law as in Eq. (6a), the absorption and emission coefficients can be calculated analytically (Rybicki & Lightman 1979). The expressions are $$\begin{array}{l}a\left(v,z\right)=\frac{\sqrt{3}{q}^{3}}{8\pi {m}_{\text{e}}}{\left(\frac{3q}{2\pi {m}_{\text{e}}^{3}{c}^{5}}\right)}^{\frac{p}{2}}\text{\hspace{0.17em}}{N}_{0}\left(z\right){B}^{\frac{p+2}{2}}\left(z\right)\hfill \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\times \text{\hspace{0.17em}\Gamma \hspace{0.17em}}\left(\frac{p}{4}+\frac{1}{6}\right)\text{\Gamma \hspace{0.17em}}\left(\frac{3p+22}{12}\right)\text{\hspace{0.17em}}{\left({m}_{\text{e}}{c}^{2}\right)}^{p1}{v}^{\frac{p+4}{2}},\hfill \end{array}$$(12)
and $$\begin{array}{l}j\left(v,z\right)=\frac{\sqrt{3}{q}^{3}{N}_{0}\left(z\right)B\left(z\right)}{4\pi {m}_{\text{e}}{c}^{2}\left(p+1\right)}\hfill \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\times \text{\hspace{0.17em}\Gamma \hspace{0.17em}}\left(\frac{p}{4}+\frac{19}{12}\right)\text{\Gamma \hspace{0.17em}}\left(\frac{p}{4}\frac{1}{12}\right)\text{\hspace{0.17em}}{\left(\frac{2\pi {m}_{\text{e}}{c}^{2}}{3qB\left(z\right)}\right)}^{\frac{1p}{2}}{v}^{\frac{1p}{2}},\hfill \end{array}$$(13)
where q and m_{e} are the absolute value of the charge and the mass, respectively, of the electron, B(z) = B_{0}(z_{0}/z) is the strength of the magnetic field at height z assuming flux freezing B_{0} = B(z_{0}), and Γ is the Gamma function.
Integrating Eq. (11) over solid angles gives the power per unit frequency radiated, $$P\left(v\right)\equiv \frac{\text{d}{E}_{v}}{\text{d}t\text{d}v}=4{\pi}^{2}\text{\hspace{0.17em}}{\displaystyle {\int}_{{z}_{0}}^{H}\text{d}z{I}_{v}\left(z\right)R\left(z\right)}.$$(14)
Since there are two outflows with opposite directions to each other, an observer at a distance d, whose line of sight makes a 90degree angle with the outflow axis, measures a flux of $$F\left(v\right)=\frac{2P\left(v\right)}{4\pi {d}^{2}},$$(15)
where d is the distance to the source. Since the observer’s line of sight is taken to be perpendicular to the outflow axis, there is no need to account for a Doppler shift.
In Fig. 1, we show the flux of the radio emission of the outflow for p = 3, B_{0} = 2 × 10^{5} G, and two values of the two main parameters, namely τ_{‖} = 8.6 and R_{0} = 110 R_{g} (the blue line, corresponding to the hard state), and τ_{‖} = 4.3 and R_{0} = 600 R_{g} (the red line, corresponding to the hardintermediate state). The slope of the partially optically thick spectrum is α ≈ 0.2.
Fig. 1 Flux per unit frequency produced by a powerlaw distribution of electrons with p = 3 and B_{0} = 2 × 10^{5} G for R_{0} = 110R_{g} and τ_{‖} = 8.6 (blue curve), and R_{0} = 600R_{g} and τ_{‖} = 4.3 (red curve). The powerlaw spectral index is α ≈ 0.2 in both curves. 
2.3 Xray observations
The Xray data analyzed in this work correspond to the 2007–2008 outburst of GX 3394 and were obtained with the Rossi Xray Timing Explorer (RXTE). The data cover the interval MJD 53769–54678. The details of the data analysis can be found in Reig et al. (2018) and Kylafis & Reig (2018). We remark that the data and the models used in this paper are the same as those used in Kylafis & Reig (2018). In this work, we restrict the analysis to the hard and hardintermediate states during the rise of the outburst. We obtained the energy spectra using the standard2 mode of the RXTE/PCA instrument in the energy range 3–9 keV to match the range used in Corbel et al. (2013). We fitted the spectra with a broken powerlaw model and a Gaussian component that represents the iron emission line at approximately 6.4 keV. We allowed for absorption at low energies. The hydrogen column density was fixed to N_{H} = 4 × 10^{21}cm^{−2} (Dunn et al. 2010).
In Fig. 2, we show the observed correlation between the 3–9 keV Xray flux and the photonnumber spectral index Γ. The blue dots correspond to the hard state and the red dots to the hardintermediate state.
Fig. 2 Relationship between the observed Xray flux in the 3–9 keV range and the photon index Γ. The blue dots correspond to the hard state and the red dots to the hardintermediate one. 
2.4 Comptonization
In our model, Compton upscattering of the soft input photons takes place in the outflow. It is assumed that the soft photons come from the inner part of the accretion disk and have a blackbody spectrum with kT_{BB} = 0.2 keV. This value of kT_{BB} is not crucial. The radiative transfer is undertaken by our Monte Carlo code, which has been in use for the past 20 yr. The soft photons are emitted isotropically upward at the base of the outflow. Typically, 10^{7} photons are sufficient to obtain robust statistics. The emergent hard Xray spectrum is a power law with photon number index Г.
In Kylafis & Reig (2018), we used this Comptonization model and reproduced the observed values of Г in the hard and hardintermediate states, along with the observed time lags of the 9–15 keV with respect to the 2–6 keV photons. The same models that reproduce the correlation between the time lag and Г (Kylafis & Reig 2018) were used for this study to compute the radio spectra (see Fig. 1) and the radio flux at 8.6 GHz (used in Fig. 3).
For each model (i.e., τ_{‖} and R_{0}) that produces, with Comptonization in the outflow, an index Г, we computed the radio flux that comes out of the outflow. In Fig. 3, we show the pairs (Г, F_{R}) from our model. We remark that Fig. 3 is entirely theoretical. In other words, it is a quantitative, theoretical prediction of our model for a future outburst of GX 3394 (and also for previous outbursts, if simultaneous radio and Xray data exist), similar to the one examined here. Figure 3 applies to the specific outburst studied here, yet we assert that the same curve (scaled up or down) should be observed in any outburst of GX 3394. It is difficult to imagine how such a correlation can be produced by the typically considered model (a hot inner flow and a relativistic jet), since Г is fixed by the hot inner flow (corona) and the radio flux from the jet, and the two do not seem to communicate with each other. While Fig. 3 depicts the rise of the outburst of GX 3394, we predict that in the declining part of the outburst, the curve is traversed backwards, that is, from high Г to low Г, although the maximum radio flux is lower than in the rising part.
The radioflux – Г prediction above is related, and qualitatively similar, to the prediction made by Kylafis & Reig (2018, see their Fig. 4) regarding the radiobreakfrequency  Г correlation in GX 3394. It is probably easier to confirm or reject the present prediction than the earlier one. As in Fig. 2, the blue dots in Fig. 3 correspond to the hard state and the red dots to the hardintermediate state.
It is interesting to look at Figs. 2–4 from a different point of view. Figure 2 is an observational correlation between F_{X} and Г. Figure 4 is an observational correlation between F_{X} and F_{R}, where we assume, for this discussion, that the sudden increase in F_{R} and its subsequent sudden decrease in the hardintermediate state (the red dots in Fig. 4) will be confirmed observationally. Therefore, it is an observational requirement that F_{R} must be correlated with Г. In other words, Fig. 3 is an observational requirement that should be obeyed by all the models. As we demonstrated above, Fig. 3 is obeyed by the outflow model. It will be interesting to see if it is obeyed by the other models also.
Fig. 3 Relationship between the theoretical radio flux at 8.6 GHz and the photon index Г. This relationship is a theoretical prediction. The blue dots correspond to the hard state and the red dots to the hardintermediate one. 
3 Results
Figures 2 and 3 show that we can find the correlation between F_{R} and F_{X}, We proceeded as follows: we binned the Xray data using the photon index Г in bins of 0.02 in size for the hard state, and of 0.1 for the hardintermediate state. We computed the average value for each bin. We then found the Monte Carlo model that produces a similar Г and took the radio flux computed by it. In this way, we have pairs (F_{R}, F_{X}) that correspond to the same Г.
In Fig. 4, we plotted the pairs (F_{R}, F_{X}). As in Figs. 2 and 3, the blue dots correspond to the hard state and the red dots to the hardintermediate state. The straight black line is not a fit to the blue dots, but represents the observational correlation found by Corbel et al. (2013). It is a powerlaw function ${F}_{\text{R}}=a{F}_{\text{X}}^{b}$, where а = 1.85 and b = 0.62. This power law breaks down when the source enters the hardintermediate state. There, the radio flux initially increases sharply and then sharply decreases within a narrow range of F_{X} values (see also Fig. 3) and this is a prediction of our model.
The radio flux depends on the value of B_{0} as F_{R} ~ B^{(p+1)/2}, where p is the exponent in the electron energy distribution (Eq. (6a)). Thus, we selected the appropriate B0 by only shifting the data points vertically, until they matched the observational correlation (the solid line in Fig. 4). In other words, the slope of the data points is correct. The shifting of the points vertically was done by the parameter B_{0}.
The careful reader may have noticed that we have used Eq. (10) for the calculation of the radio flux because of its simplicity. For an inclination angle between 40 and 60^{o}, which is thought to be appropriate for GX3394 (Shidatsu et al. 2011; Fürst et al. 2015), the calculation is much more complicated and the radio flux is lower than what we have calculated. The only effect from this is that the blue dots in Fig. 4 are not shifted vertically as much and the red dots do not reach such high values of F_{R}. In other words, the inferred value of B_{0} is slightly less than 2 × 10^{5} G.
Fig. 4 Relationship between the observed Xray flux in the 3–9 keV range and the computed radio flux at 8.6 GHz. The black line is not a fit to the blue dots, but is the observational correlation of Corbel et al. (2013). The blue dots correspond to the hard state and the red dots to the hardintermediate one. 
4 Discussion
We find it interesting that the observed slope of the F_{R}–F_{X} correlation is so adeptly reproduced with our outflow model. As mentioned in the Introduction, a correlation between the two quantities is expected, because the same electrons perform the Comptonization and the synchrotron emission in the outflow model. Of course, this could be any correlation and not necessarily the one observed.
In the hardintermediate state, the sharp drop of the radio flux has been observed in a few sources (Gallo et al. 2003; Homan et al. 2005; Fender et al. 2009; Coriat et al. 2009; van der Horst et al. 2013; Russell et al. 2020). Here, and for GX 3394, we make the prediction that as the source moves from the hard to the hardintermediate state, the radio flux first increases sharply and then sharply decreases. Since the hardintermediate state during the outburst rise lasts only for approximately five days, highcadence radio observations are needed to confirm this.
In this work, we have assumed that the observed radio emission comes from the outflowing corona. It is not clear whether there is a narrow relativistic jet in the central part of the outflow. If there is, our model makes the tacit assumption that it is not dominant. We remark that, in sources seen at high or moderate inclination (such as GX 3394), the emission from a relativistic jet (Lorentz γ ≫ 1) is expected to be weaker from the side rather than in the jet direction, because of Lorentz boosting. In such cases, the radio is expected to be dominated by the outflowing corona.
A question then naturally arises as to why sources whose radio emission comes mainly from the relativistic jet and sources whose radio emission is dominated by the outflow exhibit more or less the same F_{R}–F_{X} correlation. The answer seems to be that both the jet and the outflow are fed by the hot inner flow (ADAF). Whether the Comptonization takes place in the corona, which is rotating around the black hole (hot inner flow, ADAF), or in an outflowing corona (outflow), the hard Xray flux is more or less the same. Similarly, the scaling relations of Heinz & Sunyaev (2003) and Merloni et al. (2003) are valid whether the Blandford & Payne (1982) mechanism is active and an outflow is produced, whether the Blandford & Znajek (1977) mechanism operates and a relativistic jet is produced, or both.
Another question arises as to whether the outflow model is preferable to the typically considered model. In our opinion, the answer is unequivocally yes, and for the following reasons: (1) A model that in addition to explaining the observations also makes predictions, increases its credibility. The outflow model makes two predictions: that the radio flux should be correlated with Γ (Fig. 3) and that the radio flux in the hardintermediate state must initially suddenly increase and then suddenly decrease (see the red dots in Fig. 3) as the source goes to the soft state. A related prediction was made in Kylafis & Reig (2018) involving the radio break frequency and the photonnumber spectral index. (2) The outflow model is extremely simple, with only Comptonization in and radio emission from a parabolic outflowing corona. Nothing else is needed. (3) Despite its simplicity, the outflow model quantitatively explains many correlations and observations, some of which have not been explained by any other model. These are:
 1)
the energy spectrum from radio to hard Xrays for the source XTE J1118+480 (Giannios 2005),
 2)
the timelag – Fourier frequency correlation in Cyg X1 (Reig et al. 2003),
 3)
the correlation between the timelag and the photonnumber spectral index Γ in GX 3394 (Kylafis & Reig 2018) and other sources (Reig et al. 2018),
 4)
the fact that this correlation depends on the inclination of the source (Reig & Kylafis 2019),
 5)
the phaselag — cutoffenergy correlation observed in GX 3394 (Reig & Kylafis 2015),
 6)
the narrowing of the autocorrelation function with increasing photon energy seen in Cyg X1 (Giannios et al. 2004),
 7)
the correlation between the Lorentzian frequencies in the power spectrum and the photonnumber spectral index Γ in Cyg X1 and GX 3394 (Kylafis et al. 2008),
 8)
the photonnumber spectral index as a function of the phase of the typeB quasi periodic oscillation (QPO) in GX 3394 (Kylafis et al. 2020),
 9)
the outflow provides a natural lamppost for the hard Xray photons that return to the disk, where reflection and reverberation occurs (Reig & Kylafis 2021),
 10)
the F_{R} – F_{X} correlation in GX 3394 (this work),
 11)
and, finally, the outflow model may naturally explain the observed Xray polarization in BHXRBs (see the final remark below).
All of the above correlations are explained with only two parameters of the outflow: the radius R_{0} at its base and the Thomson optical depth τ_{‖} along its axis. For all of the observations and correlations above, these two quantities vary in our model in the same narrow ranges of 10 ≲ R_{0}/R_{g} ≲ 1000 and 1 ≲ τ_{‖} ≲ 10.
Due to the large density at the bottom of the parabolic outflow and its large size (10–10^{3} R_{g}), the soft input photons from the accretion disk, coming from below, see something resembling a “slab” at the bottom of the outflow, parallel to the accretion disk. This is because for τ_{‖} ~ 10 (appropriate for the hard state of GX 3394), the height h, at which τ_{‖} from the bottom of the outflow is 3, for example, is much less than the radius R(z_{0} + h) of the outflow there. Therefore, τ_{⊥} ≫ 3 and the bottom of the outflow appears to the soft photons as a slab. Compton scattering of Xrays in a slab parallel to the accretion disk produces polarization perpendicular to it, that is, along the outflow (Beloborodov 1998). This has been observed with IXPE (Krawczynski et al. 2022). If our picture is correct, then we predict that the Xray polarization direction in GX 3394 will change from parallel to perpendicular to the flow, as the source moves from the hard state (τ_{‖} ~ 10) to the hardintermediate state (τ_{‖} ~ 1).
Acknowledgements
We would like to thank PierreOlivier Petrucci for very useful comments on an earlier version of this paper and Iossif Papadakis, Andrei Beloborodov, Tom Russell, and Greg Marcel for useful discussions. Also, N.D.K. acknowledges useful email exchanges with Rob Fender and David Russell regarding radio observations in the hardintermediate state.
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All Figures
Fig. 1 Flux per unit frequency produced by a powerlaw distribution of electrons with p = 3 and B_{0} = 2 × 10^{5} G for R_{0} = 110R_{g} and τ_{‖} = 8.6 (blue curve), and R_{0} = 600R_{g} and τ_{‖} = 4.3 (red curve). The powerlaw spectral index is α ≈ 0.2 in both curves. 

In the text 
Fig. 2 Relationship between the observed Xray flux in the 3–9 keV range and the photon index Γ. The blue dots correspond to the hard state and the red dots to the hardintermediate one. 

In the text 
Fig. 3 Relationship between the theoretical radio flux at 8.6 GHz and the photon index Г. This relationship is a theoretical prediction. The blue dots correspond to the hard state and the red dots to the hardintermediate one. 

In the text 
Fig. 4 Relationship between the observed Xray flux in the 3–9 keV range and the computed radio flux at 8.6 GHz. The black line is not a fit to the blue dots, but is the observational correlation of Corbel et al. (2013). The blue dots correspond to the hard state and the red dots to the hardintermediate one. 

In the text 
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