© ESO 2018

1. Introduction

X-ray binaries (XRBs) display cycles of strong activity, where their luminosity increases by several orders of magnitude and their spectral shape changes drastically on long timescales, before decreasing back to quiescence. We call this entire phase an outburst. At the beginning of such an outburst, they are in the so-called hard state where their X-ray spectra are dominated by a power-law component with a hard photon index Γ < 2 (Remillard & McClintock 2006). During this state, they also show flat or slightly inverted radio spectra (e.g., Corbel & Fender 2002), interpreted as self-absorbed synchrotron emission from collimated, mildly relativistic jets (Blandford & Königl 1979). At some point, the X-ray spectrum of these objects undergoes a smooth transition from this power-law-dominated spectral shape, to a dominant blackbody of temperature ∼1 keV only. In addition, steady radio emission disappears, suggesting a quenching of the jets (Corbel et al. 2004; Fender et al. 2004, 2009). X-ray binaries remain in this soft state until a decline in luminosity makes them transit back to a hard state at the end of the outburst, along with reappearance of the jets. This surprising behavior has been observed multiple times in the past decades, and in dozens of different objects, where some have even undergone multiple outbursts (see Dunn et al. 2010, for a global overview). What is even more striking in these outbursts is that they seem to be very similar in different objects, while being different from one outburst to another in the same object.

A general scenario has been proposed by Esin et al. (1997), where changes in the accretion flow geometry provoke these spectral variations. In this view, the interplay between two different accretion flows is responsible for the spectral changes in the disk: in the outer parts, a cold standard accretion disk (SAD, Shakura & Sunyaev 2009) extends down to a given truncation (or transition) radius where an advection-dominated hot flow¹ takes place (Ichimaru 1977; Rees et al. 1982). The inner hot flow is expected to be responsible for the power-law component, while the outer cold flow produces the blackbody radiation. While the presence of a SAD in the outer regions of the disk is highly accepted to date (Done et al. 2007), the physical properties of the advection-dominated inner flow remain an open question.

Between slim disks (Abramowicz et al. 1988), advectiondominated accretion flows (ADAFs, Narayan & Yi 1994), adiabatic inflow-outflow solutions (ADIOS, Blandford & Begelman 1999), luminous hot accretion flows (LHAFs, Yuan 2001) and more peculiar models (e.g., Meyer et al. 2000; Lasota 2001), no satisfactory explanation has been provided so far (Yuan & Narayan 2014). A discussion about the major models and their current state can be found in Marcel et al. (2018). Many questions remain open but in this article we focus on: (1) reproducing the X-ray spectral shape of all the generic spectral states, (2) explaining the correlated accretion-ejection processes through their observables, that is, radio and X-rays fluxes.

In this work we consider an accretion flow extended down to the inner-most stable circular orbit, and thread by a large-scale vertical magnetic field B_z. It is well known that matter can only accrete by transferring away its angular momentum. This can be achieved by few physical mechanisms, namely internal turbulent (“viscous”) torques and magnetic torques from an outflow. When accretion is mostly due to internal (turbulent) viscosity, angular momentum is transported radially. This produces an optically thick and geometrically thin accretion disk, a SAD, which is observed as a cold multicolor-disk blackbody. The production of winds by SAD is still a debated question, but few simulations and observations have shown the possible existence of winds from standard disks (see discussion in Sect. 2.1.2). These winds cannot, however, explain the powerful jets associated with XRBs: the SAD perfectly suits for the jet-less thermal states in XRBs, that is, soft states. Alternatively, self-confined, super-Alfvénic jets can also provide a feedback torque on the disk, carrying away both energy and angular momentum in a vertical direction. This accretion mode, referred to as jet-emitting disks (JED, Ferreira & Pelletier 1995, and subsequent work), presents a supersonic accretion speed. Therefore, for the same accretion rate, this mode has a much smaller density than the SAD, leading to optically thin and geometrically thick disks. Disks accreting under this JED mode are therefore good candidates to explain power-law-dominated and jetted states in XRBs, that is, hard states.

The magnetic field strength is characterized by the mid-plane magnetization , where P_tot is the total pressure, the sum of the kinetic plasma pressure, and the radiation pressure. At large magnetization, the SAD can no longer be maintained as magneto-centrifugally driven jets are launched: a JED arises (Ferreira & Pelletier 1995; Ferreira 1997). Full MHD calculations of JEDs have shown that the transition occurs around μ ∼ 0.1 (Casse & Ferreira 2000a,b; Lesur et al. 2013).

A global scenario based on these possible dynamical transitions in accretion modes has been proposed to explain XRB cycles (Ferreira et al. 2006, hereafter Paper I; Petrucci et al. 2008). The causes of the evolution of the disk magnetization distribution μ(r) are still a matter of intense debate. The main uncertainty comes from the interplay between the magnetic field advection and diffusion in turbulent accretion disks, either geometrically thin or thick and with or without jets. Modern global three-dimensional (3D) MHD simulations do show that large-scale magnetic fields are indeed advected (Avara et al. 2016; Zhu & Stone 2018) but these simulations are always done on relatively short time scales, up to a few seconds, and it is hard to scale them to the duration of XRB cycles, typically lasting more than several months. In this paper, we assume that cycles result from transitions in accretion modes and focus on their observational consequences.

Petrucci et al. (2010) computed the thermal states of a pure JED solution and successfully reproduced the spectral emission, jet power, and jet velocity during hard states of Cygnus X-1. However, their calculations were done assuming a one-temperature (1T) plasma, but the necessity of a two-temperature (2T) plasma seems inevitable to cover the large variation of accretion rate expected during an entire outburst (Yuan & Narayan 2014).

Marcel et al. (2018, hereafter Paper II), extended this work by developing a 2T plasma code that computes the disk local thermal equilibrium, including advection of energy, and addresses optically thin-to-thick transitions in both radiation- and gas-supported regimes. For a range of radius and accretion rates, they showed that JEDs exhibit three thermal equilibria, one thermally unstable and two stable ones. Only the stable equilibria are of physical importance (Frank et al. 1992). One solution consists of a cold plasma, leading to an optically thick and geometrically thin disk, whereas the second solution describes a hot plasma, leading to an optically thin and geometrically thick disk. Due to the existence of these two thermally stable solutions, a hysteresis cycle is naturally obtained, but large outbursting cycles, such as those exhibited by GX 339–4, cannot be reproduced (Paper II). Nevertheless, JEDs have the striking property of being able to reproduce very well hard-state spectral shapes, all the way up to very luminous hard states L > 30% L_Edd. However, SAD-JED local transitions are expected to occur locally on dynamical time scales, typically ∼1 ms Kepler orbital time at 10 R_g, whereas hard-to-soft transitions involve time scales of days or even weeks. This implies that, at any given time, the disk must be in some hybrid configuration with some regions emitting jets, while others do not. It is expected that jets, namely magneto-centrifugally driven flows (Blandford & Payne 1982), are only launched from the innermost disk regions. This translates into a hybrid configuration where an inner JED is established from the last stable orbit R_in until an unknown transition radius R_J, and is then surrounded by an outer SAD until R_out. The exact location of the transition R_J depends on the global response of the magnetic field B_z to accretion rate evolution at the outer edge Ṁ_out: two unknowns. Therefore, R_J is treated here as a free parameter of the model. Such radial transition between two flows has already been studied in the context of non-magnetized accretion flows, advocating mechanisms such as evaporation or turbulent diffusion as the origin of the transition (see, e.g., Meyer & Meyer-Hofmeister 1994; Honma 1996). We study, however, configurations where the magnetization μ is large (near equipartition) and uniform in the JED region, and drops at the transition radius R_J, organizing the two-flow structure. Although the main properties of isolated JED and SAD are well understood, hybrid configurations imply mutual interactions that need to be described. For instance, part of cold radiation emitted from the SAD region must be intercepted by the geometrically thick JED and provides an additional cooling term that might change its general properties.

In this paper, we explore the observational signatures of disk configurations with an inner JED and an outer SAD. Section 2 describes this hybrid configuration, including interactions between the two regions, and explores some of its dynamical consequences. Section 3 presents the procedure followed to simulate and fit synthetic X-ray data from our theoretical spectra as well as to estimate the jet radio emission. Section 4 is devoted to the exploration of the parameter space, by varying the disk accretion rate Ṁ_in and transition radius R_J. Playing with these two parameters allows to completely cover the disk fraction luminosity diagram (hereafter DFLD, Körding et al. 2006). As an illustrative example, we apply our model and reproduce canonical states of GX 339–4, both in X-ray spectral shape and radio fluxes. We end with concluding remarks in Sect. 5.

2. Hybrid disk configuration: internal JED and external SAD

2.1. General properties

As introduced in Paper II, we consider an axisymmetric accretion disk orbiting a black hole of mass M. For simplicity, the disk is assumed to be in global steady-state so that any radial variation of the disk accretion rate Ṁ(R) is only due to mass loss in outflows. We define H(R) as the half-height of the disk, ε(R) = H/R its aspect ratio, Ṁ(R) = −4πRu_RΣ the local disk accretion rate, u_R the radial (accretion) velocity, and Σ = ρ₀H the vertical column density with ρ₀ the mid plane density. Throughout the paper, calculations are done within the Newtonian approximation. Moreover, and for the sake of simplicity, the disk is assumed to be always quasi-Keplerian with a local angular velocity , where G is the gravitational constant.

The disk is assumed to be thread by a large-scale vertical magnetic field B_z(R). We assume that such a field is the result of field advection and diffusion and we neglect thereby any field amplification by dynamo. Clearly, the existence of cycles shows that some evolution is ongoing within the disk. However, the timescales involved (days to months) are always much longer than accretion timescales inferred from the X-ray emitting regions. Therefore, as for any other disk quantity, the local magnetic field is assumed to be stationary on dynamical time scales (Keplerian orbital time).

The hybrid disk configuration is composed of a black hole of mass M, an inner jet-emitting disk from the last stable orbit R_in to the transition radius R_J, and an outer standard accretion disk from R_J to R_out. The system is assumed to be at a distance D from the observer. In the following, we adopt the dimensionless scalings: r = R/R_g, h = H/R_g = εr, where R_g = GM/c² is the gravitational radius, m = M/Ṁ, and ṁ = Ṁ/Ṁ_Edd, where Ṁ_Edd = L_Edd/c² is the Eddington accretion rate² and L_Edd is the Eddington luminosity. Since GX 339–4 appears to be an archetypal object, we decided to concentrate only on this object. We therefore use a black hole mass m = 5.8, a spin a = 0.93 corresponding to r_in = 2.1 and a distance D = 8 kpc(Miller et al. 2004)³. All luminosities and powers are expressed in terms of the Eddington luminosity L_Edd. An example of disk configuration is shown in Fig. 1 for ṁ_in = Ṁ_in/Ṁ_Edd = 0.1 and r_J = R_J/R_g = 15.

Fig. 1. Example of hybrid disk configuration in the JED-SAD paradigm. The inner disk regions are in a jet-emitting disk (JED) mode, up to a transition radius rJ, beyond which a standard accretion disk (SAD) is settled. The disk scale height H(R) is accurately displayed, while colors correspond to the central electronic temperature Te in Kelvin. The disk switches from an outer optically thick, geometrically thin jet-less disk to an inner optically thin, geometrically thick disk launching self-confined jets (not shown here). This solution has been computed for a transition radius rJ = 15 and a disk accretion rate ṁin = 0.1 at the disk inner radius rin = 2 (see Sect. 2.1 for more details). Other similar examples are shown in Fig. 11 for different pairs (ṁin, rJ).

Our goal is to compute, as accurately as possible, the radial disk thermal equilibrium from r_out to R_in by taking into account the known dynamical properties associated with each accretion mode. The inflow-outflow structure is described by the following midplane quantities:

(1)

where μ is the disk magnetization, ξ the local ejection index, m_s the sonic accretion Mach number and b the fraction of the JED accretion power that is carried away by the jets in the JED. The parameter is the magnetic shear of the magnetic configuration and is the toroidal magnetic field at the disk surface (see Ferreira 1997, for more details). From the above radial distributions, we can deduce the expressions of the vertical magnetic field B_z, the accretion speed u_R and the disk surface density Σ as a function of ṁ_in and the disk aspect ratio ε = H/R. The latter is obtained by solving the coupled energy equations for the ions and electrons in order to compute both electronic and ion temperatures (see Paper II for a full description of the method).

2.1.1. Inner JED

Jet-emitting disks solutions from a large radial disk extent are known to exist in a restricted region of the parameter space (Ferreira 1997, Fig. 2). For simplicity, we assume that any given configuration is stationary and that parameters are constant. An extensive study of the thermal structure and associated spectra of JED can be found in Paper II, where it is shown that the following set of parameters best reproduces XRB hard states, from low to very high luminosities:

• μ = 0.5: the disk magnetization μ has very little influence on X-ray spectra because the synchrotron emission does not contribute significantly to the equilibrium (Paper II). This value has therefore been chosen to lie between the two extreme values allowed for JED solutions, namely μ_min = 0.1 and μ_max = 0.8 (Ferreira 1997).

• ξ = 0.01: the smaller the ejection index, the less mass is being ejected and the larger the asymptotic jet velocity. A value ξ = 0.01 is consistent with mildly relativistic speeds (e.g., the case study for Cyg X-1 in Petrucci et al. 2010).

• m_s = 1.5: within the JED accretion mode, the jets torque is dominant and imposes m_s = m_s,jet = 2 qμ. The precise value of m_s,jet depends on the trans-Alfvénic constraint, but accretion in a JED is always at least sonic and usually supersonic m_s,jet > 1 (Ferreira 1997). In Paper II, we showed that a supersonic accretion with m_s = 1.5 allows to reproduce luminous hard states.

• b = 0.3: the fraction of the released accretion energy P_acc transferred to the jets has been computed within self-similar models and goes from almost 1 to roughly 0.2 (Ferreira 1997; Petrucci et al. 2010). The chosen value also appears as a good compromise and facilitates the reproduction of luminous hard states (Paper II).

The fact that the disk accretion rate necessarily varies with the radius was first introduced in accretion-ejection models by Ferreira & Pelletier (1993), in the context of magnetically driven jets, and later by Blandford & Begelman (1999) in the context of thermally driven outflows. In both cases, the disk ejection efficiency is characterized by the radial exponent ξ in ṁ(r) ∝ r^ξ. While it has been shown that magnetically driven jets require ξ < 0.1 (Ferreira 1997), the values measured in many simulations is usually higher, lying between 0.5 and 1 (Casse & Keppens 2004; Yuan et al. 2012, 2015, and references therein). It is somewhat troublesome that different simulations lead to a comparable value regardless of the strength of the magnetic field. Moreover, they were mostly done in the context of non-radiating hot accretion flows. But on the other hand, Zhu & Stone (2018) obtained ξ ∼ 0.003 with an isothermal equation of state. Our guess is that this issue is not settled yet, especially given the extreme sensitivity of the disk ejection efficiency to the local thermodynamics (Casse & Ferreira 2000b). As discussed above, we therefore simply assume a small value for ξ that is compatible with the existence of relativistic jets.

2.1.2. Outer SAD

As argued in the introduction to this paper, the outer disk regions are assumed to accrete under the SAD mode. This implies that the relevant torque is turbulent, probably due to the magneto-rotational instability (MRI). In this case, m_s,turb = α_νε. The Shakura-Sunyaev viscosity parameter α_ν needs to be specified and we use α_ν = 0.1 throughout this paper (Hawley & Balbus 2002; King et al. 2004; Penna et al. 2013). The magnetization must be small enough to allow for the development of the MRI; we choose μ = 10⁻³. As long as the SAD remains optically thick the value of μ does not affect our calculations of the SAD thermal equilibria.

By definition, no jets are present in a SAD. This translates into ξ_SAD = 0 (no mass loss) and b_SAD = 0 so that all released energy is either radiated or advected by the flow. Doing so, we neglect the potential presence of winds usually observed in XRBs, especially at high luminosities (Ponti et al. 2012; Tetarenko et al. 2016). This assumption appears reasonable for two reasons. First, although mass loss from turbulent disks is indeed possible and actually observed in MHD simulations (e.g., Proga et al. 2000; Bai & Stone 2013; Suzuki & Inutsuka 2014; Béthune et al. 2017; Zhu & Stone 2018), these magneto-thermally driven flows carry away a negligible fraction of the disk angular momentum and released accretion energy, thereby introducing no significant change in the disk structure. Second, the mass rate feeding the inner JED is simply ṁ(r_J) = ṁ_in (r_J/r_in)^ξ and is independent from ξ_SAD. Increasing ξ_SAD up to, for example, 0.5, would imply a strong increase of the disk accretion rate in the regions beyond r_J up to r_out. This would of course lead to a significant change of the emitted spectrum from these outer regions, but with no detectable counterpart (see, however, Chakravorty et al. 2016, and references therein) as long as the disk remains optically thick, which is the case here.

2.1.3. JED-SAD radial transition

We examine here some properties of the transition, assuming that it occurs over a radial extent of the order of a few local disk scale heights.

The first striking property is the existence of a trans-sonic critical point near R_J. Indeed, while the accretion flow is subsonic in the SAD with a Mach number m_s,SAD = m_s,turb = α_ν ε_SAD ≪ 1, it is supersonic in the inner JED with m_s,JED = m_s = 1.5. This property is a natural consequence of the transition from a turbulent “viscous” torque acting within the outer SAD to a dominant jet torque in the JED. Since the disk is assumed to be in a steady-state, the continuity of the mass flux Ṁ_JED = Ṁ_SAD must be fulfilled at the transition radius. Given the difference between the JED and SAD sonic Mach number, this implies a drastic density decrease between the SAD and the JED. The Thomson optical depth being defined by τ_T ∝ Σ, this density drop therefore implies a huge drop in the disk optical depth. Therefore, a dynamical JED-SAD radial transition naturally goes with an optically thin - optically thick transition (see Paper II).

The second striking property is the possible existence of a thin super-Keplerian layer between the JED and the SAD. In the outer SAD, the disk is slightly sub-Keplerian with a deviation due to the radial pressure gradient and of the order of . Within the JED, the much larger magnetic radial tension leads to a larger deviation of the order of μ ε (Ferreira & Pelletier 1995). This requires that the radial profile Ω(R) has two extrema (with dΩ/dR ≃ 0). Since all the disk angular momentum is carried away vertically in JEDs, there is no outward angular momentum flux into the SAD at R_J. This translates into a “no-torque” condition for the SAD. Such a situation has already been discussed in the context of a radial transition between an outer cold (SAD) and inner hot (ADAF) accretion flows, leading to a super-Keplerian layer (Honma 1996; Abramowicz et al. 1998). It is, however, not clear whether such a thin layer would still be present in our context given the existence of magnetic forces. These two extrema of the angular velocity, however, clearly define the radial end points of each dynamical (SAD or JED) solution, and the trans-sonic transition occurring in-between.

Constructing a dynamical solution describing the radial transition between a SAD and a JED is beyond the scope of the present paper and will be studied elsewhere. From now on, we assume that the two accretion modes can always be matched at a transition radius R_J. The calculation of the global disk equilibrium can subsequently be made using ṁ_in = Ṁ_in / Ṁ_Edd and r_J = R_J/R_g as independent variables.

2.2. Thermal structure of hybrid JED-SAD configurations

As described in Paper II, our accretion flow is locally described by a 2T (T_e, T_i), fully ionized plasma of densities n_e = n_i, embedded in a magnetic field B_z. We recall here the main equations used to compute the thermal equilibrium (see Paper II for full explanations). The electron and proton temperatures are computed at each radius using the coupled steady-state local energy balance equations

(2)

(3)

where the local heating term, of turbulent origin, varies according to the radial zone considered. Within the JED in R < R_J, this gives

(4)

whereas within the SAD in R > R_J it is

(5)

making use of the no-torque condition imposed at R_J and a constant Ṁ at the transition (Sect. 2.1.3). In principle, the released turbulent energy could be unevenly shared between ions and electrons by a factor δ. Throughout this paper, we use δ = 0.5 (see Paper II, Yuan & Narayan 2014, Sect. 2.3 and references therein). The other terms appearing in Eqs. (2) and (3) are the ion (electron) advection of internal energy q_adv,i (q_adv,e), the Coulomb collisional interaction between ions and electrons q_ie and the radiative cooling term due to the electrons q_rad. This term, as well as the radiation pressure P_rad term, is computed using a bridge function allowing to accurately deal with both the optically thin and the optically thick regimes (Paper II). The optically thin cooling regime q_thin is computed with the BELM code (Belmont et al. 2008), which includes Compton scattering, emission and absorption through bremsstrahlung and synchrotron processes.

The thermal equilibrium of the SAD region is well known. For large accretion rates, required for outbursting XRBs, the disk is mostly in the optically thick, geometrically thin cold regime (T_e ∼ 10⁵ – 10⁷ K). Heating of the SAD surface layers by the hard X-rays emitted by the inner JED might produce some disk evaporation (Meyer & Meyer-Hofmeister 1994; Meyer et al. 2000; Liu et al. 2005; Meyer-Hofmeister et al. 2005). This would require that we solve the 2D (vertical and radial) stratification of the disk, which is beyond our vertical one-zone approach. However, this is not expected to be crucial for the scenario depicted here. We therefore neglect the feedback of the inner JED over the outer SAD structure. In this approximation, and since the resolution is computed outside-in, the temperature of the outer SAD does not depend on any assumption made on the inner JED.

On the contrary, the effects of the outer SAD on the inner JED are twofold and cannot be neglected. The first effect is the cooling due to advection of the outer cold material into the JED. Indeed the local advection term q_adv can be either a cooling or a heating term, depending on the sign of the radial derivatives. This is self-consistently taken into account in our code (see Eq. (12) and Appendix A.2 in Paper II). The second effect results from the Compton scattering of the SAD photons on the JED electrons. This effect occurs whenever the JED is in the optically thin, geometrically thick thermal solution. This effect was not taken into account in Paper II.

Fig. 2. Effect of the external Comptonisation of the SAD photon field on the inner JED for ṁin = Ṁin/ṀEdd = 1 and rJ = 15 (green vertical line on the top figures), with different dilution factors, from left to right ω = 0, 1, 10, 50%. Top panels show the disk aspect as well as its Thomson optical depth in colors. Middle panels show the electron temperature as function of radius. Bottom panels display the local spectra emitted by each radius (dashed lines) and the corresponding total disk spectrum (black solid line). For comparison, in each panel we have overplotted the total spectra obtained in the other three panels in gray solid lines. The spectra are given in Eddington flux FEdd = LEdd/4πD2 units for GX 339–4 (see Sect. 2.1). The blue lines and dots correspond to the SAD zone, while the red lines and triangles represent the JED. The white part of the spectra shows the 3–200 keV energy range. Approximate values of the photon index Γ and energy cutoff Ecut, derived by comparison with a simple cutoff power-law model in this energy range, are indicated on each plot.

Illumination is estimated from the SAD properties, using the following geometrical prescriptions. Outside of the transition radius R_J, the disk luminosity is , where T_s = T_eff(R_*) is the effective temperature at the radius R_* where T_eff reaches its maximum (Frank et al. 2002). JED solutions being optically thin (or, at worse, slim), we assume that the radiation field in the region below R_J is well described by the average energy density

(6)

where 0 < ω < 1 is a geometrical dilution factor that describes the fraction of the SAD power that irradiates the region below R_J (see Sect. 2.3). This applies to the bolometric luminosity, but also to the luminosity in any given energy band.

This prescription allows to compute all properties of the illumination field. This radiation is then provided to the BELM code as an external source of seed photons, and the associated cooling and reprocessed spectrum are computed. More precisely, the JED is divided in many spheres of radius H in which radiation processes are computed (see Paper II for more details). Here, each sphere of radius H receives the power

(7)

where the value of ω is discussed in Sect. 2.3.

2.3. Effect of an external illumination on the JED thermal structure

Adding an outer standard accretion disk may have a colossal impact on the inner hot JED, depending on the transition radius r_J = R_J/R_g. For large values of r_J (say larger than 50–100), the power emitted by the SAD is too low to affect the global disk spectrum. But this is no longer the case when the transition radius becomes smaller. Besides, the geometrical dilution factor ω used in Eq. (6) plays an important role. It is however quite tricky to obtain an accurate estimate of its value within our framework. It depends on the solid angle under which the SAD photosphere is seen by the JED and corresponds thereby to the fraction of the SAD photons that are intercepted by the JED.

Considering a spherical hot corona of radius r_J, centered on the black hole and embedded in an infinitely thin disk, former studies (e.g., Zdziarski et al. 1999; Ibragimov et al. 2005) led to ω ∼ 2–25% depending on the dynamical (“no-torque”, “torque”) hypothesis made at r_J. The inner JED accretion flow is clearly different from a sphere of radius r_J (see Fig. 1), which could possibly suggest a value smaller than the above estimates. Moreover, Compton cooling should also be a function of the radius within the JED, and in the case of the geometry shown in Fig. 1, we could expect ω to decrease with decreasing radius.

There are however numerous effects that should magnify ω. First, the SAD is clearly flared, and therefore the infinitely thin disk approximation generally used is rather crude and tends to decrease the value of ω (see, e.g., Meyer-Hofmeister et al. 2005; Mayer & Pringle 2007). Second, although not considered in the literature, the photons emitted radially by the innermost region of the SAD are also expected to radiate directly towards the JED (see Fig. 2 top panel). Indeed, the photosphere τ_T = 1 necessarily crosses the disk midplane near r_J, allowing cold SAD photons to enter directly into the outer parts of the JED (and not only from the SAD surfaces). This should be responsible for another radiative contribution. The estimation of the corresponding photon flux emitted by the SAD and entering the JED in such a way is relatively complex; it would require us to solve the full (radial + vertical) radiative transfer problem to determine the photosphere properties of the SAD close to r_J. While this is far beyond the goal of the present paper, this effect should be similar to estimations⁴. Third, the reprocessing of the X-ray emission from the JED inside the disk will also naturally increase the SAD emission (Poutanen et al. 2018). In our model, we only take into account the intrinsic disk emission, but X-ray reprocessing can be mimicked by increasing ω. Finally, another important effect that should be taken into account is the gravitational light bending close to the black hole. This should strongly magnify the flux of disk photons impinging the JED in comparison to the Newtonian situation where they would mainly escape away from the disk. This effect should depend on the transition radius r_J as well as the radial position inside the JED. Ray-tracing simulations are required here for a rigorous computation, and again this is beyond the scope of the present paper. But this effect could be the dominant one, especially for small radii inside the JED or for small r_J, since the closer to the black hole, the stronger the light bending effect. This could result in a factor of a few to be applied to the number of SAD photons entering the JED in comparison to the absence of light bending (see, e.g., Miniutti et al. 2003). All included, we believe that ω of the order of a few tens of percent seems reasonable.

We report in Fig. 2 the JED radial temperature distribution, as well as the corresponding spectral energy distribution (SED), for a constant ω varying from 0 to 50%, the two physical extreme values for this parameter. Increasing ω obviously decreases the JED temperature and softens the SED. The variation is relatively important between ω = 0% and 50%. Using a simple power-law model, we find a spectral softening of ΔΓ ≃ 0.4 of the resulting power-law, along with a modification of its energy cutoff, from E_cut ≃ 500 keV to E_cut ≃ 200 keV.

Clearly, this effect cannot be neglected, as the value of ω has an important impact on the spectra. In this article, unless otherwise specified, we use ω = 0.2. This value appears to be close to the upper limit for previous estimations, but, considering the number of assumptions diminishing this value, we thought this was a good compromise. We note however that this does not mean that the inner JED captures 20% of the SAD luminosity, as there is still a factor (H/R_J)² in Eq. (7).

3. Synthetic observations: X-ray disk spectra and radio jet emission

3.1. From theoretical SEDs to simulated data

This work aims at providing synthetic hardness-intensity diagrams, or more precisely, disk fraction luminosity diagrams (DFLD, see, e.g., Körding et al. 2006; Dunn et al. 2010). To that purpose, our synthetic spectra must be processed in a way similar to observational data to derive the disk and power-law components from the fits, and place the corresponding points in a DFLD. This procedure, too rarely performed, is mandatory as we intend to compare our synthetic data to observations.

From our theoretical SED, an XSPEC table model was first built using the FLX2TAB⁵ command of FTOOLS⁶. Disk inclination was ignored for simplification but we add background and galactic absorption (WABS model in XSPEC with N_H = 4 × 10²¹ cm⁻², see Dickey & Lockman 1990; Dunn et al. 2010; Clavel et al. 2016). Then we simulated RXTE/PCA and HEXTE spectrum with the XSPEC simulation command FAKEIT. We use exposure times t_exp between 1 and 10 ks depending on the model flux in the 3–200 keV band in order to have a reasonably good signal-to-noise ratio (S/N). In this article, we use t_exp = 10 ks for the quiescent state (Sect. 4.4) and t_exp = 1 ks elsewhere. In Fig. 2, we plot for example the simulated spectrum from the theoretical SED produced by the JED-SAD configuration with ω = 0.1 (Fig. 2, third panel).

Each simulated spectrum is then fitted with three different models (a): WABS × (CUTOFFPL + EZDISKBB), (b): WABS × CUTOFFPL, or (c): WABS × EZDISKBB. We keep the best one according to a FTEST procedure (see, e.g., Clavel et al. 2016, Sect. 3.2). In the example shown in Fig. 2, model (a) gives the best fit with a reduced . The best fit parameters are a disk blackbody temperature T_in = 1.0 ± 0.3 keV, a photon index and a lower limit in cutoff E_cut > 400 keV. These values are consistent with simple estimations performed on the theoretical data Γ = 1.64 and E_cut ≃ 500 keV. The corresponding best fit disk and power-law components are plotted in dashed lines in Fig. 3.

The total unabsorbed disk luminosity L_Disk and unabsorbed power law luminosity L_PL are computed in the 3–200 keV range. In the example shown in Fig. 3, the powerlaw fraction, defined by PL_f = L_PL/(L_PL + L_Disk) is equal to 0.99 ± 0.01 and the total flux in the 3–200 keV band is L_tot = L_PL + L_Disk = 2.9 ± 0.1% L_Edd.

Fig. 3. Example of simulated RXTE/PCA (3–25 keV, in black) and RXTE/HEXTE (20–200 keV, in red) data sets from the theoretical SED produced by the JED-SAD configuration shown in the third panel of Fig. 2. The dotted lines are the power law and disk components corresponding to the best fit model. See Sect. 3.1 for more details.

3.2. Hard tail

It is well known that soft states show nonthermal tails generally observed above a few keV (McConnell et al. 2002; Remillard & McClintock 2006). Although uncertain, these tails are thought to be produced by a population of non-thermal electrons (see Galeev et al. 1979; Gierliński et al. 1999, for few investigations) that are not taken into account in our model. In order to reproduce such soft states, we added a power-law component to the synthetic spectra each time the fitting procedure favored a pure blackbody emission (model (c)) and the new data were re-fitted with a modified (c) WABS × (PL + EZDISKBB) model. The photon index of this power-law component is set to Γ = 2.5 and it is normalized in order to contribute to a fixed fraction of the 3–20 keV energy range(typically between 1% and 10%, Remillard & McClintock 2006)

An example is shown in Fig. 4. The theoretical model plotted at the top of this figure corresponds to a SAD extending down to r_J = r_in (no JED), with ṁ_in = 1. A fit with an absorbed disk component (model (c)) gives a disk temperature T_in = 0.97 ± 0.01 keV and a total flux L_tot = 4.1 ± 0.1% L_Edd in the 3–200 keV band. The data simulated in XSPEC include the additional power-law tail (Fig. 4, bottom panel). The best fit with new model (c) then gives T_in = 0.97 ± 0.01 keV, Γ = 2.5 ± 0.2, and L_tot = 4.5 ± 0.1% L_Edd with a reduced . We note that this procedure slightly increases the total flux detected in X-rays due to the addition of the hard power-law tail.

3.3. Jet power and radio luminosity

In a JED, the jets power available is a given fraction of the accretion power

(8)

Assuming that b is roughly a constant throughout an entire evolution, the jets power depends on both ṁ_in and r_J. We follow the computations of Heinz & Sunyaev (2003) to deduce the expected radio luminosity emitted by one jet component. It assumes that the jet emission is explained by self-absorbed synchrotron emission of nonthermal particles along the jet. We need, however, to modify their equations in order to account for the finite radial extent of the JED, imposing a finite radial extent of each jet. This leads to the following expression (see Appendix A for more details):

(9)

Fig. 4. Electron temperature (top-left) and theoretical spectrum (top-right) of the configuration ṁin = 1, μSAD ≪ 1, αν = 0.1, rJ = rin. Each annulus is displayed as a blue dot; its associated spectrum is shown in blue dashed lines and the total disk spectrum with a black solid line. The bottom panel shows final faked and fitted data after the addition of the hard power-law tail; black for PCA and red for HEXTE. Dashed lines show the best fit obtained with; see Sect. 3.2 for details.

where L_R = 4πD²F_R is the monochromatic power emitted at the radio frequency ν_R from an object at a distance D. The parameter f_R is a normalization constant, , and p is the usual exponent of the nonthermal particle energy distribution. In the standard case with p = 2 and r_J constant, one obtains the Heinz & Sunyaev (2003) dependencies, namely a radio power .

Our model therefore provides naturally both L_R and L_X. Indeed, for any given set of parameters (ṁ_in, r_J), we can compute L_X from our simulated SED, whereas an estimate of the radio luminosity can be obtained using Eq. (9). This is discussed in Sect. 4.

4. Reproducing typical XRB behavior: DFLD and canonical spectral states

In this section, we show that hybrid JED-SAD configurations can, in principle, reproduce the outbursting cycles of XRBs by varying only two parameters, the disk accretion rate ṁ_in and the transition radius r_J. This is done in two steps. First, we need to find which ranges in ṁ_in and r_J allow to cover the full DFLD. However, this diagram includes only information on X-ray emission while a cycle also deals with jet production and quenching. We therefore require the same framework to reproduce radio emission at the correct level. Then, as a second step, we define five canonical spectral states characteristic of an XRB spectral evolution during an outburst and show, more precisely, how well our framework is able to reproduce them.

Fig. 5. Total disk + power-law luminosity Ltot = Ldisk + LPL in the 3–200 keV energy range (in Eddington luminosity unit) is shown as a function of the power-law fraction LPL/Ltot. Each point within this plot corresponds to a fully computed and then XSPEC processed hybrid JED-SAD configuration. Contours (black solid lines) are for a constant disk accretion rate ṁin while the color background displays the disk transition radius rJ. Dashed black line shows the 2010–2011 cycle of GX 339–4. XSPEC fits were done with a hard tail level of 1% (left) and 10% (right). See Sect. 4.1 for a description of the figure.

4.1. Disk fraction luminosity diagram

We perform a large parameter survey for ṁ_in ∈ [0.01, 10] and r_J ∈ [r_in ∼ 50 r_in] = [2, 100]. We compute the whole thermal structure and corresponding theoretical SED for each pair (ṁ_in, r_J), and we fit as described in Sect. 3 to get the corresponding position in the DFLD. The fits⁷ are shown in Fig. 5. The smoothness of our DFLD is indicative of the absence of spectral degeneracy in our modeling (Fig. 5). In this figure, the mean transition radius is color coded and the accretion rate is shown in contours of constant values. For comparison, the 2010–2011 outburst of GX 339–4 is overplotted with a black dashed line⁸.

Figure 5 shows that we can cover the whole domain usually followed by XRBs within our framework. Concerning the hard states, we are able to reproduce their evolution up to high luminosities. Concerning the soft states, their position in the DFLD depends on the amplitude of the additional hard tail. With a hard tail representing 10% of the flux in the 3–20 keV energy range throughout the cycle, we can only reproduce soft states with L_PL/L_tot > 0.1 (Fig. 5, right). Softer states, populating the very left part of the DFLD, require a hard tail flux fraction lower than 1% (Fig. 5, left).

As expected, the accretion rate is mainly responsible for the global X-ray luminosity of the system, leading to almost horizontal isocontours for ṁ_in in the DFLD. Indeed, the higher the accretion rate, the higher the total available accretion energy P_acc ∝ ṁ_in. However, part of the energy can be advected, which explains the sharp variations of the isocontours at high luminosities (see discussion below). The effect of the transition radius r_J follows the predictions of Paper I. At large transition radii, most of the emission originates from the JED, as the outer SAD has no detectable influence. These solutions display power-law spectra for all accretion rates (see Paper II). This is the reason why they appear on the right-hand side of the DFLD. At small transition radii, two effects appear in the RXTE energy range of our simulated data. First, as its temperature and flux increase with decreasing r_J, the SAD blackbody emission starts appearing in the SED around 3 keV. Second, the closer the SAD, the stronger its illumination becomes, cooling down the inner JED and producing softer spectra. Combining these two effects leads to a disk-dominated spectrum, with a power-law fraction becoming entirely dominated by the high-energy tail when r_J → r_in.

4.2. L_X dependencies on ṁ_in

Figure 6 shows the bolometric (L_bol, top) and 3–9 keV (L_3–9, bottom) luminosity deduced from our synthetic SED as a function of the accretion rate at the inner radius ṁ_in. The colors correspond to different transition radius r_J. This figure illustrates different concerns about the radiative efficiency of accretion flows.

Fig. 6. Bolometric (top) and 3–9 keV (bottom) luminosities in function of the mass accretion rate ṁin onto the black hole. This plot is extracted from Fig. 5, done with a 10% hard tail (right). The colors are for different values of the transition radius rJ. Four different regimes are shown. Also, the rJ = rin has been drawn in dashed black to be visible at low accretion rate in the bottom panel.

In the top panel and in the SAD mode (r_J = r_in = 2), the bolometric luminosity follows the radiatively efficient regime L_bol ∝ ṁ_in as long as ṁ_in < 5. Below this accretion rate, the SAD is indeed radiating all of its available energy. At ṁ_in > 5, the SAD enters the slim domain, where more and more energy becomes advected instead of being radiated away. As a consequence, the global luminosity has a steeper slope .

As r_J increases, the JED region extends, and at any given accretion rate ṁ_in the luminosity decreases as r_J increases. This is the result of two different effects. First, as r_J increases, more and more energy is controlled by the JED and transferred to the jets b = P_jets/P_{acc, JED} = 0.3 (Papers I and II) instead of being radiated. Second, the JED thermal equilibrium is often strongly affected by advection, as f = P_adv/P_{acc, JED} ∝ ε² is no more negligible (Paper II). Combining these two effects, a more representative formulation is L_bol ∝ (1 – b – f) ṁ_in, where f ∝ ε² = ε² (ṁ_in) is also a function of accretion rate.

At low accretion rates ṁ_in < 0.5, the JED is optically thin and geometrically thick with ε ≃ 0.2–0.3 (termed thick disk solution in Paper II). In the thick disk branch, the low density of the plasma allows to be negligible. Ions are neither cooled down by radiation nor by Coulomb interactions: T_i ≫ T_e. Contrary to usual 1T plasmas, the disk thickness is then only linked to the ion pressure, P_gas,i ≫ P_gas,e + P_rad, leading to q_adv ≃ q_adv,i ≫ q_adv,e (see Eq. (13) in Paper II). In the ion thermal equilibrium from Eq. (2), advection is directly determined by ion heating, q_adv ≃ q_adv,i ≃ (1 − δ) q_turb, leading to f = q_adv/q_turb ≃ 0.5. Since q_adv,e ≪ q_adv,i ≃ (1 − δ) q_turb, we obtain q_adv,e ≪ δ q_turb. In the electron equilibrium Eq. (3), radiation is then determined by q_rad ∼δ q_turb ∝ ṁ_in, leading to the trend L_bol ∝ ṁ_in, unexpected in a thick disk⁹. In the end, combining the loss of power through jets (b) and in advection (f) reduces the JED luminosity. For instance, for r_J = 50 (yellow color) it is reduced by a factor 1/(1 – b – f) ∼ 5 compared to the SAD mode power.

At high accretion rates, ṁ_in > 2, the JED mode is optically thick and geometrically slim with ε ≃ 0.1 (termed slim disk solution in Paper II). The bolometric luminosity now has a slope (see the previous discussion on the slim SAD mode), lower in luminosity by a factor of approximately two. Indeed, in this case, f ≃ 0.15 due to the lower temperature (and then the lower ε) compared to the thick disk solution.

Between these two solutions, from the thick to the slim disks, f decreases from 0.45 to 0.15. The disk radiates more and more energy and the luminosity increases until it reaches the slope. During this transition, the slope is more abrupt and fits well with .

The bottom panel of Fig. 6 shows that the L_3–9 variation with the accretion rate is different from L_bol. In the SAD mode (r_J = r_in = 2), for ṁ_in < 1 and it slowly drops down to while approaching the slim region. The JED mode with r_J = 50 remains closer to the bolometric behavior, still with L_3–9 ∝ ṁ_in when ṁ_in < 0.5 and roughly during the transition and in the slim region. At very low accretion rates, the JED mode radiates more energy in the 3–9 keV band up to a factor of approximately four, while at higher accretion rates the SAD can radiate as high as approximately 17 times more energy in this range.

Two interesting comments can be done from Fig. 6. First a “radiatively efficient accretion flow” does not necessarily mean L ∝ ṁ_in. Indeed, as shown before in the JED-dominated mode (r_J = 50) we find L ∝ ṁ_in, while the JED radiates only a few tenths of its total available energy. The term “radiatively efficient” seems therefore inappropriate. Second, the evolution of the luminosity with the accretion rate strongly depends on the energy range used. In the SAD mode, while the disk is indeed radiatively efficient and L_bol ∝ ṁ_in, the 3–9 keV luminosity follows a regime, which could be considered as the signature of a radiatively inefficient flow. This clearly means that the interpretation of the luminosity variation with the accretion rate in terms of radiative efficiency can be strongly misleading and should be made with caution.

Finally, Fig. 6 also illustrates that the functional dependence L_3–9(ṁ_in) can be much more complex than a single power law. This is quite promising as it is known that L_3–9(ṁ_in) may need to vary from one object to another (Coriat et al. 2011, Sect. 4.3.3 and Fig. 7). However, we cannot go further without considering a proper outbursting cycle. For instance, Fig. 5 clearly shows that in order to successfully reproduce the 2010–2011 cycle of GX 339–4 (black lines in Fig. 8), one would need to (1) rise up, namely increase ṁ_in from the quiescent state until the highest hard state; subsequently (2) transit left by decreasing the transition radius r_J until the full disappearance of the JED; (3) drop down in the soft realm by decreasing ṁ_in; and finally (4) transit right back to the hard zone by increasing r_J. It can therefore be inferred from Fig. 6 that the evolution in time of the X-ray luminosity is sharper than L_3–9 ∝ ṁ_in at high luminosities, because of the necessary decrease of r_J. A detailed modeling of the actual track followed by GX 339–4 during a full cycle will be presented in a forthcoming paper.

4.3. Radio fluxes

For any given hybrid JED-SAD disk configuration, computed with a pair of parameters (ṁ_in, r_J), one can also derive an estimation of the one-sided jet radiative power P_R emitted at the radio frequency ν_R = 8.6 GHz. Assuming an electron distribution with p = 2, Eq. (9) leads to

(10)

Fig. 7. Power in the 8.6 GHz radio band in function of the 3–9 keV X-ray power (both in Eddington luminosity). Black solid lines are for constant transition radius rJ, while the color background shows the accretion rate ṁin. This figure has been made using a similar procedure to that used to make Fig. 5. The black points are the observed values for GX 339–4 during its cycles between 2003 and 2011. A possible theoretical jet line has been drawn in red (see Sect. 4.3).

Fig. 8. DFLDs for the past outbursts of GX 339–4 between MJD50290 and MJD55650 extrapolated in the 3–200 keV energy range. The spectral analysis of the 2010–2011 outburst was done in the 3–25 keV energy range (RXTE/PCA), but the models were integrated in the 3–200 keV range and not the usual ranges (e.g., Dunn et al. 2010). A typical cycle goes from Q (black), and crosses LH (orange) up to HH (red); it then transits to the HS (blue), decreases to LS (cyan) until it transits back to LH, before decreasing down to Q. The stars mark the positions of the five canonical spectral states Q, LH, HH, HS and LS defined in Sect. 4.4.

where the dimensionless factor incorporates factors such as m or R_in; see Appendix A. For the sake of simplicity, we assume that this factor is a constant throughout the whole cycle. This is far from being obvious, but our goal here is simply to show that hybrid configurations have the potential to reproduce both X-rays and jet radio emission simultaneously. To obtain an estimate of , we require that two radio observations at ν_R = 8.6 GHz, one during the quiescent state and the other during the high-luminosity hard state (see their definition below), be qualitatively reproduced. We obtain We obtain . This allows us to compute the radio power P_R as a function of (ṁ_in, r_J) and finally relate the radio power to the observed 3–9 keV power. To make Fig. 5, the same procedure as that used to make Fig. 5 was used, but with binning of the true integrated luminosity L_{3–9 keV} and radio power P_R. In addition, radio and X-ray observations of GX 339–4 are overplotted in black dots. It can be seen that most of them correspond to r_J ∼10–50 while the accretion rate spans ṁ_in ∼ 0.01 to almost 5. Therefore, reproducing the observed radio/X-ray diagram requires variations in mass accretion rate, from ṁ_in < 0.1 to almost 5 here. However, in order to describe the disappearance or reappearance of the steady radio emission, that is, the crossing of the jet line (Fender et al. 2004), one needs to invoke variations in transition radius: a decrease in r_J when the jet is quenched (bottom-right), and an increase when the jet re-appears (top-left). This is very promising, but further investigation on the model need to be done.

4.4. XRB canonical spectral states

In Sect. 4.1, it is shown that hybrid JED-SAD configurations can cover the observed DFLDs by varying ṁ_in and r_J independently. We focus here on the five typical spectral states that any given XRB needs to cross (or get close to) when making a full cycle, and detail their characteristics in our JED-SAD framework. These five states are shown in Fig. 8 and are named quiescent state (Q), low-luminosity hard state (LH), and low-luminosity soft state (LS), all at the soft-to-hard lower transition branch, and high-luminosity hard state (HH) and high-luminosity soft state (HS), both at the hard-to-soft upper transition branch.

Fig. 9. Computed radial structures of hybrid JED-SAD disk configurations associated to the canonical states defined in Table 1. From left to right: Q for quiescent state, LH for low-hard, HH for high-hard, HS for high-soft and LS for low-soft. Top: electron temperature Te at the disk mid plane. Bottom: Thomson optical depth τT. Red triangles show the JED zone, and blue dots describe the SAD zone. The vertical green line marks their separation at the transition radius rJ. The vertical yellow line marks the transition from a gas to radiation pressure supported regime within the SAD. In addition, the other two possible thermal solutions are shown in gray when present, the unstable in circles and the thin disk in triangles; see Paper II.

Fig. 10. Theoretical SED (top) and XSPEC spectral fits (bottom) for the five canonical spectral states computed in Fig. 9. It can be seen that hard state spectra are always dominated by the comptonization of soft photons, mostly due to local Bremsstrahlung and cold photons from the outer SAD. The white area in the theoretical SED corresponds to the observationally relevant 3–200 keV energy band. The value of the spectral index Γ is shown for each state, with its errors.

The quiescent state chosen is clearly not the most quiescent state that can be reached by an XRB; it is at the position in the DFLD that all objects need to cross while going up and down. Our choice of the two soft states in the DFLD is somewhat arbitrary as it depends on the chosen level of the hard tail; see Sect. 3.2. Our computed spectra are also shown from 0.5 to 500 keV but remember that the observational PCA and HEXTE data are available only from 3 to 200 keV.

The middle panel of Table 1 shows the values of the parameters ṁ_in and r_J that better characterize these five canonical states. We have also reported in this table the values (and their associated 3σ errors) of the power-law fraction PLf, X-ray luminosity, and spectral index Γ derived from XSPEC fits, as well as the expected radio flux density at 8.6 GHz, F_{8.6 GHz}. These fluxes have been computed in mJy using Eq. (10), namely F_{8.6 GHz} = 10²⁶ × P_R/(4πD²ν_R), with ν_R = 8.6 GHz and . Figures 9 and 10 illustrate the thermal state radial distribution (temperature, optical depth) and the theoretical and faked spectra. An accurate representation of the physical structure (size) and temperature (color) of the disk in those five states is shown in Fig. 11.

In the following, we discuss each of the canonical states in more detail.

– Q state. The quiescent state has an X-ray luminosity lower than 0.1% Eddington, with a typical power-law spectrum of index Γ ≃ 1.5–2.1 in the observed 3–200 keV band (Remillard & McClintock 2006). It exhibits faint but steady radio and IR luminosity fluxes (Fender 2001; Corbel et al. 2013), probing weak but detectable jets. It is located at the bottom-right of the DFLD. In our JED-SAD framework, it is characterized by a very low accretion rate ṁ_in ≲ 1 and a relatively high transition radius r_J ≫ r_in. In the example shown in this section, we choose ṁ_in = 0.06 and r_J = 100. The innermost region of the disk (from r ≃ 30 down to r_in) is optically thin with electron temperature as high as T_e ≃ 10¹⁰ K (Fig. 9, left panel). This results in a global spectrum that is the sum of multiple power-law spectra with roughly the same shape Γ ∼ 1.6–2 and E_cut ≫ 200 keV (Fig. 10, left panel), in good agreement with the observations. In addition, the power available in the jets is relatively small due to a very low accretion rate (Table 1).

– LH state. The low-luminosity hard state is characterized by a power-law-dominated spectrum, with spectral index Γ ≃ 1.5–1.6, no cutoff detected E_cut > 200 keV (Grove et al. 1998; Zdziarski et al. 2004, and references therein) and a typical luminosity L_tot ≃ 1% L_Edd. This state is also associated to steady and high radio and IR luminosities suggesting powerful jets. In this article, we choose ṁ_in = 0.4 and r_J = 50. As shown in the top panel of Fig. 9, the temperature of this flow increases quickly from T_e ≃ 10⁶ K in the outer parts of the JED to T_e ≳ 5 × 10⁹ K in the inner parts for most of the JED extension (from r = 20 down to R_in). This state does not strongly depend on r_J as long as it is larger than a few tens of R_in, as the global spectrum is the sum of similar spectra with Γ ∼ 1.2–1.8 (Fig. 10, LH-panel). This configuration is also accompanied by more powerful jets, due to a larger accretion rate compared to Q-states, in agreement with stronger observed radio emission.

– HH state. The high-luminosity hard state is also characterized by a power-law-dominated spectrum, with a spectral index Γ ≃ 1.6–1.8, but with a high-energy cutoff generally detected E_cut ≃ 50–200 keV (Motta et al. 2009) and luminosities as high as 30 % L_Edd (Dunn et al. 2010). Actually, as the luminosity increases, E_cut is observed to decrease from >200 keV to ∼50 keV, while the spectral index slightly changes from Γ = 1.6 to Γ = 1.8 before transiting to the soft state (see Fig. 6 in Motta et al. 2009). Those states also show the highest radio and IR fluxes (Coriat et al. 2009), suggesting the most powerful jets of the cycle. In our JED-SAD framework, this state is characterized by a larger accretion rate, ṁ_in > 1; we choose ṁ_in = 3. As shown in Paper II, at such a high accretion rate, the hot geometrically thick disk solution switches to a denser and cooler solution, the so-called slim disk. The disk is optically slim τ ∼ 1–10 and rather warm T_e ∼ 10^8–9 K (Fig. 9, middle panel). The spectra associated to those slim disk solutions are closer to a very hot multi-temperature disk blackbody emission (Paper II). The combination between electron temperature and optical thickness distribution with radius produces a spectral shape in agreement with observations (i.e., Γ ≃ 1.6–1.8 and E_cut ∈ [50, 200] keV). Once we choose ṁ_in = 3, the range of appropriate values for transition radius needed to reproduce the value of Γ is rather narrow; we adopt r_J = 15. The large ṁ_in and r_J result in a large jet power consistent with observations (see Table 1).

– HS state. The high-luminosity soft state is defined by a dominant multi-temperature blackbody with maximum effective temperature T_in ≲ 1 keV and total flux L_tot ∼ 5% L_Edd. In addition, many of the soft states display a minor component: the hard tail (see Sect. 3.2). These states are also characterized by the absence of steady radio emission, interpreted as the jet disappearance (Fender et al. 1999; Corbel et al. 2000). In our JED-SAD framework, this translates to r_J = r_in, that is, our accretion flow is entirely in a SAD mode. In the example shown in Fig. 9, we choose ṁ_in = 0.75 with a 10% hard tail.

– LS state. The spectral shape of the low-luminosity soft state is similar to the HS, with a typical maximum effective disk temperature T_in ≳ 0.6–0.7 keV, a Γ ≃ 2–3 hard tail, and a total luminosity approximately five times lower. We define our canonical LS state with the same level of hard tail. In our JED-SAD framework, this corresponds to lower ṁ_in but still r_J = r_in (no JED). In the example shown in this section, we choose ṁ_in = 0.45. The absence of JED means there are no jets, that is, no radio emitted.

Fig. 11. Computed geometrical shape of the hybrid disk, consistent with the dynamical resolution (Fig. 9) and SED (Fig. 10) for each of the five canonical states. The color background is the central electron temperature. We note that the X-scale is logarithmic and the Y-scale is linear.

5. Concluding remarks

In Paper II, we studied the thermal equilibrium of jet-emitting disks (JED). JEDs are assumed to be thread by a large-scale vertical magnetic field, building two jets that produce a torque responsible for supersonic accretion.

In this article, we extend the code to compute thermal equilibria of hybrid disk configurations. This configuration assumes an inner JED and an outer SAD, characterized by a highly subsonic accretion speed. The transition between those two flows is assumed to be abrupt (ΔR/R ≪ 1) at some transition radius r_J. As argued in Sect. 2.1.3, such a transition requires a discontinuity in the disk magnetization μ that can be obtained if the transition radius r_J is a steep density front. The transition radius r_J would therefore correspond to a density front advancing or receding within the disk during an outburst, as also found in the context of ADAF-SAD transitions (Honma 1996; Manmoto & Kato 2000). Why such a density front would be present is an open question, possibly answered by how matter is initially brought in towards the disk inner regions. In any case, if such a front is indeed produced, it is not clear how it would be maintained over the long duration of the outburst.

Regardless, such a density front is known to be favorable to the Rossby wave instability (Tagger & Pellat 1999; Lovelace et al. 1999; Li et al. 2000; Tagger et al. 2004; Meheut et al. 2010, and references therein), which leads to the formation of non-axisymmetric vortices within the disk. Whether or not the density front is smeared out and destroyed or simply perturbed (leading possibly to quasi-periodic oscillations) remains to be investigated. We refer the interested reader to the discussion on timing properties in Paper I, Sect. 4.

On the other hand, one might argue as well that such a discontinuity in the disk magnetization is unrealistic, and that, instead, there is a continuous increase in μ towards the disk inner regions (Petrucci et al. 2008). Assuming that such a situation were indeed possible, the transition radius r_J required in our spectral calculations would then be interpreted as the transition from the outer optically thick disk to the inner optically thin disk. Correspondingly, one could argue that the outer low-magnetized disk regions would give rise to winds, whereas jets would be launched from the inner highly magnetized disk regions (JED). The difficulty with this scenario is that it relies on the disk mass loss and the radial distribution of the large-scale vertical magnetic field, both unknown to date. Our simple approach, which assumes a sharp JED-SAD radial transition, can be seen as a first step towards addressing this difficult topic in XRB accretion disks.

The outside-in radial transition in accretion speed translates thermally, from an outer optically thick and cold accretion flow to an inner optically thin and hot flow. The soft photons emitted by the outer disk also provide a nonlocal cooling term which, added to advection of internal energy, allows a smooth thermal transition between these two regions. For a given JED-SAD dynamical solution, the corresponding spectrum depends only on the mass accretion rate onto the black hole ṁ_in and the transition radius r_J between the two flows. We explore in this article a large range in ṁ_in and r_J. Using XSPEC, we build synthetic spectra and fit them using a standard observers procedure (Sect. 3.1), allowing us to easily compare the resulting fits to observations.

We show that this framework is able to cover the whole domain explored by typical cycles in a disk fraction luminosity diagram (Fig. 5). Furthermore, five canonical X-ray spectral states representative of a standard outburst are quantitatively reproduced with a reasonable set of parameters (Figs. 9–11 and Table 1). A very interesting and important aspect of this framework is its ability to simultaneously explain both X-ray and radio emissions (Fig. 5 and Table 1). In a forthcoming paper, we will show the required time sequences ṁ_in(t) and r_J(t) needed to reproduce a full cycle within the JED-SAD paradigm.

Acknowledgments

We are grateful to the anonymous referee for his/her careful reading of the manuscript. The authors acknowledge funding support from French Research National Agency (CHAOS project ANR-12-BS05-0009, http://www.chaos-project.fr), Centre National de l’Enseignement Superieur (CNES) and Programme National des Hautes Energies (PNHE) in France. SC is supported by the SERB National Postdoctoral Fellowship (File No. PDF/2017/000841).

Appendix A

Jet radio emission

We follow here the same reasoning as Blandford & Königl (1979) and Heinz & Sunyaev (2003). Jets emit synchrotron radiation from a nonthermal power-law distribution satisfying , where γ is the particle Lorentz factor, p is the power-law index and C the normalization factor. This factor is related to the pressure of the relativistic particles and is usually assumed to follow the magnetic field pressure so that C = C_o B² where C_o is a constant. The synchrotron self-absorption coefficient α_ν and emissivity j_ν for a nonthermal particle distribution are (Rybicki & Lightman 1979)

(A.1)

where A_p and J_p are proportionality constants and weakly dependent on p. At a given distance Z from the source, the jet has a radius R_jet(Z) and a finite width ΔR_jet(Z). This allows to compute the synchrotron optical depth to self-absorption τ_ν = ΔR_jet α_ν. The local emitted spectrum is I_ν = S_ν(1 − e^−τ_ν) with a source function S_ν = j_ν/α_ν. At high frequencies in the optically thin regime, the spectrum decreases with the frequency, whereas at low frequencies, in the optically thick regime, I_ν ≃ S_ν ∝ 03BD;^5/2. At any given altitude, the self-absorbed spectrum is therefore peaked at a frequency ν_c defining the jet photosphere. It is such that τ_νc = 1, namely

(A.2)

and the jet surface brightness becomes

The monochromatic flux received at a frequency ν_c from a one-sided jet of width 2R_jet, viewed side-on at a distance D is

(A.3)

where the distances have been normalized to the gravitational radius (z = Z/R_g, r = R/R_g), b = B/B_in where B_in is a fiducial magnetic field and

(A.4)

The amplitude of the fiducial magnetic field B_in depends on the underlying jet model. In our case, we assume that it is the innermost (the largest) magnetic field within the JED, namely

(A.5)

where μ_o is the vacuum permittivity and P_* = m_pc²/σ_TR_g with m_p the proton mass, c the speed of light and σ_T the Thomson cross section.

At any given altitude z, the jet width Δr_jet is proportional to the radial extent of the jet-emitting region. Within a self-similar ansatz, namely using the self-similar variable x = z/r, one would simply write Δr_jet = (r_J − r_in) f_r(x), where f_r(x) = R(Z)/R_o is the self-similar function providing the cylindrical radius R(Z) of a field line anchored at a radius R_o within the disk. Following the same idea, the function b would be only a function of the self-similar variable x and r_jet = r_Jf_r(x). Equation (A.3) can then be written

(A.6)

where we used dz = r_inf_r(x)dx. The integral only depends on the jet dynamics and therefore, for a given frequency, the received flux scales as

(A.7)

allowing to compute the monochromatic power L_ν = 4πD²F_ν emitted by an XRB at a distance D. For a JED with μ/m_s ∼ 1, this leads to Eq. (9) with p = 2 and all proportionality constants incorporated in the dimensionless coefficient f_R. Note that this result has been obtained using an exact self-similar ansatz but it would remain valid as long as jets from black hole systems obey a more general similarity law (see discussion in Heinz & Sunyaev 2003).

This simple model also allows to derive the jet spectrum in the optically thick regime under quite generic assumptions. Equation (A.2) shows that for a given frequency ν_c, there will be a distance Z_c associated. Parts of the jet below and above Z_c will provide a negligible contribution to the overall spectrum at that frequency. As the jet width varies according to ΔR_jet(Z) ∝ R_jet(Z) and assuming as well as , after some algebra, one obtains where the power-law index is

(A.8)

Although one should not pay too much attention to this simple expression, it allows nevertheless to grasp interesting relations between the jet spectrum and the underlying physics.

The index ω describes the degree of collimation of the jet. Collimated flows require ω to be larger than unity (cone), a value of 2 (parabole) or slightly larger being acceptable. The index δ is more complex as it depends on the dominant magnetic field in the jet region. If B ∼ B_z ≫ B_ϕ, then a value δ = 2 would correspond to magnetic flux conservation in a jet with almost no toroidal magnetic field (hence no electric current). If, on the contrary, B ∼ B_ϕ ≫ B_z, then a value δ = 1 is more likely as it describes the existence of a constant asymptotic current (hence collimation). We therefore expect 1 ≤ δ ≤ 2. Now, Eq. (A.8) shows that for any ω < ω_o = ((2p + 13)δ – (p + 9))/(p + 4), one obtains α_p > 0. This shows that the less collimated the jet, the steeper the spectrum.

For the conventional value p = 2, ω_o = (17δ – 11)/6 and is always larger than unity for δ > 1. Flat spectrum sources with α₂ = 0 could then be described by a jet structure such that δ = (11+6ω)/17. This could be realized for instance with ω = 1 and δ = 1. However the presence of an asymptotic current is inconsistent with a conical jet shape (Heyvaerts & Norman 1989), meaning that it can be ruled out. The other extreme possibility, with B ∼ B_Z or δ = 2, leads to ω = 23/6 = 3.83, a highly collimated flow which usually requires the presence of an important axial current (therefore some B_ϕ). A more reasonable jet profile, such as ω = 2 (paraboloidal jet), requires δ = 23/17 = 1.35, a profile expected in a helical jet structure. This is perfectly reasonable and advocates self-confined magnetized jets as the source of the observed flat spectra in radio bands.

The monochromatic jet power is L_ν = 4πD²F_ν ∝ ν^α_p. If one considers that the jet power-law spectrum is established within a range [ν_min, ν_max], then the total bolometric jet power is

For a flat spectrum source (α_p ≃ 0), the radiative losses are dominated by the highest frequency ν_max ≫ ν_min. This translates into the convenient expression L_bol ≃ ν_max L_νmax = ν_RL_R(ν_max/ν_R)^{1 + α_p}, where L_R is the monochromatic power emitted at the radio frequency ν_R. The maximum frequency is observationally determined as the break frequency ν_B. Making use of Eq. (9) allows us to derive the radiative efficiency for a (one-sided)

which must always be (much) smaller than unity. This indeed appears to be the case if one uses the value f_R ∼ 3.4 × 10⁻⁹ (or ) derived in Sect. 4 and taking ν_B ∼ 10¹⁴ Hz, ν_R ∼ 10¹⁰ Hz with α₂ = 0.

References

All Tables

All Figures

Fig. 1. Example of hybrid disk configuration in the JED-SAD paradigm. The inner disk regions are in a jet-emitting disk (JED) mode, up to a transition radius rJ, beyond which a standard accretion disk (SAD) is settled. The disk scale height H(R) is accurately displayed, while colors correspond to the central electronic temperature Te in Kelvin. The disk switches from an outer optically thick, geometrically thin jet-less disk to an inner optically thin, geometrically thick disk launching self-confined jets (not shown here). This solution has been computed for a transition radius rJ = 15 and a disk accretion rate ṁin = 0.1 at the disk inner radius rin = 2 (see Sect. 2.1 for more details). Other similar examples are shown in Fig. 11 for different pairs (ṁin, rJ).

In the text

Fig. 2. Effect of the external Comptonisation of the SAD photon field on the inner JED for ṁin = Ṁin/ṀEdd = 1 and rJ = 15 (green vertical line on the top figures), with different dilution factors, from left to right ω = 0, 1, 10, 50%. Top panels show the disk aspect as well as its Thomson optical depth in colors. Middle panels show the electron temperature as function of radius. Bottom panels display the local spectra emitted by each radius (dashed lines) and the corresponding total disk spectrum (black solid line). For comparison, in each panel we have overplotted the total spectra obtained in the other three panels in gray solid lines. The spectra are given in Eddington flux FEdd = LEdd/4πD2 units for GX 339–4 (see Sect. 2.1). The blue lines and dots correspond to the SAD zone, while the red lines and triangles represent the JED. The white part of the spectra shows the 3–200 keV energy range. Approximate values of the photon index Γ and energy cutoff Ecut, derived by comparison with a simple cutoff power-law model in this energy range, are indicated on each plot.

In the text

	Fig. 3. Example of simulated RXTE/PCA (3–25 keV, in black) and RXTE/HEXTE (20–200 keV, in red) data sets from the theoretical SED produced by the JED-SAD configuration shown in the third panel of Fig. 2. The dotted lines are the power law and disk components corresponding to the best fit model. See Sect. 3.1 for more details.
In the text

Fig. 4. Electron temperature (top-left) and theoretical spectrum (top-right) of the configuration ṁin = 1, μSAD ≪ 1, αν = 0.1, rJ = rin. Each annulus is displayed as a blue dot; its associated spectrum is shown in blue dashed lines and the total disk spectrum with a black solid line. The bottom panel shows final faked and fitted data after the addition of the hard power-law tail; black for PCA and red for HEXTE. Dashed lines show the best fit obtained with; see Sect. 3.2 for details.

In the text

Fig. 5. Total disk + power-law luminosity Ltot = Ldisk + LPL in the 3–200 keV energy range (in Eddington luminosity unit) is shown as a function of the power-law fraction LPL/Ltot. Each point within this plot corresponds to a fully computed and then XSPEC processed hybrid JED-SAD configuration. Contours (black solid lines) are for a constant disk accretion rate ṁin while the color background displays the disk transition radius rJ. Dashed black line shows the 2010–2011 cycle of GX 339–4. XSPEC fits were done with a hard tail level of 1% (left) and 10% (right). See Sect. 4.1 for a description of the figure.

In the text

	Fig. 6. Bolometric (top) and 3–9 keV (bottom) luminosities in function of the mass accretion rate ṁin onto the black hole. This plot is extracted from Fig. 5, done with a 10% hard tail (right). The colors are for different values of the transition radius rJ. Four different regimes are shown. Also, the rJ = rin has been drawn in dashed black to be visible at low accretion rate in the bottom panel.
In the text

Fig. 7. Power in the 8.6 GHz radio band in function of the 3–9 keV X-ray power (both in Eddington luminosity). Black solid lines are for constant transition radius rJ, while the color background shows the accretion rate ṁin. This figure has been made using a similar procedure to that used to make Fig. 5. The black points are the observed values for GX 339–4 during its cycles between 2003 and 2011. A possible theoretical jet line has been drawn in red (see Sect. 4.3).

In the text

Fig. 8. DFLDs for the past outbursts of GX 339–4 between MJD50290 and MJD55650 extrapolated in the 3–200 keV energy range. The spectral analysis of the 2010–2011 outburst was done in the 3–25 keV energy range (RXTE/PCA), but the models were integrated in the 3–200 keV range and not the usual ranges (e.g., Dunn et al. 2010). A typical cycle goes from Q (black), and crosses LH (orange) up to HH (red); it then transits to the HS (blue), decreases to LS (cyan) until it transits back to LH, before decreasing down to Q. The stars mark the positions of the five canonical spectral states Q, LH, HH, HS and LS defined in Sect. 4.4.

In the text

Fig. 9. Computed radial structures of hybrid JED-SAD disk configurations associated to the canonical states defined in Table 1. From left to right: Q for quiescent state, LH for low-hard, HH for high-hard, HS for high-soft and LS for low-soft. Top: electron temperature Te at the disk mid plane. Bottom: Thomson optical depth τT. Red triangles show the JED zone, and blue dots describe the SAD zone. The vertical green line marks their separation at the transition radius rJ. The vertical yellow line marks the transition from a gas to radiation pressure supported regime within the SAD. In addition, the other two possible thermal solutions are shown in gray when present, the unstable in circles and the thin disk in triangles; see Paper II.

In the text

Fig. 10. Theoretical SED (top) and XSPEC spectral fits (bottom) for the five canonical spectral states computed in Fig. 9. It can be seen that hard state spectra are always dominated by the comptonization of soft photons, mostly due to local Bremsstrahlung and cold photons from the outer SAD. The white area in the theoretical SED corresponds to the observationally relevant 3–200 keV energy band. The value of the spectral index Γ is shown for each state, with its errors.

In the text

	Fig. 11. Computed geometrical shape of the hybrid disk, consistent with the dynamical resolution (Fig. 9) and SED (Fig. 10) for each of the five canonical states. The color background is the central electron temperature. We note that the X-scale is logarithmic and the Y-scale is linear.
In the text