Truncated accretion discs in black hole X-ray binaries: dynamics and variability signatures

For this work, we use the code BHAC equipped with adaptive mesh refinement (AMR) to solve the set of ideal as well as resistive GRMHD equations following Porth et al. (2017); Olivares et al. (2019); Ripperda et al. (2019). The basic resistive GRMHD equations are as follows,

where different symbols have their usual meaning, viz. $\rho,u^{\mu},T^{\mu\nu},F^{\mu\nu},{}^{*}F^{\mu\nu}$, and ${\cal J}^{\mu}$, stand for rest-mass density, four velocity, energy-momentum tensor, Faraday tensor, dual of the Faraday tensor, and electric 4-current, respectively. The details of these terms and explicit procedure of solving is given in Porth et al. (2017); Olivares et al. (2019); Ripperda et al. (2019). These equations are solved in a spherically symmetric, Modified Kerr-Schild (MKS) geometry in axisymmetric. By choosing an appropriate MKS stretching parameter, we ensure that the concentration of maximum resolution is near the equatorial plane of the simulation domain (McKinney & Gammie, 2004). Code BHAC employs the constrained-transport method (Del Zanna et al., 2007) to ensure a divergence-free magnetic field in the simulation domain. Throughout the work, we use a generalised unit system considering $G=M=c=1$, where $M$, $G$, and $c$ are the mass of the black hole, the universal gravitational constant, and the speed of light, respectively. In this unit system, mass, length, and time are expressed in terms of $M$, $r_{g}=GM/c^{2}$, and $t_{g}=GM/c^{3}$, respectively. Subsequently, we follow convention of sign of the metric as $(-,+,+,+)$, with four velocities satisfying $u_{\mu}u^{\mu}=-1$.

Our simulation box is extended radially from $r=r_{\rm H}$ (event horizon) to the $r_{\rm out}=500$. In the polar direction, our box is extended from $\theta=0-\pi$.

We set up a initial thin-disc following Dihingia et al. (2021); Dihingia & Vaidya (2022), the setup is based on the Novikov & Thorne (1973) model. The density profile in for the thin-disc setup in Boyer-Lindquist coordinates is given by,

Here, we choose $\alpha=2$ to maintain a thin disc configuration and $H$ is the scale height of the accretion disc, which is obtained following Riffert & Herold (1995) and Peitz & Appl (1997). The density profile on the equatorial plane $(\rho_{e}(r))$ is given by,

where $x=\sqrt{r}$, $\Theta_{0}$ is a constant related to the temperature of the initial disc, we chose $\Theta_{0}=0.0001$. Also, we consider polytropic index $\Gamma=4/3$ and entropy constant ${\cal K}=0.1$ for this study. Finally, $f(x)$ is given by,

where $x_{0}=\sqrt{r_{\rm ISCO}}$, $r_{\rm ISCO}$ is the radius of the ISCO (innermost stable circular orbit), and $s_{1},s_{2},$ and $s_{3}$ are the roots of the equation $s^{3}-3s+2a=0$. Along with the density profile, we also supply the initial azimuthal velocity, which is given as follows (Dihingia et al., 2021),

where

Here, $\Gamma^{\alpha}_{\beta\gamma}$ and $g_{\mu\nu}$ are the non-zero components of the Christoffel symbols and metric for Kerr black hole, respectively. Furthermore, we consider the thin-disc is truncated at a radial distance $r_{\rm tr}$ and extended up to the outer boundary of the simulation box $r_{\rm out}=500$.

In our study, we consider that the accretion disc is threaded by the poloidal magnetic field lines. The initial poloidal field lines are prescribed by implementing a vector potential $A_{\phi}$ following Zanni et al. (2007); Vourellis et al. (2019). The functional form of the vector potential is given by

The parameter $m(=0.1$) is related to the initial inclination of the field lines and it also determines the magnetic flux of the system. The parameter $m$ play crucial role in the launching of Blandford-Payne type wind from the accretion disc (Blandford & Payne, 1982; Dihingia et al., 2021). The initial strength of the poloidal magnetic field is determined by the choice of the plasma-$\beta$ parameter at the truncation radius $r_{\rm tr}$ on the equatorial plane as, $\beta_{\rm tr}=p_{\rm gas}^{\rm tr}/p_{\rm mag}^{\rm tr}$. Here, superscript ‘tr’ denotes quantities calculated at the $r=r_{\rm tr}$. Following our motivation, we carry out eight axisymmetric simulation models, by choosing different truncation radius $(r_{\rm tr})$, initial plasma-$\beta$ ($\beta_{\rm tr}$), effective resolution (with AMR), and magnetic resistivity. For axisymmetric models, we consider the highest effective resolution to be $2048\times 1024$ (with three refinement levels). The list of input parameters, effective resolution, and the final simulation time $(t_{\rm final})$ for all the simulation models are shown in the table 1. Out of all these models, we consider 2D40AH to be our reference model for the sake of explanation and comparison. In Fig. 1, we show the initial density distribution $(\log(\rho/\rho^{\rm tr}))$ and the initial gas pressure distribution $(\log(p_{\rm gas}/p_{\rm gas}^{\rm tr}))$ for the reference model at panels (a) and (b), respectively. In panel Fig. 1a, we also show the initial magnetic field lines in terms of gray lines. In Fig. 1b, the white line represent the boundary of plasma-$\beta=1$. In the figure, the density distribution follows equation 2 outside the truncation radius ($r_{\rm tr}=40$). Near the equatorial plane, the matter distribution is gas pressure dominated, and far from the equatorial plane and inside the truncation radius, the matter distribution is magnetic pressure dominated.

To prevent material from entering the numerical domain and interfering with accretion flow from the truncated disc, we impose no-inflow conditions at the inner radial boundary. At the polar axis, the scalar and radial components of vectors are considered symmetric, whereas the azimuthal and polar components of vectors are considered antisymmetric. Furthermore, in our simulation setup, we do not allow any inflow of matter at the outer edge of the accretion disc, which is rather a naive approximation. However, as the current study only focuses to understand the dynamics of the inner part of the truncated accretion disc, we run our simulation models up to $\sim 450-500$ inner disc orbits (at ISCO). This is about $\sim 15\%-20\%$ of the rotation time of the outer edge at $r=500$ and therefore the idealistic far out radial boundary will not affect the inner flow structure significantly within our simulation time.

We first discuss in detail our reference simulation 2D40AH, which is an axisymmetric run involving an ideal MHD approximation. Subsequently, in the later sections, a comparison is done with results obtained from resistive simulations.

The truncated accretion disc winds up the initial weak poloidal magnetic field lines and generates the toroidal component with the temporal evolution. The differential rotation of the flow triggers magneto-rotational instabilities (MRI) and drives turbulence, which helps in the transport of angular momentum. To ascertain that MRI is active in our simulations, we set the resolution such that the MRI quality factor $Q_{\theta}\gtrsim 10$ (see Appendix A for detail) throughout the simulation domain (see Sano et al., 2004). With the angular momentum transport, the Keplerian matter of the inner edge loses angular momentum and becomes sub-Keplerian. Subsequently, sub-Keplerian matter can occupy orbits of lower radii. As a result, the flow starts to accreting towards the black hole. Furthermore, the onset of disc-winds aids in transporting angular momentum outwards, which also contributes to setting up the accretion process (Dihingia et al., 2021). Eventually, the matter reaches the event horizon of the black hole.

To understand the dynamical properties in detail, in Fig. 2, we show the logarithmic normalized density distribution $(\rho/\rho^{\rm tr})$ and magnetic field lines for model 2D40AH. The panels (a), (b), (c), and (d) correspond to simulation time $t=3240,3800,4080$, and $4380$, respectively. It is interesting to note that in Fig. 2a and Fig. 2e, the accretion disc is extended up to the event horizon. In between (Fig. 2b-2d) the accretion disc is truncated. It essentially suggests that the accretion disc connects to the event horizon and disconnects from the event horizon periodically. During this process, the inner edge of the accretion disc oscillates in a quasi-periodic manner. We observe such oscillation events throughout our simulation (discussed further in section 4). In Fig. 3, we show the mass flux $\dot{m}=|\sqrt{-g}\rho u^{r}|$ at radii $r=5,10,15,$ and $20$ as a function of $\theta$ at simulation time (a) $t=3240$ and (b) $t=3800$ for the reference model. In Fig. 3a, we observe very high mass flux around the equatorial plane ($\theta\sim\pi/2$). As a result, matter advects to the black hole efficiently. In this state, matter drags the magnetic field lines along with it, and due to the high mass flux, magnetic flux efficiently accumulates around the event horizon. Further, we also find that these magnetic field lines are rooted into the ergosphere (see Fig. 2a and Fig. 2e). As the magnetic flux accumulated around the horizon reaches $\dot{\phi}_{\rm BH}\gtrsim 15$ (see section 4 for more detail), flow achieves a magnetically arrested disc (MAD) configuration (Tchekhovskoy et al., 2011). However, magnetic flux does not keep on building around the black hole and eventually loses the saturated magnetic flux in a rapid phenomenon commonly known as a magnetic eruption event (e.g., Igumenshchev, 2008; Vourellis et al., 2019; Porth et al., 2021; Dexter et al., 2020). After the eruption event, the accretion disc becomes truncated and the mass flux around the equatorial plane becomes negligible (see the black solid, blue dashed, and green dotted lines in Fig. 3b).

To understand the process of oscillation in detail, we show the density profile along with the magnetic field lines at time $t=3240$ (a) and $t=3800$ (b) for the reference model in Fig. 4. In the highly accreting state ($t=3240$), opposite polarity magnetic field lines squeezed together near the black hole, resulting in the formation of current sheets. These current sheets are prone to reconnection events. Furthermore, the turbulent $B_{\phi}$ component changes polarity and facilitates reconnection. In the ideal MHD models, the reconnection events are governed by the grid-dependent numerical resistivity. The distribution of the toroidal component of the magnetic field (a) $B_{\phi}$ and the magnetisation $(\sigma=B^{2}/\rho)$, as well as the magnetic field lines close to the black hole, are shown in Figure 5 for the reference model at time $t=4380$. The diagram clearly depicts the formation of islands of opposite polarity magnetic fields (5a), in which low-magnetised matter is trapped within highly magnetised matter (reffig-bphib). In our simulations, these figures (5a,b) confirm the formation of plasmoids and chains of plasmoids. Our simulations’ chain of plasmoids strongly suggests the presence of active tearing mode instabilities.

These reconnection events also create new poloidal field lines penetrating the equatorial (see Fig. 4b). That contributes to a strong magnetic tension force opposing the accretion of the flow. As a result, accretion is halted, and the accretion rate drops by orders of magnitude (for example, see solid black line in Fig. 3a and 3b). The increased magnetic tension force momentarily gets balanced with the force due to the ram pressure of the flow at a certain radius $r_{\rm max}$. Consequently, for the reference model, the inner edge of the accretion disc recedes up to a radius $r_{\rm max}\sim 20$, which is two times less than the initial truncation radius $r_{\rm tr}=40$. Further, the matter at the inner edge again accretes towards the central black hole due to loss of angular momentum and gravitational attraction forming an highly accreting flow (see the inner edge of the disc in Fig. 2b-2e). We observe the formation and depletion of highly accreting flow repeated till the end of our simulation. As a result, the inner part of the accretion disc keeps on oscillating between a highly accreting state and a low accreting state. Note that the radius $r_{\rm max}$ does not remain the same throughout the simulation time, $r_{\rm max}$ decreases with simulation time. For example, the edge of the accretion disc for the reference model recedes up to $r_{\rm max}\sim 17$ at simulation time $t\sim 10000$.

During the oscillation of the inner edge of the accretion disc, the inflow (high-density) and outflow (low-density) acquire shear in the transition layer. This layer may develop Kelvin–Helmholtz instability (KHI) due to shear. In Fig. 6, we plot the logarithmic density profile and the velocity streamlines for the reference model at simulation time $t=4220$. The red line in the figure corresponds to the $u^{r}=0$ contour. The regions with $u^{r}<0$ and $u^{r}>0$ are populated with inflowing and outflowing streamlines, respectively. We observe formation of velocity vortices at the boundary layers, which is a typical signature of active KHI. The inflow-outflow region does not reach a steady-state in our simulation; instead, it continues to vary with time. In Fig. 7, for example, we show the temporal evolution of the density profile and velocity streamlines at three different times for the reference model ($t=5300,5320$, and $5340$). The formation of a typical velocity vortex is depicted in the figure. The velocity vortex begins to form in the region where the flow has a high-density gradient. In addition, the inflowing and outflowing streamlines mix along the contour $u^{r}=0$ in the same region. These findings support the presence of KHI in the boundary layer. It should be noted that the only requirement for invoking KHI is a uniform velocity shear and no density difference (for more information, see Matsuoka (2014)). As a result, the boundary layer between the relativistic jet and the inflowing matter (along the $u^{r}=0$ contour) is vulnerable to the formation of KHI. As a result, KHI vortices keep on forming throughout our simulation runs. The infalling matter interacts with the fast jet ejected from close to BH through KHI vortices. These vortices formed at the interface layers can facilitate mixing and also help in mass loading to the jet.

The large-scale qualitative features of the resistive models are quite similar to those of the reference model (ideal MHD cases). We observe the accretion of matter and oscillations of the inner edge of the accretion disc within the event horizon to $r_{\rm max}$. Nonetheless, the time period of the oscillations depends on resistivity. Magnetic resistivity essentially helps in diffusing matter through the magnetic field and also impacts reconnection events in the magnetic field lines (e.g., Vourellis et al. (2019); Ripperda et al. (2019); Ripperda et al. (2020)). In Fig. 8, we show the density profile and the magnetic field lines for our resistive models (a) rl-2D40AH ($\eta=1\times 10^{-3}$) at $t=4680$ (b) rh-2D40AH ($\eta=5\times 10^{-3}$) at time $t=5190$. The times are chosen such that the accreting matter is connected to the event horizon for both models. By comparing with the reference model (Fig. 4a), for resistive cases, we do not observe the formation of plasmoids due to tearing mode instabilities. Such plasmoids form only in the domain with Lundquist number $S\gtrsim 10^{4}$, where reconnection rate is independent of $S$ and commonly known as the ‘fast’ reconnection mode or ideal tearing mode (Bhattacharjee et al., 2009; Huang & Bhattacharjee, 2010; Loureiro & Uzdensky, 2016; Striani et al., 2016; Ripperda et al., 2020, etc.). With the increase of magnetic resistivity, plasmoid instability is suppressed. A similar observation has been reported earlier by Ripperda et al. (2020). Also, the increase in resistivity suppresses the MRI-induced turbulence (for example, see Qian et al., 2017). As a result, the turbulent feature in the flow decreases with the increase of resistivity, which is evident from Fig. 4a, 8a, and 8b.

A detailed comparison of ideal and resistive MHD flow close to the black hole is shown in Fig. 8c (ideal MHD) and 8d (resistive MHD) in terms of magnetization $(\sigma=b^{2}/\rho)$ and magnetic field lines. The magnetization profile is quite similar for both models, with highly magnetized $(\sigma>1)$ regions in the off equatorial plane and the matter in the equatorial plane is low magnetized $(\sigma<1)$. The figures clearly show the turbulent nature of the flow and the formation of plasmoids due to the reconnection of opposite polar magnetic field lines for the ideal model. However, in the resistive MHD model, the flow is not turbulent, and we do not observe the formation of plasmoids. Note that we consider the resistivity $\eta$ to be constant throughout the simulation domain. In general, resistivity can be a function of space and time. Recently, with such resistivity profiles, it has been shown that resistivity plays an important role in the launching process of outflows from the accretion disc in resistive GRMHD (Qian et al., 2018), the generation of a turbulent outflow due to reconnection (Vourellis et al., 2019), or the magnetic field amplification by a mean-field $\alpha^{2}$-$\Omega$ disc dynamo (Vourellis & Fendt, 2021).

In this section, we study the slow-magnetosonic (hereafter, magnetosonic) behaviour of the truncated accretion disc. To do that, in Fig. 9, we show the radial four velocity ($u^{r}$), by following same temporal snapshots as Fig. 2. The boundary between inflow and outflow is marked by the red contour ($u^{r}=0$) in Fig. 9. As the disc is truncated, the region with $u^{r}<0$ reduces and becomes fainter (see Fig. 9b) indicating negligible inflow, whereas the region with $u^{r}>0$ increases and become darker, showing a large amount of outflow. With time the region with $u^{r}<0$ increases, also observe an increase in the radial velocity (becomes dark blue). We also observe a sharp transition of radial velocity close to the black hole (see Fig. 9c-d). Such a change in velocity indicates the presence of a shock transition in the flow. Previously, many semi-analytic hydrodynamic studies have hinted at the formation of such shocks and suggested that shock solutions are viable in understanding the radiative and timing properties of astrophysical sources (Chakrabarti (1989, 2011); Aktar et al. (2015); Kumar & Chattopadhyay (2017); Dihingia et al. (2018); Dihingia et al. (2020); Dihingia et al. (2019b); Dihingia et al. (2019a); Das et al. (2021), etc.). Similarly, semi-analytic MHD studies also suggested the formation of magnetosonic shock in the vicinity of the black hole (e.g., Takahashi et al., 2002, 2006; Fukumura et al., 2007).

We show the distribution of the slow magnetosonic Mach number ($M_{s}=u_{p}/a_{s}$; hereafter, Mach number) by following the same temporal snapshots as Fig. 2, where poloidal velocity $u_{p}^{2}=u^{r}u_{r}+u^{\theta}u_{\theta}$, slow magnetosonic speed $a_{s}^{2}=\frac{1}{2}(a_{0}^{2}+a_{f}^{2})-\frac{1}{2}\sqrt{(a_{0}^{2}+a_{f}^{2})^{2}-4a_{0}^{2}a_{f}^{2}\cos^{2}\xi}$. The sound speed and the poloidal Alfvén speed are given by $a_{0}^{2}=\gamma p/\rho h$ , $a_{f}^{2}=B_{p}^{2}/(B_{p}^{2}+\rho h)$, respectively, where $B_{p}^{2}=B_{r}B^{r}+B_{\theta}B^{\theta}$ and $h$ is the specific enthalpy of the flow. Finally, $\xi$ is the angle between the poloidal velocity vector and the poloidal magnetic field vector. In the figure, the blue line corresponds to contour $M_{s}=1$. In Fig. 10a and 10e, we observe that accretion flow around the equatorial plane is super slow-magnetosonic (hereafter, super-magnetosonic) ($M_{s}>1$), where the accretion flow is connected to the event horizon. Far from the equatorial plane, we observe a region with sub slow-magnetosonic (hereafter, sub-magnetosonic) ($M_{s}<1$) to super-magnetosonic ($M_{s}>1$) flow surrounding the rotation axis of the black hole (within $\theta\sim 25^{\circ}-45^{\circ}$ for the upper quadrant). In this region, very far from the black hole flow is super-magnetosonic ($M_{s}>1$), and also close to the black hole flow is super-magnetosonic ($M_{s}>1$). In between, we observe a sub-magnetosonic ($M_{s}<1$) region (dark violet), with a smooth transition between the layers. With time, we observe the formation of a sub-magnetosonic flow close to the black hole (Fig. 10b, see the dark violet region). As the flow moves far from the equatorial plane, the flow smoothly makes a transition from sub-magnetosonic ($M_{s}<1$) to super-magnetosonic ($M_{s}>1$) velocity. In Fig. 10c and 10d, we observe a super-magnetosonic region $(M_{s}>1)$ close to the black hole, which has a sharp transition to a sub-magnetosonic value close to the black hole $(r\sim 15)$, indicating the presence of shocks in our simulations. For the sake of clarity, we explain this in terms of a parabolic streamline $y=cx^{2.5}$, with $c=0.009$ (depicted by solid black line in Fig. 10c and Fig. 10d). Along the streamline, very close to the black hole flow is super-magnetosonic ($r\lesssim 2$), within $2\lesssim r\lesssim 15$ flow is sub-magnetosonic, and then flow is super-magnetosonic again with a shock transition (for detail see the Fig. 11e). After that, flow undergoes multiple transitions between these regimes and finally ends up as super-magnetosonic for $r\gtrsim 60$.

In Fig 11, we show the flow properties along the parabolic streamline ($y=cx^{2.5}$) for model 2D40AH at time $t=4220$. Panels (a), (b), (c), (d), and (e) correspond to the radial four-velocity $(u^{r})$, density $(\rho/\rho^{\rm tr})$, pressure $(p/p^{\rm tr})$, Bernoulli parameter $(-hu_{t})$ profile, and Mach number ($M_{s}$), respectively. The panels suggest that the flow has a shock transition in between $r_{s}=15.0-15.5$, shown in the figure by the gray vertical line. In Fig. 11a, we observe that at the shock, radial velocity sharply dropped from $u^{r}_{-}=0.2393$ to $u^{r}_{+}=0.0382$. Here, ‘-’ and ‘+’ correspond to quantities obtained before and after the shock transition. As an inset, in Fig. 11a, we show the transition of radial velocity $(u^{r})$ during the shock transition.

Away from this shock transition, the radial velocity of the flow increases closer to the black hole (smaller $r$). Whereas at a larger radius far from the shock, the radial velocity becomes zero $(r\backsimeq 58.0$, see the red dot in Fig. 11a) and increases in the opposite direction as flow moves far from the black hole. This point divides the inflow from the outflow. The red contours show a similar division between inflow-outflow in Fig. 9. Similarly, in Fig. 11b, we observe that the density profile also shows a jump across the shock front from $\rho_{-}=2.13\times 10^{-7}\rho^{\rm tr}$ to $\rho_{+}=9.08\times 10^{-7}\rho^{\rm tr}$. It indicates the presence of a highly compressed shock with a density jump $R_{\rho}=\rho_{+}/\rho_{-}=4.26$. Similar to the density, the pressure also increases across the shock with a pressure jump $R_{p}=p_{+}/p_{-}\sim 10.06$ (see Fig. 11c). Consequently, the temperature jump across the shock is $R_{\Theta}(=R_{p}/R_{\rho})=2.36$. However, the specific energy or the Bernoulli parameter $(-hu_{t})$ does not change significantly across the shock. This fact essentially suggests the non-dissipative nature of the shock (see Fig. 11d). Across the shock, the value of the Bernoulli parameter is less than unity $(-hu_{t}\sim 0.98)$, indicating that the immediate post-shock flow is gravitationally bound $(-hu_{t}<1)$.

As evident from panel (d) of Fig. 10, in Fig. 11e, the Mach number ($M_{s}$) profile along the streamline shows three magneto-sonic points at $r=2.3,43.9,$ and $65.0$, respectively. These critical points are denoted as red dots in Fig. 11e. The gray horizontal line in Fig. 11e corresponds to $M_{s}=1$, indicating the division between the sub-magnetosonic and super-magnetosonic regions. For the outgoing branch $(u^{r}>0,r>58.0)$, the sub-magnetosonic flow becomes super-magnetosonic after crossing the magnetosonic points at $r=65.0$. The in-going branch $(u^{r}<0,r<58.0)$ becomes super-magnetosonic after crossing the magnetosonic point at $r=43.9$ but becomes sub-magnetosonic after the shock transition $(r_{s})$. Finally, the flow becomes super-magnetosonic after crossing the magnetosonic point at $r=2.3$. We observe that Mach number jumps from $M_{s,-}=1.98$ to $M_{s,+}=0.49$ across the shock front.

To understand the dynamics of the shock, in Fig. 12, we show the temporal evolution of the radial velocity profile for 2D40AH from time $t=4200$ to $t=4350$. The shock locations obtained at times $t=4200,4250,4300,4310,4320,4330,4340$, and $t=4350$ are $r_{s}=15.40,14.98,14.32,13.83,13.20,12.33,11.09$ and $r_{s}=4.44$, respectively (here, we consider the mean value). Thus, shock is not steady and the shock front moves towards the black hole with time. The velocity of the shock front also increases with time. Finally, the shock disappears with a huge outflow. After the outflow, as advection starts to dominate, shock appears again. We observe the appearance and disappearance of shock throughout the simulation time in all the simulation models. This essentially suggests that the shock transition in the truncated accretion disc is a transient phenomenon.

In summary, we observe that the flow creates a hot, comparatively high-density region around the black hole during the shock transition. Also, note that the density of this post-shock region is much lower than that of the thin disc. This hot region can be associated with an extended corona surrounding the black hole. Many earlier authors anticipated the formation of such an extended corona from the truncated disc (Eardley et al., 1975; Ichimaru, 1977). With temporal evolution, the size of the corona also changes as the shock front moves towards the black hole. The variations in the size of the corona may lead to fluctuations in the electromagnetic emission. As a result, these fluctuations in emission could be useful in understanding variability in astrophysical sources.

We incorporate the GRRT post-processing module RAPTOR and calculate the near horizon emission from the reference run. RAPTOR reproduces the appearance and spectrum of black hole sources to a distant observer considering the effects due to a strong gravitational field (viz. gravitational lensing, redshift, and relativistic beaming) (Bronzwaer et al., 2018). RAPTOR also allows us to incorporate different radiative processes to render emission from the accretion disc.

In this study, we aim to study emission signatures in the hard-intermediate state. Accordingly, we consider the thermal synchrotron and Bremsstrahlung processes as the sources of emission and neglect the black body component, which may be necessary for the soft and soft-intermediate states of an outburst. We calculated the spectrum for a black hole with a mass $M=10M_{\odot}$. To model the electron temperature from the flow temperature, we use $R-\beta$ prescription following Mościbrodzka et al. (2016), where we choose $R_{l}=1,R_{h}=60$ following Mizuno et al. (2021). We collected all the emissions on a screen from $r\lesssim 60$ and fixed the line of sight angle at $i=60^{\circ}$. We fixed the distance of the screen at $r_{\rm cam}=10^{4}r_{g}$. To mimic the accretion flow around a BH-XRB, we scale the density in cgs units with the help of the rest-mass density scaling factor $\rho_{\rm unit}=M_{\rm unit}/r_{g}^{3}$, where we consider $M_{\rm unit}=10^{13}$ gm . Accordingly, we scale energy density, magnetic field strength, and number density in cgs units using, $U_{\rm unit}=\rho_{\rm unit}c^{2}$, $B_{\rm unit}=c\sqrt{4\pi\rho_{\rm unit}}$, and $N_{\rm unit}=\rho_{\rm unit}/(m_{e}+m_{p})$, respectively, where $m_{e}$ and $m_{p}$ are mass of electron and proton.

With this, the accretion rate calculated when the inner edge is attached and detached to the event horizon is of the order of $\dot{M}_{\rm acc}\sim 10^{-3}$ and $\dot{M}_{\rm acc}\sim 10^{-5}$ Eddington units ($\dot{M}_{\rm Edd}=1.44\times 10^{17}\left(M/M_{\odot}\right)$ g s^-1), respectively. Note that the spectral properties of an accretion disc depend on the mass of the black hole, accretion rate, line of sight angle, non-thermal particles, $R_{l}$, $R_{h}$ parameters, etc. (e.g., Bandyopadhyay et al., 2021; Mizuno et al., 2021; Fromm et al., 2021). For this study, we do not intend to do parametric studies. Instead, we want to study the qualitative properties of emission during the oscillation of the inner edge of the accretion disc. Therefore, we only considered one set of parameters throughout the study.

In Fig. 18, we plot the emission spectrum for the reference run at different simulation times, where the thin and thick lines present the contribution only from the thermal synchrotron process alone and that including Bremsstrahlung, respectively. In the figure, $\nu F_{\nu}$ is expressed in units of ‘Jy Hz’. Different simulation times are marked on the figure. During this simulation time range, the accretion matter moves from the radius $r_{\rm max}$ to the event horizon. The emission spectra show a peak value around $\nu\sim 10^{16}-10^{18}Hz$, which corresponds to the emission due to the thermal synchrotron process and dominates up to $\nu\lesssim 10^{19}Hz$. Whereas the high energy $(\nu\gtrsim 10^{19})$ part of the spectrum is dominated by the emission due to the Bremsstrahlung process.

The emissions from the Bremsstrahlung process show a distinct shape change with respect to frequency within $\nu\sim 10^{18-20}$Hz, which signifies the presence of regions with a sharp change in temperature and density. At the time $t=7500$ (for solid black line), the accretion disc is truncated, and the inner edge of the thin disc is also far from the black hole. Overall, the flow is cold, and the emission comes only from the thin Keplerian disc and weak disc-winds. Consequently, we observe lower emissions in the high-energy region.

As the flow gradually loses angular momentum, the inner edge of the disc approaches the black hole. During this process, the matter in the inner edge becomes hot, accumulates more magnetic flux, and disc-winds also become stronger. This results in higher intensity of thermal synchrotron and Bremsstrahlung emission. Also, we observe that peak due to thermal synchrotron emission moves towards the higher frequency range as matter approaches the black hole. As the flow reaches close to the event horizon $(t=8500$, see the green dot-dashed line), the emission within energy range $(h\nu\gtrsim 0.1KeV,\sim 10^{16}{\rm Hz})$ increases by two orders of magnitude. In comparison, the lower energy part remains unaltered, as the low magnetized outer part of the disc remains steady during this period. However, the emission due to the Bremsstrahlung increases only by one order of magnitude during this process.

To understand the radiative properties during the eruption events, in Fig. 19, we plot the emission spectrum for different simulation times (marked on the figure) when the inner edge of the disc is receding from the black hole. The thin and thick lines in the figures are drawn following a similar convention as in Fig. 18. In the figure, we observe the opposite trend to that of Fig. 18. As the truncated disc recedes, it shows strong outflow in terms of jet and disc-wind (see Fig. 13). With the outflow, accretion flow loses the advected magnetic flux. Consequently, the magnetic field strength decreases, and we observe that the thermal synchrotron emission decreases with time. Also, the synchrotron peak shifts towards the lower energy range.

We observe that most of the emission is contributed by the thermal synchrotron process during the receding, even in the high energy region $(\nu\gtrsim 10^{19}$Hz), signifying a very strong magnetic field around the black hole. The Bremsstrahlung starts to dominate again in the high energy range as the disc reaches far from the black hole ($t=9000$, see the green dot-dashed line). By then, the disc-winds leave our region of interest ($r\lesssim 60$) and do not contribute to the emission spectra.

The 2D synthetic intensity at $h\nu=1KeV$ for the reference run at different simulation times (same as Fig. 18 and 19) is shown in Fig. 20. In the figures, we plot the normalized intensity in logarithmic scale within a box $x:[-60,60]r_{g}$ and $y:[-60,60]r_{g}$. In the figure, we broadly observe two concentric rings. The size of the outer ring is about $40r_{g}$ and the inner is about $10r_{g}$. These maps essentially suggest that the inner edge of the truncated accretion disc and the disc-wind contribute to most of the emission, which constitutes the outer ring. As the inner edge moves towards the event horizon $t\sim 7500-8500$, the radius of the brightened edge decreases. At the time, $t=8000,8400$, and $8500$, we observe two extra ring-like structures surrounding the black hole due to the hot and dense post-shock matter. The post-shock region exists on both sides of the equatorial plane. Thus, the emission coming out of the post-shock region appears as two concentric rings around the black hole to the observer. The rings at time $t=8000$ are marked on the figure as ‘A’ and ‘B’. As the post-shock regions on both sides of the equatorial plane appear at slightly different inclinations to the observer. As a result, the inner rings are located more asymmetrically with respect to the equatorial plane than the outer ring.

Unlike the standard image of the black hole (Event Horizon Telescope Collaboration et al., 2019), here we observe a multiple ring structure around the event horizon. As the inner edge of the disc recedes outwards ($t\sim 8700-9000$), we observe negligible emission from the low-density flow close to the black hole. During this time, most of the emission is coming from the disc wind. As the wind leaves our area of interest $(r\lesssim 60)$, total emission drops drastically ($t=9000$). Our observation of an extended ring structure nicely fits with the GR-MHD ray-traced emission maps by Bandyopadhyay et al. (2021), who investigated how the structural components of a BH jet source - the BH, the disc, the spine jet, and the disc wind - may be disentangled by their radiative imprints.

Due to the transport of angular momentum, the sub-Keplerian matter accretes towards the black hole. Also, some of this accreting matter contributes to the outflow as jet and disc-wind. This indicates that the low angular momentum matter around the black hole plays a crucial role in the emission of hard X-rays. Consequently, the emissions coming within the energy range $h\nu\sim 1-100keV(\nu\sim 10^{17}-10^{19}Hz)$ change in two-orders of magnitude during the oscillation (see Fig. 18). Similarly, in the high energy range ($h\nu>100KeV$ or $\nu>10^{19}Hz)$, the emission increases by one order of magnitude in the oscillating phase. These features may significantly impact the understanding of spectral and timing properties in BH-XRBs. BH-XRBs often exhibit spectral variability in their outbursting phase (Belloni (2010); Belloni et al. (2011); Belloni & Motta (2016); Ingram & Motta (2019), etc.).

Note that in this study, we do not consider any Comptonization process in the calculation of the emission features. However, it is needless to mention that the inclusion of the Comptonization process is essential to understand the hard X-rays from BH-XRBs (e.g., Steiner et al. (2009); Titarchuk et al. (2014); Poutanen et al. (2018), etc.). With the inclusion of the Comptonization process, the high-energy emissions from the hot disc-wind and the post-shock region are expected to increase.

The radiation and spectral signatures calculated due to the thermal synchrotron and Bremsstrahlung depend on the considered parameters of the $R-\beta$ relation. In reality, a large number of these parameters $(R_{l},R_{h})$ need to be tested to find a suitable combination/range of parameters that match the observations (e.g., Mizuno et al., 2021; Fromm et al., 2021). Such scaling relations are often used in literature to calculate the near horizon radiative properties of AGNs (e.g., Event Horizon Telescope Collaboration et al., 2019). A more consistent approach would be to account for two-temperature fluid and include relevant heating/cooling terms to obtain the electron temperature and the emission self-consistently (e.g., Ressler et al., 2015; Sádowski et al., 2017; Ryan et al., 2017; Mizuno et al., 2021; Dihingia et al., 2022). Nevertheless, with the inclusion of these processes, the qualitative radiative feature of modulation of emission during the oscillation of accretion flow is expected to be the same.