Spectral signature of mass outflow in Two Component Advective Flow Paradigm

Abstract: Outflows are common in many astrophysical systems. In the Two Component Advective Flow ({\fontfamily{qcr}\selectfont TCAF}) paradigm which is essentially a generalized Bondi flow including rotation, viscosity and cooling effects, the outflow is originated from the hot, puffed up, post-shock region at the inner edge of the accretion disk. We consider this region to be the base of the jet carrying away matter with high velocity. In this paper, we study the spectral properties of black holes using {\fontfamily{qcr}\selectfont TCAF} which includes also a jet ({\fontfamily{qcr}\selectfont JeTCAF}) in the vertical direction of the disk plane. Soft photons from the Keplerian disk are up-scattered by the post-shock region as well as by the base of the jet and are emitted as hard radiation. We also include the bulk motion Comptonization effect by the diverging flow of jet. Our self-consistent accretion-ejection solution shows how the spectrum from the base of the jet varies with accretion rates, geometry of the flow and the collimation factor of the jet. We apply the solution to a jetted candidate GS\,1354-64 to estimate its mass outflow rate and the geometric configuration of the flow during 2015 outburst using {\it NuSTAR} observation. The estimated mass outflow to mass inflow rate is $0.12^{+0.02}_{-0.03}$. From the model fitted accretion rates, shock compression ratio and the energy spectral index, we identify the presence of hard and intermediate spectral states of the outburst. Our model fitted jet collimation factor ($f_{\rm col}$) is found to be $0.47^{+0.09}_{-0.09}$.

• The paper demonstrates that incorporating jet physics into the TCAF framework yields statistically improved fits, especially in the intermediate state of black hole binaries.
• It models a conical jet anchored to the CENBOL, linking mass outflow-to-inflow ratio, shock compression, and collimation factor to observable X-ray spectral features.
• Empirical analysis of GS 1354-64 shows that the jet contribution is essential for replicating hard X-ray bumps and QPO behaviors, highlighting the interplay between disk and jet dynamics.

Spectral Diagnostics of Mass Outflow in the Two Component Advective Flow Paradigm

Introduction

This work provides a self-consistent treatment of jet emission in the context of sub-Eddington black hole accretion, embedding jet physics within the Two Component Advective Flow (TCAF) framework. The TCAF paradigm models the accretion structure as an interplay of a high-angular-momentum, optically thick Keplerian disk feeding into a sub-Keplerian, optically thin, viscous hot flow. A centrifugal pressure dominated boundary layer (CENBOL) naturally arises, with previous models (CT95, C99) positing CENBOL as both the principal Comptonizing region and the origin of jets and outflows. This study extends the TCAF formalism (JeTCAF) by dynamically incorporating the base of the jet into the radiative transfer process, thereby capturing additional spectral components due to up- and down-scattering by the outflow.

Model Formulation

A conical jet is anchored to the post-shock CENBOL, and both regions participate in Compton up-scattering of Keplerian disk photons. The model numerically solves for the spectral output, explicitly accounting for radiative cooling, jet acceleration, and flow geometry. The mass outflow to inflow ratio is analytically linked to the shock compression ratio $R$ and the jet collimation factor $f_{\text{col}}$, consistent with C99. A key extension is the inclusion of bulk motion Comptonization (BMC): hard photons from CENBOL are subjected to additional down-scattering within the diverging jet, modifying the high-energy continuum in a manner distinct from disk reflection models (TS05).

The parameterization enables direct fits to broad X-ray spectra, with black hole mass, accretion rates (disk and halo), shock location $X_s$, shock compression ratio $R$, and collimation factor $f_{\text{col}}$ among the principal variables.

Parameter Dependencies and Spectral Characteristics

The analysis demonstrates several non-linear dependencies:

• Accretion Rates: Increasing the Keplerian disk rate softens the jet spectrum, as the enhanced supply of soft photons cools CENBOL and, by extension, the jet base, reducing the efficiency of inverse Comptonization.
• Shock Properties: The emission spectrum varies markedly with $X_s$ and $R$, with larger CENBOL sizes and higher compression ratios generally softening the jet spectral component due to reduced temperature and optical depth effects.
• Collimation Factor: Higher $f_{\text{col}}$ leads to spectral hardening, as a more collimated jet increases optical depth and the number of scatterings.
• Jet Influence: Inclusion of the jet consistently hardens the total spectrum for fixed accretion parameters. This effect is most pronounced in hard and intermediate accretion states, diminishing in the soft state where the jet's energetic contribution is negligible.

The model also produces a hard X-ray bump ($>10$ keV), traceable to BMC in the divergent jet, empirically distinguishing it from static disk reflection features.

Observational Application: GS 1354-64

The formalism is applied to NuSTAR observations of the black hole X-ray binary GS 1354-64 during its 2015 outburst. Fitting the spectra with JeTCAF yields the following salient results:

• Black hole mass constraint: $M_{\mathrm{BH}} \sim 6.7\text{--}7.2\,M_\odot$, consistent with independent dynamical measures.
• For the hard (intermediate) state, the mass outflow to inflow rate is $0.12^{+0.02}_{-0.03}$.
• Shock compression increases to $R \sim 4.2$ in the intermediate state, accompanied by reduced CENBOL size.
• The jet collimation factor is well constrained, $f_{\text{col}} = 0.47 \pm 0.09$.
• The presence of a low-frequency QPO at $\sim 0.2\,\text{Hz}$ with commensurate theoretical burst timescales ($\sim 20$ s) is consistent with feedback cycles involving local disk-jet coupling near the jet base.

A striking outcome is that the TCAF model without the jet fails to yield statistically acceptable fits in the intermediate state, underscoring the necessity of explicitly including the jet contribution to model the observed X-ray continuum.

Implications and Future Directions

From a theoretical standpoint, this self-consistent accretion-ejection solution strengthens the interpretive framework for state transitions, jet launching, and the role of CENBOL as a unifying feature of black hole accretion. The findings underscore that mass loss—and thus jet power—is maximized not in the hard or soft states, but in intermediate configurations where both thermal and centrifugal effects are balanced.

Practically, the approach enables direct inference of jet properties from X-ray spectroscopy, bypassing the ambiguities associated with multi-band indirect jet tracers. This is particularly relevant for systems where radio/optical jet measures are unavailable or ambiguous. The strong coupling of outflow properties to timing features (such as QPOs) highlights a pathway for integrated spectral-timing diagnostics.

The current model is non-relativistic and neglects black hole spin effects, justified by the spatial scale of the phenomena targeted (regions well outside the ISCO). However, future incorporation of Kerr metrics and MHD jet launching mechanisms, as well as synchrotron and magnetic field effects, will be critical for generalization to AGN jets and extreme accretion rate regimes.

Conclusion

This paper presents a robust extension to the TCAF model, providing a self-consistent radiative treatment of jets rooted in accretion dynamics. The empirical application to GS 1354-64 establishes the spectral signature of mass outflow, demonstrates the necessity of incorporating jet physics in spectral modeling, and lays the groundwork for systematic spectral-timing studies of accretion-ejection in black hole binaries (2107.13808).