Bolometric correction factor and radiative efficiency for the super-Eddington accretion flow in tidal disruption events

Technical Summary: Bolometric Correction Factor and Radiative Efficiency for Super-Eddington Accretion Flow in Tidal Disruption Events

Problem Statement
The estimation of bolometric luminosity and radiative efficiency is critical for understanding accretion flows around supermassive black holes (BHs). In Tidal Disruption Events (TDEs), a significant discrepancy exists between the accreted mass ($\Delta M$) derived from observations and the theoretical expectation of $\sim 0.5 M_\odot$ for the disruption of a solar-type star. This "missing energy" problem arises because standard calculations often assume a constant X-ray bolometric correction factor ($k_{bol} \approx 1$) and a high radiative efficiency ($\eta = 0.1$). Previous studies suggest these assumptions may be flawed, particularly during the early super-Eddington phase where emission is dominated by the unobservable extreme UV (EUV) band rather than X-rays, and where radiative efficiency is theoretically expected to be lower due to advection.

Methodology
To address these uncertainties, the authors performed two-dimensional axisymmetric radiation-hydrodynamic simulations using the ATHENA++ code. The study modeled the accretion flow resulting from a solar-type star ($M_* = M_\odot, R_* = R_\odot$) on a parabolic orbit disrupted by BHs with masses of $10^6$, $10^{6.5}$, and $10^7 M_\odot$.

• Simulation Setup: The authors utilized a pseudo-Newtonian potential to mimic Schwarzschild BH effects. Mass was continuously injected at the circularization radius ($R_c = 2R_t$) with a fallback rate profile $\dot{M}_{inject} \propto t^{-5/3}$. The computational domain spanned $2R_S \le r \le 10^5 R_S$ with a resolution of $768 \times 256$.
• Post-Processing: At $t = 16$ days post-injection (a time chosen when stable super-Eddington accretion is established), the authors performed Monte Carlo radiation transfer calculations using PYTHON to generate emergent spectra.
• Parameters: Spectra were calculated for viewing angles ($\theta$) ranging from $0^\circ$ (face-on) to $90^\circ$ (edge-on). The X-ray band was defined as 0.3–2 keV.
• Calculations: The isotropic-equivalent X-ray luminosity ($L_X$) and bolometric luminosity ($L_{bol}$) were integrated from the spectra to derive the bolometric correction factor ($k_{bol} = L_{bol}/L_X$) and the apparent radiative efficiency ($\eta = L_{bol}/\dot{M}_{in}c^2$).

Key Results

1. Dependence on Mass and Viewing Angle: Both $k_{bol}$ and $\eta$ are strongly dependent on the BH mass ($M_{BH}$) and the viewing angle ($\theta$).
   • $k_{bol}$: Values range from several tens to several thousands. For $M_{BH} = 10^6 M_\odot$, $k_{bol}$ increases from $\sim 25$ at $0^\circ$ to $\sim 1000$ at $90^\circ$. For higher masses ($10^7 M_\odot$), $k_{bol}$ remains relatively constant around $\sim 1000$ across all angles. These values are substantially larger than the commonly adopted value of unity.
   • $\eta$: The derived radiative efficiency ranges from $\sim 10^{-3}$ to $10^{-1}$. Specifically, for $10^6 M_\odot$, $\eta$ is $\sim 0.001-0.02$; for $10^7 M_\odot$, it reaches up to $\sim 0.13$ but generally remains below 0.1 for most angles. $\eta$ decreases with increasing viewing angle and is generally lower for lower BH masses (due to higher Eddington-scaled accretion rates).
2. Spectral Characteristics: The emergent spectra are dominated by the EUV band. As $\theta$ increases, X-ray emission is increasingly obscured and reprocessed by the optically thick outflow into the optical and UV bands. Consequently, $L_X$ drops by orders of magnitude (up to $\sim 1000\times$ for $10^6 M_\odot$) as $\theta$ goes from $0^\circ$ to $90^\circ$, while $L_{bol}$ decreases more mildly ($\sim 15-20\times$).
3. Correction Factor Ratio: The ratio $(k_{bol}/1.1) / (\eta/0.1)$, which scales the inferred accreted mass relative to previous estimates, ranges from $\sim 90$ to $8 \times 10^4$ across the simulated parameters. This indicates that previous estimates of $\Delta M$ were likely underestimated by 2–5 orders of magnitude.

Application to Observed TDEs
The authors applied their derived $k_{bol}$ and $\eta$ values to several specific X-ray TDEs (e.g., NGC 5905, NGC 3599, SDSS J132341.97+482701.3, ASASSN-14li). By selecting appropriate $M_{BH}$ and $\theta$ (typically small inclinations consistent with the unified model of TDEs), the recalculated accreted mass $\Delta M$ for these events falls within the range of $0.25 - 0.5 M_\odot$. This result aligns with the theoretical expectation for the disruption of a solar-type main sequence star, thereby offering a solution to the "missing energy" problem for these specific cases.

Significance and Limitations
The paper claims that the derived $k_{bol}$ and $\eta$ values provide a physically motivated correction that can significantly alleviate the missing energy problem in TDEs, provided the observer's viewing angle is small. The authors emphasize that their solution is consistent with the unified model of TDEs, where X-ray selected events are observed at low inclinations.

However, the authors maintain a modest stance regarding the universality of their findings:

• Viewing Angle Uncertainty: The solution is strongly dependent on $\theta$, which cannot be accurately determined from observations alone.
• Temporal Evolution: The calculations are based on $t=16$ days. While the authors argue that the super-Eddington phase is stable over hundreds of days, the long-term evolution of $k_{bol}$ and $\eta$ as the accretion rate drops below Eddington limits requires further study.
• Simplifications: The study assumes a solar-type star, a Schwarzschild BH (using pseudo-Newtonian potential), and ignores BH spin and magnetic fields. Factors such as the penetration factor, stellar structure, and relativistic effects were not tested and could influence the results.

In conclusion, this work provides a specific set of simulation-derived parameters that, when applied to X-ray TDEs with small viewing angles, reconciles observed radiated energy with theoretical mass disruption limits, though it acknowledges that a complete resolution to the missing energy problem requires further investigation into the full parameter space of TDE physics.