Thin accretion disks around a Schwarzschild acoustic black hole

Technical Summary: Thin Accretion Disks around a Schwarzschild Acoustic Black Hole

Problem Statement
This study investigates the phenomenological properties of thin accretion disks surrounding a Schwarzschild acoustic black hole (SABH). While analogue gravity models, such as the SABH, mathematically replicate the propagation of massless scalar fields in curved spacetime using fluid dynamics, they are not literal astrophysical black holes composed of phonons. The central problem addressed is how standard observables of a thin accretion disk—typically governed by the Schwarzschild metric—would be modified if the underlying effective geometry were replaced by the SABH line element. The authors aim to determine if the SABH tuning parameter, $\xi$, induces distinct regimes in radiative characteristics (flux, temperature, luminosity, and efficiency) that could serve as phenomenological diagnostics for distinguishing effective compact object geometries.

Methodology
The authors employ the Novikov-Thorne formalism, a standard general relativistic model for geometrically thin, optically thick, and cold accretion disks. The methodology involves the following steps:

1. Metric Definition: The SABH metric is defined with a nonzero background fluid four-velocity $v_r = \sqrt{2M\xi/r}$. The metric function $f(r)$ depends on the mass $M$ and the tuning parameter $\xi$. The event horizon structure varies with $\xi$: for $0 \le \xi < 4$, only an "optical" horizon exists; for $\xi \ge 4$, acoustic horizons appear.
2. Kinematic Analysis: The authors derive the angular velocity ($\Omega$), specific energy ($\tilde{E}$), and specific angular momentum ($\tilde{L}$) for particles in circular equatorial orbits. They solve for the marginally stable orbit radius ($r_{ms}$), or Innermost Stable Circular Orbit (ISCO), by analyzing the second derivative of the effective potential.
3. Emissivity Calculation: Using the Novikov-Thorne flux formula, the authors calculate the time-averaged radiative flux $F(r)$, temperature distribution $T(r)$ (via the Stefan-Boltzmann law), and spectral luminosity $L(\nu)$. They also compute the accretion efficiency $\epsilon$ and the Eddington luminosity $L_{Edd}$.
4. Numerical Illustration: A toy model is adopted with a central mass $M = 15M_\odot$ and an accretion rate $\dot{M}_0 = 10^{19} \text{ g s}^{-1}$. The tuning parameter $\xi$ is varied over the range $0 \le \xi \le 6$. The results are compared against the standard Schwarzschild black hole (SBH) case where $\xi = 0$.

Key Contributions and Results
The paper identifies a critical transition in the behavior of the accretion disk at $\xi \approx 2.8$, driven by a branch change in the marginally stable orbit radius $r_{ms}$. The results are categorized into two distinct regimes:

• Strengthening Regime ($0 < \xi \lesssim 2.8$):

  • As $\xi$ increases toward 2.8, the marginally stable orbit $r_{ms}$ decreases, moving closer to the horizon.
  • This proximity leads to a significant enhancement in radiative characteristics. At $\xi = 2.8$, the accretion efficiency peaks at $\approx 60.9\%$, compared to $\approx 5.7\%$ for the Schwarzschild case.
  • The maximum radiation flux, temperature, and spectral luminosity are substantially higher than in the SBH case. Specifically, the maximum spectral luminosity at $\xi = 2.8$ is approximately 207.8 times greater than that of the SBH.
  • The disk is hotter and brighter than the Schwarzschild counterpart in this regime.

• Weakening Regime ($\xi \gtrsim 2.8$):

  • Beyond $\xi \approx 2.8$, the branch of the marginally stable orbit shifts, causing $r_{ms}$ to increase rapidly.
  • Consequently, all radiative characteristics diminish relative to the Schwarzschild case.
  • At $\xi = 6$, the maximum flux is roughly 100 times smaller, and the maximum spectral luminosity is approximately 3.6 times smaller than the SBH values.
  • The accretion efficiency drops significantly, approaching the SBH value as $\xi \to \infty$.

• Eddington Luminosity:

  • Unlike the flux and temperature profiles, the Eddington luminosity analysis reveals that the SABH operates exclusively in a "weakening regime" relative to the SBH. As $\xi$ increases, the maximum Eddington luminosity decreases monotonically.
  • The authors note that for $\xi > 3$, the formal expression for Eddington luminosity yields negative values in certain radial intervals, indicating a change in the sign of the static-frame gravitational acceleration, which restricts the physical interpretation of the Eddington limit in those regions.

Significance and Claims
The authors position this work as a phenomenological investigation rather than a claim of physical astrophysical accretion onto a laboratory fluid. They explicitly state that the SABH metric is treated as an effective test geometry for massive matter, justified by the observation that infalling particles near the horizon become ultrarelativistic, making their dynamics more dependent on the background geometry than on rest mass.

The primary significance of the paper lies in:

1. Phenomenological Diagnostics: Demonstrating that the tuning parameter $\xi$ creates a bifurcation in disk observables, offering a theoretical framework to compare effective compact object geometries.
2. Quantitative Benchmarks: Providing specific numerical predictions for flux, temperature, and efficiency that deviate sharply from standard Schwarzschild predictions, particularly near the transition point $\xi \approx 2.8$.
3. Analogue Gravity Context: Extending the study of SABHs beyond quasinormal modes and gravitational lensing to include accretion disk thermodynamics, thereby enriching the theoretical toolkit for analogue gravity experiments.

The authors conclude that while the Novikov-Thorne model provides precise predictions for these effective geometries, direct comparison with real astrophysical objects (such as those observed by the Event Horizon Telescope) requires careful rescaling of mass, accretion rate, and inclination, and the current results serve as a motivation for studying disk diagnostics in non-Schwarzschild geometries rather than a direct fit to observational data.