Characterizing the Scale Height and Filamentary Structure of Radiatively Cooled MADs

Technical Summary: Characterizing the Scale Height and Filamentary Structure of Radiatively Cooled MADs

Problem Statement
Radiative cooling is a fundamental component of realistic black hole accretion disk dynamics, yet its specific impact on the structure of Magnetically Arrested Disks (MADs) remains under-explored. While previous General Relativistic Magnetohydrodynamic (GR-MHD) simulations have extensively studied MADs, most have focused on ideal, non-radiative flows suitable for extremely low-luminosity sources. Existing studies incorporating radiation have often been limited by computational complexity and parameter space dimensionality. Furthermore, a critical issue arises in characterizing the vertical thickness of MADs: the traditional definition of disk scale height assumes a density distribution symmetric about the equatorial plane. However, in MAD states, strong magnetic flux accumulation drives interchange instabilities, creating discrete, filamentary accretion streams that often deviate significantly from the equatorial plane. Consequently, standard metrics may yield misleading results regarding the true geometric thickness and dynamical state of the disk, particularly when radiative cooling alters the thermal pressure support.

Methodology
The authors performed a series of three-dimensional GR-MHD simulations using the GPU-accelerated code cuHARM around a non-spinning (Schwarzschild) black hole with mass $M = 4 \times 10^6 M_\odot$. The simulations modeled MAD states with mass accretion rates ($\dot{M}$) ranging from $10^{-7}$ to $10^{-4} \dot{M}_{\text{Edd}}$.

Key methodological components include:

• Radiative Cooling: The simulations consistently included radiative losses via synchrotron and bremsstrahlung emission. Compton cooling was excluded as it was deemed sub-dominant at these accretion rates and requires full radiation transport.
• Thermodynamics: A single-temperature prescription was adopted, assuming a fixed ratio of proton to electron temperatures ($T_p/T_e = 3$), implying strong coupling between species. The cooling rate $\Lambda$ was calculated based on local gas density, electron temperature, and magnetic field energy density.
• Initial Conditions: Simulations began with an un-magnetized axisymmetric torus in hydrostatic equilibrium, perturbed to trigger magneto-rotational instability (MRI). A specific magnetic field configuration (single-loop vector potential) was used to ensure the development of a MAD state.
• Analysis: Time-averaged diagnostics were computed over the interval $1.5 \times 10^4 t_g < t < 3.0 \times 10^4 t_g$. The authors compared the standard density-weighted angular scale height against a newly proposed metric based on the polar position of local density maxima to better capture filamentary structures.

Key Contributions and Results

1. Dominance of Synchrotron Cooling:
   The study confirms that synchrotron emission is the dominant radiative cooling mechanism across all simulated accretion rates, exceeding bremsstrahlung losses by several orders of magnitude. Synchrotron cooling peaks in the inner disk regions near the black hole and along the equatorial plane, driven by high magnetic field strengths, densities, and temperatures.

2. Radiative Efficiency and Disk Structure:
   Radiative efficiency ($\eta_{\text{rad}}$) increases non-linearly with the mass accretion rate. At low rates ($\dot{M} \lesssim 10^{-6} \dot{M}_{\text{Edd}}$), the flow behaves as a Radiatively Inefficient Accretion Flow (RIAF). However, at higher rates (approaching $10^{-4} \dot{M}_{\text{Edd}}$), cooling becomes efficient enough to significantly reduce thermal pressure. This leads to a transition where the disk becomes radiatively efficient (reaching $\approx 30\%$ efficiency at $r=5r_g$ for the highest rate) and the gas is compressed into thinner, denser filaments.

3. Failure of Traditional Scale Height Definitions:
   The authors demonstrate that the conventional density-weighted scale height definition (Equation 14) is misleading for MADs. Because MAD flows exhibit filamentary structures that are displaced from the equatorial plane due to magnetic interchange instabilities, averaging over the equatorial plane artificially inflates the measured disk height. The traditional metric fails to distinguish between the physical thickness of the accreting gas and the vertical displacement of filaments.

4. Proposal of a New Scale Height Metric:
   To address this, the paper introduces an alternative definition of disk height based on the location of local density maxima within azimuthal slices (Equation 15). This method identifies coherent "islands" of high density, excludes low-density coronal regions, and calculates the thickness of these filaments relative to their local centers.

   • Result: Using this new metric, the disk height is shown to decrease monotonically toward the black hole, approaching zero at the horizon. In contrast, the traditional metric showed an artificial increase in thickness near the horizon.
   • Cooling Impact: The new metric reveals that for accretion rates $\dot{M} \gtrsim 10^{-4} \dot{M}_{\text{Edd}}$, radiative cooling causes a substantial reduction in disk scale height within the inner $\sim 30-40 r_g$, whereas lower accretion rates show no such thinning.

5. Critical Accretion Rate:
   The authors derive an analytical condition for the critical mass accretion rate above which radiative cooling becomes dynamically significant (faster than heating). By balancing synchrotron cooling against magnetic heating (mediated by turbulence and reconnection), they estimate this transition occurs around $\dot{M} \sim 2.17 \times 10^{-4} \dot{M}_{\text{Edd}}$ at $r=30r_g$. This analytical estimate aligns with the numerical observation of structural changes in the S-4 simulation ($10^{-4} \dot{M}_{\text{Edd}}$).

Significance and Claims
The paper claims that radiative cooling fundamentally alters the internal structure of MADs by reducing thermal pressure support, thereby compressing accretion filaments and reducing the disk's scale height. The primary significance of this work lies in:

• Correcting Diagnostic Bias: It establishes that conventional measures of disk thickness are inadequate for MADs, particularly in the presence of strong cooling, and proposes a more robust, filament-aware definition.
• Identifying a Transition Regime: It identifies a critical accretion rate threshold where cooling shifts from being a minor perturbation to a dominant force reshaping the disk's geometry and dynamics.
• Observational Implications: The authors suggest that these cooling-induced structural changes (thinner, denser filaments) should imprint on observable properties, such as spectral energy distributions and light-curve fluctuations, potentially offering a new method to infer mass accretion rates in systems like Sgr A* or X-ray binaries.

The study remains modest in its scope, limiting the analysis to non-spinning black holes and assuming a fixed ion-electron temperature ratio, noting that future work is required to explore the impact of black hole spin and more complex electron-proton coupling models at higher accretion rates.