Ring Asymmetry and Spin in M87*

1. Problem Statement

Determining the dimensionless spin parameter ($a_*$) of supermassive black holes (SMBHs) is a longstanding challenge in astrophysics. While spin is a fundamental property defining the spacetime geometry of a black hole, direct observational constraints are scarce, particularly for low-accretion systems like M87* ($\dot{M}/\dot{M}_{Edd} < 0.01$).

• Current Limitations: Existing spin constraints rely heavily on X-ray techniques (limited to high-accretion systems) or indirect jet-power arguments that assume specific jet-launching mechanisms (e.g., Blandford–Znajek) and suffer from large uncertainties.
• The Gap: There is no statistically significant, direct measurement of M87*'s spin. The Event Horizon Telescope (EHT) images reveal an asymmetric ring of emission, but the relationship between this asymmetry and the black hole's spin has not been rigorously quantified across a dense parameter space using General Relativistic Magnetohydrodynamic (GRMHD) models.

2. Methodology

The authors employ a comparative approach, analyzing the brightness asymmetry of M87* in EHT observations against a comprehensive library of synthetic data generated from GRMHD simulations.

A. Simulation Library ("Illinois v5")

• Code & Physics: Simulations were performed using the KHARMA code within the PATOKA pipeline. They utilize ideal GRMHD for radiatively inefficient accretion flows (RIAF).
• Model Types: The study focuses on Magnetically Arrested Disk (MAD) models, which are favored by EHT data for M87* due to their ability to reproduce jet power and polarization constraints. (SANE models are discussed in the Appendix).
• Parameter Space:
  • Spin ($a_*$): Densely sampled in 13 steps from $-0.97$ to $+0.97$ (covering both prograde and retrograde configurations).
  • Radiative Transfer: Includes variations in inclination ($i$), and electron temperature parameters ($R_{high}, R_{low}$) which define the ratio of ion-to-electron temperatures based on plasma $\beta$.
  • Adiabatic Index: Uses $\gamma = 5/3$, considered more appropriate for non-relativistic ions in the emitting region than the $\gamma = 4/3$ used in previous libraries.
• Synthetic Data Generation: Synthetic images were ray-traced, convolved with EHT beam characteristics, and sampled with realistic $(u, v)$ coverage and thermal noise (mimicking 2017, 2018, and 2021 campaigns).

B. Asymmetry Measurement

• Metric: The authors define ring brightness asymmetry using the amplitude of the $m=1$ Fourier mode ($a_1$) of a parameterized $m$-ring model.
• Fitting Procedure:
  • Observational Data: EHT visibility amplitude and closure phase data from 2017, 2018, and 2021 were fitted using the Comrade package.
  • Synthetic Data: The same fitting pipeline was applied to synthetic $(u, v)$ data to ensure a fair comparison.
  • Resulting Values: The measured asymmetries were $a_1 = 0.41 \pm 0.04$ (2017), $0.69 \pm 0.10$ (2018), and $0.53 \pm 0.04$ (2021).

C. Statistical Analysis

• Distribution Fitting: The asymmetry distributions for each spin value were fitted with truncated Gaussian functions to determine the mode ($\mu$) and standard deviation ($\sigma$).
• Hypothesis Testing: The Kolmogorov–Smirnov (KS) test was used to calculate the probability ($p$-value) that the observed EHT asymmetry distribution is drawn from the simulated distribution for a given spin.

3. Key Contributions

1. Dense Spin Sampling: Unlike previous studies that tested sparse spin values, this work utilizes a library with 13 distinct spin values, allowing for a continuous mapping of asymmetry vs. spin.
2. Robust Methodology: The study demonstrates that asymmetry measurements are robust against changes in modeling templates (e.g., $m=3$ vs. $m=4$ rings), $(u, v)$ coverage (2017 vs. 2021), and parameter bounds.
3. Decoupling from Jet Power: The analysis provides a spin constraint based solely on near-horizon total intensity data, independent of the highly uncertain jet power estimates used in previous indirect constraints.
4. Future Forecasting: The paper quantitatively forecasts how future EHT campaigns (increasing independent samples $N$) will exponentially improve spin constraints.

4. Key Results

• Spin-Asymmetry Correlation: There is a clear correlation between the magnitude of spin ($|a_*|$) and the magnitude of ring brightness asymmetry ($a_1$).
  • Zero Spin ($a_* = 0$): Produces the minimum asymmetry ($\mu \approx 0.28$).
  • High Spin ($|a_*| \approx 1$): Produces significantly higher asymmetry ($\mu \approx 0.65 - 0.68$).
• Disfavoring Low/Zero Spin:
  • The 2018 observation ($a_1 = 0.69$) falls above the 90th percentile of the distribution for zero-spin models.
  • The 2017 and 2021 observations fall above the 50th percentile of zero-spin models.
  • Statistical Significance: The KS test yields $p \approx 0.026$ for $a_* = 0$ and $p \approx 0.049$ for $a_* = -0.25$. This marginally disfavors models with $|a_*| \lesssim 0.2$.
• Preferred Spin Range: The data are most consistent with spins in the range $|a_*| \gtrsim 0.25$, with peak probabilities around $a_* = -0.75, 0.5, 0.75$.
• Physical Mechanism: The asymmetry is driven by a complex interplay of Doppler boosting, gravitational lensing, and frame dragging. In MAD models, strong magnetic torques slow the plasma rotation, making the asymmetry sensitive to the specific spin configuration and inclination.

5. Significance and Implications

• Support for Blandford–Znajek (BZ) Mechanism: The BZ mechanism, which extracts rotational energy from a spinning black hole to power jets, predicts that M87* must have non-zero spin. The finding that low-spin models are disfavored by the ring asymmetry is consistent with the BZ model.
• Jet Launching Constraints: The results help distinguish between competing jet-launching mechanisms (e.g., BZ vs. Blandford–Payne), as different mechanisms predict different equilibrium spin distributions over cosmic time.
• Future Prospects: The paper demonstrates that the uncertainty in the mean asymmetry scales as $1/\sqrt{N}$. With future time-lapse imaging (e.g., ALMA campaigns in 2026 and beyond), the number of independent samples ($N$) could increase from 3 to 12. This would reduce uncertainty by a factor of $\sim 2$, potentially allowing for a definitive measurement of the spin magnitude and a robust rejection of zero-spin models.
• General Applicability: The methodology suggests that ring asymmetry can be used as a primary observable to constrain spin in other SMBHs (like Sgr A*) once sufficient $(u, v)$ coverage and inclination estimates are available.

Conclusion

The paper provides the first direct, statistically significant evidence suggesting that the supermassive black hole M87* possesses a non-zero spin ($|a_*| \gtrsim 0.2$). By leveraging the correlation between ring brightness asymmetry and spin in MAD GRMHD models, the authors rule out zero-spin scenarios at a marginal significance level. This result independently supports the Blandford–Znajek jet-launching mechanism and sets the stage for precise spin measurements with future EHT observations.