Millimetre Fundamental Plane of Black Hole Accretion

• The paper demonstrates that the empirical relation between mm-wavelength luminosity, X-ray luminosity, and black hole mass serves as a robust tool to probe scale-invariant coupling in accretion and jet physics.
• It employs high-resolution ALMA observations and rigorous statistical methods to achieve an intrinsic scatter of ~0.36 dex, rivaling classic mass scaling relations.
• This work introduces a novel indirect SMBH mass estimation technique, deepening our understanding of radiatively-inefficient accretion flows and compact jet models across diverse astrophysical regimes.

The millimetre fundamental plane of black hole accretion is an empirical relation that tightly links the millimetre-wavelength luminosity, X-ray luminosity, and black hole mass across a broad dynamic range in mass and accretion states. This correlation, first established for supermassive black holes (SMBHs) in active galactic nuclei (AGN), also encompasses stellar-mass black holes in X-ray binaries (XRBs), indicating a fundamental coupling between accretion physics and jet/outflow mechanisms over $\sim10^8$ in mass. The relation is physically interpreted as a manifestation of radiatively-inefficient accretion flows (RIAFs), notably advection-dominated accretion flows (ADAFs), and possibly compact jets. High-resolution millimetre and X-ray observations, predominantly from ALMA, provide the core data supporting this three-dimensional plane, establishing a new tool for indirect black hole mass estimation and energy coupling diagnostics in systems inaccessible to classical dynamical techniques.

1. Empirical Definition and Formulation of the Millimetre Fundamental Plane

The millimetre fundamental plane (mmFP) is defined by a three-parameter power-law relation connecting the nuclear 1 mm (230 GHz) luminosity ($L_{\nu,\rm mm}$), intrinsic 2–10 keV X-ray luminosity ($L_{X,2-10}$), and black hole mass ($M_{\rm BH}$). The best-fit empirical equations, unifying SMBHs and stellar-mass black holes, are:

$$\log_{10}\frac{M_{\rm BH}}{M_\odot} =(-0.22\pm0.05) \left[\log_{10}\frac{L_{X,2-10}}{10^{40}\ \mathrm{erg\,s^{-1}}}\right] +(0.97\pm0.05) \left[\log_{10}\frac{L_{\nu,\rm mm}}{10^{39}\ \mathrm{erg\,s^{-1}}}\right] +(8.32\pm0.08)$$

or, in the form isolating millimetre luminosity,

$$L_{\nu,\rm mm} = A\,M_{\rm BH}^{\alpha}\,L_{X,2-10}^{\beta}$$

with $\alpha\approx1.03\pm0.05$, $\beta\approx0.23\pm0.05$, and $$A\approx10^{21.12\pm0.10}~\mathrm{erg\,s^{-1}\,Hz^{-1}}/[M_\odot^{\alpha}\,(\mathrm{erg\,s^{-1}})^{\beta}]$$ (Elford et al., 7 Jan 2026, Ruffa et al., 2023).

The observed scatter is $\sigma_{\rm obs}\simeq0.42$ dex, with an intrinsic scatter of $\sigma_{\rm int}=0.36\pm0.07$ dex for the joint SMBH and XRB sample. The intrinsic tightness rivals or exceeds the classical radio/X-ray fundamental plane and black-hole–bulge scaling relations.

2. Observational Basis: Samples, Frequency Selection, and Methodology

Primary mmFP samples comprise 48 nearby AGN from the WISDOM and BASS surveys, spanning $M_{\rm BH}\sim10^6$–$10^{10}~M_\odot$. Nuclear mm luminosities are extracted from ALMA Band 6 (230 GHz) continuum images at $\lesssim0.1''$ (6–330 pc) resolution, isolating central emission and excluding host galaxy contamination. Contemporaneous, absorption-corrected 2–10 keV luminosities are secured from X-ray catalogs such as Chandra or Swift/XRT.

Extension to XRBs incorporates five systems (e.g., Cygnus X-1, GX 339-4), observed at 230 GHz using ALMA/ACA, JVLA, or JCMT, with X-ray data from Swift or Chandra scaled to the canonical 2–10 keV band. Consistent rest-frame frequencies, flux extraction methods, and distance calibrations ensure validity across $\sim10^8$ in $M_{\rm BH}$ (Elford et al., 7 Jan 2026).

Regression utilizes robust least-trimmed-squares and Bayesian error propagation, with intrinsic scatter folded into the fit until $\chi^2_\nu=1$. Systematic uncertainties in ALMA calibration ($\sim10\%$) dominate the formal flux uncertainties. Variability introduces up to 0.3 dex additional scatter, mitigated by multi-epoch averaging.

3. Physical Interpretation: ADAF, Corona, and Compact Jet Models

The mmFP is interpreted as the outcome of radiatively-inefficient accretion flows, primarily ADAFs, and possibly compact, self-absorbed jets (“sub-pc jet base”). ADAF models predict:

$$L_{\nu,\rm mm} \propto M_{\rm BH}^{17/16} \dot{m}^{17/16}, \quad L_X \propto \dot{m}^2,$$

implying

$$\log L_{\nu,\rm mm} \simeq 0.53 \log L_X + 1.06 \log M_{\rm BH} + \text{const}$$

in the simplest analytic treatment (Ruffa et al., 2023). The observed mmFP slopes ($\alpha\simeq1.05$, $\beta\simeq0.23$) are in broad agreement, with normalization offsets explainable by model microphysics and electron heating assumptions.

Compact jet models (e.g., Blandford–Königl type, BHJET) predict flat/inverted spectra extending to mm, with X-ray emission via SSC or external Comptonization. While these can reproduce $L_X$–$L_{\nu,\rm mm}$ correlations, the 3D ($L_X$, $L_{\nu,\rm mm}$, $M_{\rm BH}$) relation shows curvature, and the emergence of a pure power-law plane requires fine-tuning of inclination, magnetization, and particle acceleration parameters (Elford et al., 7 Jan 2026, Ruffa et al., 2023).

Coronal synchrotron models, such as the one-zone spherical hybrid thermal/non-thermal plasma model, also quantify correlated mm–X-ray–$M_{\rm BH}$ variability. Modelled spectra incorporate synchrotron self-absorption (SSA) turnovers at 100–300 GHz and reproduce mm–$M_{\rm BH}$ and mm–$L_X$ correlations for $r_c=60$–250 $R_g$, $\delta=10^{-3}$–$10^{-1}$, $B=10$–150 G (Palacio et al., 10 Apr 2025).

4. Parameterization, Model Landscape, and Astrophysical Regimes

Parameter studies in the corona model framework reveal:

Parameter	Typical Range	Physical Context
$M_{\rm BH}$	$10^6$ – $10^9~M_\odot$	Fixed per source
$r_c$	$10$–$500~R_g$	Corona size
$kT_c$	$10$–$200$ keV	Electron temperature
$\tau_T$	$0.01$–$10$	Thomson optical depth
$\delta$	$10^{-4}$–$1$	Non-thermal electron energy fraction
$p$	$2$–$4$	Power-law index of non-thermal electrons
$B$	$10$–$150$ G	Magnetic field strength (from equipartition)
$\sigma$	$0.01$–$1$	Magnetization parameter

Emergent SEDs exhibit SSA turnovers at 100–300 GHz; $\nu_{\rm p}\propto M_{\rm BH}^{-0.37}$. The $L_{\nu,\rm mm}$–$M_{\rm BH}$ slope and turnover reproduce the empirical mmFP, further underlining the explanatory adequacy of these models (Palacio et al., 10 Apr 2025).

5. Universality Across Mass Scales and Accretion States

XRBs in the hard state exhibit mm and X-ray luminosities consistent with the SMBH-defined mmFP, extending the relation by $\sim5$ dex in $M_{\rm BH}$ (Elford et al., 7 Jan 2026). Deviations are observed when sources transition to soft states (e.g., jet suppression), analogous to the breakdown of the classical radio/X-ray fundamental plane in high-Eddington regimes.

This universality suggests a scale-invariant coupling between inflow (accretion) and outflow (jet/corona) processes. The tight scatter ($\sim0.36$ dex) is evidence for homogeneous electron heating, magnetization, and energy partitioning in RIAFs and jets from XRBs to SMBHs.

6. Practical Implications and Applications

The mmFP provides an indirect SMBH mass estimator with precision $\lesssim0.5$ dex, complementing classical dynamical and reverberation mapping techniques. The relation’s intrinsic tightness approaches that of the $M_{\rm BH}$–$\sigma_\star$ relation and exceeds the classical radio-X-ray fundamental plane, particularly when mm/X data are contemporaneous and core-isolated.

The mmFP is robust to moderate variability and can be universally applied across redshifts (with ALMA, ngVLA, Athena, Lynx). This unlocks SMBH mass inference in highly dust-obscured or Compton-thick systems, and in high-redshift, lensed quasars, regimes where optical or near-IR methods fail due to extinction or faintness (Ruffa et al., 2023, Palacio et al., 10 Apr 2025).

7. Limitations and Theoretical Caveats

The mmFP holds predominantly for jet-dominated, hard/accretion-dominated states. In soft or highly radiative states, the coupling breaks down, reflecting suppression of jets or a transition to standard thin-disc phenomenology. While ADAF models match the slope and normalization with fewer tunable parameters, compact jet models require inclination, power, and spectral adjustment to recover the empirical plane.

Systematic uncertainties in mm flux isolation, X-ray absorption, and source classification (e.g., ambiguous states in XRBs, orientation in AGN) can introduce biases. Physical offsets in the normalization are sensitive to microphysical parameters (e.g., viscosity, electron heating fraction, magnetization), leaving room for refinement as more sophisticated models and broader samples are developed (Palacio et al., 10 Apr 2025, Elford et al., 7 Jan 2026).

---

In summary, the millimetre fundamental plane of black hole accretion is an empirical, physically interpretable relation demonstrably uniting the energetics of accretion and compact jet processes from stellar-mass binaries to SMBHs, providing a new window on mass estimation and inflow–outflow coupling in the low-Eddington regime (Elford et al., 7 Jan 2026, Ruffa et al., 2023, Palacio et al., 10 Apr 2025, Doi et al., 2011).