Eddington Ratio-Column Density Diagram

• The Eddington Ratio-Column Density Diagram is a quantitative framework that maps the balance between radiation pressure and gravitational binding in absorbing media.
• It employs observational X-ray fits, bolometric corrections, and black hole mass estimates to plot AGN regimes and reveal thresholds that separate stable from blow-out conditions.
• This diagram provides actionable insights on feedback mechanisms and obscuration variability, informing our understanding of AGN evolution and star-forming system dynamics.

The Eddington Ratio-Column Density Diagram encapsulates the interplay between radiative output and absorbing material in self-gravitating systems, with primary application to active galactic nuclei (AGN) and turbulent star-forming regions. The diagram provides a quantitative and predictive framework for the regulation of gas and dust by radiation pressure, the containment or ejection of circumnuclear material, and the physical mechanisms underlying observed obscuration variability, feedback, and black hole accretion cycles.

1. Formal Definitions and Physical Basis

The Eddington ratio, typically denoted as $\lambda_{\rm Edd} \equiv L_{\rm bol}/L_{\rm Edd}$, measures the bolometric luminosity $L_{\rm bol}$ of a central source relative to the Eddington limit $$L_{\rm Edd} = 1.26 \times 10^{38} (M_{\rm BH}/M_\odot)$$ erg s$^{-1}$, where $M_{\rm BH}$ is the black hole mass (Ricci et al., 2023, Nagarajan-Swenson et al., 2 Jun 2025, Thompson et al., 2014). Column density $N_{\rm H}$ is the integrated line-of-sight hydrogen column, providing the key measure of X-ray absorbing (and thus, generally, gas+dust) opacity.

The Eddington Ratio-Column Density ($N_{\rm H}$–$\lambda_{\rm Edd}$) diagram sets these axes in log-log space to visualize the relationship between radiative acceleration and gravitational binding for a self-gravitating, absorbing medium. Radiative acceleration exceeds gravity for $\lambda_{\rm Edd}$ above a critical limit determined by the absorption cross section $\sigma_d$ (dusty gas) or $\sigma_T$ (Thomson/thin gas):

$$\lambda_{\rm Edd}^{\rm eff}(N_{\rm H}) = \frac{\sigma_T}{\sigma_{\rm eff}(N_{\rm H})}$$

This critical line divides the plane into “forbidden” (short-lived, radiatively unbound) and “allowed” (long-lived, gravitationally bound) obscuration regimes (Ricci et al., 2023, Vasudevan et al., 2013).

2. Construction and Methodology

Observational and Theoretical Inputs

Key parameters are determined via observational X-ray fits for $N_{\rm H}$; $L_{\rm bol}$ is estimated from hard X-ray luminosities with bolometric corrections, and black hole mass from dynamical or virial methods (Nagarajan-Swenson et al., 2 Jun 2025, Ricci et al., 2023). On the theoretical side, local Eddington ratios can be generalized for arbitrary momentum input:

$\Gamma(\Sigma) = \frac{\dot{p}/M}{4\pi G \Sigma}$

where $\dot{p}$ is the momentum injection rate and $\Sigma$ is the gas surface density (Thompson et al., 2014).

Diagram Construction

To construct the diagram:

• Plot each system at $(\log N_{\rm H}, \log \lambda_{\rm Edd})$.
• Overlay theoretical lines: the effective Eddington limit $\lambda_{\rm Edd}^{\rm eff}(N_{\rm H})$, thresholds for Compton-thin/thick ($\log N_{\rm H}=22$, $24$), and, where relevant, curves incorporating infrared photon trapping (Ricci et al., 2023).
• Population contours or density maps may be derived from evolving space density models $\rho(\lambda, N_{\rm H}, z)$, using hard X-ray luminosity functions and black hole mass functions to simultaneously account for selection effects and accretion variability (Draper et al., 2010).

3. Key Regimes and Feedback Physics

The diagram encodes several dynamical and feedback regimes:

• Allowed X-ray/dusty absorption: $$\lambda_{\rm Edd} < \lambda_{\rm Edd}^{\rm eff}(N_{\rm H})$$, supporting stable, long-lived absorbers—characteristic of most local, moderate-luminosity AGN (Ricci et al., 2023, Vasudevan et al., 2013).
• Forbidden (blow-out) regime: $$\lambda_{\rm Edd} > \lambda_{\rm Edd}^{\rm eff}(N_{\rm H})$$, where radiation pressure expels dusty clouds on short timescales, leading to decreased covering fractions and enhanced outflows (Nagarajan-Swenson et al., 2 Jun 2025, Kudoh et al., 2024).
• Compton-thick bifurcation: Models incorporating X-ray background constraints reveal two distinct CT AGN branches: a high-$\lambda$ (merger/quasar-driven) population at $\lambda_{\rm Edd} \gtrsim 0.9$ and a low-$\lambda$ (LINER/molecular) population at $\lambda_{\rm Edd} \lesssim 0.01$ (Draper et al., 2010).

These regimes connect directly to multi-stage evolutionary frameworks, with AGN cycling through phases of fueling, obscuration build-up, radiative blowout, and unobscured decline (Nagarajan-Swenson et al., 2 Jun 2025, Ricci et al., 2023).

4. Quantitative Relationships and Physical Scalings

Hydrodynamic and analytic models underpin the diagram’s predictive content. For AGN circumnuclear dusty winds, the dusty column scales as $N_{\rm H,\,dust} \propto \gamma_{\rm Edd}^{-1/2}$, explained by the outward migration of the dust sublimation radius with increasing $\gamma_{\rm Edd}$ (Kudoh et al., 2024):

$$N_{\rm H,\,dust} = N_{\rm H,sub}\,\Gamma_{\rm Edd,d}^{-1/2}$$

where $\Gamma_{\rm Edd,d}$ is the radiative-to-gravitational force ratio for dusty gas. For dust-free (Thomson) gas, $N_{\rm H,\,gas}\approx$ constant with $\gamma_{\rm Edd}$.

Star-forming regions with turbulent, self-gravitating media exhibit a lognormal PDF of column densities, yielding a mass fraction in the super-Eddington regime given by:

$$\zeta_{-} = \frac{1}{2}\left[1 + \operatorname{erf} \left( \frac{x_{\rm crit} - \frac{1}{2} \sigma^2}{\sqrt{2} \sigma} \right) \right]$$

where $x_{\rm crit} = \ln \langle \Gamma \rangle$ and $\sigma^2$ encodes the Mach number dependence (Thompson et al., 2014). This formalism enables calculation of ejected gas mass as a function of turbulence and feedback strength.

5. Observational Manifestations and Population Studies

Major surveys, such as BASS and GOALS, apply this diagram to AGN samples spanning a wide redshift, mass, and luminosity range (Ricci et al., 2023, Nagarajan-Swenson et al., 2 Jun 2025). Key empirical results include:

• Covering factor trends: The dust and gas covering factor, as estimated from IR/bolometric ratios or X-ray obscured fractions, declines with increasing $\lambda_{\rm Edd}$: $C_{\rm f}\sim0.54$ in the obscuration build-up phase, dropping to $C_{\rm f} \sim 0.32$ post-blowout (Ricci et al., 2023).
• Locus and evolution: Across most AGN, $N_{\rm H}$ and $\lambda_{\rm Edd}$ distributions trace the same regimes, irrespective of IR selection or merger-driven triggering. No clear trend is observed for small samples, but full-population analyses support the radiation-regulated feedback cycle as the dominant mode (Nagarajan-Swenson et al., 2 Jun 2025).
• AGN variability and changing-look behavior: AGN lying near the effective limit boundary often show variable absorption (changing $N_{\rm H}$) but not catastrophic outflows, consistent with theoretical expectations (Vasudevan et al., 2013).
• Compton-thick demographics: Two-branch distribution of CT AGN is well-reproduced, with low-$\lambda$ CT AGN (LINERs) dominating at low $z$ and high-$\lambda$ CT AGN at high $z$ (Draper et al., 2010).

6. Theoretical Extensions, Limitations, and Contemporary Challenges

Analytic wind models, radiative hydrodynamics, and population synthesis approaches all inform the usage and interpretation of $N_{\rm H}$–$\lambda_{\rm Edd}$ diagrams. Critical assumptions include simplified opacity laws (constant $\kappa_d$), optically thin steady-state winds, and neglect of inner disc line/MHD-driven winds at high $\lambda_{\rm Edd}$ (Kudoh et al., 2024). Deviations from isotropy, time-dependent fueling, and multi-phase or clumpy media introduce further complexity.

A current limitation is the inability of simple models to fully account for the rapid decline in X-ray obscured fractions at the highest Eddington ratios; simulations and analytic estimates suggest the necessity of multi-component winds and varying contribution of thermal, line-driven, and MHD-driven feedback mechanisms as $\lambda_{\rm Edd}$ increases (Kudoh et al., 2024).

Increasing sample sizes, especially with direct black hole mass and high-quality hard X-ray data, are crucial for empirical discrimination between feedback models, particularly in merger-driven and IR-selected systems (Nagarajan-Swenson et al., 2 Jun 2025).

7. Summary Table: Key Zones and Physical Regimes

Regime or Zone	$N_{\rm H}$–$\lambda_{\rm Edd}$ Diagram Location	Physical Outcome / Population
Allowed absorption	$$\lambda_{\rm Edd}<\lambda_{\rm Edd}^{\rm eff}(N_{\rm H})$$	Long-lived, stable obscuration; Type-2
Forbidden (“blow-out”)	$$\lambda_{\rm Edd}>\lambda_{\rm Edd}^{\rm eff}(N_{\rm H})$$	Short-lived, blown-out gas/dust
High-$\lambda$ CT	log $\lambda_{\rm Edd}\gtrsim 0$; log $N_{\rm H}>24$	Quasar/merger mode CT AGN
Low-$\lambda$ CT	log $\lambda_{\rm Edd}\lesssim -2$; log $N_{\rm H}>24$	LINER/molecular cloud CT AGN

The Eddington Ratio-Column Density diagram thus provides an essential tool for understanding feedback-regulated evolution of AGN, star-forming regions, and their associated multi-phase gas distributions. Its framework unifies observational and theoretical perspectives, allowing robust quantitative prediction of mass ejection, covering factor evolution, and AGN outflow regimes across cosmic time (Thompson et al., 2014, Nagarajan-Swenson et al., 2 Jun 2025, Kudoh et al., 2024, Ricci et al., 2023, Vasudevan et al., 2013, Draper et al., 2010).