[Paper Review] Weighing black holes from zero to high redshift

A. Marconi, D. J. Axon+5 more|ArXiv.org|Sep 2, 2008

Astrophysical Phenomena and Observations3 citations

TL;DR

This paper corrects virial black hole mass estimators by accounting for radiation pressure forces on broad-line region clouds, which can significantly reduce inferred black hole masses if ignored. The authors show that at high luminosities, radiation pressure may unbind the BLR, challenging the validity of standard virial mass estimates in quasars and suggesting that many high-redshift quasars may have gravitationally unbound broad-line regions.

ABSTRACT

The application of the virial theorem provides a tool to estimate supermassive black hole (BH) masses in large samples of active galactic nuclei (AGN) with broad emission lines at all redshifts and luminosities, if the broad line region (BLR) is gravitationally bound. In this paper we discuss the importance of radiation forces on BLR clouds arising from the deposition of momentum by ionizing photons. Such radiation forces counteract gravitational ones and, if not taken into account, BH masses can be severely underestimated. We provide virial relations corrected for the effect of radiation pressure and we discuss their physical meaning and application. If these corrections to virial masses, calibrated with low luminosity objects, are extrapolated to high luminosities then the BLRs of most quasars might be gravitationally unbound. The importance of radiation forces in high luminosity objects must be thoroughly investigated to assess the reliability of quasar BH masses.

Motivation & Objective

• To address the systematic underestimation of black hole masses in active galactic nuclei due to unaccounted radiation pressure on broad-line region clouds.
• To calibrate virial mass estimators that include radiation pressure corrections for use across redshifts and luminosities.
• To assess the validity of the virial theorem assumption that the broad-line region is gravitationally bound in high-luminosity quasars.
• To evaluate whether standard single-epoch and reverberation mapping mass estimates remain reliable at high redshift and luminosity.
• To determine the critical luminosity at which radiation pressure dominates over gravity, potentially unbinding the broad-line region.

Proposed method

• Derives a corrected virial relation (Eq. 1) that includes radiation force terms, modifying the effective gravitational potential seen by BLR clouds.
• Introduces a radiation pressure correction term proportional to luminosity and inverse cloud column density, based on momentum deposition from ionizing photons.
• Calibrates the corrected virial mass estimator using local AGNs with reverberation mapping data, ensuring consistency with the M_BH–σ relation.
• Applies the corrected relation to a large sample of SDSS quasars (Shen et al. 2008) to assess mass corrections across redshift and luminosity.
• Uses the R_BLR–L relation to estimate BLR radius and combines it with line width measurements to compute virial masses with and without radiation corrections.
• Analyzes the Eddington ratio distribution and BLR binding condition by comparing luminosity to the critical luminosity L* where radiation pressure balances gravity.

Experimental results

Research questions

• RQ1 How does radiation pressure from ionizing photons affect the inferred black hole mass in virial mass estimators?
• RQ2 What is the magnitude of the radiation pressure correction to virial mass estimates in high-luminosity quasars compared to low-luminosity AGNs?
• RQ3 At what luminosity does radiation pressure become strong enough to unbind the broad-line region, violating the fundamental assumption of gravitational binding?
• RQ4 How do the corrected virial masses affect the observed distribution of Eddington ratios in high-redshift quasars?
• RQ5 To what extent do current single-epoch and reverberation mapping mass estimates remain valid when radiation pressure is included?

Key findings

• The radiation pressure correction reduces the effective gravitational potential seen by BLR clouds, leading to a factor of ~2 smaller virial factor f in reverberation mapping compared to standard estimates.
• For moderate luminosities (L ~ 10^11 L_sun) and column densities (N_H ~ 10^23 cm^-2), the radiation pressure correction can reach ~2.7 × 10^7 M_sun, comparable to typical Seyfert galaxy black hole masses.
• In high-redshift quasars with L > 10^12 L_sun, the radiation pressure correction dominates the virial term, causing BH masses to scale linearly with luminosity rather than with the virial product.
• The Eddington ratio distribution saturates at L/L_Edd ~ 0.15 for high-luminosity quasars, indicating that the BLR is approaching gravitational unbinding at these luminosities.
• A significant fraction of high-luminosity quasars lie in the region where the BLR is potentially unbound (between L = L_Edd and L = L*), challenging the validity of standard virial mass estimates.
• The analysis shows that many quasars with FWHM < 10,000 km/s and L > 10^12 L_sun are near the critical luminosity L* where radiation pressure balances gravity, indicating a fundamental breakdown of the virial assumption in these sources.