H$β$ line shape and radius-luminosity relation in 2.5D FRADO

TL;DR

This study tests a physically motivated 2.5D FRADO model for low-ionization BLR gas against Hβ line profiles and reverberation-like time delays. By convolving a grid of dust-driven, non-hydrodynamic cloud trajectories with photon-flux–dependent emissivity and computing transfer functions, the authors quantify how $M_{ullet}$, $ ext{dot} ext{m}$, and viewing angle shape line widths and asymmetries, and how time delays map to the R-L relation. A key finding is that peak time delays reproduce the observed R-L slope for Hβ, while average delays can misrepresent accretion-rate trends; the model also explores virial factors from FWHM and from $\sigma_{ m line}$, finding that dispersion-based measures align better with observational constraints. The work further shows that microlensing of FRADO BLRs can mimic observed distortions, offering a path to constrain BLR geometry and central engine parameters. Overall, the FRADO framework reproduces many qualitative and quantitative BLR behaviors, with selective near-side obscuration emerging as a plausible mechanism to reconcile high-$ ext{dot} ext{m}$ deviations in the R-L relation and guiding future mass-determination and cosmological applications.

Abstract

Galaxies with active galactic nuclei (AGN) exhibit broad emission lines as a key spectral feature. The shape of emission-line profiles depends on the complex dynamics of discrete clouds within a spatially extended region known as the Broad Line Region (BLR). The distribution of cloud positions within BLR, or the geometry of BLR indeed, is directly linked to measurements of time lags of BLR. In this paper, we convolve a large grid of physically-based simulations of cloud distributions in BLR with photon-flux weighted emissivity of BLR clouds to investigate the generic shape of spectral line profiles. More importantly, we extract the time-delay histograms of corresponding models to calculate the size of BLR. Our physical model is based on the assumption that the clouds are launched by the radiation pressure acting on dust in the atmosphere of the outer disk. It has very few global parameters. The model is appropriate for the low ionization part of the BLR, as it was shown by earlier model tests. It uses a non-hydrodynamical single-cloud approach to the BLR dynamics. In this way we simulate the distribution of positions and velocities of the clouds. We found that the width of line profiles gets broader with black hole mass, or with viewing angle, and gets narrower with accretion rate. The blue wing of the emission line profiles becomes more pronounced with increasing black hole mass and accretion rate, consistent with the formation and intensification of an outflow structure. We also found that the peak time-delays rather than averaged delay values better represents the observational trend and also the scatter in the radius-luminosity relation.

Figures (11)

• Figure 1: Results from a CLOUDY simulation (Ashwani Pandey; private communication) showing the H$\beta$ line emissivity as a function of the incident photon flux $\phi$ (computed from the 2.5D FRADO model) reaching a BLR cloud. The blue points represent simulation outputs, while the red curve shows a logarithmic fit, highlighting a clear nonlinear relationship.
• Figure 2: Predicted H$\beta$ line profiles grouped by black hole mass, assuming a fixed viewing angle of 39$^\circ$. Different curves within each panel correspond to representative Eddington ratios (see text for details). Note: the horizontal axis scale varies between panels. All profiles are normalized to the unit total flux.
• Figure 3: Predicted H$\beta$ line profiles grouped by Eddington ratio, assuming a fixed viewing angle of 39$^\circ$. Different curves within each panel correspond to representative black hole mass (see text for details). Note: the horizontal axis scale varies between panels. All profiles are normalized to the unit total flux.
• Figure 4: Dependence of H$\beta$ line profile shape on viewing angle. Results are shown for two representative models from our simulation grid. The left panel corresponds to a canonical AGN model and the right panel represents the mean quasar. All profiles are normalized to unit total flux.
• Figure 5: Relations between $\sigma$ and FWHM, and the predicted dependence on the black hole mass, accretion rate, and viewing angle. The black dotted line stands for an arbitrary linear fit, $\sigma = 0.34$ FWHM, to the trend of the underneath of data points in top-left panel. The solid lines in the bottom-left panel are also linear fits to the data points.