FRADO Model

Failed Radiatively Accelerated Dusty Outflow

FRADO code (in Python; Public Scientific Software)

The numerical implementation of the 2.5D FRADO model is publicly available as FRADO v 1.0.0.

The software includes numerical modelling of dusty-cloud trajectories, radiation pressure, dust sublimation, documentation, examples, and numerical validation.

Explore the 2.5D FRADO model directly in your browser via Google Colab. No local Python installation is required.

Source code on GitHub

Simulation dataset on Zenodo

The nucleus of many galaxies is highly luminous powered by the accretion of matter onto a supermassive black hole of millions to billions solar mass at the center of the host galaxy, transforming the gravitational energy into radiation, hence called active galactic nuclei (AGN). A quasar (QSO) is an extremely active and luminous type of AGN. All QSOs are AGNs, but not all AGNs are QSOs, however, we call them all QSOs throughout this content.

Quasars are thus primarily consist of a central black hole and accretion disk of material. Through the process of accretion of material onto black hole, other secondary components are formed such as a jet, dusty torus, broad and narrow line emitting regions, etc.

Quasars are characterized mainly by the mass of the central black hole, accretion rate of material, orientation of the accretion disk (with respect to observer), degree of obscuration of nucleus by torus, and presence/absence of a jet.

The spectrum of a typical quasar, as partially decomposed shown in this figure (Kovačević et al., 2017), usually consists of many different components especially Balmer emission lines which are due to reprocess and reemission of the disk radiation by circumnuclear moving material. Doppler broadening of emission lines in the spectra of AGNs as a result of velocity dispersion due to kinematics of those moving material (Δ𝜆/𝜆 = v/c ) is usually a blend of two components of narrow and broad with widths of order of few hundred km/s and up to 104 km/s, respectively. The regions where the moving material produces narrow and broad emission lines are called Narrow Line Region (NLR) and Broad Line Region (BLR).

They are frequently observed in the form of single-peaked (mostly in high accretion rates) and double-peaked (for low accretion rates) emission line profiles. They can be divided into two main types of Low Ionized Lines (LIL) like H𝞫, MgII, and FeII, and High Ionized Lines (HIL) such as CIV, and HeII.

There have been many attempts over the last 30 years to explain the origin of BLR with different scenarios. Observations indicate that the circumnuclear moving material giving rise to formation broad emission lines are (due to thermal instability most likely) clumpy, (due to observationally-implied flat geometry most likely) disk-originated, and present at all scales from the inner radius of BLR to the torus. A typical geometry of a quasar is shown in the top-right figure (Miniutti et al., 2013). The scenario of radiatively line-driving mechanism has been successful to explain the HIL part of BLR.

A decade ago, Czerny & Hryniewicz (2011) showed that dust can form at large radii of accretion disk where the disk effective temperature drops below the order of 1000 K, thus proposed a non-adhoc physically-motivated fully-analytical one-dimensional (1D) model based on radiatively dust-driving mechanism to explain the LIL part of BLR. In this model, a dusty wind is initiated from the disk due to disk radiation pressure but finally falls back onto the disk, hence the model is called FRADO (Failed Radiatively Accelerated Dusty Outflow).

The FRADO model directly predicts the inner/outer radius of LIL-BLR, asymmetry in line profiles (due to dust evaporation); observationally consistent line shape parameter FWHM/σ, the dependence of virial factor on black hole mass, and most importantly the radius-luminosity (RL) relation (Czerny et al., 2015, 2016, 2017).

The basic model of FRADO, however, is 1D fully analytical in which the flux is very local and constant with height, shielding effect is neglected, radiation from other radii is neglected in motion, the motion is only described in the vertical direction and radial motion is neglected, an wavelength-averaged dust opacity is used, and the full picture of LIL-BLR is in the form of failed wind (no signature of outflow), but the successful predictions of the model was the motivation to develop its advanced 3D version.

In order to simulate the BLR dynamics based on the radiatively dust-driving mechanism, I developed a 3D (2.5D due to azimuthal symmetry) non-hydrodynamical code which incorporates the realistic prescription of the radiation pressure of an accretion disk, advanced wavelength-dependent dust opacities, shielding effect, and dynamical dust sublimation process. The code basically solves the equation of motion (while the angular momentum is conserved) for a given set of main global parameters of a quasar, i.e. black hole mass, accretion rate, and dust-to-gas ratio (or, equivalently, metallicity).

In 2.5D FRADO, the clumpy dusty-gaseous material, initially in a Keplerian orbit around the central black hole, is launched due to irradiation by the whole extended disk acting on dust at the disk surface. Radiation is geometrically moderated by the shielding which depends on the cloud location. The full pattern of the trajectories of clumps launched at different radii of the disk is complex and depends on the main global parameters of the source. Also depending on the launching radius, as shown in the figure, clumps may lose their dust content (red) or remain dusty (blue) during the flight. The radial extension of LIL BLR is the entire radial range from which the material can be radiatively lifted from the disk surface, that sets the two ends of inner/outer radius of LIL BLR. The solid curved line in orange represents the sublimation location where the material crossing up loses the dust content. See Naddaf et al. (2021), for details.

This figure shows the result of simulation for a source at Eddington rate with black hole mass of 108 M☉ and metallicity of 5 times solar. The motion is projected onto the 2D plane of (R, z) where R is the radial distance to the black hole in the equatorial plane, and z is the vertical distance from the equatorial plane. The overall motion consists of failed wind (solid curved lines) and outflow (dashed lines) lines. Dusty and dustless situations are color-coded in blue and red, respectively. Only a fraction of trajectories is shown for a better visibility.

The 2.5D FRADO can interestingly explain the observational geometry and dynamics of AGNs. The funnel-shaped outflow is surprisingly similar to the empirical picture of AGNs proposed by Elvis (2000), and the failed part of the LIL BLR resembles the static puffed-up disk model of Baskin & Laor (2018) but in a dynamical context. See Naddaf et al. (2021), for details.

The 2.5D FRADO recently confronted against the most recent sample of reverberation measured H𝞫 time-delays and tested for the BLR size and RL relation. It was shown that the sample can be almost successfully recovered (Naddaf et al., 2020)

Points in cyan show the sample of observational data from Martínez-Aldama et al. (2019). The blue dashed line represents the RL relation predicted by 1D FRADO. The black dashed line shows the RL relation calibrated for sources with low Eddington ratios by Bentz et al. (2013).

The role of shielding effect gets more vital with the increase of the accretion rate of the source, as well as the blackhole mass.

Calculation of line profiles based on 2.5D FRADO showed that the shape of profiles not only depends on the accretion rate of the source, the black hole mass, and the viewing angle, but also it is most significantly affected by the adopted metallicity that regulates the strength of the radiation pressure (Naddaf & Czerny, 2022)

In order to test the model, the fully corrected MgII and H𝞫 emission lines in the mean composite quasar spectrum from SDSS data (Vanden-Berk et al., 2001; Shen et al., 2011) is compared to the line profile predicted by 2.5D FRADO for a case corresponding to the mean quasar physical parameters (i.e. black hole mass of 109 M☉, Eddington rate of 0.1) with metallicity of 5 times solar, viewed at 30 degrees, as shown in the right panels of the figure. In the left panels, the power-law-subtracted MgII and H𝞫 lines from SDSS data are decomposed into different components as indicated in the legends of plots. It shows that the model can appropriately explain the low ionized broad emission lines of the mean spectrum of quasars, such as MgII and H𝞫.