BLR Models

BLR Models#

brains already encapsulates several BLR dynamical models detailed below. To specify those models in running, edit the option “FlagBLRModel” in the parameter file (see Parameter File).

Broad-line regions are generally assumed to be composed of a larg number of point-like clouds. These clouds respond to the central ionizing continuum and emit broad emission lines.

BLR coordinate and observer’s coordinate#

The BLR coordinate is adopted to be left-handed (see Fig.1). The x-axis is set to be along the line of sight and the positive x-axis points to the observer. That is to say, a positive velocity in the BLR’s coordinate corresponds to a blus-shift velocity in observer’s coordinate.

../_images/fig_PA.jpg

Fig.1 The coordinate frames. X-axis is the along the line of sight and positive x-axis points to the observer. Left-handed coordinate of the BLR is preferred, which is consistent with the coordinate frame of the observer. However, for axisymmetric BLRs, the two types of coordinate frames are indistinguishable.#

BLR model 1#

This model is from Brewer et al. (2011). Clouds’ distribution has a disk-like shape (see the figure) and is axis-symetric.

  • Radial distribution: \(\Gamma\)-distribution

    \[\Gamma(r|\alpha, \theta) = \frac{1}{\Gamma(\alpha)\theta^{\alpha-1}}r^{\alpha-1}\exp\left(-\frac{r}{\theta}\right)\]

  • Dynamics: clouds’ orbital angular momentum and energy are randomly assigned following distributions

    \[\begin{split}E = \left(\frac{1}{1+\exp(-\chi)}\right)E_{\rm min},~~~ E_{\rm min}=-\frac{GM_\bullet}{r}, ~~~\chi\sim N(0, \lambda^2)\\ p(L)\sim \exp\left(-\frac{|L|}{\lambda L_{\rm max}}\right),~~~ |L| < L_{\rm max} = \sqrt{2r^2\left(E+\frac{GM_\bullet}{r}\right)}\end{split}\]

  • Emissivity: anisotropic prescription

    \[w(\phi) = \frac{1}{2} + \kappa \cos\phi\]

    where \(\phi\) is the angle between the observer’s line of sight to the central ionizing source and the cloud’s line of sight to the central source.

../_images/fig_blr_disk.jpg

Fig.2 Schematic of a disk-like broad-line region (Li et al. 2013).#

BLR model 2#

The radial distribution and emissivity are the same as model 1. The dyanmics of clouds are modeled in the orbital plane as

\[\begin{split}\rho_r \sim N(1, \sigma_r^2), ~~~~\rho_\theta \sim N(\pi/2, \sigma_\theta^2),\\ V_r = V_{\rm circ}\rho_r\cos\rho_\theta,\\ V_\theta = V_{\rm circ}\rho_r\sin\rho_\theta,\end{split}\]

where \(V_{\rm circ}=\sqrt{GM_\bullet/r}\) is the circular velocity.

BLR model 3#

  • Radial distribution: power-law distribution

    \[\rho(r|\alpha) = \rho_0 \left(\frac{r}{r_0}\right)^{-\alpha},~~~~r_{\rm in} < r < r_{\rm out}\]

  • Dynamics: Keplerian motion and inflow/outflow.

    \[\boldsymbol{v} = V_{\rm Kep}\boldsymbol{e_{\theta}} + \xi \sqrt{\frac{2GM_\bullet}{r}} \boldsymbol{e_{r}}\]

BLR model 4#

This model is the same as model 3, except for the dynamics

\[\boldsymbol{v} = \sqrt{1-2\xi^2}V_{\rm Kep}\boldsymbol{e_{\theta}} + \xi \sqrt{\frac{2GM_\bullet}{r}} \boldsymbol{e_{r}}\]

BLR model 5#

  • Radial distribution: double power-law distribution.

\[\begin{split}f(r) \propto \left\{\begin{array}{ll} r^{\alpha}, & {\rm for}~F_{\rm in}\leqslant r/R_0 \leqslant 1,\\ r^{-\alpha},& {\rm for}~1\leqslant r/R_0 \leqslant F_{\rm out}, \end{array}\right.\end{split}\]

  • The dyanmics and emissivity are the same as model 6.

BLR model 6#

This is compatible with Pancoast et al. (2014)’s model.

  • Radial distribution: \(\Gamma\)-distribution

    \[\Gamma(r|\alpha, \theta) = \frac{1}{\Gamma(\alpha)\theta^{\alpha-1}}r^{\alpha-1}\exp\left(-\frac{r}{\theta}\right)\]

  • Dynamics: A fraction \(f_{\rm ellip}\) of clouds have bound elliptical Keplerian orbits and the remaining fraction \((1-f_{\rm ellip})\) is either inflowing \((0 < f_{\rm flow} < 0.5)\) or outflowing \((0.5 < f_{\rm flow} < 1)\).

    For elliptical orbits, the radial and tangential velocities are drawn from Gaussian distributions centered around a point \((v_r, v_\phi) = (0, v_{\rm circ})\) with standard deviations \(\sigma_{\rho,\rm circ}\) and \(\sigma_{\Theta,\rm circ}\), respectively. Here, \(v_{\rm circ}=\sqrt{GM_\bullet/r}\) is the local Keplerian velocity.

    For inflowing/outflowing clouds, velocities are drawn similarly from Gaussian distributions centered around points \((v_r, v_\phi) = (\pm \sqrt{2} v_{\rm circ}, 0)`\) with standard deviations \(\sigma_{\rho,\rm rad}\) and \(\sigma_{\Theta,\rm rad}\), where “+” corresponds to outflows and “−” corresponds to inflows.

  • Emissivity: anisotropic prescription

    \[w(\phi) = \frac{1}{2} + \kappa \cos\phi\]

    where \(\phi\) is the angle between the observer’s line of sight to the central ionizing source and the cloud’s line of sight to the central source.

BLR model 7#

This is the shadowed model in Li et al. (2018).

../_images/fig_blr_twozone.jpg

Fig.3 Schematic of a disk-like broad-line region with two zones (Li et al. 2018).#

BLR model 8#

A disk wind model from Shlosman & Vitello (1993).

../_images/fig_diskwind.jpg

Fig.4 Schematic of a disk wind model (figure credit: Higginbottom et al. 2013).#

In the cylindrical coordinate, the wind stream line have an angle as

\[\begin{split}\theta = \theta_{\rm min} + (\theta_{\rm max}-\theta_{\rm min})x^\gamma,\\ x=(r_0-r_{\rm min})/(r_{\rm max}-r_{\rm min}),\end{split}\]

where \(r_0\) is the root point of the stream line. The velocity along the stream line is

\[v_l = v_0 + (v_\infty-v_0)\frac{(l/R_v)^\alpha}{1 + (l/R_v)^\alpha},\]

where \(l\) is the distance along the stream line, \(R_v\) is the scale length, \(v_0\) is the initial velocity, and \(v_\infty\) is the terminal velocity defined to be

\[v_\infty = \sqrt{\frac{2GM_\bullet}{r_0}}.\]

The velocity components are

\[v_r = v_l \sin\theta, ~~~ v_z = v_l \cos\theta.\]

The azimuthal velocity is given by assuming conservations of the angular momentum

\[v_\phi = v_{\phi, 0}\left(\frac{r_0}{r}\right) = \frac{\sqrt{GM_\bullet r_0}}{r}.\]

The density along the stream line is given by

\[\rho(l) = \frac{\dot m}{v_l} \frac{r_0 dr_0}{rdr},\]

where \(\dot m\) is the mass-loss rate at the root of the stream line.

BLR model 9#

This is the model adopted in the spectroastrometric modeling on 3C 273 by GRAVITY Collaboration (2018).

  • Radial distribution: \(\Gamma\)-distribution

    \[\Gamma(r|\alpha, \theta) = \frac{1}{\Gamma(\alpha)\theta^{\alpha-1}}r^{\alpha-1}\exp\left(-\frac{r}{\theta}\right)\]

  • Dynamics: circular Keplerian motion,

    \[V_{\rm Kep} = \sqrt{\frac{GM}{r}}.\]

  • Emissivity: isotropic prescription.

References#

  • Brewer, B. et al. 2011, ApJL, 733, 33

  • GRAVITY Collaboration et al. 2018, Nature, 563, 657

  • Higginbottom, N. et al. 2013, MNRAS, 436, 1390

  • Li, Y.-R. et al. 2013, ApJ, 779, 110

  • Li, Y.-R. et al. 2018, ApJ, 869, 137

  • Pancoast, A. et al. 2014, MNRAS, 445, 3055

  • Shlosman I., Vitello P., 1993, ApJ, 409, 372

Contents