Spectro-astrometric Analysis

With a given dynamical BLR model, it is easy to calculate its spectro-astrometric signals. Let us denote the projected coordinates of BLR clouds on the sky as \((\alpha, \beta)\), then spectro-astrometry \(s_\lambda\) along a direction \((u_\alpha, u_\beta)\) (e.g., spectral slit or interferometric baseline) is

\[s_\lambda = \frac{f_\lambda}{1+f_\lambda} (u_\alpha \times \alpha + u_\beta\times \beta),\]

where \(\lambda\) is wavelength, \(f_\lambda\) is the line flux ratio relative to the underlying continuum flux.

For spectro-inteferometry, the measured quantity is differential phase \(\phi\) (in degree), which is realted to astrometry as

\[\phi_\lambda = -2\pi \frac{B}{\lambda}s_\lambda,\]

where \(B\) is the sky-projected length of the baseline.

To compute the sky-projected position of BLR clouds, there involves a quantity called position angle \(PA\) of the BLR symmetry axis. If we create a left-handed Cartesian coordinate frame for BLR clouds, in which the \(x\)-axis is along the line of sight and positive x-axis points to the observer (see BLR Models), then the sky-projected position of a BLR cloud with coordinates \((x, y, z)\) is

\[\begin{split}\alpha &=&\,\,\,\, y\times \cos(PA) + z \times \sin(PA),\\ \beta &=& -y \times \sin(PA) + z \times \cos(PA).\\\end{split}\]

brains read input spectro-astrometric data and fit \(f_\lambda\) and \(s_\lambda\) to constrain the model parameters.

Fig.1 Rotation of the BLR coordinate (YOZ) with a position angle (PA) to the observer’s coordinate on the sky (Y’OZ’/EON).