14.6 Model fitting

Model fitting is the oldest way of analyzing interferometer data. It was effectively used in the times where the coverage of the $uv$ plane was too scarce to even think of creating an image by Fourier transform. One assumes a simple source model depending of a few parameters (source position, flux, size) and fits the visibility function of that model to the visibility data. Of course one may use a linear combination of several source models since the Fourier transform is linear. This is performed using the GILDAS task UV_FIT. The result may be displayed with the procedure PLOTFIT. Both are available in the panel ``Interferometric UV operations'' from the GRAPHIC standard menu.

Table 14.1 gives examples of a few models and their visibility functions. For source models with a circular symmetry, the visibility function is split into a radial dependent amplitude and a phase factor which depends only on the source position.

Table 14.1: A few simple source models and their visibility functions

Parameters:
$x_0$	RA position
$y_0$	DEC position
$S$	Source flux
$b$	HP size

   

Variables:
$x,y$	sky position
$r$	$\sqrt{(x-x_0)^2+(y-y_0)^2}$
$u,v$	projected spacing
$q$	$\sqrt{u^2+v^2}$

Name	Model	Visibility
Point source	$S  \delta(x-x_0,y-y_0)$	$S  e^{-2i\pi(u x_0+v y_0)}$
Gaussian	$\frac{4 S}{\pi b^2 \log{2}}  e^{-4\log{2}\frac{r^2}{b^2}}$	$S  e^{- \pi^2/4/\log{2} (bq)^2}  e^{-2i\pi(u x_0+v y_0)}$
Disk	$\frac{4 S}{\pi b^2}$  where  $\vert r\vert<b$	$S J_1(\pi b q)  e^{-2i\pi(u x_0+v y_0)}$

Some sources are actually so simple that this method may be used to a good accuracy (fig. 14.6).

Figure: $uv$-plane fit to the disk around GG Tau at 1.3mm (from [Guilloteau et al. 1999]).

Quite often this method is used for sources that are unresolved or not well resolved at a given frequency; for instance a SiO maser may consist of several point-source components at different velocities. Fitting a point source in each channel one derives a ``spot map'' (figs 14.7,14.8).

Figure 14.7: Result of a multi-channel point source fit to the Orion SiO (v=2, J=2-1) maser

Figure 14.8: ``Spot map'' of the Orion SiO (v=2, J=2-1) maser

14.6.1 Position measurement

For a source with central symmetry the task UV_CENTER determines the source position by using only the phases. Alternatively the task UV_FIT may be used to fit the amplitudes and phases at the same time, or e.g. to simultaneously fit a pair of sources.