12.6 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 12.1 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 12.1: A few source simple source models and their visibility functions

Parameters:
x0	RA position
y0	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  |r|<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. 12.6).

  

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 12.7,12.8).

  

Figure 12.7: Result of a multi-channel point source fit to the Orion SiO (v=2, J=2-1) maser

  

Figure 12.8: ``Spot map'' of the Orion SiO (v=2, J=2-1) maser