19.1 Continuum Source

Let us start with a continuum source to simplify. The basic source parameters are position (x,y), flux density S$_\nu$, and size. To determine the first 3 parameters, the best strategy is to avoid resolving the source. Since the position errors are proportional to the beam size, one should thus try to match the source size to the beam size. Having a priori information on the source position (by other observations, e.g. an optical image) will help to get a better accuracy on the source flux.

19.1.1 Flux measurement

• Accurate Position: If the position is known to better than 1/10 $^{\mathrm{th}}$ of the beam, you should then use UV_FIT with the position fixed to determine the flux and the noise. In doing so, you should use an appropriate fixed source size, based on any a priori information you have (as you are following the ``best strategy'', this source size should be smaller than the beam size). In such a case, you only have one free parameter, the flux, and a $3 \sigma$ signal is sufficient to claim a detection.

• Rough Position If the source position accuracy is not sufficient, you need to measure the position also. As you now have more parameters to derive, a higher S/N will be required to do so. When the position uncertainty is about the beam size, a 4 $\sigma$ signal will be required to get a firm detection. Use UV_FIT to measure the flux and source position, with the same fixed source size as before.

• Unknown Position When the source position is really unknown, a 5 $\sigma$ signal may become necessary to claim a detection. To locate the source position, make an image first. Cleaning is usually not required at that level, unless the sidelobe level is higher than the noise to signal ratio. Then, use UV_FIT to measure the source flux, position and their associated errors, always using the same fixed source size as before.

Note that in all cases, the source size being used should be at least equal to the effective seeing of the observations, even if the source is actually a point source.

Table 19.1 summarizes the procedure to be followed. Once you have done your best in determining the source parameters, they remain to be properly interpreted. As a rule of thumb, remember that All fluxes for detected weak sources are biased by 1 to 2 $\sigma$. The only exception is when the source position and size is known a priori. The reason for the bias is very intuitive. Assume you have observed just enough to get a $3 \sigma$ detection. A positive noise peak will bring that up to a $4 \sigma$ value, a negative noise peak down to $2 \sigma$, which you will consider as a non-detection.

Table 19.1: Recipes to use UV_FIT to measure the flux of a weak source

Rule 1	Do not resolve the source
Rule 2	Get the best absolute position before
Rule 3	Use UV_FIT to get
	the parameters and their errors
	a priori position accuracy
	$< 0.1$ Beam	$\simeq$ Beam	Any
Minimum signal	3 $\sigma$	4 $\sigma$	5 $\sigma$
Position	fixed	free	free, (make an image)
Source size	fixed	fixed	fixed

19.1.2 Other parameters

The other source parameters (position & size) require higher signal to noise to be determined. The position accuracy is the synthesized beam size divided by the S/N ratio. Hence, to get a position accuracy to 25 % of the beam size, at least a $4 \sigma$ detection is required.

The above limitations are valid for a point source. If the source is not expected to be small enough, additional complications occur. If you have performed the experiment according to the guidelines given before (i.e. avoiding resolving the source), the source size may be just about the beam size. In such circumstances, no source size at all can be estimated with current mm interferometers if the detection is less than $6 \sigma$. To convince yourself, let us perform a simple thought experiment. Assume we have detected a source at the $6 \sigma$ level. Take this $6 \sigma$ signal, and divide the observations in two equal (in sensitivity) data sets, one containing only the shortest baselines, the other ones only the longest baselines. Each subset has a $\sqrt{2}$ times higher noise level, and the error on the flux difference between these two data sets is $2$ times the original noise level. Assume that the shortest baselines give us twice more flux than the longest one. In such a case, we would in fact have a better detection ( $6.4 \sigma$) with the short baselines only, but the difference flux is only measured with $3 \sigma$. Such an experiment is not optimal from the detection point of view, since we would have obtained a better result ( $6.4 \sigma$) by observing only half of the time... Table 19.2 summarizes the corresponding numbers, and indicates that the minimum detection level to resolve a source at the $4 \sigma$ level is $7.1 \sigma$.

Table 19.2: Signal to Noise example for source size measurement. Line (1) indicate the flux measured on short baselines, line (2) on long baselines, line (3) the difference between (1) and (2), and line (4) the average. Three cases are shown: a point source, a source with size similar to the beam, and the smallest source which can be resolved at the $4 \sigma$ level.

		Point			Beam Size			Minimum Size
		Source			Source			Resolved
		Flux	Noise	S/N	Flux	Noise	S/N	Flux	Noise	S/N
(1)	Short baselines	6	1.4	4.2	9	1.4	6.4	10	1.4	7.1
(2)	Long baselines	6	1.4	4.2	3	1.4	2.1	2	1.4	1.4
(3)	Difference	0	2	0	6	2	3	8	2	4
(4)	Mean	6	1	6	6	1	6	6	1	6

The interpretation of such data is made even more difficult by the fact that if the size is unknown, the error on the total flux increases quite significantly. Fig.19.1 shows the detection of a weak high-redshift object in the Hubble Deep Field area [Downes et al 1999]

Figure 19.1: Left: $7 \sigma$ detection of the strongest source in the Hubble Deep Field. Note that the contours are visually misleading (they start at 2 $\sigma$ but with $1 \sigma$ steps, given the impression of a much better detection). Right: Attempt to derive a size. Size can be as large as the synthesized beam... Note that the integrated flux increases with the source size.

Although the detection is at the $7 \sigma$ level, the source size is not constrained by these observations, and the total flux becomes uncertain by as much as 40 % when the uncertainty on the source size is included.