Calibrating in Jy/(solid angle)

If your maps show simple source morphology and you are only interested in integrated flux densities, the simplest approach is to calibrate your map Jy/aperture for the aperture size you want to use. The listing of FLUXES also gives us the total flux, S$_{tot}$ for Uranus at 850$\mu$m is 67.9 Jy. Let us first see how we can use this value to calibrate our image in terms of Jy/arcsecond$^2$. In order to do this we need to derive a value for the Flux Conversion Factor (FCF) which is in units of Jy/arcsecond$^2$/V. To do this we first need to work out the sum of the pixel values (V$_{\rm sum}$) in an aperture of radius r. We then find the FCF is given by

$$FCF(Jy/{\rm arcsecond}^2/V)=67.9/V_{\rm sum}a$$ (5)

were a is the pixel area in square arcseconds. The easiest way to get the integrated signal in an aperture is to use KAPPA's aperadd. For our 850 $\mu$m map of Uranus we derive V$_{\rm sum}$ for a set of different circular apertures and compute the FCFs.

Radius (arcseconds)	20	30	40	60	120
V$_{\rm sum}$	45.75	60.76	64.89	70.07	77.08
FCF (Jy/arcsecond$^2$/V)	1.48	1.12	1.05	0.97	0.88

We can see from this table that the FCF is dependent on the aperture size that is used ¹ This is because there is significant signal in the sidelobes and extended error beam of the telescope. Clearly then the value of FCF can be somewhat ambiguous. What you have to remember is that if you are doing photometry of an extended object, you should use a value for the FCF derived for the same aperture.

If you need to use small apertures, i.e. the size of your HPBW, you will need to use a point source or point like source as a calibrator. Flux densities for our secondary calibrators for a 40'' aperture are given by Jenness et al. [14]. However, several of our secondary calibrators are not point sources. If you end up with, for example, IRC$+$10216 and IRAS 16293$-$2422 or Mars near perhelion as your only calibrators during your run, you are in trouble. You may be able to use a large aperture to recover all the flux and use the ratios between different apertures derived for a point source. But, you may as well bite the bullet and calibrate in Jy/beam.