# Flux Calibration

### Reference Material

• Synthesis Imaging in Radio Astronomy II, ed. Taylor, Carilli, Perley, Ch. 10 (Cornwell & Fomalont)
• Baars, J. W. M.; Genzel, R.; Pauliny-Toth, I. I. K.; Witzel, A., "The absolute spectrum of CAS A - an accurate flux density scale and a set of secondary calibrators", Astronomy and Astrophysics, vol. 61, no. 1, Oct. 1977, p. 99-106

## 1 Flux Calibration

Here we are going to go into a bit more depth on flux calibration. In a previous lecture on self-calibration, we saw the principles of solving for the internal degrees of freedom of an array by using imaging with a coarse calibration to generate a sky model, and then using the sky model to improve the calibration. However, as pertains to flux calibration, this process fails to break the degeneracy between an overall signal gain through your interferometer and the absolute brightness of the sky. That is, if you doubled the overall gain of the amplifiers in your telescope, and you made the sky half as bright, would you know the difference?

So how do you achieve absolute flux calibration?

### 1.1 Using a Reference Source

The easiest way to set an absolute flux scale is to calibrate relative to a reference source. As near as I can tell, the predominant flux scale used in radio astronomy is derived from one important work, Baars et al. (1977), which set out to measure the absolute flux scale of four of the brightest sources in the sky: Cygnus A, Cassiopeia A, Taurus A (the Crab Nebula + the Crab pulsar), and Virgo A. Unfortunately, two of these sources (Cass A and Tau A) are supernova remnants that are still expanding, and have a brightness that is decreasing with time. Therefore, Baars et al. (1977) is most often cited for the spectrum of Cygnus A – a radio galaxy 600 Mlys away with two incredible jets emanating from an AGN of a magnitude seen once in a couple billion light years. It’s not going anywhere.

Cygnus A

Unfortunately, as you can see, Cygnus A is made up of two distinct lobes, along with a central point source. This highlights one of the major difficulties in establishing an absolute flux scale, even when you have a calibrator source. For anything other than a point source, the amplitude of the signal you measure for a source will depend on the length and orientation of your interferometric baseline, because anything other than a point source has structure in the uv-plane. And this problem isn’t unique to interferometers: all sources sit on a sky that contains background sources (all the other sources behind it), as well as smooth, broad-scale emission such as galactic synchrotron emission, the cosmic microwave background, etc. Depending on how big of a beam your telescope has, you will pick up more or less of this background emission.

These are all complicated calibration issues, and can lead to major difficulties in comparing measurements made with different telescopes, even if they are operating at the same frequency. There are several solutions, but they tend to depend a lot on your telescope. For example, large telescopes with tight beams can dither on and off of a selected bright source that is far enough away as to appear to be a point source. You can then subtract your “on” and “off” measurements to isolate the source in question, and then calibrate that spectrum to a reference. Other telescopes with broader beams will have a hard time choosing an optimal point source, because they will tend to have other, much brighter sources already in their beam. For these, you may need to use detailed models of the bright sources that were chosen for you, along with models of your beam area and/or baseline lengths that account for extended structure that is attenuated by your primary beam or resolved out by your baseline fringe pattern. Suffice it to say, it can be challenging when you need an antenna calibration in order to calibrate your antenna.

### 1.2 Fluxes, Flux-Densities, and Temperatures

In radio astronomy, source fluxes are commonly reported in Jys, which is a measure of flux per Hz (i.e. a spectral flux density). This is a much more useful unit than just flux, since the flux (and therefore, the power) you measure will always depend on the bandwidth of your measurement, and if you’re going to normalize to some chosen bandwidth, it should be explicit. However, Jys convey no information about the distribution of brightness on the sky. It is a sum over all of the emission that is defined to be associated with a source. This makes Jys a convenient unit for point sources, but a poor one for describing extended structure. (And it should be noted that one telescope’s point source can be another’s extended source).

For extended structure, specific intensity ${\displaystyle I}$ (which is flux, per Hz, per solid angle), and for radio astronomy is characterized by a brightness temperature:

${\displaystyle I\equiv {\frac {2k_{B}T}{\lambda ^{2}}},\,\!}$

where ${\displaystyle k_{B}}$ is the Boltzmann constan, ${\displaystyle T}$ is the brightness temperature, and ${\displaystyle \lambda }$ is the wavelength, is a much better unit. An equivalent unit that is sometimes used is Jys/beam, where “beam” is a telescope-dependent quantity with units of solid angle. A useful conversion is:

${\displaystyle V_{\rm {Jys}}=10^{-23}{\frac {\rm {Jys}}{\rm {erg/s\cdot cm^{2}\cdot Hz}}}{\frac {2k_{B}T\Omega }{\lambda ^{2}}},\,\!}$

where ${\displaystyle \Omega }$ is the solid angle of the beam (or pixel), and ${\displaystyle V_{\rm {Jys}}}$ is the visibility measured in Jys.

Using either of these measures (temperature or Jys/beam), it is possible to explicitly include the spatial structure of the source and the telescope in calibrating to a flux scale. On the other hand, when doing this, it is important to remember that the uv-plane fundamentally has units of Jys. This is because it is the Fourier complement of image domain, and a 2-d spatial Fourier transform integrates an image (specific intensity ${\displaystyle I}$, in units temperature, or Jys/beam) against a sine wave, over a solid angle:

${\displaystyle V_{\rm {Jys}}=\int \!\!{\int {I(l,m)e^{-2\pi i(ul+vm)}dl~dm}}\,\!}$

Thus, the resulting uv-mode has units of temperature times solid angle (i.e. Jys).

### 1.3 Absolute Calibration

Absolute calibration without using a reference source is a much more involved endeavor that is somewhat beyond the scope of what can be covered here. In principle, it involves using an antenna whose gain and impedance can be calculated analytically (or measured absolutely in an anechoic chamber), and switching the antenna signal with a noise signal of known amplitude (say, a resistive load of known temperature) from which the gains of amplifiers, cables, filters, etc. can be calibrated out.