The DPFU parameter plays
havoc with EVN calibration

The basic relationship between the amplitude of the correlation coefficient r_obs measured on a baseline involving two antennas to the antenna temperature T_A (in Kelvins) and the system temperature T_sys (K) is described by the equation:

robs =   r b   =   √ TA1 TA2 Tsys1 Tsys2   ,

(1)

where the b-factor is due to losses inherent in the row coefficient r arising because of crude recording and simplified correlation.

For mapping purposes it is important that the final correlation coefficient does not depend on the antenna elevation angle at stations. We might account for it by dividing the amplitude by the geometric mean of Poly(Elev) at the stations. The Poly is a correction coefficient, a polynomial in the elevation angle (or in the zenith distance) normalized (e.g. at the zenith, but any other elevation can serve the porpose) to unity, reflecting the loss of received flux due to changing atmospheric attenuation and telescope efficiency with the elevation.

So obtained dimensionless quantity can be converted to the temperature scale (K):

or the flux density scale in Janskys (Jy)

Equation (4) is the one used in practice. It requires the use of three different quantities to correct the correlation coefficient. This document aims to reduce the number to two independent quantities. Namely, in terms of the equivalent flux density we would have:

rFlux  =  robs   √   (Fsys1/Poly1)     (Fsys2 /Poly2)   ,      (5)

where F_sys = T_sys/DPFU is the total system noise power as measured by direct comparison with a nearby sky source (in this measurement the source flux density is not scaled by Poly), or with previously calibrated diode signal, F_cal (Poly-scaled). In any case, to determine F_sys no knowledge of the DPFU is required.

The above tells us that in principle we would satisfy the final users by providing the Fsys or the total system noise power expressed in Janskys and the polynomial, Poly. Also, it seems there aren't good reasons why not to give the user the antabfs files filled just with the SEFD, i.e. Fsys already corrected with (divided by) the gain curve (Poly) evaluated at the proper elevation.

The present EVN calibration scheme may be summarized in the following steps (considerably simplified here for clarity):

Fcal = Fso × Poly × (tpical - tpi)/(tponso - tpi),   (6)

where the quantities txxx are the signal power levels of the diode (tpical), background (tpi) and source (tponso), as measured by the Field System.

2) Convert this to the noise temperature:

Tcal = Fcal × DPFU,   (7)

    User experiments
3) Use this T_cal for 'continuous' monitoring of the system noise temperature:

Tsys = Tcal × (tpi - tpzero) / (tpical - tpi').   (8)

For the VLBA racks, the AGC gain level, tpgain, is used as proxy for tpi (tpi becoming a function of tpi', tpgain and tpgain'). So obtained T_sys data constitute the ANTAB files (along with DPFU and Poly).

4) Convert the correlation coefficient to the correlated flux density using the station determined T_sys of two stations (see Eq. (4) above):

rFlux = robs { [ Tsys / (DPFU × Poly) ]1 × ×  [ Tsys / (DPFU × Poly) ]2 } 1/2

or, same as Eq. (5) above,

rFlux = robs { [Fsys/Poly]1 × [Fsys/Poly]2 }1/2   (9)

or simply

rFlux = robs { SEFD1  × SEFD2 }1/2   (10)

Conclusion: The DPFU parameter is just a scaling factor not essential for the calibration process. It is being introduced at the beginning of the process (in Tcal) and is removed (from Tsys) at the end. Thus we are free to ascribe virtually ANY nonzero value to it, not affecting the correlated flux at all.

Here are actual DPFU in LCP and Poly values at L-band taken from pipelined N07L2 (and N06L1, for DPFU only). In the following table the antenna efficiencies in the two rightmost columns have been calculated as 2760×DPFU×Poly/(π×D²/4), where the polynomial Poly has been evaluated for two extreme elevations, 0 (horizon) and 90° (zenith).

Sta- DPFU (LCP) Poly D Efficiency tion 2006 2007 0° 90° m 0° 90° Cm 0.0047 0.0047 1.0 1.0 32 0.016 0.016 Ef 1.5516 1.55 1.0 1.0 100 0.545 0.545 Hh 0.0973 0.0954 0.9638 1.0 26 0.478 0.496 Jb 1.1849 1.1849 0.7373 0.9117 76 0.532 0.657 Mc 0.1080 0.1080 0.8869 0.9831 32 0.329 0.365 Nt 0.11 0.1102 1.0 0.9239 32 0.378 0.349 On 0.0800 0.0900 0.9099 0.9887 20 0.720 0.782 Sh 0.0653 0.0765 0.5009 0.4434 25 0.215 0.191 Tr 0.1400 0.1400 1.0 1.0 32 0.480 0.480 Ur 0.0880 0.0880 1.0 1.0 25 0.495 0.495 Wb* 1.0 1.0 1.0 1.0 14×25 0.402 0.402 *Tsys values for Wb are really SEFD of the array

Note on passing that at Nt and Sh the efficiency is higher at the horizon which may be indicative of their Poly being a function of the zenith distance rather then the elevation.

The table above shows that some efficiencies are clearly unrealistic (which implies unrealistic DPFUs). In about half of the stations these values were different in 2006. Those unrealistic, however, apparently were not at all harmful for calibration of EVN experiments (otherwise Cormac would complain). Thus we see that: