PKS1413+135 Model Fitting

(September 2004; rev Oct 2005)
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Introduction

The well-known BLLAC object PKS1413+135 was observed with the VLBA during February to April of 2001. Mapping was done in three frequencies, C-band (4.995 GHz), X-band (8.421 GHz), and U-band (15.353 GHz), at three epochs: Feb 28, April 01, and April 30 of 2004.

The data were calibrated in Classic AIPS. Further cleaning and self-cal was done in difmap.

Model Fitting.

We used difmap to fit circular gaussian or elliptical gaussian model components in the UV plane, trying to account for virtually all the flux density in the map. To evaluate how well our modelling reproduces the data, we list the total integrated flux in all our model components, the total flux of all clean components from a deep clean of each map, and the RMS in the residual map in both cases.

Epoch Band ΔR(mJy) Num Comp Total model flux Num Clean Total Clean flux Peak Flux Integ Core Total Model minus Core Total Clean minus peak Total Clean minus Integ Core
Feb 28 C 0.245 12 0.869 Jy 656 0.827 0.580 0.549 0.320 0.247 0.278
Apr 01 C 0.260 12 0.891 Jy 763 0.867 0.597 0.567 0.324 0.270 0.300
Apr 30 C 0.241 12 0.921 Jy 1233 0.894 0.634 0.601 0.320 0.260 0.293
---
Feb 28 X 0.233 12 1.124 Jy 210 1.102 0.949 0.945 0.179 0.153 0.157
Apr 01 X 0.314 13 1.079 Jy 176 1.055 0.917 0.865 0.214 0.138 0.190
Apr 30 X 0.396 13 1.172 Jy 121 1.148 1.007 0.967 0.205 0.141 0.181
---
Feb 28 U 0.278 8 1.526 Jy 1979 1.497 1.339 1.328 0.198 0.158 0.169
Apr 01 U 0.212 8 1.327 Jy 1379 1.298 1.154 1.075 0.252 0.144 0.223
Apr 30 U 0.256 8 1.350 Jy 1827 1.326 1.187 1.100 0.250 0.139 0.226

Error estimation.

To estimate errors in the model parameters, we used the formulas of Fomalont from "Synthesis Imaging in Radio Astronomy II", ASP Conference Series Volume 180, 1999, page 301.

If ΔR is the post-fit rms error of the pixels in the image, then approximate errors of parameters (Z = Zero Level, P = Peak Intensity, L = Position, W = Width, F = Integrated Intensity) for one component in one dimension are:

(1) ΔZ = ΔR/3 Zero Bias
(2) ΔP = ΔR Peak Intensity
(3) ΔL = ΔRW/2P Position
(4) ΔW = ΔRW/P Width
(5) ΔF = √{ΔR² + (IΔW/W)²} Integrated Intensity

Difmap is used to construct a residual map after subtracting all the fitted components. Difmap calculates the rms flux density of the residual map and we use this rms as the ΔR (in units of mJy/beam).

Difmap component modelling provides the integrated flux density (F), x and y positions, HPBW width (H) of the major axis, and axis ratio for either a circular or elliptical gaussian. The fitted width has the beam effectively de-convolved, hence the width can be much smaller than the beam. Thus, using the fitted width in the above equations for error estimates can produce rediculously small values, thus we add the beam area in quadrature to the width from the fits.

To get the peak intensity P, given the integrated flux F, we use the relation for a one-dimensional gaussian:

  • (6) G(x) = e-4x²ln2/W²

    where W is the HPBW (=FWHM).

    For a circular 2-dimensional gaussian, the integrated flux density is given by

  • (7) F = P ∫G(x)dx ∫G(y)dy

    The integral evaluated between +/- infinity is given by:

  • (8) ∫G(x)dx = 0.5*W*sqrt(π/ln2)

    Thus equation (7) becomes:

  • (9) F = P πW²/(4ln2)
    in which P is the peak flux density per beam.

    But instead of estimating the peak from the above equation, we have measured the peak value P on the cleaned maps for each component.

    Given the model parameters from difmap F and H, and the clean beam size b1 x b2, we find the HPBW of the convolved component size:

  • (10) W = ( H² + b1b2)1/2

    the estimated peak P (Jy per beam) is found by solving equation (9) for P :

  • (11) P = 4F(ln2)/(πW²)

    Thus using the ΔR from the residual map and P and W as calculated by equations (10) and (11), we can use equations (3), (4), and (5) to estimate errors in the position, the width, and integrated flux density.

    Fomalont's equation (5) is a little puzzling because the "I" parameter is used in his article to be the intensity of any pixel. I think it would be reasonable to use "F" in place of "I", and that is what has been done.