12.1 Definition and Formalism

From Lucas lecture (Chapter 9), Eq. 9.1, the baseline-based observed visibility $\ensuremath{\widetilde{V}}_{ij}(t)$ is linked to the true visibility $V_{ij}$ of the source by:

$$\ensuremath{\widetilde{V}}_{ij}(t) = {\ensuremath{\mathcal{G}}_{ij}}V_{ij}+ \epsilon_{ij}(t) + \eta_{ij}(t)$$ (12.1)

In antenna-based calibration, ${\ensuremath{\mathcal{G}}_{ij}}$ can also be written as:

$${\ensuremath{\mathcal{G}}_{ij}}= g_i(t)g^*_j(t)=a_i(t)a_j(t)e^{i(\phi_i(t)-\phi_j(t))}$$ (12.2)

Hence, for antenna $i$, the antenna-based amplitude correction for the lower sideband $a_{i}^L$ is given by

$$a_{i}^L(t) = T_{cal_i}^L(t) G_i^L(\nu,t) \Gamma_i(t)$$ (12.3)

and for the upper sideband:

$$a_{i}^U(t) = T_{cal_i}^U(t) G_i^U(\nu,t) \Gamma_i(t)$$ (12.4)

where $T_{cal_i}^{U}$ and $T_{cal_i}^{L}$ are the corrections for the atmospheric absorption (see Chapter 10), in the upper and lower sidebands respectively. $\Gamma_i$ the antenna gain (affected by pointing errors, defocusing, surface status and systematic elevation effects). Note that Eqs.12.3-12.4 do not include the decorrelation factor $f$ (see Chapter 9 by R.Lucas) because this parameter is baseline-based. We assume here decorrelation is small enough, i.e. $f=1$; if not, a baseline-based amplitude calibration may be required.

$G_i^L(\nu,t)$ and $G_i^U(\nu,t)$ are the electronics gains (IF chain+receiver) in the lower and upper sidebands, respectively. The receiver sideband gain ratio is defined as $G_i^{UL}(\nu,t) = G_i^U(\nu,t)/G_i^L(\nu,t)$. The sideband gain ratio is to first order independent of the frequency $\nu$ within the IF bandwidth. The derivation of the receiver gains is given in Chapter 9. At Bure, the receivers and the IF chain are very stable and these values are constant with time (and equal to $G_i^{UL}$, $G_i^{U}$ and $G_i^{L}$, respectively, since we also neglected their frequency dependence within the IF bandwidth). They are measured at the beginning of each project on a strong astronomical source. Moreover in Eq.12.3-12.4, we use the fact that for a given tuning, only the receiver gains and the atmospheric absorption have a significant dependence as a function of frequency.

Section 12.2 will focus on the corrections for the atmospheric absorption ( $T_{cal_i}^U(t)$, $T_{cal_i}^L(t)$) and the possible biases they can introduce in the amplitude.

In the equations above, the amplitudes can be expressed either in Kelvin (antenna temperature scale, $T_A^*$, $\eta_f=\eta_b$) or in Jy (flux density unit, 1 Jy $= 10^{-26}$ Wm$^{-2}$Hz$^{-1}$). The derivation of the conversion factor between Jy and K, in Jy/K, $\ensuremath{\mathcal{J}}_{iS}$ (single-dish mode) and $\ensuremath{\mathcal{J}}_{iI}$ (interferometric mode) and its biases will be detailed in section 12.3 which is devoted to the flux density calibration.

Finally Section 12.4 will deal with the understanding of the terms $\Gamma _i(t)$ and $f$, the amplitude calibration of interferometric data.