10.1 Definition and Formalism

From R.Lucas lecture 7, Eq. 7.1), the baseline-based observed visibility $\ensuremath{\widetilde{V}} _{ij}(t)$ is linked to the true visibility V_ij of the source by:

$$\ensuremath{\widetilde{V}} _{ij}(t) = {\ensuremath{\mathcal{G}} _{ij}}V_{ij}+ \epsilon_{ij}(t) + \eta_{ij}(t)$$ (10.1)

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

$${\ensuremath{\mathcal{G}} _{ij}}= g_i(t)g^*_j(t)=a_i(t)a_j(t)e^{i(\phi_i(t)-\phi_j(t))}$$ (10.2)

Therefore, for antenna i, the antenna-based amplitude correction for the Lower sideband a_i^L is given by

$$a_{i}^L(t) = T_{cali}^L(t) G_i^L(\nu,t) \ensuremath{\mathcal{B}} _i(t)$$ (10.3)

and for the Upper sideband:

$$a_{i}^U(t) = T_{cali}^U(t) G_i^U(\nu,t) \ensuremath{\mathcal{B}} _i(t)$$ (10.4)

where T_cali^U and T_cali^L are the corrections for the atmospheric absorption (see lecture 8 by M.Guélin), in the Upper and Lower sidebands respectively. $\ensuremath{\mathcal{B}} _i$ the antenna gain (affected by pointing errors, defocusing, surface status and systematic elevation effects). Note that Eqs.10.3-10.4 do not include the decorrelation factor f (see lecture 7 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 7. 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). They are measured at the beginning of each project on a strong astronomical source. Moreover in Eq.10.3-10.4, 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 10.2 will focus on the corrections for the atmospheric absorption ( T_cali^U(t), T_cali^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^*, $\eta_B=\eta_F$) or in Jy (flux density unit, 1 Jy = 10^-26 Wm^-2Hz^-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 10.3 which is devoted to the flux density calibration.

Finally Section 10.4 will deal with the understanding of the terms $\ensuremath{\mathcal{B}} _i(t)$ and f, the amplitude calibration of interferometric data.