2 The gain-elevation dependence G($\epsilon$,$\epsilon$₀)

Following Ruze's (1966) antenna tolerance theory, the normalized elevation-dependent on-axis antenna gain of the reflector optimized at $\epsilon$₀ is

$${G}({\epsilon},{\epsilon}_0) = {\rm exp}[-(4{\pi}{R}{\sigma}_{\rm g}({\epsilon})/{\lambda})^{2}]$$ (3)

where the factor R ($\approx 0.8 - 0.9$) takes into account the steepness of the reflector and the illumination taper of the receiver so that R$\sigma$$_{\rm g}$ is the radio-effective surface deformation (Greve & Hooghoudt 1981). The function G($\epsilon$,$\epsilon$₀) depends exclusively on the construction-specific values $\sigma$$_{\rm g}$($\epsilon$)which do not change with time and surface adjustment. The homology-corrected telescope-independent flux [S] of a source is obtained from its calibrated flux [S'($\epsilon$)] measured at the elevation $\epsilon$ by application of the correction G^-1($\epsilon$,$\epsilon$₀) so that

$${S} = {S}'({\epsilon})\ {G}^{-1}({\epsilon}, {\epsilon}_0).$$ (4)

The literature does not provide explicit information how to deal with extended sources. We will show that for the specific case of the IRAM 30-m telescope, the on-axis gain-elevation dependence $G(\epsilon$,$\epsilon$₀) holds also for extended sources not exceeding in diameter approximately two half-power beamwidths (i.e. $\theta_{\rm S}$ 2$\theta$$_{\rm b}$); a weaker gain-elevation dependence applies for more extended sources (i.e. 2$\theta$$_{\rm b}$ < $\theta$$_{\rm S}$). The half-power beamwidth (FWHP) of the 30-m telescope is $\theta$$_{\rm b}$ = 1.16$\lambda/D$ [rad], with $\lambda$ the wavelength of observation and D the diameter of the reflector; the relevant beamwidths are given in Table 1. Note that the diameter of the full beam is $\sim$2.4$\theta$$_{\rm b}$.

  

Table 1: Beamwidth $\theta$$_{\rm b}$ of the 30-m telescope for receivers of $\sim$-13 dB edge taper, and the largest source diameter $\theta$^* [planet] suitable for determination of the gain-elevation dependence at the specific wavelength