3 Measurement and calculation of G($\epsilon$,$\epsilon$₀)

3.1 Measurement of $G(\epsilon$,$\epsilon$₀)

The experimental determination of $G(\epsilon$,$\epsilon$₀), and $\sigma$$_{\rm g}$($\epsilon$), is preferentially made at short wavelengths despite the higher atmospheric noise and less precise atmospheric correction, in particular at low elevations. Figure 1 shows the gain-elevation dependence at 1.22 mm wavelength obtained from on-off measurements (30'' wobbler throw) with the MPIfR bolometer of the quasar 0923+392 and the extended source Mars (14'' diameter). The bolometer has a taper of $\sim$-10 dB, a bandwidth of $\Delta$$\nu$ $\approx$ 60 GHz, and for observations of the planets an effective wavelength $\lambda$$_{\rm eff}$ $\approx$ 1.22 mm (245 GHz), as calculated from the spectral energy distribution of the planets and the wavelength dependent atmospheric attenuation. Figure 1 shows also the gain-elevation dependence calculated from Eq. (2) and Eq. (3) for the best-fit values $\sigma$(0) = 0.085 mm and $\sigma$(90) = 0.075 mm derived from the lsq-fit of the 245 GHz bolometer measurements, using R = 0.9. The figure shows furthermore the gain-elevation dependence derived from the FE calculations. Comparison of these curves shows that the rms-values $\sigma$$_{\rm g}$($\epsilon$) predicted from the FE calculations are $\sim$10 - 15% lower than those derived from the radiometric measurements.

Inserted in Fig. 1 is the gain-elevation dependence for 3 mm (100 GHz), 2 mm (150 GHz), and 0.86 mm (350 GHz) wavelength, derived from scaling of the empirical 1.22 mm gain-elevation dependence by using the best-fit values $\sigma$(0) and $\sigma$(90) in Eq. (2) and Eq. (3).

  

Figure 1: Gain-elevation dependence measured with the MPIfR bolometer at 1.22 mm (245 GHz) on the quasar 0932+392 [open circles] and Mars (14'') [dots] (Feb. 1995). The heavy line shows the best-fit gain-elevation dependence Eq. (3) derived from these measurements; the dashed line shows the corresponding gain-elevation dependence derived for the surface deformations predicted from the finite element calculations. The other lines show the gain-elevation dependence at 3 mm (100 GHz): a), 2 mm (150 GHz): b), and 0.86 mm (350 GHz): c), derived from scaling of the 1.22 mm gain-elevation dependence (heavy line)

  

Figure 2: Beam pattern $\cal F$($\theta$$_{\rm A}$,$\theta$$_{\epsilon}$,$\epsilon$) of the IRAM 30-m telescope calculated for $\lambda$ = 1.30 mm (230 GHz) and the elevation angle 0$\hbox{$^\circ$}$ a), 22.5$\hbox{$^\circ$}$ b), 43$\hbox{$^\circ$}$ c), 67.5$\hbox{$^\circ$}$ d), and 90$\hbox{$^\circ$}$ e). The reflector is optimized at $\epsilon$0 = 43$\hbox{$^\circ$}$ so that at this elevation the calculated reference beam is the perfect beam pattern. Heavy contours: levels at -3 dB, -6 dB, and -9 dB; thin contours: levels at -12 dB, -15 dB, -18 dB, and -21 dB

3.2 Calculation of G($\epsilon$,$\epsilon$₀)

In a diffraction calculation we derived the illumination weighted beam patterns $\cal F$($\theta$$_{\rm A}$,$\theta$$_{\epsilon}$,$\epsilon$) ($\theta$$_{\rm A}$ = azimuth direction, $\theta$$_{\epsilon}$ = elevation direction) degraded by the wavefront deformation of the homology deformations $\delta$$_{\rm g}$($\epsilon$) predicted from the FE calculations, which were scaled in amplitude by $\sim$10 - 15$\%$ to match the empirical values $\sigma$$_{\rm g}$($\epsilon$). As example, Fig. 2 shows for several elevation angles the calculated beam pattern at $\lambda$ = 1.30 mm (230 GHz), i.e. at the wavelength of the ¹²CO(2-1) line. From these and similar beam patterns we calculated for disk-like and constant surface brightness sources of diameter $\theta$$_{\rm S}$ [$\Pi$($\theta$$_{\rm A}$,$\theta$$_{\epsilon}$) = 1 for $\theta$$_{\rm A}$² + $\theta$$_{\epsilon}$² $\leq$ ($\theta$$_{\rm S}$/2)² and $\Pi$(u,v) = 0 for ($\theta$$_{\rm S}$/2)² < $\theta$$_{\rm A}$² + $\theta$$_{\epsilon}$²] the received power

$$\begin{eqnarray} {\cal P}({\epsilon},{\theta}_{\rm S}) &=& {\int}{\int}_{\theta(... ...}_{\epsilon}'){\rm d}{\theta}_{\rm A}' {\rm d}{\theta}_{\epsilon}'\end{eqnarray}$$ (5)

and the corresponding gain-elevation dependence

$$G({\epsilon},{\epsilon}_0,{\theta}_{\rm S}) = {\cal P}({\epsilon},{\theta}_{\rm S})/{\cal P}({\epsilon}_0,{\theta}_{\rm S}).$$ (6)

Table 2 summarizes the on-axis gain-elevation dependence $G(\epsilon$,$\theta$$_{\rm S}$ = 0) $\equiv$ G₀($\epsilon$) derived from the diffraction calculations, and to be applied on the IRAM 30-m telescope for observations at $\lambda$ = 3 mm (100 GHz), 2 mm (150 GHz), 1.30 mm (230 GHz), and $\lambda$ = 0.86 mm (350 GHz). For an estimated error of $\pm$10$\%$ in the values $\sigma$$_{\rm g}$($\epsilon$), the accuracy of the values G₀ is $\pm$2$\%$ at long wavelengths (3 mm - 2 mm) and $\pm$5$\%$ at short wavelengths (1.3 mm - 0.8 mm). As evident from Fig. 1 and Table 2, the gain-elevation dependence at 3 mm wavelength is only a few percent and usually below the accuracy of the measurements. The small difference between the curves of Fig. 1 and the data of Table 2 is due to the empirical determination and the calculation of $G(\epsilon$,$\epsilon$₀); however, the difference is below the accuracy of the measurements.

  

Table 2: On-axis gain-elevation dependence G₀($\epsilon$,$\epsilon$₀) for $\lambda$ = 3 mm (100 GHz), 2 mm (150 GHz), 1.3 mm (230 GHz), and 0.86 mm (350 GHz) [receivers of $\sim$-13 dB edge taper]. The accuracy of $\sigma$$_{\rm g}$ and R$\sigma$$_{\rm g}$ is $\sim$$\pm$10$\%$

An extended source is covered by a larger fraction of the beam than a point-source and hence should show a smaller gain-elevation dependence. To quantify this effect, we use the loss in gain

$${L}({\epsilon},{\theta}_{\rm S}) = 1 - {G}({\epsilon},{\theta}_{\rm S})$$ (7)

and display in Fig. 3 the relative loss in gain $\cal L$($\epsilon$,$\theta$$_{\rm S}$) between a point-source ($L(\epsilon$,0)) and an extended source ($L(\epsilon$,$\theta$$_{\rm S}$)) calculated from

$${L}({\epsilon},{\theta}_{\rm S})/{L}({\epsilon},0) = {\cal L}({\theta}_{\rm S}) .$$ (8)

The calculations show as important result that the quantity $\cal L$($\theta$$_{\rm S}$) is nearly independent of the wavelength and the elevation of the observation. Using the data of Fig. 3 and the entries of Table 2, the actual value $G(\epsilon$,$\epsilon$₀,$\theta$$_{\rm S}$) for an extended source is

$${G}({\epsilon},{\epsilon}_0,{\theta}_{\rm S}) = 1 - {\cal L... ...theta}_{\rm S})\,{L}({\epsilon},0) = 1 - {\cal L}(1 - {G}_{0}).$$ (9)

Figure 3 shows that 0.95 $\cal L$($\theta$$_{\rm S}$) for sources not exceeding $\theta$$_{\rm S}$ 2$\theta$$_{\rm b}$ so that from Eq. (9) for these sources $G(\theta$$_{\rm S}$) $\approx$ G₀, within 5 - 10$\%$. The gain-elevation dependence is weaker for more extended sources since for these $\cal L$($\theta$$_{\rm S}$) < 0.9. The gain-elevation dependence is negligible for sources of diameter 8$\theta$$_{\rm b}$ $\theta$$_{\rm S}$ since for these $\cal L$($\theta$$_{\rm S}$) 0.1 and $G(\theta$$_{\rm S}$) $\approx$ 1.

We explain in the following example how to correct observations with the 30-m telescope. Suppose a source of $\theta$$_{\rm S}$ = 30'' diameter, observed at 1.3 mm wavelength and 20$\hbox{$^\circ$}$ elevation, has a flux corrected for atmospheric attenuation of S'($\epsilon =20\hbox{$^\circ$}$) = (2 k/A)$F_{\rm eff}$$T_{\rm A}^{*}$/$\epsilon$$_{\rm ap}$ = 10 Jy. The diameter of the source is 30''/$\theta$$_{\rm b}$ $\approx$ 3 beamwidth. From Fig. 3 we find $\cal L$(3) = 0.70, so that from Eq. (9) and the value G₀(20$\hbox{$^\circ$}$) = 0.92 of Table 2 we have G(30'') = 1 - 0.70(1 - 0.92) = 0.94. The homology-corrected telescope-independent flux is S = S'(20$\hbox{$^\circ$}$)/ 0.94 = 10.6 Jy. A 10 Jy point-source observed under similar conditions has the intrinsic flux S = 10/0.92 = 10.9 Jy.

  

Figure 3: Relative gain-elevation dependence $\cal L$($\theta$S) (Eq. (8)) as function of source size $\theta$S (in units of beamwidth $\theta$b), for $\lambda$ = 2.0 mm (150 GHz): triangles, $\lambda$ = 1.30 mm (230 GHz): squares, $\lambda$ = 0.86 mm (350 GHz): circles. The dashed vertical line indicates the diameter of the full beam $\sim$2.4$\theta$b, i.e. the position of the first minimum