3 The IRAM 30-m telescope beam structure

We analyze multi-wavelength beam profiles of the 30-m telescope obtained from Moon limb scans, and demonstrate that the antenna tolerance theory of several error distributions explained above gives consistent results for the beam structure and the efficiencies.

3.1 Measurements

We applied the method of total power Moon limb scans to derive the beam at 3.4 mm, 2 mm, 1.3 mm, and 0.86 mm wavelength, using SIS receivers of 500 MHz bandwith and similar Gaussian illumination of $\sim -13$ dB edge taper. The scans of 3600'' length, across the Moon of $\sim\! 1800''$ diameter, allow an investigation of the beam to a distance of $\sim$ $\pm$ 900'' from the beam axis. After focusing the telescope, the data of a scan were taken on-the-fly in 120 s at a spatial resolution of 6'' (200 ms sampling rate). Linear baselines, determined at the outermost $\pm30''$, were subtracted from the scans. The observations were made under very stable atmospheric conditions so that the measurements are reliable to a level of approximately -30 dB, i.e. $\sim\!0.1$% of the peak intensity. The measurements were made at intermediate elevations where the homology deformations of the reflector are small (Greve et al. 1998). For the analysis we have taken scans around New Moon (mostly day time) and Full Moon (night time) which provide for the analysis the advantage of a simple brightness distribution of the Moon (Sect. 3.2, Appendix), and also the possibility to investigate the day and night time performance of the telescope.

Figure 2 shows a 3.4 mm and a 0.86 mm total power scan across the Moon, taken around New Moon. The distortion of the intrinsically sharp limb is due to the finite beam of the telescope.

  

Figure 2: Total power scans across the Moon. The 3.4 mm observation (thick line) was made 0.5 days after New Moon (6 Sep. 1994); the 0.86 mm observation (thin line) was made 4 days before New Moon (5 Mar. 1997). Note the decrease in sharpness of the limb with decreasing wavelength. (The structure of the 3.4 mm scan shows features of the Moon's surface)

  

Figure 3: Composite profiles [$f_{\rm M}(u)$] (i.e. scans across the left and right limb of the Moon and differentiated along the scan direction) observed around New Moon. For each wavelength we show two measurements (solid dots, open circles) and error bars to demonstrate the repeatability of the observations. The solid lines show the synthetic best-fit profiles [$f_{\rm S}(u)$], extrapolated somewhat beyond the measurements. The observations at 3.4 mm, 2 mm, and 1.3 mm are made simultaneously; at 0.86 mm wavelength only observations 4 days before New Moon are available. (The data refer to the situation before July 1997)

3.2 Moon scans and profiles

Within the limitations of the antenna tolerance theory, the beam pattern (Eq. (16)) is circular symmetric so that any scan direction across the Moon, around New Moon and Full Moon, may be used to derive the beam parameters. For this we use an empirical approach and compare observed scans across the limb [$P_{\rm M}(u)$, differentiated in scan direction called $f_{\rm M}(u)$]with synthetic scans [$P_{\rm S}(u)$, differentiated in scan direction called $f_{\rm S}(u)$]. The synthetic scans are calculated from the convolution of the beam $\cal F$($\theta$) $\equiv$ ${\cal F}(u,v)$(Eq. (16)) and the brightness distribution $T_{\rm M}(u,v)$ of the Moon. For measurements around New Moon and Full Moon we use the symmetric brightness distribution $T_{\rm M}$ = T₀$\Pi (u,v)$ with T₀ the average brightness and

$${\Pi}({ u,v}) = 1 + { C}_{\rm M}\,(1 - {\rho}^{2}),\,\, {\rho} = {\sqrt{({ u}^{2} + { v}^{2})}}/(15{'})$$ (17)

for $u^{2} + v^{2} \leq$ (15')², and zero outside. For New Moon we use $C_{\rm M} = 0$, for Full Moon we use $C_{\rm M}$ = 0.5 (see Fig. 10a below). With u measured near culmination approximately in East-West direction through the center of the Moon, and v perpendicular to this direction, a synthetic total power scan is

$${ P}_{\rm S}({\rm u})\ {\propto}\ { T}_{0}{\int}{\int}{\cal F} ({ u',v'}){\Pi}({ u - u',v'})\,{\rm d}u'{\rm d}v'.$$ (18)

In a similar way as applied to the observed profiles $f_{\rm M}(u)$, the synthetic composite profiles

$${ f}_{\rm S}({ u}) = {\rm d}P_{\rm S}{ (u)/{\rm d}u}$$ (19)

are constructed from sections $f_{\rm S}(u)$ with the left and right wing of the profile taken outside (15' $\leq$ ${\mid} u {\mid}$) the right and left limb of the Moon. For observations around New Moon and Full Moon, the sections $f_{\rm S}(u)$ of the left ($\sim$ East) and right ($\sim$ West) wing are symmetric and may be added to obtain a composite profile (see Figs. 10b,c below). Figure 3 shows for New Moon the observed and the best-fit synthetic profiles, $f_{\rm M}(u)$ and $f_{\rm S}(u)$, respectively; Fig. 4 shows a similar profile at 2 mm wavelength observed at Full Moon. We emphasize that the differentiation (Eq. (19)) is not a deconvolution so that the profiles $f_{\rm M}$ and $f_{\rm S}$ shown in Fig. 3 and Fig. 4 do not exactly represent the true beam pattern of the telescope.

  

Figure 4: Composite profiles $f_{\rm M}(u)$ derived from observations at 2 mm wavelength at Full Moon (dots) and New Moon (open circles, see Fig. 3). The heavy line and the thin line show the best-fit synthetic profile $f_{\rm S}(u)$ for Full Moon (night time) and New Moon (day time), respectively. The additional degradation during day time (New Moon) is a partially transient effect and is in many cases smaller than shown here. (The data refer to the situation before July 1997)

3.3 The beam parameters

We determined the best-fit beam pattern $\cal F$(u,v) (Eq. (16)) in an empirical way by minimizing the difference ${\mid} { f}_{\rm M}({ u}) - { f}_{\rm S}({ u}){\mid}$ of the observed and synthetic profiles, using the measured main beam widths $\theta$$_{\rm b}$ and measured main beam efficiencies $B_{\rm eff}$, the calculated diffraction pattern $\cal A$$_{\rm T}$ (Sect. 5), the anticipated correlation lengths and corresponding error beams $\cal F$$_{\rm e}$, and also the fact that the beam structure should scale with wavelength. Figure 3 and Fig. 4 show the best-fit synthetic profiles $f_{\rm S}(u)$; the corresponding beam parameters are shown in Fig. 5.

A consistent interpretation of the measurements shown in Fig. 3 (Fig. 4) is obtained for a beam which consists of the diffracted beam $\cal A$$_{\rm T}$ and at least two persistent error beams (EB) with correlation lengths 0.3 (^+0.2_-0.1) m and 1.5 (^+0.5_-0.1) m. These correlation lengths are identified with the anticipated panel surface errors ($L_{\rm p}$) and the panel frame adjustment errors ($L_{\rm a}$), respectively. In addition, there exists a partially transient degradation close to the main beam which may be interpreted as an error beam (1st EB) due to large-scale deformations with correlation length 3 (⁺¹_-0.5) m. This transient degradation is probably due to known transient residual thermal deformations of the reflector surface and the telescope structure (Greve et al. 1993, 1994b). When present, the transient thermal deformations are especially noticed during day time and sunshine (for instance as focus changes) and the comparison of measurements around New Moon and Full Moon illustrates this effect (see Fig. 4). From the correlation lengths mentioned above and the measured ratios $\cal F$$_{\rm e,i}$(0)/$\cal F$$_{\rm c}$(0) [Eq. (16), Eq. (11)] we derive, for the time before July 1997, the rms-values $\sigma$$_{\rm l}$(1st EB) $\approx$ 0.03 - 0.06 mm (see the footnote of Table 1), $\sigma$$_{\rm a}$(2nd EB) $\approx$0.07 mm, and $\sigma$$_{\rm p}$(3rd EB) $\approx$ 0.055 mm. The illumination weighted rms-value $\sigma$$_{\rm T}$ = R$\sqrt{{\sigma}_{1}^{2} + {\sigma}_{\rm a}^{2} + {\sigma}_{\rm p}^{2}}$ $\approx$ 0.8 $\times$ (0.095 - 0.11) $\approx$ (0.075 - 0.085) mm derived in this way is consistent with the holography measurements (D. Morris, priv. comm.) and the efficiency measurements of the planets (Kramer 1997), as it should be the case.

  

Table 1: Beam parameters of the IRAM 30-m telescope (after July 1997)

Figure 5 shows the widths $\theta$$_{\rm e}$ and amplitudes $a_{\rm e}$ of the error beams as derived from the measurements and calculated from Eqs. (11, 12) for $\epsilon$₀ = 0.62 (Greve et al. 1994c) and the correlation lengths L _i and rms-values $\sigma$_i determined above. This figure confirms the wavelength-scaling of the error beams. When present, the wavelength-scaling of the 1st error beam indicates that the associated large-scale deformations behave like random deformations with correlation length $L_{\rm l}$ = 2.5 - 3.5 m, of which there are approximately ($D/L_{{\rm l}}$)² $\approx$ 50 elements covering the aperture. This large number of deformation patches allows the application of statistical calculations (Greve 1980).

  

Figure 5: Best-fit values (solid dots) of the widths $\theta$$_{\rm e}$ and amplitudes $a_{\rm e}$ of the error beams deduced from the observed profiles of Fig. 3, and their approximations (solid lines) from Eq. (12) and Eq. (11), respectively. EB denotes the different error beams. The 1st EB is a partially transient phenomenon; the data (open circles) are shown for New Moon day time measurements. The heavy lines are the current parameters of the beam pattern. The dashed line (2nd EB) and the related measurements (solid dots) refer to the situation before the adjustment of July 1997

3.4 The empirical correlation function

The wavefront errors ($\delta$$_{\varphi}$) derived from holography measurements corrected for defocus and coma (see footnote 8) are interpreted to be due to surface errors ($\delta$) of the main reflector. This surface error distribution [$\delta$] can be used to derive the surface error correlation function [$C_{\delta}$] and from this, in an independent way, the correlation length(s) [L]. Holography measurements at 7 mm (43 GHz) wavelength are regularly made at 43$\hbox{$^\circ$}$ elevation using the geostationary satellite ITALSAT (Morris et al. 1996, 1997). At this elevation the reflector surface is optimized and free of homology deformations (Greve et al. 1998). We derived the correlation function $C_{\delta}$ (Eq. (14)), shown in Fig. 6, from $32 \times 32$ pixel holography measurements made Oct. 1993, Mar. 1994, Nov. 1994, and from a $128 \times 128$ pixel holography measurement of 0.24 m (= D/128) spatial resolution made Sep. 1996 (Morris et al. 1996). When compared with Gaussian correlation length distributions $\exp[-(d/L)^{2}$] (Eq. (9)), the empirical correlation functions show clearly the influence of the panel frame misalignment (2 in Fig. 6b), with correlation length $L_{\rm a} = 1.5 - 2.0$ m, and of the large-scale deformations (1 in Fig. 6b), with correlation length $L_{\rm l} = 2.5 - 3.5$ m. The resolution of the 128 $\times$ 128 pixel holography map with one or two measurements per correlation cell $L_{\rm p} = 0.3 - 0.5$ m is too low to clearly indicate the Gaussian component of the panel deformations (3 in Fig. 6b). However, the panel surface errors of correlation length $L_{\rm p}$ are clearly illustrated in Fig. 1.

  

Figure 6: Correlation functions derived from holography measurements, and Gaussian correlation length distributions calculated from Eq. (9). a) Full extent of the empirical correlation functions, for Oct. 1993, Mar. 1994, Nov. 1994: thin lines; for Sep 1996: thick line. b) Detailed view of the empirical correlation function of Sep. 1996 (histogram) and the individual Gaussian correlation functions calculated for $L_{\rm p}$ = 0.3 m: curve 3; $L_{\rm a}$ = 1.5 m: curve 2; and $L_{\rm l}$ = 2.5 m: curve 1. The normalization is made to curve 2 (2nd EB). The lower curve 1 is due to a low night-time rms-value of the 1st EB

3.5 The modified Ruze relation

The aperture efficiency $\epsilon$$_{\rm ap}$ of a reflector with several independent error distributions is (Ruze 1966; Shifrin 1971; Baars 1973)

$${\epsilon}_{\rm ap} = {\epsilon}({ R})_{\rm ap} + {\epsilon}({ L})_{\rm ap}$$

$$= {\epsilon}_{0}\,{\rm exp}[-{\sum}_{ i}({\sigma}_{\varphi,i... ... - {\rm exp}[-({\sigma}_{\varphi,i})^{2}])({ L}_ { i}/{ D})^{2}$$ (20)

where

$${\epsilon}({ R})_{\rm ap} = {\epsilon}_{0}\,{\rm exp}[-{\su... ...)^{2}] = {\epsilon}_{0}\,{\rm exp}[- ({\sigma}_{\varphi})^{2}]$$ (21)

is the standard Ruze relation and $\epsilon$( L)$_{\rm ap}$ a correction taking into account the scale of the surface errors. It is evident from Eq. (20) that in particular large-scale deformations (large L) contribute to $\epsilon (L)_{\rm ap}$, increasing the efficiency $\epsilon (R)_{\rm ap}$. When using in Eqs. (20, 21) the measured values ($\sigma$ _i, L _i) given in Table 1, and $\epsilon$₀ = 0.62, we find that in the wavelength region from 0.8 mm to 3 mm the quantity $\epsilon (L)_{\rm ap}$ does not exceed $1 - 2\%$, which is below the accuracy of the measurements. The difference between the Ruze relation (L = 0) and the complete expression Eq. (20) is so small so that the Ruze relation can be used for evaluation of the 30-m reflector.