Appendix A: Limb scans at various phases of the Moon

For the analysis of the beam structure we have used scans across the Moon taken around New Moon and Full Moon. In order to derive in an empirical way the parameters of the beam pattern, we have constructed from the observed total power scans $P_{\rm M}(u)$ the differentiated, composite profiles $f_{\rm M}(u)$ which we compared with synthetic profiles $f_{\rm S}(u)$ (Eq. (19)), obtained from synthetic total power scans $P_{\rm S}(u)$(Eq. (18)). There are two reasons for selecting observations around New Moon (day time) and Full Moon (night time): (a) at these phases the Moon's brightness distribution is symmetric with respect to its center and easily cast into an analytic form (Eq. (17)) for calculation of the synthetic profiles $P_{\rm S}(u)$ and $f_{\rm S}(u)$; (b) the observed total power scans $P_{\rm M}(u)$ and the differentiated scans $f_{\rm M}(u)$ are identical for any scan direction through the center of the Moon so that opposite sides of the limb can be used to construct a composite profile $f_{\rm M}(u)$. Both segments of a composite profile are normalized at their center.

We illustrate these remarks with observations. Figure 10a shows 2 mm wavelength total power scans at different phases of the Moon. In these observations, one scan passes through the subsolar point and the center ($\sim$ EW direction at culmination); the other scan is made in orthogonal direction. The figure shows the symmetric brightness distribution around New Moon (age $\approx$ 0 days) and Full Moon (age $\approx$ 15 days). As shown in Fig. 10b and Fig. 10c, at these phases the profiles $f_{\rm M}$(East limb) and $f_{\rm M}$(West limb) are comparable so that the measurements at opposite limbs can be used to construct a composite profile $f_{\rm M}$(u), as done above (see Figs. 3, 4). We find empirically that the difference of the profile sections $f_{\rm M}$(East) and $f_{\rm M}$(West) is below $\sim$ 1 dB, and hence below the accuracy of the measurements, for observations within $\sim$ $\pm$ 1 day of New Moon and Full Moon. Also, at these phases of the Moon the unbalance of the profiles because of temperature gradients across the Moon and limb darkening is smaller than the accuracy of the measurements. From observed and synthesized scans we find that the composite profiles are identical for measurements at New Moon and Full Moon (see Fig. 10c). This allows a comparison of profiles taken at New Moon and Full Moon, as done in Fig. 4. We find also that profiles taken in u-direction or v-direction are similar within $\sim$$\pm$ 1 dB when taken within $\sim$$\pm$ 1 day of a certain phase, as used in Fig. 7.

  

Figure 10: a) Observed (before July 1997) total power scans at 2 mm wavelength; West is to the right, East to the left. The age of the Moon is indicated as days after New Moon (scans of the last half of the lunar cycle are similar, with the EW direction inverted). The profiles show scans $P_{\rm M}(u)$ through the subsolar point (i.e. $\sim$ EW direction at culmination) and the center of the Moon (thick line), and scans in orthogonal direction (thin line). Only scans at New Moon and Full Moon are symmetric. b) Moon scans differentiated along $\sim$ EW direction, i.e. $f_{\rm M}(u)$ = d$P_{\rm M}(u)$du. c) Sections of the composite profile $f_{\rm M}$ with $f_{\rm M}$(West): solid dots, $f_{\rm M}$(East): open circles (folded around the beam axis)

The brightness distribution at other phases of the Moon than New Moon and Full Moon shows a more or less steep gradient and a more or less pronounced crescent (Fig. 10a). The disadvantage of using for the analysis other phases than New Moon and Full Moon lies in the fact that the available models (Krotikov $\&$ Troitskii 1964; Linsky 1966, 1973; Mangum 1993) for calculation of the Moon's millimeter wavelength brightness distribution T₀$\Pi (u,v)$ (Eqs. (17,18)) are not sufficiently precise, and mathematically cumbersome, to produce reliable synthetic profiles $P_{\rm S}(u)$ and $f_{\rm S}(u)$ for comparison with observations.

The mm-wavelength radiation from the edge of the Moon is polarized in radial direction by $\sim\!1\%$ at 1.3 mm (Barvainis et al. 1988). This is a small and systematic effect and need not be considered in the profile analysis, even when using polarized feeds.

Acknowledgements

The holography data were kindly made available by D. Morris. The holography data of 1996 and 1997 were obtained in a larger IRAM campaign including the collaboration of D. Morris, J. Lamb, B. Lazareff, M. Carter, F. Mattiocco, and the staff of IRAM-Granada. D. Teyssier of IRAM-Granada made the new measurements used in Fig. 7; A. Karpov (IRAM) provided and operated the 350 GHz receiver. The data of Fig. 1 are taken from the ARGE-Krupp-MAN documentation of the telescope. We appreciated the discussions with D. Morris and B. Lazareff, and the pointed comments of D. Downes (IRAM) and the referee, R.E. Hills.