1 Introduction

Observations with a single dish radio telescope are made with a finite size beam of which the theoretical pattern is often degraded by imperfections of the telescope, primarily deformations of the main reflector surface. A knowledge of the actual beam profile is required for the reduction of astronomical observations and estimates of a possible improvement of the telescope. The theoretical beam pattern can be calculated from the Theory of Physical Optics (Born $\&$ Wolf 1980; Love 1978; Christiansen $\&$ Högbom 1985; Kraus 1986); the beam degradation can be calculated from the Theory of Aberrations (Born $\&$ Wolf 1980) and the Antenna Tolerance Theory (Scheffler 1962; Ruze 1952, 1966; Shifrin 1971; Baars 1973), both specified by a few basic parameters which must be determined from measurements of the wavefront error topography or the beam pattern itself. The wavefront (reflector surface) error topography can be derived, for instance, from holography measurements (Morris et al. 1988; Whyborn $\&$ Morris 1995); the actual beam pattern can also be derived, for instance, from scans across a strong point-like radio source or a satellite beacon, or from scans across the limb of the Moon and the Sun (Horne et al. 1981; Lindsey $\&$ Roellig 1991). The measured beam pattern reveals, in general, the influence of spatially large-scale and small-scale wavefront deformations. Large-scale deformations distort the central part of the beam; small-scale deformations produce one, or several, underlying, extended error beams. We analyze total power scans across the Moon at 3.4 mm (88 GHz), 2.0 mm (150 GHz), 1.3 mm (230 GHz), and 0.86 mm (350 GHz) wavelength, and provide in addition to the earlier investigation of Garcia-Burillo et al. (1993) the parameters of an analytic expression of the IRAM 30-m telescope beam as required for the reduction of astronomical observations, in particular of extended sources. (For a description of the 30-m telescope and its behaviour see Baars et al. (1987, 1994) and Greve et al. (1993, 1996a, 1998)).

This publication consists of two parts. The first part explains the theory of beam degradation from several surface error distributions, and we confirm this theory with multi-wavelength beam patterns of the 30-m telescope derived from Moon limb scans observed before July 1997 (Sects. 2,3). The second part explains the result of the latest surface adjustment (July 1997), and we provide the current parameters for calculation of the 30-m telescope beam (Sects. 4-6). In detail, Sect. 2 summarizes the antenna tolerance theory for a combination of several large-scale and small-scale wavefront (reflector surface) deformations, as appropriate for the understanding of the 30-m telescope and other telescopes of similar reflector design. In this theory we use the deformation correlation length(s) to anticipate the structure of the degraded beam from details of the reflector surface construction. We explain in Sects. 3.1-3.3 how we derive in an empirical way the parameters of the degraded beam from the comparison of observed and calculated scans across the limb of the Moon, taken around New Moon (mostly day time) and Full Moon (night time). In particular we confirm the wavelength scaling of the error beam(s). In Sect. 3.4 we show the reflector surface error correlation function, derived from holography measurements, which confirms in an independent way the correlation lengths used in the analysis of the Moon scans. We explain in Sect. 3.5 in which way the standard Ruze relation is modified for the case of several error distributions. In Sect. 4 we explain the surface precision obtained from the July 1997 panel frame adjustment (Morris et al. 1996, 1997). Sect. 5 shows the current beam patterns of the 30-m telescope at 3.4 mm, 2.0 mm, and 1.3 mm, and Sect. 6 gives the current telescope efficiencies. In the Appendix we explain our choice of scans around New Moon and Full Moon. We follow the notation used by Downes (1989) and used at the 30-m telescope (see Mauersberger et al. 1989).