1 Introduction

Solar radio observations using large apertures at short microwaves, producing beams smaller than the solar disk, have shown levels of burst activity in millisecond time scales and sensitivity levels of 10^-2solar flux units ( $1~{\rm s.f.u.}=10^{-22}~{\rm W m}^{-2}~{\rm Hz}^{-1}$) which are three to two orders of magnitude smaller compared to the respective lower limits of usual patrol telescopes. (for example Kaufmann et al. 1976, 1980; Hurford et al. 1979; Butz et al. 1976; Gaizauskas & Tapping 1980).

It was shown that for active regions and for bursts occurring on them, which sizes were small compared to the antenna beamwidth, the high sensitivity attained allowed very precise beam pointing, to the level of several arcseconds (Kaufmann et al. 1982).

On the other hand, proposed models of energetic production at the origin of solar flares indicate the crucial importance of the diagnostic of rapid phenomena both in time and space, simultaneously. However, for fast solar burst time developments, producing structures shorter than one second, it is not possible to scan with a single beam the burst source in subsecond time scales, in order to obtain simultaneous informations on intensity and angular positions within the lifetime of a single rapid spike. The existing technique available to obtain high spatial resolution (few arcseconds) is based on aperture synthesis using large interferometric arrays (see for example Marsh & Hurford 1982; Kundu & Lang 1985; Kundu et al. 1990; Bastian et al. 1994). Burst time structures can be shorter than 50 ms which is of the same order of the best time resolution reported so far for VLA solar observations, being most of time much larger than this (e.g. Bastian et al. 1994).

The multiple beam technique, originally introduced by Efanov & Moiseev (1967), was addressed to this observational problem, allowing at the same time the determination of position and flux with a high time resolution. It uses a focal array system of feed-horns and radiometers placed at the focus of a large antenna. The partially overlapping beams track continuously a solar active center. As a burst occurs, with angular dimension small compared to beamwidth, the instantaneous relationship between the antenna temperatures for the different beams allows the determination of the angular position with very high accuracy, limited by tracking and radio-seeing conditions. The system development was accomplished within an agreement between the solar group operating the Itapetinga 13.7-m radome enclosed radio telescope, presently at CRAAE in Brazil, and the solar group from the Institute of Applied Physics, IAP, University of Bern in Switzerland. Five 48 GHz feed-horns/radiometers and data acquisition system were built and developed at IAP and the array was placed at the Cassegrain focus of the Itapetinga dish. The beams are partially overlapping to each other by about 1 arcmin, at the individual beams' half power levels (see Fig. 1).

A brief description of the multibeam 48 GHz feed-horns' and radiometers' setup was given by Georges et al. (1989). The first observations at Itapetinga began in 1989. Some aspects of multiple beam observations and the first results obtained were published elsewhere (Herrmann et al. 1992; Costa et al. 1995). In summary, the system achieve a 1 ms time resolution in the determination of position and flux, with a sensitivity as small as 0.04 s.f.u.

As pointed out above, the multiple beam technique was primarily conceived to study angular positions of solar bursts time structures assumed small compared to the antenna beamwidths. For burst sources components large compared to the beam sizes the multiple beam technique cannot provide positions unequivocally, unless a number of additional assumptions are made. Point-like sources were assumed in the first published analysis (Herrmann et al. 1992, 1997; Costa et al. 1995 and Correia et al. 1995) and definitions of "flux densities" assigned to equivalent point sources producing the measured antenna temperatures for each beam were adopted deriving angular positions for assumed "centroids of burst emissions".

Figure 1: Five beams (circles) arrangement observing a source with a brightness distribution $B(\theta ,\varphi )$ (thick line) which is assumed to have a Gaussian equivalent shape (dashed circle)

The multiple beams actually observe a burst source which brightness distribution is not known in advance. The antenna temperature for each beam can be expressed as:

$$% T_{{\rm A},i} \propto \int\!\!\!\!\int_{\Omega} B(\theta,\phi)P_{{\rm A},i}{\rm d}\Omega$$ (1)

where $P_{{\rm A},i}$ is the beam pattern of the i-th receiver and $B(\theta,\phi)$ is the source brightness distribution in space (see Fig. 1). The solution of the system of Eqs. (1) provide an angular position, which was called as the position of the centroid of emission. Since $B(\theta,\phi)$ is not known, when applying this method to extended sources the position obtained is just an approximate indication of the direction where the brightness is higher. The time variation of the determined position provides "dynamic maps" of the relative changes in that direction along a certain interval of time. It must be clarified that they are not instantaneous "images" of the bursting site.

The analysis is thus dependent on the assumptions made for the distribution $B(\theta,\phi)$. For example, it is not known in advance if $B(\theta,\phi)$ has a pronounced asymmetry in space, or is built up by multiple sources different in extent and position.

The goal of the present study is to develop methods to check whether the bursts sources are small or large compared to the multiple beams of the array as a necessary precondition for instantaneous position and flux determination. The only assumption made is that $B(\theta,\phi)$ is an axially symmetric function. For sake of simplicity we have adopted a Gaussian. Indeed, once the equivalent source extension is found to be smaller compared to the beams, the source actual shape becomes irrelevant for our purpose to determine position. We discuss pronounced asymmetry in Sect. 2.2. For sources small compared to HPBW, sources' positions can be determined unambiguously, and the approximation adopted for $B(\theta,\phi)$ will not produce errors in position bigger than other uncertainties intrinsic to the system. For sources large compared to HPBW, positions cannot be determined, unequivocally.

Figure 2: Schematic representation of the convolution of two identical overlapping Gaussian beams with half power beam width, HPBW (top left) and a Gaussian brightness temperature distribution with a half power width $HPW_{\rm S}$ (top right). The observed convolved patterns have a half power width $HPW_{\rm O} \ge HPBW$(bottom)

The departing point of the new method takes the realistic observed patterns which are actually the convolution between the antenna beam patterns and the source brightness temperature distribution in space. The observed patterns and the brightness distribution are not known in advance (see Fig. 2). Antenna temperatures measured by the different beams can be converted into flux density only for sources small compared to the HPBW. It will be shown that using data from at least four independent beams, assuming that the beam shapes and the sources' brightness distributions can be approximated to Gaussian functions, angular extent can be estimated for the burst sources and positions can be calculated for sources small compared to the beam size.

Data from three beams are not sufficient to define burst extent and their positions cannot be calculated unequivocally. Nevertheless we will discuss particular situations for which data obtained with three beams indicate qualitatively whether or not the burst sources are small compared to the beams. When the tests are consistent to small sources, their positions can be calculated using data from only three beams.

Finally we stress that, for sources found to be small compared to the multiple beams' HPBW, the technique has the unique advantage to determine their fluxes irrespectively from their positions with respect to the beams, or vice-versa.