4 Qualitative method with three beams observations

Data obtained from only three beams are not sufficient to determine unambiguously the radio burst position, unless the assumption of a small (compared to beam) emitting source is made.

A qualitative method was developed based on the idea that the source angular extent should be related to antenna temperature ratios obtained with different beams. These ratios lead to a contrast criterion, which should increase for smaller sources. That is to say: if an emitting source is small compared to the HPBW, the contrast should be higher than for an extended emitting region at the same location. For three beams observations, we can define a contrast parameter K as:

$$% K=\ln\left (\frac{T_{\rm H}}{T_{\rm L}} \right)+ \ln\left (... ... = \ln\left (\frac{T_{\rm H} T_{\rm I}}{T_{\rm L}^2} \right );$$ (9)

where $T_{\rm H}$, $T_{\rm I}$, and $T_{\rm L}$ are the high, intermediate and low observed antenna temperatures respectively.

Figure 10: a) Three beams disposition at 48 GHz. The dotted area shows the location of radio sources for which there is no signal exceeding -15 db from sidelobes. b) Variations of the contrast K as a function of $HPW_{\rm O}$, for different location of the emitting sources within the dotted area. The line $K=K_{\min}$ defines 2 regions of contrast values. Only K belonging to the upper part indicates a small source, compared to the HPBW. The thick horizontal line represents the range of uncertainty in $HPW_{\rm O}$ for a given $K_{\rm obs}$. The dot in this line represents the mean value

We have computed the expected contrast K for different positions within three beams (Fig. 10a) and different source sizes, assuming Gaussian source shapes. The source locations (dotted area in Fig. 10a) have been chosen such that, at each location the antenna temperatures $T_{\rm H}$, $T_{\rm I}$, and $T_{\rm L}$ satisfy the condition:

$$% 0.06 T_{\rm H} < T_{\rm L} < T_{\rm I} < T_{\rm H};$$ (10)

assuring that the data have no contamination by signals from sidelobes. The contrast K has been computed at each location, and for different values of $HPW_{\rm O}$ in the range $HPBW - 2 \times HPBW$, corresponding to a point emitting source and an emitting source of extension $HPW_{\rm S} \sim \sqrt{3} \times HPBW$, respectively. Here we have assumed that the beam size is 1.9 arcmin (see Sect. 2). The results are shown in Fig. 10b. Each curve shows, for a given emitting source location, the evolution of K versus $HPW_{\rm O}$ (in units of HPBW). The limiting curve (thick line in the plot) corresponds to the highest K for each source size and was computed analytically for the actual geometry. The condition $HPW_{\rm S}=HPBW$ ( $HPW_{\rm O}=\sqrt{2} \times HPBW$), represented by the vertical line, defines a minimum value $K_{\min}$above which the size of the emitting source is necessarily smaller than the HPBW. Figure 10b shows clearly that if $K < K_{\min}$, there is an ambiguity in comparing the source size with the HPBW. The bottom left quadrant shows small (compared to HPBW) emitting sources located near the origin 0, where the contrast is below the limit $K_{\min}$ which does not allow to distinguish between small (compared to HPBW) and extended emitting sources. In this case, source positions using three beams observations will not be computed.

In summary, the qualitative method should be used as follows with three beams observations: i) compute a contrast value $K_{\rm obs}$, ii) refer to Fig. 10b to compare $HPW_{\rm O}$ and HPBW (same as comparing $K_{\rm obs}$ to $K_{\min}$), iii) if $HPW_{\rm O} \le \sqrt{2} \times HPBW$ the position of the burst is computed, if not the position of the burst is not estimated. Our uncertainty in the half power width observed is the range defined by $HPW_{\rm O} = HPBW$ and the $HPW_{\rm O}$ which corresponds to $K = K_{\rm obs}$ in the limiting curve. In Fig. 10b we show, for a given $K_{\rm obs}$, the range of uncertainty. In order to compute positions we adopt $HPW_{\rm O}$ equals to the mean value of this range.

Figure 11: Discrepancies measured as angular distances between the positions computed with 4 beams and with beams 2-3-5 ( qualitative method) for the time structures showed in Fig. 7) of the burst on 30 December 1990 in terms of mean contrast K. Dots are the absolute discrepancies. Vertical bars show the uncertainties

Figure 12: Discrepancies measured as angular distances between positions computed by Costa et al. (1995) and the new method using four beams for the time structures labeled in Fig. 8 of the burst on 30 December 1990 in terms of mean $HPW_{\rm O}$. Larger discrepancies are found for larger source angular extents $HPW_{\rm O}$

As an illustration we have applied the qualitative method to the 30 December 1990 event, which has been observed with four beams (see previous section), allowing an unambiguous determination of the burst position as well as its instantaneous angular extent. For the qualitative test we used the antenna temperatures observed with beams 2, 3, and 5. Figure 11 shows the absolute discrepancies as large dots, computed as angular distances, between positions computed with the method described in Sect. 2 and the qualitative method for the time periods A, B, C (Fig. 7). Angular positions for peaks 1, 12, 13, 14 and 16 are not computed because have contrast lower than $K_{\rm min}$; while angular position for peak 15 is not considered because the condition given by the Eq. (10) is not fulfilled. In all the computed cases, discrepancies are less than 5 arcsec and the general trend is: for larger contrast K the discrepancy becomes smaller. In Fig. 11 vertical bars show the uncertainties due to the range in $HPW_{\rm O}$ allowed by each $K_{\rm obs}$. Here also the general trend is an inverse relation between contrast and uncertainty.