3 Comparison of the original data to the error beam corrected data

3.1 Velocity integrated maps

Figures 1 to 4 compare the velocity integrated maps of the original (uncorrected) and the corrected observations. For the uncorrected maps we use the observed spectra scaled to $T_{\rm A}^{\ast }$. This corresponds to a "0th order error beam correction'' where the forward efficiency $F_{\rm eff}$ is used as "full beam efficiency'' (cf. Downes 1989). Note that the signal to noise ratio is lower in the corrected maps because the correction subtracts signal and adds noise.

Figure 1: Integrated spectral line maps of MCLD 123.5+24.9, 12CO $J=2\rightarrow$ 1 (top) and 13CO $J=2\rightarrow$ 1 (bottom). The KOSMA observations are given in the left panels, the uncorrected and corrected key-project maps are shown in the middle and right panels. The latter two are smoothed to 22'', the angular resolution of the IRAM $J=1\rightarrow$ 0 observations. For each map, the grey scale covers the range from the minimum to the maximum of the line integrated intensity (units of [Kkms-1]) to better show the enhanced intensity contrast for the corrected map as larger (local) intensity variations. Note that for most of the positions the intensity in the corrected map (scaled to $T_{\rm mb,c}$) is larger than in the original, uncorrected map (scaled to the antenna temperature $T_{\rm A}^{\ast }$). However, if the uncorrected map is scaled to main beam brightness temperature ( $T_{\rm mb,c}$), the intensity in the corrected map will be smaller

3.1.1 CO $\mathsfsl{J}\ \mathsf{= 2 \rightarrow 1}$ maps

The pick-up by the second and third error beam accounts for a substantial fraction of the observed intensity. The map-averaged value of the additional pick-up is 50% (31%, 44%) of the intensity in the ¹²CO $J=2\rightarrow$ 1 maps of MCLD 123.5+24.9 (L1512 and L134A). For the ¹³CO $J=2\rightarrow$ 1 maps, the corresponding figures are 10%, 23% and 35%. The larger error beam pick-up is found for the ¹²CO maps because of the spatially more extended emission of the more abundant isotopomer, coupling more efficiently to the error beam pattern. For the same reason, the relative contribution of the error beam pick-up is larger in the ¹²CO $J=2\rightarrow$ 1 map of MCLD 123.5+24.9 than for L134A and L1512.

The corrected spectral line maps show an enhanced contrast. More details are visible at small angular scales which are (partially) obscured in the uncorrected maps due to the smearing with the error beam. The actual percentage of the observed intensity attributed to the error beam pick-up significantly varies with position and velocity. For positions where strong emission is found nearby, the relative contribution of the error beam is much larger than average, accounting for up to 100% of the observed emission. Examples are found in the North-West corner of MCLD 123.5+24.9, ¹³CO $J=2\rightarrow$ 1 map (right panel of Fig. 5) and the South-East corner of the L1512, ¹³CO $J=2\rightarrow$ 1 map (right panel of Fig. 6).

3.1.2 $\mathsf{^{12}}$CO $\mathsfsl{J}\ \mathsf{=1 \rightarrow 0}$map of MCLD 123.5+24.9

In the ¹²CO $J=1\rightarrow$ 0 map of MCLD 123.5+24.9, the error beam pick-up accounts (on average) for 16.5% of the observed intensity. The line profile of the error beam pick-up does not vary much across the observed area, in contrast to the CO $J=2\rightarrow$ 1 observations. This is because of the larger angular extent of the error beams for lower frequencies ( GHz), which are comparable or larger than the observed map. The correction therefore modifies the morphology of the intensity distribution only to a minor degree. Comparing the corrected ( $T_{\rm mb,c}$) and the uncorrected (scaled to $T_{\rm mb}$) spectra, we find that the latter are larger by $\sim$30%. This can be considered as an upper limit to the systematic error for CO $J=1\rightarrow$ 0 data, if no correction is done, scaling them to $T_{\rm mb}$ instead. Scaling the spectra to the antenna temperature $T_{\rm A}^{\ast }$ gives intensities which are (on average) smaller by $\sim$8%, and thus a better approximation to the corrected main beam brightness temperature if no further correction is applied.

3.2 Line profiles

Figures 5 to 8 compare the line profiles of the observed spectra ( $T_{\rm A}^{\ast }$) and the estimated pick-up in the second and third error beam. This is a crucial test for the correction method and the beam model used for the IRAM 30 m. Additionally, it provides information on the accuracy of the error beam correction method and the corrected data. An estimated error beam pick-up which systematically exceeds the observed line profile points to a systematic error in either the beam pattern model or the intensity calibration of the observations made with the smaller telescope.

Figure 5: MCLD 123.5+24.9: The observed (uncorrected) spectra (scaled to $T_{\rm A}^{\ast }$) is compared to the pick-up in the 2nd and 3rd Gaussian error beam ( $T_{\rm 2eb}+T_{\rm 3eb}$), as determined with KOSMA observations. The 12CO $J=2\rightarrow$ 1 observations are shown in the left panel, 13CO $J=2\rightarrow$ 1 observations are shown on the right. Each spectrum represents the average of $8 \times 8$ individual spectra, covering an area of 1 square arcmin

Inspection of Figs. 5 to 8 shows that this is not the case, except for a few of the ¹²CO $J=2\rightarrow$ 1 spectra observed towards L134A (left panel of Fig. 7). Here, the estimated error beam pick-up slightly exceeds the observed line profile in the red line wing (between 3 and 4 km s^-1) for positions at ( $\Delta \alpha,\Delta \delta$) $\sim$ ( $-74^{\prime \prime}$, $50^{\prime \prime}$). In addition, the estimated error beam pick-up appears to be red-shifted with respect to the observed line profile for $\Delta\delta > 600^{\prime \prime}$. Two possible explanations remain for the discrepancy. Firstly, the KOSMA observations potentially suffer from a significant error beam pick-up, which results in the error beam pick-up of the IRAM 30 m being overestimated. Secondly, the actual (error) beam pattern of the IRAM 30 m significantly differs from the model used. In either case, the thus introduced systematic error can mimic a velocity offset between the observed line profile and the estimated error beam pick-up, because of the velocity gradient observed for the line profiles south of $\Delta\delta \sim 700^{\prime \prime}$. A further, more quantitative discussion is given in Sect. 4.

The line profiles shown in Figs. 5 to 7 demonstrate that the error beam pick-up not only adds intensity, but modifies the line profile. On average, the relative contribution is larger in the line wings than in the line core. This is documented in Fig. 9, where the systematic variation of the error beam pick-up with velocity channel is shown for the map-averaged spectra. Only for the ¹²CO J= $1\rightarrow$ 0 observations made towards MCLD 123.5+24.9, the percentage of the observed intensity attributed to the error beam pick-up roughly is constant across the line profile.

Figure 9: Map-averaged line profiles of the 12CO $J=2\rightarrow$ 1 observations (first row), 13CO $J=2\rightarrow$ 1 observations (second row), and the MCLD 123.5+24.9, 12CO $J=1\rightarrow$ 0 map (lower left panel). The uncorrected spectra (dotted) are given in antenna temperature, $T_{\rm A}^{\ast }$. The pick-up by the 2nd and 3rd error beam is shown by the solid line. The fraction of the intensity attributed to the error beam pick-up is indicated by the dashed line (labels given on the right of the boxes). The bar on the velocity axis indicates the line wing region, defined in Paper I as the velocity range where significant 12CO emission is seen, but little orno 13CO

The modification of the individual line profiles by the error beam pick-up is more complex than suggested by the map-averaged profiles. It depends on the position and the velocity structure of the emission on angular scales of the error beam pattern. An inspection of individual line profiles shows that for some positions, the intensity in the line wing is effectively lowered with the error beam correction (e.g. in the South-Eastern part of the MCLD 123.5+24.9, ¹²CO $J=2\rightarrow$ 1 map), while for other positions, the line wings are found to be more pronounced in the corrected spectra. The same result is suggested by maps showing the spatial variation of the second moment, determined from the line profiles. An example is given with Fig. 10, where the intensity contrast is substantially higher for the corrected spectra, consistent with a larger spatial variation of the line profile.

Figure 10: Maps of the second moments, determined for the 12CO $J=2\rightarrow$ 1 line profiles observed toward MCLD 123.5+24.9, and smoothed to a resolution of $22^{\prime \prime }$. The second moment are determined according $(\sum _i T(v_i)\,v_i^2)/(\sum _i T(v_i))$, where T(vi) is the intensity detected in channel i (velocity vi) of the spectrum. The left panel shows the result for the original line profiles, the right panel for the corrected data