4 Self-consistency check and accuracy of the corrected data

   
4.1 Self-consistency check

The comparison of the error beam corrected key-project data, smoothed to the KOSMA resolution, with the KOSMA observations provides a check for the correction method and the intensity calibration of both data sets. For a quantification we define

$$% x = \frac{\langle T_{\rm mb,c} \rangle_{\rm KOSMA}}{T_{\rm mb}'},$$ (3)

where $T_{\rm mb}^{\prime}$ is the main beam brightness temperature of the KOSMA observations and $\langle T_{\rm mb,c} \rangle_{\rm KOSMA}$ denotes the corrected, smoothed IRAM data. For a quantitative analysis we determine the ratio x for positions within an area where KOSMA and IRAM observations are available, and for those velocity channels where the intensity exceeds a threshold of three times the rms noise level. Positions lying closer than 1arcmin to the edge of the map are excluded, because it is not possible to obtain a reliable estimate for the smoothed spectra, $\langle T_{\rm mb,c} \rangle_{\rm KOSMA}$.

For perfectly corrected IRAM 30m data and KOSMA observations with a negligible error beam pick-up, a narrow distribution, centered on 1, is expected. An average value smaller than unity suggests that the IRAM 30 m error beam pick-up is overestimated, and hence that the corrected main beam brightness temperature is underestimated. For the corrected key-project maps, the x-distributions are given in Fig. 11, together with the average ($\mu _x$), median, and width $\sigma _x$ (standard deviation) of the distribution.

Figure 11: Distribution of x, determined as the ratio of the corrected key-project data (smoothed to angular resolution of the KOSMA telescope) and the KOSMA observations (Eq. 3). The ratio x was determined for each velocity channel with an intensity larger than three times the rms in the spectrum. Positions which are located closer than 1 arcmin to the edge of the map are excluded. For each distribution, the average $\mu _x$, standard deviation $\sigma _x$, and the median (med.) is given

For the ¹²CO $J=2\rightarrow$ 1 maps, $\mu _x$ is found to be equal or smaller than unity (0.82, 1.02 and 0.88 for MCLD 123.5+24.9, L1512 and L134A, respectively), while for the ¹³CO $J=2\rightarrow$ 1 maps, $\mu _x$ is larger than unity (1.15, 1.26 and 1.06). For the ¹²CO $J=1\rightarrow$ 0 map of MCLD 123.5+24.9 we find $\mu_x=$1.02. The width $\sigma _x$ of the distributions is between 0.10 and 0.18, except for the ¹³CO $J=2\rightarrow$ 1 map of MCLD 123.5+24.9, where $\sigma _x$ is significantly larger (0.22). In general, we find larger $\sigma _x$ for the ¹³CO $J=2\rightarrow$ 1 maps, except for L134A where both maps have $\sigma_x\sim 0.13$. The influence of the finite signal-to-noise in the spectra on the $\mu _x$ and $\sigma _x$ is negligible. We obtain the same results for distributions deduced with spectra binned to a lower velocity resolution, and hence with a lower rms noise per velocity channel. Thus, the $\mu _x$and $\sigma _x$ can be used to determine the accuracy of the error beam correction method and to examine the limiting factors.

   
4.2 Accuracy of the correction method and the corrected data set

We estimate the total uncertainty in the $T_{\rm mb,c}$ scale using the self-consistency check ($\mu _x$ and $\sigma _x$) and by comparing the observed line profiles to the estimated error beam pick-up.

An average value $\mu _x$ offset from unity indicates systematic errors, which may result from

• under- or overestimation of the amplitude of the IRAM 30m error beam pattern;
• error beam pick-up in the KOSMA data;
• systematic calibration differences for both data sets.

The width of the distribution ($\sigma _x$) depends on statistical variations from position to position. These uncertainties may arise from

• axially non-symmetric error beam pattern of the IRAM 30 m;
• neglect of the first error beam in Eq. (2);
• relative photometric calibration uncertainties of the individual spectra in the map. These errors may arise from imperfectly corrected atmospheric attenuation or receiver gain drifts and are estimated to be $\sim$9% for the key-project maps (Panis 1995).

It is not possible to fully disentangle the contribution of the different effects which result in a broader $\sigma _x$ or an average $\mu _x$ offset from unity. However, a systematic analysis of the results obtained for the different sources and transitions provides an upper limit to the magnitude of the different sources of error, and allow us to estimate the accuracy of the final data set.

4.2.1 Error beam pick-up of the KOSMA telescope

The exact amount of error beam pick-up in the KOSMA observations is unknown because it depends on the large-scale structure of the emission and the (unknown) details of the KOSMA error beam pattern. If present, the KOSMA error beam pick-up is expected to play a role for the observations of the spatially extended ¹²CO $J=2\rightarrow$1 emission only. An estimate of the KOSMA error beam pick-up at 230GHz is obtained by comparing the main beam efficiency determined for bright planets ( $B_{\rm eff}\,=\,0.64$) and measured on the Moon ( $B_{\rm Moon}$ = 0.72). This shows that a KOSMA error beam of angular extent is present, and that the power radiated into this error beam is $\sim$13% of the power in the main beam. Thus, for extended sources, an error beam pick-up of $\sim$13% is expected for KOSMA observations at 230 GHz (assuming an uniform intensity distribution).

Indeed, for the maps with the spatially most extended emission (the ¹²CO $J=2\rightarrow$ 1 maps observed toward MCLD123.5+24.9 and L134A) we find $\mu _x$ which are significantly smaller than unity, suggesting that the KOSMA error beam is not negligible. The $\mu_x=0.82$ (0.88, respectively) suggest that the KOSMA error beam pick-up is 12 - 18% for the ¹²CO $J=2\rightarrow$ 1 maps, and this only if we fully attribute the $\mu_x<1$ to a KOSMA error beam. For the ¹³CO $J=2\rightarrow$ 1 maps we find that the $\mu _x$ are larger than unity. This is consistent with a negligible KOSMA error beam pick-up for the not very extended emission of the rarer isotopomer, and that other systematic errors limit the accuracy of the intensity calibration. The same applies to the ¹²CO $J=1\rightarrow$ 0 map of MCLD123.5+24.9, because of the smaller amplitude of the error beam pattern at lower frequencies.

4.2.2 Influence of the uncorrected first error beam

The correction for the first (smaller) error beam of the IRAM 30m is not possible because of the large temporal variations of the amplitude and the insufficient angular resolution of the KOSMA observations. Thus, an additional error in the intensity calibration of the corrected data set arises from the uncorrected first error beam. This error is expected to play a significant role for maps which show strong intensity variations on the angular scale of the error beam ($\sim\,$1.9' at 230GHz). Here, the x-distribution is expected to be significantly wider (lager $\sigma _x$) with an average $\mu _x$ larger than unity, because the pick-up in the small error beam strongly varies with the position. This is observed for the ¹³CO $J= 2\rightarrow 1$ map of MCLD 123.5+24.9 and L1512. Both contain cores and/or steep gradients extending over $\sim\,$1 arcmin (Figs. 1 and 2) and, at the same time, have the largest $\mu _x$ (1.15 and 1.26) and $\sigma _x$ (0.22 and 0.18). We therefore conclude that the accuracy of the intensity calibration for these maps is limited by the uncorrected first error beam, and that the thus introduced error is up to 26%. For the other maps, the first error beam accounts for less than 12% of the observed intensity, judged from the $\sigma _x$ in Fig. 11 and taking into account that the accuracy of the relative intensity calibration in the maps is 9% (Panis 1995), assuming that both errors add in quadrature.

4.2.3 Accuracy of the error beam model

In the ¹²CO $J=2\rightarrow$ 1 map of L134A the estimated error beam pick-up systematically exceeds the observed line profile for a few positions at $\sim$( $-74^{\prime \prime}$, $50^{\prime \prime}$), while both profiles appear velocity-shifted in respect to each other for positions . This cannot be reconciled unless we modify the amplitude of the error beam pattern by much more than $20\%$, which is excluded by the accuracy of the beam pattern parameters (GKW). Similarly, the possible contribution of a KOSMA error beam cannot fully account for the discrepancy because it would imply an additional pick-up in the KOSMA observations of far more than 20%, and thus extended areas with strong emission outside the maps, which is not observed (Bensch 1998).

One plausible explanation is provided by a possible deviation of the IRAM error beam pattern from the assumed axial symmetry. In the L134A maps, a velocity gradient is present South of $\delta\sim 700^{\prime \prime}$, extending over several arcmins. A slight deviation from the assumed axial symmetry can mimic a velocity shift in the line profile of the error beam pick-up. This is not excluded by the measurements of GKW, because they used cross-scans in $\sim\,$East-West direction and explicitly assume axial symmetry. Moreover, a not axially-symmetric error beam pattern is not entirely unrealistic, given the shape of the panels/panel frames of the IRAM 30m.