2 Error beam correction of the IRAM key-project data

   
2.1 Beam pattern of the IRAM 30m telescope

The beam pattern of the IRAM 30m was determined by GGC, and more recently by GKW. GKW combined a larger data set (measurements on Moon, planets and holographic data) obtained at four different wavelengths ranging from $\lambda=3.4$ mm to 0.75 mm to deduce a consistent beam pattern model. They find that the beam pattern is given by a tapered Airy pattern ( ${\cal A}_{\rm T}$) of HPBW $\theta_{\tiny\! \mbox{{${\cal A}_{\rm T}$ }}}= \theta_{\rm mb}$, and three Gaussian error beams $P_{{\rm e},i}$ of HPBW $\theta _i$ ( $i=1\dots 3$). The latter result from three independent surface error distributions in the primary reflector. Both more extended (second and third) error beams result from surface errors attributed to the panel frames and the mounting of each panel with a $3 \times 5$ array of adjustment screws. The first error beam ( $\theta_{1}\sim 10\theta_{\rm mb}$) is due to temporally variable, large-scale deformations of the reflector surface, suspected to result from residual thermal deformations (see GKW and references therein).

 

 

Table 1: Beam pattern of the IRAM 30 m at $\lambda =2.6$ mm (115 GHz) and 1.3 mm (230 GHz) before a re-adjustment of the panel frames in July 1997. The HPBW $\theta _i$ and the amplitudes a_i of the three error beams are taken from the fit shown in Fig. 5 of GKW. The p_i are determined according $p_i \propto a_i\,\theta _i^2$, using the normalization of $\sum _i p_i = 1$ (summed over main beam and the Gaussian error beams)

		$\lambda=$ 2.6mm (115 GHz)		$\lambda=$ 1.3mm (230 GHz)
component		HPBW [arcsec]	pi	HPBW [arcsec]	pi
tapered Airy patt.	${\cal A}_{\rm T}$	21	0.71	10.5	0.41
$1^{\rm st}$ error beam	$P_{\rm e,1}$	228	0.08	114	0.16
$2^{\rm nd}$ error beam	$P_{\rm e,2}$	317	0.08	158	0.16
$3^{\rm rd}$ error beam	$P_{\rm e,3}$	1900	0.13	950	0.27

In the following, we use

$$% P(\phi,\theta) = \mbox{{${\cal A}_{\rm T}$ }}(\theta) + \su... ... \ln2}{\theta_{i}^2\pi} \exp[-4\ln2\,(\theta/\theta_{i})^2],$$ (1)

for the beam pattern of the IRAM 30m, where $w_i \propto \int\!\!\!\int_{2\pi} P_{{\rm e},i} {\rm d}\Omega$. With the normalization of P(0,0)=1, the w_i are given by $w_i = p_i\,F_{\rm eff}$, where $F_{\rm eff}=\Omega_{2\pi}/\Omega_{4\pi}$ and $\Omega_{2\pi} (\Omega_{4\pi})$ is the integral of the beam pattern over the forward hemisphere ($4\pi$). Table 1 gives $\theta _i$ and p_i for the wavelengths of both lowest CO rotational transitions, applicable to the observations of the key-project.

The accuracy of the HPBW and the power in the Gaussian error beams is estimated by GKW to be % and $\sim$5%, respectively. This applies to the individual components, determined from observations at one frequency, whereas the numbers given in Table 1 are results from a consistent beam pattern model, fitted to a larger number of observations at different wavelengths. We therefore expect a higher accuracy for the latter. The exception is the integral over the first error beam, showing temporal variations of some $\pm50$% with a time constant of 1 h or more (Table 1 gives the average p₁). Therefore, a large uncertainty applies to the pick-up by the first error beam.

   
2.2 Error beam correction

The correction of the pick-up in the second and third error beam is done using KOSMA observations. Because it is not possible to trace back the contribution of the temporally variable first error beam, we count the latter as part of the main beam and use the approximation

 

$$% T_{\rm mb,c} \approx \frac{F_{\rm eff}}{B_{\rm eff} + p_1\,F_{\... ...{\ast} - \sum_{i={\rm 2,3}} p_i \langle T_{\rm mb}' \rangle_{{\rm e},i} \right]$$ (2)

for the corrected main beam brightness temperature. Here, $\langle T_{\rm mb}' \rangle_{{\rm e},i}$ denotes the KOSMA observations, smoothed to the angular resolution of the ith error beam. KOSMA observations were made in the ¹²CO and ¹³CO $J=2\rightarrow$ 1 transition towards all three sources (Bensch 1998), and a ¹²CO $J=1\rightarrow$ 0 map of MCLD 123.5+24.9 is available from Großmann et al. (1990).

The correction by de-convolution of the error beam pattern in Fourier space turned out to be not possible (Bensch et al. 2001). For each map in the data set, this correction method requires additional, fully sampled observations with the IRAM 30 m of several 10⁴ positions, and an integration time of less than 1 s per position. Currently, this is not feasible without an excessive overhead in dead-time (due to moving of the telescope, etc.) and because of the huge amount of data to be handled in a relatively short time.