4 SPECTRAL LINE CALIBRATION

The calibration efforts for spectral line data can be divided in two phases. The first phase includes the actual calibration of the data, as it is performed by the Online Calibrator software described in the next subsection. After that step the data product is delivered to the project PIs. With additional information about telescope efficiencies the calibration scale can be adjusted to be telescope-independent. At this stage, the data can be used for scientific research and - in a perfect world - should be perfectly calibrated.

In practice this is not necessarily the case. In the second phase, results from an APEX calibration monitoring program can be taken into account. The goal of this program is to verify the absolute calibration scale, to detect deviations, and to estimate absolute calibration uncertainties inherent to the data. This monitoring program consists of regular observations of well known molecular line sources, the standardized reduction and analysis of the resulting data, and the comparison of the results between various dates and with results available from other observatories or in publications. This monitoring program has only been implemented for APEX-1 and APEX-2 yet. The APEX-3 receiver has just recently been installed, and for the APEX-T2 receiver, the observations of calibration sources will be performed within the science projects, because it would require a large amount of excellent weather time otherwise.

The first phase will be covered in Sections 4.1 and 4.2, the second phase in the remainder of this section.

1 The Online Calibrator

The absolute calibration of spectral line data is based on receiver noise temperatures and system temperatures estimated by so-called SKY-HOT-COLD calibration scans which are performed immediately before the actual spectral line observations. In this section we outline the basic steps performed by the software, a detailed description of the calibration concept is available elsewhere[11, 12].

During the three phases of a calibration scan, the receiver is looking during equal amounts of time to blank sky, to a hot load at ambient temperature, and to a cold load at liquid nitrogen temperature. From these data, the Online Calibrator calculates first the receiver temperature $T_{\rm rec}$ from the signal in the HOT and COLD phases. Then it proceeds and calculates the sky temperature $T_{\rm sky}$ by taking into account the SKY signal, and correcting for spillover and forward efficiency $F_{\rm eff}$ (see section 4.2).

From the elevation of the SKY measurement, it calculates the airmass, based on a curved atmosphere, which is used by the ATM modul to determine the opacity in image and signal band, with the aid of a sophisticated atmospheric model. As a final step in the calibration scan reduction, the calibration temperature $T_{\rm cal}$ and system temperature $T_{\rm sys}$ are calculated and stored internally.

The following spectral line scans are then calibrated via

$$T_{\rm A}^* = (C_{\rm on}-C_{\rm off}) G^{-1} T_{\rm cal}$$ (2)

with $C_{\rm on}$ and $C_{\rm off}$ being the signals (counts) in the ON and OFF phase of the spectral line scan, $G$ the gain (calculated before from the calibration scan, and $T_{\rm A}^*$ the antenna temperature, corrected for $F_{\rm eff}$ and rear spillover. At this stage the data are considered calibrated, and are delivered to the PI. This calibration, however, is telescope specific.

2 Efficiencies

The spectral line data, as it is delivered to the PI, is calibrated in $T_{\rm A}^*$. In order to obtain a telescope-independent calibration, this temperature scale must be converted to e.g. main beam brightness temperature, $T_{\rm mb}$. In the calibration scheme used at APEX, these temperature scales are connected via $T_{\rm mb} = (F_{\rm eff}/B_{\rm eff})T_{\rm A}^*$, with the forward efficiency $F_{\rm eff}$ and the main beam efficiency $B_{\rm eff}$. The forward efficiency is defined as the radiation received from the forward direction relative to the radiation received from all directions. For standard radio telescopes, this number is slightly smaller than, but close to, unity. The value which is applied by default at APEX is $F_{\rm eff} = 0.95$, although recent measurements suggest values of 0.97 and 0.96 for APEX-1 and APEX-2, respectively.

Similarly, the main beam efficiency $B_{\rm eff}$ is the radiation received from the main beam relative to the radiation received from all directions. This number depends on the beam pattern of the telescope and therefore on e.g. the surface accuracy. It also depends on the beam filling factor and is therefore - strictly speaking - only correct for point sources. It can be measured by continuum scans on planets. By measurements performed during 2008 using Mars, Jupiter, and Saturn, we found a main beam efficiency of $B_{\rm eff} = 0.80$ for 230GHz, and 0.74 at 345GHz. It depends also on the observing frequency, and decreases towards higher frequencies.

3 Observing conditions

As already introduced in subsection 2.1, the receiver to be used during science observations (and hence the observing frequency) is normally selected based on the precipitable water vapor ($pwv$) present in the atmosphere. However, also the actual variation in $pwv$ has to be taken into account in this decision. An example of the effect of a strongly varying $pwv$ can be seen in Fig. 4a.: for almost all dates the scatter in the data points is comparable, except for one (2009-07-19). At that particular date not only the averaged absolute value for $pwv$ was rather high, but also its standard deviation (which basically means its variability). The $pwv$ as function of date, as well as the standard deviation $\sigma(pwv)$ as function of $pwv$, are shown in Fig. 3.

Figure: a. (left panel) Averaged precipitable water vapor for all dates when monitoring observations were performed with the APEX-1 receiver. b. (right panel) Standard deviation of the averaged pwv as function of pwv. There is no obvious dependency of σ(pwv) on <pwv>.

The calibration uncertainty is therefore not so much a function of the absolute water vapor value (which is calibrated out by the Online Calibrator), but rather of the variability of the water vapor on the timescale relevant for the calibration. This timescale is given by the frequency of HOT-SKY calibration measurements (every 10 to 15 minutes). This is demonstrated by the data from 2009-01-27, with a water vapor of $4.65 \pm 0.09\,{\rm mm}$ (see Fig. 3), where no strong variation in the calibration is seen. A low absolute $pwv$ value means less atmospheric transmission, and therefore higher signal-to-noise ratios, and will also result in smaller opacity correction factors (i.e. less amplification of any kind of noise or scatter). But also the stability of the atmospheric conditions is important for a reliable calibration. For bolometer observations this is even more crucial, because the subtraction of correlated atmospheric noise between the individual bolometer channels becomes more difficult or even impossible at the presence of short-term atmospheric instabilities.

4 Calibration uncertainty

A good estimate for the calibration stability and uncertainty can be obtained by calculating normalized line parameters for the spectral lines observed within our calibration plan monitoring program. These parameters are basically the maximum brightness temperature $T_{\rm max}$ and the velocity integrated line intensity $I = \int_{\rm line} T\,{\rm d}v$ (sometimes also nominated line area). While the first is more uncertain in the presence of spikes, the latter shows a higher dependence on the spectral baseline. It is therefore expected to show a larger variation for weak sources.

To obtain normalized parameters, reference spectra are created from all observations performed during the monitoring program by averaging the spectra for each source/line combination with an appropriate weighting for all observation dates. The normalized line parameter ($T$ or $I$) is then simply the actual value divided by the parameter obtained from the average spectrum.

Analysing $T_{\rm max}$ and $I$ for all sources and lines, we find the following standard deviations for the normalized parameters: $\sigma(T_{\rm max}) = 0.079$ and $\sigma(I) = 0.083$ for the APEX-1 receiver, and $\sigma(T_{\rm max}) = 0.113$ and $\sigma(I) = 0.141$ for the APEX-2 receiver. If we assume that the average project PI is able to perform a proper baseline subtraction for his science data, then we can estimate the overall calibration uncertainty to 8% and 12% for the APEX-1 and APEX-2 receiver, respectively. However, this uncertainty strongly depends on the actual source and/or line; it is much lower for the ``standard" lines (CO(2-1) and CO(3-2)) than for others.

Table 4 summarizes all numbers for both receivers. For lines which are monitored in LSB and USB tuning we averaged the numbers, since the differences (for the normalized parameters!) have been found to be negligible. While for the APEX-1 receiver the calibration uncertainty is $\le 10\,\%$ for all lines, for APEX-2 there are higher values for some cases. These are C$^{18}$O(3-2) and $^{13}$CO(3-2), with an uncertainty of about 13%, and even more H$_2$CO and CH$_3$OH with an uncertainty of up to $\sigma_{\rm I} = 33\,\%$ for the latter. The reason for this large scatter at these special frequencies is currently unknown. A possible explanation could be a tuning instability, with a strongly varying image band rejection. Further dedicated tests need to be performed to investigate this situation.

Table: Relative calibration uncertainty for various frequencies.

Molecule	APEX-1	H$_2$CO	$^{13}$CO	CO	CH$_3$OH	CS	HCN
$\nu$ [GHz]	all	218.2	220.4	230.5	241.8	244.9	265.9
Spectra	345	30	58	122	21	70	44
$\sigma_{\rm I}$	0.083	0.057	0.100	0.058	0.097	0.089	0.131
$\sigma_{\rm T}$	0.079	0.072	0.076	0.076	0.100	0.078	0.087

Molecule	APEX-2	CS	OCS	C$^{18}$O	$^{13}$CO	CH$_3$OH	CS	CO	HCN	HCO$^+$	H$_2$CO
$\nu$ [GHz]	all	293.9	304.1	329.3	330.6	338.4	342.9	345.8	354.5	356.7	362.7
Spectra	577	43	46	37	136	20	36	147	41	43	28
$\sigma_{\rm I}$	0.141	0.106	0.135	0.134	0.139	0.330	0.151	0.082	0.140	0.113	0.243
$\sigma_{\rm T}$	0.113	0.078	0.094	0.119	0.131	0.236	0.096	0.082	0.103	0.082	0.160

5 LSB vs. USB tuning

In general any frequency (except close to the edges of the tuning range) can be observed in lower sideband (LSB) and upper sideband (USB) tuning. Ideally, both tunings should give very similar results. For a total of five lines (two with APEX-1, three with APEX-2) we monitor line parameters using both tunings.

A comparison of the two different tunings is possible by calculating the sideband response ratio $I_{\rm USB}/I_{\rm LSB}$, the ratio of the line intensities when observed with upper and lower sideband tuning (not to be confused with the ``sideband ratio'', which is the ratio of signal to image band for one tuning). The sideband response ratio is listed for the five lines in Table 5. We find that USB tuning always results in lower intensities than LSB tuning. While this is not very significant for APEX-1, the differences are clear for APEX-2, especially for CO(3-2), which is the standard molecular line for this receiver. Fig. 4 illustrates the effect for APEX-1 and APEX-2, and also shows that this ratio does not vary significantly with time.

Table: USB/LSB sideband response ratios for various frequencies

Molecular line	CO(2-1)	CS(5-4)	OCS	13CO(3-2)	CO(3-2)
Frequency [GHz]	230.5	244.9	304.1	330.6	345.8
$I_{\rm USB}/I_{\rm LSB}$	$0.97 \pm 0.04$	$0.99 \pm 0.05$	$0.96 \pm 0.04$	$0.91 \pm 0.06$	$0.90 \pm 0.04$

Figure: The USB vs. LSB sideband response ratio for the CO(2-1) and the CO(3-2) line, the most important molecular transitions for the APEX-1 and APEX-2 receiver, respectively. For both lines USB intensities are smaller than LSB intensities. Note also the large scatter for date 2009-07-19 for the CO(2-1) line (see Section 4.3).

This result is independent of the precipitable water vapor during the observations. Thus it does not seem to be affected by differences in the atmospheric transmission between the sidebands. It rather seems that the image sideband rejection differs from the numbers which are assumed by the Online Calibrator (see Section 4.1). As we will see in Section 4.6, the spectra obtained in USB tuning seem to be in better agreement with data from other telescopes, while the line intensities obtained in LSB tuning seem to be too large. Thus the sideband rejection for LSB tuning seems to be better than assumed by the Online Calibrator.

We should note here that an improvement of the calibrator software is under discussion, e.g. by implementing a channel dependent calibration, rather than an averaged one over the whole band. Also, a better determination of the image band rejection, by line injection at the frequencies of the signal and image band for various tunings, and the application of these improved numbers in the calibrator, may improve the sideband calibration accuracy in the future.

6 Comparison with other telescopes

We compared the reference spectra obtained with the APEX-1 and APEX-2 receiver with data from other telescopes, to verify if the overall absolute calibration scales are consistent. This comparison was restricted to point-like or very compact sources, in order to properly correct for the varying beam dilutions as result of different telescope diameters (i.e. beam sizes). An extensive database is available online for the JCMT[13] $\!$, as well as for other telescopes (CSO, SEST) through published articles[14, 15, 16].

The comparison was performed for the following sources: CRL618, OH231.8, IRC+10216, IRAS15194, and CRL2866. IRC+10216 is known to be extended for the CO(2-1) and (3-2), as well as the $\rm {13}$CO(2-1) transition, but at rather low intensity levels. We treated the correction for beam dilution as if it was a point source. The inclusion or non-inclusion of these transitions in the data analysis does not change the final results described in this subsection.

The results differ depending on which APEX receiver is considered (APEX-1 or APEX-2), and also on the telescope the comparison data come from. For APEX-2, the APEX data are in good agreement with JCMT and CSO, with an average peak brightness temperature ratio of $\!\!<T_{\rm JCMT+CSO} / T_{\rm APEX2}\!\!>\:= 1.02 \pm 0.12$. Given the larger number of comparison spectra available at JCMT, we can separate data taken at APEX with LSB tuning from USB tuning, and find $<\!\!T_{\rm JCMT} / T_{\rm APEX2,LSB}\!\!>\:= 0.97 \pm 0.14$ and $<\!\!T_{\rm JCMT} / T_{\rm APEX2,USB}\!\!>\:= 1.00 \pm 0.13$, respectively. While the ratios are within the uncertainty in all cases, there is apparently a trend to overestimate brightness temperatures when using LSB tuning with the APEX-2 receiver. The difference found by comparison with JCMT data is consistent with the overall trend seen in Section 4.5, that USB tuning in general gives lower brightness temperatures than LSB tuning. We can therefore summarize that data obtained in USB tuning using the APEX-2 receiver are fully consistent with comparison data taken at other telescopes, while data taken with the same receiver in LSB tuning seem to overestimate the line temperatures.

For APEX-1, the situation is different. Separating the two sideband tunings for the APEX-1 receiver and taking into account only JCMT comparison data, we find $<\!\!T_{\rm JCMT} / T_{\rm APEX1,LSB}\!\!>\:= 0.86 \pm 0.18$ and
$<\!\!T_{\rm JCMT} / T_{\rm APEX1,USB}\!\!>\:= 0.87 \pm 0.13$ for LSB and USB tuning, respectively. If we included SEST data, the numbers would be even slightly lower (with an average of $<\!\!T_{\rm JCMT+SEST} / T_{\rm APEX1}\!\!>\:= 0.84 \pm 0.10$), but SEST data were known to slightly underestimate brightness temperatures in general (A. Lundgren, priv. comm.). Still there remains a discrepancy between APEX data taken with the APEX-1 receiver and JCMT data. These results do not claim neither that JCMT intensities are too low nor that APEX intensities are too high, still APEX project PIs should be aware of these calibration differences. Again, comparing with Section 4.5 it looks like the results using USB tuning with the APEX-1 receiver seem to be in better agreement than those from LSB tuning, although the difference is smaller than for the APEX-2 receiver.