3 CONTINUUM CALIBRATION

The calibration of bolometer observations consists of two parts: instrument calibration and atmosphere (astronomical) calibration. In this section we will mainly refer to LABOCA, since the calibration procedures for SABOCA are still under development, due to the limited amount of data available up to now.

1 Array parameters

During observations with multichannel bolometer arrays, the target source is covered by a large fraction (maybe all) of the individual bolometers in the array, leading to a large number of separate maps of the area of interest. During the data reduction process, these maps are shifted according to their location in the array, and corrected for their individual gains. These parameters - location and gain - need to be known very exactly in order to produce a combined map without artefacts.

In order to obtain these array parameters, a strong, compact continuum source (ideally Mars, Uranus, or Neptune) is observed in On-the-Fly mapping mode, with a map size large enough to cover the source with all individual bolometers of the array. A dedicated reduction routine does a two-dimensional Gaussian fit on the individual maps, and the fit results yield the positional offsets (relative to the nominal location of the channel in the array) and the relative gain of the channel. Fig. 2 shows the obtained channel locations within the array for LABOCA (left) and SABOCA (right). In order to obtain a sufficient S/N for these parameters, several parameter sets, obtained during periods without major maintenance work on the corresponding bolometer array, are averaged.

For the final array parameters, the average positional uncertainty of the individual channels is $0''\!.7$ for LABOCA and $0''\!.4$ for SABOCA. The effect of this uncertainty is similar to the effect of adding up various maps with some pointing error (i.e. an increase of the effective beam size), but much smaller, and therefore negligible.

The average error of the individual channel gains is about 3% for LABOCA and 7% for SABOCA. However, since some channels may overestimate the intensity, and others underestimate it, they cancel each other out to a certain extent. The remaining calibration error is again negligible compared to those introduced by the absolute calibration process, which is explained in the following subsections.

Figure: The positions of the individual channels of LABOCA (left) and SABOCA (right) within the bolometer array, as obtained by a fully sampled planet map covering the source with all channels. Note the different scale for the two bolometer arrays.

2 Sky opacities

The atmospheric transmission for astronomical radiation can be described by

$$I_{\rm rec} \sim I_0\,e^{-\tau_{\rm Z}\sec z}$$ (1)

with the received intensity $I_{\rm rec}$, the intensity outside the atmosphere $I_0$, the zenith opacity $\tau_{\rm Z}$ and the zenith angle $z = 90^{\circ} - El$. This formula does not take into account Earth's curvature, but is sufficiently accurate for elevations above $25^{\circ}$, where almost all science observations are carried out. While the elevation is given by the telescope position, the zenith opacity has to be measured.

At APEX, we apply a two-step process to estimate zenith opacities. The first is a so-called skydip measurement, where the actual receiver (LABOCA or SABOCA) is used. During a continuous scan from $El = 82^{\circ}$ down to $El \sim 23^{\circ}$ (this value may vary depending on the actual conditions), the atmospheric emission is measured and its dependence on the elevation fitted by a model.

The second step are continuous measurements of the precipitable water vapor in the atmosphere using a radiometer which is sensitive around a water absorption line at 183GHz. This radiometer is installed in the Cassegrain cabin of the telescope, thus it measures the same line-of-sight through the atmosphere towards the target. It uses three different IFs in order to sample the center and two different frequencies in the wings of the absorption line. Based on the ratio of the intensities measured in the different IFs, the precipitable water vapor (and therefore the opacity) is calculated. It should be noted that this method becomes somewhat unreliable for very small values of $pwv$ ( $< 0.4\,{\rm mm}$), when the absorption in the line wings becomes very small.

The usage of these two independent measurements resulted from our experience during the first two years of LABOCA operation. It was figured out that the opacities fitted from the skydips using the facility Bolometer Data Analysis Software (BoA)[9] $\!$, $\tau_{\rm sd}$, are consistently lower than the numbers from radiometer measurements. In order to obtain an agreement between observed and expected primary calibrator fluxes, an average opacity of $1.3\,\tau_{\rm sd}$ needs to be applied to the data. The reason for this discrepancy is that the BoA software uses the ambient temperature as best guess for the atmosphere temperature in the fitting process. Attempts to treat the atmosphere temperature as free parameter during the fitting process led to unrealistic high values for the opacity. Thus the BoA developers decided to fix this number (F. Schuller, priv. comm.).

Similarly it was found that the opacity calculated from radiometer measurements ($\tau_{\rm rm}$) needs to be corrected by a factor of 0.9 to yield on average a good agreement between observed and expected primary calibrator fluxes. Thus the reference opacity for a given skydip scan is calculated as the average: $\tau_{\rm ref} = 0.5 (1.3 \tau_{\rm sd} + 0.9 \tau_{\rm rm})$.

This reference opacity still contains a significant level of uncertainty, which we address by regular observations of primary and secondary calibrators during an observing run.

3 Primary and secondary calibrators

During a normal observing run, primary calibrators (Mars, Uranus, Neptune) as well as secondary calibrators (see Table 3) are observed in regular intervals. The flux densities of the planets can be well predicted based on models, taking into account distance, diameter, and illumination (phase). In practice they are estimated using the Astro program of the Gildas Software[10] $\!$. The flux density scale of the secondary calibrators is then estimated relative to the primary calibrators, with the relation mentioned in the previous subsection for the sky opacities. We are currently also investigating the use of asteroids as potential calibrators for LABOCA and SABOCA.

The main secondary calibrators used for continuum flux calibration are summarized together with their flux densities at $\lambda\,870\,\mu{\rm m}$ in Table 3. Because of the amount of SABOCA data being much less, the APEX staff is still in the process of estimating the flux densities at $\lambda\,350\,\mu{\rm m}$.

During every observing session with the bolometers, calibrators are observed regularly. Their measured flux density is corrected for the sky opacity by a linear interpolation of the reference opacities of the skydips observed closest in time. By comparison of this corrected flux density with the expected flux density (Table 3 for LABOCA) a correction factor is calibrated and can be applied to the target source flux density scale, if necessary. With these correction factors obtained by calibrator measurements, all systematic errors in the opacity calculation by skydips can be corrected for. The standard deviation for these correction factors is $\sigma(f_{\rm CalCorr}) = 0.14$ and $\sigma(f_{\rm CalCorr}) = 0.35$ for LABOCA and SABOCA, respectively. These numbers correspond to the scatter of the calibrator fluxes (and hence also of the science data) before these correction factors are applied to the data. A better estimate for the calibration uncertainty after these correction factors have been applied can be obtained when only calibrator scans over a few hours (the typical uninterrupted daily observing time of a given project) are considered. The resulting standard deviations of the calibrator correction factors are 0.05 and 0.24 for LABOCA and SABOCA, respectively, translating to relative calibration uncertainties of 5% and 24%. For LABOCA, this number is of the same order as the flux density uncertainty of the secondary calibrators themselves. Thus we estimate an absolute calibration uncertainty of 10% for LABOCA. For SABOCA currently the absolute calibration uncertainty is $25 - 30$%.

Table: Secondary calibrators for continuum observations, their positions, and flux density values at $\lambda\,870\,\mu{\rm m}$.

Source	HL Tau	CRL618	V883 Ori	N2071IR	VY CMa	CW Leo	B13134
RA[J2000]	04:31:38.45	04:42:53.60	05:38:18.24	05:47:04.85	07:22:58.33	09:47:57.38	13:16:43.15
Dec[J2000]	18:13:59.0	36:06:53.7	-07:02:26.2	00:21:47.1	-25:46:03.2	13:16:43.6	-62:58:31.6
$S_{{\rm 870}\mu{\rm m}}$	$2.0 \pm 0.2$	$4.8 \pm 0.5$	$1.4 \pm 0.3$	$9.1 \pm 0.8$	$1.5 \pm 0.2$	$4.1 \pm 0.3$	$12.9 \pm 1.3$
Source	IRAS16293	G5.89	G10.62	G34.3	G45.1	K3-50A	CRL2688
RA[J2000]	16:32:22.90	18:00:30.37	18:10:28.66	18:53:18.50	19:13:22.07	20:01:45.69	21:02:18.80
Dec[J2000]	-24:28:35.6	-24:04:01.4	-19:55:49.7	01:14:58.6	10:50:53.4	33:32:43.5	36:41:37.7
$S_{{\rm 870}\mu{\rm m}}$	$16.1 \pm 1.3$	$27.6 \pm 0.2$	$33.0 \pm 1.8$	$55.3 \pm 3.7$	$8.0 \pm 0.6$	$14.7 \pm 1.4$	$5.5 \pm 0.9$

We should close this subsection with the comment that we recently found a dependence of the correction factor introduced here on the precipitable water vapor. We are working on an understanding of this effect and will take it into account during future improvements of the calibration scheme.

4 Further calibration issues with broadband bolometers

Both facility bolometers are broadband instruments, and the spectral response is mainly defined by a set of filters. LABOCA is sensitive between 300 and 400GHz, SABOCA between 700 and 1000GHz. This wide bandpass has several consequences.

• Beam size. Over such a wide bandpass, the beam size (angular resolution) changes significantly. The half-power beam width for LABOCA is $HPBW \sim 16'' - 20''$, and for SABOCA $HPBW \sim 6''\!.5 - 8''\!.5$, while all bolometer maps are calibrated in e.g. mJy/beam with a median beam size. For extended sources, the detected flux originates in a larger area (in the source) for emission at the lower end of the bandpass than for emission at the upper end. Depending on the spectral energy distribution (SED) of the source, an over- or underestimate of the measured flux may occur.
• Effective frequency. The amount of atmospheric absorption changes over the band. For excellent weather this change is small, but for higher $pwv$ the absorption towards the higher end of the bandpass is significantly higher then towards the lower end (see Fig. 1). Thus the effective observing frequency will decrease. Besides a change in beam size (see above), this also changes the actual flux density, depending on the slope of the source SED. This effect may be more severe if the target source SED differs significantly from the SED of the used calibrators over the bolometer bandpass.

While these consequences have effects on the calibration, it is not yet known how severe they are. They are generally assumed to be rather small compared to the overall calibration uncertainty. We are currently in the process of investigating these effects for the APEX bolometers.