A&A 365, 285-293 (2001)
F. Bensch1 - J. Stutzki1 - A. Heithausen1,2
Send offprint request: F. Bensch,
1 - I. Physikalisches Institut der Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany
2 - Radioastronomisches Institut der Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
Received 27 January 2000 / Accepted 1 August 2000
The beam pattern of a single dish radio telescope is given by the main beam and additional components at larger angles, usually called error beam or stray pattern. The latter have relatively small peak amplitudes (typ. below -25 dB), depending on the rms surface error of the primary reflector. However, because of their large angular extent, they are sensitive to extended sources, and a significant fraction of the observed intensity can result from error beam pick-up. For (sub-)mm observations suffering from error beam pick-up we introduce a new temperature scale for the corrected data, the corrected main beam brightness temperature , which provides a better approximation to the intensity detected by the main beam than the commonly used antenna temperature and main beam brightness temperature. We consider two different correction methods. The first method uses complementary observations obtained with a smaller telescope. Smeared to the angular resolution of the error beam pattern they provide an estimate of the error beam pick-up in the observations of the large telescope. For the second method, the error beam pick-up is de-convolved from the observed map in Fourier space. The requirements for both correction methods and their advantages and limitations are discussed in detail. Both correction methods require additional observations, unless the full spatial extent of the emission is observed. We find that the de-convolution method is attractive for the correction of fully sampled maps with an angular extent much larger than the error beam pattern. For smaller maps and more sparsely sampled observations, the subtraction method is favorable, because the additional observations with a small telescope are less time consuming.
Key words: methods: data analysis - techniques: miscellaneous - telescopes - radio lines: ISM
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With the increasing number of large telescopes and sensitive receivers for the (sub-)mm range, the importance of an error beam correction was recognized for observations at shorter wavelengths (cf. Schneider et al. 1998). Essential pre-requisite for a correction is an accurately measured beam pattern. For the IRAM 30m telescope, this was done by Garcia-Burillo et al. (1993) and Greve et al. (1998). Based on these studies, Bensch et al. (2001) determined the influence of the error beam pick-up in the data set of the IRAM key-project, and estimated the accuracy of the applied correction method. They demonstrate that the error beam pick-up of the IRAM 30m at wavelengths of 1.3mm can account for 20-50% of the observed intensity (map average), and locally for up to 100%.
The error beam problem is well known to HI observers and large surveys are routinely corrected for the additional pick-up (cf. Hartmann et al. 1996). For observations at (sub-)mm wavelengths, however, only very few quantitative studies exist (cf. Garcia-Burillo et al. 1993; Dame & Thaddeus 1994; Schneider et al. 1998), despite the fact the error beam problem is expected to be quite common also to large (sub-)mm telescopes. In addition, the ratio of the error beam pattern to the spatial extent of the source can be very different for cm and (sub-)mm wavelengths. The correction methods and results found for HI observations therefore do not necessarily apply to the observations made at shorter wavelengths, and a more detailed study of the influence of error beam pick-up and the correction methods is needed.
This paper supplements Paper III in the series presenting the results of the IRAM key-project "Small-scale structure of pre-star forming regions''. The data set of the key-project and first results are published by Falgarone et al. (1998, Paper I) and Heithausen et al. (1998). The data reduction is presented by Panis et al. (in preparation; Paper II) and the influence and correction of the error beam pick-up is discussed by Bensch et al. (2001, Paper III). The present paper puts the error beam correction of (sub-)mm observations on firmer grounds. The relevant definitions are provided and the possible correction methods are reviewed, discussing their advantages and limitations. In addition, a guideline is provided for observers who wish to correct their own data.
In Sect. 2, we introduce the notation and give the definition of the corrected main beam brightness temperature. The error beam pattern of (sub-)mm telescopes is discussed in Sect. 3, and the error beam correction methods are presented in Sects. 4 and 5. The requirements of both correction methods are discussed and compared, their advantages and limitations are outlined (Sect. 6). A summary is given in Sect. 7.
Consider a radio telescope pointing toward a source of brightness
is the peak brightness temperature, and
varies between 0 and 1. The observed antenna
temperature, corrected for atmospheric and rear-ward losses, then reads
The main beam brightness temperature (cf. Downes 1989) is defined by , where and ( ) is the integral of the beam pattern over the forward hemisphere (main beam) of the telescope. The main beam is defined e.g. as the beam pattern enclosed by the -20 dB threshold, or the equivalent Gaussian with a half power beam width (HPBW) determined from cross-scans on calibration sources (e.g. planets). Here, we accept the former definition: for inside the -20 dB level, and elsewhere. The forward beam pattern outside the main beam, , is then denoted as error beam. Note, that this definition of the main beam is not identical to the diffraction beam of a telescope with a perfect reflector surface, because components resulting from surface irregularities partly contribute to . However, this contribution is small in most cases (Sect. 3).
For telescopes with a negligible error beam, the main beam brightness temperature gives the intensity detected by the main beam, averaged over the solid angle of the main beam, . For telescopes where a significant error beam is present, the main beam brightness temperature potentially overestimates the intensity detected by the main beam, whereas the antenna temperature underestimates it. Both, Kutner & Ulich (1981) and Downes (1989), discuss the consequences of a non-negligible error beam pick-up. Kutner & Ulich (1981) a priori define a temperature scale which gives the intensity averaged over the "diffraction pattern of the telescope'', with the components of the error beam pattern (resulting from surface irregularities) being included. Pick-up from larger angles (due to e.g. spillover past the secondary) are included in the a priori unknown source coupling efficiency . Downes (1989) points out that in case of a significant error beam, the main beam efficiency has to be replaced by the larger full beam efficiency, encompassing the main lobe and the error beam pattern. The resulting quantity is then denoted as beam averaged brightness temperature.
These definitions avoid the problem that the main beam-averaged intensity is
exceedingly overestimated because of the additional error beam pick-up.
The main draw back is, however, that the thus defined quantities
depend on the details of the error beam pattern. In addition, the error beam
pick-up might give rise to "fake'' velocity and spatial features, depending on
the angular extent and the velocity structure of the
source. Therefore, a simple scaling
of the efficiencies cannot properly correct for the error beam
pick-up. This motivates us to introduce a new quantity, the
corrected main beam brightness temperature,
For cm wave telescopes, an equivalent quantity is introduced by Kalberla et al. (1980) with the corrected brightness temperature . In fact, both ( and ) are identical if the same main beam definition is used. However, the main beam definition for cm and (sub-)mm telescopes often differ because of practical reasons. Due to the limited signal-to-noise ratio of the observations at (sub-)mm wavelengths, the dynamical range in intensity often is not sufficient to accurately measure the side-lobes of the beam pattern. Therefore, the main beam definition commonly used for cm telescopes (beam pattern between first nulls, cf. Hartmann et al. 1996) often cannot be applied to (sub-)mm observations. This motivates us to introduce as a new quantity for the (sub-)mm range.
With the observed intensity given in terms of antenna
(Eq. (1)), the
corrected main beam brightness temperature can be written as
For larger (sub-)mm wave telescopes, the error beam pattern often is dominated by surface errors arising from the limited alignment accuracy of the panels in the primary reflector. The resulting error beam pattern has an approximately Gaussian profile, centered on the main beam, with a HPBW corresponding to the correlation length of the surface error distribution (antenna tolerance theory, see Scheffler 1962; Ruze 1952, 1966; Shifrin 1971; Baars 1973).
Detailed studies of the beam pattern for several large (sub-)mm wave
telescopes have shown that the error beam can be represented
by a combination of several Gaussian error beams, arising
from independent surface error distributions. Examples are
the Itatepinga 14 m telescope (Kaufmann et al. 1987), the JCMT
15 m (Hills & Richter 1992; Prestage 1993),
and the IRAM 30 m (Garcia-Burillo et al. 1993; Greve et al. 1998).
The correlation length of the error distributions is given
by the linear dimension of the panels, panel frames, and the mounting
of the panels on the back-structure of the telescope (e.g. the array of
adjustment screws). The similar design of large telescopes, and
because the root mean square (rms) of the surface error is not
negligible compared to the wavelength for (sub-)mm
observations (as opposite to most of the present day cm-wave
telescopes), it is reasonable to assume that the error beam pattern
is dominated by one (or several) Gaussian components,
Equation (4) is only an approximation to and includes two assumptions. Firstly, the Gaussian error beams do not significantly contribute to the integral of the beam pattern on the solid angle covered by the main beam, . Secondly, the contribution of the tapered Airy pattern outside the main beam is small ( is the beam pattern of a telescope with a perfect reflector surface, with no blockage and spillover). The error introduced with both approximations is negligible in most cases. We choose the IRAM 30m to give an example. For wavelengths of 1.3mm, the ratio is 5%, while the contribution of the tapered Airy pattern to the integrated beam pattern outside the -20 dB contour is less than one percent, assuming a primary reflector illumination with a -13 dB edge taper.
For an error beam correction, additional observations are made with a
smaller telescope (main
beam and forward efficiency of
). These observations have to cover a larger area
because a part of the
pick-up originates from outside the map observed with the
larger telescope. An estimate for
in Appendix A with Eq. (A.4),
The error beam subtraction will increase the rms noise of the
corrected spectra because of the noise in the
observations made with the smaller telescope. Therefore, the
rms noise of the latter observations has
to be sufficiently small in order to not
significantly influence the corrected spectra.
being the rms
noise of the original (uncorrected) observations, we define
the rms noise of the corrected spectra as
(Appendix A) and require
to be small (of the order of a few percent). An upper limit
to the required rms noise level of the observations with the
small telescope is given by Eq. (A.2),
In the case of matching HPBWs (of the small telescope's main and the
larger telescope's error beam, i.e.
For an illustration, we assume that the N positions in the
map observed with the large telescope cover a square (a=b).
In addition, we assume equal forward efficiencies for both telescopes,
same equipment and atmospheric conditions for all observations.
Dividing Eq. (9) by
the time needed to complete the
observations in the original map,
For an order of magnitude estimate we choose
and a signal-to-noise ratio of
SNR=10. In addition, we choose
which is appropriate e.g. for
the IRAM 30 m at 230 GHz.
In addition, we assume fully sampled observations for the large telescope,
In order to give a numerical example and to compare the results obtained here to those obtained for a de-convolution of the error beam pattern in Fourier space (Sect. 5), we consider two extreme cases. For case A, we assume that the map to be corrected is small, containing up to a few 10 positions, and that the HPBW of the error beam is substantially larger than the main beam HPBW of the small telescope ( ). For case B we consider large maps with more than 1000 positions, and the case that the small telescope's main and the large telescope's error beam closely match, both giving .
For case A, we obtain . Thus, the time needed for the additional observations is of the same order than the time needed to complete the high resolution observations to be corrected. For case B, the additional time needed is . This is much smaller than unity, unless . In most cases, however, the HPBW of the error beam is larger and the time needed for the additional observations is only a small fraction of the time needed to complete the original map.
|Figure 1: Time needed for the additional observations ( error beam subtraction method): the ratio (total integration time needed for the additional observations to the integration time of original map) is shown for typical ratios = 1, 3, 6, 10, 30, 100, where and denote the error beam HPBW of the large telescope and main beam HPBW of the small telescope. We choose square geometry for the original map, (see text for the additional assumptions made)|
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For a quantification we use the same notation and assumptions as
above. The original map contains N spectra (rms noise
and covers an area of
Additional observations are made with the same
telescope to extend the existing map:
is given by the same
above (Eq. 7) with
being replaced by
Compared to the spectra in the original map, a larger rms noise
and thus a shorter integration time is sufficient for the spectra
in the extended map. In the following, we give an estimate for
Eq. (11) we obtain
For an evaluation of Eq. (15), we
use Eqs. (12, 14)
and the assumptions made in the
previous section (a=b,
same atmospheric conditions for
all observations). This gives
If we consider the limit of small maps with up to a few 10 positions ( , case A introduced in Sect. 4.2) we obtain . Here, the time needed for the additional observations is comparable to and may even exceed the time needed to complete the observations for the original map. In the limit of large maps ( , case B) we get , which is much smaller than 1% in most cases, because .
|Figure 2: Time needed for the additional observations ( error beam de-convolution method): the ratio is shown for ratios =10, 20, 40, 100. and denote the error beam and main beam HPBW of the large telescope|
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For small maps (
Sect. 4.2 as case A) we obtain
An additional limitation is given by the finite angular resolution of observations made with the small telescope. This might be a problem for the correction of observations made with large (sub-)mm telescopes (diameter of several ten meters). Here, large-scale deformations in the reflector surface potentially result in an error beam with a small HPBW, such as the first error beam of the IRAM 30m ( , see Greve et al. 1998). A correspondingly large telescope with a clean beam pattern is then needed for a correction, which may be difficult to find.
For the de-convolution method, additional observations have to be made for a large number of positions with an extremely short integration time per position (only the intensity integrated on the angular extent of the error beam pattern is of interest). This possibly results in an huge overhead in dead time due to telescope motion, data acquisition, etc. In addition, the large number of spectra in the extended map creates a comparably large amount of raw data to be handled in a relatively short time. The application of de-convolution method therefore may be impracticable or even impossible to apply. For an illustration, we give the example of the 12CO 2 1 map of MCLD 123.5+24.9, observed in the framework of the IRAM key-project. The correction for the pick-up in the second error beam requires additional observations with the IRAM 30m at positions and an integration time of less than 0.1s per position. Even more extreme numbers are obtained for the correction of the third error beam. This is currently not feasible and we apply the subtraction method to correct the key-project data.
We discuss and compare two different correction methods in detail. The subtraction method, which uses additional observations made with a smaller telescope, and the de-convolution method, where the error beam pick-up is de-convolved from the observed map in the Fourier space. Both methods have been described and successfully applied to observed spectral line maps (cf. Kalberla et al. 1980; Dame & Thaddeus 1994; Hartmann et al. 1996; Schneider et al. 1998).
The de-convolution method is attractive for the correction of fully sampled maps which cover the full spatial extent of the emission for the given noise limit. Here, the observed map contains all information required to correct for the error beam pick-up. For maps which cover only a fraction of a more extended intensity distribution, both correction methods require additional observations. Observations with a smaller telescope are required for the subtraction method, while the existing (fully sampled) map has to be extended for the de-convolution method. Typically, the additional observations require less time than was needed to complete the original observations to be corrected (in terms of total integration time), except for small maps with a few ten positions. For large maps ( positions), such as those typically observed with large surveys, the time needed is only a few percent of the time spent for the original map. Though this a minor fraction, it should be taken into account when planning large mapping projects and a corresponding fraction of the time should be allocated for the additional observations required.
For the correction of a small map or single spectra, the subtraction method is favorable because the additional observations can be done more efficiently with a smaller telescope (in this context, "small'' means that the angular extent of the original map is smaller than a few times , where is the HPBW of the error beam and is the HPBW of the small telescope). The subtraction method is limited by a potentially significant error beam pick-up in the observations made with the smaller telescope and the lower angular resolution. In particular, one has to consider that smaller telescopes tend to have cleaner beams allowing for a more accurate correction of the error beam pick-up, but possibly do not have the required angular resolution.
For the correction of large maps (angular extent error beam), the de-convolution method is attractive because the integration time for the additional observations is smaller than for the subtraction method. In practice, however, these observations are often limited by the data handling capacity and dead time (as a large amount of data has to be collected in a relatively short time) and by the limited availability of large (sub-)mm telescope for such (additional) observations.
The KOSMA 3m radio telescope at Gornergrat-Süd Observatory is operated by the University of Cologne and supported by the Deutsche Forschungsgemeinschaft through grant SFB-301, as well as special funding from the Land Nordrhein-Westfalen. The receiver development was partly funded by the Bundesminister für Forschung and Technologie. The Observatory is administered by the Internationale Stiftung Hochalpine Forschungsstationen Jungfraujoch und Gornergrat, Bern. The authors have benefitted from a joint European grant (Procope, grant# 312-pro-bmbw-gg) during the years 1995 and 1996. This research made use of NASA's Astrophysics Data System Abstract Service.
A spatially extended source is observed and the resulting map covers a rectangular area . The error beam correction is done using the subtraction method. Additional observations are made with a small telescope (main beam HPBW and efficiency of , forward efficiency ). They cover a larger area because of the angular extent of the error beam, .
The error beam pick-up is estimated using Eq. (6), with the summation being done over
only one Gaussian error beam. The corrected main beam brightness
temperature is then approximated by
With the accuracy of the estimated error beam pick-up denoted by
the accuracy of the corrected main beam brightness temperature scale
For a reliable error beam correction, the additional error
introduced by the rms noise and the finite size of the observations
made with the small telescope
should be small compared to the rms noise in the original
Therefore, we can assume
which allows us to use the approximation
Typical values ( , and ) give a ratio ( ) of the order of a few tenths.