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Astron. Astrophys. 362, 1109-1121 (2000)

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Appendix A: on-the-fly observing mode

Scanning a radio source at a constant speed while continuously integrating the received data over certain time intervals, such that a grid of data results, is the essence of the OTF observing mode. The lack of dead times between the different ON positions and less time on the OFF position makes this mode much more effective than the raster mapping mode where, for data integration, the telescope stops at each ON position. However, collecting data "on-the-fly" causes a "smearing", i.e. broadening, of the effective beam along the scanning direction.

A.1. Beam smearing and sampling

In order to quantify the "smearing" effect we neglect, for the sake of simplicity, the fact that the beam has a two-dimensional structure and consider only its one-dimensional profile along the scanning direction.

Let [FORMULA] be the respective coordinate in the plane of the sky, [FORMULA] the width of each of the integration intervals, and [FORMULA] the intrinsic beam pattern. The (normalized) effective OTF beam pattern, [FORMULA], is then given by the convolution of the original beam pattern and the box function, i.e.

[EQUATION]

where the box function, [FORMULA], is defined by

[EQUATION]

and where

[EQUATION]

The normalization is such that [FORMULA].

A.1.1. Critical sampling

As shown by antenna theory, the Fourier transform of the telescope beam [FORMULA] vanishes at the well-known cutoff frequency [FORMULA], where D and [FORMULA] denote the diameter of the antenna and the observing wavelength respectively. Hence,

[EQUATION]

vanishes at [FORMULA] as well, implying that [FORMULA] is a band limited function that, according to the Nyquist theorem, is fully specified by values spaced at equal intervals not exceeding [FORMULA]. In Eq. (A.4), the effect of the OTF mode enters only via the sinc function which does not affect [FORMULA]. Adopting [FORMULA] the first zero of the sinc function is at [FORMULA] which is already beyond the critical frequency. OTF broadening of the effective beam may be regarded as a slightly enhanced edge taper along the scanning direction. One should also note that disturbing effects due to an imperfect surface of the dish, aperture blocking by the secondary and the feed legs, or spillover at the edges of the primary and secondary, while causing changes in the details of the frequency distribution [FORMULA], do not affect the cutoff frequency. This is also true taking the two-dimensional structure of the beam into account. Thus we conclude that, from a rigorous point of view, critical Nyquist sampling provides the maximum spatial information both in the position switching modes and the OTF mode.

Usually, the Nyquist interval is expressed in units of the HPBW of the antenna. With an edge taper of -14dB, the [FORMULA] (e.g. Goldsmith 1982), and [FORMULA] HPBW/2.4.

A.1.2. Effective beam width

Employing (A.1) and (A.3), the effective width of the OTF beam, [FORMULA], corresponding to the "beam solid angle" of a real antenna, reads

[EQUATION]

where [FORMULA] denotes the effective width of the original beam. Approximating the original beam by a simple Gaussian, i.e. [FORMULA], and putting [FORMULA] HPBW/2.4, one finds a ratio of the effective widths of

[EQUATION]

Thus, for integration intervals not exceeding the critical Nyquist sampling intervals, the on-the-fly observing mode leads to an effective broadening of the beam along the scanning direction of at most four percent. One finds an even slightly smaller amount, taking the two-dimensional structure of the beam into account.

We therefore decided to sample the Cepheus B cloud at approximately the Nyquist interval at 345 GHz, i.e. [FORMULA], in scanning direction as well as perpendicular to it.

Appendix B: error beam correction

The KOSMA main beam efficiency at 345 and 330 GHz was only 55% in October 98. The telescope beam therefore detected a substantial fraction of the power with underlying error beam(s) caused by surface deviations of the telescope primary mirror from the ideal paraboloid. Imperfections are mainly caused by misalignment of the 18 primary panels, since the individual panels are nearly perfect ([FORMULA]m rms). Following the argument of Greve et al. (2000, Eq. 12) who discuss the error beams of the IRAM 30m telescope, the KOSMA telescope probably had an error beam with a width of [FORMULA] with the observing wavelength [FORMULA] and the correlation length L of surface deformations. The correlation length is roughly given by the mean panel diameter, i.e. [FORMULA] cm. The estimated error beam width thus is [FORMULA]. We rely on this estimate, since direct measurements, derived e.g. from observations of the moon edge, are missing.

Using the main beam efficiency to derive the source brightness temperature distribution leads to temperatures which are too high when extended objects are observed. In addition, the underlying error beam leads to a smearing of the intensity distribution. We corrected for the error beam using the second-order deconvolution method applied by Westerhout et al. (1973) to 21 cm data of the NRAO 300 ft radio telescope. See Bensch et al. (2000) for a discussion of other possible correction methods. The corrected brightness temperature averaged over the main beam is given by

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA] are the integral of the beam pattern over the forward hemisphere, error beam, and main beam of the telescope. [FORMULA] is the brightness temperature averaged over the error beam. [FORMULA] can be expressed in terms of [FORMULA], analogous to Westerhout et al. (1973, Eq. 10), but for only one error beam:

[EQUATION]

[EQUATION]

where [FORMULA] is the original data convolved with a Gaussian of [FORMULA] width, the width of the error beam. With [FORMULA], [FORMULA], [FORMULA], and [FORMULA], we obtain:

[EQUATION]

or [FORMULA] where E is the error beam contribution. Note that this is not only a simple scaling factor to the spectra. Instead, the varying errorbeam pickup is corrected for both in position and velocity, thus also possibly correcting line profiles. Resulting maps show an enhanced contrast (Fig. 1, Fig. 2).

The region mapped at 345 GHz is [FORMULA], a factor of 10 larger than the error beam. We therefore anticipate the correction to work in the interior parts of the map including all four positions selected for a detailed analysis, but not at the map edges.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000
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