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Subsections



5.9 Instrumental Field of View: The Beam Profile

Figure 5.16: Offset positions of the Mars beam profile observations.
\resizebox {12cm}{!}{\includegraphics{cl_MarsRasterPA.eps}}

The beam profile has been derived from a series of standard bias, grating scan observations of Mars. One observation was made on-axis and the others were distributed around the field as shown in Figure 5.16. The flux at three wavelengths in each detector, at each of the raster positions, are used to describe the radial sensitivity of the instrument and a parameterisation defines the beam profile. In addition subsets of the data along the four radial alignments, which are labelled PA30, PA75, PA120 and PA165 on Figure 5.16, have been analysed to investigate possible asymmetries in the beam profile.

Before discussing the beam profile it is important to appreciate the properties of the optical path leading to the LWS detectors as these have a profound impact on the beam profile. The optical train of the LWS consist of the contour field mirror that allows a beam of $\sim$120 $^{\prime \prime}$ to fall on the complex mirror M2, which is inclined at $\sim$22$^{\circ}$ to the incoming beam. M2 diverts the beam by $\sim$44$^{\circ}$ back through a semi-cylindrical cut-out in the contour field mirror and then presents an elliptical beam of $\sim$105 $\times$ 97 $^{\prime \prime}$ (nominal FWHM) to the collimator, the LWS entrance pupil, the re-imaging mirror and detectors. Immediately in front of each detector is a rectangular aperture with rounded ends, which when projected onto the sky is approximately elliptical with dimensions of 104 $\times$ 157 $^{\prime \prime}$ at SW1 and 138 $\times$ 131 $^{\prime \prime}$ at LW5 in the directions along and across dispersion respectively. The long and short axes of M2 projected onto the sky are PA30 and PA120 (Y and Z axes) respectively.

As the detector apertures are nominally larger than the incoming beam from M2 it is, in fact, M2 that defines the aperture of the detectors, and the character of the optics determines the instrumental profile. Perversely, the substrate that supports M2 is also reflective, particularly at longer wavelengths, and is now believed to be responsible for the fringing that is seen in off-axis targets. A second consequence of this is that the instrumental profile will have weak wings out to $\sim$120 $^{\prime \prime}$ diameter (the size of the contour field mirror).

The other feature of LWS grating spectra of objects observed off-axis is the poor stitching between adjacent detectors, which is often referred to as fracturing. The problem is apparently worse for objects in the part of the field that passes close to the cut-out in the contour field mirror. The origin of this problem is unknown but it introduces a complex, wavelength-dependent asymmetry into the instrumental profile. These problems are described in more detail in Sections 6.2 and 6.3.


5.9.1 The beam profile

The observed beam profile is the result of the convolution of the telescope PSF and the instrumental profile of each detector. Ideally the telescope PSF would be an Airy profile but the central obscuration and secondary supports, and any optical imperfections will conspire to redistribute power from the core of the profile to the Airy rings. Although a point source is being used to probe the structure of the beam, the width of the PSF, which is essentially an Airy profile, increases from $\sim$25 to 100 $^{\prime \prime}$ (FWHM) between 46$\mu $m and 178$\mu $m, and at the longer wavelengths becomes comparable with the size of the beam. Due the problems of fracturing and fringing the true shape of the instrumental profile is largely unknown.

Figure 5.17: The beam profiles for the central wavelength of each detector. The observed relative flux at each offset position is plotted against radius, with the sign taken from the RA offset in Figure 5.16. The line on each plot shows the calculated radial profile.
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To determine the effective beam size the observed fluxes at three wavelengths in each detector have been compared with those derived from a convolution of the telescope PSF with apertures of various sizes. The latest model of the telescope PSF includes the effects of the central obscuration and its supporting structure, and indicate that the power in the Airy rings is increased and that the wings of the profile contain 2-D structure. The asymmetry introduced into the profile is due to the three-legged secondary support. The aperture has been assumed to be circular with a rectangular profile.

It has previously been recognised that the effective apertures are significantly smaller that the nominal value of 100 $^{\prime \prime}$, based on the size of the beam from M2. The best value for most of the detectors lies close to 80 $^{\prime \prime}$, and for LW3, LW4 and LW5 is somewhat smaller than this. At the longest wavelengths the width of the telescope PSF is larger than the aperture itself which makes these determinations more difficult. An uncertainty of one arcsec in the radius corresponds to about 5% in the effective area of the aperture.

Also, although a simple circular aperture has been adopted, more complex shapes can provide a better description of the asymmetries, under some circumstances. However, the range of possible shapes and number of free parameters makes this approach untenable.

The relative flux at each of the observed offset positions (see Figure 5.16) is shown for each detector in Figure 5.17 with the convolution of the telescope PSF and the best fit composite aperture superimposed. At shorter wavelengths the telescope PSF is narrow enough to probe the structure of the rectangular (top hat) instrumental profile and some indication of its shape can be seen. As the telescope PSF broadens towards longer wavelengths the details of the instrumental profile become washed out and the observed profile becomes more Gaussian.

The resulting estimates for the effective beam size for each detector are given in arcsec in Table 5.9.


Table 5.9: Effective aperture of the detectors.
Detector Effective Detector Effective
  radius [ $^{\prime \prime}$]   radius [ $^{\prime \prime}$]
SW1 39.4 LW1 38.6
SW2 42.3 LW2 38.9
SW3 43.5 LW3 35.5
SW4 40.9 LW4 34.7
SW5 39.5 LW5 33.2

The effective solid angle of the detectors is required to determine the point/extended source flux correction, for the conversion of observed flux to flux per steradian for extended sources and for the calibration of sources observed in parallel and serendipity mode (see Table 5.10).


5.9.2 Asymmetry

It is already clear from Figure 5.17 there is some asymmetry in the beam profile, with groups of points lying systematically off the lines. The question of symmetry is not straightforward, and a simple analysis of the relative mean fluxes for the four radial alignments begins to show this.

Each of the radial alignments is a subset of the data which define the global mean profile. The four alignments are separated by approximately 45 $^{\circ}$ to create two orthogonal pairs, with the PA30 set aligned along the Y axis, and the PA120 set aligned along the Z axis of M2. A Gaussian profile constrained to the optical axis was fitted to each set with the on-axis point and those with $r~ >~ 65^{\prime \prime}$ excluded. Although each alignment contains nine points, only six points are used in the PA30 and PA120 solutions and only four points in the PA75 and PA165 solutions.

Figure 5.18: FWHM vs. detector for the four alignments.
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Of the parameters derived from these solutions the FWHM is probably the most reliable, and this is shown in Figure 5.18 for the different detectors. Although there is considerable scatter the behaviour is fairly consistent. Each alignment shows an increase in FWHM through the SW detectors and then a subsequent decline, which reflects the behaviour of a Gaussian fit to all the data, and the run of effective aperture sizes in the table above. In more detail it can be seen that the alignments fall naturally into two pairs with very similar behaviour. PA75 and PA120 show much more variation than PA30 and PA165, and peak at SW3 as opposed to SW5. In particular PA75 and PA120 run through the fractured region; see the spectra in Figure 6.3 of raster positions 27 and 30 in Figure 5.16.


5.9.3 Flux correction for extended sources

The LWS flux scale is based on a point source calibration, although beyond the diffraction limit at about 110 $\mu $m a substantial fraction of the flux from an on-axis point source is diffracted out of the aperture. In fact there are significant diffraction losses for all LWS detectors but provided the calibration is applied to point sources observed on-axis these losses are irrelevant because they are cancelled out in the calibration process. However, for extended sources the diffraction losses do not occur and so a correction has to be applied to correctly place the derived fluxes on the point source calibration scale. The correction factor to apply to the fluxes in case of extended sources has been calculated at three wavelengths per detector. These factors are given as $f$ in Table 5.10.

5.9.4 Extended source flux per unit area

To convert the observed flux of an extended source to flux per steradian requires both the extended source correction factor ($f$, see above) and the effective aperture in steradian explicitly.


Table 5.10: Table of extended source correction ($f$) and effective solid angle of the beam ($\omega$) for the different LWS detectors, at three different wavelengths per detector. The effective aperture radius in arcsec ($r_e$) is also given at the same wavelengths.
Detector $\lambda [\mu$m] $f$ r$_e$ [ $^{\prime \prime}$] $\omega
\times 10^6$ [sr]

43.0 0.8721 39.3 0.1140
SW1 46.2220 0.8704 39.4 0.1146
  50.0 0.8691 39.4 0.1146

50.0 0.8705 42.6 0.1340
SW2 56.2033 0.8677 42.3 0.1321
  60.0 0.8563 42.1 0.1308

60.0 0.8634 43.5 0.1397
SW3 66.1173 0.8421 43.7 0.1410
  70.0 0.8127 43.4 0.1390

70.0 0.7845 41.2 0.1253
SW4 75.6989 0.7334 40.7 0.1223
  80.0 0.7118 40.9 0.1235

80.0 0.6904 38.4 0.1088
SW5 84.7977 0.6878 39.7 0.1163
  90.0 0.6803 40.3 0.1199

90.0 0.6753 38.3 0.1083
LW1 102.425 0.6758 38.8 0.1111
  108.0 0.6757 38.6 0.1100

108.0 0.6761 39.6 0.1157
LW2 122.218 0.6734 39.1 0.1128
  130.0 0.6557 38.1 0.1071

130.0 0.6445 36.5 0.0983
LW3 141.809 0.6035 35.6 0.0935
  150.0 0.5623 34.3 0.0868

150.0 0.5727 35.3 0.0920
LW4 160.554 0.5411 35.0 0.0904
  170.0 0.4855 33.9 0.0848

170.0 0.5002 34.3 0.0868
LW5 177.971 0.4596 33.6 0.0833
  195.0 0.3749 31.6 0.0737

The corrected extended source flux for an observed flux $F$ given in Jy is $ S = F\,\times\,f/(\omega \times 10^6) $ MJy/sr, where $f$ is the extended source correction and $\omega$ is the effective solid angle of the beam in sr. These values are now given at three different wavelengths per detector (from Version 2.1 of this volume of the ISO Handbook on). As visible on Figure 5.19, for most detectors the resulting correction factor presents a gradient with wavelength. This has the positive consequence that with this correction there is a better agreement between the corrected fluxes of extended sources in the overlap regions between detectors.

Figure 5.19: Correction factor for extended sources ( $f/(\omega \times 10^6$) versus wavelength. The correction factor is given at three different wavelengths per detector. Detectors are given alternately in red and blue; red, SW1, SW3, SW5, LW2, LW4, and blue, SW2, SW4, LW1, LW3 and LW5.
\resizebox {!}{8cm}{\includegraphics{BeamParameters.eps}}

An LIA routine has been provided that applies this correction to averaged, de-fringed LSAN files called EXTENDED_FLUX. However, these corrections are derived under the assumption of a smooth and very extended flux distribution. In the real world, structured or embedded sources could produce significant discrepancies from the ideal situation, and, with it, differences in flux.

Figure 5.20: Comparison of LWS and IRAS 100 $\mu $m fluxes in the Trumpler 14 (red diamonds) and 16 (black diamonds), and Galactic Centre (blue squares) fields.
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LWS observations have been made at a number of positions in the Trumpler 14 and 16, and Galactic Centre fields and these have been compared with the IRAS 100 $\mu $m fluxes at the same positions. Both fields contain a large area of extended emission, which although relatively smooth, does change by a factor of $\sim$40 over all. Figure 5.20 shows the comparison of the converted LWS and IRAS 100 $\mu $m fluxes using the current LWS calibration. These measurements give a mean ratio, LWS/IRAS $\sim$1.0$\pm$0.1. Ideally for this comparison the extended flux should be distributed as evenly as possible, and part of the uncertainty is probably due to unresolved structure within the beam.

For fields containing multiple sources the observed flux will depend critically on the precise positions of the sources relative to the optical axis. To recover or model the observed flux will require positional information on the sources and a deconvolution with the telescope PSF and instrumental profile. Indeed, for single point sources observed off-axis a similar procedure will be required to recover the correct flux.


5.9.5 Effect of the ISO PSF at large distances: check of the straylight around Jupiter

A complete study of the beam profile of the LWS has been performed only for distances within 150" of the central source. Rasters of larger extent would have been too time-consuming. There were, however, spot checks of the flux entering the instrument at even larger distances from a very strong source: The off-position spectra for Ganymede and Callisto. They provide us with the fluxes from Jupiter, when the aperture of the LWS was pointed at distances of 5' and 9' from this planet. The results are shown in Table 5.11 and compared to the fluxes expected from a model of the PSF by Okumura 2000, [30]; see Figure 5.21.


Table 5.11: Flux at a certain distance of Jupiter normalised by the flux of Jupiter: Comparison of LWS observations of Jupiter straylight with a model of the ISO PSF by Okumura 2000, [30].

  SW2 model LW2 model
Distance Angle to S/C Z $I_{off}^{SW2}/I_{J}$ $I_{off}^{PSF}/I_{J}$ $I_{off}^{LW2}/I_{J}$ $I_{off}^{PSF}/I_{J}$
 [arcsec]  [$^{\circ}$] $\times 10^{-6}$ $\times 10^{-6}$ $\times 10^{-6}$ $\times 10^{-6}$
282 38 860 77 1100 406
291 325 440 94 890 125
530 113 84 14 300 66
532 20 38 0.1 100 25
535 2 27 5.5 52 8
537 148 73 13 350 26
545 344 53 2.8 190 52

The significance of the correlation between the measured flux and the model PSF is 2.4 $\sigma$ for detector SW2 at 56 $\mu $m and 2.6 $\sigma$ for detector LW2 at 122 $\mu $m. The correlation between flux and distance or angle alone is much weaker, hence the model PSF reflects correctly the observed flux pattern up to a distance of 9$^{\prime}$ from the source. On the other hand the measured fluxes are systematically higher than what is expected from the optical model, which could be due either to the fact that Jupiter is not a point source or to the existence of significant wings in the beam profile.

Figure 5.21: A model of the point spread function as sampled with the LWS. The contours show the drop in intensity by factors 10$^{-2.5}$ (continuous line) and 10$^{-3.5}$ (dotted line) compared to the on-source value.
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next up previous contents index
Next: 5.10 Grating Wavelength Calibration Up: 5. Calibration and Performance Previous: 5.8 Quarter-Second Processing
ISO Handbook Volume III (LWS), Version 2.1, SAI/1999-057/Dc