The image of a perfect point source obtained with a given detector or the footprint of a detector should be known in case one would like to perform:

As part of the PHT calibration programme, extensive mapping of astronomical point sources have been carried out in order to determine the footprints of the PHT-SS/SL, C100 and C200 detectors and the PHT-P apertures.

The footprints of the C100 detector array pixels were observed at 60 and 105 ${\mu}$ m (filterbands C_60 and C_105) and of the C200 array at 120 and 200 ${\mu}$ m. Footprint maps of the four C200 pixels at 120 ${\mu}$ m are presented in Fig. 4.8.

**Figure 4.8:** The measured pixel images in the C_120 filter of the planetary nebula NGC7027 which is assumed to be a point source for ISO. The footprints have been normalized to 1 at peak position. Contour levels are 2.5, 5, 10, 12.5%, 20 to 90 with step of 10%, and 98% of peak.
$\begin{figure} \begin{center} \begin{tabular} {c} {\epsfxsize=8cm \epsfbox{c120footmap.eps}} \\ \end{tabular}\end{center}\end{figure}$

The measurements were compared with a theoretical model based on the telescope point spread function measured with ISOCAM. The model assumes a two mirror f/15 telescope with radii for the primary and secondary mirror of 30 and 10 cm, respectively. It includes the stray cones of the secondary's support tripod. The model footprint is calculated from the convolution of the point spread function with the pixel surface. Comparison between the predicted and the measured footprint shows good agreement. The inferred effective solid angles of the pixels are all larger by 20 - 60% than the values given in the Observers Manual (Table 9 in [10]). This can be understood by the widening of the point source profile due to infrared light scattered by the support legs.

The effective solid angle enters the extended source calibration in the conversion from Jy/pixel to MJy/sr, see section 5.3.

4.5.1.2 PHT-P footprints

From the large number of different footprints of the PHT-P group the results for the following combinations have been analyzed:

The P2 profiles show that the central plateau of the footprint is asymmetric along the spacecraft Z-axis. It also does not peak at the central position. Cuts through the profile are plotted in Fig. 4.9.

**Figure 4.9:** Slices along the Z-axis through the footprint of P2 at 25 ${\mu}$ m using the 79'' aperture. The bold curves give the central slice, the step size between neighbouring slices is 20". The profiles are asymmetric suggesting that the point spread function is not fully centred on the aperture.
$\begin{figure} \begin{center} \begin{tabular} {c} {\epsfxsize=15cm \epsfbox{p2_25_79_zbeam.eps}} \\ \end{tabular}\end{center}\end{figure}$

The P3 profiles show less pronounced asymmetries in the central plateau. All profiles are currently compared to the theoretical model described in the previous section. First results indicate an effective solid angle to deviate $\pm$ 5-20% from the values given in the Observers Manual ([10]). In Fig. 4.10 we present a high resolution one dimensional scan over the star HR6705 ( $\gamma$ Dra). Assuming that HR6705 is a true point source and applying the same observing mode to a given astronomical target, the scan can be used to decide whether the target is pointlike or extended. The observation comprise a forward and backward going raster scan; this is achieved by setting the number of raster steps to 2 with stepsize 0'' for the cross-scan parameters.

**Figure 4.10:** High resolution scan of the star HR6705 at 60 ${\mu}$ m (P3, P_60) using the 23'' aperture and 6'' step. The observations are marked by crosses, negative values are not shown. The solid line gives the expected profile using an ISO PSF model convolved with the 23'' aperture.
$\begin{figure} \begin{center} \begin{tabular} {c} {\epsfxsize=12cm \epsfbox{psfcomp.eps}} \\ \end{tabular}\end{center}\end{figure}$

4.5.1.3 PHT-S footprints

The size of the PHT-S aperture ( $24''{\times}24''$ ) has been designed to mask the part of the sky which is directly mapped onto the PHT-S pixel at a given wavelength. In-orbit observations have shown that the PHT-S response is very sensitive to pointing variations perpendicular to the dispersion direction which is the spacecraft Y-direction.

By scanning a point source over the PHT-S aperture in Y and Z-direction, and assuming axisymmetry, PHT-S footprints have been determined for each pixel in both the SS and SL arrays. The resulting profiles show that the general shape of the PHT-S footprints are sharply peaked in non-dispersion direction (spacecraft Y-direction). The profile in dispersion direction (spacecraft Z-direction) is flatter over the aperture. These in-orbit calibrations confirm the results of ground based measurements of the beam profiles. An example of a footprint is given in Fig. 4.11.

Another feature of the PHT-S footprints is that not all peak at the centre of the aperture. For some pixels the centre of the array is on a steep flank of a footprint. As a result, the shape of the spectrum strongly depends on the exact pointing in Y-direction. For example, in case of pixel 60 (Fig. 4.11) a pointing error of $\pm$ 3 arcsec in Y directon can cause a change in response of the order of 20%.

The PHT-S spectral response function used in the current OLP version is only valid for the centre position of the PHT-S aperture. Corrections for pointing offsets can be applied using the footprints of the individual pixels. In a later version of this manual the pixel profiles in Z and Y direction will be made available.

**Figure 4.11:** Two dimensional beam profile of PHT-S pixel 60 as measured in the ISOPHOT ground calibration facility. This is an example where the intensity maxima are multiple and are off the major axes. In-orbit cross-scans along the major axes were performed which show qualitatively the same behaviour: a relatively flat and symmetrical shape in the dispersion direction and a slightly off-centre and steeply falling off profile in cross-dispersion direction.
$\begin{figure} \begin{center} \begin{tabular} {c} {\epsfxsize=12cm \epsfbox{spix602dbeam.eps}} \\ \end{tabular}\end{center}\end{figure}$

Based on the footprints of all PHT-S pixels the correction factors for the transformation from point source flux to extended source surface brightness have been derived. A homogeneous illumination of the PHT-S aperture is assumed.

4.5.2 Point source intensity fractions

The intensity fractions, or f_psf, of a point source passing through a given filter/aperture combination for PHT-P or filter/pixel combination for PHT-C has been calculated using a simplified model of the point spread function. This model assumes a uniformly illuminated round mirror of 30 cm radius with a f15 focal length and a central obscuration of 10 cm radius. The sizes of the apertures and detectors as determined on ground together with the central wavelength of the filters have been used to determine f_psf.

For C200 the fraction of light falling on the whole array is significantly different from the fraction calculated for the model described above in which the physical dimension of the detector arrays have been included. This is probably due to losses and diffraction at the inter-pixel gaps of order 100 ${\mu}$ m when the point spread function is centred on the C200 array and not on one of the pixels. Empirical correction factors obtained from calibration observations have been determined to correct for this effect.

4.5.3 Chopper vignetting/offset

Although the guaranteed unvignetted field of view for the ISO instruments is 3 arcmin, PHT can have chop throws up to $\pm$ 90'' for the largest aperture of 180'' and $\pm$ 165'' for the smallest aperture w.r.t. the CFOV. It is therefore expected that for larger chopper throws corrections should be made for possible vignetting of the chopper beams. As the chopped beam has a different ray path through the telescope-instrument system additional chopper offsets can occur which have been observed, too.

The combined effect of chopper vignetting and chopper offset can cause a relative signal difference between the on- and off- chopper beams for an otherwise flat sky. As a consequence, chopped observations of faint sources which are only a few percent brighter than the sky background can only be analyzed after corrections for these effects.

Calibration observations indicate that chopper offsets for the PHT-P detectors are small: less than 1% flux difference between the chopper positions for a flat sky. For the PHT-C detectors, the offsets depend on the chopper throw and can become substantial for the largest amplitudes (as high as 6%). Note that maps obtained with PHT32 need to be corrected for this effect.

4.5.4 Inhomogeneous FCS illumination

The detectors are not homogeneously illuminated by the FCS. For the PHT-P detectors the measured flux does not scale proportionally with the aperture area, while for the PHT-C detectors the different pixels are not receiving the same in-band power. Illumination matrices or factors for correcting the inhomogeneous illumination over the 180 arcsec field of view have been determined for each filter/detector/aperture by in-flight measurements.

For the PHT-C arrays the illumination matrices include the relative differences in responsivity among the pixels, i.e. the flat-field averaged over many observations (see also section 4.2.6). This means that an observation of a flat region of the sky should yield a flat image after calibration with the FCS measurement in the same filter.

The FCS illumination matrix is filter dependent. As a consequence, when using an FCS measurement made e.g. in the C_90 band to calibrate a measurement made in the C_60 band, the filter dependent flat-fields have to be accounted for, see section 4.2.6.

For the PHT-P detectors strong deviations from a linear scale have been measured when changing the FCS aperture while maintaining the same FCS heating power. The calibration of the FCS against astronomical standards (section 5.2) has been performed only with a few selected apertures:

The corrections using illumination matrices for C100 and C200 are included in the SPD processing, see section 13.12 for a description of the related Cal G file. It is planned for a future version of the software to include aperture corrections.

4.5 Optical Performance

4.5.1 Instrumental Footprint on the Sky

4.5.1.1 PHT-C footprints

4.5.1.2 PHT-P footprints

4.5.1.3 PHT-S footprints

4.5.2 Point source intensity fractions

4.5.3 Chopper vignetting/offset

4.5.4 Inhomogeneous FCS illumination