Astron. Astrophys. 361, 407-414 (2000)

3. The power spectra and the simulation

For each image shown in Fig. 1 the 2-dimensional angular correlation function was calculated from the brightness distribution :

where bracket represents the average over the whole area in each image shown in Fig. 1, and . Then, the coordinates were expressed in polar coordinates as , and the Fourier transforms in various radial directions were calculated (i.e. is fixed for each transform). One-dimensional power spectral density (PSD) is calculated from

where f is spatial frequency (), (), is the pixel size of the map, is the largest angular size for which the angular correlation function is evaluated, and . Note that the units of the PSDs are the same as that of the angular correlation function: . Finally, the PSDs were averaged with respect to and are shown in Fig. 2. It is noteworthy that the fluctuations at high frequencies ( arcmin^-1 for 90 µm, and  arcmin^-1 for 170 µm) are smoothed by the instrumental beam and therefore the PSDs decrease appreciably. The small error bars of PSDs represent the standard deviation among a set of the PSDs with different , showing that the PSDs are almost independent of .

Fig. 2. Fluctuation power spectral densities (PSDs) of 90 µm (top) and 170 µm (bottom) images. Open circles represent PSDs of LHEX and filled circles represent PSDs of LHNW. As well as the PSDs of the original images (Fig. 1) shown by circles connected by thin-solid lines, the PSDs of the residual images ("residual PSDs"), where the pixels containing bright sources above 150 mJy (90 µm), 250 mJy (170 µm) are masked, are also shown by circles alone. The brightest source (IRAS F10597+5723) located in LHEX significantly contributes to the PSDs of the LHEX images. The 1 error bars, shown only for the residual PSDs, represent the standard deviation in a set of the PSDs with different position angles in the sky. Thick line is an example of the IR cirrus PSD, which is an average spectrum of several IR cirrus in Ursa Major, with 100 µm brightness of 2-3 MJy/sr. Dotted lines are the IR cirrus PSDs in the Lockman Hole, estimated by assuming that the cirrus PSDs are proportional to , where is mean brightness of the cirrus cloud.

In order to check the contributions from bright sources, the PSDs are derived by masking circular regions with a 4FWHM diameter around bright sources with fluxes above : mJy at 170 µm ¹, mJy at 90 µm. In the following the resultant PSDs are called "residual PSDs", and are also shown in Fig. 2. Interestingly, the residual PSD for each image is more than half of the PSD of corresponding original image with almost the same spectral shape, indicating that there remains significant contribution from randomly distributed point sources with fluxes below .

In Fig. 2 typical IR cirrus PSDs are also compared in order to check the contribution of the IR cirrus to the PSDs of the Lockman Hole ². We examined several IRAS 100 µm maps of high-latitude clouds in Ursa Major (, ), which are reproduced from the IRAS Sky Survey Atlas (ISSA) by reducing the brightness by a factor of 0.72, following the COBE/DIRBE calibration (Wheelock et al. 1994). The average brightness of the cirrus is 2-3 MJy/sr, and each map is wide with . The derived PSDs show a power-law distribution with an index of -1.5. Gautier et al. (1992) noted that one-dimensional analysis of the IR cirrus yielded spectral indices near -2. These cirrus fluctuation spectra are much different from those obtained for the Lockman Hole images at 90 µm, which show rather flat spectra at lower spatial frequencies. The 170 µm spectra present a slope similar to the IR cirrus one, but this can be explained by the shape of the footprint power spectrum of ISOPHOT detectors. Moreover, the fluctuation powers are much larger than those estimated for the IR cirrus, which are also shown in Fig. 2. We assume that the cirrus PSD is proportional to (Gautier et al. 1992), and taking the mean brightness of the IR cirrus in the Lockman Hole as 0.33 MJy/sr at 90 µm and 1.0 MJy/sr at 170 µm. These values are estimated from the atomic hydrogen column density of in LHEX (Jahoda et al. 1990) and the COBE/DIRBE data analysis in the Lockman Hole by Lagache et al. (1999, in their Table 4). We can conclude that the IR cirrus contribution to the PSDs in the Lockman Hole is negligible over all spatial frequencies .

In the following we interpret the residual PSDs in terms of unresolved point sources, probably galaxies. We neglect the spatial correlation between galaxies, thus assuming that galaxies are randomly distributed in the images. Then the residual PSD will be the product of the footprint power spectrum of the ISOPHOT detector and the fluctuation power due to the point sources, which is a white power spectrum given by:

where is differential source counts. From the residual PSDs observed, we derive at 90 µm () and at 170 µm (), where the errors do not include systematic ones due to uncertainties in the flux calibration. Lagache & Puget (2000) reported the detection of at 170 µm for the Marano 1 field, after sources brighter than 100 mJy are removed. Contribution from detected sources with fluxes between 100 mJy and 250 mJy is estimated to be about 7000 . Thus the fluctuation power in the Lockman Hole due to the sources fainter than 100 mJy is , which is comparable to that observed in the Marano 1 field.

A simulation was performed by making 90 µm and 170 µm images made up only by galaxies with fluxes between and and calculating their PSDs. Here galaxies are treated as point sources with a PSF specific to the respective wavelength band of ISOPHOT. We used the image of the bright IRAS source (F10507+5723) seen in LHEX images as the PSF. The number of sources and their flux densities are controlled by the source counts. We examined the non-evolution count model by Takeuchi et al. (1999) and the scenario E by Guiderdoni et al. (1998). We assume because the fluctuations due to galaxies fainter than 10 mJy are negligible in case of these models. The resulted PSDs are compared with the residual PSDs in Fig. 3. These simulated PSDs are not sufficient to explain the observed PSDs although the spectral shapes are quite similar.

Fig. 3. The residual PSDs of 90 µm (top) and 170 µm (bottom) images (open circles and filled circles, same as Fig. 2) are compared with the simulated PSDs based on various number counts models: dashed lines are PSDs of the simulated images by Guiderdoni et al. (1998) scenario E, while the dotted lines are those by Takeuchi et al. (1999) no-evolution. Thick gray lines are examples of the simulated images produced by simple double power-law number count models (see text for details).

© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000
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