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

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4. Constraints on the number counts

In this section we will investigate the source counts below [FORMULA], by fitting the simulated PSDs to the residual PSDs. In Paper I we presented the number density of sources brighter than 150 mJy at both filter bands. After Paper I was published, the data processing and the source extraction technique have been improved, and now we obtained source number counts down to [FORMULA] mJy at 90 µm and [FORMULA] mJy at 170 µm, although the counts at such low fluxes may be overestimated due to the source confusion. We found that the cumulative counts above 150 mJy show quite a steep increase as the source flux decreases: [FORMULA] where [FORMULA]. These results will be described in detail in Paper III by Kawara et al. (in preparation). If [FORMULA] remains equal to -3 down to [FORMULA], then the fluctuation power ([FORMULA]) calculated by Eq. (3) exceeds the observed ones by an order of magnitude. Hence at a certain flux below [FORMULA], we expect that the slope of the counts must flatten to [FORMULA] so that the predicted fluctuations do not exceed the observed ones. We thus considered simple double power-law count models and evaluated the simulated PSDs: in the flux range [FORMULA]


and in flux range [FORMULA]


where [FORMULA], and [FORMULA], [FORMULA] are assumed to obey inequalities [FORMULA]. [FORMULA] and its uncertainty are listed in Table 1. The uncertainty in [FORMULA] includes Poisson uncertainties based on the total number of sources with flux above [FORMULA] in all images, and systematic ones originating from the incompleteness due to the source confusion, which will be discussed in paper III. If [FORMULA], the simulated PSD is not sensitive to [FORMULA] and is dominated by sources with fluxes around [FORMULA], as discussed in Lagache and Puget (2000). Various sets of parameters can be chosen so that the simulated PSDs fit to the residual PSDs. Examples of the simulated PSDs which fit well to the residual PSDs are also shown in Fig. 3, for which [FORMULA], [FORMULA] for both 90 µm and 170 µm counts, and [FORMULA] mJy for 90 µm counts and [FORMULA] mJy for 170 µm one. We found that the simulated PSDs can fit the residual PSDs over all spatial frequencies within [FORMULA] per cent for the 90 µm PSDs and [FORMULA] per cent for the 170 µm ones. These deviations of the residual PSDs from the simulated PSDs are also included as uncertainties in the residual PSDs.


Table 1. Number count parameters

We now derive the allowed regions in plots of the cumulative counts at fluxes below [FORMULA]. The bottom border of the allowed regions is derived by examining the single power-law counts ([FORMULA]3 and we obtain [FORMULA]=1.8 for 90 µm counts and [FORMULA]=1.7 for 170 µm ones. In case of double power-law counts we may choose a larger value of [FORMULA] with a set of parameter pairs ([FORMULA], [FORMULA]), and among them [FORMULA] for [FORMULA] is the largest. Thus we determine the upper borders of the allowed regions by connecting ([FORMULA], [FORMULA]) points of numerous models with [FORMULA] and [FORMULA]. The results are shown by the shaded area in Fig. 4. As for the 170 µm counts, the bottom border of the allowed region is close to the Scenario E model by Guiderdoni et al. (1998), and is consistent with the counts at 120 mJy and at 200 mJy obtained from the FIRBACK Marano 1 survey (Puget et al. 1999). On the other hand, the allowed region for 90 µm counts are significantly above those of any currently existing models.

[FIGURE] Fig. 4. The shaded areas show the allowed regions for various number count models, which are consistent with the fluctuation powers of the Lockman Hole images (Fig. 3). The bottom border is obtained from the simple number count model with a single power-law index between [FORMULA] mJy and [FORMULA] mJy (90 µm) or 250 mJy (170 µm). The upper border is determined from the allowed double power-law count models. See text for detailed information. The filled circles are observed number counts at [FORMULA]. The theoretical number count models by Guiderdoni et al. (1998) (scenario E: solid lines) and the no evolution model by Takeuchi et al. (1999) (dashed lines) are also shown. The open diamonds and triangles are observed and incompleteness-corrected source counts obtained by the FIRBACK Marano 1 survey (Puget et al. 1999). The open circles with dash-dotted line show the IRAS 60 µm counts, in which the flux is scaled by the factor [FORMULA].

The bottom border derived here cannot be applied below a certain flux [FORMULA], which is the flux at the intersection between an upper-border count model with the steepest slope ([FORMULA], [FORMULA]), and the bottom border: [FORMULA] (90 µm) or 50 mJy (175 µm). By definition, the upper border model do not require sources below [FORMULA]([FORMULA]). On the other hand, in case of the bottom border model, contribution to the fluctuations from sources below [FORMULA] is still appreciable, about half of the total. Considering the uncertainties in [FORMULA] and the residual PSDs as discussed above, we finally give [FORMULA] for 90 µm counts and [FORMULA] for 170 µm counts.

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Online publication: October 2, 2000