3 Power spectrum of the 60 and 100 $\mu$m sky

3.1 Power spectrum computation

For a given image, the power spectrum P(k) is the absolute value of the Fourier Transform (we use the Fast Fourier Transform (FFT) function of IDL) of the image, averaged over constant values of $k=\sqrt{k_x^2+k_y^2}$. To make sure that map boundaries are not contaminating the computed power spectrum, we apodize the image with a cosine function before computing the Fourier Transform. This prevents the introduction of strong discontinuities when the image is periodized in the FFT algorithm (see Appendix A).

Figure 1: Top left: the 100 $\mu$m ISSA map number 47. Top right: noise map of the 100 $\mu$m map on the left. This noise map has been obtained by subtracting two individual HCON images that were used to build the ISSA map (see text for details). Bottom left: power spectrum of the ISSA map (upper curve) and of the noise map. The power spectrum of the noise map has been fitted by a power law $\propto$k-0.83 for scales $k \leq 0.1$ arcmin-1. Above k=0.1 arcmin-1, the drop of the power spectrum is due to the instrumental function (see Sect. 3.3). Bottom right: histogram of the noise values, fitted by a sum of two Gaussian functions (Amplitude1 = 1376, $\sigma _1 = 0.04$ MJy/sr; Amplitude2 = 711, $\sigma _2 = 0.06$ MJy/sr).

The power spectrum of the 100 $\mu$m ISSA map of one of our fields is shown in Fig. 1. Since the zodiacal emission has been subtracted in the ISSA maps, there are four main contributions to the power spectrum at these wavelengths: the cirrus emission, point sources, the CIB (resolved and unresolved) and the noise. Therefore, and if the noise and the signal are not correlated, the power spectrum P(k) can be expressed in the following manner:

$$P(k) = \gamma(k) \left[P_{\rm dust}(k) + P_{\rm source}(k) + P_{\rm cib}(k) \right] + N(k)$$ (1)

where $P_{\rm dust}(k)$, $P_{\rm cib}(k)$, $P_{\rm source}(k)$ and N(k) are respectively the power spectrum of the dust emission, of the CIB, of individually detected point sources and of the noise. The factor $\gamma(k)$ represents the instrumental function.

   
3.2 Noise power spectrum

One of the main limitation of the component separation to the power spectrum is the estimate of the noise contribution. For ISSA maps, this contribution can be estimated accurately as they are the combination of up to three HCON maps. Indeed the noise power spectrum of the HCON maps, and therefore of the ISSA map, can be estimated by subtracting two HCON maps of the same region. An example of the difference between two HCON maps for a typical low brightness ISSA map is shown in Fig. 1 (top-right). The characteristic stripping of the IRAS data is seen in this difference map. The distribution of difference values is symmetric (when there is no strong point sources) and well fitted by a sum of two Gaussian functions (see Fig. 1, bottom-right).

As the distribution of the difference values is the sum of two Gaussian functions with an equivalent width $\sigma_{\rm diff}$ and assuming that the noise in the IRAS survey is stationary to a good approximation (as tested below), we can conclude that the noise of an individual HCON can also be characterized by an equivalent width:

$$\sigma_{\rm hcon} = \frac{\sigma_{\rm diff}}{\sqrt{2}}\cdot$$ (2)

Furthermore, the noise of the ISSA map, built from n HCON maps, is also Gaussian distributed with a width of:

$$\sigma_{\rm issa} = \frac{\sigma_{\rm hcon}}{\sqrt{n}}\cdot$$ (3)

Therefore, the noise level of the ISSA map is given by:

$$\sigma_{\rm issa} = \frac{\sigma_{\rm diff}}{\sqrt{n} \sqrt{2}}\cdot$$ (4)

For the example given in Fig. 1, the noise level is $\sigma_{\rm issa}=0.048$ MJy/sr.

In general three HCON maps were averaged but parts of them may be undefined. Therefore, for each ISSA map we looked at the three corresponding HCON maps and built a mask n(x,y) that gives the number of defined values that were averaged at each positions. Then, the noise map of a given ISSA map is estimated by subtracting two HCON maps and divide the result by $\sqrt{2}\sqrt{n(x,y)}$.

The power spectra of the ISSA map and of its noise map are shown in Fig. 1 (bottom-left). Both power spectra meet at small scales ( $k \sim 0.2$ pixel^-1) where the signal is noise dominated. The noise map is characterized by a $\sim$k^-0.8 power spectrum. We have investigated the variations of the statistical properties of the noise on ISSA maps where three complete HCONs exists. By looking at the three possible difference maps built from three HCONs, we found that the shape of the power spectrum of the noise is rather constant with time and that the absolute level of the noise varies by less than 15% from one HCON to the other. This is equally true at 60 and 100 $\mu$m.

Figure 2: Power spectrum (black solid line) of a typical bright ISSA field with no strong point sources. The cirrus contribution to the power spectrum has been fitted by a power law ( $P(k) \propto k^{-2.86}$ for k<0.015 arcmin-1- dash line). The dotted line is the cirrus fit multiplied by a Gaussian PSF of $\sigma =0.07$ arcmin-1. The grey line is the power spectrum of the ISSA map (black solid line) divided by the Gaussian PSF.

   
3.3 Point spread function

The multiplicative factor $\gamma(k)$ in Eq. (1) represents the effective instrumental filter function of the system (telescope, focal plane assembly, sky scanning and map making). For some ISSA maps the instrumental function can be estimated directly from the power spectrum of the map. This is the case for maps where the brightness fluctuations are dominated by the cirrus emission and where the power spectrum is simply:

$$P(k) = \gamma(k) \times P_{\rm dust}(k).$$ (5)

Following Gautier et al. (1992), the power spectrum of the cirrus dust emission follows a power law ( $P_{\rm dust}(k)=Ak^\beta$). This behavior has been confirmed through a detailed study of H I 21 cm emission by Miville-Deschênes et al. (2002). These studies show that although the power spectrum slope may vary from region to region by $\sim$20$\%$ there is no systematic break at high spatial frequencies. On the basis of these studies, we thus conclude that the power law representation for the cirrus power spectrum does not introduce any systematic effect.

To estimate $\gamma(k)$ directly on the power spectrum of ISSA maps, we have selected 20 relatively bright regions (mean brightness greater than 8 MJy/sr) where the noise and CIB contributions are negligible. To make sure that the fluctuations are dominated by the cirrus emission, point sources were filtered out using a median filtering. An example of the power spectrum of such an ISSA map is shown in Fig. 2. We found that the effective IRAS instrumental filter function in Fourier space is well describe by a Gaussian function both at 60 and 100 $\mu$m. To determine the width of the instrumental function, the power spectrum of the 20 fields were fitted by the following equation:

$$P(k) = \exp\left(\frac{-k^2}{2\sigma_k^2} \right) \times Ak^\beta$$ (6)

which represents the cirrus power law multiplied by the Gaussian instrumental function. We have found $\sigma_k=0.073 \pm 0.005$ at 60 $\mu$m and $\sigma_k=0.065 \pm 0.005$at 100 $\mu$m. In the real space, this corresponds to a Gaussian instrumental function of $\sigma=1.5\pm0.1$ arcmin at 60 $\mu$m and $\sigma=1.7\pm0.1$ arcmin at 100 $\mu$m.

The cutoff seen in Fig. 2 at frequencies larger than 0.05 arcmin^-1 is where the effect of the instrumental filter function is expected. When the power spectrum is divided by the Gaussian instrumental function (grey line) one sees that this modeling does not stand for scales smaller than 4.5 arcmin (k > 0.22 arcmin^-1). This is due to the effective resolution of the IRAS data. The power spectrum has to be cut at the Nyquist frequency corresponding to the 1.5' pixel ISSA map. However, the effective resolution of the IRAS data is about 3.8' and 4.25' respectively at 60 and 100 $\mu$m. For that reason, all the power spectrum at frequency above 0.1 arcmin^-1, which corresponds to the Nyquist frequency of the effective resolution, has to be taken with cautious.