The local dust foregrounds in the microwave sky: I. Thermal emission spectra

TL;DR

This paper investigates whether dust clouds near the Solar system could cause anomalies in the cosmic microwave background by simulating their thermal emission spectra and comparing with observational data.

Contribution

It models the thermal emission of macroscopic dust particles near the Solar system and assesses their potential impact on CMB measurements.

Findings

Dust clouds could contribute around 10 microKelvin to microwave signals.

Particles of several millimeters or smaller can produce flat, CMB-like spectra.

Such dust could explain some observed CMB anomalies.

Abstract

Analyses of the cosmic microwave background (CMB) radiation maps made by the Wilkinson Microwave Anisotropy Probe (WMAP) have revealed anomalies not predicted by the standard inflationary cosmology. In particular, the power of the quadrupole moment of the CMB fluctuations is remarkably low, and the quadrupole and octopole moments are aligned mutually and with the geometry of the Solar system. It has been suggested in the literature that microwave sky pollution by an unidentified dust cloud in the vicinity of the Solar system may be the cause for these anomalies. In this paper, we simulate the thermal emission by clouds of spherical homogeneous particles of several materials. Spectral constraints from the WMAP multi-wavelength data and earlier infrared observations on the hypothetical dust cloud are used to determine the dust cloud's physical characteristics. In order for its emissivity to demonstrate a flat, CMB-like wavelength dependence over the WMAP wavelengths (3 through 14 mm), and to be invisible in the infrared light, its particles must be macroscopic. Silicate spheres from several millimetres in size and carbonaceous particles an order of magnitude smaller will suffice. According to our estimates of the abundance of such particles in the Zodiacal cloud and trans-neptunian belt, yielding the optical depths of the order of 1E-7 for each cloud, the Solar-system dust can well contribute 10 microKelvin (within an order of magnitude) in the microwaves. This is not only intriguingly close to the magnitude of the anomalies (about 30 microKelvin), but also alarmingly above the presently believed magnitude of systematic biases of the WMAP results (below 5 microKelvin) and, to an even greater degree, of the future missions with higher sensitivities, e.g. PLANCK.

Figures

Tables

Table 1. Table 1: Geometrical optical depths of the trans-neptunian belt for an observer located in the inner Solar system. Estimates are based on debris disk models by  Krivov et al. ( 2008 ) and the total belt mass of 0.02  M ⊕ subscript 𝑀 direct-sum M_{\oplus}   (Fuentes and Holman, 2008 ) .

Particle size		$\tau$
from	to
10 $\mu$m	100 $\mu$m	$5\times {10}^{-6}$
100 $\mu$m	1 mm	$2\times {10}^{-6}$
1 mm	1 cm	$5\times {10}^{-7}$
1 cm	$\infty$	$2\times {10}^{-7}$

Equations

$$\Delta T_{\lambda }=\frac{hc}{k\lambda }/\ln \left[1+\frac{E_{\lambda }(T)E_{\lambda }(T_{\mathrm{D}})}{E_{\lambda }(T_{\mathrm{D}})+\sigma E_{\lambda }(T)}\right]-T,$$

$$T_{\mathrm{sky}}({\lambda }_i)=T+\Delta T_f({\lambda }_i),$$

$$T_{\mathrm{sky}}=\sum _{i=1}^k{w}_i\left(T+\Delta T_f({\lambda }_i)\right)=T+\sum _{i=1}^k{w}_i\Delta T_f({\lambda }_i),$$

$$\sum _{i=1}^k{w}_i=1.$$

$$\text{mbox}Var(T_{\mathrm{sky}})=\text{mbox}Var(T)+\text{mbox}Var\left(\sum _{i=1}^k{w}_i\Delta T_f({\lambda }_i)\right),$$

$${\int }_0^{\infty }\pi a^2{Q}_{\mathrm{abs}}(a,\lambda )\mathcal{F}(\lambda )\mathrm{d}\lambda =4{\int }_0^{\infty }\pi a^2{Q}_{\mathrm{abs}}(a,\lambda )B_{\lambda }(T_{\mathrm{D}})\mathrm{d}\lambda ,$$

$$T_{\mathrm{D}}=279\mathrm{K}[{\bar{Q}}_{\odot }/\bar{Q}(T_{\mathrm{D}}){]}^{1/4}{R}^{-1/2},$$

$$\frac{\int Q_{\mathrm{abs}}(a,\lambda )n(a)a^2\mathrm{d}a}{\int n(a)a^2\mathrm{d}a},$$

Full text

The Microwave Thermal Emission by Dust:

I. Thermal Emission Spectra

V. Dikarev1,2,3, O. Preuß1, S. Solanki1,4, H. Krüger1,3, A. Krivov5

11footnotetext: Max-Planck-Institut für Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany22footnotetext: The University of Bielefeld, Universitätsstraße 25, 33615 Bielefeld, Germany33footnotetext: Guest scientist at Max-Planck-Institut für Kernphysik, Heidelberg, Germany44footnotetext: School of Space Research, Kyung Hee University, Yongin, Gyeonggi 446-771, Korea55footnotetext: Friedrich-Schiller Universität Jena, Germany66footnotetext: *Corresponding author e-mail address: *vdikarev@physik.uni-bielefeld.de

Analyses of the cosmic microwave background (CMB) radiation maps made by the Wilkinson Microwave Anisotropy Probe (WMAP) have revealed anomalies not predicted by the standard inflationary cosmology. In particular, the power of the quadrupole moment of the CMB fluctuations is remarkably low, and the quadrupole and octopole moments are aligned mutually and with the geometry of the Solar system. It has been suggested in the literature that microwave sky pollution by an unidentified dust cloud in the vicinity of the Solar system may be the cause for these anomalies. In this paper, we simulate the thermal emission by clouds of spherical homogeneous particles of several materials. Spectral constraints from the WMAP multi-wavelength data and earlier infrared observations on the hypothetical dust cloud are used to determine the dust cloud’s physical characteristics. In order for its emissivity to demonstrate a flat, CMB-like wavelength dependence over the WMAP wavelengths (3 through 14 mm), and to be invisible in the infrared light, its particles must be macroscopic. Silicate spheres from several millimetres in size and carbonaceous particles an order of magnitude smaller will suffice. According to our estimates of the abundance of such particles in the Zodiacal cloud and trans-neptunian belt, yielding the optical depths of the order of $10^{-7}$ for each cloud, the Solar-system dust can well contribute $10\;\mu$K (within an order of magnitude) in the microwaves. This is not only intriguingly close to the magnitude of the anomalies (about $30\;\mu$K), but also alarmingly above the presently believed magnitude of systematic biases of the WMAP results (below 5 $\mu$K) and, to an even greater degree, of the future missions with higher sensitivities, e.g. PLANCK.

This manuscript has been accepted for publication in The Astrophysical Journal.

1 Introduction

The five-year release (Hinshaw et al., 2009) of the Wilkinson Microwave Anisotropy Probe (WMAP, Bennett et al., 2003a) data represents the current highlight of a project which provided a so far unprecedented amount of high precision cosmological data. On small angular scales the measurement of the cosmic microwave background (CMB) anisotropies led to the precise confirmation of the cold dark matter ($\Lambda$CDM) model of a nearly spatially flat universe, dominated by dark energy and non-baryonic dark matter (Spergel et al., 2006). Nevertheless at large scales a number of unexpected, potentially damaging anomalous features in the CMB have been reported, indicating that either our current understanding of standard inflationary cosmology or the data processing, including foreground removal techniques are as yet inadequate.

Among these anomalies, the expansion of the CMB sky in spherical harmonic functions resulted in an octopole which is unusually planar and oriented parallel to the quadrupole (Tegmark et al., 2003; de Oliveira-Costa et al., 2004). Three of the four planes determined by the quadrupole and octopole are orthogonal to the ecliptic at 99.1% confidence level (CL), and the normals to these planes are aligned with the direction of the cosmological dipole and with the equinoxes, inconsistent with Gaussian random, statistically isotropic skies at 99.8% CL (Schwarz et al., 2004).

The presence of non-Gaussian features in the CMB temperature fluctuations was reported by Copi et al. (2004). Land and Magueijo (2005) suggested that the presence of preferred directions in the low-order multipoles also extends to higher multipoles beyond the octopole.

Of course any of these anomalies challenges the validity of the standard scenario of inflationary cosmology which predicts scale-free, statistically isotropic and Gaussian random CMB temperature fluctuations and uncorrelated multipoles. However, although it is very unlikely that these features are just a statistical fluke, their cosmological origin is still an open debate. Therefore, besides various kinds of potential new physics (Hannestad and Mersini-Houghton, 2005; Moffat, 2005; Jaffe et al., 2005; Gordon et al., 2005), a compact cosmic topology (de Oliveira-Costa et al., 2004; Mota et al., 2004; Cornish et al., 2004) or modified inflation (Piao, 2005; Linde, 2004; Hunt and Sarkar, 2004) and also conventional effects of Galactic foreground emission (Eriksen et al., 2004; Slosar and Seljak, 2004; Naselsky et al., 2005) have been suggested as possible physical explanations.

The significance of the correlations of the quadrupole and octopole with the ecliptic plane, however, hints at the Solar system as the origin of an unaccounted bias (Schwarz et al., 2004; Copi et al., 2006). Starkman and Schwarz (2005) speculated that an unknown dark cloud of dust in the solar neighbourhood may have contributed to the microwave sky surveyed by the WMAP. Frisch (2005) proposed that the interstellar dust trapped magnetically in the heliosphere can possibly explain the CMB anomalies. Babich et al. (2007) discussed the possibility of discovering the dust of the trans-neptunian belt in the WMAP data. None of the investigations, however, gave fully convincing explanations to solve the problem of the CMB anomalies.

This paper opens a series of publications in which we simulate the microwave thermal emission of dust clouds inside or in the vicinity of the Solar system, and test whether these clouds can be responsible for the unexplained correlations or can be discovered in the CMB experiment data. Here we focus on the spectral constraints on material and size distribution of the particles forming a cloud invisible in the infrared wavelengths and imitating closely the CMB emission in the microwaves. Such cloud would indeed contribute to the CMB maps without us knowing about it. The absolute photometry and spatial geometry of candidate clouds’ contributions to the CMB maps are left for further scrutiny in follow-up studies.

The paper is organised as follows. Section 2 deals with the general constraints on an unknown cloud’s spectrum, from the infrared wavelengths to microwaves. It shows how a cloud of macroscopic particles can be noticeable in the microwaves and still avoid detection in the infrared light. Candidate clouds are proposed in the Solar system that can add non-negligible emission in the microwaves without being easily recognized in the infrared light. Estimates of the microwave temperatures of some known dust clouds are made, supporting the case for a detailed study. Constraints on the microwave spectra of foreground sources from the Internal Linear Combination (ILC) maps derived from the WMAP multi-wavelength data (Bennett et al., 2003b) are also formulated to facilitate the determination of plausible composition and size distribution of the cloud. Section 3 introduces the elements of the Mie theory of light scattering necessary for our calculations of the thermal emission from dust particles, and borrows bibliographic sources on the optical properties of several chemical compositions from the database (Henning et al., 1999). Thermal emission spectra of various sample and one natural dust clouds are generated in Sec. 4 and tested against the constraints placed by the ILC maps. Conclusions are drawn in Sec. 5.

2 The thermal emission due to a foreground cloud

2.1 In the Infrared Wavelengths

The first strong spectral constraint on the hypothetical dust cloud comes from the fact that so far it has not been recognized at other wavelengths, particularly in the infrared light, where dust is normally very bright. This is true e.g. for the Zodiacal thermal emission produced predominantly by dust with temperatures around 300 K and peaking at $\sim 10\;\mu$m, and even for the Galactic dust with temperatures around 10 K with a maximum radiance in the far infrared. However, these clouds are composed of particles much smaller than a millimetre in size, and in this section we demonstrate how a cloud of bigger particles can avoid detection in the infrared light and still shine considerably bright in the microwaves.

Figure 1 compares intensities of the CMB radiation, its average anisotropy, and several dust clouds between wavelengths of 10 to $3\times 10^{4}\;\mu$m. The emissivity of interplanetary dust (‘IPD’) is simplified to unity below 100 $\mu$m and $\lambda^{-2}$ above this wavelength (cf. Leinert et al., 1983). This is an approximation to e.g. astronomical silicate by Draine and Lee (1984) and the Grün et al. (1985) model of interplanetary dust in which the bulk of cross-section area is comprised by the meteoroids from 10 to 100 $\mu$m in size. Maris and Burigana (2007) used a similar law in their analysis of the microwave emission by interplanetary dust. The other clouds on the plot are hypothetical, with the constituent particles being macroscopic with respect to all wavelengths, i.e. having flat emissivities from the infrared to millimetre wavelength. Their geometrical optical depths $\tau$ were set so as to provide an extra temperature of 30 $\mu$K in the microwaves, i.e. close to the magnitudes of the CMB quadrupole and octopole.

The interplanetary dust at 300 K is the brightest source in the infrared light. The CMB radiation exceeds the emission from interplanetary dust above 300 $\mu$m wavelength, while the corresponding threshold wavelength for the CMB anisotropy is 600 $\mu$m. In the WMAP wavelength range (above 3 mm), the interplanetary dust is three or more orders of magnitude dimmer than the CMB anisotropy, explaining why it had not been taken into account as a serious bias. However, the steep decline of interplanetary dust in Fig. 1 with wavelength increase is due to its presumed emissivity $\propto\lambda^{-2}$ above $\lambda=100\;\mu$m. Clouds of macroscopic particles naturally do not follow this trend.

Figure 1 allows one to put constraints on the temperature and optical depth of the unknown cloud that is seen in the microwaves but as yet has not been recognized in the infrared light. Obviously it must not be brighter than the interplanetary dust in the infrared wavelengths between 10 and 100 $\mu$m, where the zodiacal thermal emission is dominant. This criterion is met by all ‘clouds’ in the plot, and even a cloud of macro-meteoroids at 300 K, i.e. an equilibrium dust temperature near Earth (see Eq. (7) in text below, or refer to Reach, 1988), would not be immediately rejected.

This criterion may look too bold: even if the cloud mimics really well the emission maps of smaller interplanetary dust particles, its addition would double the total brightness. However, Reach (1988) found that the infrared emission predictions based on the Grün et al. (1985) model are half the IRAS observations. Thus a considerable degree of freedom persists in interpreting the infrared observations using the size distribution of Grün et al. (1985), even before any exotic hypothesis is suggested, e.g. a local enhancement of dust around WMAP near the $L_{2}$ point of Sun-Earth which has been hidden from the earlier sufficiently sensitive observatories like IRAS and COBE at solar elongations $>135^{\circ}$.

Figure 1 can provide only an upper limit on the cloud’s brightness. An estimate of the lower limit is more difficult to obtain since our knowledge about big meteoroids away from the Earth orbit is rather poor. Sykes and Walker (1992) estimate the optical depth of a cometary dust trail as $10^{-8}$ (10P/Tempel 2), and provide evidence that the constituent particles are cm-sized. At 250 K, the trail is 2.5 $\mu$K bright in the microwaves. Similar estimates can be obtained for asteroid dust belts, if they are composed of particles with a unit emissivity (Low et al., 1984). These structural elements are recognized as faint but distinct features in the smooth background zodiacal emission ($\tau\sim 10^{-7}$). If there were broader structures—in space or on celestial maps—resembling the smaller interplanetary grain distribution, they could be easily confused with the Zodiacal thermal emission in the infrared wavelengths. In what follows, we present two plausible candidates, concentrated between Earth and Jupiter and in the trans-neptunian belt.

The Grün et al. (1985) model is based largely on the data acquired near Earth, most notably the lunar rock samples covered by plentiful micro-craters. They provide a vast amount of information for the model’s meteoroid mass distribution inferred over 20 orders of magnitude ($10^{-18}$ to $10^{2}$ g), however, the Moon spins, erasing memory of the direction of impacts, and it samples a very limited volume in the Solar-system space. There is just a little clue, therefore, as to where these meteoroids came from and to what extent they are representative of the entire Solar system. Populations of particles have been known to exist which are not sampled by the Moon, with the compositions and sizes not resembling those of the Earth-crossing meteoroids. Examples include the above-mentioned asteroid dust bands and cometary trails. Meteoroid streams should exist along the orbits of the comets with the perihelia outside 1 AU from the Sun, just as they exist along the comet orbits crossing the Earth’s orbit, revealed by meteor showers. Low number densities and fluxes of these particles have so far prevented their direct and remote registration outside the Earth-Moon system. Theoretical modeling is necessary to attempt to fulfill this observational gap.

Hughes and McBride (1990) simulated the number density distribution of meteoroids from short-period comets by distributing test particles uniformly in mean anomaly along each comet’s orbit. Most of these particles never cross the Earth’s orbit. Their number density peaks at 2–2.5 AU and exceeds its modest near-Earth level up to 5.5 AU (Fig. 2). The near-Earth level is about five times lower than the maximum. If their parent comets were the only source of big meteoroids ($>100\;\mu$m in size) reaching Earth and Moon, they would constitute about 10% of the total cross-section area of meteoroids in the Grün et al. (1985) model of the flux at 1 AU (see also Fig. 6 in Sec. 3.3). As the total optical depth of the zodiacal cloud is $\sim 10^{-7}$, and if its size distribution were spatially homogeneous, then these particles would contribute an optical depth of just $\sim 10^{-8}$.

However, the Poynting-Robertson drag (Wyatt and Whipple, 1950; Leinert et al., 1983; Gor’kavyi et al., 1997) causes a drift of small meteoroids toward the Sun and a depletion of these particles with heliocentric distance $\propto R^{-1}$ or steeper. The simulation by Hughes and McBride (1990) shows that big meteoroids do not follow this trend. In their model, assuming Keplerian motion, the large particles reside near the parent comet orbits. Although this assumption is only accurate if the planetary perturbations are ignored, by including a large ensemble of comets (135 in Hughes and McBride, 1990) one can initially account for the planetary perturbations already imprinted in the cometary orbit distribution (ignoring the observational selection effects on comets). More sophisticated modeling of the sources and evolution of meteoroids taking the Poynting-Robertson effect and mutual collisions into account by Ishimoto (2000) independently confirms that the number of big meteoroids is indeed considerably higher outside 1 AU.

Now let us try to estimate the microwave emission from the big particles based on Fig. 2. The total optical depth of zodiacal cloud is based on measurements confined to the vicinity of Earth’s orbit: the total thermal emission as well as visual light reflected by an interplanetary meteoroid illuminated by the Sun decreases with heliocentric distance $R$ sharply: $\propto R^{-2}$ (i.e., proportional to the incident radiation flux). The number density of small meteoroids (less than $\sim 100\;\mu$m in size) dominating the cross-section area of the zodiacal cloud according to the model by Grün et al. (1985) also decreases $\propto R^{-1}$ or steeper. Thus three quarters of the total thermal emission observed e.g. in the anti-solar direction should come from within 1 AU from the Earth. The emission-weighted cumulative numbers in Fig. 2 show that for the cometary meteoroids of Hughes and McBride (1990) one half of the radiation would come from within 1 AU outside Earth’s orbit (divide the emission-weighted cumulative number at 10 AU by that at 2 AU): their number density initially grows with distance rather than decays.

In the microwaves, the thermal emission from dust particles is in the Rayleigh-Jeans regime and is proportional to temperature. The temperature decrease with distance is rather slow, $\propto R^{-1/2}$ (see e.g. Reach, 1988, or refer to the next section of this paper for a derivation). Thus more distant particles are revealed, including virtually all of the Hughes and McBride (1990) meteoroids, whereas the small particles of the Grün et al. (1985) model are still depleted at longer distances due to the Poynting-Robertson drag.

If indeed the optical depth of the big meteoroids were $\sim 10^{-8}$ for the observations sensitive to the emission from dust within 1 AU from the Earth, the model by Hughes and McBride (1990) predicts that all meteoroids from short-period comets (including those beyond 2 AU from the Sun) would have a depth of about $5\times 10^{-8}$. A scaling factor of 5 stems from the total cumulative number plot in Fig. 2 which shows that only $1/5$ of the Hughes and McBride (1990) particles are located within 1 AU from the Earth. However, even some of these particles will be hidden to an Earth-bound microwave observer due to a slow temperature decrease. The temperature-weighted cumulative number restricts the scaling factor to three for the microwave range of wavelengths.

Note that the base value of $\sim 10^{-8}$ for the total optical depth of big meteoroids within 1 AU from Earth in the anti-solar direction is itself derived from the Grün et al. (1985) model adjusted to the meteoroid flux at 1 AU from the Sun strictly. In the Hughes and McBride (1990) model, the number density of the big meteoroids is two to three times higher, on average, between 1 and 2 AU from the Sun, than at the Earth orbit (Fig. 2). This allows one to raise the base value accordingly and to come to the total temperature-weighted depths between 6$\times 10^{-8}$ and 9$\times 10^{-8}$. This is in fact comparable with the visual optical depth of $10^{-7}$ of the zodiacal cloud! At 150 to 300 K, these meteoroids could add 9 to 27 $\mu$K emission in the microwaves, provided that they have a flat unit emissivity from the infrared wavelengths to microwaves, and without being immediately resolvable in infrared radiation (and very likely in visual light too, depending though on the particle albedo).

The assumption of Hughes and McBride (1990) of an equal meteoroid number per comet orbit can be challenged. The production rates tend to be higher for low-perihelion comets and the lifetimes tend to be longer for high-aphelion comets. One can thus argue that if a few active comets, such as 1P/Halley or 2P/Encke, produce more dust than all others, or some particular comet orbits allow for longer survival times, this in fact will only raise our estimate. We used an implicit assumption that the emission map of the big meteoroids is broadly as smooth as that of the zodiacal cloud. If a single bright comet is responsible, then the distribution is strongly biased towards its aphelion where the particles in Keplerian orbits stay much longer. They would produce a smaller and brighter spot on the sky. This is especially relevant to the Halley-type, long-period, and Kreutzer-group comets (“sun-grazers”).

There is another candidate cloud never observed in the microwaves: millimetre-sized meteoroids in the trans-neptunian belt. Based on thermal emission models of debris disks of Krivov et al. (2008), we have estimated the microwave temperatures, although an uncertainty of one order of magnitude should be borne in mind. Table 1 lists the optical depths of the trans-neptunian belt particles of four size ranges. The particles which are macroscopic with respect to the microwaves (from 1 cm and above in size) have a total depth of $2\times 10^{-7}$. At a temperature of $\sim 50$ K, they would add $\sim 10\;\mu$K if their absorption efficiency were $Q_{\rm abs}\sim 1$. When the particles of all sizes above 1 mm are included, the total extra temperature is as high as $\sim 35\;\mu$K. In the infrared light, however, the trans-neptunian belt is far too dim: taking all dust grains bigger than 10 $\mu$m into consideration, we obtain a total optical depth of $\sim 10^{-5}$, i.e. two orders of magnitude higher than that of the zodiacal cloud. However, the total blackbody emission at $\sim 30$ AU from the Sun is three orders of magnitude smaller than near 1 AU(see Eq. (7) below), hence the total thermal emission from the trans-neptunian belt seen from Earth in the anti-solar direction is no more than 10% of that of the zodiacal cloud.

Even if all or some of these particles are not the reason for the WMAP anomaly, e.g. due to their emissivities below unity, they cannot be totally disregarded, especially in the next generation CMB experiments.

2.2 In the Microwaves

Another spectral constraint comes from the microwave observations with WMAP which do not reveal any significant unknown foreground. As the CMB fluctuations are studied in terms of the effective temperature rather than radiance, let us first describe the conversion details, and introduce the notation of the temperature spectrum of a dust cloud.

A dust cloud of the temperature $T_{\rm D}$ and the column absorption area $\sigma(\lambda)$ at a given wavelength $\lambda$ adds a specific energy flux $B_{\lambda}(T_{\rm D})\,\sigma(\lambda)$ to the cosmic microwave background radiation, which is at the average temperature of $T=2.725$ K, given by $B_{\lambda}(T)$. In the CMB studies, their sum is usually described by the excess temperature $\Delta T_{\lambda}$ of an imaginary blackbody emission source that would emit the same flux, i.e. $B_{\lambda}(T+\Delta T_{\lambda})=B_{\lambda}(T)+B_{\lambda}(T_{\rm D})\,\sigma(\lambda)$. When solved for $\Delta T_{\lambda}$, this equation provides one with a handy formula

[TABLE]

where $E_{\lambda}(T)=\exp(hc/kT\lambda)-1$ comes from the Planck law, $h$ and $k$ are the Planck and Boltzmann constants (cf. Finkbeiner et al., 1999, who used an approximation working well only in the far infrared and microwaves).

In the Rayleigh-Jeans regime, i.e. when $\lambda\gg hc/kT$ at long wavelength, the spectral radiance $B_{\lambda}(T)$ is approximated by $R_{\lambda}(T)=2kT/c\lambda^{4}$ and a simple linear scaling is permitted: $\Delta T_{\lambda}=\sigma(\lambda)T_{\rm D}$. It will be used for rough estimates of the excess temperatures due to dust clouds in Sec. 2.1 only.

In general, instead of $B_{\lambda}(T+\Delta T_{\lambda})$ simply $\Delta T_{\lambda}$ is given, so that the excess temperature can be plotted in the form of a spectrum. Temperature spectra of the CMB anisotropy and several known foregrounds are shown in Fig. 3. The foregrounds are the thermal emission from galactic dust, and free-free and synchrotron emissions arising, respectively, from the electron-ion scattering and acceleration of cosmic ray electrons in magnetic fields. It is clearly seen that the CMB anisotropy is dominant between 3 and 6 mm, whereas at other wavelengths either galactic dust or free-free and synchrotron emissions are more intensive.

Even inside the range of its dominance from 3 to 6 mm, the CMB anisotropy must be cleaned of the foregrounds. Several methods have been developed to perform this task. One method is to obtain precise maps of foregrounds at the wavelengths where they dominate, refine them based on relevant physical theories, extrapolate to the WMAP wavelengths and subtract them from the WMAP data (Hinshaw et al., 2007). For example, the galactic dust emission was mapped using the IRAS and COBE observations in infrared light, cleaned from the Zodiacal dust emission by removing everything that is not correlated with the galactic hydrogen emission map, and then extrapolated to the microwaves assuming traditionally the emissivity law $\lambda^{-2}$ (Schlegel et al., 1998). This method has the caveat of not knowing every foreground or everything about the physics of known foregrounds.

In contrast, the idea of the Internal Linear Combination (ILC) method (Bennett et al., 2003b) is to reduce foreground and noise as far as possible by weighted linear combinations of multi-frequency data without using any special assumptions about particular spatial structures of the foregrounds. Suppose the thermodynamic temperature of map $i$ and wavelength $\lambda_{i}$ can be written as a sum of the CMB and the foreground temperature

[TABLE]

the CMB maps are reconstructed by co-adding the data from the five WMAP wavelengths (3.3, 5.0, 7.5, 10.0 and 13.6 mm)

[TABLE]

with the constraint that the weights $w_{i}$ add to one in order to preserve only signals with a thermal CMB spectrum, i.e.

[TABLE]

Since the emission of various foregrounds like free-free, synchrotron and dust emission shows significant spatial variation, mainly along the Galactic plane, it is clear that also the weights $w_{i}$ cannot be held constant across the whole sky. Hence, to accommodate the spectral variability of the foregrounds, the entire sky has been divided into 12 separate regions, within which the weights are constant and determined independently by the criterion that the weights have to minimize the variance

[TABLE]

so that the influence of the foreground emission is suppressed down to a minimum. The regions have been chosen such that 10 of them cover the inner Galactic plane while the outer Galactic plane as well as higher Galactic latitudes are covered each by only one region. Since the weights are constant within each region it follows that also any foreground emission, especially outside the Galactic plane, should have a constant spectrum throughout the region. Therefore, if this ILC assumption of a constant spectrum turns out to be inappropriate, e.g. because of an unknown dust cloud, the segmentation of the sky, especially outside the Galactic plane, has to be improved which could therefore also lead to a modification of the final CMB map.

Still, a comparison of the ILC and (Hinshaw et al., 2007) low-$\ell$ multipoles does not reveal any major disagreement. If the microwave spectrum of the unknown dust cloud had significant variagations, then it would have been removed, at least partially, by the ILC method and left intact by the Hinshaw et al. (2007) method, leading to substantial divergence between their results. The only way the cloud can stay hidden for both is if it has a nearly flat spectrum close to that of the CMB.

This leads us to the conclusion that if the anomalous multipoles are indeed caused by an unaccounted dust cloud, or if there is any cloud that adds an undesirable foreground, the cloud must have a relatively flat temperature spectrum between 3 and 14 mm. We return to a quantitative assessment of how flat the temperature spectrum needs to be, and what it implies for the cloud’s particle composition and sizes, in Sec. 4.

3 The thermal emission by homogeneous spherical grains

3.1 Excerpts from the theory of thermal emission

Dust particles absorb, scatter and emit electromagnetic radiation. Absorption and scattering lead to extinction of light from sources behind the particle, with the absorbed light heating up the particle. Also, scattering and emission add new light to the observer’s line of sight, with the radiative energy being transferred from other directions and wavelengths, respectively. The absorption efficiency $Q_{\rm abs}$, scattering efficiency $Q_{\rm sca}$ and extinction efficiency $Q_{\rm ext}=Q_{\rm abs}+Q_{\rm sca}$ are the ratios of the corresponding effective cross-section area to the geometric cross-section area of the particle. Knowing the radius $a$ of a spherical particle, one can easily calculate the energy absorbed, scattered and removed from the radiation flux at a given wavelength.

The thermal emission of light by a particle is described in a more elaborate way using the thermal equilibrium equation:

[TABLE]

where $\lambda$ is the wavelength, ${\cal F}$ is the incident radiance flux, $B_{\lambda}(T_{\rm D})$ is the blackbody radiance at the dust particle’s temperature $T_{\rm D}$. As the left-hand side provides the total energy absorbed from a mono-directional incident flux, the right-hand side gives the total energy emitted, omni-directionally. In the Solar system, where the solar radiation is dominant, the incident radiance flux $\cal F$ can be safely replaced by the solar spectrum, which is close to a blackbody radiating at 5700 K, reduced at the distance $R$ from the Sun by the factor $(R_{\odot}/R)^{2}$, where $R_{\odot}$ is the Solar radius.

By denoting the absorption efficiency averaged over the Solar spectrum with $\bar{Q}_{\odot}$, and the same quantity averaged over a blackbody spectrum at temperature $T_{\rm D}$ with $\bar{Q}(T_{\rm D})$, then using the Stefan-Boltzmann law and the Solar constant, one can rewrite Eq. (6) in a more concise form (cf. Reach, 1988):

[TABLE]

where $R$ is measured in AU. A perfect black body with $Q_{\rm abs}=1$ throughout the spectrum has therefore a temperature of 279 K at 1 AU from the Sun, inversely proportional to the square root of the distance. This inverse-square-root trend is often closely followed by the real dust particles.

The Mie light scattering theory allows one to calculate the efficiencies $Q_{\rm abs}$, $Q_{\rm sca}$ and $Q_{\rm ext}$ as functions of particle radius $a$ and wavelength $\lambda$ once the refractive index is provided for the particle material (Bohren and Huffman, 1983). The index of refraction is a complex number $m=n+{\bf i}k$, where the real part $n$ is the inverse phase speed in the material with respect to the speed of light in vacuum, and the imaginary part $k$ is the attenuation factor. The refractive index depends on wavelength. Under the assumption of homogeneity, it determines the propagation of electromagnetic waves inside the particle, while the assumption of spherical shape provides simplifications for determining the transformation of the waves at and near its boundary. The theory was formulated in the early 1900s in terms of the infinite series of spherical harmonic functions, its practical use was only made possible later on in the century by the development of computers. Standard Mie codes have been available to calculate the optical properties of particles based on the refractive indices.

A database of optical constants (Henning et al., 1999) provides plentiful bibliographic references and tables of the refractive indices necessary to predict the light-scattering properties of dust particles composed of various materials, in a wide range of wavelengths. Even though the microwave range is covered very sparsely, one can find a number of directions to the relevant laboratory studies and remote observations.

3.2 Particle composition and optical constants

In our selection of materials, we followed partly the rationale of Reach (1988) for materials constituting Solar system dust, largely supported by asteroid taxonomy. Carbonaceous particles are abundant in interstellar space and have been directly observed with the mass spectrometers on board Giotto and Vega-2 during the rendezvous with comet 1P/Halley in 1986 (Jessberger et al., 1988). Carbonaceous material covers the surfaces of C-type asteroids which is the dominant type in the asteroid belt, with 75% of known asteroids falling in this category. Silicate material largely constitutes the lunar and terrestrial rocks and was also revealed in the mass spectra of the 1P/Halley dust. It is abundant in circumstellar debris disks, allowing for remote studies of optical properties of silicate particles (Ossenkopf et al., 1992). 17% of known asteroids are of S-type, i.e. silicaceous according to their surface spectra. Even though the cosmic abundance of Fe is low, iron is the major component of some meteorites, magnetite (Fe3O4) is present in many of them. L-type asteroids, which constitute 7% of known asteroids show metallic surfaces. In more distant parts of the Solar system not reviewed by Reach (1988), ice grains are an important population. Ice has been reported to cover the trans-neptunian objects (see Brown et al., 1999; Jewitt and Luu, 2004). The icy moons and rings of Jupiter and Saturn further support the case for including ice in our review. Therefore, we will focus ourselves on the optical properties of carbonaceous, silicate, iron and icy particles.

Ossenkopf et al. (1992) inferred the light scattering properties of the silicate grains from the opacities of circumstellar dust disks at the wavelengths of up to 10 mm. Between 1 and 10 mm, the real part of the refractive index stays near $n\approx 3$, while the imaginary part drops from $k\approx 0.025$ at the lower boundary to $k\approx 0.0025$ at the upper boundary of the range, inversely proportional to wavelength (Fig. 4a,b). This is reported both for warm oxygen-deficient circumstellar silicates and cool oxygen-rich interstellar silicates considered in the paper.

The optical constants for silicate particles (Laor and Draine, 1993, Fig. 4a,b) differ only slightly from the values given by Ossenkopf et al. (1992), with a similar inverse phase speed $n=3.4$ and an attenuation factor decreasing with wavelength at the same rate yet from a higher $k\approx 0.05$ at 1 mm. Note, however, that the Laor and Draine (1993) constants are not provided for $\lambda>1$ mm and had to be extrapolated, assuming that the steep downward trend is continued in the microwaves.

The optical constants for olivine provided by Pollack et al. (1994), however, disagree with those of Ossenkopf et al. (1992) and a simple extrapolation of Laor and Draine (1993) beyond a wavelength of 1 mm. Ossenkopf et al. (1992) do not quote any observations beyond 1 mm. It is reasonable to assume that these authors extrapolated the trend seen in the far infrared wavelength into the microwaves. In contrast, the constants for olivine by Pollack et al. (1994) are based on laboratory measurements by Campbell and Ulrichs (1969) at 8.57 mm and 66.7 mm who show that silicates do not gain transparency in the microwaves as fast as the extrapolations suggest. Boudet et al. (2005) may have found an explanation for the disagreement. They studied the temperature dependence of the absorption by amorphous silicate grains between 10 and 300 K and found that with increasing temperatures the absorption efficiency grows considerably (by an order of magnitude) already in sub-millimetre wavelengths. Therefore, the theoretical considerations behind the extrapolation by Ossenkopf et al. (1992) may still hold true for cold dust, whereas the Campbell and Ulrichs (1969) measurements in the warmth of a ground-based laboratory turn out to be more relevant to the near-Earth dust environment. Since our main application of the Mie theory will be for a higher temperature, the Pollack et al. (1994) model is adopted as most reliable beyond 1 mm.

In contrast to silicates, the amorphous carbonaceous grains studied in the laboratory by Zubko et al. (1996) show much higher real parts of the refractive index $n$ of up to ten and more, as well as imaginary parts $k$ of several at the wavelengths near 1 mm (Fig. 4c,d). The imaginary part of the refractive index does not fall but grows with increasing wavelength for the ‘BE’ sample, i.e. amorphous carbon grains produced in burning benzene in air under normal conditions. Two other species studied by Zubko et al. (1996), those produced by arc discharge between amorphous carbon electrodes in different atmospheres (‘ACAR’ and ‘ACH2’), are characterised by the imaginary parts of the refractive indices reaching maxima between 100 and 1000 $\mu$m and then turning down. Nevertheless, as we have checked, they show qualitatively similar dependencies of the absorption efficiency on wavelength.

Additionally, the optical constants for graphite particles up to 1 mm wavelength are provided in the paper (Laor and Draine, 1993, Fig. 4c,d). The graphite particles behave similarly to the above-listed species of amorphous carbon when the electric field vector is perpendicular to the plane of graphite cleaves (their ‘$E\parallel c$’ explained in Draine and Lee, 1984), while the particles with the cleaves being parallel to the field vector are significantly different (‘$E\perp c$’). The absorption efficiency in the latter case is very low, however, making the thermal emission by a cloud of randomly oriented graphite particles well described by the first case.

We adopt for further calculations the optical constants of the ‘BE’ sample (Zubko et al., 1996) as a reasonably good representation for the absorption efficiency of the above-listed carbonaceous species.

Zubko et al. (1996) have limited their study to wavelengths up to 2 mm, while our application of the Mie theory to the WMAP data requires the optical constants for up to $\sim 10$ mm. We have not found any constants for carbonaceous particles at $\lambda>2$ mm in the literature. The optical constants of Zubko et al. (1996) are therefore extrapolated as shown in Fig. 4c,d (thin lines).

Iron particles (Fig. 4e,f) have a rather plain dependence of the optical constants on wavelength (Pollack et al., 1994). As shown in that paper, qualitatively similar dependencies are also exhibited by iron combinations with some other elements, such as iron sulfide FeS, a circumstance that allows the results of the metallic iron model application to be expanded onto a correspondingly broader range of particle composition.

Water ice is reported to have $n\approx 1.8$ and $k$ decaying from $10^{-2}$ at the wavelength of 1 mm to $10^{-3}$ at 10 mm, and at a temperature of $-1^{\circ}$C (Warren, 1984). For lower temperatures, $k$ is lower: at $-60^{\circ}$C, the values range from $3\times 10^{-3}$ at 1 mm to $3\times 10^{-4}$ at 10 mm. Solid ice can survive for a long time only far from the Sun, where the temperatures are low (see Eq. 7), e.g. $\sim 50\;{\rm K}=-223^{\circ}$C in the trans-neptunian belt. The attenuation factors for cosmic ice can therefore be even lower than $10^{-4}$. Then the grains absorb or emit virtually no microwave radiation, unless they are much bigger than the wavelength, i.e. centimetres in size. This is in agreement with Pollack et al. (1994), who show that the imaginary part decays from $3\cdot 10^{-3}$ to $3\cdot 10^{-4}$, while the real part of the refractive index is in agreement with the value found by Warren (1984). It is noteworthy that CO2 ice has a similar real part $n\approx 1.4$, whereas the imaginary part of the refractive index almost vanishes ($m<10^{-6}$) (Warren, 1986). While the attenuation factor can be raised by impurities, it is generally fair to say that cosmic ice absorbs the microwave radiation negligibly with respect to carbons and even silicates. Therefore we discuss neither water nor CO2 ice in the remainder of the paper.

3.3 Calculation of absorption efficiencies

The absorption efficiencies for the particle compositions considered above, except for ice, are shown in Fig. 5. The left panels show single-particle efficiencies, while in the right panels the efficiencies are averaged over different size distributions that can be expected in dust clouds.

Single-particle plots show a steep decrease of the absorption efficiencies of small grains with wavelength. Because the attenuation factor sharply weakens with wavelength, the absorption efficiency is proportional to $Q_{\rm abs}\propto\lambda^{-2}$ for silicate particles of 1 $\mu$m radius between 100 and 600 $\mu$m wavelengths. If the optical constants by Ossenkopf et al. (1992) were adopted, this trend would continue to the microwaves. However, the flattening attenuation of Pollack et al. (1994) changes the trend somewhat above 600 $\mu$m. As the particle size grows, the absorption efficiency flattens too, with a roughly constant value for wavelengths shorter than the particle size.

The carbonaceous particles show an analogous qualitative behavior, however, due to much stronger attenuation, their absorption efficiencies are flat for $\lambda$ less than $\sim 10a$. This is clearly seen for $\lambda<2$ mm where the actual measurements have been used by Zubko et al. (1996) to determine the optical constants.

Interestingly, the absorption efficiencies of iron particles starting from $\sim 10\;\mu$m in size are much less size-dependent over the wavelength range under consideration. They also do not completely flatten up to the biggest size considered, i.e. $\sim 10$ mm, but instead all sizes above $10\;\mu$m share a weak dependence $Q_{\rm abs}\propto\lambda^{-1/2}$. Even though this decay is extremely weak in comparison with that of like-sized silicate and carbonaceous particles, it leads to the absorption efficiency vanishing to almost $10^{-3}$ at wavelength of several millimetres, which makes iron a very unlikely candidate for the elusive cloud material. For the sake of brevity it will not be considered during subsequent deliberations.

The size distributions assumed in our calculations include power laws $n(a){\rm d}a=a^{-\gamma}{\rm d}a$ with several slopes $\gamma$, and the size distribution of interplanetary meteoroids (Grün et al., 1985) which was derived from the crater size distribution on lunar rock samples and remains the basis for the modern meteoroid environment models (Divine, 1993; Staubach et al., 1997; Dikarev et al., 2005). The absorption efficiencies of single particles were averaged over the cross-section area using

[TABLE]

where the integration limits were separated sufficiently to bracket all the dust that can emit at a given wavelength (tiny submicron-sized grains are inefficient already in infrared radiation, so e.g. for the Grün et al. (1985) distribution of interplanetary meteoroids in Fig. 6 a moderate growth of cross-section area in the grains with $s<100$ nanometres is invisible; huge meter-sized boulders are too rare).

The slopes $\gamma$ were picked around the Mathis-Rumpl-Nordsieck (MRN) distribution (Mathis et al., 1977) found for the interstellar dust and which are typical for many other dust clouds as well, i.e. $\gamma=3.5$. This slope was also derived for the steady-state size distribution of a cloud of colliding and disrupting particles (Dohnanyi, 1969). The slope can be modified by particle dynamics.

Figure 6 shows the size distributions used in our calculations as well as one of their momenta, the cross-section area distribution. Shallow slopes $\gamma$ emphasize bigger particles in the cloud, and make the average absorption efficiency accordingly flatter. The interplanetary meteoroid size distribution by Grün et al. (1985) yields a steep absorption efficiency since the bulk of its cross-section is comprised by particles from 10 to 100 $\mu$m in size. The Grün et al. (1985) distribution has different slopes in various size ranges, where different dynamics determine distinct particle lifetimes: a slope close to 3.5 for $a<1$ $\mu$m, near 2 between 1 and 100 $\mu$m, and about 5 above 100 $\mu$m.

3.4 Nonsphericity and inhomogeneity

The assumptions of sphericity and homogeneity made above for the sake of simplicity can of course be debated. Aggregates can indeed emit differently at long wavelengths, especially if they include conducting materials (carbon, iron). Interestingly, by taking the porosity into account one can come to flat absorption efficiencies for even smaller particles. Carbonaceous ballistic particle-cluster aggregates (PCAs) usually give flatter efficiencies than Mie spheres. The more realistic the method, the more pronounced is the effect: e.g. modified spectral function (MSF) predicts flatter efficiencies than discrete multipole method (DMM), and DMM flatter than effective-medium theories (EMT, see e.g. Stognienko et al., 1995, for more details and references). Therefore, we have good reasons to trust that our Mie calculations provide results rather conservative with regard to flatness, which is important for our conclusions.

4 The temperature spectra of dust clouds

The column absorption area of a dust cloud is simply the product of its column cross-section area and the absorption efficiency of constituting particles. Assuming the column cross-section area to equal $10^{-7}$, i.e. close to that of the Zodiacal cloud near Earth, and taking $Q_{\rm abs}$ from Sec. 3, we calculated the corresponding temperature spectra (see Fig. 7). The dust particle temperature was set to 300 K (cf. Eq. (7)).

The peak of the blackbody emission at that temperature is near 10 $\mu$m, while the CMB maximum is near 1 mm. This leads to very high excess temperatures due to dust over the CMB at the short wavelengths. The temperature spectra of small, micrometre-sized particles drop sharply in the far infrared and microwaves, however. In contrast, bigger particles show a pronounced flattening of temperature spectra if their size is similar to or bigger than the wavelength. The thermal emission by dust and the CMB is on the Rayleigh-Jeans side of their Planck functions above 1 mm wavelength, where they are approaching proportionality to temperature, and the temperature spectrum of dust is determined mainly by the absorption efficiency, which is flat for macroscopic particles. Silicate particles above several millimetres in radius exhibit flat temperature spectra, while the emission from carbonaceous grains larger than several hundreds of micrometres in size are already unaffected and thus not excluded by the ILC procedure.

It is important to note that the temperature of the dust cloud needs not necessarily be low to produce a flat temperature spectrum over the WMAP wavelengths. Cooling the dust particles, e.g. by moving them away to the trans-neptunian belt (to 50 K) or even further to the Oort cloud, will by itself not make the temperature spectrum flat in the microwaves. Even warm (300 K) particles can give a rather flat spectrum if they are big. Simultaneously, due to their broadly flat emissivity, they do not reveal themselves at shorter wavelengths, contrary to small micrometre-sized dust. Therefore the big particles may easily be outshone by more abundant small particles in the infrared, yet dominate the microwave emission.

A proposal by Frisch (2005) that the thermal emission of interstellar dust trapped in the heliosphere is an explanation of the CMB anomalies can be rejected since the grains are too small, sub-millimetre in size. No matter how far they are from the Sun, due to a steep decrease of their absorption efficiencies with wavelength they would be rejected by the ILC map construction procedure, and visible in the infrared wavelengths.

Power-law size distributions of dust are characterised by flat temperature spectra only if their slope $\gamma$ is weak, so that the big particles constitute the bulk of the cross-section area, according to our findings about mono-size clouds. The Grün et al. (1985) size distribution of interplanetary meteoroids gives a temperature spectrum that is quite steep for the ILC if the meteoroids are silicate, shallower if the particles are carbonaceous.

Figure 8 zooms into the WMAP wavelength range. More particle sizes are plotted in the mono-size cloud case to facilitate more accurate assessment of the temperature spectra. Note that the Grün et al. (1985) model’s size distribution, when normalised to the zodiacal cloud visual optical depth ($10^{-7}$), is brighter than 1 $\mu$K up to 6 mm wavelength under the hypothesis that all meteoroids are carbonaceous (bottom-right plot)! At the shortest wavelength of WMAP (3 mm) it can be as bright as several $\mu$K. If indeed there are five times more big meteoroids in the interplanetary space outside the Earth’s orbit (Sec. 2.1 and Fig. 2) than in the flux at 1 AU, one can get up to $\sim 10\mu$K thermal emission in the microwaves. Of course, the uncertainties of the chemical compostion and size distribution impact the accuracy of such estimates. Similarly, the trans-neptunian belt estimated to have a geometrical optical depth of $2\times 10^{-7}$ in particles bigger than 1 cm, would provide $\sim 15$ $\mu$K emission if it were fully carbonaceous ($Q_{\rm abs}\sim 0.5$ in Fig. 5, cf. Sec. 2.1). This is not the case, as there are many ice particles in the belt which are not visible in the microwaves, however, the number is intriguingly high.

The “color ratio” of mono-size clouds is plotted in Fig. 9 (left) as a function of size. It is the ratio of the excess temperatures of the clouds in the 13.6 to that in the 3.3 mm waveband, $\Delta T_{13.6}/\Delta T_{3.3}$. The color ratio of the carbonaceous particles of Zubko et al. (1996) rises to close to unity for a particle size between 1 and 2 mm, i.e. where reliable optical constants are still available. The color ratio for olivine remains low up to particle size of several mm. Obviously, in order to pass the ILC filter, one needs bigger silicate particles than carbonaceous ones. Interestingly, the color ratio of all power laws and interplanetary meteoroids of Grün et al. (1985) is too low, typically below 0.5.

We have also calculated how much emission from dust can pass through the ILC filters, assuming that the cloud is sufficiently “transparent” to leave the filter weights unaffected (i.e., determined by other foregrounds with higher brightness and/or steeper spectra). If the cloud were not “transparent”, the weights would be affected and a fraction of the cloud emission would be removed, leading to a difference between the ILC and Hinshaw et al. (2007) foreground-cleaned maps, which is too low (Sec. 2.2). Clouds with higher passthrough ratios are therefore more plausible, even though not proven, candidates for sources of large unaccounted contaminations in the WMAP data. Note that clouds with low passthrough ratios would affect the ILC weights in a way that the ratios are further lowered.

Figure 9 (right) displays the passthrough ratio for the ILC weights derived for region 1 encompassing the higher galactic latitudes and most of the ecliptic plane. This is the largest region on the sky where one could be most eager to search for an unknown dust cloud since the galactic emission is minimal inside it. It is interesting to note that the emission from the clouds composed of olivine particles is reduced by about 30% only if the particles are microscopic. Eriksen et al. (2004) had already emphasized that the ILC procedure is not very efficient in removing the emission from dust, although they found it more efficient than in our test: approximately half of the W-band (3.3 mm) simulated emission from dust was in their “cleaned” maps. They used the $\lambda^{-2}$ emissivity which is steeper than that of olivine. In our test, carbonaceous material allows us to reduce the emission of clouds of microscopic particles by half.

The ILC passthrough test confirms the results of the color ratio test: olivine particles of about 10 mm in size and carbonaceous particles of roughly 1 mm in size reach a nearly 100% “transparency” for the bias removal procedure. However, one should bear in mind that monosize clouds are very rare in nature. A broad size distribution should allow clouds composed of smaller particles to become “transparent” too, as the sharp peaks and drops of the ILC passthrough at big sizes would cancel out after an averaging. The Grün et al. (1985) size distribution of interplanetary meteoroids has ILC passthroughs of 64% and 58% for carbonaceous and olivine particles, respectively. If, however, the big meteoroids are underrepresented in the meteoroid flux at 1 AU, the percentages may be higher. Ignoring all meteoroids less than 100 $\mu$m in size, one can come to passthroughs of 95% (carbonaceous) and 55% (olivine). These numbers are directly attributable to the cloud of meteoroids from short-period comets discussed in Sec. 2.1. Note that for real non-spherical inhomogeneous particles, the percentages will likely be even closer to 100% (Sec. 3.4).

5 Conclusion

We have investigated the microwave thermal emission by dust in the Solar system. We applied the Mie theory of light scattering by spherical homogeneous particles in order to characterise the thermal emission spectra of silicate, carbonaceous, iron and icy particles. Our study is partly motivated by the WMAP observations of the CMB fluctuations that revealed large-scale structures aligned with the Solar-system geometry which are difficult to explain by the standard inflationary cosmology. One possibility to produce such fluctuations is by a dust cloud inside or in the vicinity of the Solar system. Another motivation for this study is to assess the feasibility of detection of dust in and near the Solar system in the microwaves.

We used the WMAP multi-wavelength observations and infrared surveys to constrain the physical properties of particles constituting the hypothetical cloud and to estimate the microwave emission by solar-system dust. We have found that only macroscopic, mm-sized silicate or carbonaceous grains could produce thermal emission with a spectrum compatible with that of the CMB fluctuations. Smaller dust grains, as well as iron particles, emit with a spectrum that can easily be distinguished from the CMB. The small particles would also be so bright in the infrared light that they would have been seen by the relevant telescopes, whereas the big particles have a flat emissivity throughout the spectrum, so that the abundant small grains outshine them at infrared wavelengths.

In order to attain the flat emissivity at the WMAP wavelengths, silicate particles must be several mm in size at least, whereas carbonaceous particles can be an order of magnitude smaller. This makes the carbonaceous particles the most likely candidate, as small grains are typically more likely in dust clouds than big ones. Meteoroids from short-period comets (Hughes and McBride, 1990) are a plausible candidate for such cloud. The dust needs not necessarily be cold nor remote. When the cloud is composed of particles with a broad size distribution, even smaller dust grains can contribute to an overall flat spectrum without revealing themselves. For example, all particles of the Grün et al. (1985) meteoroid flux model with sizes above 100 $\mu$m have an integral spectrum sufficiently flat to pass the ILC procedure used to clean the WMAP observations from biases, if they are composed of carbonaceous material. The trans-neptunian belt is another plausible candidate. According to our preliminary estimates, each candidate cloud can emit roughly $\sim 10~{}\mu$K in the microwaves.

We found studies (see Stognienko et al., 1995, for references) showing that the Mie theory provides in general more pessimistic estimates of compatibility between the dust and CMB spectra than advanced light scattering theories taking nonsphericity and inhomogeneity into account. It means that the real particles can be even smaller than those quoted above in this conclusion.

There is a lack of measurements of the optical constants of carbonaceous particles above a wavelength of 2 mm. Given that this material provides the best chances so far to explain the WMAP anomaly, laboratory measurements of carbonaceous particles would be very helpful for further studies of the problem.

In subsequent papers of this series, using the thermal emission model introduced here, we will test the absolute photometry and projected geometry of various known and hypothetical dust clouds in and near the Solar system, against the maps of CMB fluctuations made by WMAP.

6 Acknowledgements

Useful discussions with Dr. Harald Mutschke are greatly appreciated. Prof. Eberhard Grün’s reading and commenting the manuscript were very helpful. This work has been (partially) supported by the WCU grant (No. R31-10016) funded by the Korean Ministry of Education, Science and Technology, and by the Deutsche Forschungsgemeinschaft (grant reference SCHW 1344/3$-$1). We thank an anonimous referee whose careful reading of the manuscript has led to a pronounced improvement of the paper text.