This paper reviews the current status of theoretical modeling of electric dipole radiation from spinning dust grains. The fundamentally simple problem of dust grain rotation appeals to a rich set of concepts of classical and quantum physics, owing to the diversity of processes involved. Rotational excitation and damping rates through various mechanisms are discussed, as well as methods of computing the grain angular momentum distribution function. Assumptions on grain properties are reviewed. The robustness of theoretical predictions now seems mostly limited by the uncertainties regarding the grains themselves, namely, their abundance, dipole moments, and size and shape distribution.
Rotational radiation from small grains in the interstellar medium (ISM) has been suggested as a source of radio emission several decades ago already. The basic idea was first introduced by Erickson (1957) [
Shortly after the discovery of the anomalous dust-correlated microwave emission (AME) in the galaxy by Leitch et al. (1997) [
Understanding the spinning dust spectrum in as much detail as possible is important. First, the AME constitutes a foreground emission to cosmic microwave background (CMB) radiation. Second, it provides a window into the properties of small grains, which play crucial roles for the physics and chemistry of the ISM.
Motivated by these considerations and the accumulating observational evidence for diffuse and localized AME, several groups have since then revisited and refined the DL98 model [
The purpose of this paper is to provide an overview of the physics involved in modeling spinning dust spectra. We attempt to provide a comprehensive description of the problem at the formal level, and let the interested reader learn about the details in the various works that deal with the subject.
This paper is organized as follows: Section
Consider a grain with permanent electric dipole moment
We first consider the simplest case of a freely rotating spherical grain, with isotropic moment of inertia tensor
Here we consider an oblate axisymmetric grain with moments of inertia
Euler angles used for the description of an axisymmetric grain. The figure was reproduced from [
Between two discrete events that change its angular momentum, the grain can be considered as freely rotating. During these periods, the Euler angles change according to
Electromagnetic radiation is now emitted at the four frequencies
The case of a triaxial grain is unfortunately not analytic. In that case, power is radiated at a countably infinite number of frequencies. Hoang et al. (2011) [
The quantity of interest to us is the emissivity
We do not deal with this aspect in this paper as it does not belong,
The small grain abundance is determined primarily from observations of the wavelength-dependent extinction and the 3–25
Note that we assume the smallest grains in the ISM being mostly PAHs, but a population of ultrasmall silicate grains is not completely ruled out by observations [
PAHs may take a variety of shapes, from disk-like to nearly linear. They are not necessarily planar: for example, if one of the hexagonal carbon rings is replaced by a pentagonal ring, they are bent and become three dimensional. Above a certain size, PAHs may form irregular clusters and eventually, large three-dimensional grains.
The exact distribution of shapes is largely unknown. The lowest-frequency IR emission bands in principle carry information about the individual grains and seem to indicate that PAHs may be dominated by a few well-defined molecular structures, although not conclusively [
The smallest grains dominate the spinning dust spectrum (they can be spun up to larger frequencies and hence emit more power). It is commonly assumed that these grains are nearly planar up to a spherical-equivalent radius
In principle a consistent prescription should be given for small grains, which gives the precise nature of the grain, hence its shape (or rotational constants) and permanent electric dipole moment, which can be computed quantum-mechanically for small enough molecules. Such computations were carried out by Hudgins et al. (2005) [
Eventually, observations will hopefully allow for a more precise determination of the population of PAHs and their properties. We are currently far from having a definite handle on such refined properties of small grains, and an empirical distribution of dipole moments is required. Following DL98, more recent models assume a three-dimensional Gaussian distribution of dipole moments, with variance
In principle, one should solve for the distribution of angular momentum
The situation is much simplified if one single process is very efficient at changing the rotational configuration, on timescales much shorter than the overall timescale to change the angular momentum. If this process is characterized by an equilibrium temperature
Luckily Nature does provide us with such an efficient process to change
Following the absorption of an ultraviolet (UV) photon, small grains get heated up to large vibrational temperatures
Let us now discuss the implications of (
For example, an axisymmetric oblate grain with
S
The last integral in (
One must not forget, however, that the angular momentum distribution itself depends upon the rotational configuration
To determine the angular momentum distribution, we need to evaluate the differential transition rates
In general, transition rates are not significant for arbitrarily large values of
Most processes through which grains may change angular momentum (except for electric dipole radiation itself, to which we shall come back later on) are characterized by a damping timescale
If only one single process was interacting with the grains, their steady-state distribution would then be the Maxwellian with three-dimensional variance
Often, however, there is not a single process that dominates both excitation and damping. Since transition rates add up linearly for independent processes, so do excitation and damping rates. If all rates are of the form (
One process behaves differently from (
Accounting for the damping only for now and including this additional damping into the Fokker-Planck equation, we obtain the solution
The Fokker-Planck equation is a diffusion equation, and its validity is limited to processes that change the angular momentum by small increments. In this section, we formally discuss in which cases it may break down.
Let us now consider some stochastic interaction process
The issue of impulsive torques was addressed by Hoang et al. [
In this section, we describe the principal mechanisms that excite and damp the grains’ rotation. Since the detailed calculations are already worked out in various papers [
Collisional interactions of grains with gas atoms, molecules, or ions are perhaps the most intuitive of angular momentum transfer processes, even though the microphysical details could be very complex (see e.g., the discussion in Section 4.2 of [
The attached impactors are ejected from the grain’s surface following the absorption of UV photons that heat up small grains to large temperatures (this is the process of
In addition to a random component, ejected particles systematically decrease the angular momentum of the grain: if their ejection velocity is random in the rotating grain’s frame, they carry on average an angular momentum
For neutral impactors and neutral grains, For neutral impactors and charged grains (with charge For positively charged impacting ions, the dominant interaction is the attractive Coulomb attraction with negatively charged grains (whenever collisions with ions are relevant, a significant fraction of grains are negatively charged by colliding electrons, so the cation-
where
corresponding to a large focusing factor
Ions can exchange angular momentum with the grains at a distance, without necessarily colliding with them, by exerting a torque on their permanent electric dipole moment,
To obtain the damping rate from first principles would require accurate evaluations of the back-reaction of the grain on the ions’ trajectories, leading to a small asymmetry between trajectories increasing the magnitude of
In the case of a grain rotating about its axis of greatest inertia, and using the notation of Section 5.1.2, the damping timescale for plasma drag
Every time a small dust grain absorbs a UV photon, it gets into a highly excited vibrational state from which it decays by emitting a cascade of infrared (IR) photons, typically about a hundred per absorbed UV photon. Each one of the emitted IR photons carries one quantum of angular momentum, so its angular momentum squared is
The main difficulty in correctly evaluating (
A grain emitting electric dipole radiation also radiates away angular momentum. Classically, the radiation reaction torque is given by
The associated excitation mechanism comes from the absorption of CMB photons [
Quantum mechanically, the damping of the angular momentum is due to spontaneous decays
For a characteristic angular momentum
Draine and Lazarian [
Similarly, the rotational excitation due to photoejection of electrons following UV photon absorption is a subdominant excitation mechanism.
The relative importance of the various mechanisms described above depends upon the precise environmental conditions, that is, the gas density, temperature, ionization state, and ambient radiation field. Note that these parameters also affect the rotational transition rates through their dependence on
We list in Table
Dominant excitation and damping mechanisms for the smallest grains considered (
Phase | DC | MC | RN | PDR |
---|---|---|---|---|
Excitation | coll. (neutrals, ions) | coll. (ions) | IR | coll. (neutrals) |
Damping | e.d., coll. (neutrals) | plasma drag | e.d., IR | e.d., IR, coll. (neutrals) |
| ||||
Phase | CNM | WNM | WIM | |
| ||||
Excitation | coll. (ions, neutrals) | coll. (ions, neutrals), IR | coll. (ions) | |
Damping | e.d. | e.d | e.d |
We discussed in Section 5.1.3 how to characterize the importance of impulsive torques. In this section, we discuss specifically the case of the warm ionized medium (WIM), where collisions with ions are frequent and the rotational damping time is short.
The WIM is characterized by a large gas temperature
Hoang et al. [
Effect of impulsive torques due to collisions with ions in the WIM. Figure reproduced from [
A more important effect on the spectrum is that of increasing the characteristic internal temperature
The rotational energy of an axisymmetric grain is given by (
If interacting with a bath of characteristic temperature
Finally, the most likely angular momentum, in the case
This heuristic argument is in excellent agreement with results from detailed calculations. We show in Figure
Effect of wobbling of axisymmetric grains on the spinning dust emissivity in the WIM environment. The spectra were produced with S
Hoang et al. [
One cannot therefore neglect the fact that small PAHs are likely to be somewhat triaxial. The difficulty in properly accounting for this is that the exact distribution of ellipticities is largely unknown.
In this paper we have reviewed the current status of spinning dust modeling, and tried to summarize the recent advances in this field since the seminal papers of Draine and Lazarian [
The accuracy of theoretical predictions remains mostly limited by our poor knowledge of the properties of small grains, namely, their dipole moments, shapes and sizes, and their overall abundance, about which other observations give little information. This uncertainty can be turned into an asset, as one could potentially use the observed spinning dust emission (assuming it is the dominant AME process at tens of GHz frequencies) to constrain properties of small grains.
Such a procedure can, however, only be accomplished if environmental parameters are very well known. Indeed, the gas density, temperature, and ionization state as well as the ambient radiation field all affect the rotational distribution function of small grains in nontrivial ways. In addition, the actual observable, the emissivity, depends upon the properties of the medium along the line of sight, and an accurate modeling of the spatial properties of the environment is also required. Unless the properties of the environment are well understood, it seems very difficult to extract dust grain parameters from observed spectra, due to the important degeneracies that are bound to be present for such a large parameter space.
The view of the author is that significant advances in the field would be possible if several regions of the ISM were put under the scrutiny, not only of radio telescopes, but also of instruments at other wavelengths, in order to determine their detailed properties as much as possible and get rid of the uncertainties related to environmental dependencies.
Finally, let us mention another potentially interesting avenue to probe the properties of emitting grains, namely, the high-resolution
The author thanks Bruce Draine and Alexander Lazarian for providing detailed comments on this paper, as well as Rashid Sunyaev for his hospitality and generous financial support at the Max Planck Institute for Astrophysics during the part of summer 2012, where and when this paper was written. The author is supported by the National Science Foundation Grant no. AST-080744 and the Frank and Peggy Taplin Membership at the Institute for Advanced Study.