rotational excitation of small grains was Ref. [134], where the effect of collisions with gas atoms and absorption and emission of radiation were considered. Ref. [135] evaluated the effect of collisions with ions and “plasma drag” (torques due to the electric field of passing ions).
DL98b provided the first comprehensive study of the rotational dynamics of small grains, in- cluding all the previous effects. They evaluated, as a function of grain radius and environmental conditions, rotational damping and excitation rates through collisions, “plasma drag”, infrared emis- sion, emission of electric dipole radiation, photoelectric emission and formation of H2 molecules.
The spectra they provided are now widely used in interpreting ISM microwave emission (see for example Refs. [136, 137, 138, 139, 140, 141, 142, 143]) and for CMB foreground analyses (e.g.
Refs. [144, 145, 146, 147, 148]). Given that the DL98b models are now a decade old, and the recent surge in interest in anomalous emission, it is timely to revisit the theory of spinning dust emission, including the approximations made in DL98b. This is the purpose of the second part of this thesis.
Box 2: Standard Galactic foregrounds
Radio emission for the Milky Way is one of the major contaminants for observations of the CMB anisotropy.
Here we give a short overview of the physical origin and spectral characteristics of the standard Galactic microwave foregrounds.
• Thermal free-free emission (Bremsstrahlung)
Charged particles scatter off each other in the ISM, and in this process, radiate electromagnetic waves.
Being the lightest charged particles, free electrons get the largest accelerations and are the primary emitters of free-free radiation. The free-free emissivity in a thermal electron-proton plasma is given by (for a derivation, see Ref. [149])
jffν = 24e6 3m3/2e c3
r2π
3kTnenpe−hν/(kT)gff(T, ν), (7.1) where gff(T, ν) is the velocity averaged Gaunt factor, which is of order unity for most cases of astrophysical interest. Tabulated velocity-averaged Gaunt factors can be found in Ref. [150]. The essential property of the free-free spectrum is that it is nearly flat up to the cutoffνmax=kT /h, after which it decays exponentially.
• Synchrotron radiation
Relativistic electrons circling around the magnetic field lines also radiate energy as they are accelerated. If free electrons have a power-law distribution in energy dNe∝E−pdE, as is generally expected for Fermi acceleration in shocks, then the spectrum of synchrotron radiation is also a power law in frequency (for a derivation, see Ref. [149])
jνsynch∝ν−s, s= p−1
2 . (7.2)
The typical spectral index for Galactic synchrotron radiation iss≈1.
• “Thermal” (vibrational) dust emission
Dust grains in the ISM absorb the visible and ultraviolet starlight, which “heats them up”, i.e., puts them in excited vibrational states. The grains then spontaneously decay from the excited states and emit infrared radiation. Grains large enough to reach a steady-state configuration emit as modified blackbodies at temperature Td∼15−20 K (smaller grains undergo thermal spikes), and the far infrared dust emissivity has the form
jdustν ∝νβBν(Td)∝νβ+2, hνkTd. (7.3) The spectral indexβis measured to beβ≈1.8 [151, 152].
Below we show the characteristic antenna temperature [Tν≡Iνc2/(2kν2), whereIνis the specific intensity]
of the standard Galactic foregrounds (figure reproduced from Ref. [153]).
K Ka Q V W
CMB Anisotropy
Frequency (GHz)
( erutarepmeT annetnAM)smr ,K
1 10 100
100
20 40 60 80 200
85% Sky(Kp2) on tr ro ch yn S
ee fr e- re
FSynchrotron Dust
77% Sky(Kp0)
Figure from Bennet et al., 2003
Chapter 8
A refined model for spinning dust radiation 1
8.1 Introduction
In this chapter, we revisit and update the model of Draine & Lazarian (DL98b) for electric dipole radiation from spinning dust grains. Following DL98b, we consider either spherical grains or disk-like grains rotating about their axis of greatest inertia, leaving the discussion of non-uniformly rotating grains to the next chapter. We concentrate on the rotation rate of the grains; the size distribution has been reconsidered by other authors, and the grain dipole moment distribution should be regarded as a model parameter since one cannot compute it from first principles. We first review the calculation of DL98b for rotational excitation and damping rates. We modify the rotational excitation and damping rates by collisions with neutral species, such that it respects detailed balance in the case where the evaporation temperature is equal to the gas temperature. We include the electric dipole potential when evaluating the effect of collisions with ions. Full hyperbolic trajectories and rotating grains are used when computing the effect of plasma drag. We correct the infrared emission damping rate which was underestimated for a given infrared spectrum. Finally, we use these excitation and damping rates to calculate the grain rotational distribution function by solving the Fokker Planck equation. Updated grain optical properties and size distribution are used throughout this analysis.
An Interactive Data Language (IDL) code implementing the formulas in this chapter,SpDust, is available on the web2, and will hopefully allow for a more thorough exploration of the parameter space, as well as model fitting to observations.
This chapter is organized as follows. In Section 8.2 we remind the reader of the electric dipole radiation formula and give the resulting expected emissivity. In Section 8.3 we discuss the size
1The work presented in this chapter was reproduced from the paperA refined model for spinning dust radiation, Y. Ali-Ha¨ımoud, C. M. Hirata & C. Dickinson, Mon. Not. Roy. Astron. Soc. 395, 1055 (2009). Reproduced with permission, copyright (2009) by the Royal Astronomical Society.
2SpDustis available for download at http://www.tapir.caltech.edu/∼yacine/spdust/spdust.html.
distribution and dipole moments, along with other grain properties. We then turn to the main thrust of this study, which is the computation of the angular velocity distribution function. The theoretical formalism is exposed in Section 8.4, which presents the Fokker-Planck equation. Sections 8.5–8.9 discuss the various rotational damping and excitation processes: collisions with ions and neutral species, plasma drag, infrared emission, photoelectric emission, and random H2 formation.
The reader interested primarily in the predicted emission may wish to proceed directly to Section 8.10, where we present the resulting emissivity and the effect of various parameters and environment conditions. Our conclusions are given in Section 8.11. Appendix 8.A presents the techniques used to numerically evaluate integrals of rapidly oscillating functions involved in the plasma drag calculation.