8.3.4 Grain charge
The rotational damping and excitation rates will be dependent on the grain charge. DL98b showed that the characteristic timescale for changes in charge is much shorter than the characteristic rota- tional damping time. We will therefore average the damping and excitation rates over grain charges, as well as the electric dipole moment when computing the power radiated. We therefore need the charge distribution function4of the grains as a function of their radius and environmental conditions, fa(Z).
There are three main processes contributing to grain charging: collisional charging by electrons and ions, which rates we denote Je(Z, a) and Ji(Z, a), respectively, and photoelectric emission of electrons caused by the impinging radiation, which rate is Jpe(Z, a). For every grain radius, the steady state charge distribution function is obtained by solving recursively the following equations:
[Ji(Z, a) +Jpe(Z, a)]fa(Z) =Je(Z+ 1, a)fa(Z+ 1). (8.12)
We use the equations of Ref. [159] for collisional processes, updated with the electron sticking coefficients of Ref. [157], for Ji and Je. The photoelectric emission rate is computed according to Ref. [157]. The radiation field is taken to be a multipleχ of the average interstellar radiation field uISRF, as estimated in Refs. [160, 161].
peak of the distribution will not be affected significantly, as the variations of the rotation rate of a grain within the peak are not impulsive.
The stationary Fokker-Planck equation is given by (see for example Ref. [162])
∂
∂ωi
Di(ω)fa(ω) +1
2
∂2
∂ωi∂ωj
Eij(ω)fa(ω)
= 0. (8.13)
The coefficients are defined as:
Di(ω)≡ − lim
δt→0
hδωii
δt and Eij(ω)≡ lim
δt→0
hδωiδωji
δt . (8.14)
We assume that the medium is isotropic, and there are no physical processes that allow for a preferred direction, such as a magnetic field. As a consequence, the rotational distribution function only depends upon the magnitude ω of ω. Moreover, in a local orthonormal frame (ˆeω,eˆθ,ˆeφ), whereω, θandφare the usual spherical polar coordinates definingω, the excitation coefficient takes up the following form:
Eωˆˆω=Ek(ω) (8.15)
accounts for fluctuations along ˆω, and
Eθˆθˆ=Eφˆφˆ=E⊥(ω) (8.16)
accounts for fluctuations perpendicular toω. The components in the coordinate basis are thus:
Eωω=Ek(ω), Eθθ= E⊥(ω)
ω2 , Eφφ = E⊥(ω)
ω2sin2θ. (8.17)
Moreover, we assume there are no systematic torques, so the damping coefficient is directed along ωand we have
D(ω) =D(ω)ˆeω. (8.18)
In the spherical polar coordinate basis, the Fokker-Planck equation then becomes:
1 ω2
d dω
ω2D(ω)fa(ω) + 1
2ω2 d2 dω2
ω2Ek(ω)fa(ω)
− 1 ω2
d
dω[ω E⊥(ω)fa(ω)] = 0. (8.19) Integrating once, we get the following first-order differential equation:
dfa
dω + 2D˜
Ekfa= 0, (8.20)
where
D˜ ≡D+ 1
ω(Ek−E⊥) +1 2
dEk
dω . (8.21)
Note that ˜D is simply equal toD if the fluctuations are isotropic and independent ofω.
The coefficientsD, Ek,E⊥, and therefore ˜D from various independent rotational damping and excitation processes are additive.
A given process is said to respect detailed balance, when, if that process were the only one taking place, the grain would rotate thermally, i.e., fa(ω)∝exp(−Iω2/2kT). As one can see from the Fokker-Planck equation, this implies that this process must satisfy :
D˜ = Iω
2kTEk. (8.22)
Excitation rates are often easier to calculate than damping rates, since they are positive definite and do not rely on near-cancellation of processes that increase versus decreasing ω. Thus in some cases, we will make use of detailed balance (i.e., the fluctuation-dissipation theorem), to obtain the damping rate, knowing the excitation rate.
We can also derive Eq. (8.20) with simpler arguments5. The change of themagnitude ofω can be obtained as follows:
δω ≡ δ|ω|=ω(t+δt)−ω(t) =p
(~ω(t) +δ~ω)2−ω(t) =ω
"r
1 + 2~ω.δ~ω ω2 +δ~ω2
ω2 −1
#
= ω
"
~ ω.δ~ω
ω2 +1 2
δ~ω2 ω2 −1
8
2~ω.δ~ω ω2
2
+O(δω3)
#
=δ~ω.ˆeω+δω2⊥ 2ω +O
δω3 ω2
. (8.23)
We therefore obtain, after averaging and taking the time derivative (withE⊥(ω)≡ 12lim
δt→0 δω2⊥
δt ):
δtlim→0
hδωi
δt = −D(ω) +E⊥(ω)
ω , (8.24)
δtlim→0
hδω2i
δt = E||(ω). (8.25)
If we now consider theone-dimensional probability distribution for ω, F(ω) [such that F(ω)δω is the probability that the rotation rate is ω within δω], it satisfies the one-dimensional stationary Fokker-Planck equation:
d dω
D−E⊥ ω
F
+1
2 d2 dω2
E||F
= 0, (8.26)
which, integrated once, gives (assumingF(ω) and its derivative vanish at infinity):
D−E⊥ ω
F+1
2 d dω
E||F
= 0. (8.27)
If we know rewriteF(ω) = 4πω2fa(ω), we directly obtain Eq. (8.20).
5This paragraph is not part of the published paper.
8.4.2 Normalized damping and excitation coefficients
We will see in the next section that for collisions with neutral H atoms, at a temperature T, for a spherical dust grain at the same temperatureT, the damping and parallel excitation coefficients have the following form:
D˜H= ω τH
and E||,H =E⊥,H =2kT IτH
, (8.28)
where
τH≡
"
nHmH
2kT πmH
1/2
4πa4cx 3I
#−1
(8.29) is the characteristic rotational damping timescale for collisions with neutral H atoms. Note that they respect the detailed balance condition.
We normalize the damping and excitation coefficients of each process to those of collisions with H atoms. Taking DL98b notation, we define, for each processX :
FX(ω)≡τH
ω D˜X (8.30)
GX(ω)≡ IτH
2kTEk,X(ω). (8.31)
A special case is made of the rotational damping through electric dipole radiation (subscript ed), because of its specificω3 dependence:
d dt
1 2Iω2
ed= 2 3
µ2⊥ω4
c3 , (8.32)
so
dω dt
ed=−Ded(ω) =−2 3
µ2⊥ω3
Ic3 =−Iω3 3kT
1 τed
. (8.33)
Here we define, following DL98b:
τed≡ I2c3
2kT µ2⊥ (8.34)
Using Eqs. (8.30), (8.31) and (8.33) in Eq. (8.20), the final equation for the distribution function is dfa
dω + Iω
kT F G+ τH
τed
1 3G
I2ω3 (kT)2
fa = 0, (8.35)
where
F ≡X
X
FX and G≡X
X
GX. (8.36)
One can see that the conditions to get a thermal, Maxwellian distributionfa(ω)∝exp(−Iω2/2kT) are:
F=G and τH
Gτed →0. (8.37)
Otherwise, the general solution to this equation is :
fa(ω)∝exp (
− Z ω
0
dω0 Iω0
kT F(ω0) G(ω0)
τH
3τedG(ω0) I2ω03 (kT)2
)
. (8.38)
If allFX’s andGX’s are constant, this has a simple form :
fa(ω)∝exp
−F G
Iω2 2kT − τH
τed
1 3G
Iω2 2kT
2
. (8.39)
Note that the damping through electric dipole radiation causes the distribution to be non-Maxwellian.
In the general case, someFX’s andGX’s may depend uponωand one has to compute numerically the resulting distribution function, using Eq. (8.38).
We now turn to the calculation of the various damping and excitation coefficients, due to col- lisions, plasma drag, infrared emission, photoelectron emission, and random H2 formation. In the following microphysics sections that form the heart of the calculations, we compute excitation and damping coefficients as a function of grain radius and environmental conditions. We evaluate them numerically for a fiducial cold neutral medium (CNM) environment, defined explicitly in Eq. (8.176).