Table Of ContentDraft version January 28, 2009
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
POLARIZED INFRARED EMISSION BY POLYCYCLIC AROMATIC HYDROCARBONS RESULTING FROM
ANISOTROPIC ILLUMINATION
Lorenzo Sironi and Bruce T. Draine
PrincetonUniversityObservatory,PeytonHall,Princeton,NJ08544-1001
Draft version January 28, 2009
ABSTRACT
We study the polarized infrared emission by Polycyclic Aromatic Hydrocarbons (PAHs), when
anisotropicallyilluminatedbyUVphotons. PAHmoleculesaremodeledasplanardiskswithin-plane
9 and out-of-plane vibrational dipoles. As first pointed out by Leger (1988), infrared emission features
0 resulting from in-plane and out-of-plane modes should have orthogonal polarization directions. We
0 showanalyticallyhowthedegreeofpolarizationdependsontheviewinggeometryandthemolecule’s
2 internal alignment between principal axis of inertia and angular momentum, which gets worse after
n photon absorption. Longer wavelength features, emitted after better internal alignment is recovered,
a should be more strongly polarized. The degree of polarization for uni-directional illumination (e.g.,
J byastar)islargerthanfordiffuseillumination(e.g.,byadiskgalaxy),allelsebeingequal. ForPAHs
8 in the Cold Neutral Medium, the predicted polarization is probably too small to distinguish from the
2 contribution of linear dichroism by aligned foreground dust. The level of polarization predicted for
PAH emission from the Orion Bar is only ≈ 0.06% at 3.3µm; Sellgren et al. (1988) report a much
] largervalue,0.86±0.28%,whichsuggeststhatthesmallestPAHsmayhavemoderatelysuprathermal
A
rotation rates. Future observations of (or upper limits on) the degree of polarization for the Orion
G Bar or for dust above edge-on galaxies (e.g., NGC 891 or M82) may constrain the internal alignment
. of emitting PAHs, thus providing clues to their rotational dynamics.
h
Subject headings: ISM: dust, extinction — ISM: general — infrared: galaxies
p
-
o
r 1. INTRODUCTION of astronomical sources where PAH emission is observed
t offsetfromtheilluminatingsource. Theirupperlimitson
s The strong infrared emission features at 3.3, 6.2, 7.7,
a 8.6, 11.3 and 12.7µm have been attributed to vibra- the degree of polarization are of the order a few percent.
[ At one position on the Orion Bar they measure a linear
tional modes in planar Polycyclic Aromatic Hydrocar-
polarization of 0.86±0.28% in the 3.3µm feature, with
1 bons (PAHs) (e.g., Leger & Puget 1984; Allamandola
the polarization angle consistent with being orthogonal
v et al. 1985). Additional strong features at 16.4 and
tothelinebetweenthenebulaandthestar, aspredicted
8 ∼ 17µm (e.g., Smith et al. 2007) have also been at-
5 tributed to PAHs, although the identification is less cer- by Leger (1988). However, as we show below, the polar-
5 tain. Leger (1988) noted that a planar PAH molecule ization they report for the 3.3µm feature is much larger
4 may emit partially polarized light if anisotropically il- than expected.
. In this work we present analytic formulae for the de-
1 luminated by a source of UV photons. The basic rea-
gree of polarization of the PAH emission features, when
0 sonsarethefollowing: i)UVabsorptionisfavoredifthe
the emitting grains are anisotropically illuminated. We
9 moleculefacestheilluminatingsource;ii)spinningofthe
model PAH molecules as planar disks with in-plane and
0 molecule around its angular momentum preserves some
: memoryoftheilluminationdirection;iii)thevibrational out-of-plane vibrational dipoles. We extend the calcu-
v lations by Leger (1988) to allow for an arbitrary degree
dipoles responsible for the IR emission features oscillate
i
X either perpendicular or parallel to the molecular plane. of disalignment between the molecule’s principal axis of
inertiaaˆ (perpendiculartothemolecularplane)andits
r In particular, the C-H stretching mode (3.3µm) and the 1
a in-planeC-H bendingmode(8.6µm)oscillate parallelto angular momentum J. We discuss both the case of a
point-like illuminating source, which may be applied to
the grain plane, whereas the out-of-plane C-H bending
reflectionnebulaeliketheOrionBar,andofanextended
mode (11.3 and 12.7µm) oscillates perpendicular to the
source (e.g., a disk galaxy), which may be relevant for
molecularplane. Thestrongemissionfeaturesat6.2and
dust above NGC 891 or M82. The level of polarization
7.7µm are believed to arise from in-plane C-C stretch-
issensitivetotheanglebetweenthelineofsightandthe
ing and bending modes. In-plane modes (3.3, 6.2, 7.7,
illumination direction, and to the degree of alignment
8.6 µm) and out-of-plane modes (11.3, 12.7µm) should
between aˆ and J. Measurements of the degree of polar-
exhibit orthogonal polarization angles (Leger 1988), and 1
ization can therefore provide insight into the rotational
their electric field vector should be respectively perpen-
dynamics of PAHs.
dicular and parallel to the plane-of-sky projection of the
Thisworkisorganizedasfollows: in§2wedescribeour
illumination direction.
model for UV absorption and polarized IR emission by
Sellgrenetal.(1988)havesearchedforlinearpolariza-
PAH molecules, commenting on uncertainties regarding
tionofthe3.3and11.3µmemissionfeaturesinavariety
the alignment of aˆ with J; in §3 we present our results,
1
Electronicaddress: [email protected]; both for a star-like illuminating source and for an ex-
[email protected] tended galactic disk. The reader interested primarily in
2
the observational implications of our work may wish to 2.2. Cross section for absorption of starlight
skip§2and§3andproceeddirectlyto§4,wherewesum-
The absorption cross section for incident light with
marize our findings and discuss how future polarization
electric field E is proportional to (Leger 1988)
measurements may constrain the geometrical and rota-
(cid:88)
tional properties of PAHs. A∝ |(cid:104)j|E·d|i(cid:105)|2 , (1)
i,j
2. MODELFORPOLARIZEDEMISSIONFROMPAHS where the summation is over molecular states |i(cid:105) and |j(cid:105)
As discussed by Leger (1988), planar PAH molecules anddistheelectricdipolemomentoperator. Theπ−π∗
may emit partially polarized light as a result of electronic transitions responsible for UV absorption in
anisotropic illumination by a source of UV photons. UV PAHs have (cid:104)j|d|i(cid:105) only in the molecular plane. For a
absorption is favored if the molecular plane is perpen- diskmolecule,rapidspinningarounditsprincipalaxisaˆ
1
dicular to the illumination direction. Following UV ab- results in averaging over the angle of proper rotation φ.
sorption,in-planeandout-of-planevibrationalmodesare Thus, thegrainabsorptioncrosssectionmaybewritten,
excited, producing the observed IR emission features. for incident unpolarized light,
The grain angular momentum J stays approximately
constant duringthe wholeprocess ofUV absorption and A∝(1+cos2Θ) , (2)
IR emission (Leger 1988): first, the angular momentum
where Θ is the angle between the normal to the grain
contributed by the absorbed UV photon or removed via
plane(i.e.,theprincipalaxisaˆ ,foradiskmolecule)and
1
vibrational IR emission or rotational radio emission is
the direction of propagation kˆ of the absorbed photon.
small compared to the mean angular momentum of in-
In other words, when a planar PAH faces an unpolar-
terstellarPAHs;secondly,collisionsoftheemittinggrain
ized source, its UV absorption cross section is twice that
with interstellar atoms or ions hardly occur during the
when it is edge-on, because both components of the illu-
few seconds of IR emission; finally, Larmor precession
minating electric field can be absorbed in the first case
of J around the interstellar magnetic field takes much
and only one in the second.
longer than the IR emission burst (Rouan et al. 1992).
The angle Θ depends on the instantaneous orienta-
With J conserved, some memory of the source direction
tion of the grain with respect to the illumination direc-
is retained and the IR emission bands will be partially
tion. However, since the precession period of aˆ around
polarized. 1
J is much shorter than the time between absorption and
emission,wecanaverageovertheprecessionmotion(and
2.1. Illumination geometry the precession angle ψ) for the absorption and the emis-
sion process independently. For a point-like illuminating
We adopt the system of coordinates used by Leger
(1988) to specify the illumination geometry and the ori- source, all incoming rays have kˆ =zˆ (see Fig. 1a). The
entation of the emitting molecule (Fig. 1). The fixed corresponding ψ-averaged absorption cross section, for
coordinate system (xˆ,yˆ,zˆ) is centered on the emitting fixed J and a fixed angle β between aˆ1 and J, is
grain; the polar axis zˆ is along the illumination direc-
1
tion and xˆ is in the plane defined by zˆ and the direction A¯(cid:63)(θ,β)=1+cos2θcos2β+ 2sin2θsin2β
nˆ from the molecule to the observer (Fig. 1a). In this
≡1+C(θ,β) . (3)
frame, the grain angular momentum J has spherical an-
gles θ and ϕ (Fig. 1b). In order to describe the posi- If the illuminating source is an extended disk galaxy,
tion of the molecule with respect to J, we define a frame the angle Θ also depends on the spherical angles θ(cid:48) and
(xˆ(cid:48),yˆ(cid:48),zˆ(cid:48)) centered on the grain with zˆ(cid:48) along J and xˆ(cid:48) ϕ(cid:48) of the ray direction kˆ in the fixed (xˆ,yˆ,zˆ) frame. We
perpendicular to the plane (zˆ,zˆ(cid:48)), as shown in Fig. 1b.
assume that the emitting molecules are above the galac-
We choose the grain axes of inertia (aˆ ,aˆ ,aˆ ) so that
1 2 3 tic center; zˆ is then the direction from the center of the
aˆ is the axis of largest moment of inertia (i.e., the prin-
1 galacticdisktothegrains. Foranaxisymmetricsurface-
cipal axis); for a disk molecule, aˆ is perpendicular to
1 brightness profile, the absorption coefficient for incom-
the plane of the molecule, whereas aˆ2 and aˆ3 are in the ing rays with polar angle θ(cid:48) can be averaged over the
plane. Theirpositionintheframe(xˆ(cid:48),yˆ(cid:48),zˆ(cid:48))isdescribed azimuthal illumination angle ϕ(cid:48):
withEuler’sangles(Fig.1c): β,nutationangle,between
zˆ(cid:48) (orJ)andaˆ1; ψ,precessionangle,betweenxˆ(cid:48) andthe A¯gal(θ,β,θ(cid:48))=1+cos2θ(cid:48)C(θ,β)+sin2θ(cid:48)S(θ,β) , (4)
line of nodes, i.e., the intersection of the plane (xˆ(cid:48),yˆ(cid:48))
where we have also averaged over the precession angle
with the molecular plane (aˆ ,aˆ ); φ, angle of proper ro-
2 3 ψ. In the previous expression, C(θ,β) is the same as in
tation, between the line of nodes and aˆ . The classical
2 eq. (3) and we define
motionofarigidaxisymmetricgrainisacombinedrota-
tionarounditssymmetryaxisaˆ andaprecessionofthis 1 1
1 S(θ,β)≡ (1+cos2θ)sin2β+ sin2θcos2β . (5)
axis around the angular momentum J with constant β.1 4 2
Todescribethepolarizationoftheemittedradiation, we
For the sake of simplicity, in the following we assume a
define α, the angle between the illumination direction zˆ
uniform-brightnessdiskgalaxy;foragenericaxisymmet-
and the observer direction nˆ, and the polarization direc-
ric brightness profile B(θ(cid:48)), the absorption cross section
tionsuˆ andvˆ, respectivelyparallelandperpendicularto
in eq. (4) should be convolved with B(θ(cid:48)). As a special
yˆ in the plane of the sky (Fig. 1a).
case, we observe that if θ(cid:48) =0 for all incoming rays, i.e.
1Foraperfectlysymmetricgrain,themomentsofinertiaI2and kˆ =zˆ, we recover the absorption cross section in eq. (3)
I3 correspondingtoaˆ2 andaˆ3 areequalandnonutationoccurs. for a point-like illuminating source.
3
a) b) a z´ c)
z 1 z´
z
y´ β
n observer x´
a
u 1
a
n 3
α J β
α v θ
y a
y 2 y´
x
x´
x´ φ
ψ
ϕ
x
source
Fig. 1.— Geometry of the star-molecule-observer system, as presented by Leger (1988). In b), the molecule (dashed disk) has been
movedfarfromthecenterof(xˆ,yˆ,zˆ)forclarity. See§2.1fordetails.
2.3. Cross section for polarized emission By averaging over the azimuthal angle ϕ of the angular
momentum distribution, we simplify eqs. (9)-(10) and
The cross section for PAH emission polarized along a
obtain
direction wˆ is proportional to (Leger 1988)
(cid:88) F¯⊥(θ,β,α)=S(θ,β) , (13)
F ∝ |(cid:104)k|wˆ ·d|j(cid:105)|2 , (6) u
w
F¯⊥(θ,β,α)=cos2αS(θ,β)+sin2αC(θ,β) , (14)
j,k v
where |j(cid:105) and |k(cid:105) are vibrational states of the emitting and the corresponding ϕ-averaged cross sections for in-
molecule. We shall call F(cid:107) and F⊥ the relative emission plane modes may then be derived from eq. (8). In the
w w following, we assume that the angular momentum J is
crosssectionsforin-planeandout-of-planemodesrespec-
randomlyorientedinspace,butinprincipleeqs.(9)-(10)
tively. For out-of-plane modes, the only non-vanishing
and eqs. (11)-(12) could be convolved with any angular
terms in eq. (6) involve the component of d perpendicu-
distributionforJ,e.g.,iftheemittingmoleculesarepar-
lar to the molecule plane, and
tially aligned by the interstellar magnetic field.
F⊥ =(aˆ ·wˆ)2 . (7)
w 1
For in-plane modes, the matrix elements in eq. (6) are 2.4. Model for the alignment of aˆ1 with J
non-zero only for the components of d in the molecule The probability that a symmetric grain rotates with
plane; for a rapidly-spinning disk molecule, rotational angular momentum J ≡ |J| and nutation angle β (be-
invariance with respect to the angle of proper rotation φ tweenaˆ andJ)maybewritten(e.g.,Lazarian&Draine
1
yields 1997)
Fw(cid:107) =1−(aˆ1·wˆ)2 =1−Fw⊥ . (8) (cid:18) E (J,β)(cid:19)
By averaging over the precession angle ψ for the emis- dP(J,β)∝f(J)exp − rot sinβdβdJ , (15)
k T
sion process independently from the absorption (as dis- B ia
cussed in §2.2), the emission cross sections for out-of- where f(J)dJ is the probability that the angular mo-
plane modes with polarization along uˆ and vˆ are mentum∈[J,J+dJ],E isthegrainrotationalenergy
rot
F¯⊥(θ,ϕ,β,α)=1(cos2ϕ+cos2θsin2ϕ)sin2β apnadramTieatriiszetsheth“eindteegrrneaeloafliaglnigmnemnetn”ttbemetpweereantuaˆreawnhdicJh.
u 2 1
In the classical approximation, the rotational energy of
+sin2θsin2ϕcos2β , (9)
a symmetric oblate grain is
(cid:20) (cid:21)
F¯⊥(θ,ϕ,β,α)=cos2α 1(sin2ϕ+cos2θcos2ϕ)sin2β J2 (cid:20) (cid:18)I (cid:19) (cid:21)
v 2 Erot(J,β)= 2I 1+ I1 −1 sin2β , (16)
1 2
+cos2αsin2θcos2ϕcos2β+sin2αC(θ,β)
where I is the largest moment of inertia, corresponding
1 1
+ sin2αsin2θcosϕ(1−3cos2β) ,(10) toaˆ1,andI2 =I3 <I1;foraplanarsymmetricmolecule,
4 I = I = I /2. For the sake of simplicity, we assume
2 3 1
whereC(θ,β)hasbeendefinedineq.(3). Wehavemade that all the emitting molecules have the same moments
use of the fact that the orientation of the angular mo- of inertia.
mentum J is constant during the IR emission burst, as The absorption (eqs. (3)-(4)) and emission (eqs. (9)-
discussed at the beginning of §2. (12)) cross sections do not depend on J but only on the
The corresponding emission cross sections for in-plane angle β between aˆ and J. Therefore, we may integrate
1
modeswithpolarizationalonguˆ andvˆ canbecomputed eq. (15) with respect to J, provided f(J) is known, and
from their out-of-plane counterparts by using eq. (8): the resulting probability distribution will be used to de-
scribethedegreeofalignmentbetweenaˆ andJwhenwe
F¯(cid:107)(θ,ϕ,β,α)=1−F¯⊥(θ,ϕ,β,α) , (11) 1
u u computethePAHpolarizedemissionin§3. Inthesimple
F¯(cid:107)(θ,ϕ,β,α)=1−F¯⊥(θ,ϕ,β,α) . (12) case f(J) ∝ δ(J −J¯) (or, in general, if f(J) is strongly
v v
4
peaked at J¯), the probability distribution in eq. (15) re- with a volume-equivalent radius a ≈ 7.5˚A. We suppose
duces to thatalltheemittingmoleculesareaxisymmetricandpla-
√ nar, with I /I =2.
γexp(−γsin2β)sinβdβ 1 2
dP (β)= √ √ , (17) Following UV absorption, the alignment between the
γ πe−γerfi( γ) molecule principal axis aˆ and angular momentum J,
1
whereerfi(z)≡(2/√π)(cid:82)zexp(t2)dtistheimaginaryer- and thus the internal alignment temperature Tia, de-
0 pends on the efficacy of internal energy exchange be-
ror function. We have defined a dimensionless “internal
tween lattice vibrational modes and rotational modes.
alignment” coefficient
The Intramolecular Vibration-Rotation Energy Transfer
J¯2 (cid:18)I (cid:19) T (cid:18)I (cid:19) (IVRET) process (Purcell 1979), due to imperfect elas-
γ ≡ 1 −1 ≡ rot 1 −1 , (18) ticity of the molecule when stressed by centrifugal and
2I k T I T I
1 B ia 2 ia 2 Coriolis forces, allows energy exchange between rotation
where T ≡ J¯2/2I k . When γ → ∞, the molecule and vibrations on a timescale ∼ 10−2 s (Rouan et al.
principalroatxis tends t1oBbe perfectly aligned with the an- 1992), much shorter than the duration ∼1−10s of the
gular momentum; when γ = 0, aˆ is randomly oriented IRemissionburst. Thismeansthat,whilethemoleculeis
1
with respect to J. cooling after UV absorption, its internal alignment tem-
As discussed at the beginning of §2, J stays approx- perature Tia tends to be equal to the instantaneous vi-
imately constant between UV absorption and IR emis- brational (lattice) temperature Tvib. Following photon
sion, and so does T ; however, the molecule internal absorption, the lattice may be heated up to a tempera-
rot
alignment temperature Tia may substantially change if ture Tvib ≈ 300−1500K, depending on the grain size
part of the absorbed photon energy is transferred to ro- and the photon energy. The grain cools as IR energy is
tational degrees of freedom. We account for the uncer- radiated, and we estimate Tvib ≈ 800K when most of
tain energy exchange between vibrational and rotational the 3.3µm emission takes place, Tvib ≈ 300K for the
modesbyparametrizingthegraininternalalignmentbe- 7.7µm emission, Tvib ≈200K for the 11.3µm emission,
fore UV absorption and during IR emission respectively and Tvib ≈ 120K for the 17µm emission, unless the lo-
with γ (corresponding to T ≡T ) and γ ≤γ (corre- calradiationfieldissointensetopreventthelatticefrom
0 ia 0 r 0
spondingtoT ≥T ). Inprinciple,γ shouldbedifferent cooling down to such temperatures.
ia 0 r
for each IR emission feature, since it may be thought of The internal temperature Tia follows Tvib while the
as the internal alignment coefficient when most of the grain is cooling, with Tia ≈ Tvib as long as the vibra-
radiation in that band is emitted. tional energy levels are sufficiently closely spaced to al-
We now discuss another plausible choice for the low energy transfer between vibrations and rotation.2
parametrizationofthealignmentbetweenaˆ andJprior When the separation ∆E of vibrational levels exceeds
1
to UV absorption. If many collisions with hydrogen ∼ (cid:126)ωrot, the IVRET process ceases to operate and the
atomsoccurintheintervalbetweentwoUVabsorptions, rotational modes decouple from the lattice. The density
and there are no other torques acting, the grain will be of states can be calculated using a model normal mode
driven towards “Brownian rotation” with f(J) ∝ J2. If spectrum and the Beyer-Swinehart algorithm (Draine
there is no vibrational-rotational energy exchange (i.e., & Li 2001): for N = 200, the density of states at
C
thePAHactslikearigidrotator),theinternalalignment vibrational energy E/hc ≈ 250cm−1 is hcdN/dE ≈
temperatureT beforeUVabsorptionwillapproximately 1/(0.18cm−1). This should still allow energy exchange
0
equal the gas kinetic temperature Tgas. Integration of in quanta ∆E/hc ∼ (ωrot/2πc) ≈ 0.33cm−1 for a grain
eq. (15) with respect to J assuming f(J) ∝ J2 yields a spinningatωrot/2π ≈10GHz,sothattherotationaland
probability distribution vibrationalmodeswilldecoupleonlywhenthelatticevi-
brational energy has dropped below E/hc ≈ 250cm−1.
(cid:15)((cid:15)−1)1/2sinβdβ For our model PAH, this is the average energy if it were
dP (β)= , (19)
(cid:15) 2((cid:15)−cos2β)3/2 in contact with a ∼ 65K heat bath, which suggests
T ≈ 65K to describe alignment of aˆ with J prior to
0 1
which does not depend on Tgas but only on the geomet- photon absorption. As we discuss below, this is actu-
rical properties of the grain via (cid:15)≡I1/(I1−I2). allyalowerlimitforT0,whichispresumablyattainedin
In the following, we parametrize the disalignment be- cold interstellar clouds, whereas in bright photodissoci-
tweenaˆ1 andJbyusingeq.(17)bothbeforeUVabsorp- ation regions like the Orion Bar the radiation field is so
tion (with γ0) and during IR emission (with γr ≤ γ0). dense that PAH grains stay always hotter.
However, in the Appendix we present analytic formulae As anticipated in §2.4, we assume that the grain dis-
fortheexpecteddegreeofpolarizationifthegrainalign- alignmentbetweenprincipalaxisaˆ andangularmomen-
1
mentbeforeUVabsorptioncanbedescribedbyeq.(19), tumJmaybedescribedbytheprobabilitydistributionin
but eq. (17) still holds during IR emission. eq. (17) with “internal alignment” coefficient γ just be-
0
foreUVabsorption,andγ duringIRemission. Theratio
r
2.5. Estimating γ0 and γr T /T depends on environmental conditions, hence so
rot ia
In the dust model of Draine & Li (2007), the 3 −
13µm emission features are produced primarily by PAH 2 If aˆ1 and J are not parallel and I2 (cid:54)= I3, the molecule nu-
molecules or clusters containing between N ≈ 25 and tates. Becauseofnutation,therotationalmotionaroundaˆ1isonly
C quasiperiodic,andthecentrifugalandCoriolisstressesresponsible
∼1000 carbon atoms, and the 17µm complex is mainly fortheIVRETprocesswillhaveFouriercomponentsoverarange
duetoPAHswithNC ≈2000. Forpurposesofestimating of frequencies around the mean rotation rate ωrot (i.e., the rota-
rotational kinetic energies and the density of vibrational tion rate averaged over the nutation period). This facilitates the
couplingofrotationandvibrationsviatheIVRETprocess.
states, we will take N = 200 as a representative value,
C
5
enough (and the vibrational levels sufficiently closely
TABLE 1 spaced) that the internal alignment temperature T
Selected Cases for a Disk Molecule with NC=200 roughly equals the instantaneous vibrational temperaia-
tureT atanytime. ForaPAHmoleculewithN =200
Parameter case(a) case(b) case(c) case(d) vib C
inaradiationfieldwithχ≈3×104,thelatticecanhardly
CNM OrionBar STR aˆ1(cid:107)J
cool below T ≈ 150K (Draine & Li 2001), so that in
vib
Trot(K) 80 220 1000 ∞ a PDR like the Orion Bar T ≈150K may be appropri-
T0(K) 65 150 60 – ate prior to UV absorption.0The corresponding internal
γ0 1.2 1.5 17 ∞
γr(3.3µm) 0.10 0.28 1.3 ∞ alignment coefficient will be γ0 ≈ 1.5 for Trot ≈ 220K.
γr(7.7µm) 0.27 0.73 3.3 ∞ Values of γr for different emission bands are given in
γr(11.3µm) 0.40 1.1 5.0 ∞ Table 1; since T never drops below ≈ 150K, we take
γr(17µm) 0.67 1.5 8.3 ∞ T ≈150Kasthviebcharacteristicinternalalignmenttem-
p(cid:107)(π/2),3.3µm(%) 0.02 0.06 1.29 7.69 ia
(cid:63) perature for emission in the 17µm band.
p(cid:107)(π/2),7.7µm(%) 0.05 0.17 3.29 7.69
(cid:63)
p⊥(cid:63)(π/2),11.3µm(%) −0.14 −0.53 −8.56 −14.29 2.5.3. Case (c): Suprathermal Rotation (STR)
p⊥(π/2),17µm(%) −0.25 −0.73 −10.54 −14.29
(cid:63) As discussed in §2.5.1 and 2.5.2, PAH molecules in
does γ in eq. (18). We now discuss four possible cases,
the CNM should have moderately sub-thermal rotation
summarized in Table 1; for each case, we compute the rates (T (cid:46)T ), whereas T (cid:28)T in bright PDRs
rot gas rot gas
value of γ prior to absorption of starlight photons, and
0 like the Orion Bar. Observations of microwave emission
the values of γ during emission of radiation in the 3.3,
r from spinning grains in the diffuse ISM (e.g., Dobler &
7.7, 11.3 and 17µm bands.
Finkbeiner 2008; Dobler et al. 2008) and in PDRs (e.g.,
Casassus et al. 2008) appear to be consistent with these
2.5.1. Case (a): CNM
estimates.
In H I clouds with gas kinetic temperature Tgas ≈ However,itisofinteresttoexplorethepossibilitythat
100K (the “Cold Neutral Medium”, or CNM), the ro- somePAHsmightbesubjecttosystematictorqueswhich
tational excitation model by Draine & Lazarian (1998) couldspinthemuptohigherrotationrates. Forexample,
predicts that a PAH with N = 200 should be spinning inelastic collisions between a grain and hydrogen atoms
C
with rotational temperature Trot ≈ 80K, which corre- may be followed by ejection of H2 molecules when a C-
sponds to a rotation frequency ωrot/2π ≈10GHz. Since H bond is broken through photo- or thermo-dissociation
the molecule’s angular momentum stays approximately triggered by the absorption of UV photons. If recombi-
constant between UV absorption and IR emission, we nationofHatomspreferentiallyoccursatafewcatalytic
may assume Trot ≈ 80K during the whole process. If centers spread over the grain surface, and the H2 ejec-
T0 ≈ 65K prior to UV absorption, as discussed above, tion is systematically asymmetric, the resulting torque
the internal alignment coefficient of our model PAH will may significantly spin up the PAH molecule.
be γ0 ≈ 1.2; the values of γr for different IR emission For case (c) we assume Trot ∼ 10Tgas ≈ 1000K for
bands are listed in Table 1. PAHs in H I clouds, corresponding to a rotation fre-
We note that, if gas collisions were frequent enough to quency ω /2π ≈ 30GHz for our model grain with
rot
drive the emitting grains towards “Brownian rotation” N =200 carbon atoms. For PAHs spinning at 30GHz,
C
between two subsequent UV absorptions, then the prob- efficient energy transfer between vibrations and rotation
ability distribution in eq. (19) should be used instead of via the IVRET process is suppressed, due to insufficient
eq. (17) to describe the internal alignment prior to UV density of vibrational states, when the lattice cools be-
absorption, whereas eq. (17) may still be used during IR lowT ≈60K,hencewetakeT ≈60Kastheinternal
vib 0
emission with an appropriate choice for γr. However, as alignmenttemperaturepriortoUVabsorption,givingan
discussed in the Appendix, identical polarization results internal alignment coefficient γ ≈ 17. Values of γ for
0 r
maybeobtainedinthecurrentformalismifweemployan different PAH emission bands are listed in Table 1.
“effective” internal alignment coefficient γ ≈1.0 before
0
UVabsorption, similartothevalueγ0 =1.2adoptedfor 2.5.4. Case (d): aˆ1 ||J
case (a). The physics of internal relaxation in cold, spinning
grainsisnotwellunderstood,andperhapsthereisaslow
2.5.2. Case (b): The Orion Bar PDR
process that couples the rotational degrees of freedom
The Orion Bar, illuminated by the Trapezium stars, is to the lattice down to very low temperatures, regardless
an example of a bright photodissociation region (PDR), of the lower limit imposed by the density of vibrational
withstrongPAHemissionat3.3µm(Tielensetal.1993). states. Since in H I clouds the interval between two sub-
Physical conditions in the photodissociation zone have sequent photon absorptions is long (∼ 107 s), there is a
beendiscussedbyAllersetal.(2005),whoestimaten ≈ lot of time for a weak process to act. Such a process
H
7×104 cm−3, T ≈1000K, and χ≈3×104, where n would lead to γ (cid:29) 1, and we discuss γ = ∞ as a lim-
gas H 0 0
isthehydrogennumberdensityandχisthespecificradi- iting case.
ationenergydensityat1000˚Arelativetothevalueinthe Also, if the coupling between rotation and vibrations
localinterstellarradiationfield. Fortheseconditions,the after UV absorption took place on timescales much
rotational excitation model of Draine & Lazarian (1998) longer than the duration of the IR emission burst (∼
yields a rotational temperature T ≈ 220K for a PAH 1−10s), the internal alignment temperature during IR
rot
with N =200. emission would stay roughly equal to the value before
C
In the intense radiation field of a bright PDR, the UV absorption. If γ (cid:29) 1, this would imply γ (cid:29) 1 as
0 r
vibrational energy of our model PAH is always large well. We therefore consider γ = ∞ and γ = ∞ as the
0 r
6
limitingcaseinwhichaˆ isperfectlyalignedwithJboth where α is the angle between the line of sight and the
1
before UV absorption and during IR emission. illumination direction (see Fig. 1a), and
√ √
3. POLARIZATIONFROMPAHSILLUMINATEDBYA g1(γ)≡ πγerfi( γ) ,
STARORADISKGALAXY √ √ √
g (γ)≡6 γeγ −(3+2γ) πerfi( γ) ,
We now compute the degree of polarization expected 2
for in-plane and out-of-plane vibrational modes of PAH h(γ)≡g1(γ)/g2(γ) . (25)
molecules when anisotropically illuminated by a star or
The function h(γ) monotonically decreases from h(0)→
adiskgalaxy. Weassume,forthesakeofsimplicity,that
∞ to h(γ → ∞) = 1/4. Since h(0) → ∞, if the grain
the angular momentum of emitting grains is randomly
principalaxisofinertiaisrandomlyorientedwithrespect
orientedinspace,althoughitmaybepartiallyalignedby
to the angular momentum either before or after UV ab-
theinterstellarmagneticfield. Also,wesupposethatthe
sorption, the emitted radiation is completely unpolar-
illuminatingdiskgalaxyhasauniformsurface-brightness
ized. Since h(γ) > 1/4, we have p(cid:107) ≥ 0 and p⊥ ≤ 0 for
profile. However, the formalism developed in §2 can be (cid:63) (cid:63)
anychoiceofγ ,γ andα. Accordingtothedefinitionin
applied without such restrictions, and we discuss below 0 r
eq.(22),thismeansthatin-planeandout-of-planemodes
howtorelaxthesetwoassumptions. Thegrainalignment
are polarized respectively along uˆ and along vˆ, confirm-
between aˆ and J is described via the probability distri-
1 ingLeger’s(1988)predictionthatthepolarizationdirec-
butionineq.(17)bothbeforeUVabsorption(dP )and
γ0 tion for in-plane and out-of-plane modes is respectively
duringIRemission(dP ),with“internalalignment”co-
γr orthogonal and parallel to the plane-of-sky projection of
efficients γ and γ respectively.
0 r the illumination direction (see Fig. 1a).
For a population of PAH molecules illuminated by a
For fixed γ and γ , eqs. (23) and (24) show that the
point source, the emission polarized along a direction wˆ 0 r
IR emission bands should be maximally polarized when
(= either uˆ or vˆ – see Fig. 1a) for in-plane ((cid:107)) and out-
α = π/2, i.e., the line of sight to the emitting PAHs is
of-plane (⊥) modes is
orthogonal to the star-molecule direction. For this op-
(cid:90) 1 (cid:90) 2π (cid:90) π (cid:90) π timal viewing geometry, Table 1 reports the degree of
I(cid:107),⊥(α,γ ,γ )∝ dcosθ dϕ dP (β ) dP (β )
(cid:63),w 0 r γ0 0 γr r polarization expected for the 3.3, 7.7, 11.3 and 17µm
−1 0 0 0
emission features in different environmental conditions,
×A¯(cid:63)(θ,β0)F¯w(cid:107),⊥(θ,ϕ,βr,α) , (20) asdiscussedin§2.5;wehaveassumedthe17µmbandto
whereA¯ andF¯(cid:107),⊥ aretheabsorptionandemissioncross arisefromout-of-planemodes. Figure2showsthedepen-
sections(cid:63)computwed in §2.2 and §2.3 respectively. If the dence of p(cid:107)(cid:63)(π/2) and p⊥(cid:63)(π/2) on the internal alignment
grain angular momentum is not isotropically oriented in coefficient γ0, with the different curves corresponding to
space, it is easy to incorporate the corresponding (θ,ϕ)- different values of the ratio γr/γ0. Because γr ≤ γ0,
distribution function into the above integral. The polar- the region to the left of the solid lines is excluded. In
ized emission from PAHs above the center of a uniform- most cases, the dependence on the viewing angle α in
brightness disk galaxy is thedenominatorofeqs.(23)-(24)canbeneglected,hence
I(cid:107),⊥ (α,γ ,γ ,ω)∝(cid:90) 1dcosθ(cid:90) 2dπϕ(cid:90) πdP (β )(cid:90) πdP (β ) p(c(cid:107)(cid:63)o(mα,pγa0r,eγrso)l≈idpa(cid:107)(cid:63)n(dπ/d2o,tγt0e,dγlri)nessini2nαF,iagn.d3)s.imilarlyforp⊥(cid:63)
gal,w 0 r γ0 0 γr r
−1 0 0 0 For the extreme case of perfect alignment between aˆ
1
(cid:90) 1dcosθ(cid:48) and J during both UV absorption (γ = ∞) and IR
× A¯ (θ,β ,θ(cid:48))F¯(cid:107),⊥(θ,ϕ,β ,α) ,(21) 0
Ω gal 0 w r emission (γr = ∞), corresponding to case (d) in §2.5,
cosω eqs. (23) and (24) may be simplified:
where Ω = 2π(1 − cosω) is the solid angle subtended
by the galactic disk as seen from the emitting molecules. sin2α
A generic axisymmetric surface-brightness profile B(θ(cid:48)) p(cid:107)(cid:63)(α)= 13+cos2α , (26)
can be easily included in the previous expression.
−sin2α
The resulting degree of polarization for in-plane and p⊥(α)= . (27)
out-of-plane modes is respectively (cid:63) 7−cos2α
I(cid:107),⊥−I(cid:107),⊥
p(cid:107),⊥ ≡ u v , (22) 3.2. Polarization from PAHs illuminated by a
I(cid:107),⊥+I(cid:107),⊥ uniform-brightness disk galaxy
u v
andwillthereforebepositiveforemissionpolarizedalong Thedegreeofpolarizationforin-planeandout-of-plane
uˆ and negative for polarization along vˆ. modes from a population of PAH molecules above the
center of a uniform-brightness disk galaxy is obtained
3.1. Polarization from PAHs illuminated by a star
from eq. (21):
By integrating eq. (20), we obtain the degree of polar-
ization for in-plane and out-of-plane modes from a pop- p(cid:107) (α,γ ,γ ,ω)= 3 sin2α ,(28)
ulation of PAH molecules illuminated by a star (or a gal 0 r 1280 h(γ0)h(γr) +3 cos2α−1
generic point source) (cos2ω+cosω)
−3 sin2α
p(cid:107)(cid:63)(α,γ0,γr)=640h(γ0)h(3γrsi)n+2α3 cos2α−1 , (23) p⊥gal(α,γ0,γr,ω)=640(chos(2γ0ω)+h(cγors)ω) −3 cos2α+1 , (29)
−3 sin2α where Ω = 2π(1−cosω) is the angle subtended by the
p⊥(α,γ ,γ )= , (24)
(cid:63) 0 r 320h(γ )h(γ )−3 cos2α+1 galaxyasseenfromtheemittinggrains. ThelimitΩ→0
0 r
7
Fig. 2.— For PAHs illuminated by a star, dependence of the Fig. 3.— For PAHs illuminated by a star, dependence of the
polarization p(cid:63) on γ0, for optimal viewing geometry (α = π/2). polarization p(cid:63)(α,γ0,γr) on the viewing angle α. Dashed line:
The different lines correspond to different values of the ratio polarization for (γ0,γr)=(10,2), similar to case (c) in §2.5. The
γr/γ0: γr = γ0 (solid), γr = 0.5γ0 (dashed), γr = 0.25γ0 polarizationvaluesquotedbyLeger(1988)foradiskmoleculeare
(dotted), and γr = 0.1γ0 (dot-dashed). The polarization values shownassolidsquares,andareseentomatchthevaluescalculated
reported in Table 1 are shown as solid triangles (3.3µm), open for (γ0,γr) = (10,2). Solid line: polarization for perfect internal
triangles(7.7µm),solidcircles(11.3µm),andopencircles(17µm, alignment (γ0,γr) = (∞,∞) (see eqs. (26)-(27)), corresponding
assumed to arise from out-of-plane modes), with different values to case (d) in §2.5. The dotted line is p(cid:107),⊥(π/2)sin2α for
coafsγe0(cc)or(rγe0sp=on1d7i)n.g to case (a) (γ0 =1.2), case (b) (γ0 =1.5) or (αγ0o,fγerq)s.=(2(6∞)-,(∞27)),.whichcloselyapproximates(cid:63)thedependenceon
recovers the case of a point-like illuminating source in
eqs. (23) and (24), whereas for an infinite disk (Ω →
2π) the degree of polarization tends to zero. Figure 4
shows, for the most favorable viewing geometry (α =
π/2), the dependence of the degree of polarization on γ
0
for PAHs illuminated by a disk galaxy with Ω = π; the
different curves correspond to different values of γ /γ ,
0 r
asexplainedin§3.1. FordiffuseilluminationwithΩ=π,
wefindlevelsofpolarizationthatare∼40%ofthevalues
found for uni-directional illumination.
The extreme case of grains with perfect internal align-
ment (aˆ (cid:107) J during both UV absorption and IR emis-
1
sion, case (d) in §2.5) yields
3 sin2α
p(cid:107) (α,ω)= ,(30)
gal 80(cos2ω+cosω)−1+3 cos2α−1
−3 sin2α
p⊥ (α,ω)= .(31)
gal 40(cos2ω+cosω)−1−3 cos2α+1
4. DISCUSSIONANDSUMMARY
We have obtained analytic formulae for the degree of
polarizationexpectedforthe3.3,6.2,7.7,8.6,11.3,12.7,
16.4, and 17 µm emission features when the emitting
PAHs are anisotropically illuminated by a source of UV
photons. WemodelPAHgrainsasplanarmoleculeswith
in-plane and out-of-plane vibrational dipoles. Vibra- Fig. 4.— For PAHs illuminated by a disk galaxy with Ω = π,
tional modes oscillating in the molecular plane (respon- dependence of the polarization pgal on γ0, for optimal viewing
sibleforthe3.3,6.2,7.7,and8.6µmemissionbands)are geometry (α = π/2). The different lines correspond to different
polarized perpendicular to the plane-of-sky projection valuesoftheratioγr/γ0,asexplainedinthecaptionofFig.2.
of the illumination direction, whereas for out-of-plane
8
modes (responsible for the 11.3 and 12.7µm features) the case of uni-directional illumination, which is realized
the polarization is along the source-molecule direction, only if the galaxy’s luminosity is strongly concentrated
as originally pointed out by Leger (1988). The fact that atitscenter. Ifthegalaxyappearstotheemittinggrains
thepredictedpolarizationdirectionsforin-planeandout- asanextendedsource,thePAHpolarizationwillbeeven
of-plane modes are orthogonal provides a check on the smaller.
contributiontothepolarizationfromlineardichroismby The Orion Bar PDR is expected to be somewhat
aligned foreground dust, as this would be expected to more favorable for the PAH internal alignment because
produce polarization in a single plane. It also means of higher molecular rotation rates, but even there the
that polarization measurements could be used to diag- polarization of the 3.3µm feature is predicted to be
nosethecharacter(in-planeorout-of-plane)ofthemodes only ≈ 0.06% for our estimated (γ ,γ ) = (1.5,0.28).
0 r
responsible for the poorly-characterized 16.4 and 17µm If the polarization 0.86 ± 0.28% measured by Sellgren
emission features. et al. (1988) for the 3.3µm emission feature is correct,
Our analytic formulae show explicitly how the degree it implies that we underestimated the internal align-
of polarization depends on the viewing geometry (i.e., ment of emitting PAHs both before UV absorption (γ )
0
the angle α between the line of sight and the illumina- and during IR emission (γ ). A polarization of 0.86%
r
tion direction) and the “internal alignment” coefficients for optimal viewing angle α = π/2 would require, e.g.,
γ , describing the alignment between the grain principal (γ ,γ ) = (2.3,2.3), (3.0,1.8), (∞,0.81), etc.: all solu-
0 0 r
axisaˆ andangularmomentumJjustbeforeUVabsorp- tions must have γ ≤ γ , hence γ ≥ 2.3 and γ ≥ 0.81.
1 r 0 0 r
tion,andγ ,characterizingtheinternalalignmentduring Such good internal alignment is not expected unless the
r
the brief period of IR emission following UV excitation. smallest PAHs are rotating suprathermally. The polar-
The degree of polarization is maximal for α = π/2 and, ization at 11.3µm is predicted to be considerably larger
at fixed α, it increases with γ and γ , as the princi- thanthepolarizationat3.3µm(seeTable1). Additional
0 r
pal axis and angular momentum become more closely polarization measurements of the Orion Bar at 3.3 and
aligned. We have discussed both the case of a point- 11.3µmareneeded. Ifthelargeintrinsicpolarizationde-
like illuminating source, which may be applied to re- tected by Sellgren etal. (1988) at 3.3µm is confirmed, it
flection nebulae and PDRs such as the Orion Bar, and willcallintoquestioncurrentthinkingwithregardtothe
of an extended source (e.g., dust above a disk galaxy rotational dynamics of PAHs in PDRs, since it seems to
like NGC 891 or M82). The degree of polarization for require at least moderately suprathermal rotation rates.
uni-directional illumination is higher than for diffuse il- Comparison of our analytic results with polarization
lumination, all else being equal; however, if the galaxy’s measurements may provide useful constraints on the ge-
surface-brightnessprofileisstronglypeakedatthegalac- ometrical and dynamical properties of PAH molecules,
tic center, the galaxy would resemble a point source, concerning in particular: i) their planarity, which has
with polarization levels comparable to the case of uni- been assumed a priori in this work, although it is still
directional illumination. uncertain, especially for larger PAHs; ii) the efficacy of
The value of γ is expected to increase with increasing internal relaxation processes that can exchange energy
r
wavelengthoftheemissionfeature,sincesuchabandwill betweenvibrationalandrotationalmodes; iii)thepossi-
bemostlyemittedwhenthegrainiscoolerandaˆ andJ bility of systematic torques that may spin the grains up
1
areexpectedtobemorecloselyaligned. Therefore,when to suprathermal rotation; and iv) a determination, via
comparing two IR emission features arising from modes the polarization direction, of the character of the vibra-
of the same character (either in-plane or out-of-plane), tional modes contributing to a given IR feature, which
thebandwithlongerwavelengthshouldbemorestrongly will beof particular interest for thepoorly-characterized
polarized. Table 1 reports, in case of uni-directional il- 16.4 and 17µm bands.
lumination and optimal viewing geometry (α = π/2), Ifthepolarizationistoosmalltobedetected,theinter-
thepredicteddegreeofpolarizationforthe3.3,7.7,11.3, pretationwillnotbeclear: itcouldbetheresultofPAH
and 17µm emission features, for four exemplary choices non-planarity, or a mixture of in-plane and out-of-plane
of environmental conditions.3 contributions to the mode (this may be an issue for the
FordustintheCNM,significantdisalignmentbetween 16.4and17µmfeatures),or(morelikely)itcouldsimply
aˆ and J (γ ≈ 1, γ (cid:46) 0.5γ ) is expected to strongly derivefrompooralignmentofthegrainprincipalaxisaˆ
1 0 r 0 1
suppress the intrinsic polarization of the IR bands, with and angular momentum J. However, if a high degree of
the 3.3µm emission feature predicted to be polarized by polarization is observed, it will imply both that PAHs
only≈0.02%(forourestimatedγ =1.2andγ =0.10), are planar and that their principal axis is well aligned
0 r
presumably too small to distinguish from polarization with the angular momentum. This would be valuable
due to foreground linear dichroism. Longer wavelength informationconcerningthenatureofPAHsandtheirro-
featuresareexpectedtobemorestronglypolarized(e.g., tational dynamics.
−0.14%forthe11.3µmfeature,withγ =0.40),butthe
r
polarization levels are still observationally challenging.
BTD thanks Chris Packham and Charles Telesco for
These conditions may apply to PAHs above the disk of
helpful discussions on the possibility of polarization of
an edge-on galaxy, such as NGC 891 or M82; however,
thePAHfeatures. WearegratefultoR.H.Luptonforthe
the polarization degrees quoted above are computed for
availability of the SM graphics program. This research
was supported in part by NSF grant AST-0406883.
3 The polarization results in Table 1 assume that the PAH
angularmomentaarerandomlyorientedinspace;ifthegrainsare
partiallyoriented,e.g.,bytheinterstellarmagneticfield,wewould
expectahigherdegreeofpolarization.
9
REFERENCES
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Observatory),769-795
APPENDIX
POLARIZATION FROM PAHS IN BROWNIAN ROTATION
We have assumed above that the internal alignment between the principal axis aˆ and angular momentum J of
1
emitting PAHs can be parametrized via the probability distribution in eq. (17) both before UV absorption (with
γ ) and during IR emission (with γ ≤ γ ). However, if the interval between two subsequent photon absorptions is
0 r 0
longer than the time required for the PAH molecule to collide with its own mass of gas, and if the IVRET process is
ineffectiveatexchangingenergybetweenthegrainrotationandthe(generallycolder)vibrationalmodes, themolecule
will be driven towards “Brownian rotation” with internal alignment temperature T ≈ T prior to UV absorption.
0 gas
In this case, the probability distribution dP in eq. (19) should be used instead of eq. (17) to describe the grain
(cid:15)
internal alignment before UV absorption, whereas dP in eq. (17) with internal alignment coefficient γ may still
γr r
be appropriate during IR emission. If the molecule angular momentum is randomly oriented in space, the polarized
emission along a direction wˆ (= either uˆ or vˆ – see Fig. 1a) for in-plane and out-of-plane modes from a population of
PAHs illuminated by a point-like source is
(cid:90) 1 (cid:90) 2π (cid:90) π (cid:90) π
I(cid:107),⊥(α,(cid:15),γ )∝ dcosθ dϕ dP (β ) dP (β )A¯ (θ,β )F¯(cid:107),⊥(θ,ϕ,β ,α) . (A1)
(cid:63),w r (cid:15) 0 γr r (cid:63) 0 w r
−1 0 0 0
The corresponding expression for PAH molecules above the center of a uniform-brightness disk galaxy is
(cid:90) 1 (cid:90) 2π (cid:90) π (cid:90) π (cid:90) 1 dcosθ(cid:48)
I(cid:107),⊥ (α,(cid:15),γ ,ω)∝ dcosθ dϕ dP (β ) dP (β ) A¯ (θ,β ,θ(cid:48))F¯(cid:107),⊥(θ,ϕ,β ,α) , (A2)
gal,w r (cid:15) 0 γr r Ω gal 0 w r
−1 0 0 0 cosω
where Ω = 2π(1−cosω) is the solid angle subtended by the galactic disk as seen from the emitting molecules. The
degree of polarization can then be computed from eq. (22). Let us define
√ √
q((cid:15))≡3(cid:15)[1− (cid:15)−1arcsin(1/ (cid:15))]−1 , (A3)
where(cid:15)≡I /(I −I ). Makinguseofh(γ)asdefinedineq.(25), thepolarizationforin-planeandout-of-planemodes
1 1 2
in case of uni-directional illumination is
3 sin2α
p(cid:107)(α,(cid:15),γ )= , (A4)
(cid:63) r 320h(γ )/q((cid:15))+3 cos2α−1
r
−3 sin2α
p⊥(α,(cid:15),γ )= , (A5)
(cid:63) r 160h(γ )/q((cid:15))−3 cos2α+1
r
whereas for diffuse illumination by a uniform-brightness disk galaxy we have
3 sin2α
p(cid:107) (α,(cid:15),γ ,ω)= , (A6)
gal r 640 h(γr)/q((cid:15)) +3 cos2α−1
(cos2ω+cosω)
−3 sin2α
p⊥ (α,(cid:15),γ ,ω)= . (A7)
gal r 320 h(γr)/q((cid:15)) −3 cos2α+1
(cos2ω+cosω)
Comparison of these formulae with eqs. (23)-(24) and eqs. (28)-(29) shows that the polarization results obtained here
can be exactly reproduced by the probability distribution in eq. (17) if we adopt an “effective” internal alignment
coefficient γ before UV absorption satisfying 2h(γ ) = 1/q((cid:15)). For (cid:15) = 2, as appropriate for our model PAH with
0 0
I /I =2, the solution is γ ≈1.0.
1 2 0
