Physics and Chemistry of Purcell's Alignment

10 Pages·0.22 MB·English

by  Alex Lazarian

Mon.Not.R.Astron.Soc.000,000{000(1994)
Physics and chemistry of the Purcell's alignment
A. Lazarian
DAMTP, Silver Street, University of Cambridge, U.K.?
ReceivedAccepted
ABSTRACT
Paramagneticalignmentofsuprathermallyrotatinggrainsisdiscussed inviewofrecent
progress in understanding subtle processes taking place over grain surface. It is shown
thatintypicalISMconditions,grainswithsurfacesofamorphousH2Oice,defected sili-
cateorpolymericcarbonaceous materialarelikelytoexhibitenhanced alignment,while
those of aromatic carbonaceous material or graphite are not. The critical grain sizes
5 andtemperature forthe Purcell's alignmentare obtained andpreferential alignmentof
9 large grains is established.

n Key words: magnetic (cid:12)elds { polarization { dust extinction.
a
J

7
1
1 INTRODUCTION Mathis 1986,Chlewicki &Greenberg 1990,Greenberg1993,

Spitzer 1993).
1The paramagnetic mechanism of grain alignment (Davis &
(ii) Observations indicate that grains of di(cid:11)erent chemical
6Greenstein 1951) is believed to be too weak to account
composition exhibit di(cid:11)erent degree of alignment (Mathis
0for observations unless enhanced imaginary part of grain
1986,Whittet1992)whilethisdoesnotfollowfromthethe-
1magnetic susceptibility is assumed (Jones & Spitzer 1967,
ory (see Jones & Spitzer 1967).
0Mathis 1986). Physically this weakness means that ther-
5 (iii)Largegrainsproducemorepolarization thansmallones,
maldisorientationduetochaoticatomicbombardmentover-
9 whiletheDavis-Greenstein mechanismtendstogotheother
whelms orientation e(cid:11)ects of dielectric grains with normal
/ way (Mathis 1986, Kim & Martin 1994, Mathis 1994).
hparamagnetic susceptibility rotating in the Galactic mag-
In our recent paper on the alignment of fractal grains
pnetic (cid:12)eld. To diminish the e(cid:11)ect of thermal disorientation
(Lazarian 1994c), we were mostly concerned with the (cid:12)rst
-it was suggested in Purcell (1975, 1979) that grains rotate
o di(cid:14)culty mentioned above. This papers studies details of
suprathermally, i.e.withenergyordersofmagnitudegreater
r grain surface physics and chemistry to address the other
tthananytemperatureinthesystemtimesk(theBoltzmann
s twoproblems.Suchastudywouldnotbepossibleuntilquite
aconstant).
recently as it is based on new results obtained in the (cid:12)eld
It may be shown that among various causes of
(Duley 1993, Duley & Williams 1993).
suprathermalrotationthoseassociatedwithcorpuscularand
The structure of the paper is as follows. In section 2
radiative (cid:13)uxes result in the Gold-type alignment (Lazar-
we discuss the physics and chemistry of suprathermal ro-
ian 1994a, 1994b); its time-scale is orders ofmagnitude less
tation caused by H2 formation over grain surfaces, whereas
thanthatofparamagneticrelaxation.Therefore,`chemically
the paramagnetic relaxation is discussed in section 3. The
driven' suprathermal rotation is required by the Purcell's
ways of reconciling theory and observations are discussed
mechanism. The estimates in Spitzer & McGlynn (1979)
in section 4 and two tests for the theory are suggested in
showed that if high rates of grain resurfacing are consid-
section 5.
ered, it still improves the alignment only marginally. Thus,
straightforward attempts to explain observations through
thePurcell'smechanism(Aanestad&Greenberg 1983)had
rather limited success (Spitzer 1993). 2 `CHEMICALLY DRIVEN' SUPRATHERMAL
The major di(cid:14)culties of the paramagnetic alignment ROTATION
can be summarised as follows.
Although the interest to grain chemistry in view of grain
(i)Theparamagneticalignmentistooweakifordinarypara-
alignment can be tracked back to Purcell & Spitzer's paper
magnetism is involved and `standard' values of magnetic
of1971,itwasonly inPurcell (1975,1979)thattheconcept
(cid:12)eldandgaseousdensityareused(Spitzer&McGlynn1979,
of suprathermal rotation of grains driven by H2 formation
was introduced. These pioneering papers were based on the
level of knowledge about grain chemistry achieved at that
? Present address: Dept. of Astronomy, University of Texas, time. New discoveries in this rapidly growing area make us
Austin,TX 78712,U.S.A. to update some of the ideas put forward by Purcell. This
2 A. Lazarian
2
update does not alter the core of the Purcell's concept, but 2 (cid:13)1 6 2 2 (cid:0)1
hMzi(cid:25) l nHmH2vHE(cid:23) (1)
shows that the `chemically driven' suprathermal rotation is 32
notuniversallyapplicable,e.g.tograinsofdi(cid:11)erentchemical which spins up grains to angular velocities limited only by
y
composition. friction forces,i.e.
It is shown in Duley & Williams (1993) that H2
2 1=2td
molecules are formed over aromatic carbonaceous surfaces (cid:10)=hMzi (2)
Iz
in low-energy state as the PAH.2H complex shares all the
energy of stabilisation of H2 (4.5 eV) among its degrees of where Iz is the z component of the momentum of inertia,
freedom'.Thisprediction entailsthatsuchgrainsshouldnot and td is the rotational dumping time (Spitzer & McGlynn
rotate suprathermally due to H2 formation. 1979):
The situation is di(cid:11)erent for H atoms weakly adsorbed l%s
bywatericeorpolymerichydrocarbon.Thebinding isweak, td (cid:25)0:6tgas =0:6 (3)
nmv1
itsenergyisnotexpectedtoexceed0.1eV(Lee1972,Leitch-
Devl(cid:19)in&Williams1984,Zhang&Buch1990,Buch&Zhang where tgas is the time that takes a gain of density %s to
collide with gaseous atoms of the net mass equal to that
1991). H2 molecules formed on such sites stabilise into very
high vibrational states (Duley &Williams 1993). Aportion of the grain, n and v1 denote the density and velocity of
atoms, respectively, while m is the mass of an individual
of vibrational energy ((cid:24)0:1(cid:0)0:2 eV) is likely to be trans-
atom. To obtain both numerical values and functional de-
ferredtothecenter-of-masstranslational motion (Tielens &
7 (cid:0)1 pendencies further on we will use quantities normalised by
Allamandola 1987)atapproximately 10 s rate(Lucas&
their standard values. To tell the normalised values we will
Ewing 1981).Thusaninjection ofhigh speed H2 molecules
denote them by corresponding letters with hats, e.g. ^l is
is expected from icy or polymeric hydrocarbon grain sur-
(cid:0)5 (cid:0)5
equivalent to l=(2(cid:1)10 cm) with 2(cid:1)10 cm as a stan-
faces. Similarly using results in Duley & Williams (1986) it
is possible to show that H2 molecules formed over defected dard value of a gr5ain siz(cid:0)e.1We consider `standard' (cid:13)1 = 0:2
silicate surfaces arealso expected tohave high kinetic ener- and vH2 = 5(cid:1)10 cm s . Speaking about di(cid:11)use clouds,
we assume that all hydrogen there is in atomic form and
gies. For the hydrogenate amorphous carbon (HAC)model
of interstellar dust (Jones et al. 1990) which consists of a therefore n=nH (cid:17) 1 and m=mH2 (cid:17) 0:5. Note that some-
timesthechoice of`standard'values isarbitrary,e.g.forthe
mixture ofpolymeric, aromatic and diamond-like regions of
carbon separated by voids, the situation is more involved. time being we assume the density of active sites (cid:11)H2 to be
11 (cid:0)2
H2formationisexpectedtobee(cid:14)cientinthesenseofrecoils 10 cm .However,wewillseeinsection3that(cid:11)H2sensi-
tivelydependsonsubtleprocessesofpoisoningandtherefore
deposited withgrains forpolymeric and probably diamond-
`standard' values may be misleading.
like regions while aromatic regions are expected toeject H2
Using standard values of the parameters involved and
with low kinetic energy.
assuming that the number of active sites is proportional to
The ejection of H2 molecules with high kinetic energy 2
l , it is possible to write the following expression for the
E results in torques applied to a grain. Everywhere below
angular velocity
we assume E (cid:25) 0:2 eV. This means that our approach is
applicable to suprathermal rotation of grains with only icy, (cid:10)(cid:25)1:4(cid:1)108(cid:13)^1(cid:11)^(cid:0)H12=2(^l)(cid:0)2 s (4)
polymeric carbonaceous or silicate surfaces.
It was shown in Purcell (1979) that internal dissipa- If grains are porous and (cid:13)u(cid:11)y as those found in the
tion of energy within grains, mainly due to the Barnett stratosphere (see (cid:12)g. 7.3 in Williams 1993) the degree of
relaxation,z suppressesrotationaroundanyaxisbuttheaxis suprathermal rotation can be limited provided that H2 for-
7 mation takes place within narrow irregular pores of grains
of the greatest inertia on the time-scale (cid:24)10 =(cid:17) s, where (cid:17)
andnascentmoleculesthermalizeduetonumerouscollisions
is the ratio of grain rotational energy to the equipartition
with pore walls. These processes are discussed in more de-
energy (cid:24)kT.Thusgrains rotatearound theirmajoraxesof
tail in Lazarian (1994c), and here we mention this in view
inertia that we denote here the z-axis. Therefore the prob-
of implications that the suprathermal rotation can have on
lem becomes one dimensional as the two other components
grain structure.The tensile stresswithin thespinning grain
ofthetorquecontribute only toinsigni(cid:12)cant nutations. The
2 2 8 (cid:0)2
number of H2 molecules ejected per second from an indi- is of the order of %sl (cid:10) may reach 10 dyn cm . An in-
2 (cid:0)1 teresting hypothesis by Wright (1994) is that suprathermal
vidual site is (cid:24) (cid:13)1l vHnH(cid:23) , where (cid:13)1 the portion of H
rotation limits thefractal dimension ofgrains may havethe
atoms with velocity vH and concentration nH adsorbed by
following consequences:
the grain, while (cid:23) is the number of active sites over the
a) if grains have high fractal dimension, their degree of
grain surface and l is the characteristic grain size which co-
suprathermality is low and their fractal dimension does not
incides with the diameter for a spherical grain. Then the
change,
mean square of a residual torque from an H2 molecule of
b) grains of aromatic carbonaceous material and graphite
mass mH2 is
are not limited in their fractal dimension by suprathermal
rotation,
y We will considerrotationinducedby H2 formation,although c) grains with surfaces of defected silicate, amorphous H2O
someotherchemicalreactionsmightalsocontributetothegrain ice orpolymeric carbonaceous material withalimited num-
rotationindarkclouds,wheretheabundanceofatomichydrogen ber ofpores will have theirfractal dimension decreased asa
islow. result of suprathermal rotation.
z AquantitativetreatmentthroughsolvingFokker-Planckequa- Inshort,silicate, iceandcarbonaceous grains mayhave
tionsisgiveninLazarian(1994a). either large d ! 3 or low d ! 2 fractal dimension. One
Physics and chemistry of the Purcell's alignment 3
may also speculate that if (cid:11)H2 is proportional to grain sur- ity comestosaturation. Forlong-lived spin-up, i.e.tL (cid:29)td,
(cid:0)2
face, (cid:10) scales as l [see Eq. (4)], the tensile stress scales Eqs. (4) and (5) provide
(cid:0)2
as l as well and therefore small grains are more vulner- (cid:18) 8(cid:19)1=2
able to centrifugal disruption, which may make them more (cid:0)5 5:6(cid:1)10 1=2 (cid:0)1=4 1=2
lcr m =10 (cid:13)^1 (cid:11)^H2 v^H2 cm (6)
spherical as compared to larger grains. Adopting a model !c
of a grain as consisting of small particles jumbled together
1=4
(Mathis 1990), one may say that these conglomerates are and for short-lived spin-up, i.e. tL (cid:28) td, it is (tL=td) as
large.
more subject to disruption if they are small. However, this
problem needsfurtherinvestigation andwewill notproceed For the original Davis-Greenstein process, tr should be
compared with the time that takes a grain of mass m to
with its discussion here.
Grain dynamics depends notonly onthenumber ofac- collide with gas atoms of the same total mass tgas = 2(cid:1)
tive sites but also on their distribution over grain surface. 1012^l%^(v^Hn^H)(cid:0)1 s, where the gaseous density is normalised
(cid:0)2
by 20 cm and the `standard' thermal velocity of atoms
Both accretion of a new monomolyer and poisoning of ac-
5 (cid:0)1
tive sites cause occasional `crossovers' (Spitzer & McGlynn vH is assumed to be 1:3(cid:1)10 cm s . Note, that in many
cases the velocity v is dominated by Alfv(cid:19)enic phenomena
1979) which considerably e(cid:11)ect grain alignment. If these
(see Aron & Max 1975, Elmegreen 1992, Myers 1987, 1994,
crossovers occur on the time-scale tL, one should multiply
our estimates of (cid:10)given by Eq. (2)by t1L=2=(tL+td)1=2. Lazarian 1992, 1994a), the factthat makes tgas smaller.
Whileagrainrotatessuprathermally, chaoticbombard-
ment of atoms cannot alter its rotation. It was argued in
Purcell (1979) that a quantity tx = 1:3(tL+td) should be
compared with tr to (cid:12)nd the Rayleigh reduction factor h(cid:27)i
3 THE PURCELL'S MECHANISM
introduced in Greenberg (1968), whichrelates the measures
3.1 Paramagnetic relaxation ofpolarization andgrain alignment (seeWhittet1992).The
expression of h(cid:27)i is given in Aanestad & Greenberg (1983)
Aparamagneticgrainrotatingintheexternalmagnetic(cid:12)eld and here wejustmention thath(cid:27)idepends onthefollowing
B is subjected to dissipative torques (Davis & Greenstein quantity (Spitzer & McGlynn 1979):
1951,seealso Jones&Spitzer 1967,Purcell &Spitzer 1971,
( (cid:0)1
Roberge et al. 1993). In the grain coordinate system, the tx(tr) ; F >1
component of B perpendicular to the rotation axis appears (cid:14)e(cid:11) = (7)
(cid:0)1
as a rotating (cid:12)eld. The magnetic moment produced by the tx(Ftr) ; F <1
00
imaginary part of magnetic susceptibility (cid:31) causes rota-
whereF isthe`disorientation parameter'explicit analytical
tion of the vector of angular momentum towards the direc-
expression of which is obtained in the Appendix. One may
00
tion of B. For low frequencies, (cid:31) of paramagnetic materi-
establish a critical size lcr f after which F becomes greater
als is proportional tothecomponent oftheangular velocity
than unity [see Eq. (A11)]
7 8 (cid:0)1
perpendicular to B up to frequencies !c (cid:25) 10 (cid:0)10 s .
00 1=3
If (cid:10) (cid:29) !c, (cid:31) becomes independent of (cid:10) and approaches (cid:0)5 (v^H2(cid:23)^)
the static susceptibility. In short, large grain angular ve- lcr f =1:8(cid:1)10 %^1=4(n^Hv^H(cid:13)^1)1=12 (8)
locities increase the characteristic time of paramagnetic re-
00 where (cid:23)^ is normalised by 100. If density of active sites is
laxation as (cid:31) comes to its asymptotic values (compare
proportional to grain surface (we will see further on that
Whittet 1992). The characteristic time of this relaxation is
tr (cid:25) 1014q1(cid:0)1^l2T^sB^(cid:0)2K((cid:10))%^ s (Spitzer 1978), where grain this may not be the case):
2tegmcpme(cid:0)r(cid:0)6a3turreespTesctaivnedlyd,ewnhsiitlye m%aagrneentiocrm(cid:12)ealdlisiesdnboyrm1a0lisKedanbdy lcr f =1:7(cid:1)10(cid:0)5%^3=8(n^(cid:11)H^1H(cid:13)^=124v^H)1=8 cm (9)
2(cid:1)10 G, q1 is the enhancement factor of the imaginary
partofmagneticsusceptibility suggested inJones&Spitzer For grains greater than this size, it is important to account
(1967), and K((cid:10)) is a function with the following asymp- for partial preservation of alignment during the crossovers.
totics In the limit of tL (cid:28) tgas and l < lcr f,
(cid:26) (cid:14)e(cid:11) coincides with the parameter of alignment for the
K((cid:10))= 1(cid:10); (cid:10)(cid:28)!c (5) Davis-Greenstein mechanism and is equal to tgas=tr =
!c; (cid:10)(cid:29)!c 0:02 (^lT^sv^Hn^H)(cid:0)1B^2 K((cid:10))q1.
Up tonow we have discussed disorientation due to col-
Itwasnoted byMathis (1994) thatalowvalue ofB is used
lisions with gaseous atoms. The disorientation due to radi-
above asstandard andtheremightbeasubstantial increase
2 ation introduced in Purcell &Spitzer (1971)can be charac-
for2 B term (see e.g. Boulares & Cox 1990, (cid:12)g 3b, where terised by the time-scale trad (cid:25)5(cid:1)1013^l%^T^s(cid:0)4 s.
B =8(cid:25) isshownincgsunits). `However,manypeople would
not be willing to go beyond about 3 micro G for the stan-
dard ISM(cid:12)eld intheGalactic plane',heconcludes. Laterin
3.2 Grain spin-up
section 4we will study the possibility that enhanced values
of magnetic (cid:12)eld are responsiblefor alignment. Toobtain su(cid:14)cient alignment forq1 =1,it is required that
For simple estimates, one may consider that transition tL (cid:29) tgas. The di(cid:14)culty on this way is the (cid:12)nite life time
from one asymptotic to another is at (cid:10) = !c. Within this of active sites over grain surface.
simple model, it is possible tointroduce a critical grain size A process that can limit the life time of active sites
lcr m when the imaginary part of paramagnetic susceptibil- is resurfacing due to the accretion of a new monomolecular
4 A. Lazarian
(cid:16) (cid:17)
layer(Spitzer&McGlynn 1979).Agrainofmassm(l)grows th (cid:25) 1 exp Eh (12)
as (cid:23)0 kTs
dm(l) 1 where (cid:23)0 (cid:25) 1012 s(cid:0)1 is the characteristic frequency of vi-
= S(l)vnam(cid:24)a (10)
dt 4 bration foranadsorbed particle andEh istheenergy ofpo-
where(cid:24)a istheadsorption probability, acoe(cid:14)cientwiththe tential barrier which isapproximately one third ofthe total
range ofvalues from0to1foratoms ofdensity na,velocity binding energy (although see Jaycock & Par(cid:12)tt 1986). The
2
v and mass m; S(l) (cid:24) (cid:25)l is the geometric surface of the total binding energy foroxygen maybe assumed (cid:24)k(cid:1)800 K
grain. Assuming that accreting atoms uniformly cover the (Tielens & Allamandola 1988). In this case, the normalised
entire physical surface of grains and substituting dm(l) = E^h is equal toEh=(k(cid:1)267 K).
%Sph(l)dl,whereSph(l)isthephysical surfaceareaofgrain, Anadsorbed Hatomscans theentire surfaceofagrain
(cid:0)5 (cid:0)2
one obtains for the accreting time: ofl=2(cid:1)10 cmonthetime-scalenotexceeding10 s.Ifit
(cid:16) (cid:17)(cid:16) (cid:17)(cid:16) (cid:17)(cid:16) (cid:17) meetsanotherHatomtheyarelikelytoformH2moleculeon
4h nH mH vH 1
ta =4tgasq2 (11) atime-scalemuchshorterthanthetimeofadsorbinganother
l na m v (cid:24)a
hydrogen atom. Therefore if an oxygen atom is adsorbed
where 4h is the thickness of an adsorbed layer, and q2 after this H2 formation took place, it is free to scan the
is Sph(l)=S(l) ratio. If vH and v are due to thermal mo- grain surface unless a new H atom is adsorbed.
tions, then (cid:0)vvH(cid:1) = (cid:0)mmH(cid:1)1=2 and ta becomes proportional Consider an oxygen atom arriving at thegrain surface.
to(cid:0)mH(cid:1)1=2.However,ifnon-thermalAlfv(cid:19)enicmotionsdom- Thesites ofchemical adsorption can beeitherall (cid:12)lled with
inatem,vHbecomesequaltovandtaisproportionalto(cid:0)mmH(cid:1). nHumatboemrsoforembpetyemspitteys disuelimtoitead.reTchenetaHct2ivafotriomnatbiaornr.ieTrhoef
We will see further on that oxygen is the most impor- 3 (cid:0)8
10 K and a width of 10 cm is usually assumed for reac-
tant agent poisoning the active sites. Other chemical el-
tions betweenphysically and chemically adsorbed hydrogen
ements either form compositions that desorb more easily
atoms (Tielens & Allamandola 1988). This gives the time-
than H2O or have a negligible abundance. As all heavy ele- (cid:0)6
scaleforthereaction oftheorder1:3(cid:1)10 s.Themigration
ments, oxygen is usually depleted in the gaseous phase. For
time-scale depends on the properties of the grain surface.
instance, its depletion towards (cid:16) Persei is -0.5, which gives
(cid:0)3:7 Quantum tunneling corresponds tothe time-scale ofthe or-
n0=nH (cid:25)10 (Cardellietal.1991).Assumingthatalayer (cid:0)12 (cid:0)9
(cid:0)8 der10 s,although estimatesoftheorder3(cid:1)10 scanbe
of 4h (cid:25) 3:7(cid:1)10 cm e(cid:11)ectively covers active sites over
also found in the literature (Tielens & Allamandola 1988).
grain surface (Aanestad & Greenberg 1983), one obtains
(cid:0)3 As we will see further, the above di(cid:11)erence is crucial. In-
4h=l (cid:25) 1:85(cid:1)10 . Therefore for the accretion time-scale (cid:0)12
(cid:0)1 deed, assuming the time for migration 10 s, one obtains
for oxygen is ta (cid:25) 8tgas(cid:24)a q2. The estimate for ta can be thatbeforereacting,ahydrogenatomhaschancesofvisiting
furtherincreasedif(cid:24)a<1andthedesorptionispresent.The 6
(cid:23)cr = 10 sites of chemical adsorption, while this estimate
value of(cid:24)a (cid:24)0:1wasusedinSpitzer (1978,p.208)forillus-
falls to just (cid:23)cr (cid:25) 500 if the other value for the migration
trative purposeswhile thevalues 0.02and0.2werereported
time-scale is assumed. In the former case a grain with high
forsticking probabilities ofC oversilicate andgraphitecov-
density of active sites will have not more than one empty
eredwithamonolayerofH2Osurfaces(Williams 1993).Itis
active siteofH2 formation evenifagrain iscompletely cov-
showninJones&Williams (1984)thatthee(cid:11)ective sticking
ered by active sites; in the latter case up to (cid:24) (cid:23)=(cid:23)cr (if
probability varies as mantles begin to be deposited and as
(cid:23) > (cid:23)cr) active sites may be empty. These empty active
a consequence the time of establishing monolayer coverage
sites are the prime targets of an O atom hopping over the
may be increased by a factor of about three.
surface. If O atom is trapped in an active site, it will stay
The products of grain surface chemistry are also sub-
thereandwill beeventually hydrogenated withthereaction
jectedtodesorption whichincreases thetimeofresurfacing.
products blocking the access to the site.
According toJones&Williams (1984)theprobability ofre-
If we assume that the number of sites of physical ad-
taining the products OH and H2O on the grains surface is
sorption is Nph and among (cid:23) active sites one is empty, the
about0.7indarkquiescent clouds,whereasNHmaybepre-
probability ofan Oatom not to(cid:12)ll it as aresult ofan indi-
dominantly released onformation (Wagenblast etal. 1993). (cid:0)1
vidual hopis1(cid:0)Nph;thisprobability formhhopsdecreases
Ijemcmteeddtiaoteplhyoatfotdeersfoorrpmtiaotni.onTthhise preraoccteisosncparnodbuectdsriavreensnuobt- to (1(cid:0)Np(cid:0)h1)mh. If mh (cid:29) 1 and Nph (cid:29) 1, this expression
can be approximated:
only by UV quanta (Johnson 1982), but also by water ice
(cid:18) (cid:19)
d3i(cid:22)atmionabpseonrpettrioante(sWdeilelipaminstoetdaarl.k1cl9o9u2d)s..NTohtee,etnheartgy3(cid:22)remlearsae- (1(cid:0)Np(cid:0)h1)mh (cid:25)exp (cid:0)mh : (13)
Nph
associated with H2 formation may be another cause of des-
orbing weakly bound molecules such as CO in the vicinity The number of hops mh before an O atom hydrogenated
of active sites (Duley & Williams 1993). This means that is proportional to the ratio of the time of adsorbing a hy-
active sites can control to some extent resurfacing in their drogen atom tc = 1:5(cid:1)103((cid:13)^1n^Hv^H^l2)(cid:0)1 s to the time th
vicinity. Inbrief,wemayexpect toobtain ta oftheorder of given by Eq. (12). If a time-scale for hydrogen di(cid:11)usion is
(cid:0)12 (cid:0)5
10 tgasq2 using rather conservative estimates for the above (cid:24)10 sforagrain of10 cmwithhigh density ofactive
parameters. sites hydrogen atomwill scantheentire surface ofthegrain
However, such considerations are applicable only if and will react with an oxygen atom. If the mobility of hy-
heavy atoms form an inert monolayer rather than poison- drogenislower,theprobability tohydrogenate Oatomover
ing individual active sites. The thermal hopping of atoms agrainwillbesuppressedandmh willbeincreased byafac-
has the characteristic time (Watson 1976) tor exp((cid:0)(cid:23)=(cid:23)cr). Therefore for su(cid:14)ciently high (cid:23)=(cid:23)cr ratio
Physics and chemistry of the Purcell's alignment 5
(cid:0)3
themajorityofOatomswill eventually (cid:12)ll theemptyactive tion: even a steady-state value of nH (cid:24) 1cm (see e.g.
sites and the fact that the number of vacancies per grain Duley &Williams 1984)thecritical sizebecomes onlytwice
increases from one to (cid:23)=(cid:23)cr only contributes to the e(cid:11)ect. as large as the one given by Eq. (16). However, this is only
The exponential increase in poisoning of active sites when true if the atomic hydrogen concentration is not less than
their number exceed (cid:23)cr is likely to decrease their number that of oxygen. Otherwise, there is too little ofhydrogen to
to (cid:23)cr.The characteristic time ofpoisoning is as follows immobilise poisoning Oatoms.
(cid:16)v0 (cid:17)(cid:16)nH(cid:17) 1 Forgrains ofsize less thanlcr p,thecrossover time [see
tp (cid:25)2(cid:23)tc ; (14) Eq. (14)] is in order of magnitude tL (cid:25) (cid:23)tc(nH=n0). This
vH n0 1(cid:0)exp((cid:0)mh=Nph) 8
time is (cid:24)10 for standard values of parameters adopted in
wherev0 isthevelocityofOatomswiththenumberdensity the paper and is much less than the corresponding tgas (cid:24)
n0, the last factor is inversely proportional to the probabil- 5(cid:1)1011 s.Ifthe number of active sites is proportional to l2,
ity of O atom to (cid:12)ll an empty active site and the factor 2 than tL=tgas (cid:24) 10(cid:0)3 will not change as the grain becomes
re(cid:13)ects that in half of the cases O atoms will (cid:12)nd that all smaller. Consequently, tx becomes equaltotgas andtheim-
sitesare(cid:12)lled.Forsmallmh=Nph ratio,theexponentcanbe provement of alignment as compared to Davis-Greenstein
expanded intoseries andtp (cid:25)2(cid:23)=mh ta.Inotherwords,for process is marginal.
tp > ta, the number of active sites should not be less than
mh, which is for (cid:23)=(cid:23)cr <1 is equal to
mh = tc (cid:25)4(cid:1)103((cid:13)^1^ln^H)(cid:0)1T^g1=2expf(cid:0)E^h=T^sg (15) 4 PROBLEMS AND SOLUTIONS
th
(cid:0)5 4.1 Degree of alignment
Thereforeagrainof2(cid:1)10 cmshould havenotlessthan2(cid:1)
3
10 activesitesfortptobegreaterthanta.Thismeansthat Themoste(cid:14)cientpolarising mediumconceivablewouldcon-
the time-scale for hydrogen migration should be less than tain perfectly aligned in(cid:12)nite cylinders with their axes per-
(cid:0)9
10 s. If (cid:23) =(cid:23)cr =500 as in example discussed above, tL pendicular to the line of sight. For such a model Mie cal-
becomes 0.25ta which corresponds to(cid:24)2:5tgasq2.Although culations show that the ratio of polarization to extinction
less than our estimate for the time of spin-up limited only is of the order 0.3.At the same time, the observational up-
by monomolecular layer accretion, this estimate provides a per limit for the same ratio is (cid:20)0:064 (see Serkowski et al.
substantial increase oftL for irregular (q2 (cid:29)1) grains. The 1975).Thedi(cid:11)erence betweenthetwolimits stemsfromthe
functional dependence of tL as a function of grain fractal factthatrealistic grains areirregular, moderately elongated
dimension obtained inLazarian(1994c)alsostaysunaltered particles and it is only a fraction of grains that is aligned.
astp (cid:24)ta.However,theprocessesdiscussedheredependnot Nevertheless, it isdi(cid:14)cult toaccount fortheobservations if
only ontherelative abundance ofoxygen asit wasassumed (cid:14)e(cid:11) does not exceed unity.
in Lazarian (1994c) but also on the grain temperature Ts Our estimates of (cid:14)e(cid:11) give (cid:24) 0:3 for standard l and
and theenergy ofthe potential barrier Eh.Both values can tL =10tgas (see section 2).Although 10times greater than
be measured andthereforelong-lived spin-up canbe tested. the initial value it is not su(cid:14)cient to explain the polariza-
Consider, forinstance, grains covered by H2Oice. Sim- tion. One of the ways to increase tL is by taking into ac-
ulations in Buch & Zhang (1991) indicate that active sites count photodesorption (see Johnson 1982). Indeed, as the
(i.e. sites with binding energy in excess of k(cid:1)900 K) cover characteristic time of photodesorption is around 109 s for
(cid:0)2
around 10 of the entire grain surface that a `standard' unshielded regions of di(cid:11)use clouds, which is usually much
4
grain will have (cid:23) (cid:24)1:5(cid:1)10 active sites. Data in table 2 in less than the time of resurfacing, this may considerably in-
Tielens&Allamandola (1988)showthatmigrationscalesfor crease tL. However,the correlation of polarization with the
(cid:0)12 (cid:0)2
hydrogen and oxygen are, respectively, 10 s and 10 s. strength of 3.1 (cid:22)m water ice feature (see Joyce & Simon
For such high mobilities of hydrogen (cid:23)cr >(cid:23) and therefore 1982)testifythatthephotodesorption cannotprovideauni-
(cid:23) should beused in Eq.(14). Asubstitution ofthese values versal solution (see also Lee &Draine 1985).
5
into Eq. (15) provides mh (cid:25) 1:5(cid:1)10 and thus tL becomes One may question `standard' values for magnetic (cid:12)eld
tentimes lessthan ta.Toobtain tL (cid:29)td,oneneeds assume that is responsible for grain alignment, while leaving the
either q2 (cid:29) 1 or Ts < 10 K. For Ts = 15 K, mh becomes `standard' value of the mean magnetic (cid:12)eld intact. Indeed,
7
(cid:24)6(cid:1)10 andlong-lived spin-upbecomesimprobable. Onthe weobservethecumulative polarization andextinction along
basis of this data it is evident that the Purcell's alignment thelineofsight,whichpresumablycrossesquitedi(cid:11)erentre-
of grains with surfaces covered with amorphous H2O ice is gions. If aligned dust grains are associated with clouds, we
limited toarathernarrowrangeofgraintemperaturesifour should use the data corresponding to the clouds. Zeeman
data for (cid:23) and Eh are correct. measurements in Heiles (1989) provide B (cid:25) 1:2(cid:1)10(cid:0)5 G
The critical size of the grain corresponds to tc becom- for n (cid:25) 7 cm(cid:0)3. This provides tr (cid:25) 3 (cid:1) 1012 s, while
ing equal to the time for an oxygen atom to scan the grain tgas (cid:25)6(cid:1)1012 s,whichgivesforspherical grainstgas=tr =2.
(cid:0)1 (cid:0)2
surface. The time tc varies as (cid:13) l , the time for O atom Ifonetakesintoaccountthatgrains arenon-spherical (Pur-
2
to migrate over the grain surface as q2l ; hence the critical cell &Spitzer 1971)andsubjecttoadditional disorientation
size is through emission, (cid:14)e(cid:11) (cid:24)0:8 can characterise alignment for
lcr p (cid:25)6(cid:1)10(cid:0)6(q2n^H(cid:13)^1)(cid:0)14T^g18 exp(cid:8)(cid:0)0:25E^h=T^s(cid:9) cm (16) tL < td. This example of Heiles data for morphologically
distinct shells may be considered to be extreme. However,
If Ts increases or Eh decreases, rapid poisoning becomes numerousstudies(seeMyers1985,1987,Myers&Goodman
essential for grains of all sizes. At the (cid:12)rst glance, lcr p 1988) indicate that molecular clouds are magnetically sup-
seems to be insensitive to the atomic hydrogen concentra- ported and exhibit motions which are supersonic `in all but
6 A. Lazarian
thesmallest cloud regions'(Myers1985).Thesemotionsare having such a cluster is greater for larger grains. Alterna-
usually attributed to Alfv(cid:19)enic turbulence (see Aron & Max tively, for the Gold-type alignment (Lazarian 1994a), it is
(cid:0)5
1975, Falgarone & Puget 1986). However, the Alfv(cid:19)enic ve- possible to show that only grains larger than 10 cm are
locityisequaltothethermalvelocityofatomsfor`standard' su(cid:14)ciently inertial to be aligned under Alfv(cid:19)enic perturba-
values ofB =2(cid:1)10(cid:0)6 Gandn=20cm(cid:0)2.Thereforetorec- tions in thetypical ISM conditions.x Hereweareinterested
oncile widely accepted explanations of the molecular cloud whether the Purcell's alignment can reproduce this type of
support and non-thermal line widths withobservations, one behaviour forthe polarization curve.
2
should increase substantially B =nratio.Notethatitisthis Itwasnotedinsection2thatifthedensityofactivesites
ratio thatenters theexpression for tgas=tr and thereforean over the grain surface is low,the absolute majority ofsmall
increaseofAlfv(cid:19)enicvelocityabovethethermaloneentailsan grains would not have sites of H2 formation and therefore
increase of (cid:14)e(cid:11).Ifthe Alfv(cid:19)enic velocity is (cid:12)ve times greater would notbe aligned by the Purcell's mechanism. However,
than thethermal velocity (cid:14)e(cid:11) increases twenty(cid:12)vetimesas wedonotbelieve thatthisisaprobable explanation forthe
comparedwith`standard'values.Evenalessincreaseof(cid:14)e(cid:11) di(cid:11)erence in the alignment of small and large grains.
is adequate toexplain the polarization iftL is several times Ourcalculations insection3haveshownthatpoisoning
greater than tgas as it is discussed in section 3. intensi(cid:12)es for l less than lcr p (cid:24)0:6(cid:1)10(cid:0)5 cm. This critical
2
Note,thatthedecreaseofB =nratio asitfollows from size can further increase due to the inverse dependence of
the data in Myers & Goodman (1988) may be one of the Ts on l (Greenberg 1971):
explanations for the decrease of polarization in dark clouds
as indicated in Goodman (1994). Ts(l)(cid:25)Ts0^l(cid:0)0:2 (17)
Therefore the cumulative e(cid:11)ects of inhibited poisoning This is a result of suppressed thermal emission from small
as compared to that assumed Spitzer & McGlynn (1979) grains in infrared. Although the power-law index is just
and enhanced values of magnetic (cid:12)eld from recent observa- 0.2, this dependence may be essential in view of expo-
tions seems to favour the Purcell's mechanism to account nential dependence of lcr p on temperature [see Eq. (16)].
at least for part of observational data. Indications that the Thus,small grains areexpected tobewarmenough tohave
degree of alignment correlates with the total magnetic (cid:12)eld lcr p (cid:24)10(cid:0)5 cm.
strength (Jones1989), also favour such aninterpretation as As soon as l < lcr p, rapid poisoning of active sites in-
compared totheonewhereonly grains with superparamag- evitably causestL(cid:28)tgas.Thisforcesdownboththedegree
netic inclusions are aligned and, if aligned, their alignment
of suprathermality and the measure of alignment. One may
is perfect (Mathis 1986). erroneously conclude, thatinspite ofthisdrop,thedegreeof
2
The `standard' values of the ratio B =n appeared at alignmenthasaformallimitofunityasltendstozero.How-
the timewhen theZeeman e(cid:11)ectmeasurements weregiving ever,tr (cid:24)l2K((cid:10))andK((cid:10))(cid:24)(cid:10)[seeEq.(5)]inthelimit of
only upper limits of the magnetic (cid:12)eld strength. According high velocities, tr maybefoundtobeinversely proportional
to Heiles (1989), this was an unfortunate result of the as- to l if rotation is suprathermal and l < lcr p. At the same
sumption that the splitting is easier to discover for narrow time, if poisoning of active sites for small grains is inhib-
lines. In fact, strong magnetic (cid:12)elds usually correspond to ited, then starting from lcr m tr becomes independent of l.
broad lines. Present-day databoth onZeemansplitting and Another important factor that suppresses the alignment of
non-thermal linewidths tend to suggest that the `standard' small grains is their disorientation due tothermal emission.
2
value of B =n probably underestimate the actual value of {
In fact, one can introduce a critical size
this ratio. Having the long-lived spin-up with tL within the
(cid:0)6 (cid:0)5=4
range discussed in section 3,we feel quite comfortable with lcr t(cid:24)10 n^H cm (18)
2
less than an order of magnitude increase of B =n ratio as
starting fromwhich thedisorientation due tothermal emis-
compared with its `standard' value. 4
2 siondominates.Duetothedependenceoftrad onTs andthe
Note that we are concerned with the ratio B =n. For
(cid:0)6 change of grain temperature with l according to Eq. (17),
instance, using the `standard' values of B =3(cid:1)10 G but 1:8
(cid:0)3 the resultant dependence of tx (cid:25) trad is l . Therefore for
n = 5 cm , one obtains te(cid:11)=tr (cid:25) 0:1, which is su(cid:14)cient 2:3
small grains l<lcr p and l<lcr m, (cid:14)e(cid:11) (cid:24)l .
for explaining the polarization forthe tL=tgas ratio that we
At the same time, if rotation for small grains is not
discussed.
fastenough fortheirparamagnetic susceptibility tocometo
Tosummarise,thePurcell'smechanism seemstobead-
saturation, the degree of alignment increases with the de-
equately strong in theview ofrecent observational and the- (cid:0)0:2
creaseofsizeasl withintheinterval lcr m <l<lcr p.A
oretical results. Its implications for grains of di(cid:11)erent sizes
transition fromthelong-lived spin-up toshort-lived spin-up
and chemical composition are discussed further on.
corresponds toaconsiderable drop in(cid:14)e(cid:11) andthereforethe
alignment ofgrainswithl<lcr p isexpectedtobemarginal
2
unless very high values of the ratio B =n are present in the
4.2 Small and large grains
UsingtheMNRgrainmodel(Mathisetal.1977),itispossi- x Grainsareexpectedtobealignedwiththeirlongaxesperpen-
bletoshow(Mathis 1979)thatagood(cid:12)tofthepolarization
diculartomagnetic(cid:12)eldlines,whichcorrespondstoobservations
curve may be obtained if grains with radii above a certain (Lazarian1994a).
threshold arealigned. Theabovedependence wasaccounted { This is not an exact result as grain emissivity deviates sub-
for in Mathis (1986) by assuming that only those grains stantially from the black body approximationfor small grains.
that contain at least one superferromagnetic cluster arebe- However,computationsinPurcell&Spitzer[1971]showthatfor
ing aligned. Within the adopted model, the probability of smallgrainsdisorientationduetothermalemissiondominates.
Physics and chemistry of the Purcell's alignment 7
23=10
(cid:14)e(cid:11) starts from (cid:24)0 for the smallest grains and so as l
l>lcr m l<lcr m l>lcr m l<lcr m until l = lcr m, after which it changes to l(cid:0)1=5 A consider-
l>lcr t l>lcr t l<lcr t l<lcr t able increase of (cid:14)e(cid:11) follows at l =lcr p, which corresponds
to the transition toa long-lived spin-up. A decrease of(cid:14)e(cid:11),
(cid:0)2
(cid:0)2=3 4=3 (cid:0)2=3 4=3 which is proportional to l ,at l=lcr f becomes less rapid
l>lcr p l l l l (cid:0)2=3
((cid:24)l ) due to a partial preservation of alignment during
l>lcr f
crossovers.When thenumber ofactivesitesexceeds(cid:23)cr and
the decrease of (cid:14)e(cid:11) stops.
l>lcr p l(cid:0)2 l0 l(cid:0)2 l0 Itmaybeconsidered strange,butin viewofnewobser-
l<lcr f vational results concerning magnetic (cid:12)elds in di(cid:11)use clouds
Heiles(1989),Myers&Goodman(1988),Heilesetal.(1993)
it may become more important to explain marginal align-
1=3 17=6 17=15 109=30
l<lcr p l l l l ment ofsmall grains rather than to account for good align-
l>lcr f ment of large grains. Indeed, both the Purcell's alignment
as it is discussed here and superparamagnetic alignment as
(cid:0)1 3=2 (cid:0)1=5 23=10 it was suggested in Mathis (1986) fail to explain marginal
l<lcr p l l l l 2
alignmentofsmallergrainsiftheratioB =nislargeenough.
l<lcr f
Tosummarise,thePurcell'smechanismisabletorepro-
duce qualitatively the observed features of the polarization
Table1.Thedependenceof(cid:14)e(cid:11) ongrainsizeforvariouscombi- curve. Amore quantitative study will be done elsewhere.
nationsofthecriticalsizesifthenumberofactivesitesispropor-
tionaltothegrainsurface.lcr m = criticalsizebelowwhichthe
paramagneticsusceptibilityissaturated[Eq.(6)],lcr f =critical
4.3 Grains of di(cid:11)erent chemical composition
sizeabovewhichapartialpreservationofalignmentinthecourse
of crossovers is important [Eq. (9)], lcrp = critical size below There exist a considerable convergence in views on the di-
whichtheoxygenpoisoningofactivesitesisenhanced[Eq.(16)]
electricnatureofpolarising particles(seee.g.Whittet1993),
lcr t = critical size below which disorientation due to thermal
although the values of the imaginary part of the grain in-
emissiondominates[seeEq.(18)].
dex of refraction is a subject of controversy. Our results in-
dicate, that `H2 driven' suprathermal rotation and there-
fore the Purcell's alignment are e(cid:14)cient for grains with
surfaces of silicate, water ice or polymeric hydrocarbon.
l>lcr m l<lcr m l>lcr m l<lcr m
l>lcr t l>lcr t l<lcr t l<lcr t As the IR polarimetry shows substantial polarization in
both silicate 9.7 (cid:22)m feature and 3.7 (cid:22)m water ice band
(e.g. Whittet 1992) while graphite and amorphous carbon
0 2 0 2
l>lcr p l l l l grains are likely not tobe aligned (Mathis 1986), this may
l>lcr f serve as an argument in favour of the Purcell's mechanism.
Grains withsurfacesofaromaticcarbonaceous material and
1 21=6 27=15 129=30 graphite should rotatethermally andthereforebesubjected
l<lcr p l l l l
to the Davis-Greenstein alignment. But with theknown ra-
l<lcr f 2
tios B =n this alignment is marginal. If it is suprathermal
rotation thatcontrolsthestructureofgrains (Wright1994),
Table2.Thedependenceof(cid:14)e(cid:11) ongrainsizewhenthenumber the abovegrains areexpected tobe (cid:13)u(cid:11)y and thereforewill
ofactivesitesisstabilisedatthelevel(cid:23)cr. notbealigned byAlfv(cid:19)en-Gold processesdiscussed inLazar-
ian (1994a).
grain environment. Both the disorientation through radia-
tionandsaturationofparamagneticdissipationsuppressthe
alignment for the smallest grains. Our computations show 5 TEST CASES
that we should expect the said drop for grains less than
(cid:0)5 5.1 Cold and warm grains
10 cm.
Forlargegrains,partial preservation oforientation dur- We have seen that due to temperature increase, poisoning
ing crossovers becomes important and this improves their atoms become su(cid:14)ciently mobile to attack active sites di-
alignment: forl>lcr m andl<lcr m,(cid:14)e(cid:11) becomes propor- rectly and therefore the life time of active sites decreases.
(cid:0)2=3 4=3
tional to l and l , respectively. Other situations that Our results show that for icy grains an increase of tem-
may emerge for various combinations of four critical sizes perature up to 20 K leads to such a considerable increase
and (cid:23) < (cid:23)cr are shown in Table 1, whereas the dependen- in poisoning that long-lived spin-up becomes impossible. If
cies for (cid:23) >(cid:23)cr are given in Table 2. This information may the temperature increases further, H2 formation becomes
be used for future quantitative comparison of theory and suppressed. Indeed, for a reaction between physically and
observations. chemically adsorbed H atoms to occur, an activation bar-
Althoughpossible insomesituations,notallthecombi- rier should be overcome. A critical temperature Tcr can
nations presentedinTables 1and2areequally probable for be de(cid:12)ned for which the probability of reaction with a
di(cid:11)use clouds. We believe that it is more likely that there chemisorbed atoms is equal to the rate ofevaporation:
8 A. Lazarian
Tcr = Eh (19) and chemistry. It has been shown that grains with amor-
kln((cid:28)r(cid:23)0) phous H2Oice mantles,defected silicate andpolymeric car-
(cid:0)6 bonaceous surfaces are likely to rotate suprathermally and
where(cid:28)r isthereaction time-scale (cid:24)10 s.Estimatespro-
can be aligned by the said mechanism if the relative abun-
vide Tcr (cid:25) 50 K for H2 formation over a silicate surface (cid:0)3
dance of atomic hydrogen nH=n is not less than 10 and
(Tielens &Allamandola 1987).Above thecritical tempera-
grains are cool (Ts < 50 K). At the same time, grains cov-
ture,thereactionratedropsexponentiallyandthereduction
(cid:0)3 ered with aromatic carbonaceous material or graphite are
factor atTs (cid:24)75 Kis expected to be(cid:24)5(cid:1)10 .Such a de-
not expected to rotate suprathermally due to H2 formation
creaseine(cid:14)ciencyofH2formationentailsthesamedecrease
and therefore this mechanism is not applicable to them. It
in attainable angular velocity (cid:10)making it comparable with
has been shown that the Purcell's mechanism provides a
the velocity of thermal rotation !T.Therefore the Purcell's
preferential alignment oflarge grains, while only amarginal
mechanismbecomesinapplicable.Ourdiscussioninsection3
(cid:0)5
alignment of grains with l<10 cm is expected.
showsthatevensmallerincreaseoftemperaturemakeslong-
lived spin-up impossible for grains with mantles of amor-
phous ice. This means that, for instance, the aligned warm
(Ts (cid:24) 75 K) dust in the core regions of OMC-1 (Chrysos- Acknowledgements
tomou et al. 1994) can not rotate suprathermally due to
It is a great pleasure to express here my gratitude to
H2 formation. In general, it is unlikely that the suprather-
J.Mathis,communications withwhomstimulated mystudy
malparamagnetic alignment ofwarm(Ts >50K)dustmay
of the di(cid:11)erence in alignment of large and small grains. His
be e(cid:14)cient. Itiseither theGold-type alignment (seeLazar-
comments also contributed to substantial improvement of
ian 1994a) orDavis-Greenstein mechanism thatgovernsthe
the original version of the paper. This research also owes
alignment of warm dust. To distinguish between the two
muchtomydiscussions withP.MyersandD.Williams. En-
cases one needs to study the polarization curve. TheDavis-
couragementofN.WeissandcommunicationswithJ.Green-
Greenstein mechanism should result in e(cid:14)cient alignment
k berg & E.Wright are gratefully acknowledged. Iwould also
of small grains, while Gold-type processes preferably align
liketothanktheInstituteofAstronomy,University ofCam-
larger grains as a result of Alfv(cid:19)enic phenomena.
bridge, for the Isaac Newton Scholarship they generously
awarded me, which provided the (cid:12)nancial support for this
5.2 H-atom population work.
Itis easytoseefromEqs.(1)(cid:0)(3)that(cid:10)is proportional to
nH=n ratio and therefore becomes of the order of thermal
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567. McGlynn (1979) as cosh(cid:31)Ti = exp((cid:0)F), where (cid:31)T is the
angle of deviation of the angular. Calculations in Spitzer &
JonesA.P.&WilliamsD.A.,1984,MNRAS,209, 955.
McGlynn (1979) give
JonesR.V.&SpitzerL.,Jr,1967,ApJ,147,943.
(cid:26) (cid:18) (cid:19)(cid:27)
JonesT.J.,1989,ApJ,346, 728. (cid:25) h(4Jz)2i 3h(4J?)2i
F = 1+ (A1)
JoyceR.R.&SimonT.1982,ApJ,260, 604. 4 jJ?jh4Jzi 2h(4Jz)2i
KimS.-H.&MartinP.G.,1994,ApJ,431,783.
in the approximation of the constant angular momentum
LazarianA.,1992a,Astron.&Astrophys.Trans.,3,33.
component that is perpendicular to the axis of the greatest
LazarianA.,1994a,MNRAS,268, 713.
inertia. Note,that4Jz and 4J? areelementary changes of
LazarianA.,1994b,MNRAS(submittedpaper).
the angular momentum in an individual torque event, and
LazarianA.,1994c,A&A (inpress).
J? is the value of the component of angular momentum
LeeT.J.,1972,Nature Phys.Sci.,237, 99.
perpendicular to the axis of major inertia.
LeeH.M.&DraineB.T.,1985,ApJ,290,211.
Thesquareddispersion ofJ? att=0isgiveninSpitzer
Leitch-Devl(cid:19)inM.A.&WilliamsD.A.,1984,MNRAS, 210,577.
& McGlynn (1979, eq. 42):
LucasD.&EwingG.E.,1981,Chem. Phys.,58,385.
(cid:16) (cid:17)1=3 (cid:16) (cid:17)
MathisJ.S.,1979,ApJ,232, 747. 2 3 4 1=3 (cid:0)2=3 2=3
hJ?(0)i (cid:25) (cid:0) (AaN) h4Jz iIz
MathisJ.S.,1986,ApJ,308, 281. 2 3
2
MathisJ.S.,1990,Ann. Rev. Astron. Astrophys.,28, 37. (cid:2) h(4J?) i (A2)
MathisJ.S.,1994,privatecommunication.
where(cid:0)(x)istheGammafunction,andAa isthecoe(cid:14)cient
MathisJ.S.,RumplW.&NordsieckK.H.,1977,ApJ,217,
of the Barnett relaxation (Purcell 1979, eq. 46) and N (cid:25)
MyersP.C.,1985,inProtostarsandPlanetsII,eds.C. Black& 2
(cid:13)l nHvH is the frequency ofthe torque events.
M.Mathews,Tucson,UniversityofArizonaPress,p95.
To calculate F from Eq. (A1), one needs to know the
MyersP.C.,1987,inInterstellarProcesses, edsD.J.Hollenbach (cid:0)1 2 2
andH.A.Thronson,Jr,Kluwer,Dordrecht,Reidel,p71. ave2rageofJ? (0).Note,thathJ?(0)iisasumofhJx(0)iand
hJy(0)i and due to the symmetry inherent to the problem
MyersP.C.,1994b,privatecommunication.
2 2 2 2
MyersP.C.& GoodmanA.A.,1988,ApJ,329, 392. (cid:27)1 =hJx(0)i=hJy(0)i=0:5hJ?(0)i (A3)
NormanC.&SilkJ.,1980,ApJ,238, 138.
For a Gaussian distribution, one gets
PurcellE.M.,1975,inThe Dusty Universe,edsG.B.Field& ZZ +1
A.G.W.Cameron,New York,NealWatson,p.155. 1 1 1
PurcellE.M.,1979,ApJ,231, 404. hJ?(0)i = 2(cid:25)(cid:27)12 (cid:0)1 px2+y2
PurcellE.M.&SpitzerL.,Jr,1971,ApJ,167,31. (cid:26) 2 2(cid:27)
x +y
RobergeW.G.,DeGra(cid:11)T.A. &FlahertyJ.E.,1993,ApJ,418, (cid:2) exp (cid:0) 2 dxdy (A4)
287. 2(cid:27)1
SerkowskiK.,MathewsonD.S.& FordV.L.,1975,Astrophys. which can be calculated in polar coordinates to give
J.,196, 261. q p
1 (cid:25) 1 (cid:25)
SnellR.L.,1981,ApJ Suppl.,45, 121. = = (A5)
SpitzerL.,Jr,1978,Physical Processes in the Interstellar hJ?(0)i 2(cid:27)1 hJ?2(0)i1=2
Medium, Wiley-IntersciencePubl.,NewYork. (cid:0)2=3
Similarly theGaussian averages canbefound forh4Jz i.
SpitzerL.,Jr,1993,privatecommunication. Thedispersion ofh4Jziwhichwewilldenote(cid:27)2 isinversely
SpitzerL.,Jr&McGlynnT.A.,1979,ApJ,231,417. proportional tothesquarerootofthenumberofactivesites
StecherT.P.&WilliamsD.A.,1972,ApJ,177,L145. (cid:23) overthegrain surfaceanddepends ongraingeometryand
TielensA.G.G.M.&AllamandolaL.J.,1987,inInterstellar distribution ofactivesitesinrespecttothez-axis.Foradisc
Processes,eds.D.J.HollenbachandThronson,Reidel, shaped grain (cid:27)2 =0:25lmH2vH2(cid:23)(cid:0)1=2.Therefore
Dordrecht,p.397. Z +1 (cid:26) 2(cid:27)
WagenblastR.,WilliamsD.A.,MillarT.J.&NejadL.A.M., (cid:0)2=3 1 1 1x
1993,MNRAS,260, 420. h4Jz i= (cid:27)2p2(cid:25) (cid:0)1 x2=3 exp (cid:0)2(cid:27)22 dx (A6)
WatsonW.D.,1976,inAtomicandMolecularPhysicsandthe
which gives (see Gradstein & Ryzhik, [3.462(1)] and
InterstellarMatter,R.Balian,P.Encrenaz&J.Lequeux
[9.241(1)])
(eds.),Elsevier,Asmterdam,177.
(r )
WhittetD.C.B.,1992,Dustin the Galactic Environment, (cid:16) (cid:17)
(cid:0)2=3 1 2 1
BristolInst.Physics. h4Jz i= 2 (cid:0) D(cid:0)1=3(0) (A7)
WilliamsD.A.,1993,inDustandChemistryinAstronomy, (cid:27)23 (cid:25) 3
MillarT.J.&WilliamsD.A.(eds.),BristolInst.Physics,
where Dp(x) is the parabolic cylinder function and (cid:0)(x) is
p.71.
the Gamma function.
WilliamsD.A.,HartquistT.W.&WhittetD.C.B.,1992,
Thus substituting Eq. (A7) into Eq. (A2)
MNRAS,258, 599.
2
WrightE.L.,1994,privatecommunication. h(4Jz) i
F = (cid:20)~
Zhang,Q.&BuchV.,1990,J.Chem. Phys.,92,5004. (AaN)1=6Iz1=3h(4Jz)2i1=3h(4J?)2i1=2
10 A. Lazarian
(cid:18) (cid:19)
2
3h(4J?) i
(cid:2) 1+ (A8)
2h(4Jz)2i
where the numerical factor is equal to
(cid:20)~=(cid:16)3(cid:17)(cid:0)1=6(cid:0)(cid:0)1=2(4=3)Z 1e(cid:0)x22x(cid:0)2=3dx (A9)
4
0
2 2
Assuming h(4Jz) i=h(4J?) i,one obtains
2 1=2
5 h(4Jz) i
F = (cid:20)~ (A10)
2 (AaN)1=6Iz1=3h[4Jz]2i1=3
2 l2 2 2
Substituting h(4Jz) i = 9mH2vH2 as well as expressions
2
for Aa,N and h[4Jz] i into Eq. (A10),one gets
F (cid:25)0:8(v^H2(cid:23)^)1=3%^(cid:0)1=2^l(cid:0)2(n^Hv^H(cid:13)^1)(cid:0)1=6 (A11)
where(cid:23) isnormalised by100.Ifthenumberofactivesites(cid:23)
over the grain surface is assumed to be proportional to the
(cid:0)4=3
surfaceareaofthegrainSg(l),F decreasesasl withthe
increase of grain size. If the number of active sites is sta-
bilised due to active poisoning at the level (cid:23)cr, F decreases
(cid:0)2
as l .