Time-dependent simulations of steady C-type shocks

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by  S. Van Loo

Mon.Not.R.Astron.Soc.000,1–9() Printed21January2009 (MNLATEXstylefilev2.2)
Time-dependent simulations of steady C-type shocks
S. Van Loo,1⋆, I. Ashmore1, P. Caselli1, S. A. E. G. Falle2, and T. W. Hartquist1
9 1 School of Physicsand Astronomy, University of Leeds, Leeds LS2 9JT, UK
0 2 Department of Applied Mathematics, Universityof Leeds, Leeds LS2 9JT, UK
0
2
n Accepted -.Received-;inoriginalform-
a
J
ABSTRACT
6
1 Using a time-dependent multifluid, magnetohydrodynamic code, we calculated the
structure of steady perpendicular and oblique C-type shocks in dusty plasmas. We

] included relevant processes to describe mass transfer between the different fluids, ra-
R
diative cooling by emission lines and grain charging and studied the effect of single-
S sized and multiple sized grains on the shock structure. Our models are the first of
. oblique fast-mode molecular shocks in which such a rigorous treatment of the dust
h
grain dynamics has been combined with a self-consistent calculation of the thermal
p
- and ionisation structures including appropriate microphysics. At low densities the
o grains do not play any significant rˆole in the shock dynamics. At high densities, the
r ionisationfractionissufficientlylowthatdustgrainsareimportantchargeandcurrent
t
s carriers and, thus, determine the shock structure. We find that the magnetic field in
a
the shock front has a significant rotation out of the initial upstream plane. This is
[
most pronounced for single-sized grains and small angles of the shock normal with

1 the magnetic field.Our resultsaresimilar to previousstudies ofsteadyC-type shocks
v showingthatourmethodisefficient,rigorousandrobust.Unlikethemethodemployed
1
in the previous most detailed treatment of dust in steady oblique fast-mode shocks,
7
oursallowsareliablecalculationevenwhenchemicalorotherconditionsdeviate from
4
local statistical equilibrium. We are also able to model transient phenomena.
2
1. Key words: MHD – shock waves – ISM: dust – ISM: jets and outflows
0
9
0
:
v 1 INTRODUCTION However,for afractional ionisation below 10−7,grains play
i an important rˆole in governing the electromagnetic forces
X Molecularoutflowsassociatedwithproto-stellarobjects(e.g.
(Draine 1980). In the densest regions of molecular clouds
ar Bwahlelrye&thLeafrdaact1i9o8n3a)ldiorniviseasthioonckχsiisntloowda(χrk<m1o0le−c6u)l.aTrhcleoulodws where nH > 106 cm−3, the fractional ionisation constraint
is usually fulfilled.
valuesofχcausethegasandthemagneticfieldtobeweakly
Pilipp, Hartquist,& Havnes (1990) included the effect
coupled.Chargedparticlesarethenpushedthroughtheneu-
of dust grains on the structure of perpendicular steady C-
tralgasbyLorentzforces,givingrisetoambipolardiffusion
type shocks more rigorously than previous authors (e.g.
(Mestel & Spitzer1956).Collisionsbetweenthechargedand
DRD).Theirfour-fluidmodelspredictdifferentshockwidths
neutral particles accelerate, compress and heat the neutral
thanobtainedinsimilarmodelsofDRD.Thismodelwasex-
gasaheadoftheshockfrontandamagneticprecursorforms.
tendedto obliqueshocks byPilipp & Hartquist (1994) who
Iftheshockspeedislowenoughforthegastocoolefficiently,
found that in such shocks the tranverse component of the
theflowbecomescontinuousinallfluids(e.g.Mullan1971).
magneticfieldrotatesaboutthepropagationdirectionofthe
SuchshocksarereferredtoasC-typeshocks(Draine1980).
shock as long as there is a non-negligible Hall conductivity.
Draine, Roberge, & Dalgarno (1983, hereafter DRD)
This happens for a wide range of conditions, but Pilipp &
calculated the structures of steady perpendicular C-type
Hartquistwereunabletofindsolutionsforfast-modeshocks.
shocksindiffuseanddensemolecularmediausingasteady-
Wardle (1998) pointed out that, to find a stable shock
state multifluid magnetohydrodynamic (MHD) description.
structurefor steady oblique fast-mode shocks, one needs to
They included a detailed treatment for energy and mo-
integratethesteadystateequationsbackwardsintimefrom
mentum transfer between different particles, oxygen-based
thedownstreamboundary.However,suchamethodrequires
chemistryandradiativecooling.Theyalsoassumedthatthe
anexactthermal,chemicalandionisationequilibrium.Such
ion and electron densities exceed the grain charge density.
an equilibrium does not obtain in most shocks of interest.
Forinstance,theabundanceofwater,animportantcoolant,
⋆ E-mail:[email protected] is out of equilibrium for about 105 years or more from the
2 S. Van Loo et al.
timethegascoolsbelowseveralhundreddegree.Thishigh- (J.B)B J×B (J×B)×B
lights the necessity for time-dependent simulations to cal- E=−vn×B+r// B2 +rH B −rAD B2 ,
culatetheC-typeshockstructure.Ciolek & Roberge(2002) where r is theresistivity along the magnetic field, r the
// H
were thefirst toinclude afluid description of grain dynam- Hall resistivity and r the ambipolar resistivity. The ex-
AD
ics in multifluid MHD simulations. However, their study is pressionsoftheresistivitiesaregivenbye.g.Falle(2003)and
restricted toperpendicularshocks.Themultifluidapproach depend strongly on the Hall parameter, β =α |B|/K ρ ,
i i ni n
developed by Falle (2003) overcomes this limitation. whichistheratiobetweenthegyrofrequencyofthecharged
In this paper we study the structures of steady fast- fluid and collision frequency of the charged fluid with the
mode C-typeshocks using time-dependent multifluid MHD neutrals. Substitution of the above expression into the
simulations. In Sect. 2 we describe the numerical code and Maxwell-Faraday equation then gives the magnetic induc-
the physical processes that we include. Then, we apply the tion equation
code to perpendicular and oblique shocks with parameters
∂B ∂M ∂ ∂B
similar to those of DRD including a single-sized grain fluid + = R , (7)
∂t ∂x ∂x ∂x
(Sect.3)ormultiplegrainfluids(Sect.4).Finally,wediscuss
theresults and give ourconclusions in Sect. 5. where M = vn×B is the hyperbolic magnetic flux and R
is theresistance matrix which dependson the Hall resistiv-
ity, the ambipolar resistivity, and the resistivity along the
2 THE MODEL magnetic field. (For an expression for R, see Falle (2003).)
Equations 1 - 7 are solved using the numerical scheme
2.1 Numerical code described by Falle (2003). This scheme uses a second-order
hydrodynamic Godunov solver for the neutral fluid equa-
Astheionisationfractionwithinmolecularcloudsislow,the
tions (Eqs. 1-3). The charged fluid densities are calculated
plasma needs to be treated as a multicomponent fluid con-
usinganexplicitupwindapproximationtothemassconser-
sistingofneutrals,ions,electronsandN dustgrain species.
vation equation (Eq. 4), while the velocities and tempera-
The governing equations for theneutral fluid are given by
turesarecalculatedfromthereducedmomentumandenergy
∂ρ ∂
n + (ρ v ) = s , (1) equations(Eq.5and6).Themagneticfieldisadvancedex-
∂t ∂x n x,n Xj jn plicitly, even though this implies a restriction on the stable
time step at high numerical resolution (Falle 2003).
∂ρ v ∂ J×B
∂nt n + ∂x(ρnvx,nvn+pn1x) = c , (2)
∂e ∂ 1
n + v e +p + ρ v2 = J.E+ H(,3) 2.2 Physical processes
∂t ∂x x,n n n 2 n n j
h (cid:16) (cid:17)i Xj Thenumericalcodedescribedintheprevioussectionisquite
where e = p /(γ−1)+ρ v2/2 is the internal energy per general. In order to investigate the effects of dust grains
n n n n
unit mass, s the mass transfer rate between the charged in fast-mode shocks it is necessary to specify the source
jn
fluidj and theneutralfluid,H theenergy sinksorsources terms, i.e. the mass, momentum and energy transfer coef-
j
for the different fluids, B the magnetic field, E the electric ficients and the energy sources and sinks, to describe the
field and J thecurrent given byJ= αρ v . relevant physics. The chemical network that we adopted
j j j j
In thelimit of small mass densitiPes for thecharged flu- is the one used by Pilipp, Hartquist, & Havnes (1990) and
ids, their inertia can be neglected. Also, the charged fluids Pilipp & Hartquist (1994).
areinthermalbalanceastheirheatcapacitiesaresmall.For Mass transfer between the ion and electron fluids and
thecharged fluid j, theequations then reduceto the neutral fluid is due to cosmic ray ionisation at a rate
of10−17 s−1,electronrecombinationwithMg+,dissociative
∂ρ ∂
∂tj + ∂x(ρjvx,j) = −sjn, (4) recombination with HCO+ and recombination of ions on
charged grains. Here we have assumed that the main gas
v ×B
αjρj E+ j c +ρjρnKjn(vn−vj) = 0, (5) phase ions are Mg+ and HCO+ (Oppenheimer& Dalgarno
(cid:16) (cid:17) 1974). The cross-section for these recombination reactions
H +G = 0, (6)
j jn are taken from Pilipp, Hartquist,& Havnes(1990).
where α is thecharge to mass ratio, K the collision rate Mass is also transferred between the ion and electron
j jn
coefficient of the charged fluid j with the neutrals and G fluidsandthegrainfluidsduetoionneutralisationongrain
jn
theenergytransferrateperunitvolumebetweenthecharged surfaces. Ions and electrons stick to dust grains upon col-
fluidandtheneutralfluid.TheexpressionsforK andG liding with them. Also, they pass on their charge with unit
jn jn
aregivenbyDraine(1986).Weusethereducedenergyequa- efficiency. The grain charge then critically depends on the
tion to calculate the temperatures of the ion and electron electron-grain and ion-grain collision rates (which them-
fluids. For the grain fluids, a hydrodynamic approximation selves depends on the electron density and temperature,
with zero pressure is appropriate as the thermal velocity the ion density and temperature, and the ion-grain rela-
dispersion of the dust grains is small compared to the drift tive streaming speed). The average grain charge is evalu-
velocity with theneutrals. ated following Havnes,Hartquist, & Pilipp (1987). Individ-
The collisions of thecharged particles with theneutral ual grains can have a charge that differs from the aver-
particlesaffecttheevolutionofthemagneticfield.Usingthe age value, but it is mostly within one unit of the average
reducedmomentumequationsforthechargedfluids(Eq.5) (Elmegreen 1979).
and the expression for the current J, the electric field can Different radiative cooling processes for the neutral
beexpressed as (Cowling 1956) and electron fluids are included in the code. The elec-
Time-dependent simulations of steady C-type shocks 3
tron gas is cooled by the excitation and subsequent de- Using the upstream fluid properties, the properties far
excitation of molecular hydrogen (e.g. DRD). This process downstream of the shock are calculated from the Rankine-
only becomes important when the electron fluid temper- Hugionotrelationsforanisothermal,magnetisedshock.For
ature exceeds ≈ 1000 K. In the neutral gas, the domi- our initial shock structure we assume that the shock is a
nant cooling processes are associated with O, CO, H2 and discontinuity at x = 0 separating the upstream and down-
H2O. The radiative transitions for atomic oxygen, i.e. OI stream properties. We follow the evolution of the shock to
lines, are calculated using the procedure of DRD. The ra- steady state within the shock frame.
diative losses from excited rotational-vibrational states of
molecular hydrogen are a fit to the data presented by
Hartquist,Dalgarno, & Oppenheimer(1980),whilethoseof
CO are calculated by using Hollenbach & McKee (1979). 2.4 Computational details
Finally, we use Neufeld & Melnick (1987) to calculate the
The size of the computational box is assumed to be a few
lossesduetorotationaltransitionsinH2O.Theheatingrate time the shock width. By balancing the ion-neutral drag
oftheneutralgasbycosmicraysisfromDRD.Wehavenot
force with the Lorentz force, we can estimate the shock
included any radiative losses for the ion fluid. From Eq. 6
thickness(e.g. Draine & McKee 1993)
and using the expression for G (Draine 1986) we find that
i
theion temperature, Ti, is given by L =7×1015 vA 10−2cm−3 cm, (9)
T =T + mn(v −v )2, (8) i (cid:16)km s−1(cid:17)(cid:18) ni (cid:19)
i n 3k n i
B
wherev istheshockspeed.However,thegrain-neutralfric-
s
whereT istheneutraltemperature,m theneutralparticle
n n tion is comparable tothe ion-neutralfriction for
mass and k theBoltzmann constant.
B
ρ K
i < gn,
ρ K
g in
2.3 Initial conditions
or
The initial conditions of our simulations are adopted from χ<3×10−9 vs ,
DRD and similar studies such as Wardle (1998) and km s−1
(cid:16) (cid:17)
Chapman & Wardle(2006).Thisallowsacomparisonofour
where we substituted the expressions for the collision rates
results with those of thesestudies.
with the neutrals (Draine 1986) and assumed the relative
The neutral fluid has a number density of nH =
104 cm−3 or nH = 106 cm−3 and is composed mainly of speedbetweentheneutralsandthegrainstobeoftheorder
v /2. This constraint is fulfilled in the heated precursor for
molecular hydrogen so that mn ≈ 2mH. Small amounts bsothdensitiesandtheshockthicknessisdeterminedbythe
of other neutral species such as O, CO, and H2O are also
grain-neutral drag. The expression for the shock width is
present.TheinitialabundancesofO,CO,andH2Oaretaken
to be 4.25×10−4 nH, 5 × 10−5nH and 0 respectively. How- similar to Eq. 9 and is given by
ever,astheabundancesofOandH2Oevolvesubstantiallyas L =3×108χ vs −1L . (10)
gasisheated,theabundancesoftheimportantcoolantsare g km s−1 i
(cid:16) (cid:17)
recalculated at every time-step. The dominant ions within
thegas are assumed,as mentionedabove, tobeHCO+ and Values of χ appropriate for the precursor must be used.
Mg+. The ion mass is then mi ≈ 30mH. Also, we assume Flower, Pineau des Forˆets, & Hartquist(1985)firstpointed
out that chemistry leads toa large drop in χ in theprecur-
that the ions are only singly ionised.
sorofmanyshocks.Thenumericaldomainisthenchosento
The dust grains are all assumed to be spherical with a
radius of a = 0.4µm and a mass of 8.04 ×10−13 g (where be −10Lg 6 x6 10Lg with a free-flow boundary condition
g
thedensityofthegrainsistaken3gcm−3).Thegraintohy- at the downstream side and fixed inflow at the upstream
boundary.We usea uniform grid with 400 cells.
drogenmassratiointheunshockedregionhasthecanonical
As our initial conditions are discontinuous, the inertia
valueof0.01.Thedensitiesoftheionandelectronfluidsand
ofthechargedfluid,eventheions,cannotbeneglecteddur-
theaveragegrainchargearecalculatedfromionisationequi-
ing the initial stages of the evolution. However, the inertial
librium.Forthetemperaturesthatweadopt,i.e.T =8.4K
n
fornH =104 cm−3 andTn=26.7K fornH =106 cm−3,the phase of the charged fluid is short compared to the time
to approach a steady configuration (e.g. Roberge & Ciolek
average grain charge is quite close to −e and the electron
2007). Although the inertial phase does not affect the final
andionnumberdensitiesareroughlythesame,i.e.n ≈n .
e i
steadyshockstructurehere,thiseffectshouldnotbeignored
In the interstellar medium the magnitude of the mag-
netic field scales roughly as B ≈ 1µG(nH/cm−3)−1/2 (see for some othertime-dependent models.
DRD)sothatB=10−4 GfornH =104cm−3 andB=10−3
G for nH = 106cm−3. The Alfv´en speed of the gas is then
the same at both densities, i.e. v =2.2 km s−1. In the far
A
upstreamregionthemagneticfieldliesinthex,y-planeand
3 C-TYPE SHOCK MODELS
◦ ◦ ◦ ◦
makesanangleof30 ,45 ,60 or90 (perpendicularshock)
with the x-axis. We assume that a shock is propagating in In this section we present steady C-type shock structures
the positive x-direction with a speed of 25 km s−1. Such a for large single-sized grains. First we discuss perpendicular
shock is a strong fast-mode shock with an Alfv´enic Mach shocksandthenshockspropagatingatanobliqueanglewith
numberof M ≈11.5. themagnetic field.
A
4 S. Van Loo et al.
Figure1.TheshockstructureforaperpendicularshockpropagatinginaquiescentregionwithnH=104cm−3(left)andnH=106cm−3
(right). Fromtoptobottom weplotthevelocity inthe x-directionandz-directionfortheneutrals (solid),ions/electrons (dashed) and
grains (dotted), the tangential magnetic field and the temperatures of the ions (dashed), electrons (dotted) and the neutrals (solid)
as well as the absolute value of the grain charge (dash-dotted). The velocities are normalised with respect to the shock speed and the
magneticfieldtothetotalupstreammagneticfield.
3.1 Perpendicular shocks qualitatively similar, our shock widths are about 4 times
largerthaninDRD.Thisisduetoalowerfractionalionisa-
Figure 1 shows the structures in multiple flow variables for tion in our models which is a consequence of including the
perpendicular shocks at different densities. It is clear that chemistry governing the ionisation rather than assuming a
bothshockstructuresrepresentC-typeshocks.Asourmod- constantionfluxasDRDdid.Thedragforceofthecharged
els are similar to DRD, we can compare our results with particles on the neutrals is then lower as well and is bal-
figs.2and3ofDRD.Althoughtheshapesoftheshocksare ancedbythemagneticpressureoveralargerlengthscale.A
Time-dependent simulations of steady C-type shocks 5
Figure2.Theshockstructureforanobliqueshockpropagatingatanangleof30◦(left),45◦(middle)and60◦(right)withthemagnetic
fieldinaquiescentregionwithnH=104cm−3.Fromtoptobottomweplotthevelocityinthex-direction,y-directionandz-directionfor
theneutrals (solid),ions/electrons (dashed) andgrains(dotted), thetangential magnetic fieldinthe y-direction(solid)and z-direction
(dashed), and the temperatures of the ions (dashed), electrons (dotted) and the neutrals (solid) as well as the absolute value of the
graincharge (dash-dotted). Thevelocities arenormalisedwithrespecttothe shockspeed andthe magneticfieldto thetotal upstream
magneticfield.
6 S. Van Loo et al.
consequence of the larger shock widths is that the gas has tobebalancedbythePedersencurrent(alongE′).ThisPed-
more time to radiate away its energy. The maximum tem- ersen currentgives risetotherotation of themagneticfield
perature of the neutrals within the shock front is only 530 around the shock propagation (Pilipp & Hartquist 1994).
K for nH = 104cm−3 and 905 K for nH = 106cm−3, while Significant rotation occurs when the Hall conductivity is
DRDfound 1223 K and 1334 K respectively. larger than thePedersen conductivity (Wardle1998).
Figure 1 shows that the speed of the grains relative to For the nH = 104 cm−3 models, the Hall conductiv-
the neutrals is less than the speed of the ions relative to ity is always smaller than the Pedersen conductivity and
the neutrals. The difference in relative speeds is due to the there is no significant rotation of the magnetic field (see
differentresponseofthechargedfluidstothePedersenalong z-component of magnetic field in Fig. 2). The rotation in-
E′ (= E+vn×B) and Hall current along E′×B within creases marginally as the angle, θ, between the direction
theshock front. From Eq. 5 we find of shock propagation and the magnetic field decreases. The
othershockproperties,also,donotchangesignificantlywith
vj−vn ≈ βj2 E′×B + βj E′ , (11) θ. The temperatures of the different fluids and the average
c 1+βj2 (cid:18) B2 (cid:19) 1+βj2 (cid:18)B(cid:19) grainchargearesimilartotheonesfoundfortheperpendic-
ular shock. These results agree well with the θ-dependence
where β is the Hall parameter of the charged fluid. For
j
ofthesteadyshock structurefoundbyChapman & Wardle
electrons and ions with |β | ≫ 1, the particles are tied to
e,i
(2006).Thesimilarityoftheshockstructurefordifferentval-
themagnetic field and
uesofθ resultsfrom theminorcontributionofthegrainsto
ve,i−vn ≈ E′×B. thedragforceontheneutrals.Thegrainsdonotdetermine
c B2 thedynamics and drift with theother charged species.
At the other extreme, |β |≪1, the charged particles move Significant differences, however, are seen in the nH =
withtheneutrals.ForthejnH =106cm−3 models,thegrains 106 cm−3 models. Here, the grains are the dominant con-
tributors to the drag force on the neutrals. Furthermore,
have a Hall parameter ranging from -0.1 (the upstream
thereisasignificant chargeseparation givingrisetoalarge
value) to a minimum of -1.5 within the shock front. For
these values, the E′ term has a non-negligible contribution Hall conductivity. In Fig. 3 we see that there is a substan-
tial rotation of the magnetic field. Also the magnetic field
to the drift velocity. For the x-component of the velocity
upstream is no longer a stable node, but a spiral node as
of the grains, this term only becomes important when the
the Hall conductivity changes wave propagation upstream
grainsswitchfrommovingwiththeneutralstomovingwith
theions and electrons (see Fig. 1).The z-component of the (Wardle 1998; Falle 2003). As for nH =104cm−3, the rota-
E′ term,however,hasalargercontributiontothedriftforc- tion decreases for larger θ, while the shock width increases.
The rotation of the magnetic field has large conse-
ingthegrainstomovewith theneutrals.Thelower density
quences for the drift velocity of the charged particles. Fig-
model exhibits this effect to a lesser extent as the grain
Hall parameter lies between -1 and -20. Then the E′ term ure 3 shows that the velocity structure changes consider-
ably with θ. Also, the grains move significantly differently
is less important and provides only a small contribution to
◦
thantheionsandelectrons,especially intheθ=30 model
thedrift.
(seeFig. 3). Along thepropagation direction, electrons and
The velocity difference between the ions and electrons
ions are decelerated immediately. The grains, however, are
withtheneutralsdeterminesthetemperatureofthecharged
initially accelerated before being decelerated. Furthermore,
particlesthroughcollisionalheating.Themaximumvelocity
theylagtheionsandelectrons.Todriftinthedirection op-
differenceinthemodelsis≈v givingamaximumiontem-
s
peratureofafewtimes104 K.Electroncoolingbecomesim- posite to theions and electrons, the grains motion must be
dominated bytheE′ term inEq. 11and E′ must bein the
portantattemperaturesofabout1000Kreflectedinacon- x
siderably lower electron temperature at roughly 3×103 K. samedirectionasthex-componentofE′×B.Thelargedif-
ferences in drift velocity are expectedas therotation of the
The electron temperature within the shock front is impor-
magnetic field is induced by a charge separation (and thus
tant for the dynamics of the shock as the average grain
large Hallconductivity).Forlarger θ,theHallconductivity
charge is
androtationdecrease,sothattheions,electrons,andgrains
Z e≈−4kbTeag, (12) move more together. As expected, the velocity profile con-
g e verges to the perpendicular shock structure (see Fig. 1 and
for|Z |≫1andn |Z |≪n (e.g.Draine1980).Otherwise, 3).
g g g e
theaveragegrain chargeisroughly−1.FromFig.1wefind
thattheaveragegrainchargefollowsthisdependenceonthe
electrontemperaturewiththeaveragegrainchargereaching
4 MULTIPLE GRAIN SPECIES
maximum values of about -250.
In Sect. 3 we have assumed a single-sized grain fluid. In
molecular clouds, however, the dust has a grain-size distri-
3.2 Oblique shocks bution given by (Mathis, Rumpl,& Nordsieck 1977)
For a finite magnetic field component parallel to the shock dn ∝a−3.5, (13)
propagation andanon-zeroHallconductivity,theHallcur- da
rent along E′×B has a component parallel to the propa- wherednisthenumberdensityofgrainswithradiibetween
gation direction of the shock. As the total current in that a and a+da. To take into account the effect of multiple
directioniszeroiftheshockissteady,theHallcurrentneeds dustspecies,weincludeanadditionalsmall-grain fluidwith
Time-dependent simulations of steady C-type shocks 7
Figure 3.SameasinFig.2,butwithanupstreamdensityofnH=106cm−3.
a =0.04µm (i.e. a =a /10) and m =8.03×10−16g (or and,thus,theshockdynamicsasthecollisioncoefficientbe-
s s g s
m = m /1000). The total mass density of the grain fluid, tween grains and neutrals is proportional to a2 /m (see
s g g,s g,s
ρ +ρ ,isassumedtobe1%oftheneutralmassdensity(as Draine 1986). From Fig. 4, we see that the shock width
s g
inourpreviousmodels).Withtheadoptedgrain-sizedistri- is indeed, as expected from Eq. 10, an order of magnitude
butionmost ofthemassisinthesmallgrain fluid.Further- smaller than for themodel with single-sized large grains.
more, the small grains dominate the grain-neutral friction
8 S. Van Loo et al.
Althoughthesmallgrainshaveanaveragechargeabout
an order of magnitude smaller than the large grains (see
Eq. 12), the small grains have a larger Hall parameter and
arestronglycoupledtothemagneticfield.Thevelocityplots
inFig.4showthatthesmallgrainsmovewiththeelectrons
andions,whilethelargegrainsmovewithvelocitiesbetween
those of the neutrals and the other charged particles. The
grain charge density is dominated by the small grains and
freeelectronsaredepletedfromthegasphase(i.e.n ≫n ).
i e
Thechargeseparationbetweenthechargedparticlesisthus
small and the Hall conductivity is too. The rotation of the
magnetic field out of the plane is then negligible. Also, the
upstream magnetic field is no longer a spiral node, but a
stable node.
The inclusion of a second grain species thus changes
theshockstructureconsiderably.Includingadditionalgrain
species will further affect the shock structure as more dust
grains are loading the magnetic field lines. Then the signal
speedatwhichdisturbuncespropagatethroughthecharged
fluiddecreases.ForsmallPAHabundancesthesignalspeed
is close to 25 km s−1 (Flower & Pineau des Forˆets 2003)
leadingtothepossibilitythattheresultingshockisaJ-type
shock in which thegrain inertia cannot be neglected.
5 SUMMARY AND DISCUSSIONS
In this paper we have presented 1D time-dependent multi-
fluidMHDsimulationsofperpendicularandobliqueC-type
shocksindustyplasmas.Wehaveincludedinournumerical
code relevant mass transfer processes, such as electron re-
combinationwithMg+ anddissociativerecombinationwith
HCO+,andradiativecoolingbyO,CO,H2 andH2O.Also,
themassandchargetransferfrom ionsandelectronstothe
dust grains is taken into account to calculate the average
grain charge. Our models are the first of oblique fast-mode
shocksindensemolecularcloudsinwhichafluidtreatment
of grain dynamics has been combined with a self-consistent
calculation ofthethermalandionisation balancesincluding
appropriate microphysics. We have performed simulations
with single-sized and with two dust grains species in plas-
mas of different density.
Dustgrainsdonotchangetheshockdynamicsmuchat
nH =104cm−3andtheshockstructuresforobliqueandper-
pendicularshocks are quitesimilar. Grains, however, domi-
natetheshockdynamicsathigh densities(nH =106cm−3).
For oblique shocks, the magnetic field in the shock front
is not confined to the upstream magnetic field plane. The
amountofrotationobtaineddependsontheHallandPeder-
senconductivities.Forsingle-sizedgrainstherotationissig-
nificantandincreaseswithsmallerθastheHallconductivity
inthesemodelsiscomparabletothePedersenconductivity.
Forsomemodels,themagneticfieldattheupstreamsideof
theshock even spirals around the direction of shock propa-
gation. When we include an additional small grain species,
the rotation mostly disappears. The small grains which
carry most of the charge move with the ions and electrons
Figure 4. Similarto the 45◦ model of Fig. 3, but for two grain reducing the Hall conductivity. Our models corroborate
fluids.Thesmallgrainsareindicatedonthevelocitycomponents the main qualitative conclusions of previous studies (DRD
figureswithathin-dottedline,whiletheiraveragechargeisgiven Pilipp & Hartquist1994;Wardle1998;Chapman & Wardle
byathindashed-dotted line. 2006; Guillet, Pineau des Forˆets, & Jones 2007), showing
thatournumericalmethodisefficient,rigorous,androbust.
Time-dependent simulations of steady C-type shocks 9
In addition, we have succeeded in taking a significant step DraineB.T.,RobergeW.G.,DalgarnoA.,1983,ApJ,264,
by developing the first oblique shock code capable of pro- 485
ducing reliable results for comparisons with observations. Elmegreen B. G., 1979, ApJ, 232, 729
Previousobliqueshockmodelswererestricted byless rigor- Falle S.A. E. G., 2003, MNRAS,344, 1210
ous treatments of either the grain dynamics or the thermal Flower D. R., Pineau des Forˆets G., 2003, MNRAS, 343,
and ionisation balances. 390
However,ourmodeldoeshavesomerestrictions.Firstly, FlowerD.R.,PineaudesForˆetsG.,HartquistT.W.,1985,
wedeterminethedrift velocities ofthegrains solely bybal- MNRAS,216, 775
ancingthegrain-neutralfriction with theLorentzforce and Guillet V., Pineau desForˆets G., Jones A.P., 2007, A&A,
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sidered inertialess, large grains have a drag length, i.e. the GusdorfA.,CabritS.,FlowerD.R.,PineaudesForˆetsG.,
length scale for gas drag to decelerate the grains, that in 2008a, A&A,482, 809
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fieldcompression(Ciolek & Roberge2002).Therefore,large 2008b, A&A,490, 695
graininertiacannotalwaysbeneglected.Secondly,forsmall Hartquist T. W., Dalgarno A., Oppenheimer M., 1980,
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charge. As there is some fluctuation around the average, a Havnes O., Hartquist T. W., Pilipp W., 1987, ppic.proc,
fractionofthesmallgrainshasazeroorevenpositivecharge. 389
The neutral grains decouple from the magnetic field, while Hollenbach D., McKee C. F., 1979, ApJS,41, 555
the positive grains drift in a different direction to the neg- Jim´enez-SerraI.,Mart´ın-PintadoJ.,Rodr´ıguez-FrancoA.,
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grain charge changes the shock structure, its effect is only Lefloch B., Castets A., Cernicharo J., Loinard L., 1998,
minimal (Guillet, Pineau des Forˆets, & Jones 2007). ApJ,504, L109
A time-dependent method is not only robust (even Martin-Pintado J., Bachiller R., Fuente A., 1992, A&A,
for non-equilibrium conditions) for finding steady 254, 315
state shock structures. It is also vital for modelling Mathis J.S.,RumplW.,NordsieckK.H.,1977, ApJ,217,
transient phenomena. In the vicinity of protostellar 425
objects the abundance of SiO is significantly en- Mestel L., Spitzer L., Jr., 1956, MNRAS,116, 503
hanced (e.g. Martin-Pintado, Bachiller, & Fuente 1992; Mullan D. J., 1971, MNRAS,153, 145
Jim´enez-Serra et al. 2004).It is thought that theSiO emis- Neufeld D.A., Melnick G. J., 1987, ApJ,322, 266
sion arises from the interaction of a protostellar jet with OppenheimerM., Dalgarno A.,1974, ApJ, 192, 29
local dense clumps (Lefloch et al. 1998). Grain sputtering Pilipp W., Hartquist T. W., Havnes O., 1990, MNRAS,
withinC-typeshocksreleasestheSiOdepletedonthegrains 243, 685
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shocks indeed show good agreement with recent obser- Schilke P., Walmsley C. M., Pineau des Forˆets G., Flower
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molecular outflows (Gusdorf et al. 2008a,b). However, it is Wardle M., 1998, MNRAS,298, 507
doubtful that shocks interaction with dense clumps have
reached steady state. In subsequent papers, we will study
the interaction of C-type shocks with dense clumps and
includegrain sputteringto calculate the SiO emission.
ACKNOWLEDGEMENTS
Wethanktheanonymousrefereeforareportthathelpedto
improvethemanuscript.SVLgratefullyacknowledgesSTFC
for the financial support.
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