Dense core compression and fragmentation induced by the scattering of hydromagnetic waves

0.66 MB·English

by  S. Van Loo

Mon.Not.R.Astron.Soc.000,1–7(2006) Printed5February2008 (MNLATEXstylefilev2.2)
Dense core compression and fragmentation induced by the
scattering of hydromagnetic waves
S. Van Loo1,⋆ S.A.E.G. Falle2 and T. W. Hartquist1
1 School of Physicsand Astronomy, University of Leeds, Leeds LS2 9JT
2 Department of Applied Mathematics, Universityof Leeds, Leeds LS2 9JT
7
0
0
2 Accepted -.Received-;inoriginalform-

n
a
J ABSTRACT
We have performed 2D hydromagnetic simulations with an adaptive mesh refinement
5
code to examine the response of a pre-existing initially spherical dense core to a non-

1 linear fast-mode wave.One key parameter is the ratio of the wavelengthto the initial
v core radius. If that ratio is large and the wave amplitude is sufficient, significant
2 compression of the core occurs, as envisaged by Myers & Lazarian (1998) in their
4 “turbulent cooling flow” picture. For smaller values of that ratio, an initial value of
1 theratioofthethermalpressuretomagneticpressureof0.2,andsufficientlylargewave
1
amplitude, the scattering induces the production of dense substructure in the core.
0
This substructure may be related to that detected in the dense core associated with
7
the cyanopolyyne peak in TMC-1. Our simulations also show that short-wavelength
0
waves, contrary to large-wavelengthwaves,do not confine dense cores.
/
h
p Key words: MHD – stars: formation – ISM: clouds – ISM: individual: TMC-1 –
- ISM: molecules
o
r
t
s
a
: 1 INTRODUCTION survived as long as the amplitude that might be loosely
v
associated with the fast-mode component of the large-scale
i Ballesteros-Paredes & MacLow(2002),Padoan & Nordlund
X perturbation remained a significant fraction of its initial
(2002), Gammie et al. (2003), Li et al. (2004),
r value.
Nakamura& Li (2005), and V´azquez-Semadeniet al.
a
The interaction of fast-mode waves with large-scale
(2005) are amongst the authors who have performed 3D
density inhomogeneities produced sub-structure within the
simulations toinvestigatetheformation ofclumpsinmedia
large-scale clumps. The large-scale structures themselves
in which the value of β, the ratio of the thermal pressure
arise due to the non-linear steepening of the fast-mode
to magnetic pressure, is initially small everywhere but in
waves. The present paper is concerned with a related but
which velocity perturbations exist. These calculations have
somewhat different problem: the response of a pre-existing
included many ingredients.
dense core-like structure to a background wave. The effect
In three previous papers, we have carried
of large-amplitude hydromagnetic waves has already been
out 1D (Falle & Hartquist 2002, hereafter FH02;
studied analytically by McKee & Zweibel (1992). Using a
Lim, Falle & Hartquist 2005, hereafter LFH05) and
time-averaged form of thevirial theorem, they showed that
2D (Van Loo, Falle & Hartquist 2006, hereafter VFH06)
a turbulent medium can pressure-confine a dense clump. A
numerical studies of MHD waves in low-β plasma to elu-
limitation of their analysis is that it does not describe the
cidate the mechanism by which clumps form. We consider
effect of large-wavelength perturbations. With our numeri-
the results of particular relevance to the formation of
cal model, we can study the effect of both large and small
dense cores (e.g. Caselli et al. 2002) within translucent
wavelength perturbations.
clumps like those discovered in CO maps of the Rosette
MolecularCloud(Williams, Blitz & Stark1995).TheFH02 Numerical simulations of this problem are limited, as
and VFH06 studies concerned small-amplitude waves and most studies(e.g.Mac Low, McKee & Klein1994)only de-
involved 1D and 2D calculations, respectively. A significant scribe the effect of a shock hitting an inhomogeneity. The
difference in the results of the two studies is that dense flow initially compresses the dense core, but eventually de-
core-like structures persisted longer in the 2D simulations stroysandfragments thecore.Asimilar behaviourcan also
than in the 1D simulations. In 2D, such a structure be expected for non-linear waves. Our simulations can be
considered a natural extension and generalisation of work
on shocks interacting with inhomogeneities.
⋆ E-mail:[email protected] Theresponseofadensecoretoabackgroundwavemay
2 S. Van Loo et al.
well be relevant tothedetection of substructurein Core D, state, i.e. pg = ρa2 where a is the isothermal sound speed.
whichcoincides withthecyanopolyynepeak,intheTMC-1 In ourcalculations, we use a=0.1 unless otherwise stated.
byPeng et al.(1998).TMC-1isaridgeofdensecores,each We superpose a fast-mode magnetosonic wave on the
ofwhichhasasizeof0.1–0.2pcandamassofonetoseveral background state. We do not examine slow-mode waves, as
solar masses (e.g. Hirahara et al. 1992). The substructures theyonlygeneratehigh-densitycontrastsiftheirinitialden-
studied by Peng et al. (1998) have sizes of 0.03 - 0.06 pc sity perturbation is already large. Contrary to FH02 and
and masses of 0.01 - 0.15 M . Most are not bound by self- VFH06,wedonotonlyconsidersmall-amplitudewavesbut
gravity and are too small to⊙produce brown dwarfs. Since also finite-amplitudewaves.Then it isnolonger possible to
many of these substructures have masses below the Jeans calculate the initial state using the linear approximation of
mass,theymusthaveformedbysomemechanismotherthan the wave. Therefore, we use the approach of LFH05. For a
gravitational instability. fast-mode wave, the form of the wave written in terms of
Apossiblemechanismforformingthesubstructuremay theprimitive variables
be the same as or related to the mechanism giving rise
to dense core formation in the picture that we have pur- P=[ρ,vx,vy,Bx,By]t,
sued in FH02, LFH05 and VFH06 and which is, no doubt, must satisfy
contributing to dense core formation in the other numeri-
ccaealsseims tuolagtiivoenslarregfeerdenencesdityabcoonvetr.aHstowstervuecrt,utrheastifmβec>ha1niusnm- ∂∂Px ∝rf ≡ ρ,cf,−cfB∆xfBy,0,ρc∆2ffBy By 6=0
( (cid:16)0,0,1,0, √ρ (cid:17) By =0
less the amplitude of the wave is hyper-Alfv´enic (FH02, −
LFH02). Some dense cores may have values of β as low whererf isther(cid:0)ighteigenvecto(cid:1)rforafastwavepropagating
as 0.1 (e.g. Ward-Thompson et al. 2000, Ward-Thompson in the positive x-direction, cf the fast magnetosonic speed
2002)butsomemayhavevaluesofβgreaterthanunity(e.g. and ∆f = ρc2f −Bx2. Given the x-dependence of one of the
Kirk,Ward-Thompson & Crutcher2006).Thelinesemitted primitive variables, the others can be determined from the
byspeciessuchasCS,NH3andCOindensecoresshowsub- aboveexpression.
stantialnon-thermalcomponentstotheirbroadening,which As the x-component of the velocity disappears when
are, however, usually not highly supersonic with respect to By = 0, it is convenient to specify the profile of vy(x). We
the thermal sound speed in H2 (e.g. Fuller& Myers 1992). assumethatthey-componentofthevelocityassociatedwith
For thesubstructurein Core D of TMC-1, thenon-thermal the wave varies sinusoidally with x and with wavelength λ,
line broadening component is important but in most sub- i.e.
structures markedly subsonic in the above sense. The fact 2π(x x)
that dense cores possess substantial non-thermal internal vy =Aysin − l ,
λ
velocitiesand,insomecases,havevaluesofβ somewhatbe- (cid:18) (cid:19)
low unity supports the possibility that wave scattering on whereAy isaconstantandxl theleftboundaryofthecom-
a densecore may lead to substructurelikethat of TMC-1’s putationaldomain.Ay ischosensothatthemaximumvalue
Core D. for the total velocity ( v2+v2) is equal to a given ampli-
x y
In the present paper we examine the effect of MHD tude A. The initial state at a position x+∆x and y can
p
waves on dense cores. Additionally, we derive the proper- then becalculated using
ties of the waves that produce substructures within dense
cores. The paper is structured as follows. In Sect. 2 we de- P(x+∆x,y,0)=P(x,y,0) rf∆f ∂vy (x)∆x,
scribethenumericalmodel.Wethendescribehowthedense −(cid:18)cfBxBy ∂x (cid:19)
coreevolvesintheabsenceofMHDwaves(Sect.3)inorder where theright hand side is evaluated at x.
to understand the effect of the waves (Sect. 4). Finally, we We still need to embed a dense core in the fast-mode
summarise and discuss these results in Sect. 5. wave. We represent the core by a uniformly dense region
of radius r at the centre of the computational box. The
density ρ0 of the core is taken such that β within the
2 NUMERICAL MODEL core is between 0.1 and 1 as polarisation observations sug-
gest (Ward-Thompson et al. 2000; Ward-Thompson 2002)
WecanusedifferentapproachestostudytheeffectofMHD for some dense cores. As β = 2ρa2/B2, the core density
waves on a dense core. One way is to continuously excite must lie between 5 and 50 for a = 0.1. Note that there is
a wave at the boundary of the computational domain. The a small variation of β within a dense core as the magnetic
wavethenpropagates tothecoreandinteracts withit. An- pressure changes across thecore.
other option is to embed a dense core in a wave. In our Our model can then be specified by the isothermal
calculations weusethelatterasdensecoresarisewherethe sound speed a, the value α for background By field, the
waves are. wave properties (i.e. the wavelength λ and the amplitude
An initially uniform quiescent medium is perturbedby A), and the core properties (i.e. the density ρ0 and radius
amagnetosonicwave.Weassumethatthebackgroundmag- r).However,asweusedimensionlessunits,theradiusofthe
neticfieldisinthex y planeandthatwaveispropagating core and the wavelength of the fast-mode wave are not in-
−
in the positive x-direction. Furthermore, we work in units dependent parameters. The relevant free parameter is λ/r.
so that theunperturbedbackground state is given by Therefore, we assume a fixed core radius of r=2 in all our
calculations and only vary λ.
ρ=1, v=0, Bx =1, By =α.
To solve the 2D MHD equations we use an adaptive
The gas pressure pg is given by an isothermal equation of mesh refinement code. The basic algorithm is a second or-
Dense core compression and fragmentation 3
der Godunov scheme which uses a linear Riemann solver
(Falle1991).Weincludedthedivergencecleaningalgorithm
of Dedneret al. (2002) to stabilise the numerical scheme.
The code uses a hierarchy of grids such that the grid spac-
ing of level n is ∆x/2n where ∆x the grid spacing of the
coarsest level. A refinement criterion determines where in
the computational domain a finer grid is needed. The com-
putationaldomainis 10<x< 10and 10<y<10and
− − −
we use periodic boundary conditions. The finest grid has a
resolution of 800 800.
×
3 EVOLUTION OF A DENSE CORE IN THE
ABSENCE OF MHD WAVES
In an isothermal gas a dense core is overpressured, which
means that thecore expands.As this introduces waves and
shocks into the background medium even when the core is
not embeddedin awave, it is convenientto discuss thisex-
pansionintheabsenceofanembeddingwave.Thisproblem
isthemagneticversionofthe2DisothermalRiemannprob-
lem.
Inapurelyhydrodynamicalcase, ashock developsand
Figure1.Thedensitystructureofanexpandingdensecore.The
propagatesoutwardintotherarefiedgas.Atthesametime,
expansiondependsstronglyontheanglebetweenthepropagation
a rarefaction wave reduces the density inside the core by
directionandthemagneticfield.Perpendiculartothefieldafast-
travellinginward.Asthereisnopreferred direction,theex-
modeshock(FS)propagatesoutwardandafast-moderarefaction
pansion is isotropic. However, this is no longer true in a
(FR) inward. These waves are separated by a tangential discon-
magnetised medium because waves (and, thus, also shocks) tinuity(TD). For expansion at obliqueangles the TD breaks up
have a propagation speed which depends on the angle be- into a slow-mode shock (SS) and a slow-mode rarefaction (SR).
tween the magnetic field and the direction of propagation. ParalleltothefieldtheFSdisappears.Themodelparametersare
Figure 1 shows theexpansion of adensecore in a magnetic ρ0 = 10, a = 0.316 and α = 0. The core is not embedded in a
field. wave.
Slow-mode waves or Alfv´en waves cannot propagate
perpendicular to the magnetic field. Thus, only fast-mode
featuresdevelopinthosedirections.Aweakfast-modeshock by the slow-mode shock is comparable to that of the dense
is propagating outward into thebackgroundmedium, while core material that has passed through the slow-mode rar-
afast-moderarefactionwavepropagatesintothedensecore. efaction wave, which would make the observational identi-
The magnetic pressure increases in material through which fication of the dense-core boundary difficult. Since the slow
the shock and rarefaction wave have passed. A tangential magnetosonicspeedincreaseswhentheanglewiththemag-
discontinuity1 exists between the material that has passed neticfielddecreases,theexpansionofthecoreisfasteralong
through the front of the rarefaction wave and the mate- themagnetic field lines than perpendicularto it.
rial that has passed through the shock. The total pressure In our calculation the fast-mode shock weakens when
(pg+B2/2)iscontinuousatthetangentialdiscontinuitybut theanglebetweenitsdirectionofpropagationandthemag-
neither pg or B2/2 is. The fast-mode shock is sufficiently netic field decreases. For parallel expansion the fast-mode
weak that itdoes notproducealarge densitydiscontinuity. shockeventurnsintoalinearisedwave.Also,theslow-mode
Hence, the largest jump in density is at the tangential dis- rarefaction wavemergeswiththefast-moderarefaction (see
continuity. As it coincides with the edge of the expanding Fig. 1).
densecore,thetangentialdiscontinuitymovesoutwardwith Finally,thecoredispersesasthereisnoforceholdingit
thegas swept up bythe fast-mode shock. together. The dispersal is faster for higher initial densities
Slow-mode waves do exist for directions oblique and of the core. Also, when the thermal gas pressure becomes
paralleltothemagneticfield.Aslow-modeshockmovesout- moredominantinthebackgroundgas(i.e.β increases), the
ward and a slow-mode rarefaction wave propagates inward. densecore disappears more quickly.
These slow-mode features lie between the fast-mode shock
andrarefaction wave.Althoughthetangentialdiscontinuity
disappears, there is still a sheet in between the slow-mode
shockandrarefaction waveseparatingtheinitialcoremate- 4 A DENSE CORE EMBEDDED IN AN MHD
rial and theshocked background material. WAVE
The density of the material within a region bounded
Whenadensecoreisembeddedinafast-modewave,density
substructures can be generated by the interaction between
1 A discontinuity isa tangential discontinuity if there is neither the wave and the core. The analysis in FH02 shows that
magnetic flux nor mass flux across it, i.e. un = 0 and Bn = 0 high-densitycontrastsareassociated withslow-mode waves
(Burgess1995). when β is small. They also showed that non-linear steepen-
4 S. Van Loo et al.
∂B
= (v B).
∂t ∇× ×
More specifically, Bx increases at thetop region of thecore
anddecreases atthebottom (orinverselydependingon the
sign of thevelocity changes). The y-componentof thefield,
By,showsasimilarbehaviour,buthereleftandrightofthe
core.
Thus, the interaction with the wave causes an increase
of the magnetic pressure. While there is initially no dis-
continuity in the total pressure perpendicular to the mag-
netic field (see Sect. 3), the total pressure outside the core
can be much higher than inside the core. The magnitude
of this change depends on the amplitude of the fast-mode
wave. Small-amplitude waves are too weak to produce any
significant change and thecore then disperses asthough no
fast-modewaveispresentinthebackgroundgas(seeFig.2).
However,whentheamplitudeofthefast-modewaveislarge,
themagneticpressureincreasessignificantly.Nowtheregion
outsidethecoreisoverpressuredcomparedtothecoreitself.
Afast-modeshockarisesandpropagatesintothecore.This
shock isweak and,therefore, compresses thegas byasmall
factor as can beseen in Fig. 3a.
Althoughtheexpectedincreaseindensityfromthemo-
Figure 2. The temporal evolution of the maximum density
tionof thewaveacross thecoreissmall, Fig. 3ashowsthat
within the dense core. The thin solid line shows the evolution
in a homogeneous background gas (i.e. A = 0), while the thick the maximum density increases significantly for t>20. We
solid line represents a model with a small-amplitude wave. The also find that most mass is concentrated within a single re-
model parameters are: ρ0 =10, α=0.25, λ=20, A=0.05 and gion.Asthetime-scalecorrespondstothetimeforawaveto
a=0.1.Initially,β=0.2inthedensecore. steepen into a shock (tns 20 for λ=20 and A=0.5), we
≈
expect that this high-density structure is generated by the
interaction between the fast-mode shock and the core ex-
ing of a fast-mode wave can readily excite these slow-mode citing slow-mode waves. However, β needs to smaller than
components. unity for this process to be effective (see the analysis in
Toproducesubstructures,slow-modewavesmustbeex- FH02). Figure 3b shows that β for the fluid element as-
cited within the core itself (as their propagation speed is sociated with the maximum density obeys this constraint
low).Thismeansthatthetime-scalefornon-linearsteepen- reasonably well.
ingtns λ/2AmustbesmallerorcomparabletotheAlfv´en We can now examine how the model parameters influ-
crossing≈time tA=r/cA. Weimmediately see that for large ence the wave-core interaction. Of all parameters the wave
wavelengths tns tA even if their amplitude is compara- amplitudeis perhapsthe most important. The wave ampli-
≫
ble to the Alfv´en speed. Short-wavelength waves, however, tude determines how much the magnetic pressure changes
steepen within the core itself as long as their amplitude is at the boundary. Small-amplitude waves do not affect the
not too small. evolution of the core, while large-amplitude waves modify
TheinterplaybetweenMHDwavesandadensecoreis, the entire core structure. In the latter case, a core can be
thus, wavelength-dependent. Therefore, it is convenient to compressedtoadensityconsiderablyhigherthantheinitial
discuss the results for long and short waves separately. We one. The maximum density is limited by the β . 1 con-
typically takeλ=20 and λ=2 in ourcalculations. straint. Because of this constraint, it is also more difficult
toproducehigh-densitycontrastsinmodelswhereρ0 anda
are high. Then β is already close to unity or higher.
When the wavelength is larger than the radius of the
4.1 Results for large wavelengths
core,theinitialposition ofthecorecentrewithrespect toa
Although thetime-scale for non-linearsteepening for large- wave node becomes an additional free parameter. However,
wavelength fast-mode waves is too long for exciting slow- the fast-mode wave still propagates with a lower speed in-
mode components within the dense core, the wave strongly side the core. The variations described above always occur.
affectstheevolutionofthecore.Themostdirecteffectofthe Hence, the results are essentially independent of the initial
coreonafast-modewaveisthereductionofthepropagation position.
speedwithinthecorebecausecf B/√ρ(fora<ca).This
≈
meansthatapartofthewave-i.e.thepartwithinthecore
4.2 Results for small wavelengths
-movesslowerthantherestofthewavefront.Consequently,
the velocity of the gas must change at the boundary of the Asmall-wavelength wavesteepensmuchfaster intoashock
core. than one with a large wavelength. If the amplitude of the
However,moreimportantly,thisinducesachangeinthe small-wavelength wave is not too small, the time-scale for
magnetic field at the boundary of the core, which follows non-linear steepening is shorter than the Alfv´en crossing
directly from theinduction equation time.Thismeansthatashock arisesbefore thewaveleaves
Dense core compression and fragmentation 5
Figure 4.Similartofig.3,butnowforλ=2.
Themaximumdensityduetothenon-linearsteepening
is roughly
A2ρ0
ρmax = a2 .
Small-amplitudewavesdonotchangethedensityinthecore
significantly.Thedensecorethendispersesasinthecaseof
small-amplitudelarge-wavelengthwaves,i.e.asifthecoreis
not embedded in a wave. Large-amplitude waves, however,
generate high-density regions within the core (see Fig. 4).
These density perturbations are aligned with the y-axis as
can be seen in Fig. 5. This is because, except close to the
boundary, there is little variation in the y-direction inside
thecore.
The density perturbations generated by non-linear
steepeningsubsequentlydecay.Thishappensonatime-scale
(see FH02)
λ
te .
≈ A
Figure 3. (a) Similar to Fig. 2, but now for a large-amplitude Although the dense regions expand and disperse, high-
fastwave.(b)Theplasmabetaassociatedwiththefluidelement density regions can arise for some time due to the inter-
thathasthehighestdensity.Themodelparametersaregivenby action of the dense regions with the initial fast-mode wave
ρ0=10,α=0.25,λ=20,A=0.5anda=0.1. (seeVFH06).Thisinteraction excitesslow-mode wavesand
producesdensesubstructures.Sincethesheet-likestructures
disperse,thecorehasamorehomogeneousdensitystructure
with small-scale structures nested inside. This can be seen
the dense core. Then the generation of density substruc- in Fig. 6. Also note that, although large density contrasts
turescanbereadilydescribedwiththemodelsofFH02and are generated within the core, the core itself still exists as
VFH06. anentity.However,becauseofthefragmentationofthecore
When a fast-mode wave steepens into a shock, slow- and associated motions, the kinetic energy inside the core
mode waves are excited. This produces persistent pertur- contributessignificantlytothetotalenergy.Thedensecore,
bations with large density contrasts. The fast-mode wave therefore,expandsmorerapidlythanwhenthereisnoscat-
is most efficient in exciting the slow-mode components for tering wave.
modestvaluesofα,whichisthetangentoftheanglebetween High-densitystructuresdonotonlyariseinsidethecore,
its direction of propagation and the magnetic field. Steep- but also at the boundary.In the explanation above, we ne-
eningisslowerforverysmallvaluesofαandtheshockdoes glectedtheeffectsoccurringattheboundaryofthecore.In
not produce slow-mode waves, when α is too large (FH02). a similar way as for the large-wavelength waves, overpres-
6 S. Van Loo et al.
5 SUMMARY AND DISCUSSIONS
We have studied the interaction between magnetosonic
waves and dense cores for which the ratio, β, of thermal
gas pressure to magnetic pressure is initially 0.2. We follow
the evolution of a core embedded in a fast-mode wave. We
findthatlarge-amplitudewavescanchangetheevolutionof
thecore considerably.
The interaction between the fast-mode wave and a
dense core depends strongly on the relative size of the core
to the wavelength. When the core radius is smaller than
the wavelength, the strongest effects are induced near the
boundary of the core. Eventually, these effects disrupt the
global structure of the core, but confines it. Furthermore,
we find that a core can be compressed significantly. This
result suggests that a large-wavelength fast-mode wave can
triggerthecollapseofagravitationallyunbounddensecore,
whichisofrelevancetotheturbulentcoolingflowpictureof
Myers & Lazarian (1998). However, additional calculations
which include self-gravity are required toconfirm this.
Short-wavelengthwaves,ontheotherhand,playanim-
portantrˆoleinthegenerationofsubstructureinacorewith-
out breaking up the core. However, contrary to the large-
Figure 5. The density structure at t=3 insidethe dense core. wavelengthwaves,thedensecoreexpandsfasterthanwith-
The high-density regions are produced due to non-linear steep- outthescatteringwave.Thewavethusdoesnotconfinethe
eningof the fast-modewave. Theinitialstate isgivenby λ=2,
core. Although β is close to unity in our simulations, slow-
A=0.5,Te=0.01,α=0.25andρ0=10. modewavesexcitedbythenon-linearsteepeningofthefast-
mode wave produce high-density contrasts within the core.
Such small-scale features have been observed in the cores
of cold dense clouds such as TMC-1 (Peng et al. 1998). We
do not find the large number of microclumps inferred from
observations of Core D, but this can be readily explained
by the limitations of our model. The most important limi-
tation is perhaps that we only studied the effect of a single
wave.Arealvelocity fieldis moreaccurately described asa
superpositionofanensembleofwaves.Duetowave-wavein-
teractions,largestructuresthenbreakupintosmallerones.
Hartquist, Williams & Viti(2001)showedthatthesub-
structuresinCoreDneedtobeabout0.1Myroldtocomply
with theobserved molecular varietyand abundances.If the
substructures are being generated due to slow-mode exci-
tation, the timescales should be of the same order. For a
core radius of R= 0.1 pc and a=0.3 km s−1 and a some-
what larger Alfv´en speed, the substructures in our simula-
tions arise on timescales of a few 105 yr (a few times R/ca
whichistherelevanttime-scaleinoursimulations).Asboth
timescalesareofthesameorder,themicrostructureindense
coldcorecanindeedbegeneratedbysmall-wavelengthmag-
netosonic waves.
ACKNOWLEDGEMENTS
Figure 6. Similar to Fig. 5, but now for t = 18. Dense sub-
structuresformwithintheexpandingcoreduetotheinteraction SVL gratefully acknowledges the financial support of
betweendenseregionsandtheinitialfastmode. PPARC.
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