Braking the Gas in the beta Pictoris Disk

0.46 MB·English

by  Rodrigo Fernández

RECEIVED2005SEPTEMBER8;ACCEPTED2005DECEMBER28
PreprinttypesetusingLATEXstyleemulateapjv.6/22/04
BRAKINGTHEGASINTHEβ PICTORISDISK
RODRIGOFERNÁNDEZ1,ALEXISBRANDEKER,ANDYANQINWU
DepartmentofAstronomyandAstrophysics,UniversityofToronto.60St.GeorgeStreet,Toronto,ONM5S3H8,Canada
Received2005September8;Accepted2005December28
ABSTRACT
Metallic gas detected in the β Pictoris circumstellar debris disk raises many questions. The origin of this
gas is unclear and its very presence is difficult to explain: many constituents of the gas are expected to be
radiativelyacceleratedoutward,yettheirmotionappearstobeconsistentwithKeplerianrotationouttoatleast
300 AU. Hydrogenhas previously been hypothesizedto exist in the disk, acting as a braking agent, but the
6
amountrequiredtobrakeindividualelementsconflictedwithobservedupperlimits.
0
Toresolvethisdiscrepancy,wesearchforalternativebrakingmechanismsforthemetallicgas. Wefindthat
0
2 all speciesaffectedby radiationforceare heavily ionized. FrequentCoulombcollisionscouplethe ionsinto
a single fluid, reducing the radiation force on species feeling the strongest acceleration. For a gas of solar
n
composition,theresultingtotalradiationforcestillexceedsgravity,whileagasofenhancedcarbonabundance
a
couldbeself-braking. We alsoexploretwootherbrakingagents: collisionswithdustgrainsandneutralgas.
J
Grains surrounding βPic are photoelectrically charged to a positive electrostatic potential. If a significant
1
fractionofthedustgrainsarecarbonaceous(10%inthemidplaneandlargerathigheraltitudes),ionscanbe
1
slowed down to satisfy the observed velocity constraints. In this case, both the gas kinematics and spatial

distributionsareexpectedtocoincidewiththoseofsmallgrains,thelatterbeingindeedobserved. Forneutral
1
v gas to brake the coupled ion fluid, we find the minimum required mass to be 0.03M⊕, consistent with
≈
4 observedupperlimitsofthehydrogencolumndensity,andsubstantiallyreducedrelativetopreviousestimates.
4 Our results favor a scenario in which metallic gas is generated by grain evaporation in the disk, perhaps
2 duringgrain-graincollisions. We exclude a primordialorigin for the gas, but cannotrule out the possibility
1 of its production by falling evaporating bodies near the star. We discuss the implications of this work for
0 observationsofgasinotherdebrisdisks.
6 Subjectheadings:stars:individual(βPictoris)—circumstellarmatter—accelerationofparticles—scattering
0
—planetarysystems: formation—planetarysystems: protoplanetarydisks
/
h
p
-
o 1. INTRODUCTION
r The evolution of circumstellar disks is intimately connected to the process of star and planet formation. In the standard
t
s isolatedlow-massstarformationscenario(e.g.,Shu,Adams&Lizano1987),disksbegintheirexistenceasprotostellaraccretion
a disks, later turning into protoplanetary disks, once the newborn star reaches the T Tauri phase. The late evolutionary stages
:
v of circumstellardisks are notwell understood. Once the star has reachedthe main sequenceand the planetformationprocess
i is complete, the disk is thoughtto have beendepleted of its primordialgas, keeping small amountsgeneratedby either stellar
X
winds or evaporation of solid bodies (e.g., Lagrangeetal. 2000). In the few cases where late disks are observable, lifetimes
r of smalldustgrains(. 1µm) are shorterthan theestimated agesof those systems, implyinga dustreplenishmentmechanism
a
(Backman&Paresce 1993). These kind of late circumstellar disks are usually referredto as debris disks. The gas lifetime in
those systems, currentlyunknown,determinesthe maximumperiodoverwhich protoplanetarycorescan accretegasandform
gaseousplanets(e.g.,Bodenheimer&Lin2002),andisrelatedtheformationofterrestrialplanetsandcoresofgiantplanetsby
theincreasedefficiencyofplanetesimalaccretionduetothepresenceofgas(Rafikov2004).
So far, the best studied debris disk is that of β Pictoris, a nearby (19.3 0.2 pc, Crifoetal. 1997), A5V star. The dust
±
componentofthediskhasbeenextensivelystudiedsinceitrevealeditselfasinfraredexcessinthestellarspectrum(Aumannetal.
1985). Thefavorableedge-onorientationhasalloweddirectimaginginthermalemission(Lagage&Pantin1994),aswellasin
scattered light (Smith&Terrile 1984, Heapetal. 2000), indicatinga disk size 1000AU. The gas componenthas been more
elusive.Freudlingetal.(1995)seta3σupperlimitonthecolumndensityofHI,N∼ 1019gcm- 2,bynon-detectionofthe21cm
HI
≤
line,whileThietal.(2001)andLecavelierdesEtangsetal.(2001)searchedforsignaturesofH intheIRandUV,respectively,
2
obtainingdiscrepantresults:50M inthefirstcase(correspondingtoN 3 1021cm- 2),andanupperlimitN 1018cm- 2
⊕ H2 ∼ × H2 ≤
( 0.1M )inthesecond.Metallicelementshavebeendetectedthroughstableandvariablecircumstellarlines(Hobbsetal.1985;
⊕
∼
Ferletetal. 1987). The variable, redshiftedlines have been attributed to falling evaporatingbodies(FEBs) in highlyeccentric
orbits(Ferletetal.1987),whereasthestablecomponenthasnowellestablishedexplanationforitsorigin.Lagrangeetal.(1998)
proposedthreepossiblegasformationscenarios: stellar wind, star-grazingcometevaporation,anddustgrainevaporationnear
thestar. Recentimprovementsinthegascharacterizationincludespatiallyresolvedemissionlinesfromstablemetallicspecies,
whichrevealedthemtobeconsistentwithKeplerianrotation(Olofssonetal.2001),andspatiallyextendedouttoatleast300AU
fromthestar(Brandekeretal.2004).
Itwasrealizedearlyon(e.g.,Beustetal.1989)thatcertainelementsinthegasdiskofβPicshouldbesubjecttoaverystrong
1e-mail:[email protected]
2 Fernández,Brandeker,&Wu
radiationforcefromthestar,overcominggravitationalattraction. However,besidesKeplerianrotation,stablelinesdonotshow
anysubstantialradialvelocitiesrelativetothestar(e.g.,Lagrangeetal.1998,Brandekeretal.2004). Thisimpliesthepresence
ofabrakingmechanismactinginthedisk.
ThefirstmodelofhowgasmightbebrakedandproducethestableabsorptionlineswaspresentedbyLagrangeetal.(1998).
TheyproposedascenarioinwhichgasisinjectedintothediskduetoFEBs. Sinceneutralhydrogenisnotaffectedbythestellar
radiationforce,itwouldaccumulateintheregionofgasgeneration,intheformofanannuluslocatedafewAUfromthestar. The
elementsaffectedbyradiationforcewouldthenbesloweddowntemporarilyinthisannularregion,producingtheobservedstable
absorptionfeatures. After passing this hydrogen-richregion, elements would then be accelerated again, leaving the system at
highvelocities.Thismodelreproducesthestableabsorptionlines,andisconsistentwiththeFEBsscenarioforthevariablelines.
However,thefactthatelementsaffectedbyradiationforcereachhighvelocitiesafterpassagebytheannulus,whichextendsonly
outtoafewAUfromthestar,isincompatiblewiththeemissionfrom 300AU.Brandekeretal.(2004)calculatedtheamountof
∼
materialnecessarytoslowdownthemoststronglyacceleratedparticlestowithinobservationalconstraintsbymeansofneutral-
neutralcollisions,findingthat 50M ofhydrogenareneeded. AccordingtoThébault&Augereau(2005),therecouldbeno
⊕
∼
morethan0.4M ofgasinthedisk,asitwouldaffectthedynamicsofthedusttothedegreethatdustproductionmodelsbecome
⊕
incompatiblewithobservations.
Asanalternativebrakingmechanism,Brandeker(2004)exploredtheslowingdownofionizedspeciesbyalarge-scalemagnetic
field,concludingthatonlyatoroidalfieldcouldremovetheradialvelocitycomponentsofions. Brandeker(2004)showedthat
unlessthefieldisstrongerthan 1mGat100AU,thisconfigurationwouldbeunstabletocurrentsgeneratedbyradiationforce.
∼
Healsoraisedthequestionoftheoriginofsuchanad-hocfieldgeometry.
InthispaperweaddressthebrakingofthestablemetallicgasintheentireβPicdisk.Giventhespectraltypeofthestarandthe
factthatthediskisopticallythin,weexpectseveralgaseousspeciestobeionized.ThispromptsustoexploretheroleofCoulomb
collisions among ionized gas particles, which may be an important braking agent due to the long range of electromagnetic
interactions.Weproceedthentoexploretheinteractionofthisionizedgaswithdustgrains,whicharephotoelectricallycharged.
Finally,weextendthestudyofLagrangeetal.(1998)toexplorewhetherion-neutralcollisionsareabletobrakeparticlesaffected
byradiationforcefaroutinthedisk,andtheamountofneutralgasrequiredtosatisfytheobservedvelocityconstraints.
The paper is organizedas follows: in §2 we explore the physical conditions of the gas, and the constraints on the possible
brakingmechanismsthatthisimposes.In§3westudyion-ioncollisions,ion-graincollisions,andrevisition-neutralinteractions.
In§4wediscusstheimplicationsofourresultsforgasgenerationandextendittosimilarsystems. Ourconclusionsfollowin§5.
2. SETTINGTHESTAGEFORBRAKING
Inthissectionweexplorethephysicalenvironmentofthegaseousdisk. Thiswillenableustocharacterizetherequirements
thebrakingmechanismshavetosatisfy.
2.1. RadiationForce
TheradiationforceF actingonagivenatomoriginatesbyanetmomentumtransferduetoscatteringevents,
rad
1
F = F σ , (1)
rad λ jk
c
j<k
X
wherecisthespeedoflight,σ isthecrosssectionforthetransitionbetweenatomiclevels jandkintegratedoverwavelength,
jk
andF isthestellarfluxperunitwavelengthevaluatedatthelinecenter. Sincethestellarfluxdecaysliketheinversesquareof
λ
thedistancefromthesource,theratioβ betweenradiationandgravitationalforcesisdistance-independent:
F 1 r2 g
β rad = kA λ4 F , (2)
≡ F 8πc2GMm g kj jk λ
grav j
j<k
X
wherewehaveexpressedσ intermsofknownatomicquantities(e.g.,Hilborn1982). Here,g andg arethestatisticalweights
jk j k
oflevels jandk,A istheEinsteincoefficientforspontaneousemissionfromlevelkto j,λ thewavelengthofthecorresponding
kj jk
transition,Gisthegravitationalconstant,Mthestellarmass,andmthemassoftheatomicspeciesunderconsideration.
Theenergyofaparticleofmassminaboundorbitwithsemi-majoraxisaaroundanobjectofmassMisE =- GMm/(2a).
bind
When the contributionof radiationis added, the energyof the system increasesby an amountE =βGMm/r, where r is the
rad
distancetothestaratagivenpointintheorbit. Foracircularorbit(r=a),theparticleceasestobeboundifβ>0.5. Ifβ<0.5
theparticleremainsinaboundorbit,but“feeling"alowerdynamicalmass(1- β)M.
Values for β in this system are available in the literature for some elements, but we find it necessary to calculate a set of β
valuesforasmanyelementsaspossible. Thelowtemperaturesanddensitiesofthegasinthedisk(atdistances&fewAUfrom
thestar,see§2.2)implythatradiativede-excitationsarefasterthaneithercollisionsorradiativeexcitations,thusmostatomsare
inthegroundstate. Thisimplies j=0inequation(2),simplifyingcalculationsconsiderably.Sincewedonotexpectspeciestobe
morethantwiceionized(see§2.2),westudythefirstthreeionizationstatesofelementsfromHtoNi. Theatomicdataaretaken
fromtheNISTAtomicSpectraDatabaseversion2.0(Martinetal.1999)2. Forelementswithmultiplegroundstates(e.g.,FeI),
weassumethatthepopulationofeachlevelisproportionaltoitsstatisticalweight(e.g.,Lagrangeetal.1996). Theuncertainties
inA rangefrom<3%forthestrongestlinesto 100%fortheweakest.
k0
∼
Thestellar fluxisapproximatedwitha PHOENIX modelspectrum(Hauschildt,Allard&Baron1999)fora star ofeffective
temperatureT =8000Kandsurfacegravitylogg=4.2. GiventhatβPicisrotatingat130 4kms- 1 (Royeretal.2002),the
eff
±
2http://physics.nist.gov/asd2
BrakingtheGasinβ Pic 3
TABLE1
RADIATIONFORCECOEFFICIENTSOBTAINEDFROMROTATIONALLYBROADENEDβPIC-LIKESPECTRUM
Ion βa,b Ion βb Ion βa,b Ion βa,b
HI (1.6±0.1)10- 3 FI 0 SI 0.56±0.09 VI 72±4
HeI 0 FII (3.5±0.9)10- 6 SII (9.0±1.0)10- 5 VII 4.4±0.2
HeII ··· FIII (5.0±1.0)10- 9 SIII (2.0±1.0)10- 4 VIII 0
LiI 900±40 NeI 0 ClI (2.3±0.4)10- 3 CrI 93±5
LiII 0 NeII 0 ClII (3.7±0.4)10- 7 CrII (6.0±3.0)10- 7
LiIII ··· NeIII (9.0±2.0)10- 8 ClIII (3.0±2.0)10- 6 CrIII ···
BeI 62±7 NaI 360±20 ArI (1.7±0.3)10- 6 MnI 28±3
BeII 124±6 NaII 0 ArII 0 MnII 7±1
BeIII 0 NaIII 0 ArIII (1.5±0.2)10- 7 MnIII ···
BI 30±10 MgI 74±8 KI 200±20 FeI 27±2
BII 0.07±0.04 MgII 9±2 KII ··· FeII 5.0±0.3
BIII 19±1 MgIII 0 KIII (4.4±0.2)10- 4 FeIII (3.0±0.6)10- 7
CI (3.3±0.1)10- 2 AlI 53±6 CaI 330±40 CoI 16±1
CII (2.3±0.2)10- 3 AlII 0.36±0.05 CaII 50±10 CoII 0
CIII (8.5±0.9)10- 6 AlIII 12±1 CaIII ··· CoIII (4.0±2)10- 7
NI (2.1±0.1)10- 4 SiI 6.0±0.6 ScI 220±20 NiI 26±2
NII (7.5±0.5)10- 6 SiII 9±9 ScII (1.3±0.4)103 NiII (7.0±2.0)10- 2
NIII (7.0±1.0)10- 6 SiIII (5.8±0.6)10- 4 ScIII (9.0±3.0)10- 2 NiIII (3.0±2.0)10- 7
OI (3.3±0.2)10- 4 PI 3.4±0.6 TiI 97±5 pc 4.4×10- 11
OII (3.1±0.7)10- 9 PII (2.2±0.3)10- 3 TiII 28±2 ec 0.27
OIII (6.5±0.6)10- 7 PIII (5.0±2.0)10- 4 TiIII (5.0±0.1)10- 4 ··· ···
aβ=0meansthatnogroundstatetransitionsareknownintherange1000Å<λ<50000Å
bEmptyentriesmeanthatnoatomicdataareavailable
cProton(HII)andelectronvaluesarecalculatedusingThomsoncrosssection
spectralfeaturesaresignificantlybroadened. Aproperrotationofthespectrumrequiresdetailedresolutionofthephotospheric
absorptionprofiles,thusaspecialversionofthemodelatmospherewasusedinthiswork,whichhasconstantspacinginwave-
length,namely0.05Åintheregion1000–10000Å,and0.5Åintheregion10000–50000Å(P.Hauschildt&I.Kamp,private
communication).Arotationalbroadeningofthismodelspectrumwasperformed(e.g.,Gray1976),assumingalinearlimbdark-
ening law with ε=0.5. This choice adds an uncertainty in the relative flux level at the bottom of photospheric lines ranging
from2%inthevisualto10%intheUV.ThespectrumwasthenfluxcalibratedusingphotometricinformationfromtheTycho-2
Catalog(Høgetal.2000)andthedistancetothestar. Weestimatetheerrorofthisfluxcalibrationtobe 4%.
∼
Table1showstheresultsofourcalculationsforelementsfromHtoNi. ThemassofthestarusedisM=1.75M⊙(Crifoetal.
1997). CalculationofuncertaintiesisexplainedinAppendixA. Theagreementlevelwithpreviouslycalculatedvaluesishighly
variable. Asanexample,Lagrangeetal.(1996)calculatedβ valuesusingcalibratedHST spectra. Ourresultsagreewiththeir
valuesfor FeII, AlII and AlIII within our uncertainties, but their result for MgII differsfrom ours by a factor of 2. We take
the good agreement between their FeII value and ours as a consistency check, given the great number of transitions required
to calculateβ forthis ion. Nevertheless,whenlookingatourresults, itshouldbe keptin mindthattheywere obtainedwith a
genericmodelspectrum,thereforesubjecttopossiblesystematicdeviationsfromtheparticularβPiccase.
2.2. IonizationStateoftheGas
Since we are interested in relating the dynamics of the gas to electrostatic interactions, we need to analyze in detail the
ionizationstate ofthe gas. Giventhatthe diskis opticallythin, andthedensitiesarelow, the ionizationstate isdeterminedby
photoionizationand radiative recombination. The temperature of the gas is below 300K (Kamp&vanZadelhoff 2001), thus
recombinationratesarenotverysensitivetotheexactvalueofthetemperature,dependingmoststronglyontheelectrondensity.
Giventhespectraltypeofthestar(A5V),weexpectmostelementstobeeithersinglyionized,orneutral.
We thus wrote a simple photoionization code which calculates the densities of neutral and ionized elements from H to Ni
for an optically thin gas disk, with the corresponding electron density, and assuming solar composition. The latter choice is
motivatedbyobservationsshowingrefractoryelementshavingsolarabundancerelativetoeachother(Lagrangeetal.1995). The
stellarionizingradiationisobtainedfromourrotatedandflux-calibratedmodelspectrum,whereasinterstellarbackgroundUV
fieldandandcosmicrayionizationrate(2 10- 17s- 1atom- 1)weretakenfromWeingartner&Draine(2001)andSpitzer(1978),
×
respectively.RadiativerecombinationcoefficientsandphotoionizationcrosssectionsweretakenfromtheCloudyC96.01atomic
datalibrary(Ferlandetal.1998).Recombinationofionsondustgrainsurfaces,whichcanbethedominantrecombinationprocess
intheinterstellarmedium,isnotincludedasthedustinthedisk isexpectedto havea highlypositivecharge(AppendixB), in
contrasttothenegativelychargedgrainsoftheinterstellarmedium.
AtlowtemperaturesthechemistryofthegasisnotaffectedsignificantlyifonesetsT =T (Kamp&vanZadelhoff2001),
gas dust
whereT isthegastemperatureandT isthedusttemperature. Wethusadoptatemperatureprofilecorrespondingtograins
gas dust
4 Fernández,Brandeker,&Wu
100 1000
n
e
n
C II
10
105· n
Na I
]
3
- ]
m K
c 1 100 [
[ T

n
T
0.1
n
Fe II
0.01 10
10 100 1000
r [AU]
FIG. 1.— Plotted against cylindrical radius ρ are: average midplane dust temperature T (eq. 4, right axis), which we take to be the same as the local
gastemperature; andmidplanenumberdensities(leftaxis)ofelectrons andselectedspecies. NaIisobtainedfromobservations (Brandekeretal.2004),the
remainingspeciesareobtainedfromourphotoionizationcode. Cispartiallyionized(∼50%)throughoutthedisk(andthemaincontributorofelectrons),while
NaandFearealmostcompletelyionized.
ofsizea(e.g.,Kamp&vanZadelhoff2001)
L 1/5 AU 2/5 µm 1/5
T =282.5 K, (3)
a
L r a
⊙
(cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17)
whereListhestellarluminosity(11L⊙forβPic),andristhedistancetothestar. Toeliminatethedependenceongrainsize,we
takeameantemperatureweightedbygrainsurfaceareaandsizedistribution
T4a2dn
T4 = T4 = a a. (4)
gas h ai a2dn
R a
Weadoptdn a- 3.5andperformtheintegralovertherangea <Ra<a ,witha =1µmanda =1km. Thechosensize
a min max min max
∝
distributionimpliesthatthetotalgrainsurfacearea,andthereforeourweightedtemperature,isdominatedbyvaluescorrespond-
ingtothesmallestgrains,beinginsensitivetothechoiceofa . Theresultingtemperatureprofileinthediskmidplaneisshown
max
inFigure1(rightaxis).
TheobservationalconstraintonthegasdensitycomesfromthefittoNaIemission(Brandekeretal.2004),whichhastheform
ρ 0.94 ρ 6.32 - 1/2 z 2
n (ρ,z)=10- 5 + exp - cm- 3, (5)
NaI
ρ ρ 0.17ρ
"(cid:18) 0(cid:19) (cid:18) 0(cid:19) # " (cid:18) (cid:19) #
whereρisthedistancetothestarinthediskmidplane,ztheheightabovethemidplane(r2=ρ2+z2),andρ =117AU.TheNaI
0
profileinthe diskmidplaneis showninFigure1, scaledbya factor105. Ourphotoionizationcodenumericallysolvedforthe
densitiesofneutralandionizedspeciesthatareconsistentwiththisNaIprofile,thetemperatureprofileofequation(4),andour
assumptionofsolarcomposition.Thesensitivityofourresultstotheadoptedtemperatureprofileisdiscussedin§4.1.
OurresultsshowthatH,He,N,O,F,Ne,andAr,ifpresent,arealmostcompletelyneutral,theirmarginalionization( 10- 6)
∼
beingdeterminedbyinterstellarUVfluxandcosmicrays. Thisgroupisunaffectedbyradiationforceinitsneutralphase. The
elementsC,Cl,Be,P,andSarepartiallyionized,withionizationfractionsrangingfrom 50%forcarbonto&95%forberyllium.
∼
BrakingtheGasinβ Pic 5
TABLE2
VELOCITIESBEFOREIONIZATIONFORNEUTRAL
SPECIESWITHβ>0.5
Element Γstara vionb nI/nII+c
(s- 1) (kms- 1)
LiI 9.0×10- 6 0.11 4.4×10- 6
BeI 1.1×10- 9 58 4.4×10- 2
BI 7.5×10- 8 0.41 6.1×10- 4
NaId 1.1×10- 7 3.3 3.2×10- 4
MgI 6.8×10- 8 1.1 6.4×10- 4
AlI 1.1×10- 4 5.0×10- 4 8.5×10- 7
SiI 3.9×10- 7 0.02 1.3×10- 4
PI 4.0×10- 10 8.7 1.5×10- 1
SI 4.0×10- 10 1.5 6.4×10- 2
KI 4.4×10- 7 0.5 1.5×10- 4
CaI 1.3×10- 5 0.03 3.5×10- 6
ScI 6.4×10- 8 3.5 7.0×10- 4
TiI 1.5×10- 7 0.6 3.0×10- 4
VI 4.0×10- 7 0.2 1.2×10- 4
CrI 1.0×10- 7 0.9 4.8×10- 4
MnI 6.8×10- 8 0.4 7.6×10- 4
FeId 5.8×10- 8 0.5 9.0×10- 4
CoI 2.0×10- 8 0.8 2.9×10- 3
NiId 1.0×10- 7 0.3 4.8×10- 4
aStellarionizationrateat100AUfromthestar.
bTerminalvelocitybeforeionization,independentof
distance.
cRatioofneutral(I)toionizedphase(IIorhigher)at
r=100AUinthediskmidplane.
dSpecies with measured radial velocity by
Brandekeretal. (2004) [see §2.2 for values and
uncertainties]
Inside 50AU the stellar ionizationdominates, whereasUV backgroundtakesoveratlargerdistances. While neutralcarbon,
∼
chlorine, and sulfur feel little or no radiation pressure, the neutral phases of phosphorus and beryllium are accelerated. The
potential of these species to trace the velocity of neutrals will be discussed below. The remaining species (including Na) are
ionizedtofractions&99.9%.ElectrondensitiesinthediskmidplaneareshowninFigure1,togetherwithdensityprofilesofCII
andFeII,whicharethemostabundantspeciesbymassinthepartially-,andheavily-ionizedgroup.
The group with ionization &99.9% contains all species affected strongly by radiation force. This is crucial for explaining
the dynamics of the gas. From Table 1 we see that for most of these species, the highest β corresponds to the neutral phase
(exceptionsbeing Be and Sc), which has a very short lifetime comparedto that of the ionized phase. Once ionized, elements
feelalowerradiationforceandcanundergoCoulombinteractionswithotherchargedparticlesandlosetheexcessmomentum
acquiredfromphotons(§3.1).
Aquantityofrelevanceisthevelocityboostaneutralparticleacquiresbeforebeingionized,v . Intheopticallythincase,the
ion
ratiobetweenthenumberperunittimeofscatteringphotonstothatofionizingstellarphotonsdependsonlyontheshapeofthe
stellarspectrum,beingindependentofthedistancetothestar. Thenumberofionizingstellar photonshittinganatomperunit
timeisthestellarionizationrate
∞ F
Γ∗= λ σiondλ, (6)
(hc/λ)
Z0
whereσion istheionizationcrosssection,theionizationtime(orlifetimeasneutral)beingtheinverseofΓ∗. Thevelocityboost
beforeionizationisthen
GM 1
v β , (7)
ion≃ r2 Γ∗
whereΓ∗ isevaluatedatdistancer fromthestar. Table2showstheresultsforelementswithβ>0.5,forwhichtheionization
rate dueto interstellarUV isnegligiblerelativeto Γ . Themostrelevantfeatureis that, with the exceptionof BeI andPI, all
∗
neutrals feeling radiation force achieve velocities below 3.5kms- 1 before being ionized. Since the acceleration in the neutral
phase is nearly constant, the mean radial velocity is v /2, provided no braking mechanism acts on particles in the neutral
ion
phase. This is consistent with observed values for neutrals: v =1.2 0.3kms- 1, v = 0 0.3kms- 1, and v =0.4
NaI FeI NiI
0.4kms- 1(Brandekeretal. 2004). Hence, aslongasionizedparticlesare±effectivelybraked,the±reisno needto finda brakin±g
mechanismforhigh-βneutrals. ObservedradialvelocitiesofBeIandPIwouldbeabletotestthesepredictions. Unfortunately,
their strongest ground state transitions are in the UV, which requires observations from space. The expected line equivalent
widths,derivedusingourionizationmodel,wouldbe10mÅforPIλ1774,whileBeIλ2350wouldbehardlydetectableat3µÅ.
6 Fernández,Brandeker,&Wu
3. BRAKINGTHEIONICFLUID
We haveestablishedinthelastsectionthatthebrakingofneutralspeciesisnotrequired,onlyionsneedtobesloweddown.
Thischangesthenatureofthepuzzleandmakesitsolvable.Inthissectionweexplorecollisionalprocessesinvolvingelectrostatic
interactionsthatcanaccountforthebrakingofionizedparticles.Fromsimpledynamicalconsiderationsitisclearthatanybraking
medium needs to satisfy two requirements. First, a high collision frequency with ions, so that momentum can be effectively
exchangedwith particles accelerated by radiation. Second, the medium must provide a high inertia, so that it is not dragged
along.Wethusfocusonthreeprocesseslikelytooperateinthedisk: ion-ion,ion-dust,andion-neutralcollisions.
3.1. CollisionsbetweenIons
ThefactthatallthespeciesaffectedbyradiationpressurearehighlyionizedpromptedustoexplorefirsttheroleofCoulomb
collisions among them. For treating the problem, we assume that magnetic fields are weak enough in the regions of interest,
100AU from the star, so that the dynamics of charged particles is not influenced by them. Otherwise, complicated MHD
∼
effectsmaycomeintoplay. ThestatisticaltreatmentofcollisionsamongchargedparticleswasstudiedbySpitzer(1956),based
onresultsofChandrasekhar(1941,1943)forgravitationalinteractions. InSpitzer’sframework,particlesarecategorizedastest
andfield particles, the formerhavingan initial velocityrelativeto the centerof mass of the latter. Field particlesare assumed
tofollowaMaxwellianvelocitydistribution. Brakingoftestparticlesisthencharacterizedbythetimeittakesforthebeamof
testparticlestodiffuseinvelocityspace. Thistimecorrespondstodynamicalfrictioninthegravitationalcase, andisgivenby
Spitzer(1956)as
m2(k T)v
t (t,f)= t B f , (8)
S 8πnZ2Z2e4(m +m)H(ξ)lnΛ
f f t t f
where the subscripts t and f correspondto test and field particles, respectively. Here, v is the velocity of test particlesrelative
tothecenterofmassoffieldparticles,misthemassofindividualparticles,nisthenumberdensity,Z isthechargeinunitsof
theelectronchargee(throughoutthisworkweassumeZ =Z =1forions),T isthetemperatureofthefieldparticles,andk is
t f f B
Boltzmann’sconstant. ThequantitiesξandΛaredefinedby
v
ξ= (9)
c
th,f
2k T
c = B f (10)
th,f m
r f
λ 3 k3T3 1/2
Λ= D = B e , (11)
b 2ZZe3 πn
min f t (cid:18) e (cid:19)
wherec isthethermalvelocityoffieldparticles,λ istheelectronDebyescreeninglength,b istheminimumimpactpa-
th,f D min
rameterforCoulombcollisions,andT istheelectrontemperature.Inthiscontext,electronsandionsaresufficientlythermalized
e
toallowsettingT =T. ThelnΛfactoraccountsforthemanyweakscatteringsouttodistanceλ . ThefunctionsH andΦare
e f D
givenby3
Φ(ξ)- ξΦ′(ξ)
H(ξ)= (12)
2ξ2
2 ξ
Φ(ξ)= e- u2du. (13)
√π
Z0
Tworelatedquestionsareconsideredhere.First,foranewlyionizedparticlewithspeedv ,howlongdoesittaketoequalize
ion
itsspeedwiththerestoftheions? Second,arehigh-βionsdynamicallycoupledtolow-βions?
The first question can be answered by evaluating t with the relative velocity v=v , where v is the velocity of neutral
S ion ion
particlesjustbeforeionization,showninTable2. Theresultingt inthediskmidplaneisshowninFigure2inunitsofthelocal
S
Keplerianorbitalperiod. Ittakesaveryshortamountoftimeformostnewlyionizedparticlestobeequalizedwiththeionicsea
throughCoulombcollisions,exceptfortheelementBe. ApredictionisthereforethatBewillbepreferentiallylostfromthedisk.
Toaddressthesecondquestion,weexpressthetimescaletoradiativelyaccelerateatestionas
mv r2v
t = t ∗ = ∗ , (14)
acc F (β - 1)GM
rad t
wherev∗istheradialvelocityrelativetothestar,andβt>1isthetestparticleradiationforcecoefficient.IftS tacc,anyexcess
≪
momentumacquiredbyspeciesthatfeeltheradiationforcewillbetransferredtothefieldparticlesasaresultofmanyrandom
collisions,theionsthusbeingdynamicallycoupleddespitetheirdifferentβ values. Noticethattheratiot /t doesnotdepend
S acc
onvexceptthroughthefunctionH(ξ=v/c ),whichreachesamaximumwhenξ 1. InFigure3,weplottheratiot /t asa
th,f S D
≃
functionofthediskcylindricalcoordinatesρandz,takingξ=1,withCIIasfieldparticlesandFeIIastestparticles4. Thespecies
3In(Spitzer1956)thefunctioninequation(12)iscalledG,butwerenameitHtoavoidconfusionwiththegravitationalconstant.
4Whenξ≪1orξ≫1,tScanbecomelongerthattacc.Intheformercase,ionsmaydriftrelativetoeachother,butwithaspeedordersofmagnitudebelow
thelocalsoundspeed.
BrakingtheGasinβ Pic 7
0
Be
(v = 58 km/s)
-2
)
b
or
t
/
S-4 Na
t
(
0 (v = 3 km/s)
1
g
o
l
-6 Fe
(v = 0.5 km/s)
Ca
(v = 0.03 km/s)
-8
10 100 1000
r [AU]
FIG. 2.—Ratioofslow-downtoorbitaltimescalesalongthediskmidplane, forionswithincidentvelocities v=vion,wherevion isthevelocityofneutral
particlesjustbeforeionization,showninTable2.WeadoptCIIasfieldparticles,buttheresultsremainsimilarifotherfieldparticlesareused.
chosen are representative as they are the most abundant(in number and mass) within groupsthat feel or do not feel radiation
force(β 0andβ 5,respectively).
CII FeII
≈ ≈
Ofparticularinterestisthecouplingofionsathighaltitudesoverthemidplane.Brandekeretal.(2004)detectedCaIIemission
beingconsistentwithKeplerianorbitsevenataheightofz 80AU(ρ 120AU).Inthatzonewehavet /t 0.05forv=c ,
S acc th,f
∼ ∼ ≈
i.e.,ionsarestillcoupledthere. Theeffectofradiationforceonthecoupledensembleofionscanbequantifiedbyaneffective
radiationforcecoefficient
F β ̺
j rad,j j j j
β = = , (15)
eff
F ̺
Pj grav,j P j j
where Frad,j and Fgrav,j are the radiative and gravitationPalforces actinPg on species j within a fluid element, βj is the radiation
pressurecoefficientforspecies j,and̺ isthemassdensityofspecies j. Thusβ isaweightedaverageofthevaluesforeach
j eff
particle,theweightbeingthemassdensityofeachspecies.Ifβ <0.5,ionsbrakebythemselves.Sinceionsoutweightelectrons
eff
by a factor >2000, and there is one electron per ion, we neglectthe electronsin the estimate of β and assume that they are
eff
draggedalongtheionfluid,preservingchargeneutrality.
For solar composition, the most abundantions by mass are, in decreasing order, C(β 0), Fe(5), Si(9), Mg(9), S(0), Ni(0)
≈
and Ca(50). These particles dominate the value of β , with carbon and iron being the relevant species. If all of them were
eff
100%ionized, we would haveβ 3.5, but since the ionizationfractionof CII varieswith distance fromthe star, the actual
eff
≈
valueofβ ishigher(byafactor2atmost)andhasaweakdependenceonposition. InFigure4weplotβ averagedoverall
eff eff
ionizedspeciesasafunctionofdiskcylindricalcoordinatesρandz. Astraightforwardwayofbrakingthegasisbyinvokingan
enhancementinthecarbonabundancebyafactor 10.ObservationsofcircumstellarabsorptionlinesfromCIandCIIbyFUSE,
∼
reportedrecentlybyRobergeetal.(2005),seemtoindicatethatcarbonmayindeedbeover-abundantbythisfactor. Whetherthe
gasisproducedatthisabundanceorhasevolvedtoitis,however,notclearatthemoment.Inthefollowingsectionsweproceed
toexploreadditionalbrakingmechanismsthatactindependentlyoftherelativeabundancepatternofthegas.
3.2. CollisionswithChargedDust
Dustgrainsimmersedinaplasmaacquireanon-zerocharge(Spitzer1941),aphenomenonobservedintheinterstellarmedium
(ISM)andthesolarsystem(e.g.,Mendis&Rosenberg1994,Horányi1996). We thereforeexpectgrainsintheβPicsystemto
bechargedaswell. Thismightcontributetoslowdowntheionfluidbyenhancingtheion-graincollisionfrequencyrelativeto
8 Fernández,Brandeker,&Wu
100
t / t
S acc
80 1 -2
100 10
-4
10
60
]
U
A
[
z 10-4.5
40
-4.6
10
20
0
100 200 300 400 500
r [AU]
FIG. 3.—Contoursoftheratiobetweentheionslow-downtimescaleandtheradiativeaccelerationtimescale(equations8and14)intheβPicdisk(ρandz
arediskcylindricalcoordinates). Asanexample,weadoptCIIasfieldparticlesandFeIIastestparticleswithinjectionvelocityv=cth,f(ξ=1). Inregionsof
thediskwheremetallicgashasbeenobserved,theionslow-downtimeisordersofmagnitudeshorterthantheradiativeaccelerationtime,indicatingthationsare
dynamicallycoupledtoasinglefluid.Resultsaresimilarwhenweadoptotherelementsasfieldparticles.
thecase wheregrainsareneutral. Themassin dustgrainsisexpectedto belargerthanthatofionizedparticles, satisfyingthe
requirementofhighinertia. Wethereforefocusontherequirementofhighcollisionfrequency,whichdependsonthepotential
towhichgrainsarechargedto.
The relative importance of different grain charging mechanisms is determined by environment conditions (e.g.,
Mendis&Rosenberg 1994). In all cases, an equilibrium electrostatic potential is reached by balancing positive and negative
currents. Given the spectral type of βPic, and the fact that the disk is optically thin, the dominant source of positive charge
is the ejection of photoelectronsby stellar UV radiation, while negative charge is provided by the collection of thermal elec-
trons. The contribution of thermal ions and secondary electrons is negligible. As detailed in Appendix B, we solve for the
equilibriumpotentialof grainsin the βPic disk givenour modelspectrum, temperatureprofile, and electrondensity (§§2.1&
2.2). Due to the exponentialfall-off of photonflux in the UV wavelengths, the potentialdependsweakly (in fact, logarithmi-
cally)onallvariablesexceptfortheworkfunctionW of thegrain(eq. B9). Moredetailedcalculationsyieldthatsilicate dust
(W =8eV, Weingartner&Draine2001) ischargedto a positivepotentialeφ 0.2eV, while carbonaceousgrains(W =4.4eV,
∼
Weingartner&Draine2001)achieveamuchhigherpotentialeφ 2.5eV,bothvaluesbeingfairlyinsensitivetotheenvironment
∼
factors. Highvalueforthelattergroupresultsfromthefactthatstellarphotonswithenergy>4.4eVaremuchmorenumerous
thanthoseabove8eV.Inanycase,bothpotentialsaremuchhigherthan k T 10- 3eV,thetypicalpotentialachievedwhen
B
∼ ∼
onlythermalcollectioncurrentsareincluded. Figure5showsthepotentialofsilicatesandcarbonaceousgrainsnormalizedby
k T,asafunctionofdiskcylindricalcoordinates. Thecontributionoftheejectedphotoelectronstotheambientelectrondensity
B
isnegligible,hencewedidnotcouplegrainchargingwiththeionizationbalancecalculation.
Eachdustgrainisordersofmagnitudemoremassivethanasingleion,thereforegrainscanbeconsideredtobeimmobilein
space. Asshownin §3.1, ionsfeelradiationforceasanensemble. Theymovewith a velocityv relativetothe grains, withan
internal velocity dispersion c (equation 10). In a frame in which the center of mass of ions is at rest, dust grains drift with
th
velocity- v.Thisrelativemotionisdissipatedinadynamicalfrictiontimescale(equation8),withdustgrainsandionsastestand
fieldparticles,respectively. Sincethefrictionalforcefeltbyionsisequalandoppositetothatfeltbydustgrains,thetimescale
forslowingdownionsinthestellarframeisgivenby
̺ m k Tv
t = iont (dust,ion)= ion B , (16)
id ̺ S 8πn Z2 Z2 e4H(ξ)lnΛ
dust dust dust ion
BrakingtheGasinβ Pic 9
100
0.5
80
3
b
eff
4
60
]
U
A
[
z
4.1
40
20 4.5
4.9
0
100 200 300 400 500
r [AU]
FIG. 4.—Contoursofeffective radiationpressurecoefficientβeff (equation15),calculated byaveragingtheradiationforceoverallionizedparticles, asa
functionofdiskcylindricalcoordinatesρandz. Solarcompositionisassumed. Carbon,withβ=0,isimportantforreducingthevalueofβeff. Increasingthe
ionizationasonemovestolargerdistances(bothoutwardandupward)accountsfortheloweringofβeff.However,βeff>0.5forallregionswheremetallicgas
hasbeenobserved,indicatingthatanexternalagentisrequiredtobrakeasolar-compositionionicfluid.
100 100
1200
80 500 200 80 1000 ef /k T
B
ef /k T
100 B 1000
60 60 800
U] U]
A A
z [ z [
40 50 40
600
20 20
25
400
0 0
100 200 300 400 500 100 200 300 400 500
r [AU] r [AU]
FIG. 5.—Equilibriumelectrostaticpotentialofsphericalgrainseφ(seeAppendixBfordetailsonthecalculation)normalizedbykBT,asafunctionofdisk
cylindricalcoordinates.Leftpanelshowssilicates(W=8eV)andrightpanelcarbonaceousgrains(W=4.4eV).Duetostrongphotoelectriccurrents,bothtypes
ofgrainsarechargedtopositivepotentialsmuchgreaterthan∼kBT,thevalueexpectedifonlythermalcollectioncurrentsareresponsibleforcharginggrains.
Resultsshownherearerobustanddependonlylogarithmicallyonelectrondensity,stellarluminosity,andorbitalseparation.
where̺ isthevolumedensityofdustgrains,̺ isthatofions,andthelimitm m wasusedinequation(8). Thegrain
dust ion t f
≫
charge enters equation (16) through Z . The dynamical friction timescale (equation 8) takes only into accountthe effect of
dust
distant encounters, i.e., impact parametersmuch larger than the grain size, under the assumption that the cumulative effect of
these collisions outweights that of close encounters (including impactive ones) in determining the drag force (Spitzer 1956).
Also, this expression does not include collective effects among ions (e.g., Northrop&Birmingham 1990). Both these effects
arenegligibleforthefollowingreasons. First,impactparametersareoforderb Z e2/(k T),thusmostcollisionsarewithin
dust B
∼
10 Fernández,Brandeker,&Wu
v /c
drift th,ion
K
0
v /c
max th,ion
K - H
max
K - H(x)
0 1 2 3
x = v / c
th,ion
FIG. 6.—FunctionK- H(ξ)(equation17),whichdeterminesthesignofthenetforceactingonanion. Arrowsindicatethestableandunstablenatureofthe
equilibriumvelocitiesvdriftandvmax,respectively.ForanygivenvalueofK<Hmax,vmaxisthemaximumvelocityforwhichbrakingcanoccur,whereasvdriftis
thedriftvelocitythationsreachinatimetid(equation16).
therangea b λ , wherea is thegrainsize andλ is theDebyescreeninglength(consequently,we use b =a 1µm
D D min
inequation≪11). ≪Second,wehaveω a/c 10- 3, whereω =(4πn e2/m )1/2 istheelectronplasmafrequency,thusco≃llective
p th p e e
∼
effectsarenotlikelytocontributesignificantlytothedragforce.
Undertheseassumptions,theequationofmotionofarepresentativeionisgivenby
dv GMm m v
m =(β - 1) ion - ion
iondt eff r2 t
id
GMm K- H(ξ)
=(β - 1) ion , (17)
eff r2 K
(cid:20) (cid:21)
where
(β - 1)GMm k T
K= eff ion B . (18)
8πr2n Z2 Z2 e4lnΛ
dust dust ion
ThefunctionH(ξ)isalwayspositive,withasinglemaximumH 0.2atξ 1,andasymptoticlimitsH(ξ) 2ξ/(3π1/2)for
max
≈ ≈ ∼
ξ 1, and H(ξ) 1/(2ξ2) for ξ 1. Therefore, if K >H , no braking is possible, since the net force acting on an ion is
max
≪ ∼ ≫
alwayspositive. Inthiscase, asvincreases, therelativecontributionofthedragforcedecreasesas1/v2, leadingtoa runaway
solution. IfK<H , thereexisttwoequilibriumvelocitiesforwhichthenetforceontheionvanishes,asshowninFigure6.
max
Wedenotethemasv andv (v >v ). Ionswithinitialvelocitiesv <v<v willfeeladragforceuntiltheymove
max drift max drift drift max
with the velocity v relative to the dust, while those with initial v>v will be acceleratedoutward withoutbound. These
drift max
behaviorsreflectthenatureofchargeddustbraking.
We then calculate the value of K for the βPic dust disk, taking m =m , Z =1, and Z e2/a=eφ. The term n a2
ion Fe ion dust dust
is understood to be summed over all grain sizes. As this term also appears in the dust optical cross section, its value can be
obtained from fits to scattered and thermal emission (Artymowiczetal. 1989). In the present work we use an updated fitting
formulaforthedustprofile,whichassumesaconstantgrainalbedo0.5,andisbasedonHSTobservationsofHeapetal.(2000)