Turbulent Mixing on Helium-Accreting White Dwarfs

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by  Anthony L. Piro

ACCEPTEDFORPUBLICATIONINTHEASTROPHYSICALJOURNAL
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
TURBULENTMIXINGONHELIUM-ACCRETINGWHITEDWARFS
ANTHONYL.PIRO
CarnegieObservatories,813SantaBarbaraStreet,Pasadena,CA91101,USA;[email protected]
AcceptedforpublicationinTheAstrophysicalJournal
ABSTRACT
AnattractivescenarioforproducingTypeIasupernovae(SNeIa)isadoubledetonation,wheredetonation
5
ofanaccretedheliumlayertriggersignitionofaC/Ocore.Whetherornotsuchamechanismcanexplainsome
1
ormostSNeIadependsonthepropertiesoftheheliumburning,whichinturnissetbythecompositionofthe
0
surfacematerial.Usingacombinationofsemi-analyticandsimplenumericalmodels,Iexplorewhenturbulent
2
mixingdueto hydrodynamicinstabilitiesduringtheaccretionprocesscanmixC/O corematerialupintothe

n accreted helium. Mixing is strongestat high accretionrates, large white dwarf (WD) masses, and slow spin
a rates. The mixing would result in subsequenthelium burningthat better matches the observedpropertiesof
J
SNeIa. Insomecases,thereisconsiderablemixingthatcanleadtomorethan50%C/Ointheaccretedlayer

7 atthetimeofignition. Theseresultswillhopefullymotivatefuturetheoreticalstudiesofsuchstronglymixed
2 conditions. MixingalsohasimplicationsforothertypesofWDsurfaceexplosions,includingtheso-called.Ia
supernovae,thecalcium-richtransients(iftheyarisefromaccretingWDs),andmetal-enrichedclassicalnovae.
]
E Subjectheadings:accretion,accretiondisks—hydrodynamics—whitedwarfs
H
. 1. INTRODUCTION ble detonationscouldexplainthe majorityof normalSNe Ia
h
(Ruiteretal. 2011, 2014). Substantial theoretical work has
p TheuseofTypeIasupernovae(SNeIa)ascosmicdistance
- indicators(e.g.,Riessetal. 1998;Perlmutteretal. 1999) has alsofocusedonthedetailsoftheheliumand/orC/Oburning
o (Finketal.2007,2010;Moll&Woosley2013).
put increased emphasis on the theoretical uncertainties that
r The main issues with the double detonationscenario have
t still remain about these events. It is generally accepted
s been whether the helium ignition is robust (Holcombetal.
that SNe Ia result from the unstable thermonuclear igni-
a 2013)andwhetherthecolorsandspectraduetotheashesleft
[ tion of a degenerate C/O white dwarf (WD), but the spe-
cific progenitor systems have not yet been identified. The over from the surface helium burning are strongly inconsis-

1 two main candidates can roughly be divided into (1) sta- tent with normalSNe Ia (Woosley&Kasen 2011; Simetal.
v ble accretion from a non-degenerate binary companion un- 2012). However,itwasrecentlyarguedthatbothofthesedif-
5 ficultiesmaybeovercomebyifthereisadditionalC/Omate-
til the Chandrasekhar limit is reached (single degenerates,
1 rial in the surface helium layer prior to when the detonation
Whelan&Iben1973),(2)themergeroftwoC/OWDs(dou-
9 begins(Kromeretal.2010;Shen&Moore2014,thelatterof
ble degenerates, Iben&Tutukov 1984; Webbink 1984). An
6 whichalsoexploredamoredetailedtreatmentofthenuclear
important problem is therefore to understand which scenar-
0 network). The main argumentis that the additionalburning
iosexplainthe SNe Ia thatwe observeand whetheranyone
. decreases the hotspot size and mass needed for triggering a
1 channelisdominant.
0 How these systems ignite is also an important outstand- detonation,whichinturnleadstoless iron-peakelementsin
5 ing question. One ignitionscenario thatcould occurin both theheliumlayerthatadverselyaffectthespectra.Inaddition,
1 singleand doubledegeneratesystems isthe “doubledetona- thesiliconandcalciumleftoverfromthesurfaceburningmay
v: tion.” This happens when accretion and detonation of a he- lead to high-velocity spectral features as observed in most
SNeIa(Mazzalietal.2005;Tanakaetal.2008;Blondinetal.
i liumshellona C/O WD leadstoa promptdetonationofthe
X core(Woosley&Weaver 1994;Livne&Arnett1995). Dou- 2012; Childressetal. 2014; Maguireetal. 2014). This C/O
can get created in the surface layer by the convective he-
r bledetonationshaveanumberofattractivefeatureswhichhas
a ledtoincreasedinterestinthisscenarioinrecentyears. Ob- lium burning in the time prior to initiation of the detona-
tion (Shen&Bildsten 2009) or mixed in from the C/O core
servationally,theyareconsistentwithlimitsonshockcooling
by convectivemixing associated with this same stage. Here
of the exploding WD (Piroetal. 2010; Bloometal. 2012),
I consider in more detail whether C/O material can also be
the non-detection of a companion in pre-explosion imaging
mixed into the helium layer via turbulent mixing during the
of nearbySNe Ia (Lietal. 2011), the lack of radio emission
accretionprocess.
(Hancocketal. 2011;Horeshetal. 2012), the lackofhydro-
Additional motivations for this work are the scenarios
genemissioninnebularspectra(Leonard2007;Shappeeetal.
where the C/O core is not successfully ignited by the sur-
2013), a lack of a signature of ejecta interaction with a
face helium detonation (Shen&Bildsten 2014). In such
companion (Kasen 2010; Haydenetal. 2010), the missing
cases, the low mass of the helium would lead to a faint and
companionsinSNeIaremnants(Schaefer&Pagnotta2012),
rapidly evolving transient (Bildstenetal. 2007; Shenetal.
and the sub-Chandrasekhar ejecta masses inferred in many
2010; Waldmanetal. 2011), which may be related to var-
cases (Scalzoetal. 2014). On the theoretical side, the sub-
ious recently observed rapid transients (Peretsetal. 2010;
Chandrasekhar masses of the C/O cores when they ignite
Kasliwaletal. 2010, 2012; Foleyetal. 2013; Inserraetal.
can roughly explain the size of the width-luminosity rela-
2015). Whether or not a helium shell-burning scenario can
tion (Phillips 1993) observed for typical SNe Ia (Simetal.
explain a given observation will depend on the composition
2010) and population synthesis calculations argue that dou-
ofthesurfacelayers,whichagaincanbeaffectedbyturbulent
2
mixing. thecasesconsideredhereandIinsteadfocusonν asaturbu-
Theimpactofspinonhelium-accretingWDshasbeenthe lentdiffusivity.
subjectof previoustheoreticalwork (Yoon&Langer2004a; Equation (3) can be significantly simplified by assuming
Yoonetal.2004). Thesestudiesweremainlyfocusedonthe steady-state transport of angular momentum (also see Piro
shearinstabilitiesforprovidingtheangularmomentumtrans- 2008). This is reasonableas long as the angularmomentum
port, viscous heating, and material mixing. Since I show diffusiontimescaleatagivendepth
below that shear instability is likely subdominant to other
processes, it is worth revisiting these models. Furthermore, tvisc =H2/ν, (4)
newquestionsaboutthedoubledetonationscenarioandnew islessthanthetimescaletoaccretedowntothatsamedepth
classesofopticaltransients(bothdescribedabove)meanthat
itisusefultoaddressmixinginthecontextofthesenewpo- t =4πR2Σ/M˙ , (5)
acc
tentialapplications.
whichis easily satisfied in the WD surfacelayers. The total
In Section 2, I summarize the picture of turbulent mixing
time derivative then simplifies to d/dt ≈ V ∂/∂z, where
and angular momentum transport that I utilize. In Section adv
3, I discuss hydrodynamicmixing mechanisms and analytic Vadv = −M˙ /4πR2ρistheadvectingvelocityofthefluidin
scalingswhichdemonstratehowitdependsonaccretionrate anEulerianframe. Makingthissubstitution,integratingwith
and spin. In Section 4, I comparewith numericalmodelsto respecttoz,andtakingthelimitΩ≪ΩK,thefinalresultis
demonstratethesemainfeatures.IconcludeinSection5with
M˙ Ω
asummaryofmyresultsandadiscussionoffuturework. σ = K , (6)
4πRρν
2. TRANSPORTANDMIXINGFRAMEWORK
whereσ = dΩ/dlnr is the shear rate in the WD. Note that
Before moving into the main calculations, it is helpful to thiscanalsoberewritteninthedimensionlessform
first explain the generalstrategy utilized by this work. Both
σ t R Ω
thetreatmentoftheangularmomentumtransportandthetur- = visc K . (7)
bulent mixing are somewhat different than what is typically Ω (cid:18)tacc(cid:19)(cid:18)H(cid:19)(cid:18) Ω (cid:19)
used, so it is helpfulto explain why this approachis useful.
This shows that a non-zero shear must exist throughout the
Similar methods were used in Piro&Bildsten (2007), and
startoallowthestartospinupandthatitislargerwhenthe
thusreadersinterestedinadditionaldetailsshouldconsultthis
viscosity is small (so that t is big), the accretion rate is
work. visc
large (so that t is small), the thickness H of the layer is
acc
smaller,orwhenthespinΩissmaller.
2.1. AngularMomentumTransport
MaterialaccretingontoaWDatarateM˙ reachesitssurface 2.2. TurbulentMixingFramework
withanearlyKeplerianspinfrequency Traditionally mixing is modeled as a diffusive process
Ω =(GM/R3)1/2 =1.0M1/2R−3/2s−1, (1) whereforsomeelementiwithmassfractionXi,
K 1 8.7
dX 1 ∂ ∂X
whereM1 =M/1M⊙andR8.7 =R/5×108cmarethetyp- dti = r2ρ∂r (cid:18)r2ρD ∂ri(cid:19), (8)
ical WD mass and radiusI will be considering. Most of the
energyofthisincomingmaterialisdissipatedinathinbound- where ρ is the density and D is the diffusion coefficient for
arylayeranddoesnotreachfarintothestar(Piro&Bildsten mixing, which will vary with depth in the star. While this
2004). Nevertheless, this material adds angular momentum is well-suited for implementation in one-dimensionalstellar
atarateof(GMR)1/2M˙ ,andsoatorqueofthismagnitude evolution codes (e.g., Hegeretal. 2000), an alternative way
must be presentto transportthe angularmomentumthrough tothinkaboutthemixingislikeconvection. Justasaregion
theWDandincreaseitsspin. inthestariseitherconvectivelymixedornot,aregioninthe
SincetheseshallowsurfacelayersoftheWDhaveatypical starcanbeturbulentlymixedornot.
pressure scale height H ≪ R, where H = P/ρg and g = ThegeneralpictureIemployissummarizedinFigure1and
GM/R2 isthe surfacegravitationalacceleration,Iassumea described next. After accreting at a rate M˙ for a timescale
plane-parallelgeometryandsetR astheradiusanduse z as t ,thecolumndepthatthebaseoftheaccretionis
acc
the verticalheightcoordinateaboveR. This also allows me
tointroduceausefulvariableinthecolumndepth M˙ tacc
Σ = , (9)
acc 4πR2
dΣ=−ρdz, (2)
asshownbythehorizontaldashedlineinFigure1. Thetur-
tomeasurethedepthintheatmosphere. Intheplane-parallel bulentmixingtimeatthisdepthisestimatedas
limit, the pressure given by hydrostatic equilibrium is sim-
ply P = Σg. In thisgeometry,the transportof angularmo- tmix(Σacc)=H2/D, (10)
mentumisdescribedbyaone-dimensionaldiffusionequation
where the right-hand side is evaluated at Σ . If
(Fujimoto1993) acc
t (Σ ) > t , then the mixing takes too long to act in
mix acc acc
d 1 ∂ ∂Ω comparisonto the accretionand there is minimalmixing. If
R2Ω = ρνR4 , (3) insteadt (Σ )<t ,thenthemixingcancarrymaterial
dt R2ρ∂z (cid:18) ∂z(cid:19) mix acc acc
(cid:0) (cid:1) todepthslargerthantheaccretiondepth.
whereΩisthespinoftheWDandνistheviscosityfortrans- Evaluatingt ateverydepthinthestar,onecanthenfind
mix
porting angular momentum. In principle ν could be some- thedepthwheret = t ,whichisdenotedΣ . Thisis
mix acc mix
thinglikeamoleculardiffusivity,butthisisneverrelevantfor thedepthdowntowherematerialismixed,asshowninlight
3
Material with Y =1 accretes
.
0
at a rate M for a time t
acc
Σ (g cm-2)
Composition
of mixed
Fully mixed
material:
accumulating
material Y = Y Σ /Σ
mix 0 acc mix
.
X = X (1−Σ /Σ )
M
Σ = t 12,mix 12 acc mix
acc 4πR2 acc
X = X (1−Σ /Σ )
16,mix 16 acc mix
Σ = Σ(t =t )
mix mix acc
Unmixed C/O
X = 0.5
12
X = 0.5
16
˙
Figure1. Diagramsummarizingthemainfeaturesofmixingconsideredhere. Materialisaccretedforatimescaletacc atarateM,whichcorrespondstoa
columndepthΣacc=M˙tacc/4πR2.Aslongastmix<taccatΣacc,materialismixeddowntothedepthwheretmix=tacc.Thiscorrespondstoacolumn
ofΣmix > Σacc. Thesurfacelayerbecomescompletelymixedforalldepthsshallowerthanthis(muchlikethroughouttheconvectiveregionofastar),and
thusithasaheliummassfractionofYmix=Y0Σacc/Σmix,whereY0istheheliummassfractioninthefreshlyaccretedmaterial.Carbonandoxygenarealso
mixedupintothesurfacelayerandhavecorrespondingmassfractionsasspecifiedinthediagram.
gray in Figure 1. I assume everything is uniformly mixed like from Eddington-Sweet circulation). I hold off on con-
downtothisdepth,whichisroughlycorrectbecausethemix- sidering magnetohydrodynamicinstabilities for future work.
ing timescale t is generally smaller at shallower depths. ThiswouldlikelybesomethingliketheTayler-Spruitdynamo
mix
Thusifsomethingmixesjustalittleatthebaseofthemixed (Spruit 2002), although this may overestimate the turbulent
region,themixingisfastenoughtomixiteverywhereabove viscosityinsomecases(e.g.,Cantielloetal.2014).
this depth. In a sense, this is analogous to the widely used
assumption of complete mixing in the convectiveregions of 3.1. Kelvin-HelmholtzInstability
stars. Since materialis mixed down to Σ , the mass frac-
mix TheKelvin-Helmholtzinstability(alsoreferredtoasthedy-
tionofheliuminthemixedlayerisdecreasedtobe
namicalshearinstability)isgovernedbytheRichardsonnum-
Y =Y Σ /Σ , (11) ber
mix 0 acc mix
whereY0 istheinitialheliumfractionofthematerialwhenit Ri≡ N2, (14)
isfirstaccreted(thisvariableisintroducedjustforcomplete- σ2
ness, I always use Y = 1). Similarly, the C/O, which have
0 whereN istheBrunt-Va¨isa¨la¨frequency
massfractionsofX andX intheC/Ocore,willbemixed
12 16
upandhavemassfractions g χ dlnT
N2 = T ∇ − , (15)
ad
X12,mix =X12(1−Σacc/Σmix), (12) H χρ (cid:20) (cid:18)dlnP(cid:19)∗(cid:21)
and whereχ = ∂lnP/∂lnQ,withallotherintensivevariables
Q
set constant, ∇ = (∂lnT/∂lnP) is the adiabatic tem-
ad ad
X16,mix =X16(1−Σacc/Σmix). (13) peraturegradient,andthestarreferstoderivativesoftheen-
velope’s profile. This is just the thermal contribution to the
inthemixedlayer.
buoyancy,anditisestimatedtobe(Bildsten1998)
3. TURBULENTTRANSPORTMECHANISMS 1/2
N ≈ 3 g =2.6T−1/2s−1. (16)
To investigate the angular momentum and material trans- (cid:18)20H(cid:19) 8
port discussed above requires setting ν and D, respectively.
ForthepresentworkIfocusontransportmediatedbyhydro- Note that for the scalings presented in this section I focus
dynamic instabilities triggered by accretion at the WD sur- on the thermodynamic properties of the surface layers and
face. Inextsummarizethe Kelvin-Helmholtzinstabilityand omit the scalings with the WD mass, radius, and composi-
the baroclinic instability since these traditionally are most tion for simplicity. These scalings all assume M = 1M⊙,
effective at radial transport (rather than latitudinal transport R = 5 × 108cm, and a helium-rich composition, but I
4
consider changes from these WD properties in the numer- 3.2. BaroclinicInstability
ical investigation later. Linear analysis shows that Kelvin-
The baroclinic instability arises because surfaces of con-
HelmholtzinstabilityoccurswhenRi<1/4,whichdevelops
stant pressure and density no longer coincide if hydrostatic
intostrongturbulencethatreadilytransportsangularmomen-
balance is to be maintained when differential rotation is
tum.
present. In such a configuration, fluid perturbations along
The condition for instability of Ri < 1/4 assumes that
nearlyhorizontaldirectionsareunstable,thoughwithasuffi-
thermal diffusion can be ignored for the unstable fluid per-
cientradialcomponenttoallowmixingofangularmomentum
turbations, in other words, that the perturbations are adia-
andmaterial. Theinstabilitycanroughlybebrokenintotwo
batic. Fluid perturbations with a characteristic size L and
limits,dependingonacriticalbaroclinicRichardsonnumber
speed V become non-adiabatic when the timescale for ther-
(Fujimoto1987),
mal diffusion, L2/K, where K is the thermal diffusivity, is
less than the timescale of the perturbation, L/V. The ratio R 2 Ω 2
of these two timescales is the Pe´clet number, Pe ≡ VL/K RiBC ≡4(cid:18)H(cid:19) (cid:18)N(cid:19) =10.7T8−1Ω20.1, (22)
(Townsend 1958). The restoring force provided by thermal
buoyancyisweakenedwhenPe<1,whichrequiresthesub- whereΩ =Ω/0.1s−1. WhenRi>Ri ,Corioliseffects
0.1 BC
stitutionofN2 → PeN2 andpromotesinstability. Thermal limitthehorizontalscaleofperturbations.Thisresultsintwo
diffusion is most efficient at small lengthscales, which mo- parameterizations for viscosity estimated from linear theory
tivates setting LV/ν = Re (Zahn 1992), where ν is the (Fujimoto1993),
k c k
kinematic viscosity and Re is the critical Reynolds num-
c α 1
ber for turbulence, which is of order 1000. This gives the BC H2Ω, Ri≤Ri ,
Pe´clet number approximately related to the Prandtl number,  3 Ri1/2 BC
ν = (23)
Pr, by Pe ≈ RecPr. The turbulent perturbationsare thus BC α Ri
non-adiabaticwhen(Zahn1992) BC BCH2Ω, Ri>Ri ,
3 Ri3/2 BC
K >νkRec. (17) where I includea dimensionless factor αBC, to account for
uncertaintyinhowlineartheoryrelatestothesaturatedampli-
In the non-degeneratesurface layers the kinematic viscosity
tudesoftheinstability. Ingeneral,Ifindα .1isrequired
isdominatedbyions,andhasavalueof(Spitzer1962) BC
forsignificantmixing,whichIexplaininmoredetailbelow.
ν =1.4×10−2ρ−1T5/2cm2s−1, (18) BysubstitutingνBCintoEquation(6),Isolvefortheshear-
k 5 8 ing profile. Due to the relatively low Ri for WDs, see
BC
where ρ5 ≡ ρ/105 gcm−3, and I assume a Coulomb loga- Equation(22), itisalmostalwaysthecase thatRi > RiBC.
rithmoflnΛ= 20. SettingK =16σ T3/(3c κρ2),where Thus, using the second of the two viscosity prescriptions I
SB p
σ theStefan-Boltzmannconstant,c thespecificheat,and find
SB p
κtheopacity,thethermaldiffusivityis Ri=3.6×104α1/2ρ1/2T1/4Ω3/2M˙ −1/2, (24)
BC 5 8 0.1 −6
K =980κ−0.12ρ−52T83cm2s−1, (19) where M˙−6 = M˙ /10−6M⊙yr−1. This demonstrates di-
rectlythatRi≫Ri foraccretingWDs. Theshearis
where I approximate c = 5k /2µm and scale the opac- BC
p B p
tietyrintogκin0.h2y≡droκg/e0n.-2decfimci2engt−m1,aateprpiarol.pSriuabtestfioturtienlgecEtrqounatsicoants- σBC =1.4×10−2αB−C1/4ρ5−1/4T8−5/8Ω0−.31/4M˙−1/64s−1,
(25)
(18)and(19)intoEquation(17),Ifindthattheperturbations
arenon-adiabaticatdepthsofρ.7×106gcm−3T1/2.The This demonstrates a couple of key features of the shear.
8
new “secular” Richardson number associated with this limit First, it is higherforlargeaccretion rates, which makesper-
is, fect sense because angular momentum is being added more
quicklytothestar. Second,itissmallerforhigherspinrates.
ν Re N2 This is because the baroclinic instability becomes stronger
Ri ≡ k c . (20)
s K σ2 whentheWD is spinningfaster, whichleadsto asmoothing
out of the shearing. Also note that generally σ ≪ σ ,
BC KH
When Ris < 1/4, the so-called “secular shear instability” so that the baroclinic instability triggers before the Kelvin-
arises.
Helmholtzinstability. Thispreventstheshearratefromever
Thecompetingeffectsofaccretionincreasingσversustur- becoming large enough for the Kelvin-Helmholtz instability
bulencedevelopingwhenRis <1/4(anddecreasingσ)drive tooperateatdepthsofρ&100gcm−3,i.e.,atdepthscritical
the surface layers toward marginally satisfying Ris = 1/4 forunstableheliumignition.
(assumingforthemomentthatthesoleviscousmechanismis
the Kelvin-Helmholtzinstability). This expectation is borne 3.3. ViscousHeating
out in the white dwarf studies of Yoon&Langer (2004b). Theshearanddissipationbyviscosityalsoleadstoviscous
Thus I estimate the q due to this mechanism. Substituting heating within the accreted layer. The heating rate per unit
Ris = 1/4 into Equation (20), and assuming Rec = 1000, massis
theshearrateis
ǫ=νσ2/2, (26)
σ =2.0κ1/2ρ1/2T−3/4s−1. (21)
KH 0.2 5 8 whichfortheshearingestimatesgivenaboveresultsin
A shear rate this large is similar to the Keplerian frequency
givenbyEquation(1),butasIshownext,otherfluidinstabil- ǫBC =6.9×102αB−C1/4ρ5−5/4T8−5/8Ω0−.31/4M˙−5/64ergg−1s−1.
itieslimittheshearbeforeitcanreachsuchalargevalue. (27)
5
To put this in context, it is helpful to multiply this result by model, which extendsdown to the depth helium burning ig-
the accretion timescale t to roughly give the total energy nitedunstably.Thisconditionisgivenbytherelation
acc
pernucleonaccreted
dǫ dǫ
3α = cool, (33)
E ≈0.04α−1/4ρ−1/4T3/8Ω−3/4M˙ 1/4keVnucl−1. dT dT
BC BC 5 8 0.1 −6
(28) whereǫ3αisthetriple-αheatingrate(Fushiki&Lamb1987)
and
Incomparison,thethermalenergyatthebaseoftheaccreted
layer is about ≈ 10keVnucl−1. This means that viscous 4σSBT4
ǫ = (34)
heating is usually not important, but can be for especially cool 3κΣ2
low values of αBC and Ω (as can be see in discussions by istheradiativecoolingrate.Bothderivativesaretakenatcon-
Yoon&Langer 2004a, but in the context of shear instabili- stantpressure. I solvefor thetemperatureprofilebysolving
ties). FortheremainderofthisworkIignoreviscousheating theradiativediffusionequation,
soastofocusontheimpactofmixing,butheatingshouldbe
includedinafuller,moreself-consistentcalculation. 16σSBT3dT
F = , (35)
3κ dΣ
3.4. TurbulentMixingEstimates
whereIsetthefluxtoavaluewhichresultsinthecorrectig-
The turbulence that transports angular momentum also nitiondepthforthegivenWDmassandaccretionrate. Since
mixesmaterial,albeitwithmuchlessefficiencybecauseitre- I am ignoring viscous heating, I can use the same tempera-
quiresmoreworktoexchangefluidelementsthantojustexert tureprofilewhetherornotturbulentmixingisincluded. The
stresses. Energy argumentsgive a mixing diffusivity that is opacity is dominated by electron scattering at the relatively
relatedtotheviscosityby(Piro&Bildsten2007) hottemperaturesofthehighaccretionratesIconsider.
Whenmixingisincluded,itisincorporatedasdescribedin
D ≈ν/Ri, (29)
Section 2.2. Summarizing here, I let the WD accrete for a
wherethereisacorrectionoforderunityinthisrelation(often timet . I thensolveforthemixingdepthwithaniterative
acc
referredtoasthefluxRichardsonnumberRf),whichIignore process until the base of the layer correctly satisfies tmix =
for simplicity. I already have one free parameter in αBC to tacc. Once I know the depth of the mixing, I can compare
adjust the strength of the turbulence (although this does not thebase conditionsto Equation(33), makingsure touse the
allowmetovarytheangularmomentumandmaterialmixing correctlyfoundYmix inthestabilitycriterion. Ifdǫ3α/dT <
independently). dǫ /dT,thenthelayerisstableandaccretioncancontinue.
cool
For this mixing diffusivity, I can now evaluate the mixing Oncedǫ /dT = dǫ /dT,thenthelayerunstablyignites,
3α cool
timeasafunctionofdepthusingEquation(10),resultingin andIcanrecordtheamountofmixinginthatgivenmodel.
t =2.1×104α1/4ρ5/4T13/8Ω3/4M˙ −5/4yr. (30) 4.1. ExampleAccumulatingModel
mix BC 5 8 0.1 −6
Tomakethisschememoreconcrete,itishelpfultotocon-
Incomparison,theaccretiontimedowntoasimilardepthis
sideraspecificexampleofthetimeevolutionofanenvelope
t =4πR2Hρ/M˙ =1.9×103ρ T M˙ −1yr. (31) asitaccretestowardunstableignition. Thisexampleisfora
acc 5 8 −6 M = 1M⊙ WDaccretingatarateM˙ = 10−6M⊙yr−1. A
This shows that tmix and tacc are somewhat comparable for reasonableignitionmassforsuchconditionsisM ≈3.0×
ign
rimeapsoorntaabnlte. Fpuarrathmeermteorsr,e,msointicveattimngix/thtaactcm∝ixMin˙g−i1s/4p,ottheentmiailxly- t1h0e−m3Mixi⊙ng(I,bIense&t tTheutuWkDovs1p9i8n9t;oShbeenΩ&=Bil0d.s0te1nΩ2K00a9n)d.Fusoer
inggetsstrongerastheaccretionrateincreases. Conversely, α =0.01.
BC
thescalingswithdensityarguethatmixingisonlyeffectivein Figure2 showsfourdifferentmomentsintime asthe WD
thelayerssufficientlyshallowthattmix <taccorwhen accretesandbuildsasurfacelayerofaccretedmaterialmixed
ρ<5.9α−BC1T8−5/32Ω−0.31M˙−6gcm−3. (32) wofitthimCe/OforfrwomhicthhethWeWDDsuhrfaascbe.eeInnaecaccrehtipnagnteal,ccthisedaemnooutendt
Thisrelativelysmallnumberforρmeansthatboththeviscos- alongwiththetotalmassfractionofC/Omixedintothesur-
ityparametermustbesmallα ∼10−2andthespinmustbe facelayer. Thesolidlineshowsthetemperatureprofiledown
BC
relativelysmallΩ∼10−2Ω formixingtohappen.Thismo- to the base of the layer, with the solid circle denoting the
K
tivatesthevaluesIusefortheseparametersforthenumerical amountofmaterialthatwasactuallyaccreted(muchlikethe
calculationsinthenextsection. Inparticular,thelargeexpo- horizontaldashedline in Figure 1). The dashedcurveis the
nentofΩ−3showsthatthespinplaysanespeciallyimportant stabilitycriteriongivenbyEquation(33). Inthebottomright
role in determiningwhether mixingis occurring. Also, note panelthesurfacetemperatureprofilenowextendsdowntothis
thatthemixingIfindhereisneveraslargeaswhatYoonetal. ignitioncurvedemonstratingthatthismodelwillnowignite.
(2004)calculatewhenfocusingonthesecularshearinstabil- A number of important features and trends caused by the
ity. This means that I also do not find solutions where the mixing,thatweredescribedpreviouslyinmyanalyticexplo-
heliumismixedsufficientlyenoughtopreventunstableshell rationinSection2,arenowmademoreclearbythisnumerical
ignition. example.Theseareasfollows.
1. Mixing is strongest at earlier times and decreases the
4. NUMERICALCALCULATIONS
longer the accretion persists until ignition (as can be
I next explore the mixing in more detail with numerical seen by the mixed C/O mass fraction decreasing with
models.Forpurposesofthepresentstudy,theprofileforthese time). This because it gets more and more difficultto
conditionscan be estimated with a constantfluxatmosphere mixtolargerdepths.
6
Figure2. Foursnapshotsintimeasanaccretedlayergrowstowardunstableignition. Forthisexample,M = 1M⊙,R= 5×108cm,Ω = 0.01ΩK,and
αBC =0.01. EachpanelshowstheenvelopetemperatureT versuscolumndepthΣprofileasasolidline. Thedashedcurvesindicatewhereunstableignition
occurs,whichisfoundusingEquation(33).Thefilledcirclesindicatethedepthtowhichtheenvelopewouldextendifmixingwasnotincluded. Thetotalmass
fractionofC/Omaterialmixedintothesurfaceheliumlayerisdenotedineachpanel.NotethatIdenotethemassfractionvariablesasX12andX16eventhough
theserefertothemixedfraction(e.g.,Ihaveremovedthesubscript“mix”forbrevity).
2. Mixingcausesunstableignitionofthesurfacelayerin
alargermassofmaterialforagivenaccretionratethan
without mixing. In this case the mass of the surface
layeris3.4×10−3M⊙atthemomentofignitionafter
accreting 2.1 ×10−3M⊙ of helium. This is ≈ 13%
moremassinthesurfacelayerthantheunmixedcase.
3. Ignitingdeeperalsodecreasesthetimetoaccreteuntil
ignition, in this case by ≈ 30% from ≈ 3 × 103yrs
downto≈2.1×103yrs.
Most importantly of all, this demonstrates that it is reason-
able for ignition to occur in a fairly mixed environment. In
this case, Y = 0.62 while the mass fraction of C/O is
mix
0.38. This hopefully motivates the investigation of simi-
larlymixedconditionsinfuturesurfaceburningmodels(e.g.,
Shen&Moore2014).
4.2. TrendswithAccretionRateandSpin
Tofurtherillustrate thetrendsof themixing,I explorethe
amountofmixingasafunctionoftheaccretionrateandWD Figure3. SolidlinesarecontoursofconstantmassfractionofC/Ointhe
αspin. =In0F.i0g1uraend3,pIloutsecoMnto=urs1oMf t⊙h,eRma=ss5fr×act1i0o8ncomf,Ca/nOd. mcreixtieodnarcactreeMte˙d.lTayheisrcaaslcaufluanticotniounseosfMthe=acc1reMtio⊙n,WRD=sp5in×Ω10an8dcmthe,aancd-
BC αBC = 0.01. Thesecontoursarespacedevery0.1inmassfractionasla-
Thisshowsthereisquitearangeofmixing,fromnearlyun-
beled. Thedashedcurveshowstheboundaryatwhichtotherightlessthan
mixed on the right side of the plot to fully 90% C/O on the 0.02ofC/Oismixedupintotheaccretedmaterial.Thewigglesinthedashed
leftside. linearenumericalanddonotrepresentarealphysicaleffect.
Thegeneraltrendisthatmoremixingoccursathighaccre-
tion ratesand low WD spin. Theimpactof a highaccretion mixing. Lesscleariswhyslowerspinleadstomoremixing.
is intuitively clear, because if more angular momentum and Thefirstthingtonoteisthatitisnotbecausethereisalarger
materialisaddedmorequicklyitisreasonabletoexpectmore shear between the WD and the accreted material; the total
7
Figure4. ThesameasFigure3,butwithα=0.1. Figure5. ThesameasFigure3,butthecontours areinstead forconstant
ignitionmass. Eachcurveisspacedby2×10−3M⊙,someofwhichare
labeled.
changein spin is basically the same whether Ω = 0.001Ω
K
or0.01Ω . Itisinsteadbecausetheviscosityscaleswiththe
K
spinasshowninEquation(23). AstheWDspinshigher,sur-
facesofconstantpressureanddensitybecomemoreandmore
misaligned,whichinturndrivesthebaroclinicinstabilityand
turbulentviscosity associated with it. A higherviscosity re-
sults in a smaller shear to transport the angular momentum,
asshowninEquation(6),andlessshearleadstolessmixing.
Conversely,forsmallspinandturbulentviscosity,theshearis
largeandthusthemixingwithit. Thisisparticularlystriking
inFigure3,wheretotherightofthedashedlinethereisquite
littlemixing.
Since the analytic estimates showed a strong dependence
onα inEquation(32),IpresentthemixedC/Omassfrac-
BC
tionsforα = 0.1inFigure4. Thisroughlymovesallthe
BC
curvestotheleftandup,showinghowmuchmoredifficultit
istomixwhentheviscosityishigher. Thisissimplybecause
thereisless shearandturbulencein thiscase. For α ∼ 1
BC
thereislittlemixingexceptinthemostextremecircumstances
(of high accretion rate and low spin), and thus it is possible
thatturbulentmixingisnegligibledependingonwhatthereal
turbulentviscosityshouldbe.
Figure6. SimilartoFigure3,butnowmixedC/Omassfractionasafunction
As mentioned in the example calculation above, the mix-
oftheWDmass.
ingalso causesignitiondeeperthanwithoutmixing. Thisis
quantified further in Figure 5, where I plot contoursof con-
ers. Thisweakenedtheturbulentviscosity,leadingtoalarger
stant ignition mass. This shows that at fixed accretion rate,
shearandmoremixing.
the ignitionmass goesup forlower spin (i.e., movingto the
left). At fixed spin, the ignition mass goes down for higher
5. DISCUSSIONANDCONCLUSIONS
accretionratejustasinthenon-mixedcase,butjustnotbyas
much. I have explored mixing in the surface layers of helium-
accretingWDsviahydrodynamicinstabilities. Thesemodels
4.3. TrendswithWhiteDwarfMass highlightanumberofimportanttrendsformixing,whichare
asfollows.
Thestrengthofthemixingshouldalsodependonthemass
andradiusoftheWD.Toexplorethis,Irepeatmyanalysisof 1. MixingisgreatestatlowspinΩ,highaccretionrateM˙ ,
for the amount of mixing which is summarized in Figure 6.
andhighWDmassM.
Thisshowsthatthereisinfactastrongdependenceonmass.
Thiscanbeunderstoodfromtwomainfactors.First,theKep- 2. Mixingisstrongerforlowα ,andforα ∼1mix-
BC BC
lerianspinoftheincomingmaterialislargerwithlargermass, ingisnegligible.
whichmeansalargeramountofangularmomentumneedsto
becarriedthroughthestar. Second,thelargersurfacegravity 3. Mixingcausesunstableignitionofthesurfacelayerto
inturnresultsinasmallerscaleheightH inthesurfacelay- occursoonerandinvolvemoremass.
8
4. Depending on the parameters, ignition of the surface tion,andmaybenaturallystrongerifaccretingWDsarespin-
layer can occur in highly mixed conditions with C/O ningslowly(Livio&Pringle1998)oronONeWDs(Mason
massfractionssometimes&0.5. 2011)becauseoftheirhighermasses.
These results will hopefullymotivate studies that follow the
IthankKenShenforusefulexchangesonthedoubledeto-
evolution of the surface layers after ignition occurs (e.g.,
nationscenarioandsurfaceburningonWDs,andIthankAsh-
Shen&Moore2014),butincludingstrongmixingandlarger
leyRuiterandStuartSimforfeedbackonapreviousdraft. I
ignition depth. In particular, it will be important to under-
also thank Mark Phillips for his generosity in arranging my
standhowthenucleosyntheticproductsaredifferent. Histor-
visittoLasCampanasObservatorywheremuchofthiswork
ically, the double detonation scenario has had trouble with
wascompleted.
a high abundance of iron-peak elements near the surface of
the star which causes the colors of the resulting SN Ia to
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