Convective Properties of Rotating Two-Dimensional Core-Collapse Supernova Progenitors

2.8 MB·

by  Emmanouil Chatzopoulos

SUBMITTEDTOAPJON2016JANUARY21
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
CONVECTIVEPROPERTIESOFROTATINGTWO-DIMENSIONALCORE-COLLAPSESUPERNOVAPROGENITORS
E.CHATZOPOULOS1,9,SEANM.COUCH2,3,4,5,W.DAVIDARNETT6,7,ANDF.X.TIMMES8
1DepartmentofAstronomy&Astrophysics,FlashCenterforComputationalScience,UniversityofChicago,Chicago,IL,60637,USA
2DepartmentofPhysicsandAstronomy,MichiganStateUniversity,EastLansing,MI48824,USA
3DepartmentofComputationalMathematics,Science,andEngineering,MichiganStateUniversity,EastLansing,MI48824,USA
4NationalSuperconductingCyclotronLaboratory,MichiganStateUniversity,EastLansing,MI48824,USA
5JointInstituteforNuclearAstrophysics,MichiganStateUniversity,EastLansing,MI48824,USA
6StewardObservatory,UniversityofArizona,Tucson,AZ85721,USA
7AspenCenterforPhysics,Aspen,CO81611,USA
8SchoolofEarthandSpaceExploration,ArizonaStateUniversity,Tempe,AZ85287,USA
6
SubmittedtoApJon2016January21
1
0
ABSTRACT
2
Weexploretheeffectsofrotationonconvectivecarbon,oxygen,andsiliconshellburningduringthelatestages
n
ofevolutionina20M star.UsingtheModulesforExperimentsinStellarAstrophysics(MESA)weconstruct
a (cid:12)
1Dstellarmodelsbothwithnorotationandwithaninitialrigidrotationof50%ofcritical. Atdifferentpoints
J
during the evolution, we map the 1D models into 2D and follow the multidimensional evolution using the
5
FLASH compressible hydrodynamics code for many convective turnover times until a quasi-steady state is
2
reached. We characterize the strength and scale of convective motions via decomposition of the momentum

densityintovectorsphericalharmonics. Wefindthatrotationinfluencesthetotalpowerinsolenoidalmodes,
]
R with a slightly larger impact for carbon and oxygen shell burning than for silicon shell burning. Including
rotation in one-dimensional (1D) stellar evolution models alters the structure of the star in a manner that has
S
a significant impact on the character of multidimensional convection. Adding modest amounts of rotation to
.
h astellarmodelthatignoresrotationduringtheevolutionarystage,however,haslittleimpactonthecharacter
p ofresultingconvection. Sincethespatialscaleandstrengthofconvectionpresentatthepointofcorecollapse
- directly influence the supernova mechanism, our results suggest that rotation could play an important role in
o
settingthestageformassivestellarexplosions.
r
t Keywords: supernovae: general – hydrodynamics – convection – turbulence –stars: interiors – methods: nu-
s
a merical–stars: massive–stars: evolution
[

2 1. INTRODUCTION fidelity of 1D treatments of mixing. The properties of con-
v vectiveoxygenshell–burningintheprogenitorofSN1987A
The final years in the lives of massive stars are charac-
6 prior to collapse were studied by Baza´n & Arnett (1998) in
terized by vigorous convective shell burning, hydrodynamic
1 2D hydrodynamics simulations. One of the implications of
and convective instabilities and, in many cases, episodic
8 this study was the potential for post-explosion mixing of ra-
mass-lossevents(Meakin&Arnett2007;Quataert&Shiode
5 dioactive 56Ni throughout the SN envelope. The first three-
2012;Shiode&Quataert2014;Smith&Arnett2014;Arnett
0
dimensional (3D) simulations of massive star oxygen shell
. etal.2014)thatchangetheirthree-dimensionalstructureand
1 the initial conditions (ICs) for the core–collapse supernova convection were presented by Meakin (2006) and Meakin &
0 Arnett(2007).Thisworkshowedthattheboundariesbetween
(CCSN) explosion. In addition, rotation and magnetic fields
6 non-convective and convective regions are not stationary as
may further complicate the core-collapse process in a non-
1 standardMLTtheorypredictedbutdynamicalandthesource
linearfashion.
: ofgravitywaves. Arnett&Meakin(2011)followedupwith
v The advanced stages of burning in massive stars have
2DsimulationsofsimultaneouslyactiveC,Ne,OandSiburn-
i long been studied with 1D approximations (Clayton 1984;
X ingshellsfora23M progenitor1hrpriortocore–collapse
Woosleyetal.2002). Forconvection,themixing-lengththe- (cid:12)
finding significant departures from spherical symmetry and
r ory (MLT; Bo¨hm-Vitense 1958) remains the technique used
a strongdynamicalinteractionsbetweenshells. The3Dhydro-
and implemented in stellar evolution codes with options to
dynamics simulations together with developments in math-
choose the associated efficiency parameter, α , and the
MLT ematical methods, indicate that MLT needs significant revi-
condition determining where convection becomes active by
sionespeciallyforlatestagesofstellarevolution(Arnettetal.
using the Schwarzchild or the Ledoux criterion accounting
2015;Gilkis&Soker2015).
forsuppressionduetocompositionalgradients. Parametriza-
Theapparentdeparturesfromsphericalsymmetrythatarise
tionsfor1Dmodelsofconvectiveovershoot,semiconvection,
in the velocity fields of convective shells prior to iron core–
andthermohalinemixingarefrequentlyadopted,asisthe1D
collapse have a qualitative and quantitative impact to the
shellularapproximationtotreatrotation(Zahn1992;Meynet
susceptibility to explosion itself (Couch & Ott 2013, 2015;
&Maeder1997).
Mu¨ller&Janka2015).Couch&Ott(2013,2015)haveshown
Advances in numerical algorithms, hydrodynamic soft-
that successful explosions occur for models whose ICs in-
wareinstruments,andcomputingpowerhaveallowedmulti–
clude velocity field perturbations due to convective burning,
dimensional studies of stellar convection that can assess the
in contrast to the same models without these asphericities.
Thisisduetonon-radialmotionsintheaccretionflowexciting
9EnricoFermiFellow;manolis@flash.uchicago.edu
SUBMITTEDTOAPJON2016JANUARY21 CHATZOPOULOSETAL.
stronger post-shock turbulence that aids shock revival. Full
3D simulations of the final minutes of iron core growth and 1.5 no-rot
collapse also suggest that non-spherical progenitor structure rot-ST
]
should not be ignored (Couch et al. 2015). One might char- un
acterize convection seen in 2D or 3D hydrodynamic simula- Ms 1
[
tions and then map realizations of convective velocity fields e
F
as ICs for simulations of CCSNe. Efforts to do so include M0.5
thesphericalFourier–Besseldecomposition(Ferna´ndezetal.
2014)andnonlocalandtime-dependentaveraging(Meakin&
Arnett 2007; Arnett et al. 2009; Viallet et al. 2013). In our
analysis we use vector spherical harmonic (VSH) decompo-
sition of the momentum density field as presented by Chat-
zopoulosetal.(2014). ]
K
The stochastic nature of 1D convection algorithms has a [ 9.8
profound impact to the outcome of massive stellar evolution T c
itself. For stars of very similar mass and metallicity the end g
o
points can be dramatically different. These give remnant l
9.6
masses and explosion properties which depend strongly on
thepre–SNstellarstructureandexhibitlargevariabilityeven
innarrowintervalsofZeroAgeMainSequence(ZAMS)mass
(Uglianoetal.2012;Sukhbold&Woosley2014).
0.48
Additional uncertainty arises from the dynamical bound-
aries between convective zones (Meakin & Arnett 2007), 0.46
whichareasourceofgravitywavesandmightleadtoepisodic Ye
mass-loss events shortly before the SN explosion (Quataert 0.44
& Shiode 2012; Shiode & Quataert 2014; Smith & Arnett
2014). The circumstellar (CS) environment that is formed 0.42
around pre–SN stars due to this mass–loss history can have
a significant impact on the radiative properties of the result-
ingexplosion.
Rapid rotation has been observed for many massive stars 3] 10
(Hunter et al. 2008; Vink et al. 2010; Dufton et al. 2011; -m
Almeida et al. 2015) and can affect their pre–SN internal c
g 9
structure and composition via instabilities that alter the effi- [
c
ciencyofangularmomentumtransportandtherateofchem- ρ
ical mixing (Brott et al. 2011a,b; Ekstro¨m et al. 2012; see og 8
l
Maeder & Meynet 2012 for a review). The effects of ro-
tation on convection in the extended envelopes of red giant
stars has been studied with 3D simulations by Brun & Pala- -4×104 -2×104 0
cios(2009),whofoundthepropertiesofturbulentconvection Time before collapse [s]
to be sensitive to the rotation rate. Currently, the effects of
Figure1. Fromtoptobottom: evolutionofglobalparametersforthe“no-
rotationontheconvectivepropertiesofmassive,pre–SNstars rot”(solidblackcurves)andthe“rot-ST”(solidredcurves)modelsascal-
and their implications for CCSNe have not been thoroughly culatedinMESA.Ironcoremass(MFe), centraltemperature(Tc), central
investigated. Inthispaperwepresent2Dsimulationsofrotat- Yeandcentraldensity(ρc).Thedashedverticallinesindicatethetimesthat
correspondtotheSi-shellburningmodelsmappedinFLASH.
ingpre–SNstarsduringconvectiveC–andO–shellburning
andSi–shellburning,andusetheVSHmethodtoquantifythe
behaviorofconvectivevelocityflowsasafunctionofinternal
rotationrate.
Thepaperisorganizedasfollows.In§2wepresentthestel- Paxtonetal.2011,2013,2015).Subsequently,MESA1Dpro-
larevolutionand2Dhydrodynamicssimulationsofa20M filesareextracteddayspriortocore–collapse,duringcoreSi–
(cid:12)
pre–SN star during convective shell burning of C, O and Si. andshellC–andO–burning,and∼1hrpriortocollapsedur-
In § 3 we apply the method of VSH to decompose the con- ingSi–shellburning. Theseprofilesarethenmappedinto2D
vective velocity fields and obtain power spectra in order to usingtheadaptivemeshrefinement(AMR)multi-physicshy-
characterize the convective properties of models of different drodynamics code FLASH version 4.3 (Fryxell et al. 2000;
rotationrates. Finally,in§4wediscussourconclusionsand Dubey et al. 2012) including rotation perpendicular to the
implicationsforCCSNe. plane of the simulation (“2.5D” approach). The 2D simula-
tions are run for > 3 convective turn-over timescales, long
2. STELLAREVOLUTIONANDHYDRODYNAMICS enough to diminish the effects of the initial dynamical tran-
SIMULATIONS sient resulting from mapping 1D “convective” profiles to a
Ouranalysisof2Dconvectioninrotationpre–SNprogeni- multi-dimensional hydrodynamic grid. Lastly, FLASH sim-
torshasthreedistinctsteps. First,weevolvemodelsofastar ulation output at three different times is extracted and post-
with ZAMS mass of 20 M , solar metallicity but different processedusingtheVSHmethod(Chatzopoulosetal.2014)
(cid:12)
rotationratesusingversion7503ofthestellarevolutioncode toobtainthepowerspectraoftheconvectivemotionsandthe
Modules for Experiments in Stellar Astrophysics (MESA; energycascade.
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SUBMITTEDTOAPJON2016JANUARY21 CHATZOPOULOSETAL.
2.1. MESApre–SNevolution.
Our MESA models use initial rigid rotation rates of zero 1013
(T“hneo–inroitti”a)larontdat5io0n%porfofithleescrimitipcoalseKreipgliedr–iablovdaylureot(a“triootn–SwTh”e)n. 1-1 s] 11001121 nroot--rSoTt
the model first lands on the ZAMS. The rotating model as- -g 10
sumesthetransportofangularmomentumandchemicalmix- g 10
r 9
ingviatheSpruit–Taylormechanism(ST;Spruit1999,2002). e 10
[
For MLT convection in MESA we adopt the Schwarzchild uc 108
criterion and αMLT = 1.6. We use an automatically ex- εn 107
tending nuclear reaction network starting from a basic 8–
6
10
isotope network and reaching a 21–isotope network by the
end of the run. We note ≈ 100-150 isotopes are required to 0
10
accurately represent core neutronization and neutrino cool-
ing (Arnett & Meakin 2011; Arnett et al. 2015). The 10-1
“Helmholtz”equationofstate(EOS;Timmes&Swesty2000) He
-2
is used. Standard mass–loss prescriptions appropriate for Xi 10 C
massive stars are adopted (Vink et al. 2001; Glebbeek et al. -3 N
10 O
2009). After performing a resolution study, we choose a
Si
spatialresolutionparameter(mesh delta coeffinMESA -4
10
terminology) equal to 0.5 and a temporal resolution factor
(varcontrol target) of 10−3 where good convergence 10-5
((cid:39) 10−2 level) in terms of final carbon-oxygen core mass
and iron core mass is achieved. The chosen grid resolution
resulted in final output models with 1200–1600 Lagrangian
zones.
Figure1showstheevolutionofironcoremass(M )and
Fe 9
centraldensity(ρ ),temperature(T )andaveragechargeper 10
baryon (Y ) for bcoth the “no–rot” cand “rot–ST” models for K] Shell
thelast50e,000secofevolutionandtheformationoftheiron T [ C & O-burning
core. Theverticaldashedlinesshowthestagewhereweex-
tractedtheMESAmodelscorrespondingtoSi–shellburning,
at∼1.3hourspriortocore–collapse, whenthepeakof(cid:15)
nuc
profilereachedthemaximumvalue(3−4×1018ergg−1s−1). 108
Thebuild–upoftheFe–coreuptotheChandrasekharmassis 107 108 109 1010
smoother for the “rot–ST” model predominantly due to the r [cm]
effectsofSTandrotationalmixing.
Figures 2 and 3 show the distributions of nuclear energy Figure2. From top to bottom: radial profiles of nuclear energy genera-
generationrate((cid:15) ), composition(X)andtemperaturefo- tion rate ((cid:15)nuc), composition, and temperature (T) for the “no-rot” (solid
nuc i blackcurves)and“rot-ST”(solidredcurves)MESAmodelsattheC&Oshell
cused in the convective, shell-burning regions. The MESA burningphasepriortomappingtoFLASH.Inthecompositionplotthe“no-
modelswecalculatedonotexhibitaphaseofstronglyevident rot”modelisrepresentedbysolidcurvesandthe“rot-ST”modelbydashed
shellNe–burning,incontrasttothethemoremassive(23M ) curves.Theverticaldashedlinesdenotetheradiallimits,Rin,shandRout,sh
(cid:12) oftheconvectiveshellthatwasdecomposedwiththeVSHmethod.
modelusedinArnett&Meakin(2011). Theverticaldashed
linesindicatetheinnerandouterradialboundaries(R and
in,sh
R )thatwerechosenfortheconvectiveshellstobeana-
out,sh shows the rotational velocity profiles for all models mapped
lyzedwiththeVSHmethod(see§3fordetailsonhowtheir inFLASH.
valuesweredetermined). Theeffectsofenhancedmixingin
the“rot–ST”modelareclearlyillustratedinthecomposition 2.2. 2DFLASHhydrodynamics.
panels.
Thesix1DMESAmodelsimportedintoFLASHarelisted
To further isolate the effects of rotation, we impose an ar-
in Table 1. The C & O–shell burning FLASH models are
tificalrotationalvelocityprofileonthe“no–rot”models:
run for 1000 sec (simulated time), and the Si–shell burning
rΩ models for 500 sec. The dynamical effects of rotation and
v (r)= , (1)
rot 1+(r/A)2 angular momentum conservation are handled by the unsplit
piecewise parabolic method (PPM) hydrodynamic solver in
where r is the radial (spherical) coordinate, Ω is the angular FLASH(Leeetal.2009). Wenotethatthenewestimplemen-
velocity and A the characteristic radius where the rotational tationoftheunsplitsolverinFLASHhandlesspeciesadvec-
velocity peaks as in Chatzopoulos et al. (2013). We denote tion in a way that is nearly identical to consistent multifliud
modelsthatusethisrotationlawas“rot–2”. Wecarefullyse- advection scheme methods (Plewa & Mu¨ller 1999). The 1D
lectedthevaluesofAandΩinordertocaptureafewfullro- rotational velocity profiles were mapped on the 2D grid as-
tations(4–8)fortheconvectiveshellmaterialwithinthesim- suming“shellular”rotationasvectorswithdirectionperpen-
ulatedtimescalesinFLASHandassessthedynamicaleffects diculartotheR–zplaneofthesimulationandtherotationaxis
ofthecentrifugalforce. ForC&O–shellburningwepicked coincidentwiththepolaraxis.Inthisapproximationtheangu-
A=3.46×108cmandΩ=0.1s−1andforSi–shellburning lar velocity, Ω(r), is constant for a particular spherical shell
A = 2.2×108 cm and Ω = 0.1 s−1 respectively. Figure 4 or on equipotential surfaces (Zahn 1992; Meynet & Maeder
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Table1
PropertiesofMESAmodelsmappedinFLASH.
Model Rin,sh(108cm) Rout,sh(108cm) vrot,sh(kms−1) vconv(kms−1) τrot(s) t1(s) t2(s) t3(s)
C&Oshellburning
no–rot 3.46 36.94 0. 136 - 545 749 1000
rot–ST 3.46 36.94 5.5 368 3957 545 749 1000
rot–2 3.46 36.94 185.6 142 126 545 749 1000
Sishellburning
no–rot 1.64 36.00 0. 445 - 300 400 500
rot–ST 1.64 36.00 7.1 362 1505 300 400 500
rot–2 1.64 36.00 110.0 449 138 300 400 500
Note. —“no-rot”: non-rotatingmodel. “rot-ST”:rotatingmodelthatincludesthemagneticfieldeffectsoftheSpruit-Taylor
dynamo. “rot-2”: rotatingmodelproducedbyintroducingarotationalvelocityprofileto“no-rot”uponmappingtoFLASH.The
rotationaltime-scale,τrot,correspondstotherotationalperiod(onefullrevolutionaroundtherotationaxis)forthegivenrotational
speedsinthecenteroftheconvectiveshell.
18
10
] no-rot
1 16
-1-g s 110014 rot-ST rrrooottt---SS2 TT(C ((CS &i )& O O))
erg 1012 100 rot-2 (Si)
[
c 10
u10
n
ε 108 -1m s]
k
v [rot 10
0
10
-1
10
He
-2
Xi 10 C 1
N
-3
10 O 107 108 109 1010
Si r [cm]
-4
10 Fe
-5 Figure4. Rotationalvelocityprofilesforthe“rot-ST”(solidcurves)andthe
10
“rot-2”(dashedcurves)models.BlackcurvesdenotetheC&Oshellburning
phaseandredcurvestheSishellburningphase. Theformoftherotational
profileforthe“rot-2”modelsisgivenbyEquation1.
]
K 9 ployed (Couch et al. 2015) and the “Helmholtz” EOS was
10
T [ Shell used. The main inconsistency is the treatment of convec-
Si-burning tion in transition from 1D to 2D: in 1D the convective en-
ergy transport and cascade is treated via approximate MLT
prescriptions, while in 2D and 3D convective flow naturally
develops in the fluid within unstable regions. The mapping
8
10 from 1D to 2D triggers an initial dynamical transient that
107 108 109 1010 lasts for a ∼ 100–300 sec before a quasi-steady state is re–
r [cm] established. For our VSH analysis presented in § 3 we use
theFLASHoutput(“snapshots”)atthreedifferenttimes,well
Figure3. SameasFigure2butfortheSishellburningphase.
after(>300secforC&O–shellburningand>200secfor
Si–shellburning)theinitialtransienthastransversedthecom-
putational domain. Table 1 also details the properties of the
1997). Morespecifically,eachgridcellwasgivenarotational convectiveshellsandsimulationoutput.
velocityv = Ω(r)RwhereΩ(r) = v (r)/rwithv (r) All 2D FLASH simulations were run on the Texas Ad-
rot rot rot
beingMESA1Drotationalvelocityandrthesphericalradial vancedComputingCenterStampedesupercomputer.Thesize
coordinate(r =(x2+y2)0.5). of all simulation domains was chosen to be 1010 cm includ-
The transition from 1D MESA to 2D FLASH was smooth ingboththecoreandtheconvectiveshellsofallmodels. The
in terms of the important physics involved; a hardcoded 21– maximumresolutionchosenwas9kmcorrespondingtocon-
isotope’networkidenticaltotheoneusedinMESAwasem- vergenceintotalenergyandmassata∼ 10−7 leveloverthe
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SUBMITTEDTOAPJON2016JANUARY21 CHATZOPOULOSETAL.
Speed (x 106 cm/s) Speed (x 106 cm/s) Speed (x 106 cm/s)
13.4 36.9 14.5
7.2 4 15.3 4 7.6 4
3.9 6.4 3.9
2.1 2.6 2.1
1.1 2 1.1 2 1.1 2
m) m) m)
9Distance (x 10 c0 9Distance (x 10 c0 9Distance (x 10 c0
O16 -2 O16 -2 O16 -2
0.81 0.81 0.81
0.57 0.57 0.57
0.40 0.40 0.40
-4 -4 -4
0.28 0.28 0.28
0.20 0.20 0.20
-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4
Distance (x 109 cm) Distance (x 109 cm) Distance (x 109 cm)
Figure5. Velocitymagnitudeand16Omassfractionforthe“no–rot”(leftpanel),“rot–ST”(rightpanel)and“rot–2”(rightpanel)C–&O–shellburningmodels
attheendofthesimulation(t=t3).
Speed (x 106 cm/s) Speed (x 106 cm/s) Speed (x 106 cm/s)
44.6 35.8 44.8
17.6 4 15.0 4 17.7 4
7.0 6.2 7.0
2.7 2.6 2.8
1.1 2 1.1 2 1.1 2
m) m) m)
9Distance (x 10 c0 9Distance (x 10 c0 9Distance (x 10 c0
Si28 -2 Si28 -2 Si28 -2
0.48 0.50 0.48
0.18 0.18 0.18
0.07 0.07 0.07
-4 -4 -4
0.03 0.03 0.03
0.01 0.01 0.01
-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4
Distance (x 109 cm) Distance (x 109 cm) Distance (x 109 cm)
Figure6. Velocitymagnitudeand28Simassfractionforthe“no–rot”(leftpanel),“rot–ST”(rightpanel)and“rot–2”(rightpanel)Si–shellburningmodelsat
theendofthesimulation(t=t3).
course of the simulation. At that resolution we are able to smallerstructuresandhighvelocityvorticesinteractingwith
resolve Eddy sizes covering a considerable range of the tur- eachother. Mildmixingisalsoseen,withinstabilitiesdevel-
bulentenergycascade(from∼30,000kmdownto∼10km). oping at the interfaces of the convective shells. This mixing
Weshouldcaution,however,thattheturbulentenergycascade can be due to the process of turbulent entrainment also seen
in2Disinherentlyinverted(Porter&Woodward1994;Taki- byMeakin&Arnett(2007). Inallsimulations,convectiveel-
waki et al. 2014; Couch & Ott 2015) and full 3D treatment ementsinteractwiththeinner(core)boundaryandbreakinto
isrequiredtoaccuretlyreproduceit. Themorphologyofthe smallerstructuresthatthensubsequentlyreunitewhilerising
flow changes significantly from 2D to 3D, and the velocity upwards. Artificalflowsareseenneartheaxisofthesimula-
scale is moderately higher in 2D (Meakin & Arnett 2007). tion(∼6deg),acommonissueof2Dcylindricaltreatment.
We consider our simulations as an initial, exploratory step.
Eachsimulationwasrunon128coresandthewallclocktime 3. VSHDECOMPOSITIONOFCONVECTIVESHELLS
rangedfrom14to22hoursforatotalof>12,000core–hours
Output from the FLASH simulation is taken at three in-
used. Figures5and6showtheO16orSi28massfractionand
stances,t ,t andt (theendofthesimulation);allaregiven
the2D(x and y)velocity magnitudeattheend ofeachsim- 1 2 3
in Table 1. A total of 18 snapshots for all 6 cases, are post–
ulation. Theprevalenceofconvectionisapparentinallcases
processed using the VSH analysis implemented in the code.
withcharacteristicvelocitiesreaching∼150kms−1 forC&
These time–scales are chosen to represent different evolu-
O–shellburningand∼450kms−1forSi–shellburninginthe
tion phases during shell convection, well after the initial dy-
“no–rot”and“rot–2”cases.Itisnoteworthythatbytheendof
namic transient and several convective turnover time–scales
allsimulationsthevelocitiesinthe“rot–ST”casesweremuch
afterthat((cid:39)10-100forC&O–shellburningand(cid:39)10-50for
higherforC&O–shellburning(∼370kms−1)butslightly
Si–shellburning).
lowerforSi–shellburning(∼360kms−1).
ThegoaloftheVSHanalysisistodecomposethemomen-
The convective elements seen in the velocity magnitude
tum density field within the selected shell into radial (A )
panels span a range of sizes, with dominant large–scale mo- nlm
andsolenoidal(B andC )modesandthencalculatethe
tions that cover more than half of the size of the convective nlm nlm
powerspectrumforeachofthemodestodeterminetheglobal
shells,aswellassmallerscalevorticeswhicharejustvisible
propertiesof2Dor3Dfluidmotion. Inourcaseweanalyze
(for example in the “rot–ST” C & O–burning shell). Con-
a2Dmomentumdensityfield,thereforethesolenoidalB
vection is established ∼ 200-300 sec after the start of each nlm
modesareirrelevantbecausetheycanceloutandwillnotbe
simulation, after the initial transient exits the computational
discussed further. Also, because our simulations were con-
domain. The evolution past 200 sec shows large scale con-
ducted in a full domain possessing reflection symmetry, we
vectivecurrentsthatcovertheentireshell, breakingdownto
expectthepresenceofanodd-eveneffectinthepowerspec-
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SUBMITTEDTOAPJON2016JANUARY21 CHATZOPOULOSETAL.
tra, a feature inherent to the 2D treatment. The first step is addition of rotation in otherwise equal stellar structure has
todeterminetheradiallimitsoftheconvectiveregionswhere little effect on the global properties of convection. A small
VSHisapplied.Forthis,volume–weightedradialmomentum reduction of total power is apparent, possibly related to the
densityprofilesarecalculated(Equation22ofChatzopoulos effectsofcentrifugalforcesandtheexistenceofanextrade-
etal.2014). Locationswheretheradialcomponentreachesa gree of freedom (movement perpendicular to the simulation
minimumareusedasourfinalchoiceforR andR ; domain). Ontheotherhand,thealgorithmforcalculatingthe
in,sh out,sh
thesearealsopresentedinTable1. effects of rotation during stellar evolution leads to a pre–SN
The next step is to declare the maximum radial and angu- star with clearly stronger convection during the C&O–shell
larresolutionfortheVSHcomponents,bothofwhichcanbe burning phase (“rot–ST” model, upper middle panel in Fig-
expressed as a length scale, λ . The chosen resolution scale ure 9). Indeed, the final peak 2D-velocity magnitude in the
r
then determines the number of radial (n) and angular (l and “rot–ST” model is more than double those of the “no–rot”
m)modesrequiredfortheexpansionintheVSHcomponents, and“rot–2”models. Wediscussthisinmoredetailin§4.
whicharegivenbythefollowingformulae:
3.2. ShellSi–burning.
n = 2(R2−R1), (2) The VSH power spectra for the cases of shell Si–burning
max λ are shown in Figure 8. The evolution of the total power is
r
showninthebottomrowofFigure9. Thereducedspectrare-
and
vealthatlargescalesalsodominateoversmallerscalesduring
π(R +R )
l = 1 2 , (3) shell Si–burning, with l peaking in the range 4–6 through-
max 2λ
r out the evolution. The corresponding angular length scales
where the total number of modes in 2D momentum density rangefrom1.5×109 cmto2.0×109 cm. Theradialscales
fielddecompositionis arealsointhesamerangeimplyingnearlycircularshapefor
the convective elements, which are at about half the size of
N =(n +1)(l +1). (4)
total max max theconvectiveshell. Secondarypeaksoccuratsmallerscales
For the C & O–shell burning models we choose λ = throughouttheVSHspectralevolution(∼109cm). Asinthe
r
1.585×108cm,correspondingtol =40andn =42, caseofC&O–shellburning,thespectralslopesremainconsis-
max max
while for the Si–shell burning models we choose λ = tentovertimeandsolenoidalmodelsdominateradialmodels
r
1.495×108 cm, l = 40 and n = 46. We choose to byafactorof∼10,000forl<5.
max max
truncateradialn >20modesbecausewefindthattheircon- The evolution of the total power in the solenoidal (C)
tributiontothetotalpowerisminimal(10−5−10−4level)and modesshowsthatconvectionduringSi–shellburningisabout
we can thus reduce computation time. With these choices, a 10timesstrongerthanconvectionduringC&O–shellburning,
totalof861modeswerecaclulatedforeachofthe18FLASH asexpectedfromthehigherratesoflocalenergygeneration.
snapshots. For the purposes of our study, we calculate re- Thetotalpowerdoesnotseemtovarysignificantlyovertime
duced VSH spectra by firstly summing over all the “phase” forallrotationrates. Asmallreductionoftotalpowerinthe
(m)components. Then, wecalculatethereducedangular(l) C–modes is seen by the end of the simulations for the “rot–
andradialnpowerspectrabysummingovereithernorl,re- ST”and“rot–2”cases. Formodelsinitiallyidentical,modulo
spectively, for instance α(cid:48) ≡ (cid:80) α , α(cid:48) ≡ (cid:80) α , and theinclusionofrotation(“no–rot”versus“rot–2”),thiseffect
l n nl n l nl
similarly for the solenoidal modes. In our presentation of maybeduetothedynamicalimpactofcentrifugalforcesand
VSH power spectra later, we will simply refer to the radial theextradegreeoffreedom,asarguedforthecaseofC&O–
andsolenoidalmodesasAandC,respectively(fortheexact shellconvection. Wedo,howeverobserveaqualitativediffer-
definitionsofAandCconsultEquations6-8ofChatzopoulos encebetweenthetwostagesofshellburning: duringSi–shell
etal.2014). burning,the“rot–ST”modelexhibitsnearlyidenticaland,at
latetimes,somewhatlowerconvectivepowerascomparedto
3.1. ShellO–andC–burning.
the“no–rot”model,whiletheoppositebehaviorwasobserved
Figure7showsthereducedVSHpowerspectraforallmod- forC&O–shellburning. Wereturntothisissueinthefollow-
elsundergoingC&O–shellburning.Theevolutionofthetotal ingsection.
power,summedoverallcomponents,isshowninthetoprow
4. DISCUSSIONANDCONCLUSIONS
ofFigure9. Forallcasesthebulkofconvectivepoweriscon-
centratedinlargescales(l<10,n<5),withthepeakvalues In this paper we explored the effects rotation on the con-
implyingangularscalesof∼1.6−3.2×109 cmandradial vective properties of a 20 M pre–SN progenitor star, more
(cid:12)
scales of approximately the same range, revealing a nearly specifically during C&O–shell burning and Si–shell burning
circularcharacteristicshapefortheconvectiveeddies. These inthehourstomonthspriortocore–collapse. Westudiedro-
eddiescanbecomparableinsizetotheshellitself. Forsome tationbothbyincludingitself–consistentlyduringtheevolu-
spectra,asecondarypeakofpowerisobservedatsmallerval- tionofthestarbutalsobyimposingittoanothewiseidentical,
uesofl(∼15-20),indicatingthatasmallfractionofthetotal non–rotating model in order to better isolate its effects. We
powerispossessedbysmallerscales(∼ 4.3×108 cm). The run 2D FLASH simulations with the rotational velocity field
overallslopeofthespectraremainsnearlyfixedovertime.As mapped and pointing inwards perpendicular to plane of the
expectedforaconvectivevelocityfieldconfinedinashell,the simulationdomain. Weemployedvectorsphericalharmonic
power in the solenoidal modes is clearly dominant over that decompositionofthemomentumdensityfieldtocharacterize
intheradialmodes. convectionatdifferenttimesduringthehydrodynamicsimu-
Regardlessofthedegreeofrotation,thetotalpowerindom- lations.Ourinitialexplorationoftheinfluenceofrotationon
inantsolenoidalmodesdeclinesoverthecourseofthesimu- thepropertiesofconvectiondeepinthecoreofamassivestar
lations as the convective energy cascade settles. A compar- can(andshould)beimprovedbyusing3Dsimulations,where
ison between the “no–rot” and “rot–2” case shows that the theturbulentcascadegoesfromlargescalestosmallscales.
6
SUBMITTEDTOAPJON2016JANUARY21 CHATZOPOULOSETAL.
C & O burning (no-rot)
t t t
1 2 3
1050
-2s]1049
-1m 1048
c1047
2er [g 11004465 AC
ow1044
P 1043
5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35
l l l
2]1050
-s
-1m 1049
2 c1048
g
er [1047
w
Po1046
5 10 15 5 10 15 5 10 15
n n n
C & O burning (rot-ST)
t t t
1 2 3
2]
-s
-1m 1050
c
2
g
er [1049
w
o
P
1048
5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35
l l l
2]
-s
1
-m
c
2
er [g1050
w
o
P
5 10 15 5 10 15 5 10 15
n n n
C & O burning (rot-2)
t t t
1 2 3
2]
1- s1050
-m
c
2g 1049
er [
ow1048
P
5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35
l l l
2]
1- s1050
-m
c
2
g
er [1049
w
o
P
5 10 15 5 10 15 5 10 15
n n n
Figure7. EvolutionofreducedVSHpowerspectrainlandnforthe“no–rot”(upperpanel),“rot–ST”(middlepanel)and“rot–2”(lowerpanel)C&O–shell
burningmodels.A(irrotational)andC(solenoidal)modesareshownwithblac7kandredcurvesrespectively.SincetheA<<Calways,weshowthepowerin
theirrotationalmodesonlyforthe“no–rot”case.Thecyancurvesshowtime–averagedCspectraforthethreesnapshots.
SUBMITTEDTOAPJON2016JANUARY21 CHATZOPOULOSETAL.
Si burning (no-rot)
t t t
1 2 3
-2s]11005501
-1m 1049
c1048
2g 1047
wer [11004465 AC
o1044
P 1043
5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35
l l l
2] 1051
1- s1050
-m
2 c1049
g
er [1048
w
o1047
P
5 10 15 5 10 15 5 10 15
n n n
Si burning (rot-ST)
t t t
1 2 3
-2s] 1051
-1m 1050
c
2 1049
g
er [1048
w
o1047
P
5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35
l l l
2]
-s
1
-m 1051
c
2
g
wer [1050
o
P
5 10 15 5 10 15 5 10 15
n n n
Si burning (rot-2)
t t t
1 2 3
1052
2]
1- s 1051
-cm1050
2g 1049
wer [1048
Po1047
5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35
l l l
2]
-s
1
-m
c 1051
2
g
er [
w
Po1050
5 10 15 5 10 15 5 10 15
n n n
Figure8. SameasFigure7butfortheSi–shellburningmodels.
8
SUBMITTEDTOAPJON2016JANUARY21 CHATZOPOULOSETAL.
no-rot rot-ST rot-2
2] 1052 1052 1052 A
-s C
-1m 1051 1051 1051 ning
2[g c 1050 1050 1050 O bur
er 1049 1049 1049 &
w C
o 1048 1048 1048
P
600 700 800 9001000 600 700 800 9001000 600 700 800 9001000
] 52 52 52
2 10 10 10
-s
-1m 1051 1051 1051 ng
c ni
2 1050 1050 1050 ur
g b
er [ 1049 1049 1049 Si
w
o 1048 1048 1048
P
300 350 400 450 500 300 350 400 450 500 300 350 400 450 500
t [s] t [s] t [s]
Figure9. EvolutionoftotalVSHpowerinirrotationalmodes(A;blackcurves)andsolenoidalmodes(C;redcurves)foralltheprogenitormodelsofTable1.
TheC&OshellburningmodelsareshownintheupperpanelsandtheSishellburningmodelsinthelowerpanels.
Wefindthat,regardlessofthedegreeofrotation,thechar- whichmayhaveasmallimpactonthesusceptibilitytoasuc-
acteristic size of the convective elements is somewhat larger cessfulSNexplosionfollowingironcorecollapse(Couch&
duringtheC&O–shellburningphasecomparedtotheSi–shell Ott2013).
burning phase, while the characteristic convective velocities We emphasize that, to answer the question of whether the
aremorethantwiceaslargeduringSi–shellburningascom- inclusionofrotationsignificantlychangestheICstothecore–
pared to C&O–shell burning. The dominant scales implied collapseSNmechanism,self–consistentevolutionwiththeef-
by the resulting VSH power spectra span more than 50% of fectsofrotationincludedistheproperapproachtotake.Inour
thesizeoftheconvectiveshells,withsecondarytypicalscales analysiswehavefollowedtheevolutionofa20M SNpro-
(cid:12)
as small as 0.5−1×109 cm. The slope of the VSH power genitor with both the effects of rotation and magnetic fields
spectra stays consistent over time regardless of the presence includedinthetransportofangularmomentumandchemical
or absence of rotation. In all cases we find nearly circular mixingusingtheMESAcode(“rot–ST”models).Wefinddif-
shapesforthecharacteristicconvectiveelements. Intermsof ferences in the sign of the effect depending on the nature of
the 2D convective flow properties, our results are in agree- convectiveshell–burning: duringC&O–shellburningthereis
mentwiththoseofMeakin&Arnett(2007)(seealsoMeakin more power stored in the solenoidal components than in the
(2015); Cristinietal.(2015)). Inparticular,wefindahighly case of no rotation while during Si–shell burning the effects
intermittentflowandmixingduetoturbulententrainment. are very small with hints of even reduction of the solenoidal
Intermsoftheeffectsofrotationonthestrengthandprop- modepowerbytheendofthesimulation.Weattributethisef-
ertiesofconvectionalonewefindour2Dsimulationssuggest fecttodifferencesintheinitialMESAmodelsforthetwodis-
minimalimpact,andgenerallyleadtoanoverallsmallreduc- tinctshell–burningstages. Acarefullookintheupperpanel
tionofthetotalconvectivepowerstoredinsolenoidalmotions ofFigure2showsthatthe“rot–ST”(cid:15) profileduringC&O–
nuc
(“no–rot” versus “rot–2” models) regardless of the nature of shellburninghasasecondarypeak(duetoO–burning)thatis
convectiveburning(C,O,orSishell). Wesuggestthisisdue nearly an order of magnitude greater than the corresponding
to the effects of the centrifugal forces mildly expanding the oneforthe“no–rot”model. Onthecontrary, inFigure3we
star, andthuschangingthelocationsoftheconvectiveshells see that during Si–shell burning the peaks in the (cid:15) pro-
nuc
overtime,subjectingtheshellmaterialtolowertemperatures file for the “no–rot” model in the region 4 − 6 × 108 cm
that, in turn, trigger lower nuclear burning rates and weaker aregreaterthanthecorrespondingonesfor“rot–ST”.Thisis
convection. Thatalonemay leadtoinitial velocityperturba- due to the enhanced chemical mixing by the ST mechanism
tions of smaller amplitude than in the case of zero rotation duringtheC&O–shellburningphasethateffectivelyrecycles
9
SUBMITTEDTOAPJON2016JANUARY21 CHATZOPOULOSETAL.
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