Active Carbon and Oxygen Shell Burning Hydrodynamics

0.23 MB·English

by  Casey Meakin

Draftversion February5,2008
PreprinttypesetusingLATEXstyleemulateapjv.11/26/04
ACTIVE CARBON AND OXYGEN SHELL BURNING HYDRODYNAMICS
Casey A. Meakin1 & David Arnett1
Draft versionFebruary 5, 2008
ABSTRACT
We have simulated 2.5×103 seconds of the late evolution of a 23M⊙ star with full hydrodynamic
behavior. We present the first simulations of a multiple-shell burning epoch, including the concur-
rent evolution and interaction of an oxygen and carbon burning shell. In addition, we have evolved
a 3D model of the oxygen burning shell to sufficiently long times (300 seconds) to begin to assess
the adequacy of the 2D approximation. We summarize striking new results: (1) strong interactions
6
occur between active carbon and oxygen burning shells, (2) hydrodynamic wave motions in noncon-
0
0 vective regions,generatedat the convective-radiativeboundaries, are energetically important in both
2 2D and 3D with important consequences for compositional mixing, and (3) a spectrum of mixed p-
andg-modesareunambiguouslyidentifiedwithcorrespondingadiabaticwavesinthesecomputational
n
domains. We find that 2Dconvective motions are exaggeratedrelative to 3D because of vortexinsta-
a
bility in 3D. We discuss the implications for supernova progenitor evolution and symmetry breaking
J
in core collapse.
6
1 Subject headings: hydrodynamics, turbulence, stars: interiors, core collapse

1
v 1. INTRODUCTION 177species versionusedto evolvethe 1Dmodel. The 25
8 Numerical simulations of stellar evolution are gener- nucleus network contains electrons, neutrons, protons,
4 ally based upon restrictive assumptions regarding dy- 4He, 12C, 16O, 20Ne, 23Na, 24Mg, 28Si, 31P, 32S, 34S,
3 35Cl,36Ar,38Ar,39K,40Ca,42Ca,44Ti,46Ti,48Cr,50Cr,
namics(e.g.,hydrostaticbalanceandmixing-lengthcon-
1 52Fe, 54Fe, 56Ni, and all significant strong and weak in-
vection), because the dynamic timescales are so much
0 teraction links. The reaction rates, including 12C(α,γ),
shorter than the nuclear burning timescale. Neutrino
6 are from Rauscher & Thielemann (2000).
cooling acceleratesthe last burning stages so that direct
0 Atwodimensionalmodelhasbeencalculatedona90◦
dynamic simulation is feasible (Bazan & Arnett 1998;
/
h Asida & Arnett 2000), at least for the oxygen burning wedge which is embedded in the equatorial plane of a
p shell. We are extending this workto longerevolutionary spherical coordinate system and has radial limits which
- encompass both the oxygen and carbon burning convec-
times, larger computational domains, and three dimen-
o
tive shells. Table 1 lists some additional details of the
r sional flow (3D). In this letter, we summarize new re-
t sults with a discussion of the hydrodynamics underlying simulated model. A three dimensional model including
s just the oxygen shell and bounding stable layers is par-
a importantsymmetrybreakingandcompositionalmixing
tially evolved at present (300 seconds).
: processes which may significantly affect progenitor and
v
i core-collapse supernova models. A detailed discussion 3. RESULTS
X of these results will appear separately; we indicate the
r wider implications here. FlowTopology withTwoBurningShells. Followingthe
a readjustment of the outer boundary due to small incon-
2. ADOUBLESHELLMODEL:ACTIVEOXYGENAND sistencies in the initial 1D model, a quasi-steady state
CARBONBURNING flow develops, shown in Fig.2. The top half of the figure
shows the velocity magnitude; the lower shows energy
Previously we have evolved a 23M⊙ model with the
generation. Velocities are significant even in the non-
one-dimensional TYCHO code to a point where oxy-
convective regions, but have different morphology. The
gen and carbon are burning in concentric convective
convective regions have round patterns (vortices) with
shells which overlay a silicon-rich core; details may be
occasionalplumes, while the nonconvective regions have
found in (Young & Arnett 2005). The resultant stel-
flattenedpatterns(mostlyg-modes). Theflowfluctuates
lar structure is presented in Figure 1. For subse-
strongly. New fuel is ingested from above; the oxygen
quent hydrodynamic evolution we use PROMPI, a ver-
flame shows “feathery” features corresponding to such
sion of the PROMETHEUS direct Eulerian PPM code
fuel-rich matter flashing as it descends. This was previ-
(Fryxell, Mu¨ller, & Arnett 1989) that has been ported
ouslyseen(Bazan & Arnett1998;Asida & Arnett2000).
to multi-processor computing systems via domain de-
A new feature appears in the movie version of Fig. 2,
composition. Ourmulti-dimensionalcalculationsusethe
which shows a pronounced, low order distortion of the
same physics as the one-dimensional TYCHO code, in-
comoving coordinate, squashing and expanding the ap-
cluding nuclear reactionrates, equationof state, and ra-
parent circles on which carbon burning proceeds. This
diative opacities. We use a 25 nucleus reaction network
is due to the coupling of the two shells by waves in the
in these models, tuned to capture the oxygen and car-
nonconvectiveregionbetweenthem. Thisbehaviorseems
bonburning energy generationratesto within 1%of the
robust; we expect it to persist, so that at core collapse
1StewardObservatory,UniversityofArizona,Tucson,AZ85721 this part of the star (at least) will have significant non-
Electronicaddress: [email protected],[email protected] spherical distortion.
2
Fig. 1.— The 23M⊙ TYCHO model used as initial conditions
forthehydrodynamicsimulationisshownhere. (top)Densityand Fig. 2.— Themagnitude oftheflowvelocity(top) andthenet
temperature profiles are shown illustrating the large number of energy generation rate, ǫnet =ǫnuc+ǫν (bottom), is shown for a
scale heights simulated as well as the complex entropy structure snapshotofthesimulationwhichincludesbothanoxygenandcar-
duetotheburningshells. (middle)Themixinglengthvelocitiesare bonburningconvectionzone. Thecarbonburningconvectiveshell
shownanddelineatetheextentsofthecarbonandoxygenburning extends to a radius of ∼4.5×109 cm whilethe figure is truncated
convection zones. The dashed vertical lines mark the boundaries at a radius of ∼2.9×109 cm for clarity. A weak silicon-burning
of the simulationdomain. (bottom) Theonion skincompositional convection zone develops at the inner edge of the grid due to a
layering of the model is shown here for the four most abundant small boundary-zone entropy error which accumulates during the
species. courseofthecalculation.
∂lnρ dlnρ
TABLE 1 ν2 =−g − , (1)
ModelParameters B (cid:16) ∂r (cid:12)(cid:12)s dr (cid:17)
(cid:12)
whereg isthegravity,andtheterminparenthesesisthe
Quantity Value
difference between the fractional density gradient of the
stellarstructureandthefractionaldensitychangedueto
Stellarmass(M⊙) ...................... 23 a radial (adiabatic) Lagrangian displacement. Regions
Stellarage(yr) ......................... 2.3×106
Oxygenshellconvective timescalea (s)... ∼102 whereνB iszeroareunstabletoconvectivemotions. The
Carbonshellconvective timescalea (s) ... ∼103 spikes in νB in our model are due to steep, stabilizing
Hydrosimulationtime(s) ............... 2.5×103 composition gradients which separate fuel from ash and
Inner,outergridradius(109 cm) ........ 0.3,5.0 lead to sharp gradients in density. Convection excites
Pressurescaleheights acrossdomain .... ∼9 wave motions in the adjacent stable layers which give
Angularextentofgrid(rad) ............. π/2
Gridzoning,nr×nφ×nθ................. 800×320×1 rise to the density perturbations. Similar internal wave
Numeroftimesteps ..................... ∼1.5×106 phenomenacanbe observedinlaboratoryice-watercon-
vection experiments where the largest temperature fluc-
aThese convective timescales are based on the mixing tuationsaremeasuredimmediatelyabovetheconvecting
lengththeoryvelocities. layer where the buoyancy frequency is large (Townsend
1966), highlighting the generality of this phenomenon.
Resonant Modes. The underlying stellar structure de-
termines the set of discrete resonant modes that can be
excited. The narrow stable layers which bound the con-
Stiffness and theSourceof DensityPerturbations. An- vectiveshellsinoursimulation,includingthe(truncated)
other asymmetry, nonspherical density perturbations, corelayer,areisolatedenoughfromotherwavepropaga-
was found by Bazan & Arnett (1998); Asida & Arnett tion regions to act as resonating cavities. These modes
(2000). The fluctuations in density and temperature, are the deeper, interior counterparts to the modes ob-
presented in the top panel of Figure 3 as root mean served in helio and asteroseismology studies of milder
square deviations from an angular mean, reach values evolutionary stages.
as large as ∼10% and are localized at the nonconvective Eachmodecanbe uniquelyidentifiedbyits horizontal
regionjust beyond the convectiveboundary (top panel). wavenumberindex,l,anditsoscillationfrequency,ω. We
The fluctuations are coincident with regions where the identifyexcitedmodesinoursimulationbyisolatingspa-
buoyancy frequency, νB, is large, which can be seen in tial and time components of the motion throughFourier
the bottom panel of Figure 3. Here, ν2 is a measure transforms. In Figure 4 we present a power spectrum at
B
of the “stiffness” ofthe stratification(Turner 1973), and eachradius inthe simulationfor motions with l=4, the
is proportional to the restoring buoyancy force on per- largest horizontal scale that can fit into the 90◦ wedge
turbed stellar matter, simulated. A direct comparison between modes identi-
3
termediate nonconvective region, and 6.9×1046ergs in
the carbon burning shell. The kinetic energy is small in
0.1
the outer stable region, but is still increasing by the end
0.05 of the simulation.
In our simulations, the importance of the excited g-
0 modes in the stable layers lies primarily in the role they
play in mediating mixing at the convective boundaries
-0.05
andin the stably stratified layers. We identify a cycle in
which kinetic energy builds up in the stable layer until
-0.1
the underlying wavemodes reachnon-linear amplitudes,
breakdown and drive mixing. This process is analogous
1
to the physical picture which underlies semiconvective
0.8
mixing(Stevenson1979;Langer et al.1983;Spruit1992)
0.6
but is driven on a hydrodynamic rather than a ther-
0.4 mal timescale. The growth time for the fastest growing
0.2 modes in our simulation is ∼200 seconds and leads to
0 anaveragemigrationspeedofouteroxygenshellbound-
ary of ∼4×104 cm s−1, entraining mass into the con-
vection zone at a rate of ∼10−4 M⊙ s−1, significantly
affecting the evolution. Identifying the spectrum of ex-
Fig. 3.— (top) Nonspherical density and temperature per-
cited modes in numerical simulations, including ampli-
turbations (rootmeansquarefluctuation aboutthe angularmean
dividedbytheangularmean)areshownfortheinner1.2×109 cm tudes and waveforms, provides guidance for developing
of the simulation domain. (bottom) In this propagation diagram and testing a quantitative model of this mixing mecha-
(showing oscillation frequency versus stellar radius) the gravity nism. Animportantparametercontrollingtheboundary
wave and (l=4) acoustic wave propagation zones are indicated by entrainmentrateisthegradientRichardsonnumbersuch
the horizontal and cross-hatched shading, respectively. The grav-
that steeper density (and composition gradients) will
itywavecavityisboundedabovebythebuoyancyfrequency(solid
line),whiletheacousticcavityisboundedbelowbytheLambfre- leadtolowermixingratessothatsharpgradientsareex-
quency(dashedline). pected to form and persist (Peltier 2003; Alexakis et al.
2004). In addition to the mixing associated with wave-
fied in the simulation and those calculated from the lin- breaking,enhancedcompositionaldiffusioncanbedriven
earized (non-radial) wave equation of stellar oscillations by the presence of the oscillatory flow setup by g-modes
(Unno et al. 1989) is presented in the right hand panel (Press & Rybicki1981;Knobloch & Merryfield1992). It
fortwomodeswithsignificantpower. Althoughthesim- has been demonstrated that the structure of presuper-
ulation data has additional features, including “noise” nova iron cores is very sensitive to how mixing is han-
in the convection zones, the mode shapes in both veloc- dled at convective boundaries with significant implica-
itycomponentsarestrikinglysimilarbetweensimulation tions for both the explosion mechanism and nucleosyn-
and the wave equation; the identification is unambigu- thetic yields (e.g. Woosley & Weaver 1988). If the am-
ous. Gravity waves evanesce (exponentially attenuate) plitudesofthewavemotionsidentifiedinoursimulations
beyond the boundaries of the stable layers but still con- remain robust to the numerical limitations (e.g., resolu-
tain significant power in the convection zones. Acoustic tion, domain size) then neglecting the mixing processes
waves are free to propagate in the acoustic cavity which associatedwith these wavesconstitutes a large source of
overlaps the carbon burning convection zone. error in progenitor models.
Duringthelateevolutionaryepochsimulatedhere,the
g-mode and p-mode propagation zones are not widely 4. DISCUSSION
separatedin radius,allowingwavemodes of mixed char- Differences in 2D and 3D.While these 2Dsimulations
acter to couple (Unno et al. 1989). The modes in the allow us to see the interaction of carbon and oxygen
acoustic cavity are trapped by the boundary conditions shells, and show wave generation at convective bound-
of the calculation but would otherwise propagate into aries,they impose an unphysicalsymmetry on the prob-
the stellar envelope where they would deposit their en- lem. Our 3D simulations have only been carried to 300
ergythroughradiativedamping,providinganadditional seconds of stellar time, but show that the wave genera-
channel for energy transport out of the burning region. tionis a robustresult. The flow inthe convectionzones,
The good agreement of the numerical modes with the however, are qualitatively different between the 2D and
analytic modes indicates that our numerical procedures 3D models: the 2D flow is dominated by vortices which
give an excellent representation of the hydrodynamics span the convection zone, while the 3D flow is charac-
of waves, even of very low mach number. We note that terized by smaller scale plumes. Quantitatively, the am-
anelasticcodeswillnotreproducethep-modeandmixed plitude of the 2D convective motions are largerthan the
mode waves properly, as we have ascertained by direct 3Dbyafactorof8,whilethe3Dmotionsarelargerthan
integration of the anelastic wave equation. mixinglengthvaluesbyafactorof1.5. Two-dimensional
Kinetic Energies and Wave Induced Mixing. During simulations of gravitational collapse will be misleading at
the simulations, convective motions excite waves and least tothe extentthat convective motions are important.
build up significant kinetic energy in nonconvective re- Presupernova Models. Perhaps the most important
gions. The integrated kinetic energies are 2.1×1045ergs impact that internal waves have on stellar structure in
in the inner nonconvective region, 5.2×1047ergs in the the late stages of massive star evolution is the degree to
oxygenburningconvectiveshell,9.7×1045ergsinthe in- which they drive compositional mixing. In our simula-
4
Fig. 4.— (left)The frequency power spectrum of the l=4component of the radial velocity (vx) isshown as a function of radius. The
dashed horizontal lines indicate the frequencies of two waves with significant power that are compared to linear theory in the adjacent
panel. (right)Twol=4waveformsareshown. Thewaveformsextractedfromthesimulationdata(dottedline)andcalculatedwithlinear
theory(solidline)areshownforcomparison. Foreachmodethehorizontal(vy)andradialvelocity(vx)componentsarepresentedinunits
ofcms−1. Theshadedareascorrespondtostablystratifiedregions. Bothofthesemodeshavesomep-andg-modecharacter. Themode
inthebottom panelispredominatelyofg-modecharacter,whilethemodeinthetoppanelismoreclearlyofmixedtype.
tionsinternalwavemodesgrowtonon-linearamplitudes The conclusion that internal wave modes do not grow
and mix material at convective boundaries on a hydro- to largeamplitudes during corecollapsethroughnuclear
dynamical timescale. Given the strong dependence of driven overstability (Murphy et al. 2004), is based upon
presupernova structure on the rate at which mixing oc- an analysis that ignores (1) the dynamics of convective
curs at convective boundaries we see the incorporation motionand(2)theshell-shellinteractions,bothofwhich
of internal wave physics into stellar evolution codes as a are expected to become more violent as collapse is ap-
neccesary refinement. proached. Asymmetries in core collapse have implica-
Symmetry Breaking. Spherical symmetry in presuper- tions for pulsar birth kicks, explosion mechanisms, and
nova models is broken by (1) the density perturbations for gravitational wave generation. Talon & Charbonnel
induced by turbulence within the convection zone, (2) (2005)haveshownthatinternalgravitywavescantrans-
the wave interactions between burning shells, and (3) portangularmomentumataratesufficientto beimpor-
rotationally induced distortions. The perturbations by tantinthe evolutionofsolarmassstars;we suggestthat
waveswhicharetrappedbetweentheoxygenandcarbon theyareimportantforevolutionofmoremassivestarsto
burning shells are correlated on large angular scales, as core collapse, and for plausible prediction of the angular
is rotation, while the turbulent perturbations have both momentum distribution in that collapse.
asmallerscaleandamplitude. Ourrestrictedsimulation
domain filters out wave modes with l < 4, so it is likely
that even larger scale perturbations exist in real stars. The authors would like to thank Jeremiah Murphy,
Symmetry breaking will seed instabilities in an outward Christian Ott, Patrick Young, Adam Burrows, Ed Ol-
propagating supernova shock (Kuranz et al. 2005), and szewski, Luc Dessart and Philip Pinto for useful discus-
in the collapsing core. The converging case has been in- sions and comments. We would also like to acknowledge
tensely studied for inertial confinement fusion (Lindl, J. the referee, Raph Hix, for comments that have greatly
1998). The diverging case has implications for the prob- improved the manuscript. This work was supported in
lem of 56Ni and56Co decay in SN1987A (Herant & Benz part by the University of Arizona and by a subcontract
1991; Kifonidis et al. 2003). from the University of Chicago ASCI Flash Center.
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