RADIAL STELLAR PULSATION AND THREE-DIMENSIONAL CONVECTION. III. COMPARISON OF TWO-DIMENSIONAL AND THREE-DIMENSIONAL CONVECTION EFFECTS ON RADIAL PULSATION 4 1 Christopher M. Geroux1 and Robert G. Deupree 0 Institute for Computational Astrophysics and Department of Astronomy and Physics, Saint Mary’s 2 University, Halifax, NS B3H 3C3 Canada n [email protected] a J 5 1 ABSTRACT ] R We have developed a multidimensional radiation hydrodynamics code to simulate the inter- S action of radial stellar pulsation and convection for full amplitude pulsating models. Convection . h is computed using large eddy simulations. Here we perform three-dimensional simulations of RR p Lyrae stars for comparison with previously reported two-dimensional simulations. We find that - o the time dependent behavior of the peak convective flux on pulsation phase is very similar in r both the two-dimensional and three-dimensional calculations. The growth rates of the pulsation t s in the two-dimensional calculations are about 0.1% higher than in the three-dimensional calcu- a lations.The amplitude of the light curve for a 6500 K RR Lyrae model is essentially the same for [ our 2D and 3D calculations, as is the rising light curve. There are differences in slope at various 1 times during falling light. v 2 Subject headings: convection — hydrodynamics — methods: numerical — stars: oscillations — stars: 4 variables: general — stars: variables: RR Lyrae 6 3 1. INTRODUCTION regions for models near the red edge would be . 1 important (Christy 1966b; Cox et al. 1966b), but 0 RR Lyrae and classical Cepheid variables have neither models in which mixing length convection 4 long played an important role as standard can- was frozen in at static model levels (e.g. Tuggle & 1 dles in the development of our understanding of Iben1973)normodelsinwhichconvectioninstan- : v the structure and evolution of our galaxy and taneously adjusted to the static flux for the cur- i of nearby galaxies. Their importance has led to X rent state variables (Cox et al. 1966b) predicted a long history of trying to model the pulsation a return to stability at the red edge. This led to r a of these variables (e.g. Christy 1964, 1966a; Cox the development of more sophisticated time de- et al. 1966a; Stellingwerf 1975; Bono & Stellingw- pendentmixinglengthapproaches(e.g.Stellingw- erf1994)withone-dimensionalhydrodynamicsim- erf1982a,b,1984a,b,c;Kuhfuss1986;Xiong1989), ulations. These were successful in computing a and one dimensional hydrodynamic simulations number of full amplitude light curves that agreed using convective models such as these were able insomedetailwiththoseobserved,atleastaslong to compute a red edge (e.g. Bono & Stellingw- as the models were not chosen too close to the erf 1994; Gehmeyr 1992a,b, 1993). Further cal- red edge of the instability strip. It was specu- culations (e.g. Bono et al. 1997a,b; Marconi et al. lated early on that convection in the ionization 2003;Marconi&Degl’Innocenti2007)wereableto produce full amplitude light curves of RR Lyrae 1Now at Physics and Astronomy, University of Exeter, variables which somewhat resemble what is ob- StockerRoad,Exeter,UKEX44QL 1 served, although the agreement between the ob- spherical shell. This is a comparatively simple served and computed light curves for low ampli- version of techniques where the computational tude RR Lyrae variables near the red edge re- meshisallowedtomoveaccordingtocertainrules mains relatively poor (Marconi & Degl’Innocenti (e.g. Gehmeyr 1992a; Dorfi & Feuchtinger 1991; 2007). The general conclusion appears to be that Feuchtinger&Dorfi1996). Thehorizontalmotion the treatment of convection in pulsating stars re- is determined by the conservation laws, and mass mains unsatisfactory (e.g. Buchler 2009; Marconi can flow into and out of a spherical shell; there 2009) as evidenced by the relatively poor agree- just cannot be any net mass flow out of the shell. ment between the observed and computed light This has allowed the calculation of full amplitude curves near the red edge of the RR Lyrae gap. pulsationmodelsin2D(Geroux&Deupree2013). The problems with the local mixing length ap- The primary results of these calculations are that proach, includingthenecessityofassumingvalues the light curves resemble those produced by one- of free parameters which can significantly change dimensional codes for models not close to the red the solution, led Deupree (1977a) to approach the edge, although the amplitudes tend to be some- problemoftheinteractionbetweenconvectionand what lower, while models near the red edge agree radial pulsation in RR Lyrae variables in a differ- muchbetterwiththeobservedlightcurvesthando ent way. He performed two-dimensional (2D) hy- those of Marconi & Degl’Innocenti (2007). These drodynamicsimulationsfollowingthelargesteddy 2D calculations did not produce a red edge, how- developed in two dimensions by the convective in- ever,becausetheconvectiveregionwantedtogrow stability in the hydrogen ionization zone. This into the regions well below the ionization regions allowed him to determine the red edge location asthemodelsbecamecoolerandthepulsationam- (Deupree 1977b) and to show that the first over- plitude became larger. This significantly changed tone red edge was to the blue of the fundamental the structure and potential energy and thus ther- red edge (Deupree 1977c). The reason for the red mal energy content in regions of the model just edge is that convection essentially allows energy interior to the ionization regions. The thermal transport out of the hydrogen ionization region relaxation time of these deeper regions is suffi- when pulsational instability needs to store it and cientlylongthatitisimpracticaltofollowtheevo- stops transporting energy near maximum velocity lution to a full amplitude solution corresponding when it needs to be released to drive the pulsa- to the newer structure with an explicit hydrody- tion. However, he was not able to compute full namic calculation. amplitude models because the algorithm he used We appreciate that convection is not a 2D to determine the radial flow of his mesh (by forc- phenomenon. The argument made by Deupree ing the mesh to move at the horizontal average (1977a) and by Geroux & Deupree (2013) is that radial velocity at each radial mesh point) could thetimedependenceofconvectionispossiblymore notkeeptheverynarrowhydrogenionizationzone important in determining the pulsation behav- resolved in the mesh for more than about twenty ior than the details of the convective flow. This periods. There have been some other 2D calcula- is clearly an assumption, and we are now in a tions undertaken recently to study the interaction position to examine this by the computation of of convection and pulsation (Mundprecht et al. three-dimensional (3D) convection and pulsation 2012; Gastine & Dintrans 2011), but these have forcomparisonwiththeGeroux&Deupree(2013) not yet led to a comparison of full amplitude so- 2D results. The physics and model input in this lutions with observations. paper are the same as for the 2D calculations. The problem with the mesh propagation in The calculations are made with the OPAL opaci- multi-dimensional calculations has been success- ties (Iglesias & Rogers 1996) in conjunction with fully solved by Geroux & Deupree (2011), who thelowtemperature(Alexander&Ferguson1994) devised a radial mesh flow algorithm in which opacities. Radiation is treated with the diffusion the mass in a given spherical shell does not vary approximation everywhere. The OPAL equation during the course of the calculation. This does of state (Rogers et al. 1996) is used throughout. not mean the calculation is Lagrangian; it merely Convection is treated as a large eddy 2D or 3D means that there is no net mass flow out of a flowsimulationdependingonthecalculation,with 2 a subgrid scale eddy viscosity approach to mimic vective cells and good resolution. The 6◦ simula- the effects of the small scale convective flow that tion was the smallest angular coverage for which cannot be resolved in the mesh. The equations we found more than one distinct convective cell. and more details are given by Geroux & Deupree We have performed short 3D simulations with the (2013). Eachcalculationinthispaperuses16pro- number of θ and φ zones of 5×5 up to zonings cessors. of 40×40. Simulations with the largest number In this paper we will compare 2D and 3D mod- of angular zones had very large computational re- els both during the pulsational growth for several quirementsandtheamountofcomputationaltime models and at full amplitude for one specific cal- required to reach full amplitude would have been culation. As one can imagine, the 3D calculations prohibitivelylong. Asacompromisewechose20θ arequitetimeconsuming,anditwillbeafewmore and 20 φ zones, zoning which is still quite compu- months before all models are complete to full am- tationally demanding (these calculations require plitude. In the next section we compare the 2D several months). It should be emphasized that and 3D convective flow patterns. In section 3 we the calculation time per time step is quite reason- examine how the difference between 2D and 3D able; however, the number of time steps required convectioneffecttheradialpulsationgrowthrates to obtain a full amplitude solution is large. and how the convective strength depends on pul- We begin by comparing the flow patterns asso- sation amplitude. In Section 4 we consider the ciatedwiththeconvectivemotion. Figure1shows differences in time dependent behavior of full am- thetop16%byradiusofthe6300Keffectivetem- plitude pulsation with 2D and 3D convection. perature2Dsimulation. Thecolorshowsthetem- perature of the material and the vectors show the 2. CONVECTIVE FLOW PATTERNS convective velocity. The white lines show the hor- izontal periodic boundaries. Figure 2 is similar to We have performed simulations of RR Lyrae Figure1butshowsaslicethroughthecomparable pulsation with 3D convection at effective temper- 3Dsimulation. The convective flowpattern ofthe aturesof6200,6300,6400,6500,6700,and6900K. 2D simulation at first glance appears similar to a Theinitialparametersofthesemodelsmatchtheir slice through a comparable 3D simulation. How- 2D counterparts presented by Geroux & Deupree ever,therearesomedifferencesintheflowpattern. (2013)–L = 50L , M = 0.7M , X = 0.7595, and (cid:12) (cid:12) In particular the circular flow pattern clearly vis- Z = 0.0005. The difference is that these mod- ible in the 2D simulation is not as noticeable in els have the extra dimension for fluid flow. Given the slice through the 3D simulation. This may be the highly turbulent nature of convection in the aresultofthefactthattheextradimensionallows surface ionization regions of RR Lyrae stars, the someofthereturnflowtotakeplaceinadifferent convectivemotionshouldbe3D.These3Dsimula- plane. tions have the same radial and θ-zoning (140×20) WhileFigure2providesinformationabouthow as the 2D calculations but also have 20 φ zones the 3D convective flow pattern behaves in the ra- covering 6◦, producing a 3D version of a pie slice dial and θ directions, it is more informative to see subtending 36 square degrees. The choice of 6◦ the flow pattern in both the horizontal directions. coverage in each direction comes from relatively Figures 3 and 4 show the temperature isosurface short 3D simulations with angular zoning which at 104 K spanning the full horizontal extent of subtended total angles from 2◦×2◦ to 10◦×10◦ the 6300 K effective temperature 3D simulation with both θ and φ stepping simultaneously in in- during the pulsation compression and expansion crements of 2◦ between the two extremes. These phases, respectively. One can see that the con- short simulations were for a 5700 K effective tem- vection truly is 3D in nature. The reduction in perature model with strong convection and were convective strength from compression to expan- carried out until convection had finished growing sionisclear,withlargervelocityvectorsandlarger from machine round off errors and at least two variations in the temperature isosurface showing additional pulsation cycles had been completed. stronger convective flows during compression and The 6◦×6◦ configuration was found to be a good smaller velocity vectors and a flatter temperature compromisebetweentheinclusionofmultiplecon- isosurface during expansion. Figures 5 and 6 are 3 Fig. 1.—Upper16%byradiusofa2Dsimulation ofa6300KeffectivetemperatureRRLyraemodel. Temperature is indicated by the color scale, and Fig. 3.—Temperatureisosurface(T =104 K)and the vectors show the direction of the convective convectivevelocityvectorsforpointsonahorizon- flow. Note the relatively narrow, high velocity talplaneabovetheisosurface. Thecoloroftheiso- downward convective flow in comparison to the surface indicates upward convective motion in red slower moving wider area upward flow. The white and downward convective motion in blue. This radiallinesindicatethehorizontalperiodicbound- “snapshot” is taken during radial pulsation con- aries of the calculation. tractionfora6300Keffectivetemperaturemodel. Fig. 2.— Upper 16% by radius of an r–θ slice through a 3D simulation of a 6300 K effective temperature RR Lyrae model. Temperature is in- dicated by the color scale, and vectors show the Fig. 4.— Similar to Figure 3 except during ra- motion in the r–θ plane. Note that, in contrast dial pulsation expansion instead of radial pulsa- to Figure 1, the downward motion covers a wider tioncontraction. Noticethattheconvectiveveloc- area and that there is little evidence of upward itiesarelargerduringcontractionthanexpansion. flow in this particular plane. 4 thesameasFigures3and4exceptforthe6700K effective temperature model. Comparing the fig- ures for the 6300 K effective temperature model to the figures for the 6700 K effective temper- ature model, it is clear that the change in con- vective strength from compression to expansion is smaller for the hotter model, although convection remains stronger during contraction than expan- sionforbothmodels. Itisinterestingtonotethat the convective patterns show some similarity to the solar granulation pattern, in that they have largeslowmovinghotupflows,surroundedbyfast narrow cool down flows. A possible criticism of these calculations is of the small angular extent (6◦×6◦) only containing two (T =6300 K model) to four (T =6700 K eff eff model)convectivegranulesaswellasthepoorres- olution (20×20 horizontal zones). In an attempt Fig. 5.— Similar to Figure 3 except for a model to help validate that our simulations are getting with an effective temperature of 6700 K during the large scale flows correct we can compare the radial pulsation contraction. granulesizesandtheupflowfillingfactortoother 3D simulations of convection in stars. Recently workbyMagicetal.(2013)hasmappedoutgran- ulediametersusinghighresolution3Datmosphere models and derived a simple relation between the granule diameter and the two parameters T and eff logg. UnfortunatelytheT andlogg ofourmod- eff els (around logg =2.8 and T =6000−7000 K) eff havenotbeenmodelledbyMagicetal.. Theclos- est effective temperature of their models which bracket our gravities is 5500 K. Though the au- thors caution against extrapolation of their rela- tions it may still give some indication of the size one might expect for granules in our simulations. From Magic et al.’s relations we obtain 10.20 and 10.18forlogd (withd incm)forthelargest gran gran granules in the 6300 K and 6700 K effective tem- perature models respectively. A simple way to es- timate the diameter of the granule in our simula- tions is to divide the horizontal area of our com- putational domain by the number of granules and calculating the diameter of the granule from this area. Doingsoresultsinlogd of10.5and10.3 gran for the 6300 K and 6700 K models respectively. Our simulations have granules that are slightly Fig. 6.— Similar to Figure 5 except during radial largerthanpredictedfromtheworkbyMagicetal. pulsation expansion instead of contraction. but are the same order of magnitude. We have also explored the filling factor of up- flows (the fraction of the horizontal area with v −v >0),f ,andfoundthatthereisatimede- r 0 up 5 pendenceonthepulsationphase. Thefillingfactor was measured at the temperature where the tem- perature gradient is the steepest (104 K). When the star is fully expanded f ≈ 0.7, while when up thestarisfullycontractedf ≈0.4. Thetimeav- up erageofthefillingfactoroverfourpulsationphases results in (cid:104)f (cid:105)≈0.6. These values are quite sim- up ilarforboththe6300Kand6700Kmodelsexam- ined. Thetimeaveragedvalueof0.6isreasonably close to that found by other authors of about 2/3 (Magic et al. 2013; Stein & Nordlund 1998). The similarity of the granule sizes and filling factor to those obtained by higher resolution 3D studies of stellar convection, covering larger horizontal ex- tents provides some confidence that we are cal- culating the largest scale structures of convection correctly. Despite this order of magnitude agree- ment, we do not argue that our horizontal zoning issufficientforourcalculationstoadequatelyrep- resent the details of turbulent convection. Given theamountofcomputertimerequiredforthecur- rent 3D calculations, full amplitude 3D calcula- tions with, say, an order of magnitude more zones ineachhorizontaldirectionisstillsometimeaway. 3. 2D AND 3D DEPENDENCE OF PUL- SATION GROWTH RATES AND OF CONVECTIONONPULSATIONAM- PLITUDE Geroux&Deupree(2013)showedthatthepeak Fig. 7.— Six period average of the peak convec- convectivefluxforapulsatingmodeldependedon tive flux during a pulsation period and three pe- thepulsationamplitude. Herewewishtoseethat riod average of the log of the peak kinetic energy thisremainstruein3D.Figure7showshowthesix per period versus the time since the beginning of period average of peak convective flux per period the calculation. 2D calculations are denoted by varies with pulsation amplitude. This average is the dashed curves, and 3D by the solid curves. determinedinthefollowingway–firstwefindthe Although the effects of convection on the growth convective flux for every zone in a given model rate are small, they become apparent over many and select the largest value. We then compare periods. These results are for a 6500 K effective this peak convective flux for all models within a temperature model. given period and again select the largest value. Thesixperiodaverageisthentheaverageofthese single period peak fluxes. The peak convective fluxaveragedoverthesixperiodsclearlyincreases as the peak kinetic energy of the radial pulsation increases. Figure 7 also shows that the corresponding 3D simulations have a larger peak maximum convec- tive flux for a given peak kinetic energy than for the 2D simulations. This does have an effect on 6 the pulsational growth rates. We determine the growth of the stellar pulsation by using the three period average of the peak kinetic energy per pe- riod. The three period average reduces the varia- tion introduced by the first overtone in these fun- damental mode calculations. Figure 7 indicates thatthe3Dpulsationalgrowthrateislessthanthe 2D growth rate. We have calculated the growth rates for the 2D and 3D simulations and find the 2D growth rates are larger than the 3D growth rates by about 0.07-0.09% per period, with larger differences for cooler models. This suggests that the relative behavior of convection in 2D and 3D, intermsofitsinteractionwithpulsation,doesnot vary much across the fundamental mode region of the instability strip. In the calculation of the pulsational kinetic energy it is assumed that the pulsation is given by the radial motion of the co- ordinate system. Note that this assumption does not affect the numerical simulation; it only affects ourinterpretationoftheresults. Infactthepulsa- tionalkineticenergyvastlyexceedstheconvective kinetic energy so that some error in this assump- tion should not alter our conclusions. We note in this regard that all our calculations are pulsation dominated not convectively dominated, including those near the red edge. This suggests that the density stratification does not strongly limit the pulsation as found in some cases by Gastine & Dintrans(2011). The2Dand3Dgrowthratesare muchclosertoeachotherthanarethe1Dand2D growth rates given in Geroux & Deupree (2013), emphasizingthatconvectionineitherits2Dor3D framework helps to slow the pulsational growth. Fig. 8.— From top to bottom: peak convective flux, peak of the ratio of convective luminosity to 4. FULL AMPLITUDE TIME DEPEN- total luminosity, peak radial convective velocity, DENT BEHAVIOR maximum variation of the horizontal temperature variationfromthehorizontallyaveragedtempera- Wehaveindicatedthatthetimedependentbe- ture, and surface pulsation velocity for 3D (solid) havior of convection as a function of pulsation and 2D (dash) calculations of the 6500 K effec- phaseisgenerallythesamein2Dand3D,basedon tive temperature model as functions of pulsation general trends in the convective velocity and the phase. Peak values are the maximum value of a warping of isothermal surfaces. Here we wish to quantity throughout the 2D or 3D computational examinetherelativestrengthofconvectionalittle mesh at any given time. Note that these peak more quantitatively in both types of calculations. quantities are generally smaller for the 2D calcu- Thisisnotoverlystraightforwardbecauseweneed lation than for the 3D calculation. a definition of the convective strength which can account for the differences in the flow patterns in 2D and 3D. This comparison will be made for a 6500 K model at full amplitude in both 2D and 3D. To proceed, we need to compute the convec- 7 tive flux for the two cases. There is no explicit expression for the convective flux in the conserva- tion equations, only expressions for the total en- ergy advection, the PdV work, the conversion of thesubgridscalekineticenergyintoheat, andthe radiation terms (see equation 8 of Geroux & De- upree 2013). Thus, the energy equation includes theenergybalanceofthepulsation,radiation,and convectionwithoutexplicitlydividingtheflowinto thatassociatedwithpulsationandthatassociated withconvection. However,wewouldliketoexam- ine the behavior of the (radial) convective flux, which we will approximate by F =c ρ(v −v )∆T. (1) conv. P r r0 where v is the radial velocity of a given zone, v r r0 is the velocity of the coordinate system, and ∆T is the difference between the temperature in the zone from the horizontal average temperature at that radial zone. Recall that the velocity of the coordinate system is that required for the mass in the spherical shell to remain constant throughout the calculation (see discussion in Geroux & De- upree 2011). One possible comparison is between the maxi- mumconvectivefluxesanywherethroughthecom- putational mesh at a particular time. Once con- vectionhasdevelopedsufficiently, thiswillbeina downwardmovingcolumnineither2Dor3D.This maximum convective flux is merely the maximum value of the flux given in equation (1) over all the zones in the calculation. However, one could ar- guethatthestrengthofconvectionshouldbemea- sured by the amount of convective energy trans- port through a spherical surface. Here we must add up all the convective fluxes from all the zones atagivenradius, withtheindividualzonesurface areastakenintoaccount. Forconvenience,weturn thisintoaconvectiveluminosity. Acomparisonof these two convective flux related quantities in 2D and 3D will not necessarily yield the same result Fig. 9.— The top panel shows the comparison of because the fraction of the surface area taken up the 2D and 3D light curves for the 6500 K model. by the downward moving material is quite differ- Thebottompanelshowsthe3Dlightcurvedcom- ent in the differing dimensions. Specifically, the pared to observations of v93 in M3 by Cacciari 2D extension into 3D would have the downward et al. (2005) convectiveflowsmovinginalongtrenchnotshown in the 3D simulations. We present the results of such a 2D – 3D com- parisoninFigure8foraneffectivetemperatureof 6500 K. The top panel shows the comparison of 8 the 2D and 3D maximum convective flux, and the slope between phases 0.2 and 0.6. The 2D calcu- second panel shows the maximum convective lu- lationactuallyagreesbetterwiththeobservations minositycomputedasdescribedabove. Themaxi- during declining light, although the 3D slope be- mumconvectivefluxishigherin3Dthan2D,while tween phases 0.2 and 0.6 is closer to that of V93 the opposite is true for the convective luminos- than is the 2D slope. The reasons for these differ- ity. Clearly, the fraction of the surface taken up ences are unknown, and more 3D calculations are by the large convective flux situated in the down- required to determine the sensitivity of the light ward flow makes the difference. Having said that, curvestoparametersofthemodelandthezoning. wenotethattherelativetimedependenceofeither Thecompletionofthe3Dmodelsatothereffective themaximumconvectivefluxortheconvectivelu- temperatureswillindicatewhetherthisisaglobal minosity is quite similar in 2D and 3D. The con- problem or confined to this one model. vective energy transport increases markedly dur- ingthelatterphasesofpulsationalcontractionand 5. DISCUSSSION decreasesduringearlyexpansion. AsnotedbyDe- We have computed a number of 3D hydrody- upree (1977a) and also by (Gastine & Dintrans namicmodelsofRRLyraevariables,oneofwhich 2011) this is the type of behavior which leads to has now reached full amplitude. The convective a decrease in the pulsational driving by the ion- flow pattern, of course, is genuinely 3D and thus ization zones. This is not to say that this time differentfromthatfoundinourprevious2Dmod- dependent behavior is the sole property affecting els (Geroux & Deupree 2013). However, the dif- pulsationalgrowthordecay. Forexample,Gastine ferences of the effects between 2D versus 3D con- & Dintrans (2011) present an example in which vection on pulsation appear to be comparatively the static model density stratification can appre- modest. The phase dependent behavior of the ciablyaffectthepulsationamplitude. Whilewedo peak convective flux is quite similar between the not believe this is an issue in this particular case 2D and 3D models, and the 3D models decrease because the pulsational kinetic energy is so much the pulsational growth rate by only about 0.1% larger than the convective kinetic energy, we have per period compared to the 2D models. The com- not done a suite of calculations covering the pos- parison between light curves from the 2D and 3D sible range in the physical and model properties calculations for the one 3D model at full ampli- to determine if any of these affect the pulsation tude are somewhat different during falling light, amplitudes. although the amplitude of the pulsation and the Thepeakconvectivevelocity,showninthethird rising part of the light curve are quite similar. As panel of Figure 8, appears to be somewhat higher full amplitude 3D calculations are completed, we in 3D than 2D. The largest difference is at maxi- should be able to determine how pervasive these mum convective flux, but the velocity differences differencesare, particularlyclosertotherededge. are not that large. The maximum horizontal tem- Also, very little has been done in terms of param- peraturevariationalsoappearstobealittlelarger eter studies in the 3D calculations. These remain in 3D than 2D at maximum convective flux. The difficultsimplybecause3Dcalculationstofullam- combination of these two differences are responsi- plitude require so much time. ble for the increased maximum convective flux in Therelativelysmalldifferencesbetweenthe2D 3D. and 3D calculations in terms of the effects of con- Of course, the crucial test in modelling RR vection on pulsation should be considered good Lyrae stars at full amplitude is the light curve. news. This suggests that the effects of different We compare the 2D light curve to the 3D light masses, luminosities, and compositions can prob- curve for the 6500 K full amplitude model and ably be mapped out in 2D instead of the full the 3D light curve to that of V93 in M3 (Cacciari 3D. Thus, the time can be shortened consider- et al. 2005) in Figure 9. We first note that the ably because the 2D calculations take only days amplitude of the light curve and the rising light toweeks,whereasthe3Dcalculationsrequiresev- segments of all three light curves agree well. The eral months to reach full amplitude. 2D and 3D light curves differ in the rate of de- cline from maximum light and then in the new 9 The authors gratefully acknowledge the sup- Dorfi, E. A., & Feuchtinger, M. U. 1991, A&A, port of ACEnet, both for providing high perfor- 249, 417 mance computing in Atlantic Canada and for an Feuchtinger, M. U., & Dorfi, E. A. 1996, A&A, ACEnet Research Fellowship to CMG. We also 306, 837 thank ACEnet for the use of the Data Cave in vi- sualizing the 3D calculations. As anyone who has Gastine, T., & Dintrans, B. 2011, A&A, 528, A6 performed significant 3D simulations knows, visu- alizing the results is almost as difficult as the cal- Gehmeyr, M. 1992a, ApJ, 399, 265 culations themselves, and the Data Cave anima- —. 1992b, ApJ, 399, 272 tionsmadethispossible. ACEnetisfundedbythe Canada Foundation for Innovation and provincial —. 1993, ApJ, 412, 341 funding agencies of Nova Scotia, New Brunswick, and Newfoundland and Labrador. CMG received Geroux, C. M., & Deupree, R. G. 2011, ApJ, 731, partial financial support during writing from a 18 Consolidated STFC grant (ST/J001627/1). —. 2013, ApJ, 771, 113 REFERENCES Iglesias, C. A., & Rogers, F. J. 1996, ApJ, 464, Alexander, D. R., & Ferguson, J. W. 1994, ApJ, 943 437, 879 Kuhfuss, R. 1986, A&A, 160, 116 Bono, G., Caputo, F., Cassisi, S., Incerpi, R., & Magic, Z., Collet, R., Asplund, M., Trampedach, Marconi, M. 1997a, ApJ, 483, 811 R., Hayek, W., Chiavassa, A., Stein, R. F., & Bono, G., Caputo, F., Castellani, V., & Marconi, Nordlund, ˚A. 2013, ArXiv e-prints M. 1997b, A&AS, 121, 327 Marconi, M. 2009, in American Institute of Bono, G., & Stellingwerf, R. F. 1994, ApJS, 93, Physics Conference Series, Vol. 1170, Ameri- 233 can Institute of Physics Conference Series, ed. J. A. Guzik & P. A. Bradley, 223–234 Buchler, J. R. 2009, in American Institute of Physics Conference Series, Vol. 1170, Ameri- Marconi, M., Caputo, F., Di Criscienzo, M., & can Institute of Physics Conference Series, ed. Castellani, M. 2003, ApJ, 596, 299 J. A. Guzik & P. A. Bradley, 51–58 Marconi, M., & Degl’Innocenti, S. 2007, A&A, Cacciari, C., Corwin, T. M., & Carney, B. W. 474, 557 2005, AJ, 129, 267 Mundprecht, E., Muthsam, H. J., & Kupka, F. Christy, R. F. 1964, Reviews of Modern Physics, 2012, ArXiv e-prints 36, 555 Rogers, F. J., Swenson, F. J., & Iglesias, C. A. —. 1966a, ApJ, 144, 108 1996, ApJ, 456, 902 —. 1966b, ARA&A, 4, 353 Stein, R. F., & Nordlund, A. 1998, ApJ, 499, 914 Cox,A.N.,Brownlee,R.R.,&Eilers,D.D.1966a, Stellingwerf, R. F. 1975, ApJ, 195, 441 ApJ, 144, 1024 —. 1982a, ApJ, 262, 330 Cox, J. P., Cox, A. N., Olsen, K. H., King, D. S., & Eilers, D. D. 1966b, ApJ, 144, 1038 —. 1982b, ApJ, 262, 339 Deupree, R. G. 1977a, ApJ, 211, 509 —. 1984a, ApJ, 277, 322 —. 1977b, ApJ, 214, 502 —. 1984b, ApJ, 277, 327 —. 1977c, ApJ, 215, 232 —. 1984c, ApJ, 284, 712 10