How the merger of two white dwarfs depends on their mass ratio: orbital stability and detonations at contact

5.7 MB·English

by  Marius Dan

Mon.Not.R.Astron.Soc.000,1–13(2011) Printed13January2012 (MNLATEXstylefilev2.2)
How the merger of two white dwarfs depends on their
mass ratio: orbital stability and detonations at contact
Marius Dan1(cid:63), Stephan Rosswog1,2, James Guillochon2, Enrico Ramirez-Ruiz2
1School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany
2TASC, Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064
2
Accepted?.Received?;inoriginalform?
1
0
2
ABSTRACT
n Despitetheiruniqueastrophysicalrelevance,theoutcomeofwhitedwarfbinarymerg-
a
ers has so far only been studied for a very restricted number of systems. Here we
J
presenttheresultsofasurveywithmorethantwohundredsimulationssystematically
1
scanning the white dwarf binary parameter space. We consider white dwarf masses
1
rangingfrom0.2to1.2M andaccountfortheirdifferentchemicalcompositions.We
(cid:12)
findexcellentagreementwiththeorbitalevolutionpredictedbymasstransferstability
]
E analysis.Muchofoureffortinthispaperisdedicatedtodeterminingwhichbinarysys-
H temsarepronetoathermonuclearexplosionjustpriortomergeroratsurfacecontact.
We find that a large fraction of He-accreting binary systems explode: all dynamically
.
h unstable systems with accretor masses below 1.1 M and donor masses above ∼ 0.4
(cid:12)
p M arefoundtotriggeraheliumdetonationatsurfacecontact.Asubstantialfraction
(cid:12)
-
ofthesesystemscouldexplodeatearliertimesviadetonationsinducedbyinstabilities
o
r intheaccretionstream,aswehavedemonstratedinourpreviouswork.Wedonotfind
t definitive evidence for an explosion prior to merger or at surface contact in any of the
s
a studied double carbon-oxygen systems. Although we cannot exclude their occurrence
[ if some helium is present, the available parameter space for a successful detonation

in a white dwarf binary of pure carbon-oxygen composition is small. We demonstrate
1
thatawidevarietyofdynamicallyunstablesystemsareviabletypeIacandidates.The
v
6 next decade thus holds enormous promise for the study of these events, in particular
0 with the advent of wide-field synoptic surveys allowing a detailed characterization of
4 their explosive properties.
2
. Key words: supernova: general – white dwarfs – nuclear reactions, nucleosynthesis,
1
abundances – hydrodynamics
0
2
1
:
v
1 INTRODUCTION that the companion was a red giant (Li et al. 2011), and
i
X restrictsthepre-explosionmasstransferratefromanynon-
r Type Ia supernovae are lynchpins in measuring distances giant companion to a narrow range of values (Horesh et al.
a on cosmological scales (Riess et al. 1998; Perlmutter et al. 2011; Chomiuk et al. 2012). Observations indicate that the
1998). These events are bright enough to be seen from a eventisafairlystandardtypeIa(Brownetal.2011;Bloom
great distance such that many thousands have been cata- etal.2011),andthustheparticularprogenitorchannelthat
loguedbymoderntransientsurveyssuchasPTF(Rauetal. led to this event is likely to be similar to a significant frac-
2009),CfA(Hickenetal.2009a,b),andLOSS(Leamanetal. tion of the type Ia events observed. The strong restrictions
2011).Despitethelargenumberofdetectedevents,thepos- and/or elimination of non-degenerate companions compels
sibleprogenitor(s)oftypeIasupernovaearestillnotknown us to explore the possibility that the companion is also a
(Howell 2011). degenerate star. This double degenerate (DD) scenario for
A strong constraint comes from the type Ia event re- type Ia would also be consistent with the observed delay
cently discovered in M101, which placed an upper limit of time distribution, which is approximately proportional to
0.02R onthesizeofstarthatexploded(Nugentetal.2011; t−1(Totanietal.2008;Maozetal.2010;Ruiteretal.2010).
(cid:12)
Bloom et al. 2011). This all but eliminates the possibility This progenitor channel would further explain why hydro-
gen has never been observed in a type Ia (Howell 2011).
Moreover,therecentlyobservedsuper-Chandrasekharmass
events(Howelletal.2006;Scalzoetal.2010;Taubenberger
(cid:63) E-mail:[email protected]
(cid:13)c 2011RAS
2 M. Dan, S. Rosswog, J. Guillochon, E. Ramirez-Ruiz
et al. 2011) give further credence to the idea that at least
some fraction of type Ia may be attributed to the DD sce-
nario.
Because white dwarfs (WDs) represent the final evolu-
ONe - CO
tionarystateofthevastmajorityofstars,theyarealsothe
most common constituents of binaries containing compact
stellarremnants.WDsarerepositoriesofthermonuclearfuel CO - CO
anditisnotsurprisingthatthereareseveralpathwaysthat
can produce a powerful explosion. In fact, it seems that all
but the lowest mass WDs (<0.2 M ) are capable of being ONe
(cid:12) CO - He+CO -
ignitedbyimpactsthatareofordertheirownescapespeeds H e +-CO He+CO
He+CO
(Rosswog et al. 2009; Raskin et al. 2009). Moreover, the
resulting theoretical lightcurves and spectra show excellent
agreementwithobservedtypeIa.Unfortunately,thesecolli- He - He He+-CO CO - He ONe - He
sion events are most likely too rare to explain a substantial He
fraction of the observed type Ia events.
More frequently, a WD will find itself as a companion
to another WD after both stars have evolved off the main
sequence or after a dynamical exchange in a dense stellar
environment (Shara & Hurley 2002; Lee et al. 2010). Bi- Figure 1. The initial chemical composition of double WD sys-
nary population modeling, which is calibrated by the cur- tems with donor masses M2 from 0.2 to 1.05 M(cid:12) and accretor
rently observed systems, estimates around 108 double WDs masses M1 between 0.2 and 1.2 M(cid:12). See text for additional in-
formation.
(DWDs) in the Milky Way (Nelemans et al. 2001b). The
ratesatwhichthesesystemscomeintocontact,althoughof
great interest for the formation of interacting binaries and
merger(Danetal.2011),simulationswithapproximateini-
possibly explosive mergers, remain uncertain, because they
tial conditions start with too little angular momentum and
dependprimarilyonthedistanceofthetwoWDsafterleav-
predictamergerthatoccursmuchmorerapidlythanwhatis
ing the poorly understood common envelope phase (Woods
realizedinnature.Ourpreviousresultsindicatedthatmany
et al. 2011). Once the WDs have exited this phase, the two
DD systems with a donor composed largely of helium (He)
WDs are driven together via gravitational wave radiation,
areexpectedtobeignitedbyaninteractionbetweentheac-
the signature of which is expected to be a dominant com-
cretion stream and the torus of material acquired from the
ponent of the ambient gravitational wave sky (e.g. Lipunov
donor(Guillochonetal.2010).Herewegreatlyexpandupon
etal.1987;Hilsetal.1990;Nelemansetal.2004;Liuetal.
ourinitialstudywith225newsimulationsthatfullyexplore
2010).
thepossiblecombinationsofWDdonorsandaccretorsinthe
A critical stage in the evolution of a compact WD bi- mass range between 0.2 and 1.2 M . We demonstrate that
(cid:12)
naryistheperiodjustaftermasstransferhasbeeninitiated. oursystematicstudyexhibitstheorbitalstabilitytrendsex-
Forthebinarytosurvive,themasstransfermustbestable, pectedformergingDDsystems,thusenablingustoevaluate
which depends sensitively on the internal structure of the which systems are expected to produce strong thermonu-
donor star, the binary mass ratio, and the angular momen- clear events in the lead up to or at the point of merger. We
tumtransportmechanisms(e.g.Marshetal.2004;Gokhale findthatmanysystemswithHedonorsmayproduceexplo-
et al. 2007). The torques applied by a disk that may form sions,butthatsystemswithcarbon-oxygen(CO)donorsare
fromaccretedmaterial,andthetidesraisedontheaccretor, incapableofexplodingwithouttheinclusionofasignificant
arecrucialforthebinary’sorbitalstability(Ibenetal.1998; He layer.
Piro2011).Ifthesemechanismsareunabletowidentheor-
In Section 2, we recap our numerical approach and de-
bit,thesystemwillmergequickly.Thesubsequentevolution
scribehowoursimulationscovertheparameterspaceofDD
canproduceaRCoronaBorealisstar(Webbink1984;Clay-
systems.InSection3wegiveabriefoverviewofhowdynam-
ton et al. 2007; Longland et al. 2011), trigger an explosion
icalorbitalstabilityisdeterminedandcomparetheseexpec-
asatypeIa(Webbink1984;Iben&Tutukov1984;Piersanti
tationstooursimulationresults.Section4discussesdetona-
et al. 2003a,b; Yoon et al. 2007), or undergo an accretion-
tionsthatareeithertriggeredbytheaccretionstreamorat
induced collapse to a neutron star (Saio & Nomoto 1985;
themomentofmergerforbothHeandCOmass-transferring
Nomoto & Kondo 1991; Saio & Nomoto 2004; Shen et al.
donorstars.Wesummarizeourfindingsandrelateourmod-
2011).
elswithtypeIaobservationsinSection5.Forreference,we
While theoretical advances detailing the possible out- provide fitting formulae bounding the region of parameter
comesofWDmergershavebeenmadeinthelastdecade(see space where detonations are expected in Appendix A.
reviewbyPostnov&Yungelson2006),simulationshaveonly
probedtheparameterspaceofDDmergersforaverysmall
number of systems, with the principle emphasis being on
systems with a total mass greater than the Chandrasekhar 2 NUMERICAL APPROACH
mass. We have previously stressed the vital role of the nu-
merical initial conditions in determining the outcome of a The simulations of this investigation are performed with a
3Dsmoothedparticlehydrodynamics(SPH)code(Rosswog
(cid:13)c 2011RAS,MNRAS000,1–13
How the merger of two WDs depends on the mass ratio 3
Macc Macc Macc
Figure 2. Remnant profiles both with accurate (solid-red line) and approximate (dashed-green line) initial conditions (ICs) for the
0.81+0.9 M(cid:12). Shown are the spherically enclosed mass (left) and spherically averaged density (center) and temperature (right) vs.
radiusfromthecenteroftheaccretor.
et al. 2008), for recent reviews of the SPH method see e.g. libriumbyaddingavelocity-dependentdampingtermtothe
Monaghan (2005)and Rosswog (2009). Theorbital dynam- momentumequation.Subsequently,weplacethestarsinthe
ics of a binary system is very sensitive to the redistribution corotatingframeataninitialseparationthatislargeenough
ofangularmomentumduringmasstransferandthereforeac- to avoid any immediate mass transfer. The orbital separa-
curatenumericalconservationiskeytoareliablesimulation. tionisthenadiabaticallyreducedinawaythatthetimescale
SPHisaverypowerfultoolforthistypeofstudybecausethe over which the separation changes is much longer than the
equationsconserveangularmomentumbyconstruction,see dynamicaltimescaleofthedonor.Duringthisprocess,spuri-
e.g. Section 2.4 in Rosswog (2009). In practice, the quality ousoscillationsareminimizedbyapplyingadampingforce.
ofconservationisdeterminedbytheforceevaluation(inour When the first SPH particle reaches the inner Lagrangian
case by the opening criterion of our binary tree Benz et al. point L , we stop the relaxation process, transform the ve-
1
1990)andbythetimesteppingaccuracy,bothofwhichcan locities to the fixed frame and set this moment as the time
bemadearbitrarilyaccurate.Ourcodeusesanartificialvis- origin (t = 0) of the simulation. Most of the previous SPH
cosity scheme (Morris & Monaghan 1997) that reduces the WD merger calculations (Benz et al. 1990; Segretain et al.
dissipation terms to a very low level away from discontinu- 1997; Guerrero et al. 2004; Yoon et al. 2007; Lor´en-Aguilar
ities, together with a switch (Balsara 1995) to suppress the et al. 2009; Pakmor et al. 2010, 2011) started from rather
spurious viscous forces in pure shear flows. The system of approximate ICs. In Dan et al. (2011) we have shown in
fluid equations is closed by the Helmholtz equation of state detail that the initial conditions have a strong impact on
(Timmes & Swesty 2000). It accepts an externally calcu- all major aspects of the simulations. Here, we only briefly
lated nuclear composition and allows a convenient coupling illustrate the differences between the approximate and our,
to a nuclear reaction network. We use a minimal nuclear accurate ICs in the remnant structures resulting from the
reaction network (Hix et al. 1998) to determine the evolu- mergerofa0.81+0.9M systeminFigure2.Whenusing
(cid:12)
tionofthenuclearcompositionandtoincludetheenergetic theaccurateICthereislessmassinthediskandmoremass
feedback onto the gas from the nuclear reactions. A set of inthetrailingarmthanwhenstartingfromtheapproximate
only seven abundance groups greatly reduces the computa- IC. The approximate IC also overestimates the density and
tional burden, but still reproduces the energy generation of temperature in the region surrounding the accretor. As we
all burning stages from He burning to NSE accurately. We explain in Section 5, the pollution of the ambient environ-
use a binarytree (Benz et al.1990) to search for the neigh- mentpriortoanysupernova-likeeventmayhaveimportant
borparticlesandtocalculatethegravitationalforces.More observational consequences.
details about our SPH code can be found in Rosswog et al.
The compositions used in this study are shown in Fig-
(2008) and the provided links to the literature.
ure 1. Below 0.45 M the stars are made of pure He. In
(cid:12)
principle, WDs with masses below 0.45 M could harbor
(cid:12)
heavier elements in their cores (Iben & Tutukov 1985; Han
2.1 Initial conditions et al. 2000; Moroni & Straniero 2009), but Nelemans et al.
(2001b)haveshownthattheprobabilityforhybridWDsin
The WD donor mass distribution in our parameter space thisrangeis4−5timessmallerthanforHeWDs.Basedon
startsat0.2M(cid:12) andreachesupto1.05M(cid:12) instepsof0.05 Rappaport et al. (2009), who found that a 0.475 M(cid:12) WD
M(cid:12). For accreting WDs we investigate those with masses has a He envelope of about 30% of its total mass, between
between 0.2 and 1.2 M(cid:12), again in steps of 0.05 M(cid:12). Accu- 0.45 and 0.6 M(cid:12) we adopt a hybrid He-CO composition
rate initial conditions (ICs) were constructed following the with a ∼ 0.1 M He mantle and pure CO core. WDs be-
(cid:12)
proceduredescribedinDanetal.(2011),hereweonlybriefly tween 0.6 and 1.05 M are assumed to be made entirely of
(cid:12)
summarize it. CO.AccordingwithIben&Tutukov(1985),thereisonlya
For all our simulations the stars are initially synchro- small amount of He at the surface of the CO core for this
nized, cold and isothermal (T =105 K). In a first step, the rangeofmasses.IthasbeenshownthattinyamountsofHe
stars are individually driven to an accurate numerical equi- canbeimportantfordetonationsofCOmatter(e.g.Seiten-
(cid:13)c 2011RAS,MNRAS000,1–13
4 M. Dan, S. Rosswog, J. Guillochon, E. Ramirez-Ruiz
1.05 28
1 Norbs
26
0.95
Definitely Unsta0b.l1e M200.8.95 2242
0.8
20
10
100 DirDeicst kI Amcpcarcettion 00.7.75 18
0.65 16
0.7 0.8 0.9 1 1.1 1.2
M
1
Figure3.NumberoforbitsofmasstransferfortheHe(left)andCO(right)donors.ForHemass-transferringsystems,themasstransfer
stability limits are over-plotted. The region below the continuous lower line indicates where a disk forms, above this line the accretion
stream directly impacts on the accretor. The mass transfer is guaranteed to be unstable above the upper continuous line (Marsh et al.
2004).Dashedlinesrepresentstabilitylimitsassumingasynchronizationtimescaleoftheaccretorwiththeorbitof0.1,10and100yr.
AllsystemswithCOdonors(right)belongtotheunstable,directimpactregimeofmasstransfer.
zahl et al. 2009), but for the numerical resolution that we et al. 2001a)
canaffordinthislargeparameterstudy(20,000SPHparti-
r
cles/star)wecannotproperlyresolvetheouterHelayers.For min = 0.04948−0.03815logq
a
thisrangeofmassesthereshouldbemoreOxygenthanCar-
+0.0475log2q−0.006973log3q, (1)
bon(Iben&Tutukov1985),andwehavechosenmassfrac-
tionsX(12C)=0.4andX(16O)=0.6uniformlydistributed
whereaistheorbitalseparation.Thetwovery-shortperiod
throughthestar,similartoLor´en-Aguilaretal.(2009).For
binariesHMCncandV407Vulbelongtothissocalleddirect
WDs with masses above 1.05 M we use X(16O) = 0.60,
(cid:12) impact category as they both satisfy r < R (Marsh &
X(20Ne) = 0.35 and X(24Mg) = 0.05 (similar to the com- min 1
Steeghs 2002; Ramsay et al. 2002).
position found by Gil-Pons & Garc´ıa-Berro (2001) for a 1.1
Tidal coupling can also return angular momentum to
M WD),distributeduniformlythroughouttheentirestar.
(cid:12)
the orbit, heating both WD members in the process. This
tidal heating can influence the donor’s mass transfer re-
sponse and, as a result, affect the stability of the system.
Forthecouplingtoeffectivelystabilizethesystem,thesyn-
chronization torques must act on a timescale shorter than
thetimescaleoverwhichtherateofmasstransferexponenti-
3 ORBITAL STABILITY ates.SomeWDbinariesclearlyavoidmerger,asweseetheir
survivorsintheformofAMCanumVenaticorum(AMCVn)
After one or more common envelope phases, a white dwarf
binaries (see reviews by Warner 1995; Nelemans 2005; Sol-
binarycanemergecloseenoughforgravitationalradiationto
heim 2010). Survival is however not guaranteed even when
drive the binary towards shorter orbital periods until mass
masstransferisexpectedtobestable.Masstransfercan,for
transfer is initiated. This occurs at orbital periods between
example, exceed the Eddington rate or spark a thermonu-
2 and 15 minutes depending on the characteristics of the
clear explosion that might destroy the binary.
donor. For the binary to survive, the mass transfer must
In Figure 3 we show the number of orbits each system
bestable, which depends sensitively on the binary mass ra-
survivespriortomerger.Themasstransferratesweareable
tio and the size of the donor star. Dynamical instability is
to numerically resolve are so far above the Eddington limit
guaranteedforsystemsinwhichthemassratioofthedonor
thatphotonsareeffectivelytrappedand,asaresult,nosig-
star to the accretor star q ≡ M /M exceeds 2/3 (see e.g.
2 1
nificant mass lossisanticipatedto occur.Whileallsystems
Nelemansetal.2001a).Forq<2/3,thesystemcanstillsur-
in our survey merge, we expect that binaries belonging to
vive as an interacting binary if orbital angular momentum
the disk accretion regime would survive if we were able to
lossesarebalancedbyothermechanisms.Inmanyinstances,
resolve the initial rate of mass transfer. In our simulations,
the formation of a tidally truncated accretion disk can effi-
the degree to which the donor star overfills its Roche lobe
cientlyreturntheadvectedangularmomentumtotheorbit.
is artificially enhanced by our finite resolution. This results
The system can then survive and evolve to longer periods
in a mass transfer rate that increases too quickly and pro-
asaresultofmasstransfer.Ontheotherhand,masstrans-
duces an under-massive accretion disk that cannot return
fer can proceed directly into the surface of the accretor. As
sufficient angular momentum to the orbit to avoid merger.
a result, the system is destabilized by transforming orbital
angularmomentumintotheaccretor’sspin.Adirectimpact Additionally, dynamically unstable systems are ex-
will occur if the radius of closest approach of the accretion pectedtobedesynchronizedattheinitialseparationswhere
stream r is less than the radius of the accretor R . An numericallyresolvablemasstransfersetsin(New&Tohline
min 1
estimate of r is given by (Lubow & Shu 1975; Nelemans 1997).Constructingsuchinitialconditionsischallengingas
min
(cid:13)c 2011RAS,MNRAS000,1–13
How the merger of two WDs depends on the mass ratio 5
17
Figure 4. Temperature (left) and density (right) evolution over the last 8 orbital timescales for the 0.6+0.9 M(cid:12) system. Tmax is the
maximumtemperature,ρ(Tmax)itscorrespondingdensityand(cid:104)T(cid:105)istheSPH-smoothedtemperature,seeEq.(2).(cid:104)T(cid:105)iscomputedusing
theparticleswithinthesmoothinglengthofparticlewithTmax.Coloredbandsshow±1,2σ(standarddeviations)distributionsaround
themeanvalues,thewhitelineindicatesthearithmetictemperaturemeanoftheneighborsofthehottestparticle.Timeismeasuredin
unitsoftheinitialorbitalperiod,τ .
orbit
desynchronicity is a strongly destabilizing effect. While all 2008) “all rather loosely decide whether detonation con-
oursimulationswerestartedwithsynchronizedstars,theac- ditions are reached based solely on the peak temperature
cretor becomes desynchronized after only a few orbital rev- reached in the simulation above a certain density” (Seiten-
olutions,andthusthesystemsquicklyevolvetoastatethat zahl et al. 2009). Apart from local density and tempera-
isclosetowhatisrealizedinnature.Despitethequickbuild- ture, the temperature profile and the size of the region be-
up to this desynchronized state, our simulations are able to ing heated determines whether a detonation is triggered or
capturethebasicangularmomentumtransportmechanisms not.Dependingonthesefactors,thecriticalradiihavebeen
correctlyasillustratedbythedramaticincreaseinthenum- found to vary between centimeters and hundreds of kilo-
beroforbitsasweapproachthediskaccretionboundary.In meters (Seitenzahl et al. 2009). In general, these values are
comparisonwithpreviousworkathigherresolutioninwhich substantiallybelowtheresolutionlengthsintherelevantre-
thewhitedwarfbinariesmergewithinonlyafeworbits,our gionsthatwecanaffordinthisstudy(∼107 –108 cm)with
resultsillustratetheimportanceofcarefullyconstructedini- its more than 200 simulation runs.
tial conditions.
Given the delicacy of the exact detonation conditions,
we resort in the bulk of this work to a simple yet robust
timescaleargument.Weassumethatdynamicalburningsets
4 DETONATIONS AT CONTACT in when the thermonuclear timescale, τnuc, drops below the
local dynamical timescale τ . When this happens matter
dyn
In this section we investigate which binary systems could heats up more rapidly than it can expand, cool and quench
produce immediate detonations at contact, first during the the burning. To compute τnuc we use τnuc =cpT/(cid:15)nuc (e.g.,
masstransfer,whichinthefollowingwerefertoas“stream- Taam1980;Nomoto1982),wherethenuclearenergygener-
induced detonations”, and, if detonations from the stream ation rate, (cid:15)nuc, is taken directly from the reaction network
interactionareavoided,duringthefinalcoalescencethatwe and cp is the specific heat at constant pressure, which we
call“contactdetonations”.InSec.4.1webrieflysummarize calculate from the Helmholtz equation of state. For the lo-
the detonation criteria that we use in Sects. 4.2 and 4.3 cal dynamical timescale we use τdyn = H/cs, where cs is
to discuss the detonation prospects for both He- and CO- the sound speed and H the local pressure scale height. We
transferring systems. estimate the H as P/ρg, where g is the local gravitational
acceleration.
In our SPH formulation, the energy variable that is
4.1 Detonation criteria evolvedforwardintimeistheinternalenergy.Thetempera-
tureateachparticlepositionisfoundateverytimestepbya
In its simplest form, a detonation is a shock wave that ad-
Newton-Raphsoniteration.ThistemperatureT ,alabelling
a
vancessupersonicallyintoanunburnt,reactivemedium.Be-
theparticle,isthebestestimateforthelocalmattertemper-
hind the shock wave exothermic reactions take place that
ature,butsmallnumericalfluctuationsintheinternalenergy
continuously drive the shock to wear down possible dissi-
(sayduetofinitenumericalresolution,finiteSPHneighbour
pative effects. Whether a detonation forms or not depends
number,finitetimesteppingaccuracyetc.)canleadtolarger
delicatelyontheexactlocalconditions.Thedetonationcon-
fluctuationsintheSPHparticletemperatureestimate.This
ditionsthathavebeenappliedinpreviouswork(Finketal.
isespeciallysoinregionswherematterisdegenerateandes-
2007; Ro¨pke et al. 2007; Townsley et al. 2007; Jordan et al.
(cid:13)c 2011RAS,MNRAS000,1–13
6 M. Dan, S. Rosswog, J. Guillochon, E. Ramirez-Ruiz
Figure 5. Particle temperatures within a cross-section slice with a thickness equal with four smoothing lengths for a 0.3+0.6 M(cid:12)
systemusing4×104 (left)and2×105 (center)SPHparticles.Therightpanelshowstheprobabilitydistributionfunctions(PDF)ofthe
temperaturewithintheredcircle(interactionradiusoftheparticlewithminimum(τnuc/τdyn)(T))shownintheleftandcentralpanels.
sentially independent of temperature. To some extent, this cal resolution that can be afforded in this parameter study,
couldbealleviatedbydesigningaFehlberg-typeintegration theoveralldynamicsiscapturedaccurately,seeforexample
scheme that rejects and re-takes time steps if their temper- the excellent agreement of the numerical results with the
ature changes are too large. In addition to being computa- analytical estimates for the orbital evolution shown in Fig-
tionally expensive, this approach may be ineffective in the ure3.Nevertheless,atthecurrentlowresolutiontheshown
case the fluctuations are not associated with the choice of interactionregionscancontainparticlesfromdifferentmor-
time-step. Therefore, one should be very cautious in inter- phologicalpartsoftheflow.Intheshownexamplethemost
pretingtemperaturesofveryfew(substantiallylessthanthe promising particle has neighbours in the cold (T <5×107
typicalneighbournumber)hotparticles,whichmayalsobe K) incoming accretion stream, in the hot interaction region
due to small internal energy fluctuations. (∼2×108 K) and in the moderately hot inner disk regions
(∼108K).AsaconsequencethePDFscanbemulti-peaked.
AlthoughwestressthatT isthebestlocaltemperature
a
As resolution increases and the neighbour particles come
estimate, we also calculate the SPH-smoothed temperature
from a smaller and smaller local volume, the PDFs will be-
value
come narrower, single-peaked and T and (cid:104)T(cid:105) will become
(cid:104)T(cid:105) =(cid:88)mbT W(|(cid:126)r −(cid:126)r |,h ), (2) increasingly similar.
a ρ b a b a
b b In the remainder of the paper we explore the M1 −
M plane to determine the regions in which the detonation
2
which is a robust lower limit. Here, ρ and m are the den-
b b conditions we have described are fulfilled.
sities and masses of neighbour particles b, W is the cubic
spline kernel and h the smoothing length of particle a. In
a
the limit of infinite resolution, both temperature estimates
would coincide. Although it is plausible that a detonation
would be triggered when (τ /τ )(T) < 1, we only con-
nuc dyn
sider a detonation as inescapable consequence beyond rea-
4.2 Helium mass transferring systems
sonabledoubtifthemoreconservativeratio(τ /τ )((cid:104)T(cid:105))
nuc dyn
is also below unity.
As discussed in Section 2.1, some of the donors we consider
AsamatterofillustrationweshowinFigure4theevo- are composed of He in their outermost layers. For those
lutionoftemperatureanddensityofa0.6+0.9M system systems with either He cores or thick He shells, the mass
(cid:12)
duringthelaststagesofitsmerger.Theleftpanelshowsthe transferred to the accretor will be pure He until either the
maximum particle temperature (green) together with the donor is disrupted, or until the He layer on the donor is
SPH-smoothed value (red) and the arithmetic mean value exhausted. As donors with a significant fraction of He may
within the interaction radius (= 2h) of the hottest particle exist for 0.2 (cid:54) M (cid:54) 0.6 M , the mass ratio of systems
2 (cid:12)
(white). The shaded regions indicate the 1 and 2 σ devi- with He donors can vary quite widely, and thus both dy-
ations. The corresponding quantities for the densities are namically stable and unstable systems with He donors are
plotted in the right panel. To illustrate the geometry of the expected to exist. Those dynamically stable systems that
potential ignition region we show in Figure 5 (left and cen- are close enough to one another to initiate mass transfer
tral panel) the particle temperatures within a slice around will appear as AM CVn systems, whereas the dynamically
the orbital plane. For a fair comparison we chose the slice unstable systems, which experience an exponential growth
thickness in each case to be 4 times the smoothing length in the accretion rate, are only likely to be observed prior
of the particle with the minimum ratio of (τ /τ )(T). to contact. As WDs are supported by degeneracy pressure,
nuc dyn
The interaction region of this particle is indicated in both the mass-radius relationship of WDs only weakly depends
panels by the red circles. The probability distribution func- on the WDs’ composition, and thus the ratio of He to CO
tions (PDFs) of the particles in these interaction regions in WDs is largely unknown, aside from those WDs that are
are shown in the right panel. Despite the very low numeri- too massive to contain any significant He.
(cid:13)c 2011RAS,MNRAS000,1–13
How the merger of two WDs depends on the mass ratio 7
0.6 −1.2 4.2.2 Stream-induced detonations prior to contact
0.55
−1.4 As we described in Guillochon et al. (2010), detonations of
the He torus can be triggered by the accretion stream sev-
0.5
−1.6 eralorbitspriortothemergerevent.Usingoursignificantly
0.45 largersetofaccretingsystems,wehaveprovidedanupdated
−1.8 versionofthefigureoriginallypresentedinDanetal.(2011)
2
M 0.4 showing the systems that are expected to have surface det-
−2 onations triggered by the accretion stream, see Figure 6. A
0.35
thermonuclearrunawaywilloccurpriortothemergingevent
−2.2 ifτ /τ ∼1,aconditionthatdependsonthethermody-
0.3 nuc dyn
namic state of the He torus acquired through the accretion
0.25 −2.4 process. The conditions for burning are more easily met for
systems in which the temperature and density of the mate-
0.2 rial acquired pre-merger is large to begin with, leading to
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
M conditions favorable for uncontrolled burning once the ma-
1 terial is compressed by the dynamical interaction, whether
Figure 6.TorusmassesoftheHedonorsystems.Ifeverτnuc(cid:54) by the stream or by the donor itself.
τdyn, we measure the masses when the timescales for the first The pre-interaction conditions in the He torus depend
time become equal, otherwise we take the mass when the ratio onthreeprimaryfactors:themassoftheaccretor,thecom-
τnuc/τdyn is minimal. The white contours show τnuc/τdyn when ponent of the stream velocity normal to the accretor’s sur-
theratioisataminimum,withthethickcontourshowingwhere face at impact, and the evolution of M˙ with time. For in-
theratioisunity.AcomparisonwithFigure8showsthatalarge
creasing accretor mass, the larger gravitational pressure at
fraction of WDs with He donors undergo accretion stream trig-
the accretor’s surface leads to larger hydrostatic pressures,
geredHe-detonationspriortocontact.Thetorusmassesatdeto-
nationcanbeaslargeas∼0.1M(cid:12),butalsosignificantlysmaller and thus greater equilibrium densities and temperatures.
forlargerMtot.SystemswithamassiveHedonorandsmalldisk Conversely, an increasing accretor mass shrinks the accre-
massesatdetonationmaybecapableofproducingmultiplesub- tor’s radius relative to the circularization radius, resulting
luminouseventspriortofinalcoalescence. in more “disk-like” accretion and thus a less-concentrated
torusthatismoresupportedbyrotationthanfluidpressure.
A shrinking accretor also reduces the normal component of
thestreamvelocityrelativetotheaccretor’ssurface,andas
a result the degree of compression achieved as a result of
thestream’simpactontotheaccretor’ssurfaceismorelim-
ited.Theseeffectsaremanifestasthelargerτ /τ ratios
nuc dyn
4.2.1 He detonations in stable systems toward the highest accretor masses in Figure 6.
Anecessaryconditionforsuccessfulstreamdetonations
Observed AM CVn systems are primarily expected to ac-
isthatthestream’srampressureisconsiderablylargerthan
crete through a disk (Nelemans 2005), although a few sys-
thehydrostaticpressureofthetorus,whichisonlypossible
tems are likely undergoing direct impact accretion (Marsh
ifthemassofthestreamiscomparableinmasstothetorus
& Steeghs 2002; Ramsay et al. 2002). While the tempera-
that has already accumulated. This implies that the sec-
turenecessaryfordynamicalHeburningissomewhathigher
ond derivative of the amount of mass transferred, M¨, must
than the virial temperature of a ∼ 0.6 M WD, slow He
(cid:12) be comparable to M˙ itself. Therefore, stream detonations
burningcanproceedintheaccretor’sacquiredHeshellover
are only possible when an exponentially growing M˙ can be
a timescale shorter than the thermal timescale, raising the
sustainedforseveralorbits,aconditionthatistypicalofsys-
entropy of the accreted material before the excess heat can
temswithabruptorbitalevolution.Becauseofthestochastic
be radiated away into space. This is in fact the reason why
nature of the triggering event, a detonation can be realized
no single WDs with a significant He layer are expected for
M (cid:38)0.6M ,aseveniftheWDbeginsitslifewithalayerof atanypointintimewheretheHetorusmassislargerthan
(cid:12)
the minimum shell mass required for a successful detona-
He,thislayerwillbeconvertedintoCOonatimescalethat
tion (Bildsten et al. 2007; Shen & Bildsten 2009; Woosley
is significantly shorter than the Hubble time. If the density
& Kasen 2011), which we find can precede the final merger
at the base of the He layer is large enough (i.e., if the He
event by many orbits.
shellismassiveenough),thetemperatureeventuallyrisesso
much that the burning timescale becomes shorter than the
convective timescale in the envelope, leading to a run-away
process that produces a detonation (Bildsten et al. 2007).
4.2.3 Contact detonations
However, a detonation can only occur in these kinds of
systemsifthedonorismassiveenoughsuchthatitcanpro- If detonations from the stream interaction are avoided, the
videtheminimumrequiredmass.Itturnsoutthatthemin- system has another chance to initiate a He detonation dur-
imum mass necessary is larger than ∼0.2 M for accretors ingthefinalcoalescenceofthetwosystems.ExceptforWDs
(cid:12)
with mass M < 0.6 M (Shen & Bildsten 2009), systems that are almost equal in mass, the dynamical interaction
1 (cid:12)
that may or may not be dynamically stable depending on duringthefinalcoalescenceinvolvesthedonorbeingripped
the synchronization timescale. apart by tides, while the accretor remains relatively unaf-
(cid:13)c 2011RAS,MNRAS000,1–13
8 M. Dan, S. Rosswog, J. Guillochon, E. Ramirez-Ruiz
Figure 7. Column density snapshots corresponding to the moment when the system first crosses (τnuc/τdyn)(T)<1, or, if that does
notoccur,when(τnuc/τdyn)(T)isminimum.ThetensnapshotsshownherearerepresentativeofallDDchemicalcompositionandmass
combinations shown in Figure 1. In this parameter space we fully explore the possible combinations of WD donors and accretors and
investigatetheirorbitalstabilityandwhetherdetonationspriororatthemergermomentarepossible.Systemswithamassratiocloseto
one(diagonaldashed-line)showaquickdisruption,withinabout20orbits,whilethoseawayfromthislineshowaslowdepletionofthe
donor(ie.lowM˙)overdozensoforbitsofmasstransferandtheformationofanextendedatmospheresurroundingtheaccretor.Wefind
thatalargefractionofthesystemswithHedonors(belowthehorizontaldashed-line)areexpectedtoexplodepriortooratthepoint
ofmerger.Incontrast,theCOaccretingsystems(abovethehorizontaldashed-line)areunlikelytoexplodeatorpriortothemerger.
fected. Approximately half of the donor’s remaining mass thepointofcollisionis(cid:112)2G(M +M )/(R +R )=3×108
1 2 1 2
falls on top of the previously accreted He, but at a speed cms−1,whichisapproximatelyone-thirdoftheescapespeed
that’s only a fraction of the escape speed from the accretor from a 0.8 M accretor.
(cid:12)
astheshreddeddonorispartlyheldaloftbyitsangularmo-
At the point of disruption, half of the donor star falls
mentum(Figure7).Thusthefinalcoalescencecanbeviewed
ontotheaccretorasthedonor’sRocheloberapidlyshrinks.
asalow-speedcollision(Rosswogetal.2009).Insimulations
Oncethishasoccurred,thedonorstarcanbeviewedasma-
of WD-WD head-on collisions, collisions between pairs of
terial whose orbital dynamics are primarily determined by
WDswithmassaslowas0.4M HeWDswereseentopro-
(cid:12) the surviving accretor. This means that the common cen-
ducedetonations.Themutualescapespeedoftwo0.4M at
(cid:12) terofmotion,thebarycenter,shiftsfrombeinglocatedata
(cid:13)c 2011RAS,MNRAS000,1–13
How the merger of two WDs depends on the mass ratio 9
M2
Figure 8.Logarithmicratioofthethermonucleartimescaleτnuc tothelocaldynamicaltimescaleτdyn atthemomentofmergerovera
gridspanningtheentireparameterspaceofHe(left)andCOdonors(right)masses.Theratiowascomputedusingboththeindividual
particle temperature T (upper panels) and the SPH-smoothed temperature (cid:104)T(cid:105) (lower panels, see eq. 2). Thermonuclear runaway is
expectedforconfigurationsabovethecontourswithτnuc/τdyn=1.
pointlyingbetweenthetwoWDstolyingwithinthecenter prone to produce detonations. Systems that do not satisfy
oftheaccretor.Fromtheplungingdonormaterial’spointof the criteria for detonating either by the stream mechanism
view, which originally moved on a circular orbit, the point or at the point of merger may be progenitors of He-novae
around which it orbits moves, placing the material on a instead (e.g., Yoon & Langer 2004).
highly eccentric orbit with a large r˙. This results in a col-
The ratio τ /τ is very large in the right-lower cor-
nuc dyn
lision that is more violent, with a plunging speed that can
nerofFigure8,leftpanels,closetotheregionofdiskaccre-
be almost as large as the accretor’s escape speed. In sys-
tion. As described in Section 3, we expect that the systems
tems with a large q, the barycenter is usually already close
shouldbedynamicallystable,astheydonotshowanysign
to the accretor’s center of mass, and thus there is no rapid
of a merger after 70−90 orbits of mass transfer.
acceleration within the corotating frame, and thus the final
coalescence is much more gentle. This behavior is clearly
visible in the right panel of Figure 9, which shows that the
4.3 Carbon-Oxygen mass transfer
temperature at minimum (τ /τ )(T) tends to increase
nuc dyn
with increasing q for a fixed total mass. While we find that He accreting WDs yield detonations,
The optimal conditions for ignition are reached when systems accreting CO are unlikely to explode at or prior to
the temperature is at its peak, close to the time of the merger,seetherightpanelsofFigure8.Figure9showsthe
donor’s disruption, provided that the accretor is below 1.1 region of ρ−T plane where a CO detonation can be suc-
M andthedonorisabove0.4M ,seetheupper-leftpanel cessfully ignited (Seitenzahl et al. 2009), together with the
(cid:12) (cid:12)
of Figure 8. For all systems with M <0.9M , individual temperature, both T and (cid:104)T(cid:105), and density of all CO mass-
tot (cid:12)
particletemperaturesT of108 Kandgreaterarereachedin transferring systems at the moment when (τnuc/τdyn)(T) is
the torus formed via the mass transfer, but the density is minimal. Overall, there is a clear tendency to increase the
toolowtoleadtoadynamicalburningevent.Ifweconsider temperatureatminimum(τnuc/τdyn)(T)withthemassratio
the more conservative smoothed temperature criteria, only for a fixed total mass, see the right-panel of Figure 9.
systems with Mtot > 1.3M(cid:12) are capable of detonating at Based on the detonation criteria of Seitenzahl et al.
contact. Under both criteria, systems with q ∼ 1 are more (2009),Pakmoretal.(2011)havefounddetonationsintheir
(cid:13)c 2011RAS,MNRAS000,1–13
10 M. Dan, S. Rosswog, J. Guillochon, E. Ramirez-Ruiz
Figure 9. Left panel: The filled diamonds show (ρ,T) at minimum (τnuc/τdyn)(T) and are connected by black lines to the (ρ,(cid:104)T(cid:105)),
representedbyfilledsquares.Colorcodedisthetotalmassofthesysteminsolarunits.(cid:104)T(cid:105)iscomputedatthelocationofparticlewith
temperatureT.NoneoftheCOmass-transferringsystemsreachestheCOdetonationcriteriaofSeitenzahletal.(2009)(shadedregion)
or the τnuc/τdyn < 1 condition (see the right panels of Figure 8). Right panel: Temperature at minimum (τnuc/τdyn)(T) in q−Mtot
plane.Thetemperatureshowsacleartrendtowardshighervalueswithincreasingmassratioand/orfixedtotalmass.
mergerstudyof(nearly)equalmassWDs.BecausePakmor from 0.2 to 1.2 M and accounted for their different initial
(cid:12)
et al. (2011) used approximate ICs that result in higher chemical compositions.
temperatures and densities during the evolution of mass
While such a broad study is necessarily limited in the
transfer (Dan et al. 2011), we run a direct comparison with
numericalresolutionthatcanbeaffordedineachsimulation,
their0.81+0.9M system.Wecomparedtheminimumob-
(cid:12) wehavefoundexcellentqualitativeagreementwithprevious
tained value of τ /τ with accurate (Dan et al. 2011)
nuc dyn numericalstudies(Danetal.2011)andsemi-analyticalwork
and approximate (same as in Pakmor et al. 2011) ICs for
(Marsh et al. 2004) on the orbital stability of white dwarf
non-rotating stars and found a difference of only 5%, with
binary systems. One of the main conclusions of our previ-
τ /τ ≈ 7. The difference is much larger however, for
nuc dyn ous study (Dan et al. 2011) was that the accuracy of the
systems with lower mass ratios. For the corotating system
numerical initial conditions has a major impact on all sub-
with 0.6+0.9 M components, τ /τ for the approxi-
(cid:12) nuc dyn sequentevolutionarystagesandcanbemoreimportantthan
mateICsisfourordersofmagnitudegreaterthanthevalue
numerical resolution. All our simulations were started from
obtained for using the accurate ICs. The two ICs also lead
carefullyconstructed,initiallytidally-lockedbinaryconfigu-
to important differences in the merger remnant profiles at
rations.Thedynamicallyunstablesystemsinnatureshould
themomentwhenτ /τ isminimum:thetemperatureis
nuc dyn –attheinitialseparationswherenumericallyresolvablemass
overestimatedwhenusingtheapproximateICsandtheden-
transfer sets in– have an accretor star that is desynchro-
sity is larger in the hot envelope surrounding the accretor,
nized with the orbit (New & Tohline 1997). In practice,
where the dynamical burning takes place (Figure 2).
these initial conditions require a cumbersome approach as
desynchronicity is a strong de-stabilizing effect, and there-
fore constructing a relaxation scheme that does not lead to
a premature merger is difficult. However, in all of our sim-
ulations the accretor star becomes desychronized with the
5 DISCUSSION
orbitalperiodafterabout5to10orbitalrevolutions.Wefind
Although small patches of the white dwarf merger parame- systemsthatarenearthedividinglinebetweendiskanddi-
terspacehavebeenexploredpreviouslyinamoderatenum- rect impact accretion can still persist for many dozens of
ber of simulations (Benz et al. 1990; Rasio & Shapiro 1995; orbitspriortomerging,despitethefactthattheaccretoris
Segretain et al. 1997; Guerrero et al. 2004; D’Souza et al. already desynchronous. We also find that as the binary ap-
2006;Yoonetal.2007;Motletal.2007;Lor´en-Aguilaretal. proaches merger, the orbital period changes so quickly that
2009; Pakmor et al. 2010; Fryer et al. 2010; Pakmor et al. corotation cannot be maintained. In other words, while our
2011; Dan et al. 2011; Raskin et al. 2011), the basic ques- initial conditions do not represent the physical state at the
tion:Whichwhitedwarfsystemsproduceexplosionspriorto onsetofresolvablemasstransfer,thesystemsquicklyevolve
ordirectlyatmerger?hassofarremainedunanswered.This to the state that is likely realized in nature.
seriousgapinourunderstandinghasinparticularhampered As expected, systems of a given mass ratio lead to
ourabilitytoconfidentlyjudgethepotentialofDDmergers higherpeaktemperaturesasthetotalsystemmassincreases.
as type Ia progenitor systems. In this paper we close this We additionally find that there is a clear trend to produce
gap by methodically covering the whole parameter space of higherpeaktemperaturesasthemassratioapproachesunity
white dwarf mergers using more than 200 simulation runs. (Figure9).Itcomesasaninterestingandmaybesomewhat
Wehavesystematicallyscannedthewhitedwarfmassrange
(cid:13)c 2011RAS,MNRAS000,1–13