Mon. Not. R. Astron. Soc. 000, 1–?? (2007) Printed 20 August 2019 (MN LATEX style file v2.2)
Remnant evolution after a carbon-oxygen white dwarf merger
S.-C. Yoon1,2⋆, Ph. Podsiadlowski3 and S. Rosswog4
1Astronomical Institute ”Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands
2Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA95064, USA
3Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
4School of Engineering and Science, Jacobs University Bremen†, Campus Ring1, Bremen 28759, Germany
Accepted/ Received
ABSTRACT
We systematically explore the evolution of the merger of two carbon-oxygen (CO) white
dwarfs. The dynamical evolution of a 0.9 M⊙ + 0.6 M⊙ CO white dwarf merger is followed
by a three-dimensional SPH simulation. The calculation uses a state-of-the art equation of
state that is coupled to an efficient nuclear reaction network that accurately approximates all
stages from helium burning up to nuclear statistical equilibrium. We use an elaborate pre-
scription in which artificial viscosity is essentially absent, unless a shock is detected, and a
much larger number of SPH particles than earlier calculations. Based on this simulation, we
suggest that the central region of the merger remnant can, once it has reached quasi-static
equilibrium, be approximated as a differentially rotating CO star, which consists of a slowly
rotating cold core and a rapidly rotating hot envelope surrounded by a centrifugally supported
disc. We construct a model of the CO remnant that mimics the results of the SPH simulation
using a one-dimensional hydrodynamic stellar evolution code and then follow its secular evo-
lution, where we include the effects of rotation on the stellar structure and the transport of
angular momentum. The influence of the Keplerian disc is implicitly treated by considering
mass accretion from the disc onto the hot envelope. The stellar evolution models indicate that
the growth of the cold core is controlled by neutrino cooling at the interface between the core
and the hot envelope, and that carbon ignition in the envelope can be avoided despite high
effective accretion rates. This result suggests that the assumption of forced accretion of cold
matter that was adopted in previous studies of the evolution of double CO white dwarf merger
remnants may not be appropriate. Specifically we find that off-center carbon ignition, which
would eventually lead to the collapse of the remnant to a neutron star, can be avoided if the
following conditions are satisfied: (1) when the merger remnant reaches quasi-static equilib-
rium, the local maximum temperature at the interface between the core and the envelope must
be lower than the critical limit for carbon-ignition. (2) Angular-momentum loss from the cen-
tral merger remnant should not occur on a time scale shorter than the local neutrino cooling
time scale at the interface. (3) The mass-accretion rate from the centrifugally supported disc
must be sufficiently low (Ṁ . 5 × 10−6...10−5 M⊙ yr
−1). Our results imply that at least
some products of double CO white dwarfs merger may be considered good candidates for
the progenitors of Type Ia supernovae. In this case, the characteristic time delay between the
initial dynamical merger and the eventual explosion would be ∼ 105 yr.
Key words: Stars: evolution – Stars: white dwarf – Stars: accretion – Supernovae: general –
1 INTRODUCTION
The coalescence of two carbon-oxygen (CO) white dwarfs with
a combined mass in excess of the Chandrasekhar limit has long
been considered a promising path towards a Type Ia supernova (SN
Ia; Iben & Tutukov 1984; Webbink 1984). Indeed, in the last few
⋆ E-mail: scyoon@science.uva.nl (SCY); podsi@astro.ox.ac.uk (PhP);
s.rosswog@iu-bremen.de (SR)
† formerly International University Bremen
years, a few massive double CO white dwarf systems have been
found that have periods short enough for them to merge within
a Hubble time (e.g. Napiwotzki et al. 2002, 2004). This double-
degenerate (DD) scenario can also easily explain the lack of hy-
drogen and helium lines in most SN Ia spectra and the occur-
rence of SNe Ia both in old and young star-forming systems (e.g.
Branch et al. 1995).
Theoretically, the final fate of double CO white dwarf merg-
ers has been much debated. Previous studies assumed that the dy-
namical disruption of the Roche-lobe filling secondary should lead
c© 2007 RAS
http://arxiv.org/abs/0704.0297v2
2 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 1. Schematic illustration of the configuration of the remnant of a
double CO white dwarf merger once quasi-static equilibrium has been es-
tablished.
to the formation of a thick disc around the primary white dwarf
(Tutukov & Yungelson 1979; Mochkovitch & Livio 1989, 1990).
Therefore, accretion of CO-rich matter from the thick disc onto
the central cold white dwarf has been studied for investigating the
evolution of such mergers by many authors (Nomoto & Iben 1985;
Saio & Nomoto 1985, 1998, 2004; Piersanti et al. 2003a,b). As ac-
cretion rates from the thick disc should be close to the Edding-
ton limit (Ṁ ≈ 10−5 M⊙yr
−1), most of those studies concluded
that carbon ignition in the envelope of the accreting white dwarf
is an inevitable consequence of such rapid accretion of CO-rich
matter. Once carbon ignites off-center, the burning flame propa-
gates inwards on a relatively short time scale (∼ 5000 yr), and
the CO white dwarf is transformed into an ONeMg white dwarf
(Saio & Nomoto 1985, 1998). When the mass of the ONeMg white
dwarf approaches the Chandrasekhar limit, electron capture onto
Ne and Mg is expected to lead to the gravitational collapse of
the white dwarf to a neutron star (Nomoto & Kondo 1991; see
Dessart et al. 2006 and Kitaura, Janka & Hillebrandt 2006 for re-
cent studies of such collapse).
However, the evolution of the remnants of double CO white
dwarf mergers is not yet well understood. For instance, it has
been debated whether the accretion rate decreases when the ac-
creting white dwarf reaches critical rotation (Piersanti et al. 2003a;
Saio & Nomoto 2004). More importantly, the canonical descrip-
tion of the merger remnant as a primary white dwarf + thick
disc system is clearly an oversimplification. In previous three-
dimensional smoothed particle hydrodynamics (SPH) simula-
tions (Benz et al. 1990; Segretain, Chabrier & Mochkovitch 1997;
Guerrero et al. 2004; see also Sect. 2), a large fraction of the dis-
rupted secondary and the outermost layers of the primary form an
extended hot envelope around the cold core containing most of the
primary mass. The rest of the secondary mass becomes a centrifu-
gally supported disc in the outermost layers of the merger rem-
nant. Interestingly, the merger remnant reaches a state of quasi-
static equilibrium within a few minutes from the onset of the dy-
namical disruption of the secondary. As the structure of the cold
core plus the hot envelope appears to have a fairly spheroidal shape
(see below) rather than the toroidal shape obtained with a zero-
temperature equation of state (Mochkovitch & Livio 1989, 1990),
the merger remnant may be better described as a differentially ro-
tating single CO star consisting of a slowly rotating cold core and
a rapidly rotating hot extended envelope surrounded by a Keple-
rian disc, as illustrated in Fig. 1, than the previously adopted pri-
mary white dwarf + thick disc system. The further evolution of the
merger must therefore be determined by the thermal cooling of the
hot envelope and the redistribution of the angular momentum in-
side the central remnant, and accretion of matter onto the envelope
from the Keplerian disc.
With this new approach to the problem in mind, we here re-
visit both the dynamical and the secular evolution of double CO
white dwarf mergers. In the following section (Sect. 2), we present
the numerical results of an SPH simulation of the dynamical evolu-
tion of the coalescence of a 0.9 M⊙ WD and a 0.6 M⊙ CO white
dwarf up to the stage of quasi-hydrostatic equilibrium, and we care-
fully investigate the structure of the merger remnant. In Sect. 3, we
construct models of the central remnant in quasi-static equilibrium
state (primary + hot extended envelope) which mimic the SPH re-
sult and calculate the thermal evolution of the merger remnant using
a hydrodynamic stellar evolution code. In particular, the conditions
for avoiding off-center carbon ignition are systematically explored.
In Sect. 4, we conclude this work by discussing uncertainties in our
assumptions, the implications for Type Ia supernovae and future
work.
2 DYNAMICAL EVOLUTION OF THE MERGER
Before discussing the subsequent thermal evolution after the co-
alescence of a double CO white dwarf coalescence binary, we
investigate the configuration of the remnant in quasi-static equi-
librium in some detail. For this purpose, we have carried out
a SPH simulation of the dynamical process of the coalescence
of two CO white dwarfs of 0.9 M⊙ and 0.6 M⊙, respec-
tively. Our simulation uses a 3D smoothed particle hydrodynam-
ics (SPH) code that is an offspring of a code developed to simu-
late neutron star mergers (Rosswog et al. 2000; Rosswog & Davies
2002; Rosswog & Liebendörfer 2003). It uses an artificial viscos-
ity scheme with time-dependent parameters (Morris & Monaghan
1997). In the absence of shocks, the viscosity parameters have a
very low value (α = 0.05 and β = 0.1; most SPH implemen-
tations use values of α = 1...1.5 and β = 2...3); if a shock is
detected, a source term (Rosswog et al. 2000) guarantees that the
parameters rise to values that are able to resolve the shock prop-
erly without spurious post-shock oscillations. To suppress artificial
viscosity forces in pure shear flows, we additionally apply a switch
originally suggested by Balsara (1995).
To account for the energetic feedback onto the fluid from
nuclear transmutations, we use a minimal nuclear reaction net-
work developed by Hix et al. (1998). It couples a conventional α-
network stretching from He to Si with a quasi-equilibrium-reduced
α-network. Although a set of only seven nuclear species is used,
this network reproduces the energy generation of all burning stages
from He-burning to NSE very accurately. For details and tests we
refer to Hix et al. (1998). We use the HELMHOLTZ equation of
state (EOS), developed by the Center for Astrophysical Thermonu-
clear Flashes at the University of Chicago. This EOS allows to
freely specify the chemical composition of the gas and can be cou-
pled to nuclear reaction networks. The electron/positron equation
of state has been calculated without approximations, i.e. it makes
no assumptions about the degree of degeneracy or relativity; the
exact expressions are integrated numerically to machine precision.
The nuclei in the gas are treated as a Maxwell-Boltzmann gas,
the photons as blackbody radiation. The EOS is used in tabular
form with densities ranging from 10−10 6 ρYe 6 10
11 g cm−3
and temperatures from 104 to 1011 K. A sophisticated, biquin-
tic Hermite polynomial interpolation is used to enforce thermody-
namic consistency (i.e. the Maxwell-relations) at interpolated val-
ues (Timmes & Swesty 2000).
We use a MacCormack predictor-corrector method (e.g.
Lomax et al. 2001) with individual particle time steps to evolve
the fluid. With our standard parameters for the tree-opening cri-
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 3
Figure 2. Dynamical evolution of the coalescence of a 0.6 M⊙ + 0.9 M⊙ CO white dwarf binary. The panels in the left column show the density in the
orbital plane, the panels in the right column the temperature in units of 106 K. Lengths are in code units (= 109 cm).
c© 2007 RAS, MNRAS 000, 1–??
4 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 3. Dynamical evolution of the coalescence of a 0.6 M⊙ + 0.9 M⊙ CO white dwarf binary. Continued from Fig. 2.
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 5
Figure 4. The evolution of the local peak of temperature during the merger
of two CO white dwarfs of 0.6M⊙ and 0.9M⊙, respectively, as a function
of time after the onset of the simulation.
terion and the integration, this time marching implementation con-
serves the total energy to better than 4 × 10−3 and the total an-
gular momentum to better than 2 × 10−4. Note that this could,
in principle, be improved even further by taking into account the
so-called “grad-h”-terms (Springel & Hernquist 2002; Monaghan
2002; Price 2004) and extra-terms arising from adapting gravita-
tional smoothing terms (Price & Monaghan 2006).
To avoid numerical artifacts, we only use equal mass SPH
particles. For the initial conditions, we therefore stretch a uniform
particle distribution according to a function that has been derived
from solving the 1D stellar structure equations. This technique is
described in detail in Rosswog, Ramirez-Ruiz & Hix (2007). This
particle setup is then further relaxed with an additional damping
force (e.g. Rosswog, Speith & Wynn 2004) so that the particles can
settle into their true equilibrium configuration. The calculations are
performed with 2×105 SPH particles, a much larger particle num-
ber than could be afforded by previous calculations, and run up to
a much longer evolutionary time (5 minutes) than previous calcu-
lations (see Table 1).
Figs. 2 and 3 show the dynamical evolution of the merging
process of the double white dwarf system considered in this study.
The panel in the left columns show the densities and the panel in
the right columns the temperatures (in units of 106 K) in the orbital
plane. The secondary is completely disrupted within 1.7 minutes,
and mass accretion onto the primary induces local heating near the
surface of the primary. Fig. 4 shows the evolution of the maximum
temperature as a function of time. The peak in temperature reaches
1.7×109 K at t ≃ 1.0 min, where t = 0.0 marks the moment when
the simulation starts. Carbon ignites when T & 109 K, but nuclear
burning is quenched soon due to the local expansion of the hottest
region, as is also observed in the simulations of Guerrero et al.
(2004). The total amount of energy released due to nuclear burn-
ing is about 1045 erg.
Segretain, Chabrier & Mochkovitch (1997) considered the
same initial white dwarf masses as in the present study. But
they adopted the original artificial viscosity prescription of
Monaghan & Varnas (1988), which is known to introduce spuri-
ous forces in shear flows, and they did not include nuclear burning
(Table 1). By the end of their calculation (t = 1.56 min), Tmax
reached 8×108 K, while in our simulation, it decreases to 8×108 K
only when t ∼ 1.7 min. Interestingly, Tmax decreases further af-
Figure 5. Top: Density contour of the merger remnant in the x− z plane at
t = 5.3 min. Here one code unit corresponds to 109 cm. Middle: Thermo-
dynamic structure of the merger remnant at t = 5.31 min: shown are the
temperature and the density as a function of distance from the centre, along
the positive x- and z-axis, as indicated. Bottom: Angular velocity in units
of the local Keplerian value at t = 5.31 min, along the positive/negative
x- and y-axis of the merger remnant.
terwards in our calculation, as shown in Fig. 4, and reaches a steady
value at Tmax ≃ 5.6 × 10
8 K when t & 2.5min. In the other cal-
culations by Benz et al. (1990) and Guerrero et al. (2004), the dy-
namical evolution of the merger was not followed for more than 2
minutes either, and we cannot directly compare our results to theirs.
However, we suspect that Tmax would also decrease further in the
systems they considered if they had continued their calculations for
a longer evolutionary time. It should also be noted that energy dis-
sipation by artificial viscosity might lead to overheating, and that
c© 2007 RAS, MNRAS 000, 1–??
6 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
thermal diffusion – which may play an important role in the outer-
most layers – is not considered in the present study. It is thus likely
that Tmax in the quasi-static equilibrium state may be even lower
in reality than in our simulation.
Fig. 5 shows the structure of the merger remnant at quasi-static
equilibrium. The central region with R . 109 cm (Mr . 1.1 M⊙)
has a fairly spheroidal shape, and a centrifugally supported disc
appears at R & 109 cm where the angular velocity is close to
the Keplerian value. The fraction of the secondary mass contained
in the Keplerian disc is larger in our simulation (about 67%) than
in Segretain, Chabrier & Mochkovitch (1997) (about 41 %). The
innermost core (R . 3 × 108 cm; Mr . 0.6 M⊙) is essen-
tially isothermal, and the temperature has its peak value (T ≃
5.6 × 108 K) at R ≃ 5 × 108 cm and Mr ≃ 0.85 M⊙. The
disc material extends over 4× 109 cm along the z-axis as the tem-
perature is still high; if thermal diffusion were included, the disc
would become much thinner on a short time scale of a few hours.
Therefore, our simulation confirms the remnant structure at quasi-
static equilibrium that is illustrated in Fig. 1. In the next section, we
investigate the secular evolution of the merger from such a quasi-
static equilibrium state.
3 SECULAR EVOLUTION OF THE MERGER REMNANT
3.1 Physical assumptions and methods
Our SPH simulation shows that the remnant of the merger of two
CO white dwarfs (0.9 M⊙ + 0.6 M⊙) in the state of quasi-static
equilibrium has the following features (see Fig. 6):
(i) The core is cold and nearly isothermal.
(ii) The local peak of temperature (Tp) is located at a mass co-
ordinate slightly less than the primary mass.
(iii) A steep gradient in temperature appears at the interface be-
tween the core and the local peak of temperature.
(iv) The interface is rather widely extended into the primary
(∆Minterface ≈ 33 % of the primary mass), and the mass of the
quasi-isothermal cold core (Mcore) is about 77 % of the primary
mass.
(v) The mass of the outer envelope above the local peak of tem-
perature contains about 33 % of the mass of the secondary, and the
rest of the secondary forms a Keplerian disc.
Let us define Tp as the local peak of temperature at quasi-static
equilibrium, MCM as the mass of the central remnant (cold core +
hot envelope), and Mp as the location of Tp in the mass coordi-
nate (i.e., Mp = Mcore + ∆Minterface; see Fig. 6). To construct
models of the central remnant, we use a one-dimensional hydrody-
namic stellar evolution code which incorporates the effects of ro-
tation on the stellar structure, transport of angular momentum due
to the shear instability, Eddington-Sweet circulation, and the Gol-
dreich, Schubert and Fricke instability, and dissipation of rotational
energy due to shear motions. The effects of magnetic fields are ne-
glected (see Sec. 4). More details about the code are described in
(Yoon & Langer 2004; hereafter YL04) and references therein.
In order to mimic the temperature profile of the central rem-
nant as obtained from the SPH simulation, we artificially deposit
energy in the envelope, using the following prescription for a white
dwarf with M = MCR:
e(Mr) = A(T
(Mr)− T (Mr)) [erg g
], (1)
where
Figure 6. Initial model of the central remnant for sequences Sa1 – Sa11.
The top and middle panels show temperature as a function of the mass co-
ordinate and radius, respectively. The solid curve in the bottom panel gives
the angular-velocity profile as a function of radius. The dashed curve de-
notes the angular velocity in units of the local Keplerian value.
T ′(Mr) =
3 · 107 K+ (7 · 107 K− 3 · 107 K)
Mcore
if Mr < Mcore,
Tp − (Tp − 7 · 10
Mp−Mr
Mp−Mcore
if Mcore 6 Mr 6 Mp,
C − (C − Tp)
log[ρ(Mr)/ρs]
log[ρ(Mp)/ρs]
if Mr > Mp.
In this way, the temperature profile in the central remnant model
follows T ′(Mr). Here, A and C are constants. We use A =
105 erg g−1 s−1 K−1 and C = 2× 108 K in most cases.
A rotational profile is imposed as
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 7
Table 1. Comparison of SPH simulations of double CO white dwarf mergers. The columns list: M1 and M2: the masses of the primary and the secondary,
respectively; NoP: the total number of particles used; νsph: the type of artificial viscosity employed, ‘std.’ refers to Monaghan & Varnas (1988); Network:
type of nuclear network employed; tsim: evolutionary time that has elapsed by the end of the calculation; Tmax: maximum temperature obtained during the
simulation; and Tp: the local peak of temperature at the end of the calculation
Ref.∗ M1 M2 NoP νsph Network tsim Tmax TP
1 1.2 M⊙ 0.9 M⊙ ∼ 7× 103 std. None 51 sec. ? ∼ 109 K
2 0.8 M⊙ 0.6 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 50 sec. 1.4× 109 K ?
2 1.0 M⊙ 0.6 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 65 sec. 1.6× 109 K ?
2 1.0 M⊙ 0.8 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 65 sec. 2.0× 109 K ?
3 0.9 M⊙ 0.6 M⊙ ∼ 6× 104 std None 1.56 min. ? ∼ 7× 108 K
4 0.9 M⊙ 0.6 M⊙ 2× 105 see Rosswog et al. (2000) QSE-alpha network 5.3 min. 1.7× 109 K 5.6× 108 K
∗1: Benz et al. (1990), 2: Guerrero et al. (2004), 3: Segretain, Chabrier & Mochkovitch (1997); 4: Present Study
Table 2. Merger remnant model sequences. Each column lists the following: No.: sequence label, MCR: mass of the central remnant, Mcore: mass of the
quasi-isothermal core, Mp: location of the local peak of temperature in the mass coordinate, Tp: the local peak of temperature, ρp: density at Mr = Mp,
τJ: adopted time scale for angular momentum loss according to Eq. (4), Ṁacc: adopted mass accretion rate from the Keplerian disc, C-ig: off-center ignition
of carbon, MWD,ig: total mass of the central remnant when off-center carbon ignition occurs, Mr,ig: location of off-center carbon ignition in the mass
coordinate.
No. MCR Mcore Mp Tp ρp τJ Ṁacc C-ig MWD,ig Mr,ig
[M⊙] [M⊙] [M⊙] [108 K] [106 g/cm3] [yr] 10−6 M⊙/yr [M⊙] [M⊙]
Sa1 1.11 0.6 0.84 5.6 0.8 ∞ 0.0 No - -
Sa2 1.11 0.6 0.84 5.6 0.8 102 0.0 Yes 1.11 0.80
Sa3 1.11 0.6 0.84 5.6 0.8 103 0.0 Yes 1.11 0.80
Sa4 1.11 0.6 0.84 5.6 0.8 104 0.0 Yes 1.11 0.85
Sa5 1.11 0.6 0.84 5.6 0.8 105 0.0 No - -
Sa6 1.11 0.6 0.84 5.6 0.8 105 10.0 Yes 1.34 1.09
Sa7 1.11 0.6 0.84 5.6 0.8 105 5.0 Yes 1.34 1.20
Sa8 1.11 0.6 0.84 5.6 0.8 105 2.0 No - -
Sa9 1.11 0.6 0.84 5.6 0.8 105 1.0 No - -
Sa10 1.11 0.6 0.84 5.6 0.8 5 · 105 5.0 No - -
Sa11 1.11 0.6 0.84 5.6 0.8 5 · 105 1.0 No - -
Aa1 1.25 0.6 0.93 5.0 2.3 ∞ 0.0 No - -
Aa2 1.25 0.6 0.93 5.0 2.3 102 0.0 Yes 1.250 0.90
Aa3 1.25 0.6 0.93 5.0 2.3 103 0.0 Yes 1.250 0.92
Aa4 1.25 0.6 0.93 5.0 2.3 104 0.0 Yes 1.250 1.12
Aa5 1.25 0.6 0.92 5.0 2.3 105 0.0 No - -
Aa6 1.25 0.6 0.92 5.0 2.3 106 0.0 No - -
Aa7 1.25 0.6 0.92 5.0 2.3 105 10.0 Yes 1.360 1.20
Aa8 1.25 0.6 0.92 5.0 2.3 105 5.0 No - -
Aa9 1.25 0.6 0.92 5.0 2.3 105 1.0 No - -
Aa10 1.25 0.6 0.92 5.0 2.3 106 10.0 Yes 1.382 1.22
Ab1 1.25 0.7 0.92 5.0 3.1 ∞ 0.0 No - -
Ab2 1.25 0.7 0.92 5.0 3.1 103 0.0 Yes 1.250 0.97
Ab3 1.25 0.7 0.92 5.0 3.1 104 0.0 No - -
Ab4 1.25 0.7 0.92 5.0 3.1 105 0.0 No - -
Ab5 1.25 0.7 0.92 5.0 3.1 105 10.0 Yes 1.344 1.21
Ab6 1.25 0.7 0.92 5.0 3.1 105 5.0 No - -
Ac1 1.25 0.5 0.88 5.9 1.6 ∞ 0.0 Yes 1.250 0.84
Ad1 1.25 0.6 0.92 6.0 1.8 ∞ 0.0 Yes 1.250 0.90
Ad2 1.25 0.6 0.92 6.0 1.8 106 5.0 Yes 1.252 0.90
Ae1 1.25 0.6 0.90 6.8 1.5 ∞ 0.0 Yes 1.250 0.87
Ba1 1.363 0.82 0.95 5.0 12.2 ∞ 0.0 No - -
Ba2 1.363 0.82 0.95 5.0 12.2 102 0.0 Yes 1.363 0.95
Ba3 1.363 0.82 0.95 5.0 12.2 103 0.0 Yes 1.363 1.12
Ba4 1.363 0.82 0.95 5.0 12.2 104 0.0 No - -
Ba5 1.363 0.82 0.95 5.0 12.2 105 0.0 No - -
Ba6 1.363 0.82 0.95 5.0 12.2 105 10.0 Yes 1.398 1.34
Ba7 1.363 0.82 0.95 5.0 12.2 105 5.0 Yes 1.483 1.43
Ba8 1.363 0.82 0.95 5.0 12.2 105 1.0 No - -
Ta1 1.25 0.60 0.86 5.0 28.8 ∞ 0.0 No - -
c© 2007 RAS, MNRAS 000, 1–??
8 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 7. (a) Evolution of a non-rotating white dwarf accreting with a constant accretion rate of Ṁ = 10−5 M⊙ yr−1 with an initial mass of 0.9 M⊙
(Seq. N0.9) in the density – temperature plane. (b) The local effective accretion rate (Ṁeff,r := 4πr
2ρv) as a function of density in Seq. N0.9, at different
evolutionary epochs as indicated by the labels. (c) – (f) The rates of energy loss/production due to neutrino (ǫν ) cooling, compressional heating (ǫcomp),
nuclear energy generation (ǫnuc) and thermal diffusion (ǫth) at different evolutionary epochs. Note that here ǫν , ǫcomp and ǫnuc represent the values which
are used in the evolutionary calculations, while ǫth is an order-of-magnitude estimate according to Eq. (7).
Ω(Mr) =
ΩO, if Mr < Mcore,
ωO + (1− ωO)
Mr−Mcore
MCR−Mcore
if Mr > Mcore,
where ΩO = 0.2
GMCR/R3 and ωO = ΩO/
GMcore/r
core.
As shown in Fig. 6, this simple assumption gives a rotational ve-
locity profile that is morphologically similar to that found in the
SPH simulation: a steep gradient at the interface between the core
and the envelope, and a local peak in the envelope. Within our 1-D
approximation of the effects of rotation, the exact shape of the ro-
tational velocity profile does not affect the main conclusions of the
present work for the following reasons. Firstly, the velocity gradient
at the interface is adjusted to the threshold value for the dynamical
shear instability on a very short time scale (see below, and discus-
sions in YL04). Secondly, our 1-D approximation underestimates
the effect of the centrifugal force on the stellar structure in layers
which rotate more rapidly than about 60 % critical (YL04), and un-
certainties due to this limit are much greater than due to the shape
of Ω(r) in the outer layers of the envelope. Possible uncertainties
due to this limitation are critically discussed in Sect. 4.
The central remnant may lose angular momentum by out-
ward angular momentum transport into the Keplerian disc
(Popham & Narayan 1991; Paczyński 1991) and/or by the gravita-
tional wave radiation, e.g., due to the r-mode instability (Andersson
1998; Friedman & Morsink 1998). Our code cannot properly de-
scribe any of these effects, and here we consider them simply by
assuming a constant time scale for the angular momentum loss (τJ;
see Knaap 2004; cf. Piersanti et al. 2003a), such that the specific
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 9
Table 3. Accreting white dwarf model sequences with a constant accretion
rate (Ṁ = 10−5 M⊙/yr). The columns list: No: sequence label, Minit:
initial mass, logLinit/L⊙ : initial luminosity, MWD,ig: the total mass of
the white dwarf when carbon ignites off-center, and Mr,ig: location of car-
bon ignition in the mass coordinate. Sequences with ‘N’ are for non-rotating
models, and ‘R’ for rotating models.
No Minit logLinit/L⊙ MWD,ig Mr,ig
N0.7 0.7 −2.118 0.999 0.793
N0.8 0.8 −2.128 1.010 0.862
N0.9 0.9 −2.188 1.039 0.939
N1.0 1.0 −2.137 1.087 1.024
N1.1 1.1 −2.170 1.150 1.114
N1.2 1.2 −2.119 1.225 1.207
R0.8 0.8 −2.114 1.297 1.038
R0.9 0.9 −2.119 1.249 1.050
R1.0 1.0 −2.082 1.207 1.069
R1.1 1.1 −2.050 1.205 1.127
angular momentum of each mass shell decreases over a time step
∆t by an amount
∆ji = ji [1− exp(−∆t/τJ)] . (4)
Mass accretion from the Keplerian disc is also considered in
some model sequences, with different values for the accretion rate
(Ṁacc). The angular-momentum accretion is treated in the same
way as in YL04: the accreted matter is assumed to carry angular
momentum at a value close to the Keplerian value if the surface
velocity of the central remnant is below critical, while no angular-
momentum accretion is allowed otherwise.
Model sequences with different sets of MCR, Mcore, Mp, τJ,
Ṁ , and Tp are calculated, as summarized in Table 2. The initial
model in Seqs S is intended to reproduce the result of our SPH sim-
ulation, where MCR = 1.10 M⊙ and Mp ≈ 0.84 M⊙ are adopted.
We also assume MCR = 1.25 M⊙ and Mp ≈ 0.9 M⊙ in Seq. A,
and MCR = 1.364 M⊙ and Mp = 0.95 M⊙ in Seq. B, to simulate
mergers of 0.9 − 1.0 M⊙ + 0.7 − 1.0 M⊙ white dwarf binaries.
At a given MCR, different sets of Mcore, Mp, and Tp are marked
in the sequence label by minor characters (a, b, c, d, e), while dif-
ferent sets of τJ and Ṁacc are indicated by Arabian numbers. For
instance, sequences Sa1 – Sa11 have the same initial merger model,
but different values for τJ and Ṁacc. Rotation is neglected in a test
sequence Ta1 (i.e., the models are non-rotating). The temperature
and angular-velocity profiles in the initial central remnant model of
Seqs Sa1 - Sa11 are shown in Fig. 6. The temperature (a few to sev-
eral 108 K) and the size (∼ 109 cm) of the envelope appear to be
comparable to those obtained from the SPH simulation (see Fig. 5).
For comparison, we also ran model sequences for classi-
cal cold-matter accretion with a constant accretion rate of Ṁ =
10−5 M⊙/yr, for both non-rotating and rotating cases, as summa-
rized in Table 3.
3.2 Results
3.2.1 Classical models of cold-matter accretion
Before discussing the central remnant models, let us first investi-
gate the evolution of classical cold-matter accreting white dwarf
models in detail. In these models, the accreted matter is assumed to
have the same entropy as the surface value of the accreting white
dwarf. As shown in previous studies (e.g. Nomoto & Iben 1985),
the thermal evolution of rapidly accreting white dwarfs is deter-
Figure 8. Local effective mass accretion rate (Ṁeff,r ≡ 4πr
2ρv) as a
function of density in the models of sequence R0.9, at different evolutionary
epochs.
mined by the interplay of compressional heating and thermal dif-
fusion. Fig. 7 shows an example of the evolution of such accret-
ing white dwarf models for an initial WD mass of 0.9 M⊙ and
a constant accretion rate of Ṁacc = 10
−5 M⊙ yr
−1 (Seq. N0.9;
Table 3).
Fig. 7a shows that the temperature increases continuously in
the envelope (ρ . 106 g cm−3), and finally carbon burning be-
comes significant at ρ ≃ 5.6 × 105 g cm−3 and T ≃ 6× 108 K
when t ≃ 1.3 × 104 yr. In Figs. 7c –f, the rates of compressional
heating (ǫcomp), neutrino cooling (ǫν ), nuclear energy generation
(ǫnuc) and thermal diffusion (ǫth) are shown. In our stellar evolu-
tion code, the compressional heating rate is calculated according
ǫcomp =
(Kippenhahn & Weigert 1990). Neutrino cooling rates are obtained
following Itoh et al. (1996). While ǫcomp, ǫν , and ǫnuc in the figures
correspond to the values that are used for the evolutionary calcula-
tions, the thermal diffusion rate (ǫth) – which is only calculated
implicitly in the code – can only be estimated to within an order-
of-magnitude from
ǫth ≈ TCP/τth . (6)
Here CP denotes the specific heat at constant pressure, and τth the
local thermal diffusion time scale defined as
τth ≡ H
P/K , (7)
where HP is the pressure scale height, and K
[(4acT 3)/(3CPκρ
2)] is the thermal diffusivity. It is clear
from Fig. 7 that the local peak of temperature is located where the
compressional heating rate begins to dominate over the thermal
diffusion rate (ρ ≈ 105 g cm−3), as expected. The neutrino
cooling rate also increases as the temperature in the envelope
becomes higher, but nuclear energy generation becomes significant
before neutrino cooling dominates the thermal evolution, inducing
a carbon-burning flash around ρ ≃ 5.6 · 105 g cm−3.
As Table 3 shows, and consistent with the findings of
Nomoto & Iben (1985), such off-center carbon flashes occur re-
gardless of the initial mass of the white dwarf, if Ṁacc ≈
10−5 M⊙ yr
−1. The results with models including rotation show
that carbon ignition may be delayed if the effect of rotation is in-
cluded (Table 3; see also Piersanti et al. 2003a and Saio & Nomoto
2004). The reason is that the local effective mass accretion rate
c© 2007 RAS, MNRAS 000, 1–??
10 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 9. (a) Evolution of the central remnant in Seq. Sa1 in the log ρ − T plane. The dotted curve gives the critical temperature where the nuclear energy
generation rate due to carbon burning equals the energy loss rate due to neutrino cooling. (b) The local effective accretion rate (Ṁeff,r ≡ 4πr
2ρvr) as a
function of density in the merger remnant model of Seq. Sa1, at different evolutionary epochs as indicated by the labels. (c) – (f) The rates of energy loss/gain
due to neutrino (ǫν ) cooling, compressional heating (ǫcomp), nuclear energy generation (ǫcomp) and thermal diffusion (ǫth) as a function of density in the
central remnant models of Seq. Sa1 at different evolutionary epochs. Note that here ǫν , ǫcomp and ǫnuc represent the values which are used in the evolutionary
calculations, while ǫth is an order-of-magnitude estimate according to Eq. (7).
(Ṁeff,r ≡ 4πr
2ρv) inside the white dwarf at a given mass is lower
because of the centrifugal force. For instance, in Seq. N0.9, we
have Ṁeff,r ≈ 10
−5 M⊙ yr
−1 at around ρ = 5 × 105 g cm−3
when t ≃ 104 yr (Fig. 7b), but Ṁeff,r is lowered by a factor of
two in the corresponding rotating model at a similar epoch (i.e.,
Ṁeff,r ≈ 5 × 10
−6 M⊙ yr
−1), as revealed in Fig. 8. However,
carbon ignition occurs well before the white dwarf reaches the
Chandrasekhar limit, in all model sequences considered. Thus, ro-
tation by itself cannot change the conclusion of the previous work
that the coalescence of double CO white dwarfs should lead to
accretion-induced collapse rather than a thermonuclear explosion,
unless the accretion rate is significantly lowered, as was also shown
by Piersanti et al. (2003a) and Saio & Nomoto (2004).
3.2.2 Sequences without angular-momentum loss and mass
accretion
Having understood the physics of the thermal evolution of CO
white dwarfs which accrete cold matter with a rate close to the Ed-
dington limit, we now investigate the evolution of the central rem-
nant model consisting of a cold core and a hot envelope as described
in Sect. 3.1. First, we examine the results of the model sequences
where both angular-momentum loss and mass accretion from the
Keplerian disc are neglected (i.e., τJ = ∞ and Ṁ = 0; Seqs Sa1,
Aa1, Ab1, Ac1, Ad1, Ae1, Ba1, & Ta1).
Fig. 9a illustrates the evolution of the central remnant for
MCR = 1.10 M⊙ in Seq. Sa1 in the density – temperature
plane. Note that the local peak of temperature at t = 0.0 (Tp =
5.6× 108 K) is significantly below the critical temperature for car-
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 11
bon ignition (TC−ig; dotted curve in Fig 9a). It is shown in Fig. 9b
that the local effective accretion rate (Ṁeff,r) remains relatively
high (5×10−6−10−5 M⊙ yr
−1) around ρ = 106 g cm−3, where
the local peak of temperature is located, for about 5000 yrs. Despite
such high effective accretion rates, the temperature peak continu-
ously decreases, although the inner core becomes somewhat hotter
due to compression, and the central remnant finally becomes a cold
white dwarf. A few remarkable differences compared to the stan-
dard accreting white dwarf models are found in this regard. Firstly,
since the envelope is very hot, neutrino cooling – in particular by
photoneutrinos – is significant from the beginning, and even domi-
nant over the thermal diffusion at the interface between the core and
the envelope as shown in Fig. 9c. In cold-matter accreting white
dwarfs, neutrino cooling becomes important only after a significant
amount of mass has been accreted (Fig. 7). Secondly, the compres-
sional heating rate is slightly lower than the neutrino cooling rate
around the local peak of temperature. As the contraction of the cen-
tral remnant is mainly determined by the thermal evolution of the
envelope, the local accretion rate is in fact controlled by the cool-
ing process. This explains why we have ǫcomp ≈ ǫν around the
local peak of temperature for the initial ∼ 104 yrs, and why the
local peak of temperature continuously decreases despite the rela-
tively high effective accretion rate. This conclusion is the same for
all other sequences with a TP that is significantly lower than TC−ig
(Seq. Aa1, Ab1, & Ba1) including the non-rotating case (Seq. Ta1;
Table 2).
We find that, in Seq. Sa1, the differentially rotating layers at
the interface between the core and the envelope in the initial central
remnant model are stable against the dynamical shear instability
(DSI). They are, however, unstable to the DSI in other sequences,
where the interface is more degenerate (see YL04 for discussions
on the DSI). Consequently, in Seq. Aa1 for example, the rate of
rotational energy dissipation (ǫrot) appears to be very high initially
(Fig. 10). The differentially rotating layers are rapidly smeared out
by the dynamical shear instability (see the discussion in Sect. 2 in
YL04), and ǫrot falls below the thermal diffusion and/or neutrino
cooling rate only within 20 yrs. Hence we conclude that the rota-
tional energy dissipation does not play an important role for the
long-term evolution of the central remnant.
Fig. 11 shows how the evolution of the central remnant
changes if the local peak of temperature in the initial model (Tp) is
close to or above the critical limit for carbon burning (TC−ig), with
Seqs Ac1 and Ae1 as examples. In contrast to Seq. Sa1 or Aa1, car-
bon burning dominates the evolution very soon in both sequences,
and the temperature increases rapidly. Although the further evolu-
tion has not been followed in the present study, it is most likely that
the carbon-burning flame propagates inward such that the central
remnant is converted into an ONeMg white dwarf within several
thousand years as shown by Saio & Nomoto (1998).
As summarized in Table 2, all other sequences follow the same
evolutionary pattern: off-center carbon ignition is avoided in Seqs
Ab1, Ba1, and Ta1 where Tp is significantly below TC−ig, while
carbon ignites off-center in the other sequences where Tp & TC−ig.
It is thus remarkable that the thermal evolution of the central rem-
nant is sensitively determined by the local peak of temperature in
the quasi-static equilibrium state.
In conclusion, in the absence of angular-momentum loss and
mass accretion from the Keplerian disc, the thermal evolution of
the central remnant is roughly controlled by neutrino cooling at the
interface between the core and the envelope, and off-center carbon
burning may be avoided as long as Tp < TC−ig, while it seems
inevitable if Tp & TC−ig.
Figure 10. Upper panel: The angular velocity relative to the local maximum
as a function of the mass coordinate in the central remnant models of Seq.
Aa1 at different evolutionary epochs. Lower panel: the rate of rotational
energy dissipation (ǫrot; see YL04) as a function of the mass coordinate in
the corresponding models shown in the upper panel.
3.3 Effect of angular momentum loss
In Seqs Sa2 – Sa5, the central remnant has the same initial condi-
tions as in Seq. Sa1, angular momentum loss from the white dwarf
with different time scales τJ is considered according to Eq. (4).
Note that off-center carbon ignition occurs in Seqs Sa2, Sa3 &
Sa4, where τJ . 10
4 yr, while it is avoided in Seq. Sa5 where
τJ = 10
5 yr. These results indicate that off-center carbon igni-
tion should be induced if the angular-momentum loss occurs too
rapidly for neutrino cooling or thermal diffusion to control the ef-
fective mass accretion. For instance, Fig. 12 shows that in Seq. Sa4,
where τJ = 10
4 yr, the effective mass accretion rate reaches a few
10−5 M⊙ yr
−1 at the interface between the core and the envelope
(ρ ≈ 106 g cm−3), and the compressional heating rate exceeds the
neutrino cooling rate.
It is shown that the critical angular-momentum-loss time
scale, τJ, for off-center carbon ignition (τJ,crit) is smaller for Seqs
Ab and Ba than for Seqs Sa and Aa: τJ,crit ≈ 10
3 for Seqs Ab
and Ba, and τJ,crit ≈ 10
4 for Seqs Sa and Aa. This is due to the
different local thermodynamic properties at the interface between
the core and the envelope in different central remnant models. As
shown in Fig. 13, higher density and/or temperature at the interface
result in a shorter neutrino cooling time, making it possible to avoid
local heating for a smaller τJ. In other words, τJ,crit roughly cor-
responds to the time scale for neutrino cooling at the local peak of
temperature (τν,p).
From this experiment, we conclude that, in the absence of
c© 2007 RAS, MNRAS 000, 1–??
12 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 11. Evolution of the central remnant in Seqs Ac1 (upper panel) and
Ae1 (lower panel), in the log ρ−T plane. The dotted curve gives the critical
temperature where nuclear energy generation rate due to carbon burning
equals to energy loss rate due to neutrino cooling.
mass accretion from the Keplerian disc, carbon ignition may be
avoided in the central remnant, if Tmax,init < TC−ig, and if
τJ > τν,p.
3.4 Mass accretion from the Keplerian disc
In reality, mass accretion from the Keplerian disc onto the central
remnant is expected. The accretion rate is determined by the vis-
cosity of the disc, which is not well known. However, we expect
the accretion rate from a Keplerian disc may be significantly lower
than from a pressure-supported thick disc that was assumed in pre-
vious studies. Our results, as summarized in Table 2, indicate that
even with mass accretion, the central remnant with Tp < TC−ig can
avoid off-center carbon ignition if the accretion rate is sufficiently
low (i.e., Ṁ < 5× 10−6...10−5 M⊙ yr
−1), and if τJ > τν,p (see
Table 2).
The thermal history of the central remnant in those sequences
where carbon ignites off-center is similar to that of the white dwarf
in classical accretion model sequences. However, as the central
remnant has a rapidly rotating hot envelope, carbon ignition is sig-
nificantly delayed compared to the case of classical accretion. In
Seq. N1.2, where Minit = 1.2 M⊙ and Ṁacc = 10
−5 M⊙ yr
carbon ignites only when about 0.025 M⊙ is accreted, while in
Seq. Aa7 more than 0.15 M⊙ have to be accreted to induce carbon
ignition at the same accretion rate, despite its higher initial mass.
On the other hand, the comparison of Seq. Aa7 with Seq. Aa10
indicates that off-center carbon ignition is delayed if the central
remnant keeps more angular momentum. The critical accretion rate
for inducing off-center carbon ignition is thus difficult to precisely
determine, as our 1-D models significantly underestimate the effect
of the centrifugal force, especially in the envelope where carbon
ignites. In addition, the physics of angular momentum loss/gain is
not well understood yet, as discussed in Yoon & Langer (2005).
Note that MWD,ig in Seqs Aa10, Ba6 and Ba7 is already very
close, or even above the Chandrasekhar limit. However, the cen-
tral density in those models is still smaller by an order of mag-
nitude than the critical limit for carbon ignition due to the effect
of rotation. As the carbon-burning flame will propagate inwards
within several thousand years (Saio & Nomoto 1998), only about
∼ 0.05 M⊙ may be further accreted by the time the burning flame
reaches the center, and the central density may not become high
enough to induce a thermonuclear explosion before the whole cen-
tral remnant is converted into an ONeMg white dwarf. (Super-)
Chandrasekhar mass ONeMg white dwarfs produced in this way
will eventually collapse to a neutron star (see Yoon & Langer 2005;
Dessart et al. 2006).
On the other hand, the white dwarf continuously grows
to/above the Chandrasekhar limit (≈ 1.4 M⊙) without suffering
carbon ignition (neither at the center nor off-center) in Seqs Sa8,
Sa9, Sa10, Sa11, Aa8, Aa9, Ab6, and Ba8. The outcome in these
cases is thus the formation of a (super-) Chandrasekhar mass CO
white dwarf, which will eventually explode as a Type Ia super-
nova. The mass of the exploding white dwarf should depend on
the amount of angular momentum (Yoon & Langer 2005) and can-
not exceed the mass budget of merging white dwarfs. Fig. 14 shows
the evolutionary paths of the central remnant for Seqs Sa8, Sa9 &
Sa11 as examples in the mass – angular momentum plane. Note that
the central remnant initially has a large amount of angular momen-
tum (J = 1.11×1050 erg s), such that without loss/gain of angular
momentum, it should accrete matter until it reaches M ≃ 1.68 M⊙
where it explodes in a SN Ia explosion. In Seqs Sa8 and Sa9, the ac-
cretion time scale (τacc) is longer than the angular momentum loss
time scale, and the total angular momentum of the white dwarf con-
tinuously decreases while the total mass increases. Consequently,
carbon ignites at the center when the white dwarf grows to 1.50M⊙
and 1.42 M⊙ for Seqs Sa8 and Sa9, respectively. In Seq. Sa11, on
the other hand, both mass and angular momentum of the central
remnant continuously increase, given that τacc . τJ, and a SN Ia
explosion is expected only when M ≃ 1.70 M⊙. Note that this
is even larger than the mass budget of the binary system consid-
ered for this sequence (i.e, 0.9 M⊙+0.6 M⊙). In nature, the white
dwarf must stop growing in mass when M = 1.5 M⊙, and a SN Ia
explosion will be induced only when a sufficient amount of angular
momentum has been removed, e.g. via gravitational wave radiation,
as illustrated by the path Sa11-B in Fig. 14.
4 CONCLUSION AND DISCUSSION
We have explored the dynamical and secular evolution of the
merger of double CO white dwarf binaries whose total mass ex-
ceeds the Chandrasekhar limit. Based on our new SPH simula-
tion of the coalescence of two CO white dwarfs of 0.9 M⊙ and
0.6 M⊙, we suggest that the immediate post-merger remnant is
best described as a differentially rotating CO star consisting of a
slowly rotating cold core and a rapidly rotating hot envelope that is
surrounded by a Keplerian disc rather than as “cold white dwarf +
thick disc” system, as in previous investigations. The evolution of
such a CO star is determined by the thermal evolution of the en-
velope, and the growth of the core is controlled by the cooling due
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 13
Figure 12. Same as in Fig. 9, but for Seq. Sa4.
Figure 13. Contour lines of the neutrino cooling time scale (τν ≡
TCp/ǫν ) in the log ρ − T plane. The level at each line gives log τν in
units of years. The dashed line denotes the critical temperature for carbon
ignition. The local peak of temperature and the corresponding density in the
initial model of Seqs Sa, Aa, Ab, Ac, Ae, and Ba are marked by the filled
symbols as indicated by the labels.
to neutrino emission and thermal diffusion, which is fundmentally
different from the assumption of “forced accretion of cold matter”.
Our 1-D stellar evolution models of the central remnant, i.e.
the cold core and the hot envelope, which include the effects of
rotation, indicate that there are three necessary conditions for the
merger remnant to avoid off-center carbon ignition such that a SN
Ia may be produced:
(i) The local peak of temperature of the merger remnant at the
interface between the core and the envelope must be lower than the
critical temperature for carbon ignition (Tp < TC−ig).
(ii) The time scale for angular-momentum loss from the central
remnant by must be larger than the neutrino cooling time scale at
the interface (τJ > τν,P).
(iii) Mass accretion from the Keplerian disc onto the cen-
tral remnant must be sufficiently slow (Ṁacc . 5 ×
10−6...10−5 M⊙ yr
Our new SPH simulation confirms that at least the first condition
(Tp < TC−ig) should be fulfilled in the CO white dwarf binary
considered.
As emphasized in Sect. 3.1, our 1-D models significantly un-
derestimate the effect of the centrifugal force on the stellar structure
in the rapidly rotating outermost layers. However, since thermal
diffusion always dominates over both neutrino cooling and com-
pressional heating in the outer envelope (ρ . 105...106 g cm−3)
above the interface, as shown in Figs. 7 and 12, the detailed struc-
ture of the rapidly rotating outermost layers above the interface may
not significantly affect our results on the thermal evolution of the
merger remnant, as long as the angular momentum of the envelope
is not lost faster than the local neutrino cooling time scale at the
interface. On the other hand, mass accretion from the Keplerian
disc should occur preferentially along the equatorial plane of the
envelope. As shown in the SPH simulation, the envelope is more
extended along the equatorial plane, where most angular momen-
tum is deposited, than along the polar axis, and the resultant com-
pressional heating must be much weakened, compared to the case
of our 1-D models. The enhanced role of rotation must thus help
c© 2007 RAS, MNRAS 000, 1–??
14 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
to increase the critical mass accretion rate for inducing off-center
carbon ignition, in favor of producing a Type Ia supernova.
We have concluded that the loss of angular momentum on a
short time scale (τJ . τν,p ≈ 10
4...105 yr) may induce off-center
carbon ignition even when Tmax,init < TC−ig. Rapidly rotating
compact stars may experience loss of angular momentum by gravi-
tational wave radiation, due to either the bar-mode instability or the
r-mode instability. The onset of the dynamical or secular bar-mode
instability requires a very high ratio of the rotational energy to the
gravitational energy: Erot/Egrav & 0.2 for the dynamical bar-
node instability, and Erot/Egrav & 0.14 for the secular bar-mode
instability (e.g. Shapiro & Teukolsky 1983). As both our 1-D mod-
els and SPH simulation give a value of Erot/Egrav that is much
lower (about 0.06 – 0.07) than those critical limits, the bar-mode in-
stability may not be relevant. The r-mode instability may operate,
in principle, even with such a low Erot/Egrav (Andersson 1998;
Friedman & Morsink 1998). However, we estimate that the growth
time of the r-mode instability (τr), using our central remnant mod-
els and following Lindblom (1999), is & 106 yr, which is much
longer than the local neutrino cooling time scale (τν,p ≈ 10
4 yr).
Alternatively, angular momentum might be transported from the
accreting star into the Keplerian disc when the accretor reaches
critical rotation. Calculations by Saio & Nomoto (2004) indicate,
however, that the decrease of the total angular momentum due to
such an effect is not significant in accreting white dwarfs. In con-
clusion, neither gravitational wave radiation nor outward angular-
momentum transport is likely to lead to a rapid loss of angular
momentum from the central remnant such that τJ < τν,p, unless
magnetic torques are important.
The central remnant may be enforced to rotate rigidly on a
short time scale in the presence of strong magnetic torques (cf.
Spruit 2002). The central remnant in both our SPH simulation and
1-D models has Jtot > 10
50 erg s, which is significantly higher
than the maximum limit a rigidly rotating white dwarf can retain,
as shown in Fig. 14. This means that if magnetic torques led to rigid
rotation, a large amount of angular momentum should be trans-
ported into the Keplerian disc (Case a in Fig. 14), or mass shed-
ding of super-critically spun-up layers should occur from the cen-
tral remnant (Case b in Fig. 14). In Case a, the local density around
the interface should increase by several factors by the time when
the central remnant reaches rigid rotation as implied by Fig. 15.
Off-center carbon ignition might be inevitable in this case due to a
resultant high effective accretion rate, if the time for angular mo-
mentum redistribution were shorter than the local cooling time due
to neutrino losses. In Case b, on the other hand, the local density
at the interface might not increase if mass shedding from the cen-
tral remnant occurred at a sufficiently high rate. Therefore, the role
of magnetic fields in the merger evolution remains uncertain at the
current stage and is a challenging subject for future work.
The coalescence of more massive double CO white dwarf bi-
naries is likely to result in a higher maximum temperature due to the
enhanced role of gravity. Consequently, given the important role of
the maximum temperature in the merger remnant for its final fate,
less massive binary CO white dwarfs may be favored for the pro-
duction of SNe Ia from such a channel.
We note that there are number of potentially important factors
that have not been included in either the present study or previous
simulations. These include the following points:
(i) The previous and present simulations assumed that white
dwarfs are cold prior to the merging process. However,
Iben, Tutukov & Fedorova (1998) point out that tidal interactions
Figure 14. Evolution of the central remnant in the mass – angular momen-
tum plane. The thick solid curve shows the angular momentum of a rigidly
rotating white dwarf with critical rotation at the surface as a function of the
white dwarf mass. The thick dashed curve and the thick dot-dashed curve
give the critical angular momentum for a differentially rotating CO white
dwarf to reach carbon ignition at the center (ρc = 2 × 109 g cm−3), and
electron-capture induced collapse (ρc = 10×1010 g cm−3), respectively,
according to Yoon & Langer (2005). A SN Ia explosion is expected in the
hatched region. The filled circle denotes the initial model of the central rem-
nant in Seq. Sa. The evolution of the central remnant in Seqs Sa 8, Sa9 and
Sa11 is shown by the thin dotted curves, as indicated. The thin solid curves
denote possible evolutionary paths of the central remnant with strong mag-
netic torques that may enforce rigid rotation, with loss of angular momen-
tum but without mass shedding (Case ’a’), and with both loss of angular
momentum and mass shedding (Case ’b’). See the text for more details.
Figure 15. The density profile in the initial model of the central remnant
in Seqs Sa (solid curve), and in a corresponding hot (Tc = 108 K) white
dwarf model that rotates rigidly at critical rotation at the surface (dashed
curve).
might heat up the white dwarfs as the orbit shrinks, which could
weaken the gravitational potential of the primary. Furthermore, as
the temperature of white dwarfs is a function of their age, younger
progenitors should have more extended envelopes, which may re-
sult in a lower Tp.
(ii) A thin hydrogen/helium envelope must be present initially in
both the primary and the secondary. As hydrogen or helium should
ignite at a much lower temperature than carbon, the influence of
the release of nuclear energy during the merger process may be
even more important than shown in the existing SPH simulations,
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 15
which is likely to lower Tp. Furthermore, neutrino losses, which
were neglected in the present study, would also tend to reduce Tp.
(iii) At a given total mass (Mtot = Mprimary + Msecondary),
different mass ratios of the white dwarf components (q ≡
Msecondary/Mprimary) must result in different merger structures.
(iv) A lower q at a given Mtot may not only lead to a stronger
gravitational potential of the primary, but also to a lower mass
accretion rate during the dynamical mass transfer (Guerrero et al.
2004). As the former and the latter will tend to increase and de-
crease Tp, respectively, quantitative studies are necessary to predict
how Tp will change with q.
Finally, another important ingredient that needs to be consid-
ered is thermal diffusion during the dynamical evolution. As shown
above, the mass-accretion rate from the Keplerian disc onto the en-
velope of the central remnant is one of the most important factors
that critically determine the final fate of double CO white dwarf
mergers. The accretion rates depend on the structure of the Kep-
lerian disc at thermal equilibrium, which can be only understood
by including thermal diffusion in future simulations. But here we
emphasize again that the accretion rates from a centrifugally sup-
ported Keplerian disc should be significantly lower than those from
a pressure-supported thick disc that was previously assumed, which
opens the possibility for at least some double CO white dwarf
mergers to produce SNe Ia.
ACKNOWLEDGMENTS
We are grateful to Norbert Langer and Ken’ichi Nomoto for many
useful suggestions and comments. SCY is supported by the VENI
grant (639.041.406) of the Netherlands Organization for Scientific
Research (NWO). The computations have been performed on the
JUMP supercomputer at the Höchstleistungsrechenzentrum Jülich.
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c© 2007 RAS, MNRAS 000, 1–??
http://arxiv.org/abs/astro-ph/0610872

  We systematically explore the evolution of the merger of two carbon-oxygen
(CO) white dwarfs. The dynamical evolution of a 0.9 Msun + 0.6 Msun CO white
dwarf merger is followed by a three-dimensional SPH simulation. We use an
elaborate prescription in which artificial viscosity is essentially absent,
unless a shock is detected, and a much larger number of SPH particles than
earlier calculations. Based on this simulation, we suggest that the central
region of the merger remnant can, once it has reached quasi-static equilibrium,
be approximated as a differentially rotating CO star, which consists of a
slowly rotating cold core and a rapidly rotating hot envelope surrounded by a
centrifugally supported disc. We construct a model of the CO remnant that
mimics the results of the SPH simulation using a one-dimensional hydrodynamic
stellar evolution code and then follow its secular evolution. The stellar
evolution models indicate that the growth of the cold core is controlled by
neutrino cooling at the interface between the core and the hot envelope, and
that carbon ignition in the envelope can be avoided despite high effective
accretion rates. This result suggests that the assumption of forced accretion
of cold matter that was adopted in previous studies of the evolution of double
CO white dwarf merger remnants may not be appropriate. Our results imply that
at least some products of double CO white dwarfs merger may be considered good
candidates for the progenitors of Type Ia supernovae. In this case, the
characteristic time delay between the initial dynamical merger and the eventual
explosion would be ~10^5 yr. (Abridged).

Mon. Not. R. Astron. Soc. 000, 1–?? (2007) Printed 20 August 2019 (MN LATEX style file v2.2)
Remnant evolution after a carbon-oxygen white dwarf merger
S.-C. Yoon1,2⋆, Ph. Podsiadlowski3 and S. Rosswog4
1Astronomical Institute ”Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands
2Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA95064, USA
3Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
4School of Engineering and Science, Jacobs University Bremen†, Campus Ring1, Bremen 28759, Germany
Accepted/ Received
ABSTRACT
We systematically explore the evolution of the merger of two carbon-oxygen (CO) white
dwarfs. The dynamical evolution of a 0.9 M⊙ + 0.6 M⊙ CO white dwarf merger is followed
by a three-dimensional SPH simulation. The calculation uses a state-of-the art equation of
state that is coupled to an efficient nuclear reaction network that accurately approximates all
stages from helium burning up to nuclear statistical equilibrium. We use an elaborate pre-
scription in which artificial viscosity is essentially absent, unless a shock is detected, and a
much larger number of SPH particles than earlier calculations. Based on this simulation, we
suggest that the central region of the merger remnant can, once it has reached quasi-static
equilibrium, be approximated as a differentially rotating CO star, which consists of a slowly
rotating cold core and a rapidly rotating hot envelope surrounded by a centrifugally supported
disc. We construct a model of the CO remnant that mimics the results of the SPH simulation
using a one-dimensional hydrodynamic stellar evolution code and then follow its secular evo-
lution, where we include the effects of rotation on the stellar structure and the transport of
angular momentum. The influence of the Keplerian disc is implicitly treated by considering
mass accretion from the disc onto the hot envelope. The stellar evolution models indicate that
the growth of the cold core is controlled by neutrino cooling at the interface between the core
and the hot envelope, and that carbon ignition in the envelope can be avoided despite high
effective accretion rates. This result suggests that the assumption of forced accretion of cold
matter that was adopted in previous studies of the evolution of double CO white dwarf merger
remnants may not be appropriate. Specifically we find that off-center carbon ignition, which
would eventually lead to the collapse of the remnant to a neutron star, can be avoided if the
following conditions are satisfied: (1) when the merger remnant reaches quasi-static equilib-
rium, the local maximum temperature at the interface between the core and the envelope must
be lower than the critical limit for carbon-ignition. (2) Angular-momentum loss from the cen-
tral merger remnant should not occur on a time scale shorter than the local neutrino cooling
time scale at the interface. (3) The mass-accretion rate from the centrifugally supported disc
must be sufficiently low (Ṁ . 5 × 10−6...10−5 M⊙ yr
−1). Our results imply that at least
some products of double CO white dwarfs merger may be considered good candidates for
the progenitors of Type Ia supernovae. In this case, the characteristic time delay between the
initial dynamical merger and the eventual explosion would be ∼ 105 yr.
Key words: Stars: evolution – Stars: white dwarf – Stars: accretion – Supernovae: general –
1 INTRODUCTION
The coalescence of two carbon-oxygen (CO) white dwarfs with
a combined mass in excess of the Chandrasekhar limit has long
been considered a promising path towards a Type Ia supernova (SN
Ia; Iben & Tutukov 1984; Webbink 1984). Indeed, in the last few
⋆ E-mail: scyoon@science.uva.nl (SCY); podsi@astro.ox.ac.uk (PhP);
s.rosswog@iu-bremen.de (SR)
† formerly International University Bremen
years, a few massive double CO white dwarf systems have been
found that have periods short enough for them to merge within
a Hubble time (e.g. Napiwotzki et al. 2002, 2004). This double-
degenerate (DD) scenario can also easily explain the lack of hy-
drogen and helium lines in most SN Ia spectra and the occur-
rence of SNe Ia both in old and young star-forming systems (e.g.
Branch et al. 1995).
Theoretically, the final fate of double CO white dwarf merg-
ers has been much debated. Previous studies assumed that the dy-
namical disruption of the Roche-lobe filling secondary should lead
c© 2007 RAS
http://arxiv.org/abs/0704.0297v2
2 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 1. Schematic illustration of the configuration of the remnant of a
double CO white dwarf merger once quasi-static equilibrium has been es-
tablished.
to the formation of a thick disc around the primary white dwarf
(Tutukov & Yungelson 1979; Mochkovitch & Livio 1989, 1990).
Therefore, accretion of CO-rich matter from the thick disc onto
the central cold white dwarf has been studied for investigating the
evolution of such mergers by many authors (Nomoto & Iben 1985;
Saio & Nomoto 1985, 1998, 2004; Piersanti et al. 2003a,b). As ac-
cretion rates from the thick disc should be close to the Edding-
ton limit (Ṁ ≈ 10−5 M⊙yr
−1), most of those studies concluded
that carbon ignition in the envelope of the accreting white dwarf
is an inevitable consequence of such rapid accretion of CO-rich
matter. Once carbon ignites off-center, the burning flame propa-
gates inwards on a relatively short time scale (∼ 5000 yr), and
the CO white dwarf is transformed into an ONeMg white dwarf
(Saio & Nomoto 1985, 1998). When the mass of the ONeMg white
dwarf approaches the Chandrasekhar limit, electron capture onto
Ne and Mg is expected to lead to the gravitational collapse of
the white dwarf to a neutron star (Nomoto & Kondo 1991; see
Dessart et al. 2006 and Kitaura, Janka & Hillebrandt 2006 for re-
cent studies of such collapse).
However, the evolution of the remnants of double CO white
dwarf mergers is not yet well understood. For instance, it has
been debated whether the accretion rate decreases when the ac-
creting white dwarf reaches critical rotation (Piersanti et al. 2003a;
Saio & Nomoto 2004). More importantly, the canonical descrip-
tion of the merger remnant as a primary white dwarf + thick
disc system is clearly an oversimplification. In previous three-
dimensional smoothed particle hydrodynamics (SPH) simula-
tions (Benz et al. 1990; Segretain, Chabrier & Mochkovitch 1997;
Guerrero et al. 2004; see also Sect. 2), a large fraction of the dis-
rupted secondary and the outermost layers of the primary form an
extended hot envelope around the cold core containing most of the
primary mass. The rest of the secondary mass becomes a centrifu-
gally supported disc in the outermost layers of the merger rem-
nant. Interestingly, the merger remnant reaches a state of quasi-
static equilibrium within a few minutes from the onset of the dy-
namical disruption of the secondary. As the structure of the cold
core plus the hot envelope appears to have a fairly spheroidal shape
(see below) rather than the toroidal shape obtained with a zero-
temperature equation of state (Mochkovitch & Livio 1989, 1990),
the merger remnant may be better described as a differentially ro-
tating single CO star consisting of a slowly rotating cold core and
a rapidly rotating hot extended envelope surrounded by a Keple-
rian disc, as illustrated in Fig. 1, than the previously adopted pri-
mary white dwarf + thick disc system. The further evolution of the
merger must therefore be determined by the thermal cooling of the
hot envelope and the redistribution of the angular momentum in-
side the central remnant, and accretion of matter onto the envelope
from the Keplerian disc.
With this new approach to the problem in mind, we here re-
visit both the dynamical and the secular evolution of double CO
white dwarf mergers. In the following section (Sect. 2), we present
the numerical results of an SPH simulation of the dynamical evolu-
tion of the coalescence of a 0.9 M⊙ WD and a 0.6 M⊙ CO white
dwarf up to the stage of quasi-hydrostatic equilibrium, and we care-
fully investigate the structure of the merger remnant. In Sect. 3, we
construct models of the central remnant in quasi-static equilibrium
state (primary + hot extended envelope) which mimic the SPH re-
sult and calculate the thermal evolution of the merger remnant using
a hydrodynamic stellar evolution code. In particular, the conditions
for avoiding off-center carbon ignition are systematically explored.
In Sect. 4, we conclude this work by discussing uncertainties in our
assumptions, the implications for Type Ia supernovae and future
work.
2 DYNAMICAL EVOLUTION OF THE MERGER
Before discussing the subsequent thermal evolution after the co-
alescence of a double CO white dwarf coalescence binary, we
investigate the configuration of the remnant in quasi-static equi-
librium in some detail. For this purpose, we have carried out
a SPH simulation of the dynamical process of the coalescence
of two CO white dwarfs of 0.9 M⊙ and 0.6 M⊙, respec-
tively. Our simulation uses a 3D smoothed particle hydrodynam-
ics (SPH) code that is an offspring of a code developed to simu-
late neutron star mergers (Rosswog et al. 2000; Rosswog & Davies
2002; Rosswog & Liebendörfer 2003). It uses an artificial viscos-
ity scheme with time-dependent parameters (Morris & Monaghan
1997). In the absence of shocks, the viscosity parameters have a
very low value (α = 0.05 and β = 0.1; most SPH implemen-
tations use values of α = 1...1.5 and β = 2...3); if a shock is
detected, a source term (Rosswog et al. 2000) guarantees that the
parameters rise to values that are able to resolve the shock prop-
erly without spurious post-shock oscillations. To suppress artificial
viscosity forces in pure shear flows, we additionally apply a switch
originally suggested by Balsara (1995).
To account for the energetic feedback onto the fluid from
nuclear transmutations, we use a minimal nuclear reaction net-
work developed by Hix et al. (1998). It couples a conventional α-
network stretching from He to Si with a quasi-equilibrium-reduced
α-network. Although a set of only seven nuclear species is used,
this network reproduces the energy generation of all burning stages
from He-burning to NSE very accurately. For details and tests we
refer to Hix et al. (1998). We use the HELMHOLTZ equation of
state (EOS), developed by the Center for Astrophysical Thermonu-
clear Flashes at the University of Chicago. This EOS allows to
freely specify the chemical composition of the gas and can be cou-
pled to nuclear reaction networks. The electron/positron equation
of state has been calculated without approximations, i.e. it makes
no assumptions about the degree of degeneracy or relativity; the
exact expressions are integrated numerically to machine precision.
The nuclei in the gas are treated as a Maxwell-Boltzmann gas,
the photons as blackbody radiation. The EOS is used in tabular
form with densities ranging from 10−10 6 ρYe 6 10
11 g cm−3
and temperatures from 104 to 1011 K. A sophisticated, biquin-
tic Hermite polynomial interpolation is used to enforce thermody-
namic consistency (i.e. the Maxwell-relations) at interpolated val-
ues (Timmes & Swesty 2000).
We use a MacCormack predictor-corrector method (e.g.
Lomax et al. 2001) with individual particle time steps to evolve
the fluid. With our standard parameters for the tree-opening cri-
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 3
Figure 2. Dynamical evolution of the coalescence of a 0.6 M⊙ + 0.9 M⊙ CO white dwarf binary. The panels in the left column show the density in the
orbital plane, the panels in the right column the temperature in units of 106 K. Lengths are in code units (= 109 cm).
c© 2007 RAS, MNRAS 000, 1–??
4 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 3. Dynamical evolution of the coalescence of a 0.6 M⊙ + 0.9 M⊙ CO white dwarf binary. Continued from Fig. 2.
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 5
Figure 4. The evolution of the local peak of temperature during the merger
of two CO white dwarfs of 0.6M⊙ and 0.9M⊙, respectively, as a function
of time after the onset of the simulation.
terion and the integration, this time marching implementation con-
serves the total energy to better than 4 × 10−3 and the total an-
gular momentum to better than 2 × 10−4. Note that this could,
in principle, be improved even further by taking into account the
so-called “grad-h”-terms (Springel & Hernquist 2002; Monaghan
2002; Price 2004) and extra-terms arising from adapting gravita-
tional smoothing terms (Price & Monaghan 2006).
To avoid numerical artifacts, we only use equal mass SPH
particles. For the initial conditions, we therefore stretch a uniform
particle distribution according to a function that has been derived
from solving the 1D stellar structure equations. This technique is
described in detail in Rosswog, Ramirez-Ruiz & Hix (2007). This
particle setup is then further relaxed with an additional damping
force (e.g. Rosswog, Speith & Wynn 2004) so that the particles can
settle into their true equilibrium configuration. The calculations are
performed with 2×105 SPH particles, a much larger particle num-
ber than could be afforded by previous calculations, and run up to
a much longer evolutionary time (5 minutes) than previous calcu-
lations (see Table 1).
Figs. 2 and 3 show the dynamical evolution of the merging
process of the double white dwarf system considered in this study.
The panel in the left columns show the densities and the panel in
the right columns the temperatures (in units of 106 K) in the orbital
plane. The secondary is completely disrupted within 1.7 minutes,
and mass accretion onto the primary induces local heating near the
surface of the primary. Fig. 4 shows the evolution of the maximum
temperature as a function of time. The peak in temperature reaches
1.7×109 K at t ≃ 1.0 min, where t = 0.0 marks the moment when
the simulation starts. Carbon ignites when T & 109 K, but nuclear
burning is quenched soon due to the local expansion of the hottest
region, as is also observed in the simulations of Guerrero et al.
(2004). The total amount of energy released due to nuclear burn-
ing is about 1045 erg.
Segretain, Chabrier & Mochkovitch (1997) considered the
same initial white dwarf masses as in the present study. But
they adopted the original artificial viscosity prescription of
Monaghan & Varnas (1988), which is known to introduce spuri-
ous forces in shear flows, and they did not include nuclear burning
(Table 1). By the end of their calculation (t = 1.56 min), Tmax
reached 8×108 K, while in our simulation, it decreases to 8×108 K
only when t ∼ 1.7 min. Interestingly, Tmax decreases further af-
Figure 5. Top: Density contour of the merger remnant in the x− z plane at
t = 5.3 min. Here one code unit corresponds to 109 cm. Middle: Thermo-
dynamic structure of the merger remnant at t = 5.31 min: shown are the
temperature and the density as a function of distance from the centre, along
the positive x- and z-axis, as indicated. Bottom: Angular velocity in units
of the local Keplerian value at t = 5.31 min, along the positive/negative
x- and y-axis of the merger remnant.
terwards in our calculation, as shown in Fig. 4, and reaches a steady
value at Tmax ≃ 5.6 × 10
8 K when t & 2.5min. In the other cal-
culations by Benz et al. (1990) and Guerrero et al. (2004), the dy-
namical evolution of the merger was not followed for more than 2
minutes either, and we cannot directly compare our results to theirs.
However, we suspect that Tmax would also decrease further in the
systems they considered if they had continued their calculations for
a longer evolutionary time. It should also be noted that energy dis-
sipation by artificial viscosity might lead to overheating, and that
c© 2007 RAS, MNRAS 000, 1–??
6 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
thermal diffusion – which may play an important role in the outer-
most layers – is not considered in the present study. It is thus likely
that Tmax in the quasi-static equilibrium state may be even lower
in reality than in our simulation.
Fig. 5 shows the structure of the merger remnant at quasi-static
equilibrium. The central region with R . 109 cm (Mr . 1.1 M⊙)
has a fairly spheroidal shape, and a centrifugally supported disc
appears at R & 109 cm where the angular velocity is close to
the Keplerian value. The fraction of the secondary mass contained
in the Keplerian disc is larger in our simulation (about 67%) than
in Segretain, Chabrier & Mochkovitch (1997) (about 41 %). The
innermost core (R . 3 × 108 cm; Mr . 0.6 M⊙) is essen-
tially isothermal, and the temperature has its peak value (T ≃
5.6 × 108 K) at R ≃ 5 × 108 cm and Mr ≃ 0.85 M⊙. The
disc material extends over 4× 109 cm along the z-axis as the tem-
perature is still high; if thermal diffusion were included, the disc
would become much thinner on a short time scale of a few hours.
Therefore, our simulation confirms the remnant structure at quasi-
static equilibrium that is illustrated in Fig. 1. In the next section, we
investigate the secular evolution of the merger from such a quasi-
static equilibrium state.
3 SECULAR EVOLUTION OF THE MERGER REMNANT
3.1 Physical assumptions and methods
Our SPH simulation shows that the remnant of the merger of two
CO white dwarfs (0.9 M⊙ + 0.6 M⊙) in the state of quasi-static
equilibrium has the following features (see Fig. 6):
(i) The core is cold and nearly isothermal.
(ii) The local peak of temperature (Tp) is located at a mass co-
ordinate slightly less than the primary mass.
(iii) A steep gradient in temperature appears at the interface be-
tween the core and the local peak of temperature.
(iv) The interface is rather widely extended into the primary
(∆Minterface ≈ 33 % of the primary mass), and the mass of the
quasi-isothermal cold core (Mcore) is about 77 % of the primary
mass.
(v) The mass of the outer envelope above the local peak of tem-
perature contains about 33 % of the mass of the secondary, and the
rest of the secondary forms a Keplerian disc.
Let us define Tp as the local peak of temperature at quasi-static
equilibrium, MCM as the mass of the central remnant (cold core +
hot envelope), and Mp as the location of Tp in the mass coordi-
nate (i.e., Mp = Mcore + ∆Minterface; see Fig. 6). To construct
models of the central remnant, we use a one-dimensional hydrody-
namic stellar evolution code which incorporates the effects of ro-
tation on the stellar structure, transport of angular momentum due
to the shear instability, Eddington-Sweet circulation, and the Gol-
dreich, Schubert and Fricke instability, and dissipation of rotational
energy due to shear motions. The effects of magnetic fields are ne-
glected (see Sec. 4). More details about the code are described in
(Yoon & Langer 2004; hereafter YL04) and references therein.
In order to mimic the temperature profile of the central rem-
nant as obtained from the SPH simulation, we artificially deposit
energy in the envelope, using the following prescription for a white
dwarf with M = MCR:
e(Mr) = A(T
(Mr)− T (Mr)) [erg g
], (1)
where
Figure 6. Initial model of the central remnant for sequences Sa1 – Sa11.
The top and middle panels show temperature as a function of the mass co-
ordinate and radius, respectively. The solid curve in the bottom panel gives
the angular-velocity profile as a function of radius. The dashed curve de-
notes the angular velocity in units of the local Keplerian value.
T ′(Mr) =
3 · 107 K+ (7 · 107 K− 3 · 107 K)
Mcore
if Mr < Mcore,
Tp − (Tp − 7 · 10
Mp−Mr
Mp−Mcore
if Mcore 6 Mr 6 Mp,
C − (C − Tp)
log[ρ(Mr)/ρs]
log[ρ(Mp)/ρs]
if Mr > Mp.
In this way, the temperature profile in the central remnant model
follows T ′(Mr). Here, A and C are constants. We use A =
105 erg g−1 s−1 K−1 and C = 2× 108 K in most cases.
A rotational profile is imposed as
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 7
Table 1. Comparison of SPH simulations of double CO white dwarf mergers. The columns list: M1 and M2: the masses of the primary and the secondary,
respectively; NoP: the total number of particles used; νsph: the type of artificial viscosity employed, ‘std.’ refers to Monaghan & Varnas (1988); Network:
type of nuclear network employed; tsim: evolutionary time that has elapsed by the end of the calculation; Tmax: maximum temperature obtained during the
simulation; and Tp: the local peak of temperature at the end of the calculation
Ref.∗ M1 M2 NoP νsph Network tsim Tmax TP
1 1.2 M⊙ 0.9 M⊙ ∼ 7× 103 std. None 51 sec. ? ∼ 109 K
2 0.8 M⊙ 0.6 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 50 sec. 1.4× 109 K ?
2 1.0 M⊙ 0.6 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 65 sec. 1.6× 109 K ?
2 1.0 M⊙ 0.8 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 65 sec. 2.0× 109 K ?
3 0.9 M⊙ 0.6 M⊙ ∼ 6× 104 std None 1.56 min. ? ∼ 7× 108 K
4 0.9 M⊙ 0.6 M⊙ 2× 105 see Rosswog et al. (2000) QSE-alpha network 5.3 min. 1.7× 109 K 5.6× 108 K
∗1: Benz et al. (1990), 2: Guerrero et al. (2004), 3: Segretain, Chabrier & Mochkovitch (1997); 4: Present Study
Table 2. Merger remnant model sequences. Each column lists the following: No.: sequence label, MCR: mass of the central remnant, Mcore: mass of the
quasi-isothermal core, Mp: location of the local peak of temperature in the mass coordinate, Tp: the local peak of temperature, ρp: density at Mr = Mp,
τJ: adopted time scale for angular momentum loss according to Eq. (4), Ṁacc: adopted mass accretion rate from the Keplerian disc, C-ig: off-center ignition
of carbon, MWD,ig: total mass of the central remnant when off-center carbon ignition occurs, Mr,ig: location of off-center carbon ignition in the mass
coordinate.
No. MCR Mcore Mp Tp ρp τJ Ṁacc C-ig MWD,ig Mr,ig
[M⊙] [M⊙] [M⊙] [108 K] [106 g/cm3] [yr] 10−6 M⊙/yr [M⊙] [M⊙]
Sa1 1.11 0.6 0.84 5.6 0.8 ∞ 0.0 No - -
Sa2 1.11 0.6 0.84 5.6 0.8 102 0.0 Yes 1.11 0.80
Sa3 1.11 0.6 0.84 5.6 0.8 103 0.0 Yes 1.11 0.80
Sa4 1.11 0.6 0.84 5.6 0.8 104 0.0 Yes 1.11 0.85
Sa5 1.11 0.6 0.84 5.6 0.8 105 0.0 No - -
Sa6 1.11 0.6 0.84 5.6 0.8 105 10.0 Yes 1.34 1.09
Sa7 1.11 0.6 0.84 5.6 0.8 105 5.0 Yes 1.34 1.20
Sa8 1.11 0.6 0.84 5.6 0.8 105 2.0 No - -
Sa9 1.11 0.6 0.84 5.6 0.8 105 1.0 No - -
Sa10 1.11 0.6 0.84 5.6 0.8 5 · 105 5.0 No - -
Sa11 1.11 0.6 0.84 5.6 0.8 5 · 105 1.0 No - -
Aa1 1.25 0.6 0.93 5.0 2.3 ∞ 0.0 No - -
Aa2 1.25 0.6 0.93 5.0 2.3 102 0.0 Yes 1.250 0.90
Aa3 1.25 0.6 0.93 5.0 2.3 103 0.0 Yes 1.250 0.92
Aa4 1.25 0.6 0.93 5.0 2.3 104 0.0 Yes 1.250 1.12
Aa5 1.25 0.6 0.92 5.0 2.3 105 0.0 No - -
Aa6 1.25 0.6 0.92 5.0 2.3 106 0.0 No - -
Aa7 1.25 0.6 0.92 5.0 2.3 105 10.0 Yes 1.360 1.20
Aa8 1.25 0.6 0.92 5.0 2.3 105 5.0 No - -
Aa9 1.25 0.6 0.92 5.0 2.3 105 1.0 No - -
Aa10 1.25 0.6 0.92 5.0 2.3 106 10.0 Yes 1.382 1.22
Ab1 1.25 0.7 0.92 5.0 3.1 ∞ 0.0 No - -
Ab2 1.25 0.7 0.92 5.0 3.1 103 0.0 Yes 1.250 0.97
Ab3 1.25 0.7 0.92 5.0 3.1 104 0.0 No - -
Ab4 1.25 0.7 0.92 5.0 3.1 105 0.0 No - -
Ab5 1.25 0.7 0.92 5.0 3.1 105 10.0 Yes 1.344 1.21
Ab6 1.25 0.7 0.92 5.0 3.1 105 5.0 No - -
Ac1 1.25 0.5 0.88 5.9 1.6 ∞ 0.0 Yes 1.250 0.84
Ad1 1.25 0.6 0.92 6.0 1.8 ∞ 0.0 Yes 1.250 0.90
Ad2 1.25 0.6 0.92 6.0 1.8 106 5.0 Yes 1.252 0.90
Ae1 1.25 0.6 0.90 6.8 1.5 ∞ 0.0 Yes 1.250 0.87
Ba1 1.363 0.82 0.95 5.0 12.2 ∞ 0.0 No - -
Ba2 1.363 0.82 0.95 5.0 12.2 102 0.0 Yes 1.363 0.95
Ba3 1.363 0.82 0.95 5.0 12.2 103 0.0 Yes 1.363 1.12
Ba4 1.363 0.82 0.95 5.0 12.2 104 0.0 No - -
Ba5 1.363 0.82 0.95 5.0 12.2 105 0.0 No - -
Ba6 1.363 0.82 0.95 5.0 12.2 105 10.0 Yes 1.398 1.34
Ba7 1.363 0.82 0.95 5.0 12.2 105 5.0 Yes 1.483 1.43
Ba8 1.363 0.82 0.95 5.0 12.2 105 1.0 No - -
Ta1 1.25 0.60 0.86 5.0 28.8 ∞ 0.0 No - -
c© 2007 RAS, MNRAS 000, 1–??
8 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 7. (a) Evolution of a non-rotating white dwarf accreting with a constant accretion rate of Ṁ = 10−5 M⊙ yr−1 with an initial mass of 0.9 M⊙
(Seq. N0.9) in the density – temperature plane. (b) The local effective accretion rate (Ṁeff,r := 4πr
2ρv) as a function of density in Seq. N0.9, at different
evolutionary epochs as indicated by the labels. (c) – (f) The rates of energy loss/production due to neutrino (ǫν ) cooling, compressional heating (ǫcomp),
nuclear energy generation (ǫnuc) and thermal diffusion (ǫth) at different evolutionary epochs. Note that here ǫν , ǫcomp and ǫnuc represent the values which
are used in the evolutionary calculations, while ǫth is an order-of-magnitude estimate according to Eq. (7).
Ω(Mr) =
ΩO, if Mr < Mcore,
ωO + (1− ωO)
Mr−Mcore
MCR−Mcore
if Mr > Mcore,
where ΩO = 0.2
GMCR/R3 and ωO = ΩO/
GMcore/r
core.
As shown in Fig. 6, this simple assumption gives a rotational ve-
locity profile that is morphologically similar to that found in the
SPH simulation: a steep gradient at the interface between the core
and the envelope, and a local peak in the envelope. Within our 1-D
approximation of the effects of rotation, the exact shape of the ro-
tational velocity profile does not affect the main conclusions of the
present work for the following reasons. Firstly, the velocity gradient
at the interface is adjusted to the threshold value for the dynamical
shear instability on a very short time scale (see below, and discus-
sions in YL04). Secondly, our 1-D approximation underestimates
the effect of the centrifugal force on the stellar structure in layers
which rotate more rapidly than about 60 % critical (YL04), and un-
certainties due to this limit are much greater than due to the shape
of Ω(r) in the outer layers of the envelope. Possible uncertainties
due to this limitation are critically discussed in Sect. 4.
The central remnant may lose angular momentum by out-
ward angular momentum transport into the Keplerian disc
(Popham & Narayan 1991; Paczyński 1991) and/or by the gravita-
tional wave radiation, e.g., due to the r-mode instability (Andersson
1998; Friedman & Morsink 1998). Our code cannot properly de-
scribe any of these effects, and here we consider them simply by
assuming a constant time scale for the angular momentum loss (τJ;
see Knaap 2004; cf. Piersanti et al. 2003a), such that the specific
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 9
Table 3. Accreting white dwarf model sequences with a constant accretion
rate (Ṁ = 10−5 M⊙/yr). The columns list: No: sequence label, Minit:
initial mass, logLinit/L⊙ : initial luminosity, MWD,ig: the total mass of
the white dwarf when carbon ignites off-center, and Mr,ig: location of car-
bon ignition in the mass coordinate. Sequences with ‘N’ are for non-rotating
models, and ‘R’ for rotating models.
No Minit logLinit/L⊙ MWD,ig Mr,ig
N0.7 0.7 −2.118 0.999 0.793
N0.8 0.8 −2.128 1.010 0.862
N0.9 0.9 −2.188 1.039 0.939
N1.0 1.0 −2.137 1.087 1.024
N1.1 1.1 −2.170 1.150 1.114
N1.2 1.2 −2.119 1.225 1.207
R0.8 0.8 −2.114 1.297 1.038
R0.9 0.9 −2.119 1.249 1.050
R1.0 1.0 −2.082 1.207 1.069
R1.1 1.1 −2.050 1.205 1.127
angular momentum of each mass shell decreases over a time step
∆t by an amount
∆ji = ji [1− exp(−∆t/τJ)] . (4)
Mass accretion from the Keplerian disc is also considered in
some model sequences, with different values for the accretion rate
(Ṁacc). The angular-momentum accretion is treated in the same
way as in YL04: the accreted matter is assumed to carry angular
momentum at a value close to the Keplerian value if the surface
velocity of the central remnant is below critical, while no angular-
momentum accretion is allowed otherwise.
Model sequences with different sets of MCR, Mcore, Mp, τJ,
Ṁ , and Tp are calculated, as summarized in Table 2. The initial
model in Seqs S is intended to reproduce the result of our SPH sim-
ulation, where MCR = 1.10 M⊙ and Mp ≈ 0.84 M⊙ are adopted.
We also assume MCR = 1.25 M⊙ and Mp ≈ 0.9 M⊙ in Seq. A,
and MCR = 1.364 M⊙ and Mp = 0.95 M⊙ in Seq. B, to simulate
mergers of 0.9 − 1.0 M⊙ + 0.7 − 1.0 M⊙ white dwarf binaries.
At a given MCR, different sets of Mcore, Mp, and Tp are marked
in the sequence label by minor characters (a, b, c, d, e), while dif-
ferent sets of τJ and Ṁacc are indicated by Arabian numbers. For
instance, sequences Sa1 – Sa11 have the same initial merger model,
but different values for τJ and Ṁacc. Rotation is neglected in a test
sequence Ta1 (i.e., the models are non-rotating). The temperature
and angular-velocity profiles in the initial central remnant model of
Seqs Sa1 - Sa11 are shown in Fig. 6. The temperature (a few to sev-
eral 108 K) and the size (∼ 109 cm) of the envelope appear to be
comparable to those obtained from the SPH simulation (see Fig. 5).
For comparison, we also ran model sequences for classi-
cal cold-matter accretion with a constant accretion rate of Ṁ =
10−5 M⊙/yr, for both non-rotating and rotating cases, as summa-
rized in Table 3.
3.2 Results
3.2.1 Classical models of cold-matter accretion
Before discussing the central remnant models, let us first investi-
gate the evolution of classical cold-matter accreting white dwarf
models in detail. In these models, the accreted matter is assumed to
have the same entropy as the surface value of the accreting white
dwarf. As shown in previous studies (e.g. Nomoto & Iben 1985),
the thermal evolution of rapidly accreting white dwarfs is deter-
Figure 8. Local effective mass accretion rate (Ṁeff,r ≡ 4πr
2ρv) as a
function of density in the models of sequence R0.9, at different evolutionary
epochs.
mined by the interplay of compressional heating and thermal dif-
fusion. Fig. 7 shows an example of the evolution of such accret-
ing white dwarf models for an initial WD mass of 0.9 M⊙ and
a constant accretion rate of Ṁacc = 10
−5 M⊙ yr
−1 (Seq. N0.9;
Table 3).
Fig. 7a shows that the temperature increases continuously in
the envelope (ρ . 106 g cm−3), and finally carbon burning be-
comes significant at ρ ≃ 5.6 × 105 g cm−3 and T ≃ 6× 108 K
when t ≃ 1.3 × 104 yr. In Figs. 7c –f, the rates of compressional
heating (ǫcomp), neutrino cooling (ǫν ), nuclear energy generation
(ǫnuc) and thermal diffusion (ǫth) are shown. In our stellar evolu-
tion code, the compressional heating rate is calculated according
ǫcomp =
(Kippenhahn & Weigert 1990). Neutrino cooling rates are obtained
following Itoh et al. (1996). While ǫcomp, ǫν , and ǫnuc in the figures
correspond to the values that are used for the evolutionary calcula-
tions, the thermal diffusion rate (ǫth) – which is only calculated
implicitly in the code – can only be estimated to within an order-
of-magnitude from
ǫth ≈ TCP/τth . (6)
Here CP denotes the specific heat at constant pressure, and τth the
local thermal diffusion time scale defined as
τth ≡ H
P/K , (7)
where HP is the pressure scale height, and K
[(4acT 3)/(3CPκρ
2)] is the thermal diffusivity. It is clear
from Fig. 7 that the local peak of temperature is located where the
compressional heating rate begins to dominate over the thermal
diffusion rate (ρ ≈ 105 g cm−3), as expected. The neutrino
cooling rate also increases as the temperature in the envelope
becomes higher, but nuclear energy generation becomes significant
before neutrino cooling dominates the thermal evolution, inducing
a carbon-burning flash around ρ ≃ 5.6 · 105 g cm−3.
As Table 3 shows, and consistent with the findings of
Nomoto & Iben (1985), such off-center carbon flashes occur re-
gardless of the initial mass of the white dwarf, if Ṁacc ≈
10−5 M⊙ yr
−1. The results with models including rotation show
that carbon ignition may be delayed if the effect of rotation is in-
cluded (Table 3; see also Piersanti et al. 2003a and Saio & Nomoto
2004). The reason is that the local effective mass accretion rate
c© 2007 RAS, MNRAS 000, 1–??
10 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 9. (a) Evolution of the central remnant in Seq. Sa1 in the log ρ − T plane. The dotted curve gives the critical temperature where the nuclear energy
generation rate due to carbon burning equals the energy loss rate due to neutrino cooling. (b) The local effective accretion rate (Ṁeff,r ≡ 4πr
2ρvr) as a
function of density in the merger remnant model of Seq. Sa1, at different evolutionary epochs as indicated by the labels. (c) – (f) The rates of energy loss/gain
due to neutrino (ǫν ) cooling, compressional heating (ǫcomp), nuclear energy generation (ǫcomp) and thermal diffusion (ǫth) as a function of density in the
central remnant models of Seq. Sa1 at different evolutionary epochs. Note that here ǫν , ǫcomp and ǫnuc represent the values which are used in the evolutionary
calculations, while ǫth is an order-of-magnitude estimate according to Eq. (7).
(Ṁeff,r ≡ 4πr
2ρv) inside the white dwarf at a given mass is lower
because of the centrifugal force. For instance, in Seq. N0.9, we
have Ṁeff,r ≈ 10
−5 M⊙ yr
−1 at around ρ = 5 × 105 g cm−3
when t ≃ 104 yr (Fig. 7b), but Ṁeff,r is lowered by a factor of
two in the corresponding rotating model at a similar epoch (i.e.,
Ṁeff,r ≈ 5 × 10
−6 M⊙ yr
−1), as revealed in Fig. 8. However,
carbon ignition occurs well before the white dwarf reaches the
Chandrasekhar limit, in all model sequences considered. Thus, ro-
tation by itself cannot change the conclusion of the previous work
that the coalescence of double CO white dwarfs should lead to
accretion-induced collapse rather than a thermonuclear explosion,
unless the accretion rate is significantly lowered, as was also shown
by Piersanti et al. (2003a) and Saio & Nomoto (2004).
3.2.2 Sequences without angular-momentum loss and mass
accretion
Having understood the physics of the thermal evolution of CO
white dwarfs which accrete cold matter with a rate close to the Ed-
dington limit, we now investigate the evolution of the central rem-
nant model consisting of a cold core and a hot envelope as described
in Sect. 3.1. First, we examine the results of the model sequences
where both angular-momentum loss and mass accretion from the
Keplerian disc are neglected (i.e., τJ = ∞ and Ṁ = 0; Seqs Sa1,
Aa1, Ab1, Ac1, Ad1, Ae1, Ba1, & Ta1).
Fig. 9a illustrates the evolution of the central remnant for
MCR = 1.10 M⊙ in Seq. Sa1 in the density – temperature
plane. Note that the local peak of temperature at t = 0.0 (Tp =
5.6× 108 K) is significantly below the critical temperature for car-
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 11
bon ignition (TC−ig; dotted curve in Fig 9a). It is shown in Fig. 9b
that the local effective accretion rate (Ṁeff,r) remains relatively
high (5×10−6−10−5 M⊙ yr
−1) around ρ = 106 g cm−3, where
the local peak of temperature is located, for about 5000 yrs. Despite
such high effective accretion rates, the temperature peak continu-
ously decreases, although the inner core becomes somewhat hotter
due to compression, and the central remnant finally becomes a cold
white dwarf. A few remarkable differences compared to the stan-
dard accreting white dwarf models are found in this regard. Firstly,
since the envelope is very hot, neutrino cooling – in particular by
photoneutrinos – is significant from the beginning, and even domi-
nant over the thermal diffusion at the interface between the core and
the envelope as shown in Fig. 9c. In cold-matter accreting white
dwarfs, neutrino cooling becomes important only after a significant
amount of mass has been accreted (Fig. 7). Secondly, the compres-
sional heating rate is slightly lower than the neutrino cooling rate
around the local peak of temperature. As the contraction of the cen-
tral remnant is mainly determined by the thermal evolution of the
envelope, the local accretion rate is in fact controlled by the cool-
ing process. This explains why we have ǫcomp ≈ ǫν around the
local peak of temperature for the initial ∼ 104 yrs, and why the
local peak of temperature continuously decreases despite the rela-
tively high effective accretion rate. This conclusion is the same for
all other sequences with a TP that is significantly lower than TC−ig
(Seq. Aa1, Ab1, & Ba1) including the non-rotating case (Seq. Ta1;
Table 2).
We find that, in Seq. Sa1, the differentially rotating layers at
the interface between the core and the envelope in the initial central
remnant model are stable against the dynamical shear instability
(DSI). They are, however, unstable to the DSI in other sequences,
where the interface is more degenerate (see YL04 for discussions
on the DSI). Consequently, in Seq. Aa1 for example, the rate of
rotational energy dissipation (ǫrot) appears to be very high initially
(Fig. 10). The differentially rotating layers are rapidly smeared out
by the dynamical shear instability (see the discussion in Sect. 2 in
YL04), and ǫrot falls below the thermal diffusion and/or neutrino
cooling rate only within 20 yrs. Hence we conclude that the rota-
tional energy dissipation does not play an important role for the
long-term evolution of the central remnant.
Fig. 11 shows how the evolution of the central remnant
changes if the local peak of temperature in the initial model (Tp) is
close to or above the critical limit for carbon burning (TC−ig), with
Seqs Ac1 and Ae1 as examples. In contrast to Seq. Sa1 or Aa1, car-
bon burning dominates the evolution very soon in both sequences,
and the temperature increases rapidly. Although the further evolu-
tion has not been followed in the present study, it is most likely that
the carbon-burning flame propagates inward such that the central
remnant is converted into an ONeMg white dwarf within several
thousand years as shown by Saio & Nomoto (1998).
As summarized in Table 2, all other sequences follow the same
evolutionary pattern: off-center carbon ignition is avoided in Seqs
Ab1, Ba1, and Ta1 where Tp is significantly below TC−ig, while
carbon ignites off-center in the other sequences where Tp & TC−ig.
It is thus remarkable that the thermal evolution of the central rem-
nant is sensitively determined by the local peak of temperature in
the quasi-static equilibrium state.
In conclusion, in the absence of angular-momentum loss and
mass accretion from the Keplerian disc, the thermal evolution of
the central remnant is roughly controlled by neutrino cooling at the
interface between the core and the envelope, and off-center carbon
burning may be avoided as long as Tp < TC−ig, while it seems
inevitable if Tp & TC−ig.
Figure 10. Upper panel: The angular velocity relative to the local maximum
as a function of the mass coordinate in the central remnant models of Seq.
Aa1 at different evolutionary epochs. Lower panel: the rate of rotational
energy dissipation (ǫrot; see YL04) as a function of the mass coordinate in
the corresponding models shown in the upper panel.
3.3 Effect of angular momentum loss
In Seqs Sa2 – Sa5, the central remnant has the same initial condi-
tions as in Seq. Sa1, angular momentum loss from the white dwarf
with different time scales τJ is considered according to Eq. (4).
Note that off-center carbon ignition occurs in Seqs Sa2, Sa3 &
Sa4, where τJ . 10
4 yr, while it is avoided in Seq. Sa5 where
τJ = 10
5 yr. These results indicate that off-center carbon igni-
tion should be induced if the angular-momentum loss occurs too
rapidly for neutrino cooling or thermal diffusion to control the ef-
fective mass accretion. For instance, Fig. 12 shows that in Seq. Sa4,
where τJ = 10
4 yr, the effective mass accretion rate reaches a few
10−5 M⊙ yr
−1 at the interface between the core and the envelope
(ρ ≈ 106 g cm−3), and the compressional heating rate exceeds the
neutrino cooling rate.
It is shown that the critical angular-momentum-loss time
scale, τJ, for off-center carbon ignition (τJ,crit) is smaller for Seqs
Ab and Ba than for Seqs Sa and Aa: τJ,crit ≈ 10
3 for Seqs Ab
and Ba, and τJ,crit ≈ 10
4 for Seqs Sa and Aa. This is due to the
different local thermodynamic properties at the interface between
the core and the envelope in different central remnant models. As
shown in Fig. 13, higher density and/or temperature at the interface
result in a shorter neutrino cooling time, making it possible to avoid
local heating for a smaller τJ. In other words, τJ,crit roughly cor-
responds to the time scale for neutrino cooling at the local peak of
temperature (τν,p).
From this experiment, we conclude that, in the absence of
c© 2007 RAS, MNRAS 000, 1–??
12 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
Figure 11. Evolution of the central remnant in Seqs Ac1 (upper panel) and
Ae1 (lower panel), in the log ρ−T plane. The dotted curve gives the critical
temperature where nuclear energy generation rate due to carbon burning
equals to energy loss rate due to neutrino cooling.
mass accretion from the Keplerian disc, carbon ignition may be
avoided in the central remnant, if Tmax,init < TC−ig, and if
τJ > τν,p.
3.4 Mass accretion from the Keplerian disc
In reality, mass accretion from the Keplerian disc onto the central
remnant is expected. The accretion rate is determined by the vis-
cosity of the disc, which is not well known. However, we expect
the accretion rate from a Keplerian disc may be significantly lower
than from a pressure-supported thick disc that was assumed in pre-
vious studies. Our results, as summarized in Table 2, indicate that
even with mass accretion, the central remnant with Tp < TC−ig can
avoid off-center carbon ignition if the accretion rate is sufficiently
low (i.e., Ṁ < 5× 10−6...10−5 M⊙ yr
−1), and if τJ > τν,p (see
Table 2).
The thermal history of the central remnant in those sequences
where carbon ignites off-center is similar to that of the white dwarf
in classical accretion model sequences. However, as the central
remnant has a rapidly rotating hot envelope, carbon ignition is sig-
nificantly delayed compared to the case of classical accretion. In
Seq. N1.2, where Minit = 1.2 M⊙ and Ṁacc = 10
−5 M⊙ yr
carbon ignites only when about 0.025 M⊙ is accreted, while in
Seq. Aa7 more than 0.15 M⊙ have to be accreted to induce carbon
ignition at the same accretion rate, despite its higher initial mass.
On the other hand, the comparison of Seq. Aa7 with Seq. Aa10
indicates that off-center carbon ignition is delayed if the central
remnant keeps more angular momentum. The critical accretion rate
for inducing off-center carbon ignition is thus difficult to precisely
determine, as our 1-D models significantly underestimate the effect
of the centrifugal force, especially in the envelope where carbon
ignites. In addition, the physics of angular momentum loss/gain is
not well understood yet, as discussed in Yoon & Langer (2005).
Note that MWD,ig in Seqs Aa10, Ba6 and Ba7 is already very
close, or even above the Chandrasekhar limit. However, the cen-
tral density in those models is still smaller by an order of mag-
nitude than the critical limit for carbon ignition due to the effect
of rotation. As the carbon-burning flame will propagate inwards
within several thousand years (Saio & Nomoto 1998), only about
∼ 0.05 M⊙ may be further accreted by the time the burning flame
reaches the center, and the central density may not become high
enough to induce a thermonuclear explosion before the whole cen-
tral remnant is converted into an ONeMg white dwarf. (Super-)
Chandrasekhar mass ONeMg white dwarfs produced in this way
will eventually collapse to a neutron star (see Yoon & Langer 2005;
Dessart et al. 2006).
On the other hand, the white dwarf continuously grows
to/above the Chandrasekhar limit (≈ 1.4 M⊙) without suffering
carbon ignition (neither at the center nor off-center) in Seqs Sa8,
Sa9, Sa10, Sa11, Aa8, Aa9, Ab6, and Ba8. The outcome in these
cases is thus the formation of a (super-) Chandrasekhar mass CO
white dwarf, which will eventually explode as a Type Ia super-
nova. The mass of the exploding white dwarf should depend on
the amount of angular momentum (Yoon & Langer 2005) and can-
not exceed the mass budget of merging white dwarfs. Fig. 14 shows
the evolutionary paths of the central remnant for Seqs Sa8, Sa9 &
Sa11 as examples in the mass – angular momentum plane. Note that
the central remnant initially has a large amount of angular momen-
tum (J = 1.11×1050 erg s), such that without loss/gain of angular
momentum, it should accrete matter until it reaches M ≃ 1.68 M⊙
where it explodes in a SN Ia explosion. In Seqs Sa8 and Sa9, the ac-
cretion time scale (τacc) is longer than the angular momentum loss
time scale, and the total angular momentum of the white dwarf con-
tinuously decreases while the total mass increases. Consequently,
carbon ignites at the center when the white dwarf grows to 1.50M⊙
and 1.42 M⊙ for Seqs Sa8 and Sa9, respectively. In Seq. Sa11, on
the other hand, both mass and angular momentum of the central
remnant continuously increase, given that τacc . τJ, and a SN Ia
explosion is expected only when M ≃ 1.70 M⊙. Note that this
is even larger than the mass budget of the binary system consid-
ered for this sequence (i.e, 0.9 M⊙+0.6 M⊙). In nature, the white
dwarf must stop growing in mass when M = 1.5 M⊙, and a SN Ia
explosion will be induced only when a sufficient amount of angular
momentum has been removed, e.g. via gravitational wave radiation,
as illustrated by the path Sa11-B in Fig. 14.
4 CONCLUSION AND DISCUSSION
We have explored the dynamical and secular evolution of the
merger of double CO white dwarf binaries whose total mass ex-
ceeds the Chandrasekhar limit. Based on our new SPH simula-
tion of the coalescence of two CO white dwarfs of 0.9 M⊙ and
0.6 M⊙, we suggest that the immediate post-merger remnant is
best described as a differentially rotating CO star consisting of a
slowly rotating cold core and a rapidly rotating hot envelope that is
surrounded by a Keplerian disc rather than as “cold white dwarf +
thick disc” system, as in previous investigations. The evolution of
such a CO star is determined by the thermal evolution of the en-
velope, and the growth of the core is controlled by the cooling due
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 13
Figure 12. Same as in Fig. 9, but for Seq. Sa4.
Figure 13. Contour lines of the neutrino cooling time scale (τν ≡
TCp/ǫν ) in the log ρ − T plane. The level at each line gives log τν in
units of years. The dashed line denotes the critical temperature for carbon
ignition. The local peak of temperature and the corresponding density in the
initial model of Seqs Sa, Aa, Ab, Ac, Ae, and Ba are marked by the filled
symbols as indicated by the labels.
to neutrino emission and thermal diffusion, which is fundmentally
different from the assumption of “forced accretion of cold matter”.
Our 1-D stellar evolution models of the central remnant, i.e.
the cold core and the hot envelope, which include the effects of
rotation, indicate that there are three necessary conditions for the
merger remnant to avoid off-center carbon ignition such that a SN
Ia may be produced:
(i) The local peak of temperature of the merger remnant at the
interface between the core and the envelope must be lower than the
critical temperature for carbon ignition (Tp < TC−ig).
(ii) The time scale for angular-momentum loss from the central
remnant by must be larger than the neutrino cooling time scale at
the interface (τJ > τν,P).
(iii) Mass accretion from the Keplerian disc onto the cen-
tral remnant must be sufficiently slow (Ṁacc . 5 ×
10−6...10−5 M⊙ yr
Our new SPH simulation confirms that at least the first condition
(Tp < TC−ig) should be fulfilled in the CO white dwarf binary
considered.
As emphasized in Sect. 3.1, our 1-D models significantly un-
derestimate the effect of the centrifugal force on the stellar structure
in the rapidly rotating outermost layers. However, since thermal
diffusion always dominates over both neutrino cooling and com-
pressional heating in the outer envelope (ρ . 105...106 g cm−3)
above the interface, as shown in Figs. 7 and 12, the detailed struc-
ture of the rapidly rotating outermost layers above the interface may
not significantly affect our results on the thermal evolution of the
merger remnant, as long as the angular momentum of the envelope
is not lost faster than the local neutrino cooling time scale at the
interface. On the other hand, mass accretion from the Keplerian
disc should occur preferentially along the equatorial plane of the
envelope. As shown in the SPH simulation, the envelope is more
extended along the equatorial plane, where most angular momen-
tum is deposited, than along the polar axis, and the resultant com-
pressional heating must be much weakened, compared to the case
of our 1-D models. The enhanced role of rotation must thus help
c© 2007 RAS, MNRAS 000, 1–??
14 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog
to increase the critical mass accretion rate for inducing off-center
carbon ignition, in favor of producing a Type Ia supernova.
We have concluded that the loss of angular momentum on a
short time scale (τJ . τν,p ≈ 10
4...105 yr) may induce off-center
carbon ignition even when Tmax,init < TC−ig. Rapidly rotating
compact stars may experience loss of angular momentum by gravi-
tational wave radiation, due to either the bar-mode instability or the
r-mode instability. The onset of the dynamical or secular bar-mode
instability requires a very high ratio of the rotational energy to the
gravitational energy: Erot/Egrav & 0.2 for the dynamical bar-
node instability, and Erot/Egrav & 0.14 for the secular bar-mode
instability (e.g. Shapiro & Teukolsky 1983). As both our 1-D mod-
els and SPH simulation give a value of Erot/Egrav that is much
lower (about 0.06 – 0.07) than those critical limits, the bar-mode in-
stability may not be relevant. The r-mode instability may operate,
in principle, even with such a low Erot/Egrav (Andersson 1998;
Friedman & Morsink 1998). However, we estimate that the growth
time of the r-mode instability (τr), using our central remnant mod-
els and following Lindblom (1999), is & 106 yr, which is much
longer than the local neutrino cooling time scale (τν,p ≈ 10
4 yr).
Alternatively, angular momentum might be transported from the
accreting star into the Keplerian disc when the accretor reaches
critical rotation. Calculations by Saio & Nomoto (2004) indicate,
however, that the decrease of the total angular momentum due to
such an effect is not significant in accreting white dwarfs. In con-
clusion, neither gravitational wave radiation nor outward angular-
momentum transport is likely to lead to a rapid loss of angular
momentum from the central remnant such that τJ < τν,p, unless
magnetic torques are important.
The central remnant may be enforced to rotate rigidly on a
short time scale in the presence of strong magnetic torques (cf.
Spruit 2002). The central remnant in both our SPH simulation and
1-D models has Jtot > 10
50 erg s, which is significantly higher
than the maximum limit a rigidly rotating white dwarf can retain,
as shown in Fig. 14. This means that if magnetic torques led to rigid
rotation, a large amount of angular momentum should be trans-
ported into the Keplerian disc (Case a in Fig. 14), or mass shed-
ding of super-critically spun-up layers should occur from the cen-
tral remnant (Case b in Fig. 14). In Case a, the local density around
the interface should increase by several factors by the time when
the central remnant reaches rigid rotation as implied by Fig. 15.
Off-center carbon ignition might be inevitable in this case due to a
resultant high effective accretion rate, if the time for angular mo-
mentum redistribution were shorter than the local cooling time due
to neutrino losses. In Case b, on the other hand, the local density
at the interface might not increase if mass shedding from the cen-
tral remnant occurred at a sufficiently high rate. Therefore, the role
of magnetic fields in the merger evolution remains uncertain at the
current stage and is a challenging subject for future work.
The coalescence of more massive double CO white dwarf bi-
naries is likely to result in a higher maximum temperature due to the
enhanced role of gravity. Consequently, given the important role of
the maximum temperature in the merger remnant for its final fate,
less massive binary CO white dwarfs may be favored for the pro-
duction of SNe Ia from such a channel.
We note that there are number of potentially important factors
that have not been included in either the present study or previous
simulations. These include the following points:
(i) The previous and present simulations assumed that white
dwarfs are cold prior to the merging process. However,
Iben, Tutukov & Fedorova (1998) point out that tidal interactions
Figure 14. Evolution of the central remnant in the mass – angular momen-
tum plane. The thick solid curve shows the angular momentum of a rigidly
rotating white dwarf with critical rotation at the surface as a function of the
white dwarf mass. The thick dashed curve and the thick dot-dashed curve
give the critical angular momentum for a differentially rotating CO white
dwarf to reach carbon ignition at the center (ρc = 2 × 109 g cm−3), and
electron-capture induced collapse (ρc = 10×1010 g cm−3), respectively,
according to Yoon & Langer (2005). A SN Ia explosion is expected in the
hatched region. The filled circle denotes the initial model of the central rem-
nant in Seq. Sa. The evolution of the central remnant in Seqs Sa 8, Sa9 and
Sa11 is shown by the thin dotted curves, as indicated. The thin solid curves
denote possible evolutionary paths of the central remnant with strong mag-
netic torques that may enforce rigid rotation, with loss of angular momen-
tum but without mass shedding (Case ’a’), and with both loss of angular
momentum and mass shedding (Case ’b’). See the text for more details.
Figure 15. The density profile in the initial model of the central remnant
in Seqs Sa (solid curve), and in a corresponding hot (Tc = 108 K) white
dwarf model that rotates rigidly at critical rotation at the surface (dashed
curve).
might heat up the white dwarfs as the orbit shrinks, which could
weaken the gravitational potential of the primary. Furthermore, as
the temperature of white dwarfs is a function of their age, younger
progenitors should have more extended envelopes, which may re-
sult in a lower Tp.
(ii) A thin hydrogen/helium envelope must be present initially in
both the primary and the secondary. As hydrogen or helium should
ignite at a much lower temperature than carbon, the influence of
the release of nuclear energy during the merger process may be
even more important than shown in the existing SPH simulations,
c© 2007 RAS, MNRAS 000, 1–??
Remnant evolution after a carbon-oxygen white dwarf merger 15
which is likely to lower Tp. Furthermore, neutrino losses, which
were neglected in the present study, would also tend to reduce Tp.
(iii) At a given total mass (Mtot = Mprimary + Msecondary),
different mass ratios of the white dwarf components (q ≡
Msecondary/Mprimary) must result in different merger structures.
(iv) A lower q at a given Mtot may not only lead to a stronger
gravitational potential of the primary, but also to a lower mass
accretion rate during the dynamical mass transfer (Guerrero et al.
2004). As the former and the latter will tend to increase and de-
crease Tp, respectively, quantitative studies are necessary to predict
how Tp will change with q.
Finally, another important ingredient that needs to be consid-
ered is thermal diffusion during the dynamical evolution. As shown
above, the mass-accretion rate from the Keplerian disc onto the en-
velope of the central remnant is one of the most important factors
that critically determine the final fate of double CO white dwarf
mergers. The accretion rates depend on the structure of the Kep-
lerian disc at thermal equilibrium, which can be only understood
by including thermal diffusion in future simulations. But here we
emphasize again that the accretion rates from a centrifugally sup-
ported Keplerian disc should be significantly lower than those from
a pressure-supported thick disc that was previously assumed, which
opens the possibility for at least some double CO white dwarf
mergers to produce SNe Ia.
ACKNOWLEDGMENTS
We are grateful to Norbert Langer and Ken’ichi Nomoto for many
useful suggestions and comments. SCY is supported by the VENI
grant (639.041.406) of the Netherlands Organization for Scientific
Research (NWO). The computations have been performed on the
JUMP supercomputer at the Höchstleistungsrechenzentrum Jülich.
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http://arxiv.org/abs/astro-ph/0610872
