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NUCLEAR REACTIONS IN TYPE IA SUPERNOVAE:

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NUCLEAR REACTIONS IN TYPE IA SUPERNOVAE: EFFECTS OF
PROGENITOR COMPOSITION AND DETONATION ASYMMETRY

By
David A. Chamulak

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirmnents
for the degree of

DOCTOR OF PHILOSOPHY
Physics

2009

ABSTRACT

NUCLEAR REACTIONS IN- TYPE IA SUPERNOVAE: EFFECTS OF
PROGENITOR COMPOSITION AND DETONATION ASYMMETRY

By
David A. Chamulak

Type Ia supernovae go through three distinct phases before their progenitor star is
obliterated in a thermonuclear explosion. First is “simmering,” during which the
12C + 12C reaction gradually heats the white dwarf on a long (~ 103 yr) timescale.
Next is a period of subsonic burning. Finally, a detonation is thought to occur that
ﬁnishes unbinding the star. This thesis investigates the nuclear reactions that take
place in these three phases and considers what that may be able to tell us about the
progenitor systems and the mechanics behind the detonation. First, we investigate
the nuclear reactions during this simmering with a series of self-heating, at constant
pressure, reaction network calculations. As an aid to hydrodynamical simulations of
the simmering phase, we present ﬁts to the rates of heating, electron capture, change
in mean atomic mass, and consumption of 12C in terms of the screened thermally
averaged cross section for 12C + 12C. Our evaluation of the net heating rate includes
contributions from electron captures into the 3.68 MeV excited state of 13C. We com-
pare our one-zone results to more accurate integrations over the white dwarf structure
to estimate the amount of 12C that must be consumed to raise the white dwarf tem-
perature, and hence to determine the net reduction of Ye during simmering. Second,
we consider the effects of 22Ne on ﬂame speed. Carbon-oxygen white dwarfs contain
22Ne formed from a-captures onto 14N during core He burning in the progenitor star.
In a white dwarf (type Ia) supernova, the 22Ne abundance determines, in part, the
neutron-to—proton ratio and hence the abundance of radioactive 56Ni that powers the
lightcurve. The 22N e abundance also changes the burning rate and hence the laminar

ﬂame speed. We tabulate the ﬂame speedup for different initial 12C and 22Ne abun-

dances and for a range of densities. This increase in the laminar ﬂame speed ~—about
30% for a 22Ne mass fraction of 6%——affects the deﬂagration just after ignition near
the center of the white dwarf, where the laminar speed of the ﬂame dominates over
the buoyant rise, and in regions of lower density ~ 107 gem“3 wl'tere a transition to
distributed burning is conjectured to occur. The increase in ﬂame speed will decrease
the density of any transition to distributed burning. Finally, we look at how a sur-
face detonation affects the composition of nuclides across the supernovae remnant.
Several scenarios have been proposed as to how this delayed detonation may actually
occur but careful nucleosynthesis calculations to determine the isotopic abundances
produced by these scenarios have not been done. The surface detonation produces a
clear compositional gradient in elemental Ni in layers of the white dwarf that do not
burn to nuclear statistical equilibrium (N SE). A number of nuclides show a gradient
but when combined into elemental abundances Ni shows the largest change over the
face of the star. The Ni abundance varies by as much as an order of magnitude across

the star. This may be a way to observationally test detonation models.

DEDICATION

To the men who understand—or think they do

Allen W. Jackson, The Half Timber House

iv

ACKNOWLEDGMENT

Special thanks goes to my advisor Ed Brown; my committee members, Tim Beers,
Kirsten Tollefson, Mark Voit, and Romeo Zegers; collaborators, Hank Timmes, Casey
A. Meakin, Ivo Seitenzahl, James Truran; For all the Physics and Astronomy graduate
students at Michigan State, and for every one else. who was there for me over these

last 10 years in College.

\7

TABLE OF CONTENTS

List of Tables ................................. vii
List of Figures ................................ viii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Compositional Effects ........................... 3
1.2 Detonation ................................ 5
Methodology..........................6
2.1 Thermonuclear cross sections and reaction rates ............ 6
2.2 Reaction networks ............................ 9

Electron captures in simmering . . . . . . . . . . . . . . . . . 11

3.1 Introduction ................................ 11
3.2 The reduction in electron abundance during the explosion ....... 13
3.2.1 The role of neon-22 and other trace nuclides .......... 14
3.2.2 The reaction 13N(e_, Ve)13C ................... 16
3.2.3 Production and subsequent captures of neutrons ........ 18
3.3 Reaction network calculations ...................... 19
3.3.1 The reactive ﬂows ......................... 20
3.3.2 The effective Q—value of the 12C + 12C reaction ........ 26
3.3.3 Heating of the white dwarf core and the end of simmering . . 29
3.4 Discussion and conclusions ........................ 34

Theeffectof22Neonﬂamespeed ................ 39

4.1 Introduction ................................ 39
4.2 The laminar ﬂame ............................ 40
4.3 Results ................................... 43
4.4 Discussion ................................. 47

Compositional effects of surface detonation . . . . . . . . . . . 49

5.1 Introduction ................................ 49
5.2 The Explosion Model ........................... 49
5.3 Results ................................... 52
5.4 Discussion ................................. 57

Conclusion............................58

Bibliography..........................61

vi

3.1

3.2

4.1

L"!
5.4

5.2

LIST OF TABLES

430-Nuclide Reaction Network ...................... 19

Change in electron abundance per carbon consumed during the pre-

explosion convective burning ...................... 26
Laminar ﬂame speed and width ..................... 44
493—Nuclide Reaction Network ...................... 52

Partial table of nuclides that do and do not display a separation because
of asymmetric detonation. ........................ 53

vii

3.1

3.2

3.3

3.4

3.6

4.1

4.2

5.1

5.2

Q31
C0

LIST OF FIGURES

Images in this dissertation are presented in color

Thermally averaged cross-sections as a function of temperature .

Reaction ﬂows at p = 109 gcm‘3

Reaction ﬂows at p = 3 x 109g CHI—3
Change in electron abundance as a function of carbon consumed . . .

Convective and electron capture timescales as a function of temperature

Correlation between 12+ log(O / H) and Am15(B) induced by the elec-
tron abundance

Flame speed as a function of density
Abundances of selected nuclides during a burn at p = 2.0 x 109 gcm—J
Final abundance of elemental Ni as a function of central angle

Thermal expansion time. scale as a function of initial position .

Final abundance of elemental Ni as a function of radial velocity

viii

17

22

23

30

33

36

42

45

Chapter 1

Introduction

In the pantheon of astronomical explosions few are as bright as the Type Ia supernova
(hereafter SNe Ia). SNe Ia are, by deﬁnition, a category of variable stars that lack
hydrogen and present a singly ionized silicon line (in the rest frame) at 615.0 nm
in their spectrum near peak light. There are several ways by which a supernova of
this type could form, but all share a common underlying mechanism. Stars with a
main sequence mass between approximately 0.8-8 solar masses ﬁrst consume hydrogen
and form helium. Helium is built up in the core of the star until it too starts to
burn, at which time the star enters its giant phase of evolution. During the giant.
phase the star fuses the helium in the core. to carbon and oxygen. Since the mass
of the star is less than 8 solar masses it is not able to continue to burn carbon to
neon and magnesium and becomes a carbon-oxygen white dwarf, once the helium is
consumed. Stars between 8 and 10 solar masses can burn carbon and become oxygen-
neon-magnesium white dwarfs. When a slowly-rotating, carbon-oxygen (C—O) white
dwarf accretes matter from a companion, it cannot exceed the Chandrasekhar limit of
approximately 1.38 solar masses, beyond which it would no longer be able to support
itself through electron degeneracy pressure and begin to collapse. In the absence of

a countervailing process, the white dwarf would collapse to form a neutron star, as is

thought to occur in the case of an oxygen-neon-magnesium white dwarf.

As described in Hillebrandt 8.: Niemeyer (2000), the increase in pressure and den-
sity due to the increasing mass raises the temperature of the core. Once the temper-
ature and density of the core is such that the heating from carbon burning is greater
then the energy loss from thermal neutrinos, a period of convective burning ensues,
lasting approximately 1,000 years. This phase of burning is known as the “simmer-
ing” phase. Once burning has begun, the temperature of the white dwarf starts to
rise. A star, like the sun, supported by thermal pressure would expand and cool in
order to counter-balance an increase in thermal energy. However, since degeneracy
pressure is independent of temperature; the white dwarf is unable to regulate the
burning process in the same way. When the heating rate from carbon burning be-
comes faster then the sound crossing time of the star a subsonic deﬂagration ﬂame
front is born. The details of the ignition are still unknown, including the location and
number of points where the ﬂame begins. The ﬂame accelerates dramatically, due to
hot ash being Rayleigh-Taylor unstable to the surrounding cold, and therefore denser,
fuel. The deﬂagration is is hot enough to burn the C—O to iron group elements. At
some point, it is thought, that the burning changes form a subsonic deﬂagration to a

supersonic detonation, but the method by which happens is not well understood.

Regardless of the exact details of nuclear burning, it is generally accepted that a
substantial fraction of the carbon and oxygen in the white dwarf is burned into heavier
elements within a period of only a few seconds, raising the internal temperature to
051

billions of degrees. This energy release from thermonuclear burning (~ 1 ergs) is

more than enough to unbind the star.

Near the time of maximum luminosity, the spectrum contains lines of intermediate-
mass elements from oxygen to calcium; these are the main constituents of the outer
layers of the star. Months after the explosion, when the outer layers have expanded

to the point of transparency, the spectrum is dominated by light emitted by material

2

 

near the core of the star, heavy elements synthesized during the explosion; most
prominently iron peak elements. The radioactive decay of 56N i through 56Co to 56Fe
produces high-energy photons which dominate the energy output of the eject-a at
intermediate to late times.

Type Ia supernovae have a characteristic light curve. The typical visual absolute.
magnitude of Type Ia supernovae is All, 2 —19.3 (Hi11ebrandt & Niemeyer, 2000).
Because of this similarity, in the past decade SN e la have become the premier standard
candle for measuring the geometry of the universe. Current observations are sampling
the SNe Ia population out to a red shift, 2 z 1.6 (Riess et al., 2004), and future
missions will push this limit even farther to .3 S, 2. The larger sample of SNe Ia
carries with it the prospect for discovering the progenitors of these events and their
evolution towards ignition. Numerical models (for a sampling of recent work, see
Gamezo et al. 2004; Plewa et a1. 2004; Ropke et al. 2006) are steadily becoming more
reﬁned and can begin to probe the connection between the properties of the progenitor
white dwarf—its birth mass, composition, and binary cornpani011---a11d the outcome

of the explosion.

1.1 Compositional Effects

Despite the advances in modeling the post-ignition ﬂame evolution (for a sampling of
recent work, see Gamezo et al., 2004; Plewa et al., 2004; Ropke et al., 2006; Jordan
et al., 2008), we still lack a complete understanding of which subset of the binary
white dwarf population become SNe Ia, and how differences in the progenitor map
onto differences in the outcome of the explosion.

The composition of the white dwarf at the time of the explosion should have an
effect on the nucleosynthesis that takes place during the explosion and the isotopic

abundances of the final composition. The observable properties of SNe Ia resulting

3

from Chandrasekhar-mass explosions are chieﬂy determined by their final composi-
tion, the velocity profiles of key spectral lines at early- and late-times (e.g., P-Cygni
proﬁles of Si II at 615.0 nm at early times), the opacity of the material through which
the photons from radioactive decay must propagate, the kinetic energy of ejecta, and
its interaction with the density proﬁle of the surrounding circumstellar or interstellar
medium (Filippenko, 1997; Pinto & Eastman, 2000; Hillebrandt & Niemeyer, 2000;
Leibundgut, 2001; Mazzali 81. Podsiadlowski, 2006; Marion et al., 2006; Blondin et. al.,
2006; Badenes et al., 2007; Woosley et al., 2007). The dominant parameter in setting
the peak brightness, and hence width, of the light curve is widely believed to be the
mass of 56Ni ejected by the explosion. Timmes et a1. (2003) showed the mass of 56N i
produced depends linearly on the electron fraction, Ye, at the time of the explosion,

and that Ye itself depends linearly on the abundance of 22Ne in the white dwarf.

The C:O ratio is set by the mass of the. progenitor main-seqence star, although
Ropke & Hillebrandt (2004) find that the C:O ratio is of secondary importance in set-
ting the explosion energetics. After 12C and 16O, the next most abundant nuclide is
22Ne, which is synthesized via 14'N(Ct, 7)19F(,t’3+)180(a, 7)22Ne during core He burn-
ing. The abundance of 22Ne is therefore proportional to the initial CNO abundance of
the progenitor main sequence star. Timmes et a]. (2003) showed that the mass of 56N i
synthesized depends linearly on the abundance of 22Ne at densities where electron
capture rates are much slower than the explosion timescale, ~ 1 8. Simulations with
embedded tracer particles (Travaglio ct al., 2004; Brown et al., 2005; Ropke et al.,

2006) have confirmed this dependance.

These simulations studied the effect. of adding 22Ne by 1.)ost—processing the (p, T)
traces, and as a result did not account for variations in either the progenitor struc-
ture or the sub—grid ﬂame model caused by changes in the 22Ne abundance. One-
dimensional studies that did attempt to incorporate different progenitors self-consistently

(Hoﬂich et al., 1998; Dominguez et. al., 2001) found a much smaller dependence of the

4

56N i yield on metallicity.1

1.2 Detonation

Besides effects from the progenitor white dwarf there is also a lack of understanding
of the mechanics of how the star actually detonates. It is unclear how a SN Ia
transitions from a subsonic deﬂagration to a supersonic detonation but simulations
ﬁnd a transition must occur since pure deﬂagration models do not burn enough of the
white dwarf to account for any but the most subluminous events (Reinecke et al., 2002;
Gamezo et al., 2004; Ropke et al., 2007a). Different models have been proposed to
handle detonations in simulations (Khokhlov, 1988; Gamezo et al., 2004; Ropke et al.,
2007b; Plewa et al., 2004; Plewa, 2007), however careful nucleosynthesis calculations

to determine exact isotopic yields have not been done.

 

1It is unclear whether these studies allowed [0 / Fe] to vary as a function of [Ft / H]

5

Chapter 2

Methodology

Since all of the work covered in this thesis was completed using reaction networks
this chapter will review reaction network metl‘iodology. This chapter is meant to only
be primer in reaction network calculations, for a more detailed review of reaction

network calculations see Timmes (1999).

2.1 Thermonuclear cross sections and reaction rates

Consider a reaction that takes the form j + k —) Z. This reaction has a number of
quantities associated with it; one being the cross section of the reaction, a. The
cross section gives the number of reactions per second per target per ﬂux of incoming
projectiles and may be directly measured through experiment, which can be difﬁcult
depending on the. energy measured or the stability of the reactants. In such cases
where a can not be measured directly 0 may be evaluated following a theoretical

approach. A rate, rJ-‘k, for the reaction j + k —> I can be described by the equation

3 3
rj.k = [OJ-k Ivj — ka d njd 72k, (2.1)

6

where lvj — vk| is the relative velocity of j and k and nj and ink. are the number
densities of j and k respectively. This integral can be evaluated by specifying a

distribution function. For example, take a l\-’Iaxwell-Boltz1nann distribution for j and

 

k
. 3/2 mzv-
3, __,_ m] _, _ 11 3...
“Pnjfapbrl "X" 2pr d1“ (2'2)

where mj is the mass of j, k}, is the Boltzmann constant, and T is absolute tem-
perature. If we transform from velocity to energy we can write the reaction rate
as

rjfk 2: (ov)jqk nj-nk, (23)

where (an) is the thermally averaged cross section defined as

on 2 <av>jp = (35-) (km—W [Ox Evy-p (E) p (11%;) dB. (24)

mt

where. p is the reduced mass in the center of I'nass system. I have deﬁned (j, k) E
(011)3- k to eliminate confusion later when talking about one and three body reactions

which I will designate by (j) and (j, k, 1) respectively.

A similar technique can be used to ﬁnd the reaction rate for a pl’iotodissociation of
the form j + '7 —-> 1+ 777.. The velocities of the nuclei are always insigniﬁcant to that of
light therefore the integrand in equation 2.1 is not dependent on d3nj and d3nk can
be represented by a Planck distribution of photons. It is, however, very difﬁcult to
directly measure the cross section for a photodissociation and even if a cross section
was somehow measured the inherent error would mean that equilibrium abundances
would be off. It is therefore easier to calculate the photodissociation rate from the

forward rate using detailed balance. This gives an equation 'rj = ( j) nj where

. Glam AlAm 3/2 "111ka 3/2 le
_— —— —— l, n i.’ —— . 2.:—
<]) < Gj > ( Aj ) 271712 < I I) (\1) k1,] ( a)

7

 

 

Here (I, m) and Q1", are the thermally averaged cross section and energy release of
the reaction I + m —* j + '7, A1 is the atomic mass of l, mu is the mass of a nucleon,

and the partition functions G = 2,- (2J,- + 1) exp (—E.l-/kT).

For weak reactitms. like for the case of photodissociation above, the velocity of
the nuclei is insigniﬁcant since in the center of mass system, vj << 've, I'vj — up] 2:
Incl. Depending on the astrophysical conditions that one. is interested in d3'ne is
either given by a Boltzmann distribution, partially degenerate distribution or Fermi
distribution of electrons. It then follows that the electron capture rate is just a

function of temperature, T, and number density, n]-

rj = (j) (Tppn) M (215)

Here Ye is related to the number density of electrons, 726 by the equation Y6 =

ne/ (pNA), where p is the density and NA = 6.022 x 1023 g“1 = (1 amu)_1.

A special consideration to take into account when calculating reaction rates is
the effect of electron screening. In plasmas at high density or low temperature, the
presence of background electrons and nuclei cause the reactant nuclei to feel a different
Coulomb repulsion then they would if they were not surrounded by charged particles.
If the change of the electrostatic potential, (25, is such that Ad << kT the generalized
reaction rate integral can be separated into the traditional expression (eq. 2.4) and
a screening factor. fsm— (for a review of why this is see Salpeter &. Van Horn, 1969),

resulting in a new reaction rate (j, k)* such that,
(j‘k)* : fSF‘I (Zj,Zk,,0,T,I/,) <j2k> (2'7)

The screening factor is a function of the charges of the interacting nuclei, Z J- and

Z k, the density p, the temperature T and the composition of the plasma Yi. For self

8

consistency a screening factor should be applied to a measured rate and the reverse
rate should be calculated through detailed balance (Calder et al., 2007). This is the
only way to be sure that material in equilibrium will have. the right abundances when
calculated with the network. This, however, is not always done and a screening factor
often will be applied to a given forward and reverse. rate resulting in a network run

to equilibrium not preserving detailed balance.

2.2 Reaction networks

The number density of each nuclide. in an astrophysical plasma at constant density
varies with time according to the number of reactions per (31113 per second an expersion

given by

an ’. v.‘ 1. '
(all) __. 2 NJ, 73 + 2 ”$ka + Z N}.i~..17“j.k.z- (2.8)
p j M

j.k.l
here ni is the number density of nuclide i and r is the reaction rate deﬁned in § 2.1.
The ﬁrst sum is over all one body reactions such as decays, photodissociation, elec-
tron / positron captures and neutrino induced reactions. The second sum is over two
body reactions like 12C+0z. The third sum is over three body reactions like the

triple-a reactions. The N 7's are deﬁned as

N} = Ni, N11,, 2 N,/ [1 |ij|!, and Ajk‘, = N,/ [I [NJ-ml! (2.9)
m m

Here N,- can be positive or negative and speciﬁes how many particles of nuclide i
are created or destroyed in a reaction. The products in the denominators, which
include the factorials, run over the number of particles destroyed in a reaction and
avoid double or triple counting reactions when like particles react with each other,
like 1l‘zC-l-12C or the triple-a reactions.

For ﬂuid equations the. mass density is approximated as p = Z,- 77,in n-1, where

9

,u is the mass of one nucleon. The mass fraction of nuclide 2' is then X, = niAip—1p_1

and we can write n,- = (X,-/A,-);rp E Yinp. This allows us to rewrite equation (2.8),
in order to exclude changes in the number density which are only due to expansion

or contraction of the gas, as

Y,— ZNj( j)Y- +ZN MAPAA<j>k>Y7Yk+ZNij1P9 NA(j,k, l)YY,,Y,. (2.10)
1' k. I
Here equation (2.10) follows directly from equation (2.8) when the deﬁnition for 7‘
and Y are introduced.
Therefore, it follows that the total energy generation per unit mass, due to reac-

tions in a given time step which changes abundances by AY is

5 = 4;, c2Z (AL-5%) — 5V, (2.11)

Here 6,, is the energy loss from neutrinos, A1,: is the mass of nuclide i and c is the
speed of light. Equations (2.10) form a series of stiff coupled ordinary differential

equations, for which standard numerical solvers exist (sec Press et al., 1992).

10

Chapter 3

Electron captures in simmering

3. 1 Introduction

In this chapter we explore, using a reaction network coupled to an equation for self-
heating at constant pressure, the reduction in Ye that occurs after the onset of the
thermonuclear runaway (when the heating from 12C + 12C reactions is faster than
coolng by thermal neutrino emission), but before the burning becomes so fast that
local regions can thermally run away and launch a ﬂame. This “simmering” epoch
lasts for ~ 103 yr, long enough that electron captures onto products of 12C burning
can reduce the free electron abundance Ye. A similar, but independent calculation,
of the reduction of 1’} during simmering was performed by Fire & Bildsten (2008).
Their calculation did not use a full reaction network, but it did take into account the
change in energy of the white dwarf due to the growth of the convective zone. Our
calculation agrees with their ﬁndings, in particular that there is a maximum Ye at
the time of the explosion, and that the reduction in Y8 is linear in the amount of 12C
consumed prior to when the rate of 12C burning outpaces that of the weak reactions.
This chapter expands on their work in three ways. First, by using a full reaction

. 9 . . .
network, we are able to quantify the role of 2“Ne and trace nuclides 1n setting the

11

change in electron abundance with 120 consumption, dYe / all/12. Second, we calculate
the heating from the electron capture reactions and include the contribution from an
excited state of 13C. Third, we provide tabulated expressions for the rate of heating
a, the rate of change in electron abundance dY8/dt, and the rate of change in the
mean atomic mass d(A) /dt in terms of the reaction rate for the 12C + 12C reaction.
These expressions are useful input for large-scale hydrodynamical simulations of the

simmering phase which do not resolve such microphysics.

We ﬁrst give, in § 3.2, a simple estimate for the reduction in Y8 during the pre-
explosion simmering and describe the role of 22Ne and other trace nuclides. In § 3.3
we describe our numerical calculations, explain the reactive ﬂows that occur (§ 3.3.1),
and give simple approximations to the heating rate and carbon consumption (§ 3.3.2).
We detail, in § 3.3.3, some of the limitations of our approach. We evaluate the energy
required to raise the white dwarf central temperature, and hence the amount of
12C that must be consumed, and compare it against the one-zone calculation. We
also estimate the central temperature at which convective mixing becomes faster
than electron captures. This convective mixing advects electron capture products
to lower densities where they can ﬁ—-decay: the convective Urca process. Because
each electron capture-decay cycle emits a neutrino anti-neutrino pair, there is energy
lost from the white dwarf, and our calculation underestimates the amount of 12C
consumed prior to the ﬂame runaway. The convective Urca process (Paczyr’iski, 1972)
reduces the rate of heating by nuclear reactions (thereby increasing the amount of
120 that must be consumed to raise the temperature), but cannot result in a net
decrease in entropy and temperature for constant or increasing density (Stein et al.,
1999; Stein & Wheeler, 2006). The Urca reactions also tend to reduce the effects of
buoyancy, and in degenerate matter have a direct influence on the convective velocity

(Lesaffre et al., 2005).

12

3.2 The reduction in electron abundance during

the explosion

The demise of an accreting white dwarf begins when the. central temperature and
density are such that the heating from the 12C + 12C reaction becomes greater than
the cooling from thermal neutrino emission. For a density p = 2.0 x 109 gem—3
this requires a temperature T a: 3.0 x 108 K (see Gasques et al., 2005, for a recent
calculation). Initially the heating timescale is long, tH E T(ciT/dt)“1 ~ 103 yr; as
the temperature rises and the reaction rate increases, tH decreases. Woosley et a1.

(2004) estimate that when T > 7.6 x 108 K, fluctuations in the temperature are
sufﬁcient to ensure that a local patch can run away and the flame ignites.

The basic reactions during 12C burning were ﬁrst worked out in the context of
core carbon burning in evolved stars (Reeves & Salpeter, 1959; Cameron, 1959). Dur-
ing simmering, 12C is primarily consumed via 12C(12CL’, a)20Ne and 12C(12C, p)23Na.
These reactions occur with a branching ratio 0.56/0.44 for T < 109 K (Caughlan
&, Fowler, 1988). At temperatures below z 7 x 108 K, neutronization—that is, a
reduction in Y€—occurs via the reaction chain 12C(p,'y)13N(e”,1/(3)13C. This elec-
tron capture implies that there is a maximum Y; at the time of the explosion, as
ﬁrst pointed out by Piro & Bildsten (2008). One can readily estimate the change in
electron abundance, AYE. For simplicity, take the branching ratio for 12C +12C to be
1:1 for producing 19 + 23Na and 4He + 20Ne. Thus there is one 17 produced for every
four 12C consumed. Two additional 12C are consumed via 12C(n,'y)13C(4He, n)160,
which also destroys one 4He nucleus, and via 12C(19,7)13N((3_,1/e)13C, which also
destroys one p. Thus for every 6 12C consumed there is one electron capture, so
that dYe /dY12 z 1 / 6, where Y12 is the molar abundance of 120. As an estimate for
the heating from this reaction sequence, summing over the q-values for the strong

reactions gives a net heat release of (16 MeV) / 6 = 2.7 MeV per 12C nucleus con-

13

sumed. Note that at densities above 1.7 x 109 gem—3, the reaction 23Na(e‘,1/(,)23Ne
contributes to the rate of decrease in Ye, so that dYe /dY12 z 1 / 3 at those densities.
The total effective rate of dYe /dY12 depends on the rate of convective mixing and the
size of the convective core (see § 3.3.3) but is always at least as large as the contri-
bution from 13N(e‘, Ve)13C. In the following sections, we investigate these reactions

in detail.

3.2.1 The role of neon-22 and other trace nuclides

Reactions on 22Ne, 23Na, and other trace nuclides also occur during shell-burning in
asymptotic giant branch (AGB) stars, and we briefly summarize their role in that
context before describing the very different environment in a simmerng white dwarf
core. In AGB stars more massive than about 4 Mg, the hydrogen burning shell,
at a temperature of 60-100 MK, extends into the convective envelope. The enve-
lope composition is then directly affected by the various hydrogen-burning cycles:
CNO, NeNa, and MgAl (Lattanzio 81. Boothroyd, 1997; Herwig, 2005). The hydrogen-
burning shell is also disturbed by thermal pulses due to ignition of the helium layer.
At each pulse, dredge-up of material may occur, in which helium-burnt material is
mixed into the stellar envelope, polluting it with 4He, 12C, 22Ne, and heavy s-process
elements (Izzard et al., 2007). Thus, the fate of 22Ne is either to contribute to hydro-
gen burning via 22Ne(p,7)23Na or to become a neutron source for the s-process via
the 22Ne(a,n)251\’1g reaction. The 22Ne(a,n)25Mg reaction requires the high tem-
peratures (T > 2.5 x 108 K) that can be found at the bottom of the pulse-driven
convective zone during the helium shell ﬂashes. The neutrons are released with high
density [log(Nn/cm—3) ~ 10] in a short burst (Gallino et al., 1998; Busso et al., 1999).
These peak neutron densities are realized for only about a year, followed by a neutron
density tail that lasts a few years, depending 011 the stellar model assumptions. These

neutrons are the genesis of the classic high-temperature s-process in AGB stars.

14

Should 23Na be present, there are two usual possibilities for subsequent nuclear
processing in AGB stars via either the 23Na(p, (1)20Ne reaction or the 23Na(p, 7)24Mg
reaction. The former reaction gives rise to the classical NeNa cycle (Marion & Fowler,
1957; Rolfs & Rodney, 1988; Rowland et al., 2004), whereas the competing (p, '7)
reaction transforms 23Na to heavier isotopes and bypasses the N eNa cycle. How much
material is processed through the (p, a) reaction on 23N a as opposed to the competing
(p, 7) reaction is of interest for AGB star (and classical novae) nucleosynthesis. New
measurements of the 23Na + 1) cross section (Rowland et al., 2004) suggest that for
T = (20—40) MK 23Na(p, 7)24Mg competes with the (p, a) branch, disrupts the NeNa

cycle, and produces a ﬂow into the MgAl hydrogen burning cycle.

Caution about intuition developed for reaction sequences on 22Ne and 23Na in
AGB star environments seems prudent when applied to the dense, carbon-rich en-
vironments of white dwarfs near the Chandrasekhar mass limit. During the slow
12C simmering preceding the explosion, the large 1’12 ensures that p liberated by
the 12C(12C,p)23Na branch will capture preferentially onto 12C, rather than 22Ne or
23N a. Figure 3.1 shows the ratio of thermally averaged cross-sections, /\ E NA(av), to
that for 12C(p,'y)13N for three reactions: 22Ne(p,7)23Na (solid line), 23Na(p,a)20Ne
(dashed line), and 23Ne(p,n)23Na (dotted line). In addition to the Coulomb pene—
tration, there are numerous resonances that determine how the cross-sections change
with temperature. When the ratio of the thermally averaged cross-sections is of or-
der unity, as it is for 22Ne(p,'y)23Na, then the proton capture is determined by the

relative abundances of 22Ne and 12C.

Note that for the latter two reactions, we plot the largest proton-consuming branch
rather than the (p, '7) branch. We include screening in all reactions (Yakovlev et al.,
2006) with the plasma taken to consist of 12C and 160 with mass fractions of 0.3 and
0.7, respectively. For T > 108 K, A[22Ne(p, 'y)23Na] is well-constrained experimentally

(Iliadis et al., 2001). For a white dwarf with a central density p = 2 x 109 gem—3

 

the ignition temperature (where nuclear heating dominates over cooling via thermal
neutrino emission) is z 3 X 108 K (Gasques et al., 2005); the burning timescale
becomes less than the timescale for 23Na(e_, ue)23Ne once the temperature increases
beyond T ,2 6 X 108 K. Over this range, the thermally averaged cross-sections for
22Ne(p,’y)23Na and 12C(p,*y)13N are comparable, but the abundance of 12C is far
greater; having p capture preferentially onto 22Ne would therefore require it to have
a mass fraction roughly twice that of 12C.

At densities greater then 1.7 x 109 g CID—3, the reaction 23Na(e—, V6)23N8 produces
roughly one 23Ne nucleus for every six 12C nuclei consumed. The screened, ther-
mally averaged cross-section for 23Ne(;0, n.)23Na is 2, 30 times that of 12C(p,'y)l3N
(Fig. 3.1), so that 23Ne could become a competitive sink for protons. For our self-
heating burn (see § 3.3.1) starting at p = 3 x 109 gcn‘1‘3, the abundance of 23Ne
reaches 1’23 = 3 x 10—4 (Y23 z 0.015Y12) by the point the heating timescale TH
becomes shorter than the timescale for electron captures onto 23Na. Although our

f 12C that must be consumed to

one-zone approximation overestimates the amount 0
raise the central temperature (see § 3.3.3), should enough of the convective core lie
above the threshold for electron capture onto 2'3N a, the abundance of 23Ne can be—

come large enough to choke off the 12C(1),’}")13N reaction, as noted by Piro & Bildsten

(2008).

3.2.2 The reaction 13N(e‘, 1/8)13C

As described above, the large 12C abundance ensures that protons prrxluced by
12C(12C, p) 23Na lead to the formation of B+-uristal)le 13N via 12C(p,7)13N un-
less an appreciable abundance of 23Na or 23Ne can build up. At these densities, the
rate for electron capture is substantially greater than the rate of 13+-decay for 13N.
The electron Fermi energy is z 5.1 MeV(pYe/109 gem-3)1/3, and the q-value for

the 3+ decay of 13N is 2.2 MeV. As a result, there are several excited states of 13C

16

 

l
—d
—
—1
.l
—(
1
—
l

l
l

E l “a...“ ._"._.,,,,..23Ne(p,n) E

.. ’1’ “\\“ ’,,23Na(p,a) -
2 10 :— T“----""" -_-__
>~ : E
s : :
U _ _
9! 1 ”N609,”
\ _ ..
‘S I I

l llllnll
l lJlllllI

 

 

—
—
—
-
—
_
I-n
—
—
—
—
—
-
_
-
—
—
_

 

0.2 0.4 0.6 0.8 1.0
T (GK)

Figure 3.1: Ratio of thermally averaged cross-sections, /\ E NA(0v), to that for
12C(p,'y), for three reactions: 22Ne(p,’y) (solid line); 23Na(p,a) (dashed line); and
23Ne(p,n) (dotted line). Screening is included in A; we evaluate the ratio at p =
3 x 109 gem"3 for 23Ne(p, n) and at 109 gcm‘3 for the other two. In the temperature
range where the heating timescale is slow enough for weak interactions to reduce Ye
(cf. Fig. 3.4), the thermally averaged cross—sections for 22Ne(p,y) and 12C(pxy) are
of similar magnitude. For 22Ne to compete with 12C for p—captures at T Z, 3 x 108 K
requires a 22Ne mass fraction X22 2; (22/ 12)X12. At densities less than the electron
capture threshold on 23Na a small ﬂow of 23Na(p,a)2ONe can occur (cf. Fig. 3.2).
At higher densities 23N a electron captures to form 23Ne; the large cross-section for
23Ne(p, n)23Na allows it. to compete with captures onto 12C if 1’23 / Y12 2, 1%.

into which the electron can capture. Of tl'iese, the transition to the excited state
Eexc = 3.68 MeV, with spin and parity J 7‘ = 3 / 2‘, is an allowed Gamow-Teller tran-
sition. We computed the electron capture rate using the experimental log ft for the
ground—state transition (Ajzenberg-Selove, 1991). Gamow-Teller strengths to excited
states were calculated with the shell-model code OXBASH (Brown et al., 2004) em-
ploying the Cohen-Kurath II potential (Cohen & Kurath, 1967) in the p—shell model
space. A quenching factor of 0.67 (Chou et al., 1993) was applied to this strength,
and the resulting ft values were used with the analytical phase space approxima-
tions of Becerril Reyes et a1. (2006) to obtain the capture rate. These shell-model
calculations agree well with recent (3He, t) scattering data (Zegers et. al., 2007). At

3, captures into the excited state at Eexc = 3.68 MeV account for

pi}; = 109 gem"
,2 0.3 of the total rate (Rec 2 12 s"1); this fraction increases with density. Because
Rec > (Gm—V2, this capture does not freeze out during the simmering. unlike cap—
ture onto 23Na. Although the capture into the excited level does not increase the
capture rate substantially beyond that for the ground-state—to—ground-state transi-

tion, it does increase the heat deposited into the white dwarf from this reaction. We

ﬁnd the heat deposited from this reaction, at pY, = 109 gem—3, to be 1.3 MeV.

3.2.3 Production and subsequent captures of neutrons

Finally, we consider the contribution from heavier nuclei, such as 56Fe, inherited from
the main-sequence star. In AGB stars the reaction 13C(a. n)mO (during a He shell
ﬂash, 22Ne(a,n)251\/Ig also contributes) acts as a neutron source for the s-process. In
contrast, the large 12C abundance of the white dwarf core prevents a strong .s-process
ﬂow during the pre—explosion simmering. The cross-section for 12C(n, 7)13C is 63.5
times smaller than the cross-section for 56Fe(-n, 'y)57Fe (Bao 85 Kappeler, 1987) at an
energy of 30 MeV, which is not sufﬁcient to overcome the vastly larger abundance of

12C nuclei (for a progenitor with solar metallicity, the 12C:56Fe ratio [for a 12C mass

18

 

 

El. A El. A El. A El. A

 

 

n

H 1~3 F 1524 C1 31 44 Mn 46-63
He 3—4 Ne 17-28 Ar 31 47 Fe 46- 66
Li 6 -8 Na 20— 31 K 35 46 Co 50 -67

Be 7, 9 ‘11 Mg 20~33 Ca 35 53 Ni 50-73
B 8, 10-14 Al 22~35 Sc 40 53 Cu 56- 72
C 9~16 Si 22’ 38 Ti 39 --55 Zn 55 72
N 12- 20 F 26-40 V 4357 Ga 60~75
O 13—20 S 2742 Cr 437—60 Ge 59* -76

 

Table 3.1: 430-Nuclide Reaction Network

fraction of 0.3] is 140021).

3.3 Reaction network calculations

In this section we investigate the reactions that occur during simmering in more detail
using a “self-heating” reaction network. Under isobaric conditions the temperature

T evolves with time according to

 

dT 5’
.2 . (.1
(it Cp (3 )

Here Cp is the. speciﬁc heat at constant pressure, and the heating rate a is given by
equation (2.11). The neutrino loss rate, per unit mass, from the weak reactions is
taken from Fuller et al. (1982) and Langanke & Martinez-Pinedo (2001). We neglect
thermal neutrino emission processes; this is an excellent approximation over most of
the integration. Our reaction network incorporated 430 nuclides up to 7b‘Ge (see Table
3.1). We use the reaction rates from the library REACLIB (Rauscher & Thielemann,
2000; Sakharuk et al., 2006, and references therein). We incorporated screenng using
the formalism of Graboske et al. (1973). The matter does not reach nuclear statistical

equilibrium or quasi-nuclear statistical equilibrium. Thus, although our treatment of

19

 

 

screening does not preserve detailed balance (Calder et al., 2007) this does not affect
our calculation. Our equation of state has contributions from electrons, radiation, and
strongly coupled ions. We include thermal transport by both degenerate electrons and
photons (for a complete description of our thermal routines, see Brown et al. 2002,
and references therein). At conditions of p = (1 3) x 109 gem"3 and T = 5 x 108 K,
the speciﬁc heat Cp is dominated by the ions, which are in a liquid state (plasma
parameter I‘ E (25/3) (eQ/kBT) (pYeJ\7A)1/3 z 10), and have Cp z 2.9kBNA/(A),

where (A) is the average atomic mass.

3.3.1 The reactive ﬂows

In this section we refine our estimate of dYe /d.Y12 made in § 3.2.2. We. integrate
equation (3.1) starting from the temperature at which heating from the 12C + 12C
reaction equals the heat loss from thermal neutrino losses (this determines the onset of
thermal instability). For simplicity, we split the solution of the thermal and network
equations. That is, for each time-step dt we solve the thermal equations to obtain T
and p, integrate the reaction network at that T and p to compute Y,- and e, and use

5 to advance the solution of equation (4.1).

In the initial phases of the simmering, the convective timescale is slow, and our
one-zone calculations give an adequate description of the heating (when corrected for
the gradient in temperature). As the temperature of the white dwarf increases, the
heating timescale tH decreases; moreover, the convective mixing becomes more rapid,
and one must treat the hydrodynamical ﬂows in order to calculate the nucleosynthesis
properly (see § 3.3.3). In this section, we will restrict our integration to where T <
0.6 GK, for which the heating timescale is Z 104 s, so that electron captures onto

23Na are not frozen out.

To explore the. reaction channels that link 12C consun'iption with the reduction in

20

 

Y}, we deﬁne the reactive ﬂow between nuclides 2' and j as

“Has/t

where the integral is for the reactions linking nuclide 2' with j. We differentiate

de:

-— dt 3.2
d, , < >

 

 

i—>j

between F (2 —+ j) and F ( j —> 2'), i.e., we treat inverse rates separately. Figures 3.2

and 3.3 show the reactive ﬂows for p = 109 gcm‘3 and 3 x 109 gcm’3

, respectively.
In both cases the composition is {X12,X16,X22} = {O.3,0.7,0.0}. Each row of the
chart is a different element (Z), with the columns corresponding to neutron number.
For viewing ease, we only plot those ﬂows having F > 0.01 — max(F), and we indicate
the strength of the ﬂow by the line thickness. We highlight the ﬂows of the initial
12C fusion reactions in red. For illustrative purposes, we only show the ﬂows for

when the temperature is not sufﬁciently hot for photodissociation of 13N. Finally, by

convention and to avoid cluttering the plot, we do not show ﬂows into or out of p, n,

and 4He.

To facilitate con'iparisons between different. runs, we deﬁne a normalized ﬂow as

~ . . _ F(i“’jl
F“ —+ J) : F(12C —> 23Na) + F(12C —> 20Ne)°

 

(3.3)

At a density of 109 gcm‘3 we ﬁnd F(12C —+ 23Na) 2 0.43 and F(12C —> 20Ne) =
0.57, which reﬂects the branching ratio (Caughlan & Fowler, 1988). In agreement with
the arguments in § 3.2, most of the protons liberated by 12C(12C, p)23Na capture onto
12C, with F(12C —+ 13N) z F(13N —+ 13C) = 0.20. Note that if all 19 were to capture
onto 12C, we would have F(12C —+ 13N)/F‘(12C ——> 23Na) 2 0.5, because the reaction
12C(12C,p)23Na consumes two 12C. The a-particle released by the 12C(12C,0z)20Ne
reaction captures onto 13C to form 160 and a neutron, which in turn destroys another

12C via 12C(71,7)13C. The ﬂow F(12C ——> 13C) = 0.26 and nearly matches the number

21

WMDDDD EDD
Na(11)|:]|:] DD
MWDDC DUDE
F®DDE GDDDDD
0(8)l:ll:l . DD 14 15
NmD D D
z, “mama El 11 12 13
Hem DEED
Rem DDD8
Li(3)DDD 6 7
He(2)[:“:}3 4 5
maﬁa
n<0)[j 2 NH
1

Figure 3.2: Flows during a constant-pressure self-heating burn at p = 109 gcm‘3
with an initial composition {X 12, X 16, X22} 2 {0.3, 0.7, 0.0}. We integrate over the
period of the burn with T < 6 X 108 K. Each row contains the isotopes of a particular
element Z, with the columns containing different. neutron numbers N. The width of
the arrow is proportional to the magnitude of the ﬂow, with only those ﬂows having
magnitude > 0.01 - max(F) being shown. The primary 12C + 120 reactive ﬂows are
indicated with a lighter shading and are in red.

 

 

\

 

 

 

 

sDDDD
SEED

22

mwmmmmm
Naanmmrus
momma:
F®DDT,U
0®DDD1 DD
NWD DD
NC©DDD DD‘
Bmm DEEDS”
mam DDD89
Li(3)DDD 6 7
He(2)|:“:] 3 4 5
HmDDD
n(0)[:| 2 N—+
1

 

 

 

 

 

 

 

 

 

 

L
D
D
D

12 13

—I

Figure 3.3: Same as in Fig. 3.2, but at p = 3 X 109gcm‘3. The reaction
2'lNa(e*,1/(,)23Ne is now the dominant destroyer of 2‘lNa instead of (17,7), (p,a),
and (n,’7) reactions.

23

of a-particles produced by 12C(12C, (1)20Ne; n—captures onto 23N a and 20Ne. account
for the difference. At p = 109 gcm”3 the only electron captures are onto 13N, so
AYC = —13‘(13N —+ 13C). Dividing by AY12 == —[13‘(12C —+ 23Na.) + F(12C —+
20Ne) + F020 -+ 13N) + 15020 ——) 130)] gives did, mm = 0.14 (Table 3.2). This is
slightly less than the estimate of 1 / 6 (§ 3.2.2) because of the lower branching ratio of

120 _* 23Na.

At p = 3 x 109 gem—3, 23Na is consumed by the reaction 23Na(e_,1/(,)23Ne rather
than by p- or n-capture (cf. Figs. 3.2 and 3.3). The ﬂow is much larger than that
from 13N, because the reaction 23Ne(p,n)23Na competes for p and produces more
23Na. Indeed, F(12C ——> 13C) = 2.5F(12C —> 13N) because of extra 72. produced
by the reaction 23Ne(p,n)23Na. There is an additional contribution from electron
captures onto 17F produced via 16O(1r),’y)17F, but this ﬂow is only about 4% of the
23Na(e—,Ve)23Ne ﬂow. At both densities, 22Ne plays a small role in reducing Y6,
which we veriﬁed by comparing the ﬂows for a burn with X22 = 0.06 with those
for a ﬂow with X22 = 0.0 (Fig. 3.3). We ﬁnd that, for a burn starting at p = 3.0 x
109 gcm"3, F(22Ne —i 23Na)/F(12C —> 13N) = 0.08 and F‘(22Ne —> 23Ne)/F(12C —>
13C) -—- 0.05. Although A[22Ne(p,’y)23Na] S, /\[12C(p,’y)13N] at T S 6 x 108 K
(Fig. 3.1), the abundances are in a ratio of Y22/Y12 = 0.11. Below the electron
capture threshold for 23N a, the reaction 22Ne(p, 7)23Na will cause a slight reduction

in dYe/leg equal to the ratio of F(22Ne ——+ 23Na) to F(12C ——> 13N).

Finally, we conﬁrmed the lack of an s-process ﬂow (§ 3.2.3) by performing a run
with nuclides from 20Ne to 56Fe present at up to 3 times their solar abundances
(Anders &. Grevesse, 1989). The threshold for electron capture onto 56Fe is 1.5 x

3, and so at higher densities carbon ignition occurs in a more neutron-rich

109 gem“
environment. We therefore start the calculation by artiﬁcially suppressing the strong
interactions and allowing the mixture to come into ,B-equilibrium. We then turn on

the strong reactions and let the runaway commence. In all cases the heavier nuclides

24

(lid not have a substantial impact on the reactive ﬂows. The value for dYe /dY12 is
somewhat larger than 0.3 at densities p 2 1.2 x 109 g cm'3, the threshold density for
25Mg(e", Ve)25Na(p, 72), because of the reactions 2J‘Mg(p,a,/)25Al(e"',1/(,)25lVIg(e",1/6.)
2F’Na. Of these two captures, 25Al is [3+-unstable, and hence the electron capture is
fast enough to proceed throughout simmering; the capture onto 25Mg has a timescale,
at p = 2 x 109 gcm“3, of z 900 s and will therefore freezeout during simmering, just
as captures onto 23N a freezeout.

Our runs span a range of initial densities, from 109 g cur—3 (for which the electron
Fermi energy is too low to induce electron captures onto 23Na) to 6 x 109 gcm‘3,
which represents an extreme case for accretion onto a cold, initially massive white
dwarf (Lesaffre et al., 2006). In all cases we took the initial 12C mass fraction to be
X12 = 0.3. As noted above, the mass fraction of 22Ne would have to exceed that of
12C to change the nucleosynthesis during simmering appreciably. At higher densities,
p—captures onto 24Mg can also play a role (§ 3.3.1), but our results will not change

appreciably so long as X12 is not substantially less than 0.3.

The calculations, being a single reaction network integration, do not incorporate
the effects of mixing in the white dwarf core. Our focus here is to elucidate the
reactions that set Ye. These calculations do not determine the total amount of carbon
consumed (although see § 3.3.3 for an estimate) or the total mass of processed material
that lies at a density greater than the electron capture threshold. We list our one-zone
results in terms of the change in electron abundance per carbon consumed, dYe /dY12.
Table 3.2 summarizes our numerical ﬁndings of dYe / dY12 for densities 109, 3 x 109,
and 6 x 109 gem—3, for 22Ne mass fractions X22 2 0 and 0.06, and ﬁnally a run
(denoted as 3231,”, in Table 3.2) with elements heavier than 20Ne present at 3 times
solar abundance. We use this value of 3231”; as representing a rough upper limit
based on the z 0.5 dex scatter in [F(/ H] present in local ﬁeld stars (Feltzing et. al.,

2001)

 

 

 

composition density d1}, /dY12 Z,- in /dY12
109 gem—3

ng = 0.00 1.0 0.136 0.340
- - 3.0 0.297 0.340
--- 6.0 0.302 0.342
X22 = 0.06 1.0 0.125 0.347
~-- 3.0 0.301 0.344
~- 6.0 0.305 0.346
32., 1.0 0.130 0.361
~° - 3.0 0.349 0.370
6.0 0.355 0.380

 

Table 3.2: Change in electron abundance per carbon consumed during the pre-
explosion convective burning

3.3.2 The effective Q-value of the 12C + 12C reaction

The scope of this work is to elucidate the nuclear reactions that occur during the
pre-explosion simmering, and including their effects in large-scale hydrodynamics
simulations of the entire white dwarf is advisable. As an aid to such simulations,
we present ﬁts for the carbon depletion rate and effective heat deposition, which
improve on previous expressions (Woosley et al., 2004). Since the reaction chain
is controlled by the reaction 12C + 12C, we write the rate of carbon consumption,
leg/dt as being proportional to the thermally averaged cross-section, A = Mﬁav)

leg 1 2 .
-—— = -]l" — . “ .
d, 112 (,Yii pi) o 4)

This deﬁnition is such that the quantity in parenthesis is the reaction rate per pair
of 12C nuclei and Mn = 2 if the only 12C-destroying reaction present is 12C + 12C.
One can determine Mm from summing the normalized reaction ﬂows (eq. [33]) out

of 120.
In a similar fashion, we can deﬁne the effective heat. evolved, (Jeff, per reaction

26

 

 

12C + 12C by the equation

1 . . r
E : gaffA/A (51/122 pA) . (3.0)

With this deﬁnition, one has 5 = (qeﬂng /1’l[12) x leg/dt; this may be compared
with equation (2.11) to relate qeff to the binding energy of the nuclei. To compute
these quantities, we integrated the reaction network over a grid of p and T, both of
which were held ﬁxed for each run. We found in all cases that the instantaneous
energy generation rate 5 would, after some initial transient ﬂuctuations, settle onto
a constant value until a signiﬁcant (> 10%) depletion in 12C had occurred. We used
this plateau in E to obtain qeff and Mn. For the densities of interest, the values of
both get; and 3112 obtained this way are nearly independent of temperature. We ﬁnd
Mlg = 2.93; this value is accurate to within 2% over all our runs. The value of qeff
increases slightly with density, but is nearly constant over the temperature range of

interest. We ﬁnd that qeff = 8.91 MeV, 9.11 MeV, and 9.43 MeV for p = 109 gcm—3,

3 3

2 x 109 gcm- , and 3 x 109 gem" , respectively. At each of these densities, the
quoted value of qeﬂ‘ is accurate to within 3% over the temperature range of 3 x 108 K

to 7 x 108 K.

For use in large-scale hydrodynamic models, one ﬁrst computes the screened ther-
mally averaged cross-section A (for the most recent rate, see. Gasques et al., 2005)
at the thermodynamic conditions of a given cell. Combining A and the cell’s carbon
abundance Y12 with our estimates of 11112 and qeff, one computes dY12 / dt and 5 from
equations (3.4) and (3.5). In effect, this procedure incorporates the results of a large
reaction network and careful treatment of the detailed nuclear physics into simple
expressions. We caution that these ﬁts were obtained in the regime —AY12 < 0.003,
for which proton captures onto 23Na are not competitive with proton captures onto

12C. Note that the heat released per l2C nucleus consumed is (Jeff/.0112 z 3.1 MeV,

27

slightly higher than our simple estimate (§ 3.2), and also somewhat higher than those
used by Piro & Bildsten (2008). This is because of our inclusion of heating from the
13N(e”‘, ue)13C and 23Na(e—, ue)23Ne reactions. Our estimate of the heat evolved is
less than that computed under the assumption that the products of 12C burning are
a 3:1 20Nez24Mg mixture (see, e.g. Woosley et al., 2004), which releases 5.0 MeV per
12C nucleus consumed.1 Our net heating rate, per 12C + 12C reaction, is about 10%
less than that used by Woosley et al. (2004), but we effectively consume 2 3 12C
nuclei per reaction.

The change in electron abundance is related to A via

dYe de 1 9 ,
— z — M ’ —Y“ 3.6
6175 ( 1200/12) X (2 12 p3), ( )

where (11/6/de is taken from Table. 3.2. Finally, we may compute the change in

 

the mean atomic mass, (A), as a function of carbon consumed. On differentiating

(A) = (Z, dY; / (1112)"1, and substituting equation 3.4, we have

 

35%) = (A)27l112 (% Y122pA) (Z ddl’lg) (37)

where. the quantity 2.,- d1”,- /dY12 is computed from the ﬂows (eq. [33]) and is listed
in Table 3.2. Horn the simple description of the reactions (§ 3.2; see also Piro &
Bildsten 2008) we have, for every six 12C destroyed, one each of 13C, 160, 20Ne, and
23Na (or 23Ne), so that Z, in /dY12 % one third. This results in a smaller change
in (A) than would result from burning 12C + 12C to a 3:1 2ONez23Na mixture (in
that case, 2,- dY. /dY12= 0. 42). We advocate using equations (3.4)-(3.7) and the
computed values (Table 3.2) of dYe /dY12 and Z,- in /dY12 in numerical sinmlations

of simmering.

 

1Note that in eq. (1) of \Voosley et al. (2004), the factor of one. half is subsumed
into their quantity A1212.

28

3.3.3 Heating of the white dwarf core and the end of sim-

mering

In previous sections, we evaluated the heating and neutronization of the white dwarf
core in terms of the rate of 12C consumption. We now estimate the net amount
of 12C that is consumed in raising the white dwarf temperature and evaluate the
temperature at which electron captures onto 23Na freeze out. In the one-zone isobaric
calculations, using equations (3.4) and (3.5), we have dT/d 1’12 = (qeffNA) / (M12019),
so that AT z 0.15 GK (AX12/0.()1), where AX12 is the change in mass fraction of
12C. The change of 12C abundance required to raise the temperature from 3 x 108 K
to 8 x 108 K is then |AY12| a: 2.8 x 10‘3. This is about 11% of the available 12C,
for an initial 12C mass fraction of X12 = 0.3. Figure 3.4 shows the decrement of the
e‘ abundance, Ye(t = 0) —— Y6(t) = ~AY8 as a function of 12C consumed, Y12(t =
0) — Y12(t) = —AY12. Note that we are plotting the decrement in abundance. The
calculation was started at an initial density and temperature of p = 3.0 x 109 gem—3
and T = 1.9 x 108 K. As tH shortens, the electron captures onto 23Na “freeze out”
and dYe / dY12 decreases to z 1/6. When —AY12 2, 0.003, the abundances of 23Na
and 23Nehave increased sufficiently that they compete with 1BC to consume protons,

and thereby halt the neutronization, in agreement with Fire & Bildsten (2008).

As noted by Fire & Bildsten (2008), a one-zone model will overestimate the heat
required to raise the central temperature by a given amount, and hence overestimate
the amount of 12C that must be consumed during simmering. We perform a calcula-
tion similiar to theirs: starting with an isothermal white dwarf with a given central
density and with a temperature set by equating heating from the 12C + 12C reaction
with neutrino losses, we then raise the central temperature to 8 x 108 K, keeping the
total white dwarf mass ﬁxed, and follow an adiabatic temperature gradient until we

intersect the original isotherm at radius rmnv, which we then follow to the stellar sur-

29

 

T = 0.85 GK; tH= 10’7yr -
T = 0.67 GK; 1H: 10'5yr

     

1ch = 0) - Ye(t)1>< 104

 

 

 

l
0 1 2 3 4
[Y12(t= 0)‘ Y12(t)]x 103

Figure 3.4: Change in electron abundance, Ye(t = 0) — Y6(t), as a function of carbon
consumed, Y12( = 0) — Y12(t). The break in the slope, at 1712(0) — Y12 z 2 X 10’3,
occurs when the heating timescale tH = Cp/E becomes less than the timescale for
electron capture onto 23Na, which is z 2700 s at p = 3.0 X 109 gem—3. We indicate
this point with the thin vertical line. To guide the eye, the short dotted line indicates
a slope of 1 / 3.

30

face. We compute the total stellar energy, gravitational and thermal, in both cases,
and take the difference to ﬁnd the heat required to raise the central temperature of
the star to 8 X 108 K. The temperature of 8 x 108 K is chosen as a ﬁducial temperature
representing the point at which the heating of a ﬂuid element proceeds faster than
the convective timescale (Woosley et al., 2004). The evolution of the white dwarf core
is not exactly isobaric: the expanding convective zone heats the white dwarf. As the
entropy of the white dwarf increases, it expands and reduces the core pressure. For

3, we ﬁnd that in raising the central

an initial central density Pinit = 3.0 X 109 gem—
temperature from 3 X 108 K to 8 X 108 K the radius expands by a factor of 1.1 and
the central pressure decreases to 0.59 of its initial value.

For initial central densities Pinit. = 109 g cm—3, 3 X 109 g cm‘3, and 6 X 109 g cm‘3,
corresponding to white dwarf masses of 1.35 Me), 1.38 MG), and 1.39 MG), the initial
temperatures deﬁned by the onset of thermal instability are 3.8 X 108 K, 1.9 X 108 K,
and 1.0 x 108 K, respectively. The energy required to raise. the central temperature
to 8 X 108 K is EC = 2.11, 4.12, and 5.58 keV per nucleon, respectively. When the
central temperature has reached 8 X 108 K, the masses of the convective zone for these
three cases are 0.69 MG), 1.10 Mg, and 1.29 Mg, respectively. The spatial extent of
the convective zone, rconv z 1000 km for the three cases, is in agreement with the
ﬁndings of Kuhlen et al. (2006). We checked our computation of EC by computing
E, = fconv. C'p [Tﬁnal — Tinitiall dM, as was done by Piro & Bildsten (2008). Both
methods give comparable estimates, but E; slightly underestimates the change in
energy (by z 10%), because it does not account for the expansion of the white dwarf.
Neglecting the change in Ye as the white dwarf heats introduces a small correction to

EC: for an adiabatic 1.38 Mg white dwarf with a central temperature of 8 X 108 K,

a reduction in Y6 by 1.66 X 10‘3 reduces EC by only 3.3%.

If the white dwarf were entirely mixed, raising the central temperature 8 X 108 K

would require, for the three Pinit cases here, that |AY12| = 1.36 X 10‘3, 1.66 X 10‘3,

31

and 1.87 X 10‘3, respectively. Because the changes in composition are only mixed over
the convective zone, the decrement in Y12, and hence Ya is more pronounced there.
Using our estimate of rmnv, we estimate that within the convective zone |AY12| =
3.0 X 10'3, 2.1 X 10‘3, and 2.01 X 10‘3, respectively, for {Jinn/(109 gcm‘3) = 1.0,
3.0, and 6.0. Should the radial extent of the convective zone be smaller than our
estimate, for example because of convective Urca losses (Stein et al., 1999; Stein &
Wheeler, 2006; Lesaffre et al., 2005), the abundance of 12C will be further reduced in
the white dwarf core. A lower 12C abundance reduces the laminar speed of the ﬂame

launched at the end of simmering (Timmes & Woosley, 1992; Chamulak et al., 2007).

Finally, we estimate at what point the convective mixing timescale. becomes shorter
than the timescale for electron captures onto 23Na. Using our adiabatic temperature-
gradient white dwarf models, we compute the typical convective velocity vrms, and
hence a characteristic turnover time tconv = rmnv/vrms, using the mixing length
formalism (see the discussion in Woosley et al. 2004) with the total luminosity and
evaluating thermodynamical quantities at their central values. Figure 3.5 shows tconv
(thick lines) for the three cases of Pinit considered above, as well as the electron cap-
ture timescale, tee for 23N a (thin lines) for those cases with a density above threshold.
To estimate the effect of the 23Na/23Ne pairs on the convective zone, we evaluated
the mass fraction of these pairs for the case pinit = 3.0 X 109 gcm‘3 when the tem-
perature had risen to T = 3.5 X 108 K and tec a: imm- (Fig. 3.5, solid line). At this
point the convective core has a mass 0.5 Me) which is comparable to their calcula-
tion. We estimate, from the energy required to heat the white dwarf to this point,
that. the mass fraction of 23Na/23Ne pairs in the convective zone will be X23 2 0.004
at this time. We note that such a large number of Urca pairs will have a dramatic
effect on the properties of the convection zone (Lesaffre et al., 2005). As is evident
from Figure 3.5, there is a range of temperatures for which tconv < too < tH. In

this region, the effective d Y6- /dY12 will depend on the fraction of mass with densities

32

above the capture threshold and on the effects of Urca losses on the convective ﬂows,
but will still be larger than the minimum value set by 13N(e_,1/e)13C. Incorporating
the effects of the neutrino luminosity from the 23Na/23Ne reactions in this regime
is numerically challenging (Lesaffre et al., 2005, 2006) and beyond the scope of this
work; for now, we just note that this is the primary uncertainty in determining the

9 . . .
amount of 1“C consumed during the pre—explos1ve phase, and a better treatment 18

 

 

needed.
_fl 1 1 1 In 1 1 l 1 1 1T Ti 1 I 1 1 1 l 1 l 1 l 1 _
104 :— ‘5
Z 1
I 1
A h -
m
V 3
0 l:- -:
J, 10 E E
A " :
:9, C _
> — _
8
+9 102 :— 1
10 F 1
F1 1 I 1 1 1 l 1 1 1 l 1 1 1 l 1 1 1 l 1 1 1 l 1 1 1 l 1 fl

 

 

 

o
N
.o
w
.o
4;

0.5 0.6 0.7 0.8
T(GK)

Figure 3.5: Mixing length convective timescales (thick lines) and electron capture
timescales (for densities above threshold) for 23Na (thin lines), as a function of
the central temperature during the simmering. Three initial densities are shown:
109 gcm“3 (dotted line), 3 X 109 gcm’3 (solid lines), and 6 X 109 gcm‘3 (dashed
lines). The electron capture timescales increase. as the white dwarf heats because the
density decreases during simmering.

33

3.4 Discussion and conclusions

Using a nuclear reaction network coupled to the. equation for self-heating at constant
pressure (eq. [3.1]), we have investigated the change in Y8 induced by electron captures
on nuclei produced by 12C fusion during the pre—explosion simmering of the white
dwarf. We conﬁrm that there is a maximum Ye at ﬂame ignition (Piro & Bildsten,
2008). We quantiﬁed the role of 22Ne and other trace nuclides in setting the change
in electron abundance with 12C consumption by using a full reaction network, and
we included the heating from electron captures into an excited state of 13C. We gave
simple formulae (eq. [3.4] —[3.7]) for the energy generation rate, the rate of change in
electron abundance, and the rate of change in the mean atomic mass to include the

detailed nuclear physics into large-scale hydrodynamical simulations.

Our estimates of the maximum Ye at the. time of the explosion are roughly similar
to those of Piro & Bildsten (2008). If we neglect the effect of thermal neutrino losses
on the evolution of the white dwarf, then Ye is reduced by 2.7 X 10—4—63 X 10‘4
within the convective zone. This reduction in Y6 depends predominantly on the
amount of 12C consumed prior to ignition. The electron captures during simmering
reduce Ye below the value set by neutron-rich 22Ne inherited from core He burning
by the white dwarf ’s progenitor star. Reducing Ye in the explosion depresses the
yield of 56Ni and increases the amount of 54Fe and 58Ni synthesized (Iwamoto et al.,
1999; Timmes et al., 2003), even in the absence of further electron captures onto the
Fe-peak isotopes in nuclear statistical equilibrium (NSE) in the densest portion of
the white dwarf. As a result, any correlation between host system metallicity and
white dwarf peak luminosity will be weakened for Z 5 2(5) (for which the reduction in
1}, due to captures during simmering is greater than the change due to initial white

dwarf composition).
To illustrate how the simmering electron captures affect the light curve, we recon-

34

struct the comparison made by Gallagher et al. (2005), who compiled a set of SNe Ia
with measured Am15(B), deﬁned as the change in B over 15 days post-peak, and
host galaxies with measured abundances of oxygen to hydrogen, denoted O/ H. We
construct an expression for M55, the mass of 56N i produced in the explosion, that
depends on dYe / dY12 (Table 3.2), AY12, and the host galaxy composition (we assume
that the white dwarf has the same 0/ H ratio as the galaxy). The trace nuclide that
predominantly sets Ye in the white dwarf is 22Ne, which traces the aboriginal abun-
dance of CNO nuclei in the main-sequence star from which the white dwarf evolved.
We therefore recast the linear formula for M56 (Timmes et al., 2003) in terms of O / H.
For simplicity, we ﬁx the 1H:4He ratio, as well as the ratio of heaver elements to 160,
to their solar system values (Asplund et al., 2005) and neglect corrections from the
change in [O/ Fe] with [Fe/ H] (Ramirez et al., 2007) and the increase in [N/ O] with
[O/ H] (Liang et al., 2006). With these. assumptions, the mass of 56Ni ejected in the
explosion is

O

A4156 = 3156.0 [I — 72.7 (TI) + 58

dYe
leg

 

Ayn] , (3.8)

where M550 is the total mass of N SE material synthesized at densities where electron
captures during the explosion are negligible, dYe /dY12 is the change in electron abun-
dance with carbon consumption given in Table 3.2, and —AY12 is the total amount

of 12C consumed during the pre—explosion convective phase.

Relating this to an observable such as Am15(B) requires an explosion model that.
follows the radiation transfer. Our interest here is to isolate how M56 changes with
the CN O abundance of the progenitor when the other parameters, such as the ejecta
kinetic energy and the total mass of iron-peak nuclei, are held ﬁxed. This is important,
because the relation between light curve width and peak brightness depends on these
other parameters as well (see Woosley et al., 2007, for a lucid discussion). We use the

model MO70103 (W'oosley et al., 2007), in which the total mass of iron-peak ejecta is

35

 

 

 

 

 

 

 

2-0 _ --- are: —7.5><10—4 [ —
' — AYe = 0 r
_ """ Gallagher et a1. (2005) ﬁt a
s 1'5? i
1.0 - —
- I I l l I l I l I l l l I l l I I I l I I I l '-

8.4 8.6 8.8 9.0 9.2 9.4

12 +log10 (g)

Figure 3.6: Correlation between 12+log(O / H) and Am15(B) induced by the electron
abundance. The points in the plot are taken from Gallagher et al. (2005). We show
the ’maximal’ case for the reduction in Ye during simmering (dashed line), in which
dYe /dY12 = 0.30 in the early stages of the burn, which is an upper limit to the
derivative. In this case AYe = —7.5 X 10‘4. For comparison, we also show the case.
in which there is no reduction of Ye during simmering (solid line). Note that the
relation between 171156 and Am15(B) is shallower than that used by Gallagher et al.
(2005, dotted line); see the text for an explanation. To the left of the vertical bar,
the decrease in M56 due to electron captures during simmering will exceed that due
to enriclunent by 22Ne, and hence any correlation between 0/ H and Am15(B) will
be masked.

36

0.8 Aim: of that, the innermost 0.1 AL; is stable iron formed m situ from electron
captures during the explosion, with the remainder being a mix of radioactive 56Ni and
stable Fe. Note that this model follows the peak luminosity-light curve width relation
(Phillips, 1993), whereas Mazzali & Podsiadlowski (2006) suggest that varying the

ratio of 56N i to stable Fe may create dispersion about this relation.

Using the model of Woosley et a1. (2007), we set M550 2 0.7 111.3), use equa-
tion (3.8) to compute M56 as a function of O / H for different AYlg, and interpolate
from Woosley et al. (2007, Fig. 22) to ﬁnd Am15(B). Figure 3.6 displays this result.
We plot here a maximal case (dashed line) with dYe /dY12 = 0.30 in the initial part
of the simmer, appropriate for the one-zone calculation with electron captures onto
23Na (Fig. 3.4); for this case A}; = —7.5 x 10’4. For comparison, we also plot a
case (solid line) with AYe = 0 during simmering. This gives a sense of how large the
variation in Ye might be. The vertical bar indicates the value of 12 + log10(O/ H) at
which the change in Am15(B) from the 22Ne abundance equals that from the elec-
tron captures during simmering for this maximal case. To the left of this curve the
linear correlation between 22Ne abundance and Am15(B) will be masked by varia—
tions in the simmering of the white dwarf. For comparison, we also Show the data
from the compilation of Gallagher et al. (2005) and Hamuy et al. (2000) and plot the
relation between 12 + log(O/ H) and A7n15(B) used by Gallagher et al. (2005, dotted
line). This trend is much steeper than our ﬁnding. The difference is due to how the
56Ni mass was varied; whereas the models we use (Woosley et al., 2007) hold the
kinetic energy and total mass of iron-peak ejecta ﬁxed, Gallagher et al. (2005) based
their peak brightness on delayed detonation models (Hoﬂich et al., 2002) for which
a variation in 56N i also produced a change in the relative amounts of iron—peak and

intermediate n'iass-elements, as well as a different explosion kinetic energy.

It is evident from Fig. 3.6 that the scatter in the data points is larger than the

expected trend due to progenitor composition, esper‘rially at sub-solar metallicities.

37

Both AYl-Z and dYe /dY12 depend on the central density, which is not obviously cor-
related with metallicity, and hence the correlation between peak brightness and O/ H
will be masked by differences in the pre—explosion simmering. Indeed, if the variation
in AYe were as large as the two cases we plot in Fig. 3.6, then variations in 56N i would
be determined more by AY12 than by stellar composition for galaxies with sub—solar
O/H. There is a general trend that SNe Ia are systematically brighter in galaxies
with active star-formation (Hamuy et al., 2000; Gallagher et al., 2005; Sullivan et al.,
2006; Howell et al., 2007). Sullivan et al. (2006) showed that the SNe Ia rate increases
linearly with the speciﬁc star formation rate, and that SNe Ia associated with actively
star forming galaxies were intrinsically brighter than those associated with passive
galaxies. Although many of these passive galaxies are more massive, and hence more
metal-rich (Tremonti et al., 2004), the observed scatter in SNe Ia peak brightnesses
remains much larger than the expected trend with metallicity (Piro & Bildsten, 2008;
Howell et al., 2009; Gallagher et al., 2008). This suggests that the correlation with
the chemical abundances of the host galaxy is a secondary effect in setting the peak

brightness of SNe Ia.

38

Chapter 4

The effect of 22Ne on ﬂame speed

4.1 Introduction

The possibility that type Ia supernovae might evolve with the abundance of 0-
elements in the host population, combined with questions about whether this intro-
duces systematic variations in the Phillips relation, motivates further investigation
of how the progenitor composition influences the explosion. As a ﬁrst step, we in-
vestigate in this chapter how the abundance of 22Ne. affects the laminar ﬂame speed
Slam and width 618m of a 12C-wO-22Ne mixture. Our principal conclusion is that
Slam increases roughly linearly with the 22Ne mass fraction X22. At X22 2 0.06,
the speedup varies for carbon mass fraction X12 2 0.5, from 2 30% at densities
2, 5.0 x 108 gem"3 to z 60% at lower densities.

These calculations are relevant for two regimes: 1) the initial burn near the center
of the white dwarf where the gravitational acceleration is small and the laminar
ﬂame speed dominates the evolution of a bubble of ignited material (see, e.g., Zingale.
& Dursi, 2007), and 2) the burning at densities ~ 107 gcm‘3 where the Gibson
length scale becomes 6G ~ 61am- The Gibson scale KG is deﬁned by v(€G) = Slain

where 22(8) is the eddy velocity for a lengthscale 6 (see Hillebrandt & N iemeyer,

39

2000, for a succinct review). The region where 61am = 6G is conjectured to be a
possible location for a deﬂagration-to—detonation transition (N iemeyer & Woosley,
1997). Our calculation does not apply in the ﬂamelet regime, where the buoyancy of
the hot ashes generates turbulence via the Rayleigh-Taylor instability. In this regime,
the effective front speed becomes independent of the laminar ﬂame speed (Khokhlov,
1995; Reinecke et al., 1999; Zhang et al., 2006), and the composition affects the front
speed only through the Atwood number, At :—: (pfud — p,,_.,h)/(pfue1 + Pash)» where

pfueMash) is the density in the unburned (burned) material.

In § 4.2 we describe our computational method and benchmark our calculations
against earlier results of Timmes & Woosley (1992). Section 4.3 presents the com-
puted flame speeds as functions of pfuel, X12, and X 22. We provide a ﬁtted expression
for Slam as a function of these parameters. We also give a physical explanation for
the speedup before concluding, in § 4.4, with a discussion of how the transition to
distributed burning would occur at a lower density if the 22Ne abundance were in-

creased.

4.2 The laminar ﬂame

To solve. for the conductive ﬂame speed Slam: we used the assumption of isobaric
conditions to cast the equation for the energy as two coupled equations for the tem-

perature and flux (Timmes & Woosley, 1992; Bildsten, 1995),

dT F

__ —_— __ 4.1
dz K ( )
(IF pCp

If; : p5 " SlaInTF- (4-2)

40

Here F is the heat flux and Cp is the specific heat.1 The heating rate 5 is given
by equation (2.11) and d/dt = Slam(d/d:1:). Our reaction network incorporated 430
nuclides from n to 76Ge and is the same network used in § 3.3. We use the reaction
rates from the library REACLIB (Rauscher & Thielemann, 2000; Sakharuk et al., 2006,
and references therein). On the timescale of the ﬂame passage, electron captures are
unimportant, and Ye is essentially ﬁxed; we found that Slam was unchanged when
weak reactions were removed from the network, so we used only strong rates for
computational efﬁciency. We incorporated screening using the formalism of Graboske
et al. (1973). Across the ﬂame front, the matter does not reach nuclear statistical
equilibrium or quasi-nuclear statistical equilibrium until most of the 12C is depleted.
Thus, although our treatment of screening does not preserve detailed balance (Calder
et al., 2007) this does not affect our calculation of the ﬂame speed. Our equation
of state has contributions from electrons, radiation, and strongly coupled ions. We
include thermal transport by both degenerate electrons and photons (for a complete
description of our thermal routines, see Brown et al. 2002, and references therein).
Equations (4.1)—(4.2), when combined with appropriate boundary conditions, have
Slam as an eigenvalue. Ahead of the ﬂame the material is at an arbitrary cold tem-
perature Tfuel = 108K; in this region we set dT/da: to a small positive value and
integrate equations (4.1) (4.2). For simplicity, we split the solution of the'thermal
and network equations; that is, for each step dcr we solve the thermal equations, to
obtain T and p, integrate the reaction network at that T and p to compute Y,- and
5, and use that to advance the solution of the thermal equations. For our choice of
slam: the second boundary condition is that F ——> 0 asymptotically behind the front.
and that F is peaked where 5 is maximum. We iterated until Slam had converged to

within 0.01%. We ﬁnd that Slam is insensitive to T fuel for pfuel 2, 5 x 108 gem—3.

 

1We neglect here terms such as aE/BXi, which account for the change in the
thermal properties as the abundance of nuclide 1' changes. These terms are much
smaller than 5 for matter not in NSE.

41

At lower densities this is no longer true, but the relative increase in Slam with X22

remains robust.

 

10000 ﬁftv—r1 r ft11r1l] I I it?
.

 

 

. , A
'T/\ 100.0 ’. _ :o‘"~‘.
(I) .v‘SI-I".
E ’.'o"$,t'
v .45.? .......
FD 1'",(--"°
8 10.0 .’.f."’
0.. °’- "
m “I..- -
a) I". :
E .. {I’ +....+....+ TW92 _
CU .
c 1.0 ... - 4. _ + 130 nuclide _E,
+._+.-+ 430 nuclide :
Fit ‘
0.1 J I I I I I I I I I I L I I I I I I I I
108 109

density (g cm‘3)

Figure 4.1: Flame speeds computed with an 130-nuclide network (dashed line) and a
430—nuclide network (dash-dotted line). We compare these with the results of Timmes
85 Woosley (1992, dotted line), and our fit (eq. [4.3]; solid line). Our 130-nuclide
network uses the same nuclides as Timmes &: Woosley (1992).

Figure 4.1 shows a comparison between our ﬂame speeds for a 1:1 C:O mixture
and those of T immes & Woosley (1992). Here we adopt the same 130-nuclides as
used in that paper. Although in this case we are using the same nuclides and starting
composition, the rates, equation of state, and thermal conductivities are not identical.
Overall, our ﬂame speeds differ by no more than 25%; the largest. discrepancy is at
pfuel = 108 g cm’3. Most of this discrepancy is due to different opacities used by the
two codes. For pfuel S, 7 x 108 gem—3, photons become more efﬁcient than electrons

at transporting heat within the ﬂame front, with the dominant opacity being free-

42

free. Timmes & Woosley (1992) included a fit to electron-ion scattering in the semi—
degenerate regime (Iben, 1975). At these densities where T > 2 x 109 K and the
free-free opacity dominates, the contribution from electron-ion scattering decreases
the total opacity. We compared our opacities along a (p, T) trace generated for a
run at pfue] = 108 gcm‘3. We found that non-degenerate electron-ion scattering can
lower the opacity by z 24%, depending on how the opacity is interpolated between
the two limiting ﬁts. In addition, our free-free opacities differ by 30% at the location
along the (p, T) trace where IF I is maximum.

Finally, we also investigated the effect of reaction network size: increasing the
network from 130 to 430 nuclides resulted in a 25% increase in Slam at pfuel :
2.0 X 109 gcm‘3; further increases in the size of the reaction network did not yield

any appreciable increases in Slam;

4.3 Results

We now present the results of our ﬂame calculations for different initial mixtures
of 12C, 16O, and 22Ne and different. ambient densities. Table 4.1 lists Slam and
the ﬂame width deﬁned by 613”, = (T851, — Tﬁlcl)/ max ldT/dx|, with T fuel(ash) being
the temperature in the unburned (burned) matter. We tabulate these quantities
for p9 E mud/109 gcm’3 ranging from 0.05 to 6.0, and X12 = 0.3—0.7, with the
remaining composition being 1GO and 22Ne. For each choice of p9 and X12, we use 3
different 22Ne abundances, X22 = 0.0, 0.02, and 0.06. Over most of the range in p9,
X12, and X22 in Table 4.1, we ﬁnd that an increase in X22 from 0 to 0.06 causes Slam
to increase by approximately 30%. We confirmed several of the table entries using an
independent diffusion equation solver that uses adaptive grids (Timmes & Woosley,
1992) and a different reaction network and opacity routine. From this, we estimate

that the uncertainty in the ﬂame speeds listed in Table 2 are m 30%, with about 10%

43

 

X 12 X16 X 22 10 Slam 61am

 

 

(109 g ("m—3) (km s‘l) (cm)
0.30 0.70 0.00 0.1 1.20 1.64x10—2
0.30 0.64 0.06 0.1 1.23 1.52x10_2
0.30 0.70 0.00 0.5 14.9 5.63x10—4
0.30 0.64 0.06 0.5 18.4 4.48x10-4
0.30 0.70 0.00 2.0 49.4 7.64x10—5
0.30 0.64 0.06 2.0 66.9 5.60x10’5
0.30 0.70 0.00 6.0 124 1.88x10-5
0.30 0.64 0.06 6.0 163 1.39x10-5
0.70 0.30 0.00 0.1 6.22 3.07x10-3
0.70 0.24 0.06 0.1 9.59 1.80x10-3
0.70 0.30 0.00 0.5 49.4 1.81x10-4
0.70 0.24 0.06 0.5 67.4 1.32x10-4
0.70 0.30 0.00 2.0 131 3.00x10—5
0.70 0.24 0.06 2.0 171 2.27x10-5
0.70 0.30 0.00 6.0 304 7.93x10-6
0.70 0.24 0.06 6.0 379 6.13x10—6

 

Table 4.1: Laminar flame speed and width

coming from numerical uncertainty and about 20% from different physics treatments
as described above. We emphasize, however, that both codes ﬁnd the same trends,
e.g., an increase in Slam with X22.

We ﬁt the tabulated Slam with the. approximate expression

' .4 X
= 1 mi . [mt—02.2)]
X X 3
x 0.3883 —1—2 +0.09773 ——12 kins—1 (43)
0.5 0.5

which has ﬁt errors, as compared against speeds calculated using the 430-nuclide
network with X22 2 0, that average 33%, with a maximum of 70%, for X12 2 0.5
and p9 = 0.07, and with a minimum of 0.1%, for X12 2 0.5 and p9 = 6. For accurate
work, interpolation from Table 4.1 is preferred. At p S 108 gem—3, the speedup is

negligible for X12 = 0.3 but increases to z 50% for X12 2 0.5.

44

 

 

0.0101..,.......,...,,

11
.0
8
A
I
0

0.008
__X = 0.06

0.006

.......
-----
o
o

:—

I I I I I I I I I I I l I I I I I I I l

0 .004

 

0 .002

 

I I I I I I I l I I I I I I I I I I I I

 

 

 

I ............................... ..
0.000 .1 . I . 1. . . N6

0.0 0.2 0.4 0.6 0.8 1.0
1-[Y12/max(Y12)]

 

Figure 4.2: Abundances of selected nuclides during a burn at p = 2.0 x 109 gem—3

and with an initial 12C mass fraction of 0.3. We Show runs with an initial 22Ne
abundance of 0.06 (solid lines) and 0.0 (dashed lines).

To understand how the addition of 22Ne increases Slam, we plot in Fig. 4.2 some
selected abundances Y = X /A for a ﬂame with an initial X12 = 0.3 and pfuel =
2.0 x 109 gem-3. We use the fraction of 12C consumed, 1 — Ylg/ max(Y12), as our
coordinate and plot the region where this value is monotonic with distance. In a C / O
deﬂagration, the ﬂame speed and width are set by the initial burning of 12C. The
buildup of Si-group nuclides and then establishment of nuclear statistical equilibrium
occur on longer timescales, so that the peak of the heat ﬂux is reached as 12C is
depleted via the reactions 12C(12C,a)20Ne and 12C(12C,p)23Na(p,a)20Ne. For the
case with X22 = 0.06 (solid lines) one sees that 22Ne is depleted before the 120 is
even half-consumed. The 22Ne lifetime becomes less than the 12C lifetime once the
a abundance is Y4 2, 104. At temperatures in the ﬂame front the uncertainty in
the 22Ne(o:, n)25Mg rate is estimated to be about 10% (see Karakas et al., 2006, and
references therein). This is unlike the case in AGB stars, for which the uncertainty
at T < 3 x 108 K is approximately a factor of 10. Note that signiﬁcant burning does
not occur until the 12C lifetime becomes of order the time for the ﬂame front to move
one ﬂame width. This requires temperatures in excess of 2 X 109 K for the densities
of interest, and so 22Ne is preferentially destroyed in a ﬂame via (61,71) rather than

by p—capture (cf. Podsiadlowski et al., 2006).

The neutrons made available from the destruction of 22Ne capture preferentially

3, successive (n, (1)

onto 20Ne formed during 12C burning. At pfuel Z 5 x 108 gcm—
reactions build up 170 and 14C, the latter of which then undergoes 14C(p,n)14N
(n,a)11B(p, 2a)4He. At densities of pfuel S 5 x 108 gcm’3 and carbon abundances
X12 = 0.5, the ﬂow 2ONe(n, 'y) 21Ne(n,()z)180(p,(1)15N(p,(1)12C also contributes. The
net effect of having 22Ne in the fuel mixture is that during 12C burning, the abundance
of protons is depressed and the abundance of 4He elevated, as illustrated in Fig. 4.2.

The fact that these ﬂows require two neutron captures onto the products of 12(3+12C

suggests that the increase in 5 should scale roughly as X32. Since Slam o< 51/2, this

46

implies that the increase in ﬂame speed will be. linear in X22, which agrees with the
numerical solution of equation (4.1). Because these ﬂows are initiated by n-capture
onto 20Ne, we tested our sensitivity to the reaction rate by recomputing the case.
X12 2 0.5 and pfue] = 7.0 x 107 gcm‘3. A decrease in the 20Ne + n rate by a factor
of 10 produced a decrease in the ﬂame speedup, from 70% to 20%. At higher densities
there was no difference in the speedup. The only case for which there was no increase
in Slam was for Pfuel S 108 gcin—3 and X12 2 0.3 (see Table 4.1). For this case, the
slower consumption of 22Ne relative to 12C and (in, y) captures 011 25Mg suppress the

generation of a-particles early in the burn.

4.4 Discussion

We have computed the laminar ﬂame speed in an initially degenerate plasma consist-
ing of 12C, 160, and 22Ne. We ﬁnd that, over a wide range of initial densities and 12C
abundances, the ﬂame speed increases roughly linearly with 22Ne abundance, with
the increase being 5:: 30% for X22 2 0.06, although there are deviations from this
rule at lower densities. These studies are relevant to the initial burning at the near-
center of the white dwarf, and at late times where the ﬂame may make a transition
to distributed burning. To see how the increase in laminar ﬂame speed changes the
density where the burning becomes distributed, we write Slam z p" (1 + 6X22) and
find from our table that at pflwl = 7.0 x 107 gcm’3 (the lowest density for which
our numerical scheme converged) and X12 2 0.5, 7) z 1.6 and 6 z 0.7/0.06. Re-
cent numerical studies (Zingale et al., 2005) ﬁnd that the Rayleigh-Taylm instability

drives turbulence that obeys Kolmogorov statistics, so that [G (x 8'3 N umeri-

lam '

—1.5

hm , so solving for

cally, we ﬁnd that the ﬂame width scales roughly as (51am oc 5'
61am MG 2 1 implies that increasing X22 from 0 to 0.06 would lower the transition

density by z 30%. A reduction in the density of this transition will lead to a reduc-

47

tion in the mass of 56Ni synthesized (Hoﬂich et al., 1995). We conjecture that if a
deﬂagration-to—detonation occurs, the addition of 22Ne decreases the overall mass of
Ni—peak elements, in addition to lowering the isotopic fraction of 56Ni.

Our results can be improved in several ways. First, the 22Ne may be partially
consumed as 12C burning gradually heats the core of the white dwarf (Podsiadlowski
et al., 2006) some a: 103 yr prior to ﬂame ignition. This may further reduce the
electron fraction of the white dwarf, but will also change the reaction ﬂows in the ﬂame
front. At low densities the morphology of the ﬂame becomes more complicated, as the
ﬂows responsible for reaching quasi-statistical equilibrium are no longer fast enough
to keep up with the carbon burning. Indeed, at p S 108 gcm73, the eigenfunction for
the ﬂux begins to show two maxima and the ﬂame speed becomm more dependent
on the ambient fuel temperature. Further studies with more realistic compositions
and at lower ambient densities are ongoing and will be reported in a forthcoming

publication.

48

Chapter 5

Compositional effects of surface

detonation

5. 1 Introduction

In this chapter we explore the nuclear-synthetic yields that result from a surface det-
onation on a carbon-oxygen white dwarf. In § 5.2 we describe the type of detonation
we are modeling and our computational method. Section 5.3 presents the computed
gradient. We provide a list of nuclides that Show a clear gradient. We also Show that
there is also a gradient in velocity space for elen’iental Ni before concluding, in § 5.4,

with a discussion of how our results can be improved.

5.2 The Explosion Model

A promising candidate for the SNe Ia explosion mechanism is the gravitationally
conﬁned detonation (GCD) model (Plewa et al., 2004). In this model, the carbon
burning runaway within the convective core is postulated to occur in a small region

displaced from the stellar center, which results in a highly buoyant ﬂame bubble that

49

quickly rises to the stellar surface after burning only a few percent of the star during its
ascent (e.g. Livne et al., 2005). When the buoyant ash, as well as unburned material
pushed ahead of the rising ﬂame bubble, erupts forth from the stellar core it is largely
conﬁned to the surface of the white dwarf by gravity, and becomes a strong surface
ﬂow which sweeps completely over the star, eventually converging at the antipode
of the breakout location. The high temperatures and densities reached within the
converging surface ﬂow are sufficient to trigger a detonation which incinerates the

mostly unburned C / 0 white dwarf.

The resulting nuclear-synthetic yield consists almost entirely of detonation burning
products, and depends on how much the star has expanded by the time the detonation
initiates. More highly expanded (hence lower density) cores at detonation result in
a smaller fraction of Fe peak nuclei, less 56Ni, and consequently a larger fraction of
intermediate mass elements (IMEs) due to incomplete relaxation to N SE. Therefore,
lower luminosity (less 56N i producing) explosions are accompanied by a larger yield

of IMEs.

The expansion of the star prior to detonation is a result of the work done on the
star by the rising ﬂame bubble. Deﬂagrations which burn more mass prior to reaching
the stellar surface excite higher amplitude pulsations and hence more expanded stars
at the time of detonation. It has been found that the expansion of the star due to the
deﬂagration is very well represented by the fundamental radial pulsation mode of the
underlying white dwarf (see Fig. 2 and Fig. 12 in Meakin et al., 2009). Therefore,
while it is essential to understand the nature of the deﬂagration so as to better
understand the mapping between initial conditions and final outcomes, the range of
outcomes due to the deﬂagration can be parameterized by the pulsation amplitude,
resulting in a one parameter family of models. While the phase of the pulsation at
the time of detonation is a second parameter, it plays a. lesser role because the time

for the detonation wave to completely incinerate the star is short compared to the

50

 

pulsation period.

In the explosion model discussed in this chapter, we initiate a detonation in the
surface layer of a cold (T = 4 x 107 K) white dwarf of mass 1.365 MG which has been
expanded according to its fundamental radial-pulsation mode by such an amplitude
that it has a central density of 2 x 108 gcm‘3. This expansion results in ~ 0.711173,
of high density material (p > 1 x 108 gcm'3), which burns to N SE in the detonation
(primarily as 56N i). The remaining 0.6 Mg of material burns to IMEs, e.g. Si, S,
Ca, and Ar, resulting in only a small amount of unburned C/O in the outermost
layers. The detonation is initiated by heating a small spherical volume (~ 4 km in
radius) within the surface layer of the expanded white dwarf where the density is
107 gem—3. The reactive—hydrodynamic simulation of the explosion was conducted
using the FLASH code (Fryxell et al., 2000). The code framework and the included
physics is identical to that described in Meakin et al. (2009), and a maximum resolu-
tion of 4 km is used. An in depth description of the detonation phase, which results
in a homologously expanding remnant, can be found in Meakin et al. (2009). Detailed
yields are calculated by post processing Lagrangian tracer particles included in the
explosion calculation and are the primary focus of this chapter.

Our reaction network incorporates 493 nuclides from n to 86Kr (Table 5.1). We
use the reaction rates from the Joint Institute for Nuclear Astrophysics REACLIB
Database1 (Rauscher & Thielemann, 2000; Sakharuk et al., 2006, and references
therein); the light-element rates are mostly experimental and are from compilations
such as Caughlan & Fowler (1988) and Iliadis et al. (2001). Weak reaction rates are
taken from Fuller et al. (1982) and Langanke & Martinez-Pinedo (2001). Screening
is incorporated using the formalism of Graboske et a1. (1973). Although our treat-
ment of screening does not preserve detailed balance (Calder et al., 2007) this does

not affect our calculation since we are concerned only with nuclides that never reach

 

1http://groups.nscl.msu.edu/jina/reaclib/db/

51

 

El. A El. A El. A El. A

 

 

1— 3 Ne 17—28 K 35—46 Ni 50- 73
He 3- 4 Na 20—31 Ca 35—53 Cu 54—70
68 Mg 20—33 Sc 40—53 Zn 55- 72

7, 9 ‘11 Al 22—35 Ti 39—55 Ga 58—73
8, 10-14 Si 22—38 V 4357 Ge 59— 76
9 16 P 26—40 Cr 43—60 As 62 —76

12 20 S 27—42 Mn 46 --63 Se 62 82
13—20 Cl 31—44 Fe 46- 66 Br 71— 81
15 —24 Ar 31—47 Co 50—67 Kr 71 86

mozowg

 

Table 5.1: 493-Nuclide Reaction Network

nuclear statistical equilibrium.

5.3 Results

We now present the results of our reaction network calculations for a surface det—
onation in a SN Ia model. We ﬁnd that a number of nuclides exhibit a gradient
throughout the stellar remnant. We determined this by plotting the ﬁnal abundances
of the various nuclides verses the ﬁnal abundance of 28Si. 288i is one of the most
abundant species in the regime that does not reach nuclear statistical equilibrium.
By plotting the ﬁnal abundances of the nuclides versus the ﬁnal abundance of 28Si
we can separate out the particles we are interested in. We are interested in material
that never reaches equilibrium and therefore is most affected by its thermal history.
We then separate the particles into groups by the ejection angle. For these groups we
chose particles ejected between +90° and +600, between +15° and between —15°,
and between -60° and —90°. In this system the detonation was prescribed to happen
on the surface of the star in the +900 direction. Any nuclide that showed greater
then a 10% separation in the abundance between the different ejection angles was

considered as having a gradient. Table 5.2 shows a partial listing of the nuclides that

52

 

separation no separation

 

 

o7Ni SON
“Moo 2831
55 N1 328
"8N1 52 F0
()0le 48C1
34Ar 24110
308 39K
29 1')
531:: 27?1
"8Cu
59Cu

 

Table 5.2: Partial table of nuclides that do and do not display a separation because
of asymmetric detonation.

show a separation.

()ur simulation is not run sufficiently past freeze out to allow all beta unstable
nuclei to decay. If we force the unstable nuclei to decay, however, and group nuclei
in elemental abundances, we find Ni to have the greatest gradient over the star. In
material not burned to equilibrium we ﬁnd the abundance of elemental Ni to increase
by an order of magnitude between +900 and —90°. Figure 5.1 shows this dependence

elemental Ni has on central angle.

We ﬁnd that the con‘iposition gradient. arises because of the extreme off center
nature of the detonation. At ﬁrst the shock wave from the detonation works with
gravity compressing the fuel and leading to a longer thermal expansion time. We
deﬁne a thermal expansion time texp as the time it takes a particle trajectory to go
from it’s maximum temperature Tmax to a. temperature of Tmaxe’l. When the shock
wave reaches the opposite side of the star the shock is now acting against the force of
gravity helping to eject the material and leading to a shorter thermal expansion time.
This difference in the thermal expansion time scales means material in the ejecta

has different thermal histories as function of position (see Fig. 5.2.). These different

53

 

 

theta

Figure 5.1: Final abundance of elemental Ni for each tracer, represented by a dot, as a
function of the ejection angle relative to the center of the star for the tracer particles.
The detonation occurred on the surface of the star in the theta = 90 direction. The
particles with a Ni abundance above 10“4 are particles that have burned to NSE.

thermal histories lead to the compositional gradient in the outer layers of the ejecta

since the material never reaches NSE.

 

.0

U»)

G

U)
lllllll'lllllllllllllll

 

 

p—d
O
p—A
LII

X (108 cm)

Figure 5.2: Initial spatial position of the tracer particles. Color represents each
particles thermal expansion time scale. The center of the star is at Y:0, X=0 The
detonation was initiated at Y=5, X=0. Particles on the side of the star where the
detonation starts have higher thermal expansion time scales than the particles on the
opposite side of the star.

An interesting side effect of the different expansion times is that material on
opposite sides of the star will be expanding at different velocities. This leads to
another effect where there is a gradient in velocity space. Figure 5.1 shows how
elemental Ni varies with radial velocity. In material not burned to equilibrium the
part of the remnant with the most Ni will also be the part with the highest radial
velocity. This is self consistent since the Ni abundance is greater on the side of
the remnant with the shortest expansion time therefore it follows it should have the

highest radial velocity.

 

10‘2

 

 

 

 

103:— 2
A 104 E— 2}: I '. E
>_‘ -— It, _.
10‘s:— " ‘2
1045:r 1.
: 1
10_7 - I I I I l I I I I I I I I I'rll'.l. I‘ I I I I I I I I I I I I _

0 1-10 2-10 3-109

Vr(cm s")

Figure 5.3: Final abundance of elemental Ni as a function of the ﬁnal radial velocity.
The particles with a Ni abundance above 10‘4 are particles that have burned to NSE.
Particles with a Ni abundamre less then 10‘5 show greater Ni abundance for larger
radial velocities.

56

5.4 Discussion

We have computed the abundances and positions of nuclides created in a surface
detonation in a SN Ia. We ﬁnd a compositional gradient produced by the detonation
in the ejecta. This compositional gradient is connected with the thermal expansion
time scale since the burning never reaches nuclear statistical equilibrium and therefor
the reactions that take place are dependent on the thermal history of the material.
The different expansion timescales also result in there being a compositional gradient
in velocity space.

Our results can be improved in several ways. First, light curves and spectra need
to be generated to test the observational signature in more detail. It is currently
unclear how much of an observation effect this compositional gradient has. A series
of synthetic spectra generated over a range of time allows for direct comparison with
observed supernovae. Even if any compositional effects are obscured, we conjecture,
the spectra will see some dependance on observing angle since the side of the remnant
that. expands at higher velocities will also be at a lower density making it more
transparent. The surface ﬂow, which consists partly of deﬂagration ash, which was
excluded from our the present model, needs to be considered. The surface ﬂow might
also have a spectral signature itself such as Fe peak material at high velocity and

should be compared with the underlying compositional gradient.

Chapter 6

Conclusion

The principal use of this work will be in developing better large scale simulations of
SNe Ia for comparison with observation. Simulating an entire Type Ia event from
accretion to detonation using nothing more than ﬁrst principals is currently too time
intensive to be done on modern computers and it will be many years before this can
even be considered. In the past SNe Ia models were often constructed with tunable
parameters that could be adjusted until something resembling reality was produced.
Whether or not these parameters corresponded to anything that physically happens
is still an open question. More recent work has focused on simulating the physics
that occurs on scales smaller then can be resolved by current SNe Ia models, so called
sub—grid models. Sub-grid models tabulate much of the physics taking place and
can be incorporated into the SNe Ia simulations in order for them have an improved

theoretical robustness.

The simmering work in chapter 3 can be used to improve models of the SNe Ia
prior to deﬂagration. Currently, doing nucleosynthesis calculations in a simulation
of a simmering white dwarf is too computationally expensive. Given table 3.2 any
individual wishing to simulate a simmering white dwarf needs only to keep track

of how much carbon is burned in the simulation. Carbon burning requires only

58

one differential equation to evaluate and is much less computationally intensive then
evaluating an entire reaction network. This will aide in research being done on the
convective Urca process and on the start of deﬂagration. Energy loss from neutrinos
in the convective Urca process may introduce a cooling term that is comparable to
the heating term from the consumption of 12C. The amount of cooling that happens,
and hence the amount of 12C burned, from convective Urca has varied considerably
from publication to publication making a universally accepted prescription difficult
(Stein et al., 1999; Lesaffre et al., 2005). Convective Urca may also effect. the size and
turnover time scale of the convective zone changing how much neutron rich material
is made, and hence Y}; varies throughout the star. Future research will have to model
how neutrino losses change the overall heating rate of the star as well as affect the
convective zone. Knowing the exact composition at the end of simmering is ultimately
necessary for exact modeling of the nucleosynthesis that happens afterwords, and

therefore the ﬁnal abundance of nuclides.

The work on ﬂame speed in chapter 4 has been used to better understand the
deﬂagration phase. SNe Ia simulations require a sub-grid model of the ﬂame as an
input parameter. This work is the ﬁrst to provide a such a parameterization of a flame
with 22Ne present. This parameterization can be incorporated into SNe Ia simulations
to test compositional effects on SNe Ia events. In fact, this parameterization has
already been included into code to test that very thing (Townsley et al., 2009). Studies
recently done by Townsley et al. (2009) have looked at how 22Ne effects the dynamics
of the expansion that happens before detonation. Their ﬁndings suggest that the
presence of 22Ne does not effect the amount of 56Ni produced beyond that already
predicted by Timmes et al. (2003) However all of the models in Townsley et al. (2009)
assumed a transition to detonation at the exact same density. Depending on how a
transition to detonation is deﬁned in a model the transition may be dependent on the

properties of the ﬂame. At lower densities these properties become harder to deﬁne.

59

At densities above the expected detonation transition density the carbon and oxygen
burning timescales signiﬁcantly separate creating two ﬂame like regions, one where
carbon burns and another where everything else burns. At this point our original
concepts of ﬂame speed and width need to be reevaluated. Future work will have to
focus on better understanding the ﬂame at these low densities and as it transforms to
a detonation. Ultimately, knowing exactly how the progenitor effects 56Ni production
and hence the light curve will allow SNe Ia to be better used as standard candles.
Finally, the tracer particle calculations in chapter 5 can be used to determine the
ﬁnal outcome of a simulation in far better detail then before. SNe Ia simulations do
not keep track of all nuclides. Knowing the distribution of all the nuclides relative
to the ones kept track of in simulations will allow for publication of quick look-up
tables for the composition of ejecta. Having an accurate composition of the eject will
also all for better simulated light. curves and spectra to be plotted allowing for better
comparisons between simulation and observation. This work can be extended to test
the observational properties of different explosion models in more detail. Light curves
and spectra need to be generated from the data to better compare with observation.
The effects that surface ﬂow has on the spectra and light curve also needs to be
considered. If these steps are taken it may be possible in the future to determine the.
exact mechanism by which SNe Ia detonate, removing one of the unknowns behind

these bright explosions in the sky.

60

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