Helium accreting white dwarfs
An accretion mass study
Masterarbeit in Astrophysik
von
Michael Fuchs
angefertigt im
Argelander Institut für Astronomie
vorgelegt der
Mathematisch-Naturwissenschaftlichen Fakultät
der
Rheinischen Friedrich-Wilhelms-Universität Bonn
November 30, 2015
1. Gutachter:
Dr. Richard J. Stancliffe
2. Gutachter:
Prof. Dr. Norbert Langer
III
IV
Although supernovae Ia are important observed events—for instance because of
their role as distance indicators in cosmology or as producers of intermediate and
heavy elements—their progenitors are still not fully understood. Helium accreting
white dwarfs are one of these possible progenitors of supernovae Ia. The accretion
mass—the accreted mass needed for ignition of the helium layer—is an important
parameter for the determination of whether or not helium accreting white dwarfs
will end up as a supernova Ia.
In this study we performed one-dimensional simulations with the stellar evolution code STARS, in which helium is accreted onto carbon/oxygen white dwarfs with
a constant accretion rate. We investigated the behaviour of the accretion mass in dependency on the accretion rate (in a range between 10−9 M⊙ yr−1 and 10−5 M⊙ yr−1 ),
the initial mass of the white dwarf (in a range between 0.65 M⊙ and 1.2 M⊙ ), the
composition of the accreted material and the initial temperature of the white dwarf.
Two initial temperatures were chosen: one that is above and one that is below the
ignition temperature of helium.
We will show that the accretion mass decreases with an increasing accretion rate
and/or a decreasing initial mass of the white dwarf. For the models with a large
initial temperature, the accretion mass drops by roughly one order of magnitude
for accretion rates between 3 × 10−8 M⊙ yr−1 and 9 × 10−8 M⊙ yr−1 . While a lower
initial temperature increases the accretion mass for accretion rates larger than 3 ×
10−8 M⊙ yr−1 , a replacement of the composition with heavier elements than helium
has no effect on the accretion mass.
V
VI
Contents
1. Introduction
1.1. The life and death of stars
1.2. The physics of a SN Ia . .
1.3. Mass transfer . . . . . . .
1.4. Progenitor models . . . . .
1.5. Aim of this work . . . . .
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2. Method
2.1. The STARS stellar evolution code . .
2.2. Modifications to the STARS code . . .
2.3. Input models . . . . . . . . . . . . . .
2.4. Assumptions for the accretion phase
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3. Accretion mass
3.1. Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Analytical Description . . . . . . . . . . . . . . . . . .
3.3. Internal structure of a helium accreting 0.81 M⊙ WD .
3.4. Degeneracy at the time of ignition . . . . . . . . . . .
3.5. Temperature Dependency Of The Accretion Mass . .
3.6. Different compositions of the accreted material . . . .
3.7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Summary
53
A. Tables
57
Bibliography
63
VII
VIII
1. Introduction
Although thermonuclear supernovae—or better known as type-Ia supernovae (SN
Ia)—play an important role as distance indicators for cosmology (Perlmutter et al.
1999) and the enrichment of the interstellar medium with heavy elements (Hillebrandt et al. 2013), their progenitor models are still poorly understood. It is widely
accepted that the progenitor of a SN Ia is a white dwarf (WD) that accretes material
from a companion. Thus, the WD can grow in mass, increase its temperature and
eventually explode as a SN Ia. One possible scenario for such a progenitor model
is a WD that accretes helium from an appropriate companion star. In this thesis the
mass that has to be accreted to ignite the accumulated helium is explored in dependence on the accretion rate, the initial mass of the WD and the inital temperature of
the WD.
In this chapter a brief overview over the evolution of single and binary stars, and a
summary of the physics of supernovae Ia are presented. Afterwards, the importance
of mass transfer for the progenitor systems of possible supernovae Ia explosions are
described. At the end the major progenitor models that can lead to a supernova Ia
are summarized.
1.1. The life and death of stars1
If a main sequence (MS) star evolves without a companion, its fate is determined by
its initial mass. More massive stars exert more pressure on their cores, producing
larger temperatures and densities. Therefore the fusion rate is increased and hence
more massive stars are brighter, evolve faster and ultimately have a shorter lifetime
than lower mass stars. Stars with masses smaller than 0.5 M⊙ never reach core
temperatures that are sufficiently high to ignite helium. Stars with masses between
1 If
not quoted otherwise, this section refers to Kippenhahn et al. (2012).
1
1. Introduction
0.5 M⊙ and roughly 8 M⊙ will produce elements up to carbon and oxygen through
burning, while stars with even larger masses will be able to produce an iron core at
the end of their lifetime.
Because the internal stellar temperature decreases from the centre to the surface,
the various burning stages are ignited in inner layers earlier than in outer ones.
Thus, hydrogen burning will start occurring in the stellar core at a phase when the
outer stellar regions are not yet sufficiently hot. After the exhaustion of hydrogen in
the core, stars undergo a phase of shell hydrogen burning, during which the layers
above the burning shell will expand to huge dimensions. Stars in this phase are
known as red giants and at the beginning of the expansion they are classified as
subgiants (Fig. 1.1).
Figure 1.1. – Sketch of the evolution of a star with a main-sequence mass of 1 M⊙ . The different
stages of stellar evolution are labelled and the grey shaded regions indicate the main
sequence, the horizontal branch and the location of the white dwarfs.
2
1.1. The life and death of stars
The cores of stars with a solar metallicity and with masses lower than 2.3 M⊙ 2 be-
come degenerate before igniting helium, which leads to an unstable burning event—
the core helium flash. During the flash the temperature in the burning region increases, which will eliminate the degeneracy. When the burning region behaves
like an ideal gas, the core of the star can expand and cool, and finally burn helium
in a stable manner to carbon and oxygen. As long as the star burns helium in the
core, the star is located on the horizontal branch (HB) in the Hertzsprung-Russell
diagram (HRD). For initial masses larger than 2.3 M⊙ helium will fuse to carbon
and oxygen in a stable reaction without a prior core helium flash. In both scenarios,
helium burning will occur in a second burning shell in addition to the still existing
hydrogen burning shell. Stars undergoing this phase are called asymptotic giant
branch (AGB) stars.
At the beginning of the AGB phase, the star burns helium in a shell above the carbon/oxygen core. Such a star is called an early AGB (E-AGB) star. Similarly to the
hydrogen shell burning phase the envelope of the star expands and the AGB star
reaches dimensions similar to a red giant. As helium burning continues, the helium
and the hydrogen burning shell approach each other. If the intershell region between the burning regions becomes small enough, a runaway process—the helium
shell flash—may occur. The layers above the flash region expand and cool again until hydrogen burning ceases. Afterwards, the helium shell also cools and eventually
the outer layers start to contract and heat up. When the hydrogen shell becomes sufficiently hot, it can start burning its material to helium. Thus, a new cycle can begin.
These cycles are called thermal pulses and a star undergoing these pulses is called a
thermal-pulsing AGB (TP-AGB) star (Stancliffe 2005). The ongoing contraction and
expansion cycles change the luminosity periodically and cause a strong stellar mass
loss of up to 10−4 M⊙ yr−1 . During this phase the envelope above the hydrogen
shell will be ejected, leaving behind a degenerate carbon/oxygen core with a helium envelope and a tiny hydrogen layer. In the HRD, the star moves to the left and
when any nuclear energy source ceased, the star starts to cool down, i.e. it becomes
a carbon/oxygen white dwarf (CO WD).
Stars with main sequence masses larger than 8 M⊙ reach temperatures that are
2 For
stars with lower metallicity the mass limit becomes smaller.
3
1. Introduction
large enough to fuse carbon and oxygen through various chemical reactions to the
most stable element iron. For iron cores exceeding the Chandrasekhar mass, electron degeneracy pressure is insufficient to support the gravitational pressure. This
is the beginning of the core-collapse supernova (CCSN) during which the core implodes and afterwards releases its energy in an explosion.
Observations show that at least half of the visible stars live in binary systems, i.e.
systems consisting of two stars orbiting each other. Mass, energy and angular momentum transfer from one star to its companion can change the chemical composition and the lifetime of both stars. The evolution of these stars is mostly determined
by their initial mass ratio, initial metallicity and separation.
The fate of CO WDs can be drastically altered if they are assumed to belong to
binary systems. In contrast to a WD without a companion such a WD can grow in
mass by accreting material from its companion. Thus, the WD can exceed its Chandrasekhar mass, which will trigger a thermonuclear explosion known as a Type Ia
supernova.
SN Ia show no hydrogen lines in their nearly uniform spectra, but they have silicon, iron and calcium features around the time when the SN radiates the maximum
amount of light. There is a tight correlation between the peak luminosity and the
speed of luminosity decrease after maximum light radiation—the so-called Phillips
relation (Phillips 1993). Due to this relation SNe became excellent distance indicators, providing the groundwork for the discovery of the accelerated expansion of
the universe (Perlmutter et al. 1999).
Every SN Ia explosion enriches its environment with roughly 0.7 M⊙ of nickel—
which decays into iron—and 0.7 M⊙ of intermediate-mass elements3 (major con-
tributors are oxygen, magnesium, silicon, sulphur and calcium) (Hillebrandt et al.
2013). Therefore SN Ia play an important role in the chemical evolution of the universe.
3 Elements
4
between oxygen and calcium.
1.2. The physics of a SN Ia
1.2. The physics of a SN Ia
Nowadays it is believed that a SN Ia evolves from carbon-oxygen white dwarfs that
have reached their Chandrasekhar mass limit (MCh ≈ 1.37 M⊙ ), which induces the
thermonuclear explosion (Arnett 1969). The explosion is triggered by an increase in
temperature and density. Since the matter of a white dwarf is highly degenerate,
a change in temperature does not affect the structure of the WD, because pressure
and density are independent of it. Thus, in contrast to a classical gas, the WD cannot expand and cool when the fuel is ignited and heat is produced. Therefore the
additional heat directly results in a temperature increase. Because nuclear reaction
rates highly depend on temperature, these will be drastically enhanced, which in
turn speeds up heat generation resulting in a thermonuclear runaway.
According to our current understanding, the explosion itself starts with the ignition of the carbon/oxygen mixture near the center by creating a flame that propagates subsonically. During this deflagration, parts of the WD can expand which
leads to densities that are low enough for intermediate elements such as silicon to
be produced, which are needed to explain observed SN Ia spectra (Fink et al. 2014).
To reach the observed explosion luminosities, the burning flame has to turn into a
supersonically moving shock wave at some point. This is the beginning of the detonation, whose explosive burning front will burn through the whole WD and disrupt
it, leaving no remnant behind (Hillebrandt et al. 2000).
While half of the WD material is burned to intermediate elements, the other half is
burned to nickel, which decays to cobalt with a half-life of 6 days, which in turn decays to iron with a half-life of 77 days (Nadyozhin 1994). The γ-rays that are created
during these radioactive decay processes interact with the ejecta of the supernova
which emit optical and infrared photons. These photons are responsible for the observed lightcurve, which declines from its peak luminosity due to the decreasing
amount of radioactive elements in time.
The energy necessary to unbind a 1.4 M⊙ WD is approximately 0.5 × 1051 erg
(Kippenhahn et al. 2012). The nuclear energy provided by burning 0.7 M⊙ of carbon
and oxygen to nickel is roughly 1051 erg and burning the same amount of carbon
and oxygen into intermediate elements delivers the same amount of energy. Thus,
the nuclear energy delivered by the explosion is sufficient to unbind the WD and
5
1. Introduction
give the ejected material an additional amount of kinetic energy of 1051 erg, which
corresponds to ejecta velocities of about 104 km/s (Maoz and Mannucci 2012).
1.3. Mass transfer
One of the key physical processes to understand the evolution of a binary system is
mass transfer from one stellar object to another. It is a complicated mechanism with
a lot of individual physical processes going on. Today the physics of accretion are
still not fully understood, but a general picture has emerged.
Mass transfer requires mass loss and mass loss begins if the outer layers of a star
lose their gravitational binding to that star. Two ways for how this may happen
are known today: mass loss via Roche-lobe overflow (RLOF) or via a strong stellar
wind.
The Roche lobe is the area of a binary
system in which material is in gravita-
Roche Lob
tional equilibrium between both stars,
i.e. it is a gravitational equipotential.
The Roche lobe of each star has the
shape of a raindrop (Fig. 1.2). These
are connected by a point located on
the line connecting both stars, the so-
Companion
called L1 Lagrangian point of the sys- Figure 1.2. – Schematic diagram of the Roche lobe
of a star with mass M1 and its comtem. Normally the radii of the stars
lie well within their Roche lobe, but if
panion with mass M2 . The sketch is
shown for M1 > M2 .
one star increases in size and exceeds its
Roche lobe, material flows into the second star’s Roche lobe via the L1 Lagrangian
point—a process called stable Roche lobe overflow. The material is now gravitationally bound to the second star and can be accreted.
If the transferred matter is larger than the mass that can be accreted by the companion star, a common envelope (CE) will be formed engulfing the companion star
and the compact core of the donor star (Iben and Livio 1993). Due to drag forces
between the compact objects and the gas, the gas heats up, expands and finally
6
1.4. Progenitor models
loses its gravitational bond to the system4. The CE phase leads to the possibility
of forming helium stars—a non-degenerate star consisting of helium—or helium
WDs, thus providing the theoretical evolutionary background for possible helium
accreting SNe Ia progenitor models.
The second mass-loss mechanism is the possibility of a star having a strong stellar
wind, a process in which particles are blown from the star’s surface by radiation
pressure. The strong stellar wind suppresses the formation of a CE, although the
conditions would be suitable to form a CE if mass was lost through Roche-lobe
overflow (Li and van den Heuvel 1997). Thus, a strong wind changes the stability
conditions for stable mass transfer. There are different ways of how a stellar wind
can be created, depending on the mass and the evolutionary stage of the donor.
Massive stars lose material in form of hydrogen while staying on the main sequence.
Intermediate mass stars lose a large fraction of their mass during their RG or AGB
phase.
Before exploding as a SN Ia, a binary system can undergo several mass transfer
stages. Due to the variety of initial binary models (different initial masses and separations), there are a lot of different paths for such systems to evolve to a possible SN
Ia (Wang and Han 2012). By combining these evolutionary paths with the probability for the occurrence of the assumed binary systems, two out of these models have
been extracted that are predominantly used to explain the majority of the observed
SN Ia: the single- and double-degenerate models.
1.4. Progenitor models
Classical models assume that a WD has to grow in mass until it exceeds its Chandrasekhar mass to trigger the SN explosion. Depending on the type of the donor
star, the composition of transferred matter and the accretion rate, different models that can lead to a SN Ia event have emerged, two of which are the previously
mentioned single- and double-degenerate models.
4 For
further reference see Ivanova et al. (2013).
7
1. Introduction
The double-degenerate model
Double-degenerate systems consist of two WDs with a total mass larger than or
equal to the Chandrasekhar mass orbiting each other in a close binary system. They
move on nearly circular orbits and lose energy and angular momentum by gravitational waves. If their distance is small enough, gravitational forces become so strong
that the heavier WD will start to accrete matter from its companion by forming an
accretion disk (Iben and Tutukov 1984).
A system consisting of two WDs can be created by a main-sequence binary system
that undergoes two common envelope phases. There are two channels that lead to
such a binary system (Iben and Tutukov 1984). For an initial separation of the two
MS stars that is smaller than 460 R⊙ , a CE is formed when one of the two MS stars
becomes a red giant. The following CE ejection leaves behind a helium core and
the second MS star. After the helium core evolves into a CO WD, the second, more
massive star also becomes a red giant and the second common envelope is created.
After the repulsion of the envelope, the helium core of the secondary evolves into
the second CO WD. If the initial separation of the two MS stars is between 460 R⊙
and 1500 R⊙ , the first CE occurs after one MS star becomes a TP-AGB star. After the
envelope is stripped away, the binary system consists of one CO WD and one MS
star. The MS star evolves similarly to the stars of the first channel. The creation of a
CO WD and helium WD system can be accomplished if the mass of the secondary
becomes smaller than 0.5 M⊙ after the second CE phase.
The single-degenerate model
The second classical progenitor model is called the single-degenerate (SD) model
(Whelan and Iben 1973). In the single-degenerate model the companion of the WD
is a non-degenerate star: either a main sequence star (van den Heuvel et al. 1992),
a subgiant (Han and Podsiadlowski 2004), a helium star (Tutukov and Yungelson
1996) or a red giant (Patat et al. 2011).
In this scenario the accreted material can either be hydrogen or helium. In either
case, the accretion rates have to be suitable for accreting the material in a stable
manner such that the WD can grow in mass and finally exceed its Chandrasekhar
mass.
8
1.4. Progenitor models
Hydrogen accretion case
As we will see in the following, hydrogen cannot be ac-
creted in an abitrary manner for SN progenitors. Only for a very narrow range of
accretion rates, the accretion of hydrogen can result in burning hydrogen to helium
in a stable way—which is the prerequisite for mass growth to and beyond the Chandrasekhar mass. For CO WD masses between 0.7 M⊙ and 1.4 M⊙ stable burning
occurs for rates ranging from Ṁ = 7 × 10−8 M⊙ yr−1 to Ṁ = 7 × 10−7 M⊙ yr−1 ,
which also depends slightly on the initial mass of the WD (Nomoto 1982).
If the accretion rate is larger than the stableburning rate, the accreted material that cannot be
burned settles on top of the burning region. The
non-burning material expands in such a way that a
red giant-like configuration will be created (Nomoto
1982). Such a configuration may lead to a CE in
which the hydrogen will be ejected, i.e. the mass
grow of the WD is strongly suppressed and the WD
cannot reach its Chandrasekhar mass.
If the accretion rate is lower than the range for
stable accretion, shell burning becomes thermally
unstable (Starrfield et al. 1972). Then, a series of
shell flashes5 will occur and subsequently a certain
amount of accreted matter—depending on the accretion rate and mass of the CO core—is blown away
from the WD. The system can still explode as a SN
Ia if the amount of lost matter is small enough and
the companion star has enough material to supply Figure 1.3. – Possible binary evolutionary scenarios of
the WD with new material until the latter reaches its
Chandrasekhar mass.
Thus, the model parameters, i.e. initial mass ratio
and separation, are strongly limited by the required
hydrogen
accreting
WDs with wide orbits
(from Wang and Han
(2012)).
resulting accretion rate. Due to these restrictions in parameter space, the number of
5A
shell flash occurs, if the accreted matter is ignited in a shell on top of the core. The burning region
will expand and cool the outermost layers until burning ceases. Due to the ongoing accretion, the
accreted matter heats up again and a new flash may occur.
9
1. Introduction
possible single-degenerate models is heavily diminished.
The evolutionary path that can form a CO WD - RG system is sketched in Fig.
1.3. Here, two star form a CE, after one MS star evolved into a TP-AGB star. The
ejection of the CE leaves behind a CO WD and a MS star that will further evolve to
a RG. To be able to transfer hydrogen to the CO WD, the system must not create a
CE, because in this case the hydrogen envelope of the RG would just be ejected. The
initial parameter space for such a system is small. Therefore the SN Ia rate created
by the RG channel is much smaller than the rate created by the MS channel (Han
and Podsiadlowski 2004).
The three different paths that can form a CO WD - MS star system are shown Fig.
1.4. A brief description of the evolution of these paths is given in the caption.
Helium accretion case
If the accreted material consists of pure helium instead of
hydrogen, helium burns stably to carbon and oxygen for accretion rates ranging
from Ṁ = 6 × 10−7 M⊙ yr−1 to Ṁ = 6 × 10−6 M⊙ yr−1 , again, with a slight dependency on the mass of the WD (Iben and Tutukov 1989). A WD accreting helium with
a suitable rate can stably grow in mass until it reaches its Chandrasekhar mass.
As for the hydrogen accretion case, accretion rates that are too large lead to a
red-giant-like configuration. Insufficiently high accretion rates cause a sequence of
flashes, which can also result in a SN Ia explosion (Nomoto 1982). The evolutionary paths that can lead to a helium accreting CO WD are sketched in Fig. 1.5 and
described in the caption.
The double-detonation scenario
If the accretion rate of helium is lower than roughly Ṁ = 4 × 10−8 M⊙ yr−1 , no
flashes will occur. Instead, a thick layer of non-burning helium will build up on
top of the WD (Nomoto 1982). While the WD contracts, density and temperature
in the helium layer rise until the conditions for helium burning are fulfilled. At
this point, the largest part of the helium layer is degenerate. The WD cannot react
to the temperature change by increasing its radius like other stars, which would
decrease the temperature. Therefore a thermonuclear runaway at the bottom of the
helium layer—the densest and hottest region therein—occurs. The detonation of the
10
1.4. Progenitor models
helium layer creates a pressure wave, which propagades to the center. Under the
right circumstances the pressure wave will trigger a second detonation after igniting
the carbon/oxygen mixture off-center 6 (Livne 1990). The second detonation will
Figure 1.4. – Possible binary evolutionary scenarios of hydrogen accreting WDs with close orbits
(from Wang and Han (2012)).
The main-sequence binary evolves until the primary becomes either a subgiant (A),
an E-AGB star (B) or a TP-AGB star (C). The following dynamically unstable RLOF
leads to a CE phase. After the ejection of the CE, the binary system consist of a MS
star and either a helium star (A), a helium RG (B) or a CO WD (C). In channel C, the
CO WD will accrete hydrogen from its companion star. In channel B, the helium RG
will transfer its envelope via RLOF to the MS star, leaving behind a CO WD - MS
star system, in which the CO WD accretes material from the MS star. The helium
star sketched in channel A will first evolve to a helium RG. Afterwards, the system
will evolve in the same way as the system in channel B.
6 If
the second detonation cannot be triggered, the explosion of the helium layer will look like a SN
“.Ia” (Bildsten et al. 2007), i.e. a supernova that has a lower total luminosity than regular SN Ia
events have. Additionally the lightcurve declines faster than for regular SN Ia events.
11
1. Introduction
Figure 1.5. – Possible binary evolutionary scenarios of helium accreting WDs (from Wang and
Han (2012)).
Channel A: One MS star becomes a subgiant and loses its hydrogen envelope via stable RLOF. The remaining helium star evolves into a helium red giant, which transfers its helium envelope to the secondary MS star via stable RLOF, leaving behind a
CO WD. When the secondary becomes a subgiant, a CE may form, leaving behind a
CO WD and a helium star after the CE ejection. Eventually, the CO WD can accrete
matter from its accompanying helium star.
Channel B: The more massive MS star evolves to an E-AGB star without interfering
the secondary. When the E-AGB star expands, a CE may be created. After the repulsion of the CE, the remaining helium RG transfers its envelope to the secondary
via stable RLOF. As for channel A, the MS star of the created CO WD - MS star system evolves to a subgiant, loses its envelope in the following CE phase and finally
becomes a helium star, which will serve as the donor star for the accreting CO WD.
Channel C: While one of the MS stars becomes a TP-AGB star, the second one starts
burning helium in its core. Due to the expansion of the TP-AGB star, the system
forms a CE. The ejection of the CE leaves behind a CO WD - helium star binary
system, in which helium can be transferred to the CO WD.
12
1.5. Aim of this work
disrupt the WD and may lead—depending on the thickness of the helium layer and
the initial temperature of the WD—to a SN Ia (Woosley and Kasen 2011).
There are some challenges in explaining the normal SN Ia events with the doubledetonation model. A thick helium layer prevents the production of heavy elements,
thus the theoretically predicted light curves and spectroscopy of SN Ia do not coincide with the observed ones (Hoeflich and Khokhlov 1996). However, recent studies
showed that the double-detonation event could also lead to normal SN Ia brightnesses if the mass of the helium layer at the time of the explosion is small (Bildsten
et al. 2007; Shen and Bildsten 2009; Fink et al. 2010). By comparing nucleosynthesis
and light curves of theoretical models with observations Woosley and Kasen (2011)
showed that only a helium shell mass lower than 0.05 M⊙ creates the right amount
of nickel to reproduce a SN Ia-like event.
To ignite the helium layer, the accreting WD does not necessarily have to reach its
Chandrasekhar mass, why this scenario is also called a sub-Chandrasekhar mass SN
Ia. The importance of these models is supported by the study of the delay time distribution (DTD) with binary population synthesis (BPS) models (Ruiter et al. 2011).
In this study the authors assumed that a 0.1 M⊙ helium layer will trigger a sub-
Chandrasekhar mass SN Ia. With this assumption, the number of binary systems
that can produce a sub-Chandrasekhar mass model is large enough to explain the
observed SN Ia rates.
1.5. Aim of this work
As was mentioned in the last section, the mass of helium that needs to be accreted
by the CO WD before the helium layer ignites7 —in the following this quantity is
named the accretion mass Macc —is a crucial parameter to determine whether helium accretion can lead to a SN Ia or SN “.Ia”. It is also a parameter that has to be
set initially when performing complex three-dimensional simulations of the explosion process in the double-detonation scenario and thus needs to be constrained if
productive 3D work is to be done in the future. Finally, Macc can be used for setting
up BPS models which can statistically explore to what extent the helium accretion
7 The
ignition is defined as the point when the total energy generated by helium burning exceeds the
total energy lost by neutrinos.
13
1. Introduction
channel contributes to the total SN Ia rate.
In this work, we investigate the range of conditions that influence Macc with the
aim of providing constraints for 3D simulations of the explosion process and for
BPS models. We will also compare the results of Macc with analytical studies of this
topic.
14
2. Method
Numerical calculations for this work were made using the STARS one dimensional
evolution code (Eggleton 1971). This code solves the equations of stellar structure
simultaneously using the Henyey method with an adaptive mesh point and time
step control. In the following the basic computational algorithms of the code are
described.
2.1. The STARS stellar evolution code
The stellar structure equations
The structure and evolution of stars in hydrostatic equilibrium are described by
well-known differential equations: the four structure equations (Eggleton 1971)
∂P
Gm
=−
∂m
4πr4
,
(2.1)
T ∂P
∂T
=
∇
∂m
P ∂m
,
(2.2)
∂r
1
=
∂m
4πr2 ̺
,
(2.3)
∂L
= ǫ − ǫν + ǫ g
∂m
,
(2.4)
and the composition equations (Eggleton 1972)
∂
∂m
σ
∂Xi
∂m
=
DXi
+ R i − Si
Dt
.
(2.5)
15
2. Method
The first equation is the momentum equation, which is derived by the assumption
that the gravitational force balances the pressure, i.e. the equation for hydrostatic
equilibrium. Here, P is the pressure, m the mass coordinate, r the radius and G the
gravitational constant.
The second equation describes the thermal structure of a star with temperature T.
If the energy transport is driven by radiation (e.g in the interiors of low-mass stars
or the envelopes of high-mass stars on the MS), the temperature gradient ∇ equals
∇r (Eggleton 1972), i.e.:
∇r =
3κPL
16πacGr2 T 4
.
(2.6)
In this equation, L is the luminosity, κ the opacity, a the Stefan-Boltzmann constant
and c the speed of light. If the energy cannot be transported away fast enough
by radiation (e.g. interiors of massive stars or envelopes of low-mass stars on the
MS), convection sets in. A region of a star becomes convective if the Schwarzschild
criterion is violated, namely
∇r < ∇ a
.
(2.7)
∇ a , the adiabatic temperature gradient1 , generally lies in the range of 0.05 ≤ ∇ a ≤
0.40 (Eggleton 1972). In convective zones the temperature gradient ∇ is described
by
∇ = ∇ a + F (∇r − ∇ a )
, if
∇r ≥ ∇ a
.
(2.8)
Here, the transfer of energy is described by using mixing length theory (MLT) (BöhmVitense 1958), which determines the function F. In MLT, convective motion is accomplished by moving bubbles of matter. After these bubbles have traveled a characteristic distance, the mixing length l, they dissolve in the surrounding environment and adopt the physical properties of the surrounding matter.
The third structure equation describes the law of mass conservation (̺ is the mass
density) and the last structure equation describes the conservation of energy. ǫ is
1 This
16
is the temperature gradient that a star would have if all the energy is transported by radiation.
2.1. The STARS stellar evolution code
the total energy produced by nuclear burning of all elements, ǫν the energy lost via
neutrinos and ǫg the energy produced or lost by gravitational expansion or compression.
The composition equations describe the temporal evolution and spatial distribution of the abundance (by mass fraction) Xi of a chemical element i. The equations
are solved for the elements hydrogen, helium, helium-3, carbon, neon, oxygen and
nitrogen. Since helium burning is the farthest burning stage reached in our simulation, only elements created by the pp-chain, CNO-chain or alpha capture process
can occur. Thus, other elements are not included to save computational time.
In the composition equations Ri and Si are the rates at which element i is burned
or created due to nuclear reactions. The diffusion coefficient σ characterises the
efficiency of mixing of the elements. In radiative zones it is zero, in convective
regions σ is calculated by the MLT approach and is related to the linear diffusion
coefficient D by
σ = 4πr2 ̺
2
D
(2.9)
(Stancliffe et al. 2004).
The finite-difference equations
To be able to solve the partial derivatives of the differential equations numerically,
first Eq. (2.1) - (2.4) and Eq. (2.5) have to be rewritten as finite-difference equations
(FDE). In order to do this, a grid of finite mesh points has to be set up. For each
mesh point k the FDEs of the structure equations can be written as
Vk − Vk−1 = 0.5 (mk − mk−1 ) ( gk + gk−1 )
⇒ 0 = Vk − Vk−1 − 0.5 (mk − mk−1 ) ( gk + gk−1 )
.
(2.10)
In these equations Vk refers to P, T, r or L. gk is an abbreviation of the right-hand
side of the respective equation. Since in the FDEs the partial derivatives become finite differences of two consecutive mesh points, the right-hand side of the structure
equations have to be approximated by a value that lies between those of the two
17
2. Method
surrounding mesh points. The simplest approach is to choose the mean value, i.e.
0.5 × ( gk + gk−1 )2 .
The rewritten composition equations take the following form (Stancliffe et al.
2004):
X
− Xk
X − Xk+1
σk+ 1 k+1
− σk− 1 k
=
2
2
δmk+ 1
δmk− 1
2
⇒0=
Xk − Xk0
+ R X,k
∆t
2
!
δmk
!
Xk − Xk0
+ R X,k δmk
∆t
− σk+ 1
2
Xk+1 − Xk
X − Xk+1
+ σk− 1 k
2
δmk+ 1
δmk− 1
2
.
(2.11)
2
Here, ∆t is the current time step, Xk are the abundances at mesh point k at the
current and Xk0 the abundances at the previous time step. R X,k is the rate at which
an element X is burned due to nuclear reactions, δmk± 1 the mass between mesh
2
point k and k ± 12 . σk± 1 is the diffusion coefficient between mesh point k and k ± 21 .
2
For each time step the equations are solved for every mesh point with special
treatment of the outermost and innermost mesh point: here the boundary conditions3 of the stellar structure equations are inserted. Thus, for every mesh point
a matrix that contains the FDE of each variable is set up. In principle a solution is
found if Eq. (2.10) and Eq. (2.11) are (nearly) fulfilled. To realize this the code sets up
initial values and varies them until it finds a solution, which is called a converged
model.
The initial values have to be “guessed” in advance. The STARS code handles this
by recording the changes made during the last time step. The “guessed” values are
received by adding these changes to the values of the last converged model.
2 The
STARS code uses an adaptive mesh point control, thus the mass coordinates of Eq. (2.10) are
substituted with the mesh point k. In consequence the FDEs take a slightly different form and
another FDE that handles the distribution of the mesh points (the mesh-spacing function that will
be discussed at the end of this section) has to be added.
3 The central boundary conditions are: m = 0, r = 0 and L = 0. The surface boundary conditions
are: m = M, P (m = M ) = 0, r (m = M ) = R and T (m = M ) = Teff , where Teff is the effective
temperature.
18
2.1. The STARS stellar evolution code
After inserting the “guessed” parameters into the finite-difference equations, the
difference from zero of the right-hand side of the FDEs (Eq. (2.10) and Eq. (2.11))
are calculated. If these differences are lower than a certain limit, the model has
converged and a new model for the next time step can be computed. Otherwise
the corrections of the “guessed” parameters that will set the FDEs to zero have to be
evaluated. This is done by a linearization of the derivatives of the FDEs with respect
to the parameters Vk and solving the corresponding matrix (the Henyey matrix) for
the corrections (Kippenhahn and Weigert 1994). The corrected values are inserted
into the FDEs and it is verified again if the FDEs equal zero4 . This process is repeated until the model converges or a certain number of iteration attempts fail, in
which case the calculation is repeated with a smaller timestep or aborted, if a preset
number of iteration attempts failed.
Adaptive time-step and mesh-point control
After a model has successfully converged, the time step is modified depending on
the extent of changes of all variables that were made during the last iteration process. Thus, in evolutionary phases with large changes in a short time (e.g. the core
helium flash) the time step is reduced, thereby increasing the temporal resolution
for this kind of phase. In contrast to this, for stars residing in evolutionary phases
with little changes (e.g. hydrogen burning on the main sequence) time steps become
large, which saves a lot of computational time.
The number of mesh points is fixed during the whole simulation. In order to improve the resolution of regions in the star with the largest variation in the variables,
the mesh points change their position in mass for each model. This behaviour is regulated by the mesh-spacing function (MSF), which tells the code how it should distribute the mesh points depending on the pressure, temperature, radius and mass
4 Because
of the linearization, the FDEs do not necessarily have to equal zero.
19
2. Method
coordinate of each mesh point. In the STARS code, the MSF is realized by
Q = ln
!
2
r
T
(m/M )2/3
+ 1 + c2 ln
+ 1 + c4 ln
c1
c3
T + c5
+ c6 ln P + c7 ln
P + c8
P + c9
+ c10 ln
P + c11
P + c9
.
(2.12)
The mesh points have to be distributed in such a way that
dQ
= const
dk
(2.13)
is fulfilled. The coefficients c1 - c11 in Eq. (2.12) are constants that can be used to
change the weight of mesh points for a certain variable.
Physics implemented in STARS
The equation of state (EOS) connects pressure with density and temperature. The
first implementation of an EOS in the STARS code included the physics of partial
degeneracy and special relativity of the electron gas (Eggleton et al. 1973).
To cover the evolution of low-mass stars and the intitial WD cooling phase Pols
et al. (1995) updated the STARS code with new formulas for the EOS. Their formalism added the physics of coulomb interaction (which becomes important for high
mass and low density models), pressure ionization and the dissociation of molecular hydrogen (both important in the envelope) to the EOS.
Nuclear burning rates, neutrino loss rates and opacities are provided by tables
and the required values are found by interpolating on these grids. The last update of
the opacity tables were made by Eldridge and Tout (2004) who implemented OPAL
tables (Iglesias and Rogers 1996) including mixtures that are enhanced in carbon
and oxygen.
The neutrino loss rates are taken from Itoh and Kohyama (1983), Munakata et al.
(1987), Itoh et al. (1989) and Itoh et al. (1992). The nuclear reaction rates are mainly
taken from Caughlan and Fowler (1988), for a more detailed description we refer to
Stancliffe (2005), chapter 2.1.
20
2.2. Modifications to the STARS code
2.2. Modifications to the STARS code
Accretion at a constant accretion rate
To simulate accretion for this work the STARS code was extended by the functionality
of applying a constant accretion rate for different accretion material compositions.
This was achieved by simply adding material to the outermost mesh point. No
energy is released during this process and thus the only energy gain the WD can get
due to accretion is supplied by the additional matter which will compress the WD
and release heat.
Accreting matter that has a different composition than the outermost layer of the
WD can cause convergence problems in this region, because many variables are altered in the outermost mesh points simultaneously. We found that this becomes
relevant if the accretion rate is larger than 10−8 M⊙ yr−1 . To avoid these prob-
lems at the beginning of the accretion phase, the accretion rate is initially set to
10−11 M⊙ yr−1 and in the following time steps increased according to
Ṁi = Ṁin
1− αi
Ṁ
αi
(2.14)
until the final accretion rate is reached (Fig. 2.1). Here Ṁin is the initial, Ṁ is the final
and Ṁi the current (at model number i) accretion rate. The parameter α determines
the number of models for which the accretion rate should be increased. Setting this
parameter to low values makes the code more vulnerable to convergence problems,
setting it to large values would distort the results because a large amount of matter would be accreted with the wrong accretion rate. Setting this value to 70 is a
reasonable compromise for our calculations.
Another parameter that affects the amount of accreted matter in this initial phase
is the time step ∆t. To reduce the amount of matter accreted with the wrong accretion rate, ∆t has to be set to a value as small as possible, but values lower than 10 yr
produced convergence problems at the beginning of the accretion phase. Therefore,
as long as the accretion rate is being increased we limit ∆t to a value of 102 yr.
During the “increasing accretion rate phase” a certain amount of matter ∆M is accreted with an accretion rate that deviates from the constant accretion rate in which
we are interested in. For a 0.81 M⊙ model with an accretion rate of 2 × 10−8 M⊙ yr−1
21
2. Method
log10 Ṁ [M⊙ yr−1 ]
8 · 10−5
Ṁ
∆M/MWD
-8
6 · 10−5
-9
-10
4 · 10−5
-11
2 · 10−5
-12
0
0
20
40
60
80
100
120
∆M/MWD
-7
140
model number
Figure 2.1. – The accretion rate Ṁ and the accreted mass ∆M divided by the initial WD mass
MWD during the phase of increasing the accretion rate for a MWD = 0.81 M⊙ model
with Ṁ = 2 × 10−8 M⊙ yr−1. Accretion starts after 30 time steps and the final
accretion rate is reached after 100 time steps.
the ratio of ∆M and the total accreted mass MWD during this phase is of the order
of 10−5 , thus negligible (Fig. 2.1). For larger accretion rates this ratio can grow to
values that would distort our results. The following calculations will estimate ∆M
and explore for which parameters ∆M becomes large in comparison to Macc .
Although the time step changes due to the adaptive time-step control, we calculate ∆M with the constant time step limit ∆t, which gives us an upper limit of ∆M.
The total time of the WD spent in the “increasing accretion rate phase” at a model
number i can be approximated by
t(i ) = ∆t · i
.
(2.15)
During that time the WD accretes the following amount of matter:
∆M =
=
Z t=∆t α
t =0
Z i= α
i=0
Ṁi dt
Ṁin
1− αi
Ṁ
αi
∆t di
= Ṁin ∆t α Ṁ/ Ṁin − 1 / ln Ṁ/ Ṁin
22
2.2. Modifications to the STARS code
≈ ∆t α Ṁ/ ln Ṁ/ Ṁin
.
(2.16)
The last approximation is valid for Ṁ ≫ Ṁin . The term ln Ṁ/ Ṁin is roughly of
the order of 10, thus Eq. (2.16) can be further simplified to
∆M ≈ 10−1 ∆t α Ṁ ≈ 7 × 102 yr Ṁ
.
In the last equation α = 70 and ∆t = 102 were inserted. The ratio of ∆M and
the accretion mass Macc is a quantity that measures the error ǫ of Macc due to the
increasing accretion rate. Although it is physically reasonable that the accretion rate
increases before accreting at a nearly constant rate, we are only interested in constant
accretion rates. For this purpose we take this error into account and assume that the
simulations can be trusted if ǫ < 10−2 . Eventually, a ratio of Macc and Ṁ can be
calculated that fulfils this condition, namely
10−2 > ǫ =
⇒
∆M
7 × 102 yr Ṁ
≈
Macc
Macc
Ṁ
< 7 × 10−4 yr−1
Macc
.
(2.17)
This inequality is violated for small accreted masses and large accretion rates. As
we will see later, Macc increases for larger Ṁ and smaller initial WD masses MWD .
Thus, this inequality has to be particularly checked for WDs with a large initial mass
that accrete matter with a large accretion rate.
Modification of the mesh-spacing function
To ensure numerical stability during the ignition phase, the developing burning region should be provided with a significant number of the available mesh points.
The distribution of mesh points is managed by the MSF (Eq. (2.12)), which originally only depended on pressure P, temperature T, radius r, mass coordinate m
and the total mass M. At the beginning of the ignition of helium, of those five variables only the temperature changes significantly in the burning region. Given that
the maximum temporal increase in temperature in the burning region is around
23
2. Method
three order of magnitudes smaller than the maximum temperature difference over
all mesh points in a WD model, the MSF will increase the number of mesh points
only slightly in the burning region. Thus, to increase the resolution in this region,
the MSF has to be extended.
A suitable parameter that characterises the burning region is the helium abundance. The energy generation rate of burning helium at a temperature of roughly
108 K can be approximated (Kippenhahn et al. 2012) by
ǫHe ∝ Y 3 ̺2 T 40
,
(2.18)
where Y is the helium abundance and ̺ the density.
Since the energy created by burning helium depends mostly on the temperature
the burning region will always originate in a helium-rich region where the temperature is largest. In a WD the temperature increases from the surface to the center,
thus the burning region will be created at the interface between the carbon/oxygen
core and the helium envelope, i.e. the region where the helium abundance drops
significantly.
The individual terms of the MSF have to be smooth functions to work properly.
Since the helium abundance is nearly a step function for helium accreting WDs,
it cannot be added as an additional term directly. Instead the helium abundance
profile is coupled to the pressure term in the MSF (Eq. (2.12)), which is replaced by
P − P0.75 He
Q P = c6 ln P + c7 arctan c8
P0.75 He
.
(2.19)
P0.75 He defines the pressure coordinate at which the helium abundance reaches a
env , where X env is the mean helium abundance in the region above
value of 0.75 XHe
He
env will
the CO core (defined as XHe < 0.001). While at the beginning of accretion XHe
be one, it will decrease after the burning of helium has started. The free parameter
c8 determines the slope of the second pressure term. Thus, a larger c8 increases the
number of mesh points around P0.75 He within a certain pressure range.
The transition region between convective and radiative zones is responsible for
most of the numerical instabilities. Therefore, the resolution of the burning region
should be spread over a pressure range that includes this transition region. An
24
2.2. Modifications to the STARS code
appropriate value for the parameter c8 that fulfils this condition and maximises the
resolution in the burning region was empirically determined to be 4.
Table 2.1. – Coefficients of the MSF used for the accretion phase.
c1
c2
c3
c4
c5
c6
c7
c8
2.5
1.00
10−4
10−2
105
8 × 10−3
2 × 10−1
4
The coefficient c7 determines the weight of the second pressure term relative to
the remaining MSF terms. c7 was chosen in such a way that 50 mesh points are reserved for the area below and 150 above the combination of burning and convective
region. While decreasing the number of mesh points outside this area would lead to
numerical instabilities in the core and envelope, a larger number would not improve
the numerical behaviour of the code. A list of the coefficients that were chosen for
the simulations can be found in Tab. 2.1 and an example of the distribution of the
mesh points with the new MSF can be found in Fig. 2.2.
T [K ]
107
1025
106
1024
105
T
P
104
0
50
100
150
200
250
300
350
400
450
P [g/cm s2 ]
1026
108
1023
500
k
Figure 2.2. – Temperature and pressure plotted against mesh point k with the modified meshspacing function applied. The grey region corresponds to the convective region, the
red-patterned area to the burning region. Due to the arctan term in the MSF the
pressure is flattened around mesh point k = 275. Therefore the MSF puts roughly
300 points in the combination of the burning and convective region.
25
2. Method
2.3. Input models
The CO WDs were created from main sequence stars with initial masses between
3.0 M⊙ and 8.0 M⊙ , an initial hydrogen abundance of X = 0.7 and a metallicity
of Z = 0.02. The evolutionary track of the creation of these CO WDs is shown as
an example in Fig. 2.3. The main-sequence stars evolved to red giants and after-
4
2
log10 ( L [L⊙ ])
3
1
2
3
1
4
0
-1
5
5.4
5.2
5
4.8
4.6
4.4
4.2
4
3.8
3.6
3.4
log10 ( T [K])
Figure 2.3. – Evolutionary track of a M = 6.0 M⊙ main-sequence star from which a M = 0.81 M⊙
CO WD model was created by applying an artificial mass loss rate.
The marks 1 - 5 indicate the following events:
1: Beginning of the evolution at the main sequence. No mass loss or gain is applied
between point 1 and 2.
2: A constant mass-loss rate of 0.5 × 10−5 M⊙ yr−1 is switched on
3: The mass-loss rate is switched off after the hydrogen and helium layers are removed
4: The “hot” CO WD model
5: The “cold” CO WD model
wards to AGB stars with no mass-loss mechanism applied. After producing CO
26
2.3. Input models
cores the outer layers of the AGB stars were continuously reduced by adopting an
artificial constant mass loss rate until the shells above the CO core were removed.
The maximum allowed impurities of helium in the CO WD were limited to a total
mass fraction of 10−3 . Altogether 12 CO WDs with masses between 0.66 M⊙ and
1.21 M⊙ in steps of approximately 0.05 M⊙ were created.
After removing the helium and hydrogen shells, mass loss was turned off and the
WDs were cooled until they reached a maximum temperature of log10 ( Tmax ) = 8.2.
At this point the central temperature takes values between log10 ( Tc ) = 7.9 and
log10 ( Tc ) = 8.1, which depends on the initial WD mass. Thus, for these “hot” models the temperature profile is not isothermal and the temperature at nearly every
point of the WD is larger than the ignition temperature of helium (Fig. 2.4). The
non-isothermal structure is caused by the former mass-loss process, after which the
WD had insufficient time to thermally relax.
1.6 × 108
hot model
cold model
T [K ]
1.2 × 108
8.0 × 107
4.0 × 107
0
0
0.2
0.4
0.6
0.8
1
m [M ⊙ ]
Figure 2.4. – Temperature structure of a hot model in comparison to a cold model (MWD =
0.81 M⊙ for each model).
To investigate the temperature dependency of the accretion mass, WDs with lower
temperatures were also created. This was done by evolving the “hot” models in time
without any accretion, so that they were able to cool. The evolution was stopped
when the core temperature reached log10 ( Tc ) = 7.5. In comparison to the “hot”
models, these “cold” WDs have—except for their surfaces—an isothermal structure
27
2. Method
(Fig. 2.4). In contrast to the thermal structure, the mechanical structure is almost the
same for the “hot” and “cold” models (Fig. 2.5). A summary of five of these initial
models can be found in Tab. 2.2.
̺ [g cm−3 ]
1.2 × 107
hot model
cold model
8.0 × 106
4.0 × 106
0
0
0.2
0.4
0.6
0.8
1
m [M ⊙ ]
Figure 2.5. – Density structure of a hot model in comparison to a cold model (MWD = 0.81 M⊙
for each model).
Matter was accreted onto both the “hot” and “cold” WD models with a constant
accretion rate between Ṁ = 10−9 M⊙ yr−1 and Ṁ = 5 × 10−6 M⊙ yr−1 . To compare
the behaviour of small pollutions in the accreted helium, different simulations with
helium abundances of Y = 1.00, Y = 0.90 and Y = 0.10 were performed, where
the missing helium was replaced by a carbon/oxygen mixture with a mass ratio of
1 : 1.
2.4. Assumptions for the accretion phase
In our simulations the accretion of helium is treated in a simplified way to decrease
the computational effort. The WD is assumed to be non-rotating and spherically
symmetric. The accreted matter is deposited directly on top of the WD and also
treated in a spherically symmetric way. Thus, the usage of a one-dimensional stellar
evolution code is sufficient.
All energy and momentum transfer processes that happen within a possible ac-
28
2.4. Assumptions for the accretion phase
cretion disk or by the interaction of the gas that is transferred by the companion to
the WD is neglected. Therefore, the thermal response of the WD to the accretion is
only caused by the additional weight of the accreted matter.
Table 2.2. – Selected quantities of the created WDs. Tc and ̺c are central temperature and density,
Ts and ̺s are surface temperature and density. Tmax denotes the maximum temperature, L the surface photon luminosity and XHe the total mass fraction ofhelium. The
unit of the temperatures is one Kelvin, the unit of the densities is one g/cm3 and
the unit of the luminosity is one L⊙ .
XHe 10−3
log( Tc )
log( Tmax )
log( Ts )
log(̺ c )
log(̺s )
log( L)
0.66
0.04
7.92
8.21
4.94
6.60
1.20
7.91
8.19
4.97
6.99
0.94
3.80
7.97
8.23
5.05
7.33
−6.26
−6.07
1.06
0.81
1.05
0.03
8.03
8.23
5.09
7.64
1.21
0.45
8.14
8.21
5.12
8.21
0.66
0.04
7.52
7.52
4.34
6.68
0.81
1.20
7.53
7.53
4.40
7.04
0.94
3.80
7.51
7.51
4.44
7.37
1.05
0.03
7.52
7.52
4.48
7.68
1.21
0.45
7.52
7.52
4.46
8.25
M [M ⊙ ]
hot models
cold models
0.97
−5.92
−5.80
1.12
−5.60
0.95
−5.80
−5.85
−1.54
−1.43
−5.52
−1.32
−5.78
−5.66
1.10
−1.43
−1.36
29
30
3. Accretion mass
In this chapter the results of the simulations of the helium accreting WDs are given.
The dependency of the accretion mass on the accretion rate, the initial mass of the
WD, the initial temperature of the WD and the composition of the accreted material
will be presented and compared to analytical descriptions of this topic.
3.1. Results
Macc [M⊙ ]
10−1
10−2
MWD
MWD
MWD
MWD
MWD
10−3
10−9
= 0.66
= 0.81
= 0.94
= 1.05
= 1.21
10−8
10−7
10−6
Ṁ [M⊙ yr−1 ]
Figure 3.1. – Accretion mass at the onset of helium ignition Macc as a function of Ṁ for five different initial WD masses.
In Fig. 3.1 the accretion mass is plotted against the accretion rate for CO WDs
with different initial masses (0.66 M⊙ , 0.81 M⊙ , 0.94 M⊙ , 1.05 M⊙ and 1.21 M⊙ ). For
31
3. Accretion mass
accretion rates larger than 10−6 M⊙ yr−1 the uncertainties due to the initial increas-
ing accretion rate become larger. Only those results that accrete less than 1% of their
mass with the “wrong” accretion rate—and thus fulfil inequality (2.17)—are plotted.
The results are also listed in table A.1 in the appendix.
In general the accretion mass is larger for the less massive CO WDs. More massive WDs have higher densities than the less massive ones, so they need less time to
reach the conditions for helium ignition, which depends on density, helium abundance and temperature (Eq. (2.18)). For a constant accretion rate less time means
that less material can be accreted.
For a given initial mass the accretion mass decreases slightly from 10−9 M⊙ yr−1
to 3 × 10−8 M⊙ yr−1 , then falls more steeply for rates between 3 × 10−8 M⊙ yr−1
and 9 × 10−8 M⊙ yr−1 , after which the decrease flattens again. In the regime of
steep decline the accretion mass drops roughly by one order of magnitude.
For the following calculations three regimes of the accretion mass are defined: the
slow accretion ( Ṁ ≤ 3 × 10−8 M⊙ yr−1 ), the fast accretion ( Ṁ > 9 × 10−8 M⊙ yr−1 )
and the transition regime (3 × 10−8 M⊙ yr−1 < Ṁ ≤ 9 × 10−8 M⊙ yr−1 ).
3.2. Analytical Description
The analytical description of the physics of helium accreting WDs was primarily
described by Nomoto (1982). Based on his calculations, an approximation for the
accretion mass will be calculated, which will then be compared to the results from
the simulations.
Compressional heating and heat conduction
During accretion two physical processes play an important role: heating of the WD
via compressional heating and the heat flow via conduction. The additional material accreted on top of the carbon/oxygen core compresses the WD and therefore
releases energy in its interior. The expression for the energy released due to accretion is split into two parts (Nomoto 1982). The first term results from the increase in
density at a fixed mass shell as a result of the increase in the total mass of the WD.
32
3.2. Analytical Description
The extra luminosity due to this effect can be approximated by
M
M⊙
( M)
4
≈ × 10−2 L⊙
3
( M)
= (∂ ln̺/∂M ) q Ṁ
Lg
T
T7
( M)
λ̺
−
8
10 M⊙ yr−1
!
,
(3.1)
where
λ̺
(3.2)
is the compression rate and T7 is the temperature in units of 107 K. The first term in
Eq. (3.2) is the derivative of the density with respect to the total mass within a given
( M)
mass shell. λ̺
( M)
and therefore L g
is almost constant for every mass shell, because
the change in density with the total mass is almost homologous.
Figure 3.2 – Simulation of WDs with different masses and a central temperature of Tc = 3 × 107 that
accrete helium with Ṁ = 7 ×
10−10 M⊙ yr−1 (Fig. 3 of Nomoto
(1982)).
Top panel: Compression rate and
energy release as a function of the
total mass MWD .
Bottom panel: Central density ̺c
and radius R as a function of the
total mass MWD .
33
3. Accretion mass
The central density as a function of total mass and the corresponding compression rate for an accretion rate of 7 × 10−10 M⊙ yr−1 is shown in Fig. 3.2. In the
( M)
mass range between 1.0 M⊙ and 1.2 M⊙ the energy release L g increases by roughly
( M)
λ
2 × 10−2 L⊙ . 10−8 M̺ yr−1 becomes unity for Ṁ = 7 × 10−10 M⊙ yr−1 and M =
⊙
1.3 M⊙ .
( M)
By replacing λ̺
( M)
Lg
( M)
in Eq. (3.1) with Eq. (3.2), L g
T
Ṁ
102
M
f ( M)
M⊙
T7
7
10−8 M⊙ yr−1
T
Ṁ
M
f ( M)
.
L⊙
M⊙
T7
10−8 M⊙ yr−1
4
= × 10−2 L⊙
3
= 2 × 10−1
( M)
can be rewritten as
(3.3)
(3.4)
102
7
× Ṁ. f ( M ) is an increasing function in M only and was chosen such that it becomes unity for M = 1.3 M⊙ . For
lower masses the energy release decreases.
The second term contributing to the heating of the WD follows from the compression as the matter moves to the centre in mass-space. This energy release is given
by
In the first step λ̺
(q)
Lg
was replaced by f ( M ) ×
10
≈
× 10−1 L⊙
7
T
T7
10−8
( M)
This luminosity is comparable to L g
Ṁ
M⊙ yr−1
.
(3.5)
, but the energy is released only in a small
mass range Msh in the outer part of the WD, while the energy released due to Eq.
(3.4) is distributed over all mass shells of the WD.
The energy released by compressional heating can be transported to the inner
mass shells by heat conduction (Kippenhahn et al. 2012). Heat conduction is the
main mechanism for heat transportation in highly degenerate materials such as in
the interior of WDs. The time after which the core becomes isothermal is the time
scale at which heat can be transported to the interior of a WD. This conduction time
scale τcon is typically of the order of 106 yr to 107 yr.
The accretion time scale τacc can be defined as the time needed to accrete 1 M⊙ of
34
3.2. Analytical Description
material
τacc =
1 M⊙
.
Ṁ
(3.6)
Accretion rates that are of the order of Ṁ = 10−7 M⊙ yr−1 correspond to accretion
time scales that equal the conduction time scale. If the accretion rate is much smaller
than this value, heat transportation to the interior is faster than heating the outer
part of the WD. Thus, the heat gained due to accretion is distributed uniformly over
the whole WD. In this case the total luminosity increase due to accretion Lslow
acc can
be described by
(q)
( M)
Lslow
acc = L g + L g
.
(3.7)
On the other hand, if the accretion rate is much larger than Ṁ = 10−7 M⊙ yr−1 ,
compressional heating increases the temperature in the small mass shell near the
surface much faster than conduction can transport the additional heat to the centre.
The total additional luminosity in the small mass shell Msh can also be described by
( M)
Eq. (3.7), but the total mass has to be replaced by Msh . If Msh ≪ M, L g
much smaller than
(q)
Lg ,
becomes
thus
(q)
Lfast
acc ≈ L g .
(3.8)
Approximation of the accretion mass
In a WD heat can either be transported by the thermal movement of ions or by heat
conduction via electrons, so their different contributions to the entire specific heat
capacity have to be added up according to
−
e
cν = cion
ν + cν
.
(3.9)
The specific heat capacity of ions is given by
cion
ν =
2 kb
3 Amu
(3.10)
35
3. Accretion mass
(Kippenhahn et al. 2012), where kb is the Boltzmann constant, mu is the atomic mass
unit and A the mass number. By evaluating the constants and setting the mass
number to 14 (which is the mean mass number of a 50% carbon and 50% oxygen
mixture), cion
ν can be approximated as
7
cion
ν ≈ 10
erg
gK
.
(3.11)
The electrons contribute to the specific heat capacity by
−
ceν
π 2 k2b Z
=
me c2 Amu
√
1 + X2
T
X2
(3.12)
(Kippenhahn et al. 2012). Here, me is the electron mass, Z is the atomic number and
X the degree of relativity, namely
X=
pF
me c
.
(3.13)
p F —the Fermi momentum—is given by
pF = h
r
3
s
eV s √
3ne
3̺
=h3
≈ 1.72 × 10−7 1/3 3 ̺
8π
8m p π
g
,
(3.14)
where ne is the number density of electrons and m p the mass of a proton. Inserting
the Fermi momentum into Eq. 3.13, the degree of relativity can be approximated as
X=
pF
cm
cm √
≈ 5.87 × 104
p F ≈ 10−2 1/3 3 ̺
me c
eV s
g
.
(3.15)
If X is much greater than 1, the gas will be relativistic, if it is much smaller than
1, the gas will be non-relativistic. Except for their very thin outer envelopes, accreting WDs always have densities of at least 106 g cm−3 . For densities
larger than
√
106 g cm−3 , X will be greater than 1 (Eq. 3.15), i.e. the term
1+ X 2
X2
in Eq. (3.12)
behaves roughly as 1/X.
Combining all these equations and setting
36
Z
A
= 12 , the specific heat capacity due
3.2. Analytical Description
to electrons can be approximated by
−
ceν
7
≈ 10
T
108 K
̺
107 g cm−3
−1/3
erg
gK
.
(3.16)
If the mean temperature reaches 108 K and at the same time the mean density is of
the order of 107 g cm−3 , the two contributions of the specific heat capacity become
comparable. During the heating process due to accretion, the temperatures are in
a range between 5 × 107 K and 108 K. The mean density is a function of the total mass, where a larger mass corresponds to a higher mean density. For WDs of
masses between 0.4 M⊙ and 1.2 M⊙ the mean density varies between 106 g cm−3
and 108 g cm−3 . Thus, for WDs lighter than 1.2 M⊙ the specific heat capacity of the
electrons is nearly of the same order of magnitude as that of the ions. Therefore the
total specific heat can be approximated by
7
cν ≈ 2 × cion
ν = 2 × 10
erg
gK
.
(3.17)
Knowing cν , the amount of energy ∆E that is needed to heat up the WD by a certain
temperature difference ∆T can be calculated:
∆E = cν Msh ∆T
.
(3.18)
For small accretion rates the energy can be distributed over the whole WD, thus
Msh ≈ MWD . For large accretion rates the energy is only distributed over a small
mass range in the outer part of the WD. In the latter case the energy needed to
increase the temperature by a certain amount is lowered by a factor of MWD /Msh in
comparison to small accretion rates.
By setting ∆T to the temperature difference that is needed to reach the ignition
conditions for helium burning, ∆E gives the amount of energy that has to be delivered to the WD to ignite helium. This energy is gained by compressional heating for
which the luminosities are given by Eq. (3.4) and Eq. (3.5). The time for heating up
37
3. Accretion mass
the WD due to accretion until the point of ignition ∆tacc can be calculated by
∆E
L acc
∆tacc =
,
(3.19)
where Lacc is either given by Eq. (3.7) or Eq. (3.8). Finally the accretion mass Macc is
given by
Macc = Ṁ ∆tacc
.
(3.20)
Combining Eq. (3.20), Eq. (3.18) and Eq. (3.19) delivers a formula for the accretion
mass, namely
Macc =
erg
2 × 10
gK
7
Msh ∆T
Ṁ
L acc
.
(3.21)
3.3. Internal structure of a helium accreting 0.81 M⊙ WD
The behaviour of the accretion mass in the slow and fast accretion regime and the
transition regime can be understood by examining the internal structure of the accreting WDs. Fig. 3.3 shows the temperature structure for the three different regimes
of accretion of a 0.81 M⊙ WD at three different times, namely at the begin of accre-
tion, at the ignition of helium and at a point in time in between. For the same
accretion rates and times, Fig. 3.4 illustrates the temperature-density plots. In these
plots, the ignition curve is indicated (black solid line), which is defined as the curve
in the T-̺ plot at which the nuclear energy released by the triple alpha process ǫnuc
equals the neutrino losses ǫν , i.e. above this line more energy is released from nu-
clear reactions than is lost by neutrinos. Therefore, the emerging thermal runaway
fusion will eventually trigger the explosion. In Fig. 3.5 the evolution of the central
and the maximum temperature as well as the evolution of the neutrino, surface and
helium luminosity is shown.
38
3.3. Internal structure of a helium accreting 0.81 M⊙ WD
T [K ]
1.5 × 108
1.0 × 108
t = 0 yr
t = 8.34 × 106 yr
t = 3.26 × 107 yr
5.0 × 107
0
0
0.2
0.4
0.6
0.8
1
1.2
m [M ⊙ ]
T [K ]
1.5 × 108
1.0 × 108
t = 0 yr
t = 1.83 × 105 yr
t = 4.22 × 105 yr
5.0 × 107
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
m [M ⊙ ]
T [K ]
1.5 × 108
1.0 × 108
t = 0 yr
t = 3.62 × 104 yr
t = 1.59 × 105 yr
5.0 × 107
0
0
0.2
0.4
m [M ⊙ ]
Figure 3.3. – Temperature structure of a 0.81 M⊙ WD at the beginning of accretion, shortly after
ignition and between those times. (Top panel: Ṁ = 10−8 M⊙ yr−1 ; Mid panel: Ṁ =
5 × 10−8 M⊙ yr−1 ; Bottom panel: Ṁ = 10−8 M⊙ yr−1 )
39
3. Accretion mass
3
Ψ=0
T 108 K
Ψ = −1
Ψ = 10
ǫnuc > ǫν
1
t = 0 yr
t = 8.34 × 106 yr
t = 3.26 × 107 yr
ǫnuc < ǫν
0.3
103
3
104
Ψ=0
T 108 K
Ψ = −1
105
106
̺ g cm−3
107
108
Ψ = 10
ǫnuc > ǫν
1
t = 0 yr
t = 1.83 × 105 yr
t = 4.22 × 105 yr
ǫnuc < ǫν
0.3
103
3
104
Ψ=0
T 108 K
Ψ = −1
105
106
̺ g cm−3
107
108
Ψ = 10
ǫnuc > ǫν
1
t = 0 yr
t = 3.62 × 104 yr
t = 1.59 × 105 yr
ǫnuc < ǫν
0.3
103
104
105
106
̺ g cm−3
107
108
Figure 3.4. – Temperature-density plot of a 0.81 M⊙ WD at the beginning of accretion, shortly
after ignition and between those times. The dot indicates the location of the border
between the helium layer and the carbon/oxygen core (Y < 0.90). The ignition
curve as well as the curves for a constant degeneracy parameter Ψ = −1, Ψ = 0
and Ψ = 10 are also shown. (Top panel: Ṁ = 10−8 M⊙ yr−1 ; Mid panel: Ṁ =
5 × 10−8 M⊙ yr−1 ; Bottom panel: Ṁ = 10−7 M⊙ yr−1 )
40
3.3. Internal structure of a helium accreting 0.81 M⊙ WD
102
L [L ⊙ ]
T 108 K
1.4
1
Tc
Tmax
0.6
2000
101
100
Lν
Lph
LHe
10−1
4000
6000
2000
model number
4000
6000
model number
(a) Ṁ = 10−8 M⊙ yr−1 : The core becomes isothermal after 3 × 106 yr, the minimum temperature is
reached after 8 × 106 yr and ignition starts after 3 × 107 yr.
102
L [L ⊙ ]
T 108 K
1.4
101
1
Tc
Tmax
0.6
2000
100
3000
4000
2000
model number
(b) Ṁ =
Lν
Lph
LHe
5 × 10−8
3000
4000
model number
M⊙
yr−1
: Ignition starts after 4 × 105 yr.
102
L [L ⊙ ]
T 108 K
1.4
101
1
Tc
Tmax
0.6
2000
Lν
Lph
LHe
100
3000
model number
2000
3000
model number
(c) Ṁ = 10−7 M⊙ yr−1 : Ignition starts after 2 × 105 yr.
Figure 3.5. – Left panel: Evolution of the maximum and central temperature a 0.81 M⊙ WD.
Right panel: Photon, nuclear and neutrino luminosity evolution of the same WD.
Note that the model number is not a linear function of time.
41
3. Accretion mass
Slow accretion regime
In the slow accretion regime, the temperature of the WD first decreases in time (see
text below). The core becomes isothermal after 3 × 106 yr, which is the time when
the central temperature Tc equals the maximum temperature Tmax (Fig. 3.5a). It
drops significantly under the ignition temperature until a minimum temperature
Tmin is reached. The initial decrease in temperature is driven by neutrino losses (see
Fig. 3.5a). The energy gained due to accretion is
Lslow
acc ≈
10
× 10−1 L⊙
7
T
T7
10−8
Ṁ
M⊙ yr−1
1 + 1.4
M
f ( M)
M⊙
(3.22)
(see Sec. 3.2, Eq. (3.7)). For an accretion rate of 10−8 M⊙ yr−1 and a 0.81 M⊙ WD
with an initial temperature of approximately 108 K, the accretion luminosity Lslow
acc
becomes unity at the beginning of accretion, which is much smaller than the energy
loss by neutrinos (≈ 30 L⊙ ) and therefore cannot impede cooling. The point when
heating balances cooling defines the minimum temperature Tmin of the core of the
WD during the accretion process, which is given by
Lslow
acc ( Tmin ) = L ( Tmin ) + L ν ( Tmin )
.
(3.23)
The neutrino luminosity is given by
Lν ∝
Z2 6
T
A
(3.24)
(Kippenhahn et al. 2012) and the photon luminosity by
L = 4πR2 σT 4
(3.25)
(Kippenhahn et al. 2012), where σ is the Stefan-Boltzmann constant. Canceling Tmin
in Eq. (3.23) and ignoring constants shows the proportionality
2
3
const. + Tmin
M · Ṁ ∝ Tmin
42
.
(3.26)
3.3. Internal structure of a helium accreting 0.81 M⊙ WD
Therefore a larger Ṁ and/or M causes a larger Tmin . This is consistent with the
data from our simulations (see Tab. 3.1). For larger Ṁ the duration of the accretion
process is shorter. Due to the shorter evolution time the mean density of the WD
will increase less, which will finally lead to an increase in the central temperature
at the time of ignition Tcacc to fulfil igniton conditions. This explains the increase of
Tcacc for larger Ṁ.
Table 3.1. – Numerical results of the minimum temperature Tmin and the central temperature at
the point of ignition Tcacc . The results are shown for different initial WD masses and
for two different accretion rates. Tcacc and Tmin increase with increasing mass and
accretion rate. ∆T is the difference between Tcacc and Tmin .
Ṁ 10−8 M⊙ yr−1
0.1
1.0
0.66
Tmin 107 K
2.9
Tcacc 107 K
4.2
1.3
0.81
3.0
4.2
1.2
0.94
3.2
4.2
1.0
1.05
3.4
4.4
1.0
1.21
4.3
5.0
0.7
0.66
5.4
6.9
1.5
0.81
5.8
7.1
1.3
0.94
6.4
7.6
1.2
1.05
7.1
8.0
1.1
1.21
9.2
9.4
0.2
M [M ⊙ ]
∆T 107 K
For an accretion rate of 10−8 M⊙ yr−1 , Tmin is reached within a time of about
8 × 106 yr. The time ∆tcool it takes the WD to cool to Tmin can be approximated by
comparing the total energy that is gained by accretion until the minimum temperature is reached
slow
∆Eacc
= Lslow
acc ∆tcool
.
(3.27)
with the amount of energy that is needed to cool the WD (Eq. 3.18). For M =
43
3. Accretion mass
0.81 M⊙ , ∆T = 4 × 107 K and L = 1 L⊙ the cooling time ∆tcool can be calculated as
∆tcool = (cν MWD ∆T ) /Lslow
acc
(3.28)
= 1.1 × 107 yr ,
(3.29)
which is almost identical to the cooling time calculated in the simulation.
When the WD cools, neutrino losses decrease until heating due to accretion exceeds neutrino losses and the now more important photon losses on the surface of
the WD (Fig. 3.5a). As a consequence, the WD heats up again until the heliumrich region of the WD reaches ignition conditions. The densest part of the helium
mantle needs the lowest ignition temperature, thus helium ignites in the transition
region between the core and the envelope (Fig. 3.4). The heating phase lasts roughly
3 × 107 yr. Because heating the WD takes a much longer time than cooling, most of
the mass is accreted during the heating phase.
The simulated accretion masses can be compared with the analytically calculated
ones by setting Msh = M and inserting Eq. (3.22) in Eq. (3.21). During accretion, M
increases continuously, but for simplicity we assume that M is the initial mass of the
CO WD MWD . The temperature difference ∆T is roughly 1.0 × 107 K and the mean
temperature of the WD during accretion can be approximated to be T ≈ 5 × 107 K
(Tab. 3.1). With these numbers the analytically calculated accretion mass for the
slow is
slow accretion regime Macc
slow
Macc
= 2 × 1014
= 5 × 10
erg
g 7 × 107
−2
M⊙
L⊙
M⊙ yr−1
1.4 f ( M ) +
h
M
i
1.4 MM⊙ f ( M ) + 1
M
M⊙
−1 ! −1
.
(3.30)
(3.31)
By setting M = 1.3 M⊙ , f ( M ) becomes unity and therefore Macc = 0.02. This value
is lower than the numerical result by a factor of three. The analytically calculated
slow differs by alaccretion mass stays nearly constant for lower initial masses. Macc
most two order of magnitudes from the simulation results of the 0.66 M⊙ WD. The
difference can be explained by the simplifications that were made. In reality, the
44
3.3. Internal structure of a helium accreting 0.81 M⊙ WD
temperature does not stay constant in time and space, the mass increases in time,
∆T is a declining function in MWD and the specific heat capacity, which is proporslow , decreases with M
tional to Macc
WD .
Eq. (3.30) also shows that the accretion mass does not depend on Ṁ, which is
comparable with the numerical result, where Macc decreases only slightly with increasing Ṁ. This decrease can be explained by the fact that energy losses by neutrinos were neglected in the analytical calculation of the accretion mass. The total
energy loss by neutrinos within a given time ∆t is
∆Eν =
Z t1
t0
Lν ∝
Z2 6
T ∆t
A
,
(3.32)
and the energy gained by accretion is
slow
∆Eacc
∝
Z t1
t0
Ṁ = Macc
,
(3.33)
where Macc is the accreted mass within ∆t. For larger Ṁ the accretion time is lower.
Therefore ∆Eν becomes smaller, which means that less accretion energy is needed to
heat up the WD to ignition conditions. Because Macc is proportional to the accretion
energy, the accretion mass will be lower for larger Ṁ.
Fast accretion regime
In the fast accretion regime the temperature structure of the WD barely changes and
the temperature stays above the ignition temperature. For every accretion rate value
examined, the whole evolution takes place within 106 yr, which is shorter than the
conduction time scale. Thus, neither can the core become isothermal, nor can heat
be transported from the outer heated part to the interior of the WD (see Fig. 3.3
and Fig. 3.5c). In contrast to the slow accretion regime, here the energy gained by
accretion (Eq. (3.8)) is much larger than the energy lost by neutrinos. Thus, the
WD cannot cool and the outer half of the WD (except for the very cool envelope)
fulfils the temperature and density conditions for helium burning (Fig. 3.4). During
accretion, more and more helium rich material moves from the cool and less dense
outer part of the WD to the region where helium ignition conditions are fulfilled.
45
3. Accretion mass
When enough material is accumulated ignition will start.
As in the slow accretion regime, the accretion mass decreases with increasing Ṁ.
Again, this is a consequence of the decreasing neurino losses due to the shorter
evolutionary time.
Transition regime
Between the fast and the slow accretion regime, the accretion mass drops by more
than one order of magnitude. As stated above, in the fast accretion regime, the WD
cannot cool and redistribute its energy. By decreasing the accretion rate the WD
becomes able to cool and the energy can be transported to the interior (Fig. 3.5b).
The energy generated by accretion is still sufficiently high so that the WD can only
cool for a short period. After that the WD heats up until the helium layer ignites.
3.4. Degeneracy at the time of ignition
Whether the burning region develops into a core-helium flash like event or into a
more stable burning event mostly depends on the degeneracy of the material in
the burning region at the time of ignition. The degeneracy is characterized by the
degeneracy parameter Ψ. If Ψ is much larger than zero, the material is degenerate,
else it is not. If Ψ is roughly zero, the material is said to be partially degenerate.
The degeneracy of the accreting WDs are indicated in Fig. 3.4. In the slow accretion regime the WD ignites in a highly degenerate environment. With increasing
accretion rates, the burning region moves to regions with smaller degeneracy parameters. For rates larger than 10−7 M⊙ yr−1 , Ψ becomes smaller than one. There-
fore, for large accretion rates stable burning becomes possible.
3.5. Temperature Dependency Of The Accretion Mass
The time when mass transfer from the companion onto the WD sets in is not fixed
and depends on the specific evolutionary path. Ultimately, it is determined by the
main-sequence configuration of the binary. During the time period prior to accretion, the WD cools down. Thus, a delay of the beginning of accretion decreases the
46
3.5. Temperature Dependency Of The Accretion Mass
Macc [M⊙ ]
initial temperature of the WD at this point.
10−1
10−2
Tmax = 1.6 × 108 K
Tmax = 3.4 × 107 K
10−9
10−8
10−7
10−6
Ṁ [M⊙ yr−1 ]
Figure 3.6. – Accretion mass at the onset of helium ignition as a function of Ṁ for two different
initial maximum temperatures of a 0.81 M⊙ WD.
Fig. 3.6 shows how the accretion mass changes when the initial temperature of
the accreting CO WD is lowered1 . For accretion rates lower than 3 × 10−8 M⊙ yr−1 ,
the accretion mass is independent of the initial temperature. In this regime the WD
will adjust its temperature as a result of the additional accretion energy. In contrast
to the “hot” models, for the “cold” models the energy lost by neutrinos is smaller
than the accretion energy. Thus, the WD heats up from the beginning of accretion.
When the temperature of the WD equals the minimum temperature that a “hot”
model WD reaches during accretion (see Tab.3.1), the evolution of the “hot” and
“cold” models becomes nearly identical. The initial evolutionary period, where the
two models approach the minimum temperature is much shorter than the following
heating period. Thus, the time needed to ignite the WD and therefore the accreted
mass should be equal for both models.
For larger accretion rates the accretion mass depends on the initial temperature.
1 The
simulation data of all “cold” WD models can be found in table A.2 in the appendix.
47
3. Accretion mass
While the accretion masses of the “hot” WDs decrease only slightly after the strong
drop around Ṁ = 5 × 10−8 M⊙ yr−1 , the ones for the initially cooler WDs decline
almost linearly (in log-log) with increasing Ṁ. As mentioned before, the constant
behaviour of the “hot” WDs can be explained by their hot temperatures in combination with the large accretion rates. In consequence, for all accretion rates helium
ignites almost at the same temperature with the same accretion mass.
The initially cooler WD has an isothermal core with a temperature that lies below 108 K, i.e. the WD has to be heated by accretion until the ignition temperature
is reached. Macc is larger for a smaller initial temperature, because the temperature difference needed to heat up the WD to the ignition temperature is also larger.
Hence more heat is required which results in a longer period from the beginning of
accretion until ignition. The final consequence is an increase in Macc .
In the fast accretion regime, Macc can be described by inserting the equation for
the accretion luminosity of the fast regime Eq. (3.8) into the equation for the accreted
mass Eq. (3.21). The initial temperature of the “cold” models is 3 × 107 K and the
ignition temperature is roughly 8 × 107 K. Therefore we assume an average temper-
ature of 5 × 107 K and a temperature increase ∆T of 5 × 107 K. With these numbers
fast can be
the analytically calculated accretion mass for the fast accretion regime Macc
approximated as
fast
Macc
= 2.4 × 10−6
= 24 Msh
1
K
T
T7
−1
Msh ∆T
.
(3.34)
(3.35)
The accretion mass only depends on the mass of the shell that is heated by accretion. This is not in agreement with the simulation results, where the accretion mass
is a decreasing function of Ṁ. Again, neutrino losses were neglected in the calculations as well as the increasing amount of transported heat with decreasing Ṁ, which
could be the reason for this discrepancy.
48
3.6. Different compositions of the accreted material
3.6. Different compositions of the accreted material
The accreted material does not necessarily have to be pure helium. The initial metallicity of the binary system’s donor star impurifies the composition of the envelope.
This will also change the composition of the material that is accreted from the WD.
To check whether these impurities have an impact on the accretion mass, additional
simulations with different compositions of the accreted material were performed.
The impurities were simulated as a carbon/oxygen mixture with a mass ratio of
Macc [M⊙ ]
1 : 1.
10−1
10−2
Y = 1.00
Y = 0.90
Y = 0.10
10−9
10−8
10−7
10−6
Ṁ [M⊙ yr−1 ]
Figure 3.7. – Accretion mass at the onset of helium ignition as function of Ṁ for three different
abundances of helium of the accreted material Yacc . The initial mass of the WD is
0.81 M⊙ .
We assumed that elements heavier than oxygen would behave in the same way as
the carbon/oxygen mixture, because their ignition temperature is much larger than
every temperature in the WD reached during accretion, i.e. they would not be involved in nuclear reactions (or at least wouldn’t contribute significant energy). The
possibility of a hydrogen impurity was also ignored, because it would ignite near
the surface due to its low ignition temperature. This leads to numerical instabilities
49
3. Accretion mass
in the stellar evolution code.
The accretion mass for a 0.81 M⊙ WD for three different compositions are shown
in Fig. 3.72 . For realistic helium abundances (Y ≫ 0.9), the composition has no effect
on the accretion mass. Just for the theoretical case of a very large carbon/oxygen to
helium ratio (Y = 0.10), Macc increases significantly.
Changing the composition affects the nuclear energy generation rate ǫnuc , because
it is proportional to Y 3 (see Eq. (2.18)). Reducing the amount of helium by one order
of magnitude will reduce ǫnuc by a factor of 103 . Thus the heating of the ignition
zone until the runaway process starts lasts longer and more material can be accreted.
3.7. Discussion
The fate of a helium accreting WD depends mainly on the accretion rate. The possible behaviour after ignition as well as a comparison with prior studies of the accretion mass will be discussed in this section.
Expected behaviour after ignition
In the slow accretion regime a thick layer of helium can be accreted on top of the
WD until ignition starts. The helium burning region is highly degenerate (Fig. 3.4).
Therefore, conditions for the double-detonation scenario, which can lead to a SN
Ia, are fulfilled. Unfortunately, a thick helium layer prevents the production of
heavy elements, which are needed to explain observed spectra of SN Ia events (Hoeflich and Khokhlov 1996). The limits on the mass of the accreted helium layer that
can still produce a SN Ia event were recently investigated. Studies performed by
Woosley and Kasen (2011) suggested that only WDs that accrete less than 0.05 M⊙
helium can become a SN Ia. By comparing this limit with our results, a SN Ia explosion can only be realised with WDs that have an initial mass larger than 1.15 M⊙
(Fig. 3.1). This constraint limits the possible SN Ia events, because only a small
fraction of all WDs in the universe have masses larger than 1.15 M⊙ (Kepler et al.
2007).
2 The
simulation data from WDs with different initial masses and temperatures can be found in table
A.1 and table A.2.
50
3.7. Discussion
As explained by Piersanti et al. (2014), the models that accrete faster than 4 ×
10−8 M⊙ yr−1 undergo a series of shell flashes, where a fraction of the accreted matter is expelled due to the rapid shell expansion above the burning region. During
each flash the total mass of the WD grows until it reaches its Chandrasekhar mass.
Independently from its inital temperature structure, the WD temperature profile
becomes similar to those of our “hot” models after the first flash. Therefore, the differences of Macc between our “hot” and “cold” models only have to be taken into
account for the first flash. Depending on Ṁ, Macc of the “cold” models can be larger
than Macc of the “hot” models by up to a factor of 4. Thus, the number of flashes
needed to reach the Chandrasekhar mass will also be reduced.
Two of our accretion models evolved beyond the point of helium ignition. One
“hot” 0.94 M⊙ model produced five recurrent shell flashes before the evolution code
could not create a new converged model. Helium was accreted with an accretion
rate of 10−7 M⊙ yr−1 , which lies well within the range where Piersanti et al. (2014)
found their shell-flash models (between 4 × 10−8 M⊙ yr−1 and 4 × 10−7 M⊙ yr−1 ).
Another model accreted material steadily until it reached its Chandrasekhar mass.
The mass of the modelled star was 0.66 M⊙ and the accretion rate was 10−6 M⊙ yr−1 ,
which has a slightly larger Ṁ than the steady accretion regime found by Piersanti
et al. (2014) (between 2 × 10−7 M⊙ yr−1 and 9 × 10−7 M⊙ yr−1 ). However, the model
fits into the steady accretion regime calculated analytically by Iben and Tutukov
(1989) (between 6 × 10−7 M⊙ yr−1 and 6 × 10−6 M⊙ yr−1 ).
Comparison with prior studies
Fig. 3.8 shows the accretion mass of two of our “hot” models as a function of
Ṁ in comparison with results from Piersanti et al. (2014) and Woosley and Kasen
(2011). While our “hot” models have an initial maximum temperature of about
1.6 × 108 K, the models made by Piersanti et al. (2014) have a maximum temperature of 2.8 × 108 K and the ones made by Woosley and Kasen (2011) have temperatures of roughly (7 − 8) × 107 K. Because the initial temperature influences Macc ,
our results seem to coincide with those of the other authors. The models calculated
by Piersanti et al. (2014) have a lower accretion mass, because their models are initially hotter than our models. The models made by Woosley and Kasen (2011) have
51
10−1
MWD = 0.81
Pir = 0.81
MWD
Woo = 0.80
MWD
10−2
10−9
10−8
Macc [M⊙ ]
Macc [M⊙ ]
3. Accretion mass
10−1
MWD = 1.05
Pir = 1.02
MWD
Woo = 1.00
MWD
10−2
10−9
Ṁ [M⊙ yr−1 ]
10−8
Ṁ [M⊙ yr−1 ]
Figure 3.8. – Comparison of Macc with results from Piersanti et al. (2014) and Woosley and Kasen
Pir is the initial mass of the model from Piersanti et al., MWoo is the initial
(2011). MWD
WD
mass of the model from Woosley and Kasen.
a larger Macc , because their models are initially colder.
In the right-hand plot of Fig. 3.8 the accretion masses differ in the slow accretion
regime. This can be explained by the slightly different initial masses of the models.
The small deviations in the accretion masses in the slow accretion regime confirm
the results for Macc . The differences become larger in the fast and transition regime
which can be ascribed to different initial temperatures of the models. The initial
temperature is determined by the point when accretion starts, which is ultimately
determined by the initial separation, masses and compositions of the two stars of
the binary system.
52
4. Summary
In this study we investigated the amount of helium that has to be accreted on top of
a CO WD to ignite the accumulated helium layer. The analysis was performed for
different initial masses, accretion rates and initial temperatures of the CO WD. The
dependency of the accreted mass on those parameters can be concluded as:
• Larger accretion rates lead to lower accretion masses.
• For the “hot” models the accretion mass drops between 3 × 10−8 M⊙ yr−1 and
9 × 10−8 M⊙ yr−1 . For the “cold” models the accretion mass falls more steeply
after Ṁ ≈ 3 × 10−8 M⊙ yr−1 .
• The accretion mass decreases with larger initial masses.
• For accretion rates lower than 3 × 10−8 M⊙ yr−1 , the accretion mass is independent of the initial temperature of the WD. For larger accretion rates the
accretion mass increases with decreasing initial temperature.
• Pollutions in the form of heavier elements than helium in the accreted material
have no influence on the accretion mass.
While for the largest accretion rates that were applied in this study the environment of the burning region can be treated as an ideal gas, for lower accretion rates
the burning region becomes more and more degenerate. Therefore the explosion
becomes more unstable and “flash-like” for lower accretion rates.
As described in section 3.7, for the double-deonation scenario, Woosley and Kasen
(2011) constrained the maximum mass that can be accreted to produce a spectrum
that looks like a SN Ia event. This constraint can only be fulfilled for WDs with initial
masses larger than 1.15 M⊙ . Therefore, the SN Ia events from the double-detonation
channel seem to be a very rare event.
53
4. Summary
The simplifications that were made for this study included the non-rotating, spherical WD and the constant accretion rate. Adding rotation to the WD will decrease
the gravitational pressure exerted on the WD. Therefore more material can probably
be accreted until ignition occurs, which will lead to an increase in the accretion mass
for larger rotation velocities.
To study the accretion mass with more detail, the time dependency of the accretion rate has to be known. To accomplish this, the full binary evolution of possible
helium-accreting models including the right mass transfer rates has to be simulated.
During the evolution of a helium-accreting model, the binary stars undergo one or
more common-envelope phases. Due to the lack of knowledge of the mass loss efficiency in this phase, helium-accreting models with variable accretion rates can only
be simulated with large uncertainties at the moment.
With the accretion mass known for certain MCO - Ṁ combinations, 3-dimensional
simulations of the helium-ignition phase and the possible following explosion can
be set up. Setting up the simulation right before the explosion in comparison to
starting the simulation at the beginning of accretion has the advantage of saving
a lot of computational time. The light curves and spectra of the explosion event
calculated by those 3-dimensional simulations can be compared to observational
data of SN Ia. With those information a final statement can be made, which models
will end up as a supernova Ia.
In the future the different accretion regimes have to be explored in more detail.
For accretion rates lower than 4 × 10−8 M⊙ yr−1 , a thick helium layer will build up
on top of the CO core. The following shell explosion may lead to a supernova “.Ia”
or a supernova Ia, if a double detonation can be triggered and the accreted mass
is small enough. The accretion regime for rates larger than 4 × 10−8 M⊙ yr−1 can
be subdivided into three regimes: the flash, the steady accretion and the red giantlike regime. The exact boundaries of these regimes can be explored by following
the evolution of helium accreting WDs beyond the point of ignition. In the steady
accretion regime the accreted helium is directly burned into carbon/oxygen (Iben
and Tutukov 1989). Thus, the WD can reach its Chandrasekhar mass if the companion star is able to transfer enough helium. For larger accretion rates, the material
that cannot be converted to carbon/oxygen will be accumulated on top of the burning region. Due to the helium burning shell, the additional material will expand to
54
4. Summary
red giant-like dimensions (Nomoto 1982). Whether such systems can accumulate
enough mass or if they lose their mass in a common envelope phase has to be investigated in future studies. If the accretion rate is smaller than the accretion rates for
the steady accretion regime, a series of shell flashes will occur (Starrfield et al. 1972).
Here, a tiny helium layer will build up on top of the CO WD. When the temperature is sufficiently high, the helium layer will be ignited. The burning region will
expand and cool until it becomes cold enough that the burning ceases. Due to the
ongoing accretion, these layers are heated again and a new flash may occur. During
these recurrent flashes, a certain amount of matter will be ejected and lost from the
binary system. Whether or not such systems can reach their Chandrasekhar mass
crucially depends on the amount of lost matter, which also has to be investigated in
the future.
The outcome of the different accretion regimes can finally be used for setting up
population synthesis codes. The results of these codes can ultimately show if the
helium channel contributes to the total supernova Ia or supernova “.Ia” rate significantly.
55
56
A. Tables
Table A.1. – The accretion mass Macc and the ignition mass Mign of the “hot” models for accretion rates between 10−9 M⊙ yr−1 and 10−6 M⊙ yr−1.
MWD [M⊙ ]
0.656
0.809
Ṁ M⊙ yr−1
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
4.0 × 10−8
3.5 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
3.5 × 10−8
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
Mign [M⊙ ]
Macc [M⊙ ]
Y = 1.00
6.01 × 10−1
5.49 × 10−1
5.05 × 10−1
4.36 × 10−1
4.03 × 10−1
3.67 × 10−1
1.93 × 10−1
5.93 × 10−2
3.46 × 10−2
4.07 × 10−2
2.94 × 10−2
2.54 × 10−2
2.30 × 10−2
2.05 × 10−2
1.90 × 10−2
1.60 × 10−2
1.38 × 10−2
1.20 × 10−2
1.09 × 10−2
4.75 × 10−1
4.30 × 10−1
3.90 × 10−1
3.28 × 10−1
2.99 × 10−1
2.67 × 10−1
2.07 × 10−1
8.82 × 10−2
3.91 × 10−2
2.69 × 10−2
1.99 × 10−2
1.57 × 10−2
1.35 × 10−2
1.10 × 10−2
9.83 × 10−3
7.64 × 10−3
6.17 × 10−3
5.09 × 10−3
1.257
1.205
1.161
1.092
1.059
1.023
0.849
0.715
0.690
0.697
0.685
0.681
0.679
0.676
0.675
0.672
0.670
0.668
0.667
1.284
1.239
1.199
1.137
1.108
1.076
1.016
0.897
0.848
0.836
0.829
0.825
0.823
0.820
0.819
0.817
0.815
0.814
Mign [M⊙ ]
Macc [M⊙ ]
Y = 0.90
6.02 × 10−1
5.57 × 10−1
5.14 × 10−1
4.38 × 10−1
4.10 × 10−1
3.73 × 10−1
1.98 × 10−1
5.98 × 10−2
3.73 × 10−2
4.15 × 10−2
3.00 × 10−2
2.58 × 10−2
2.32 × 10−2
2.05 × 10−2
1.91 × 10−2
1.63 × 10−2
1.39 × 10−2
1.20 × 10−2
1.12 × 10−2
4.86 × 10−1
4.38 × 10−1
3.99 × 10−1
3.37 × 10−1
3.00 × 10−1
2.70 × 10−1
2.07 × 10−1
8.82 × 10−2
3.91 × 10−2
2.69 × 10−2
1.99 × 10−2
1.57 × 10−2
1.37 × 10−2
1.10 × 10−2
9.86 × 10−3
7.69 × 10−3
6.19 × 10−3
5.11 × 10−3
1.258
1.213
1.170
1.093
1.066
1.029
0.854
0.716
0.693
0.697
0.686
0.682
0.679
0.676
0.675
0.672
0.670
0.668
0.667
1.296
1.247
1.208
1.146
1.109
1.079
0.989
0.876
0.841
0.833
0.828
0.824
0.823
0.820
0.819
0.817
0.815
0.814
Mign [M⊙ ]
Macc [M⊙ ]
Y = 0.10
6.49 × 10−1
6.08 × 10−1
5.77 × 10−1
5.28 × 10−1
5.11 × 10−1
4.97 × 10−1
4.85 × 10−1
6.38 × 10−2
4.56 × 10−2
3.97 × 10−2
3.45 × 10−2
3.05 × 10−2
2.82 × 10−2
2.61 × 10−2
2.46 × 10−2
2.27 × 10−2
2.01 × 10−2
1.70 × 10−2
1.45 × 10−2
5.12 × 10−1
4.75 × 10−1
4.48 × 10−1
4.06 × 10−1
3.90 × 10−1
3.78 × 10−1
3.66 × 10−1
3.53 × 10−1
5.78 × 10−2
3.27 × 10−2
2.41 × 10−2
1.94 × 10−2
1.69 × 10−2
1.47 × 10−2
1.34 × 10−2
1.10 × 10−2
9.40 × 10−3
8.17 × 10−3
1.305
1.264
1.233
1.184
1.166
1.153
1.141
0.720
0.701
0.696
0.690
0.686
0.684
0.682
0.680
0.678
0.676
0.673
0.670
1.321
1.284
1.257
1.215
1.199
1.187
1.175
1.162
0.867
0.842
0.833
0.829
0.826
0.824
0.823
0.820
0.819
0.817
57
A. Tables
Table A.1. – The accretion mass Macc and the ignition mass Mign of the “hot” models for accretion rates between 10−9 M⊙ yr−1 and 10−6 M⊙ yr−1 .
MWD [M⊙ ]
0.809
0.942
1.053
1.211
58
Ṁ M⊙ yr−1
5.0 × 10−6
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
3.5 × 10−8
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
3.5 × 10−8
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
3.5 × 10−8
Mign [M⊙ ]
Macc [M⊙ ]
Y = 1.00
4.48 × 10−3
3.68 × 10−1
3.30 × 10−1
2.94 × 10−1
2.40 × 10−1
2.14 × 10−1
1.86 × 10−1
1.41 × 10−1
7.62 × 10−2
2.92 × 10−2
1.66 × 10−2
1.15 × 10−2
8.79 × 10−3
7.38 × 10−3
5.85 × 10−3
4.99 × 10−3
3.90 × 10−3
3.14 × 10−3
2.57 × 10−3
2.25 × 10−3
2.80 × 10−1
2.49 × 10−1
2.20 × 10−1
1.73 × 10−1
1.50 × 10−1
1.24 × 10−1
9.05 × 10−2
5.45 × 10−2
2.61 × 10−2
1.26 × 10−2
7.54 × 10−3
5.26 × 10−3
4.30 × 10−3
3.36 × 10−3
2.91 × 10−3
2.16 × 10−3
1.70 × 10−3
1.37 × 10−3
1.19 × 10−3
1.51 × 10−1
1.27 × 10−1
1.07 × 10−1
7.25 × 10−2
5.61 × 10−2
4.12 × 10−2
2.79 × 10−2
1.73 × 10−2
9.81 × 10−3
0.814
1.310
1.271
1.236
1.182
1.156
1.128
1.083
1.018
0.971
0.958
0.953
0.951
0.949
0.948
0.947
0.946
0.945
0.944
0.944
1.333
1.302
1.273
1.226
1.203
1.177
1.143
1.107
1.079
1.066
1.061
1.058
1.057
1.056
1.056
1.055
1.055
1.054
1.054
1.362
1.339
1.319
1.284
1.267
1.253
1.239
1.229
1.221
Mign [M⊙ ]
Macc [M⊙ ]
Y = 0.90
4.51 × 10−3
3.69 × 10−1
3.32 × 10−1
2.98 × 10−1
2.41 × 10−1
2.18 × 10−1
1.86 × 10−1
1.43 × 10−1
7.71 × 10−2
3.01 × 10−2
1.70 × 10−2
1.15 × 10−2
8.82 × 10−3
7.39 × 10−3
6.00 × 10−3
5.09 × 10−3
3.97 × 10−3
3.14 × 10−3
2.59 × 10−3
2.27 × 10−3
2.86 × 10−1
2.50 × 10−1
2.25 × 10−1
1.77 × 10−1
1.53 × 10−1
1.27 × 10−1
9.19 × 10−2
5.56 × 10−2
2.67 × 10−2
1.28 × 10−2
7.62 × 10−3
5.38 × 10−3
4.31 × 10−3
3.36 × 10−3
2.95 × 10−3
2.19 × 10−3
1.70 × 10−3
1.38 × 10−3
1.20 × 10−3
1.54 × 10−1
1.30 × 10−1
1.08 × 10−1
7.45 × 10−2
5.69 × 10−2
4.22 × 10−2
2.87 × 10−2
1.77 × 10−2
9.87 × 10−3
0.814
1.311
1.274
1.240
1.183
1.160
1.128
1.085
1.019
0.972
0.959
0.953
0.951
0.949
0.948
0.947
0.946
0.945
0.944
0.944
1.339
1.303
1.278
1.230
1.206
1.180
1.145
1.109
1.080
1.066
1.061
1.058
1.057
1.056
1.056
1.055
1.055
1.054
1.054
1.365
1.341
1.319
1.286
1.268
1.254
1.240
1.229
1.221
Mign [M⊙ ]
Macc [M⊙ ]
Y = 0.10
7.44 × 10−3
3.95 × 10−1
3.63 × 10−1
3.39 × 10−1
3.03 × 10−1
2.89 × 10−1
2.78 × 10−1
2.66 × 10−1
2.52 × 10−1
2.22 × 10−1
2.37 × 10−2
1.50 × 10−2
1.15 × 10−2
9.70 × 10−3
7.85 × 10−3
7.15 × 10−3
5.89 × 10−3
5.00 × 10−3
4.31 × 10−3
3.90 × 10−3
2.99 × 10−1
2.71 × 10−1
2.51 × 10−1
2.20 × 10−1
2.07 × 10−1
1.96 × 10−1
1.84 × 10−1
1.68 × 10−1
1.40 × 10−1
5.07 × 10−2
1.15 × 10−2
8.21 × 10−3
6.70 × 10−3
5.07 × 10−3
4.19 × 10−3
3.35 × 10−3
2.80 × 10−3
2.38 × 10−3
2.14 × 10−3
1.58 × 10−1
1.38 × 10−1
1.24 × 10−1
1.02 × 10−1
9.24 × 10−2
8.32 × 10−2
7.27 × 10−2
5.98 × 10−2
4.35 × 10−2
0.817
1.337
1.304
1.281
1.245
1.231
1.220
1.208
1.194
1.164
0.965
0.957
0.953
0.951
0.950
0.949
0.948
0.947
0.946
0.946
1.352
1.324
1.304
1.273
1.260
1.249
1.237
1.221
1.193
1.104
1.064
1.061
1.060
1.058
1.057
1.056
1.056
1.055
1.055
1.369
1.349
1.336
1.313
1.304
1.295
1.284
1.271
1.255
A. Tables
Table A.1. – The accretion mass Macc and the ignition mass Mign of the “hot” models for accretion rates between 10−9 M⊙ yr−1 and 10−6 M⊙ yr−1.
MWD [M⊙ ]
1.211
Ṁ M⊙ yr−1
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
Mign [M⊙ ]
Macc [M⊙ ]
Y = 1.00
4.07 × 10−3
2.88 × 10−3
2.21 × 10−3
1.83 × 10−3
1.36 × 10−3
1.09 × 10−3
7.60 × 10−4
5.80 × 10−4
4.60 × 10−4
3.90 × 10−4
1.215
1.214
1.214
1.213
1.213
1.212
1.212
1.212
1.212
1.212
Mign [M⊙ ]
Macc [M⊙ ]
Y = 0.90
4.12 × 10−3
2.95 × 10−3
2.27 × 10−3
1.87 × 10−3
1.39 × 10−3
1.11 × 10−3
7.60 × 10−4
5.80 × 10−4
4.60 × 10−4
3.90 × 10−4
1.215
1.214
1.214
1.213
1.213
1.212
1.212
1.212
1.212
1.212
Mign [M⊙ ]
Macc [M⊙ ]
Y = 0.10
1.94 × 10−2
6.98 × 10−3
4.49 × 10−3
3.45 × 10−3
2.41 × 10−3
1.86 × 10−3
1.24 × 10−3
1.00 × 10−3
8.30 × 10−4
7.30 × 10−4
1.231
1.218
1.216
1.215
1.214
1.213
1.213
1.212
1.212
1.212
59
A. Tables
Table A.2. – The accretion mass Macc and the ignition mass Mign of the “cold” models for accretion rates between 10−9 M⊙ yr−1 and 10−6 M⊙ yr−1 .
Models that could not be evolved until the point of ignition was reached, are indicated by a dash “—”.
MWD [M⊙ ]
0.656
0.809
0.942
60
Ṁ M⊙ yr−1
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
3.5 × 10−8
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
3.5 × 10−8
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
Y = 1.00
Mign [M⊙ ]
Macc [M⊙ ]
6.01 × 10−1
5.49 × 10−1
5.05 × 10−1
4.36 × 10−1
4.03 × 10−1
3.68 × 10−1
2.73 × 10−1
1.68 × 10−1
1.36 × 10−1
1.18 × 10−1
9.63 × 10−2
7.36 × 10−2
5.68 × 10−2
3.61 × 10−2
2.70 × 10−2
1.59 × 10−2
1.16 × 10−2
9.10 × 10−3
7.83 × 10−3
4.75 × 10−1
4.30 × 10−1
3.90 × 10−1
3.28 × 10−1
2.99 × 10−1
2.68 × 10−1
2.20 × 10−1
1.54 × 10−1
1.20 × 10−1
9.97 × 10−2
7.66 × 10−2
5.42 × 10−2
3.93 × 10−2
2.25 × 10−2
1.57 × 10−2
8.66 × 10−3
6.06 × 10−3
4.61 × 10−3
3.89 × 10−3
3.68 × 10−1
3.30 × 10−1
2.95 × 10−1
2.41 × 10−1
2.17 × 10−1
1.92 × 10−1
1.63 × 10−1
1.30 × 10−1
1.257
1.205
1.161
1.092
1.059
1.024
0.928
0.823
0.792
0.774
0.752
0.729
0.713
0.692
0.656
0.672
0.667
0.665
0.664
1.284
1.239
1.199
1.137
1.108
1.077
1.029
0.963
0.929
0.909
0.886
0.863
0.849
0.832
0.825
0.818
0.815
0.814
0.813
1.310
1.272
1.236
1.183
1.158
1.134
1.105
1.071
Y = 0.90
Mign [M⊙ ]
Macc [M⊙ ]
6.19 × 10−1
5.58 × 10−1
5.20 × 10−1
4.46 × 10−1
4.08 × 10−1
3.73 × 10−1
2.73 × 10−1
1.68 × 10−1
1.38 × 10−1
1.19 × 10−1
9.69 × 10−2
7.39 × 10−2
5.83 × 10−2
3.65 × 10−2
2.74 × 10−2
1.63 × 10−2
1.16 × 10−2
9.13 × 10−3
7.88 × 10−3
4.75 × 10−1
4.38 × 10−1
3.98 × 10−1
3.31 × 10−1
3.06 × 10−1
2.71 × 10−1
2.21 × 10−1
1.59 × 10−1
1.22 × 10−1
1.02 × 10−1
7.76 × 10−2
5.48 × 10−2
3.94 × 10−2
2.30 × 10−2
1.59 × 10−2
8.85 × 10−3
6.07 × 10−3
4.64 × 10−3
3.93 × 10−3
3.78 × 10−1
3.37 × 10−1
2.96 × 10−1
2.45 × 10−1
2.23 × 10−1
1.98 × 10−1
1.65 × 10−1
1.32 × 10−1
1.275
1.214
1.176
1.102
1.063
1.029
0.929
0.823
0.794
0.775
0.753
0.730
0.714
0.692
0.683
0.672
0.667
0.665
0.664
1.285
1.248
1.207
1.140
1.116
1.080
1.031
0.968
0.931
0.911
0.887
0.864
0.849
0.832
0.825
0.818
0.815
0.814
0.813
1.320
1.278
1.238
1.187
1.165
1.140
1.106
1.073
Y = 0.10
Mign [M⊙ ]
Macc [M⊙ ]
6.49 × 10−1
6.08 × 10−1
5.77 × 10−1
5.28 × 10−1
5.10 × 10−1
4.97 × 10−1
4.85 × 10−1
4.72 × 10−1
2.06 × 10−1
1.49 × 10−1
1.12 × 10−1
8.39 × 10−2
6.56 × 10−2
4.44 × 10−2
3.42 × 10−2
2.23 × 10−2
1.74 × 10−2
1.44 × 10−2
1.29 × 10−2
5.12 × 10−1
4.75 × 10−1
4.48 × 10−1
4.05 × 10−1
3.90 × 10−1
3.78 × 10−1
3.66 × 10−1
3.53 × 10−1
3.29 × 10−1
1.61 × 10−1
1.01 × 10−1
6.67 × 10−2
4.79 × 10−2
2.91 × 10−2
2.11 × 10−2
1.27 × 10−2
9.54 × 10−3
7.69 × 10−3
6.75 × 10−3
3.95 × 10−1
3.63 × 10−1
8.48 × 10−2
3.03 × 10−1
2.89 × 10−1
2.78 × 10−1
2.67 × 10−1
2.53 × 10−1
1.305
1.264
1.233
1.184
1.166
1.153
1.141
1.128
0.862
0.804
0.768
0.740
0.721
0.700
0.690
0.678
0.673
0.670
0.669
1.321
1.284
1.257
1.215
1.199
1.187
1.175
1.162
1.138
0.970
0.910
0.876
0.857
0.838
0.830
0.822
0.819
0.817
0.816
1.337
1.304
1.027
1.245
1.231
1.220
1.209
1.195
A. Tables
Table A.2. – The accretion mass Macc and the ignition mass Mign of the “cold” models for accretion rates between 10−9 M⊙ yr−1 and 10−6 M⊙ yr−1.
Models that could not be evolved until the point of ignition was reached, are indicated by a dash “—”.
MWD [M⊙ ]
0.942
1.053
1.211
Ṁ M⊙ yr−1
3.5 × 10−8
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
3.5 × 10−8
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
1.0 × 10−9
2.0 × 10−9
3.5 × 10−9
1.0 × 10−8
1.5 × 10−8
2.0 × 10−8
2.5 × 10−8
3.0 × 10−8
3.5 × 10−8
4.0 × 10−8
5.0 × 10−8
7.0 × 10−8
1.0 × 10−7
2.0 × 10−7
3.5 × 10−7
1.0 × 10−6
Y = 1.00
Mign [M⊙ ]
Macc [M⊙ ]
1.03 × 10−1
8.45 × 10−2
6.22 × 10−2
4.09 × 10−2
2.77 × 10−2
—
—
—
—
—
2.08 × 10−3
2.80 × 10−1
2.50 × 10−1
2.22 × 10−1
1.78 × 10−1
1.59 × 10−1
1.41 × 10−1
1.21 × 10−1
1.01 × 10−1
8.27 × 10−2
6.81 × 10−2
4.87 × 10−2
3.00 × 10−2
1.90 × 10−2
—
—
—
—
—
1.15 × 10−3
1.52 × 10−1
1.34 × 10−1
1.19 × 10−1
9.55 × 10−2
8.46 × 10−2
7.43 × 10−2
6.45 × 10−2
5.50 × 10−2
4.61 × 10−2
3.84 × 10−2
2.70 × 10−2
1.52 × 10−2
8.63 × 10−3
—
—
—
1.044
1.026
1.004
0.983
0.969
—
—
—
—
—
0.944
1.333
1.303
1.275
1.231
1.212
1.194
1.174
1.154
1.136
1.121
1.102
1.083
1.072
—
—
—
—
—
1.054
1.364
1.345
1.330
1.307
1.296
1.286
1.276
1.266
1.257
1.250
1.238
1.227
1.220
—
—
—
Y = 0.90
Mign [M⊙ ]
Macc [M⊙ ]
1.05 × 10−1
8.59 × 10−2
6.40 × 10−2
4.20 × 10−2
2.83 × 10−2
—
—
—
—
2.51 × 10−3
2.11 × 10−3
2.87 × 10−1
2.53 × 10−1
2.28 × 10−1
1.79 × 10−1
1.62 × 10−1
1.44 × 10−1
1.22 × 10−1
1.01 × 10−1
8.42 × 10−2
6.97 × 10−2
5.01 × 10−2
3.07 × 10−2
1.93 × 10−2
—
—
—
—
1.39 × 10−3
1.16 × 10−3
1.54 × 10−1
1.35 × 10−1
1.21 × 10−1
9.84 × 10−2
8.71 × 10−2
7.60 × 10−2
6.63 × 10−2
5.51 × 10−2
4.71 × 10−2
3.93 × 10−2
2.78 × 10−2
1.55 × 10−2
8.84 × 10−3
—
—
—
1.046
1.028
1.006
0.984
0.970
—
—
—
—
0.944
0.944
1.340
1.306
1.281
1.232
1.215
1.197
1.175
1.154
1.137
1.123
1.103
1.084
1.072
—
—
—
—
1.054
1.054
1.365
1.347
1.332
1.310
1.298
1.287
1.278
1.267
1.259
1.251
1.239
1.227
1.220
—
—
—
Y = 0.10
Mign [M⊙ ]
Macc [M⊙ ]
2.29 × 10−1
1.65 × 10−1
9.31 × 10−2
5.44 × 10−2
3.56 × 10−2
1.93 × 10−2
1.32 × 10−2
7.47 × 10−3
5.44 × 10−3
4.30 × 10−3
3.74 × 10−3
2.99 × 10−1
2.71 × 10−1
2.51 × 10−1
2.21 × 10−1
2.09 × 10−1
2.00 × 10−1
1.90 × 10−1
1.77 × 10−1
1.59 × 10−1
1.32 × 10−1
8.23 × 10−2
4.38 × 10−2
2.62 × 10−2
1.27 × 10−2
8.29 × 10−3
4.41 × 10−3
3.15 × 10−3
2.46 × 10−3
2.12 × 10−3
1.61 × 10−1
1.43 × 10−1
1.32 × 10−1
1.16 × 10−1
1.09 × 10−1
1.03 × 10−1
9.64 × 10−2
8.89 × 10−2
8.04 × 10−2
7.10 × 10−2
5.14 × 10−2
2.63 × 10−2
1.34 × 10−2
5.44 × 10−3
3.29 × 10−3
1.59 × 10−3
1.171
1.107
1.035
0.996
0.977
0.961
0.955
0.949
0.947
0.946
0.946
1.352
1.324
1.304
1.274
1.262
1.253
1.243
1.230
1.212
1.185
1.135
1.097
1.079
1.066
1.061
1.057
1.056
1.055
1.055
1.372
1.354
1.343
1.327
1.321
1.314
1.308
1.300
1.292
1.282
1.263
1.238
1.225
1.217
1.215
1.213
61
A. Tables
Table A.2. – The accretion mass Macc and the ignition mass Mign of the “cold” models for accretion rates between 10−9 M⊙ yr−1 and 10−6 M⊙ yr−1 .
Models that could not be evolved until the point of ignition was reached, are indicated by a dash “—”.
MWD [M⊙ ]
1.211
62
Ṁ M⊙ yr−1
2.0 × 10−6
3.5 × 10−6
5.0 × 10−6
Y = 1.00
Mign [M⊙ ]
Macc [M⊙ ]
1.00 × 10−5
4.60 × 10−4
1.00 × 10−5
1.211
1.212
1.211
Y = 0.90
Mign [M⊙ ]
Macc [M⊙ ]
6.30 × 10−4
4.68 × 10−4
1.00 × 10−5
1.212
1.212
1.211
Y = 0.10
Mign [M⊙ ]
Macc [M⊙ ]
1.11 × 10−3
8.50 × 10−4
7.30 × 10−4
1.212
1.212
1.212
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I hereby declare that the work presented here was formulated by myself and that
no sources or tools other than those cited were used.
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