Final Report for CSUMS 2010

Kevin Jumper

August 4, 2010

Contents

1 Abstract 2

2 Background 2

3 Method 33.1 Using bubblerise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Varying the Initial Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 Progenitor Mass Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.5 Helmholtz’s Equations of State and Pressure Root-Finding . . . . . . . . 53.6 Including Isobaric Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 63.7 Pre-Expansion and the Virial Theorem . . . . . . . . . . . . . . . . . . . . 6

4 Data and Analysis 74.1 Varying the Initial Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.1 Figure 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1.2 Figure 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Progenitor Mass Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3.1 Figure 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3.2 Figure 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.4 Pressure Root-Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Including Isobaric Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.5.1 Figure 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.5.2 Figure 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5.3 Figure 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5.4 Figure 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.6 Pre-Expansion and the Virial Theorem . . . . . . . . . . . . . . . . . . . . 17

5 Conclusion 19

6 Appendix: bubblerise Code 19

1

7 Acknowledgements 34

8 References 34

1 Abstract

During CSUMS, I used the bubblerise.F program written by Dr. Robert Fisher to modelthe deflagration phase of a single-degenerate type Ia supernova. I gradually modified theprogram to include additional physics, such as flame turbulence and isobaric expansionin nuclear statistical equilibrium (NSE). Dr. Fisher also asked me to change the fit forour model’s Atwood numbers to account for more recent data. In the final phase ofmy research, I attempted to determine a governing equation for pre-expansion of thesupernova progenitor based on the virial theorem.

2 Background

Supernovae are the detonations of stars at the end of their lifespans, and are classifiedinto three types: Ia, Ib, and II (Hoyle & Fowler, 1960). Of these, type Ia is characterizedby a lack of hydrogen and the production of silicon ions in its spectra (Wheeler & Hark-ness, 1990). Type Ia supernovae are of interest to astrophysicists because they appearto have nearly uniform luminosities, suggesting that they could be used as ”standardcandles” that would enable the gauging of distances in space, among other measurements(Branch & Tammann, 1992). However, their use in this capcity is hindered because themechanisms by which they occur are not fully understood (Phillips, 1993).

It is generally thought that white dwarves are the most likely candidates for progen-itor stars (Hoyle & Fowler, 1960). However, scientists have not reached a consensus onwhat mechanism causes these stars to explode. Two proposed channels in which type Iasupernovae could occur are single-degenerate and double-degenerate systems.

In the single-degenerate channel, a white dwarf is in a binary system with anotherstar, typically on the main sequence. The white dwarf is electron-degenerate, whichmeans that the matter is so dense that all of the lower electron energy states becomeoccupied (Whelan & Iben, 1973). In accordance with the Pauli Exclusion Principle, theremaining electrons exert great pressure on their surroundings resisting further collapse,as they are not permitted to share quantum numbers with another electron in the sameatom. Typically, the white dwarf will accrete matter off of its companion for some timeprior to supernova until it approaches the Chandrasehkar mass, about 1.4 solar masses(Hillebrandt & Niemeyer, 2000). Just before this point, it will sufficiently increase thestar’s temperature to ignite a nuclear flame that will lead to a supernova (Hoyle &Fowler, 1960).

In contrast, the double-degenerate channel involves the merger of a pair of degen-erate white-dwarf stars. Typically, each star has a mass significantly less than theChandrasehkar mass (Webbink, 1984).

2

This paper will henceforth consider the single-degenerate channel for type Ia su-pernovae. Still, there are many models even for this case. We have focused on thegravitationally-confined detonation (GCD) model, first proposed by Plewa et. al (2004).Following the ignition of the flame, it will buoyantly rise through the star, fusing itscarbon and oxygen nuclei (the white dwarf will tend to lack lighter elements exceptfor those it has accreted, as it will have fused them long ago). The burning of fuel isknown as deflagration, and initially occurs with subsonic speeds (Buchler & Mazurek,1975). Eventually, the flame will move through the surface of the star, an event known asbreakout, and eject the fusion by-products, which I will refer to as nuclear ash, into theatmosphere of the star. This ash is gravitationally pulled into a collision with the far sideof the star, which destabilizes it, causing a supersonic detonation front to gravitationallyunbind and destroy the star (Plewa et al, 2004).

The progenitor may also undergo pre-expansion, an increase in its radius prior todetonation, which some think may bring about the conditions necessary for certainelements in the supernova’s spectra to form, such as sillicon (Byckhov & Liberman,1995).

While he had more advanced simulations for the overall GCD model, Dr. Fisherwanted to gain more insight into the deflagration phase. Therefore, he wrote a simpleFORTRAN 90 program, bubblerise (Fisher, 2006).

3 Method

3.1 Using bubblerise

bubblerise is a semi-analytic model that reads in information from a .dat file aboutconditions at given points throughout a white dwarf of a given mass and interpolates tofind the values it needs at each timestep. bubblerise accepts interactive input from theuser, two three-digit numbers given on the command line. The first is the inital radius ofthe spherical flame bubble in kilometers. The second number is the initial offset radiusof the bubble, the distance from the center of the star to the center of the flame bubble,in kilometers. While the program accepts inputs and gives outputs in kilometers, itsinternal calculations are done in cgs units, centimeters, grams, and seconds.

The program can be configured to provide a variety of outputs, but its defaults are thecurrent offset radius and the fractional mass of the star burned so far. It will continueto calculate the previous values for each timestep until either a specified number ofiterations have occured, the bubble breaks out of the star, or the program’s logic detectscertain unphysical conditions.

3.2 Varying the Initial Radii

Dr. Fisher had me begin by determining how the fractional burned mass changed as Ivaried one input from 10 km to 100 km in 10 km intervals and held the other constant.This was to give me practice working with a LINUX operating system and input from

3

the command line. After piping the outputs into files, I learned how to make graphs onMATLAB to display the data.

3.3 Progenitor Mass Problem

After this, Dr. Fisher asked me to determine how varying the initial mass of the pro-genitor affected the fractional burnt mass for the aforementioned conditions. This wasslightly more complicated, as the arrays of bubblerise were configured for arrays with464 lines, but the the .dat files for other conditions had different numbers, requiring meto manually change the arrays.

3.4 Turbulence

Dr. Fisher next decided that we needed an area correction factor to account for the effectsof turbulence as the bubble rose. There is a flame-polishing scale, λc, below which thesimulation will ”smooth out” any turbulent structures that would form, significantlydecreasing the area exposed to burning.

λc =(Sl)

2

Ag

Sl is the laminar rise speed, the rate at which the flame bubble ascends in the absenceof turbulence. A is the Atwood number, a dimensionless quanity related to the differencein densities of two fluids, and g is the local value of gravitational acceleration.

The Atwood number is defined as follows:

A =ρ2 − ρ1ρ1 + ρ2

Where:ρ2 > ρ1

The following was used in bubblerise as an area correction factor and was based onAlexei Khokhlov’s power law (1995):

areafac = (rbλmin

)1/2

rb is the radius of the bubble and:

λmin =2π(Sl)

2

Ag

Scientists are not entirely certain of what the correct value for λ is, and many modelsuse multiples of the λc shown above, as did Khokolov’s.

4

3.5 Helmholtz’s Equations of State and Pressure Root-Finding

Following the successful inclusion of turbulence into the bubblerise program, Dr. Fisherwanted me to consider the expansion of the flame bubble through adiabatic processes. Iwas told to use Helmholtz’s equations of state as shown below:

Ptot = Prad + Pion + Pele + Ppos + Pcoul

εtot = εrad + εion + εele + εpos + εcoul

As Paul Rich of the University of Chicago wrote,“the subscripts “rad,” “ion,” “ele,”“pos,” and “coul” represent the contributions from radiation, nuclei, electrons, positrons,and corrections for Coulomb effects, respectively.” (2009)

Dr. Fisher also said that the pressure inside the bubble,P , should equal the pressureoutside the bubble,Po. This was actually in error for an adiabatic process, but consistentwith an isobaric process.

The pressure inside the bubble is a function of the fuel density, ρfuel, backgroundtemperature of the star, T , and the composition of the star, xi. Thus we obtained thefollowing equation:

P (ρfuel, T, xi) − Po = 0

Dr. Fisher told me that in an electron-degenerate star the pressure is nearly inde-pendent of temperature, so the T argument is less important than the others.

Given the above equation, Dr. Fisher asked me to write a root-finder using thebisection method so that we could eventually solve for pressure. In this method onesurrounds the root with a value that is smaller than it and another value that is larger,a procedure known as bracketing. The subtleties of bracketing are discussed in Presset al., 2001. Next, one bisects the bracketed interval, and determines if the sign ofthe function’s value changes on a given half. If this condition is met, the root mustbe somewhere on that half of the bisected interval. Continuing the bisection processenables the value of the root to be estimated as accurately as we desire but becomescomputationally expensive.

As it would be redundant to have me write a bisection method code when manyalready exist, Dr. Fisher asked me to download the appropriate function from NumericalRecipes and to integrate it into a simple program, bisection (Press et al., 2001). As theNumerical Recipes function lacked inputs and printed output, this involved writing codeto pass arguments from the main program to the function. To test if it worked, he toldme to use a simple polynomial function and to find one of the roots. I selected y = x2−5.

When this was completed, Dr. Fisher had me modify F. X. Timmes’ program forHelmholtz’s equations of state to reflect the conditions in the white dwarf we werestudying. I set the background temperature of the star to 5.0 ∗ 107 Kelvin, the centraldensity to 2.0 ∗ 109 grams per cubic centimeter, and the composition to fifty percentoxygen and fifty percent carbon by mass. Consequently, we were able to obtain a valuefor the total pressure and input it into my bisection method code as an additional checkon its accuracy.

5

In spite of our progress, Dr. Fisher soon realized that we had an incorrect approachand needed a new method.

3.6 Including Isobaric Expansion

Dr. Fisher suggested that it may be possible to instead interpolate the density of thebackground as a function of pressure if we used data that was known to be for isobaricconditions. He directed me to the work of Alan Calder et al., who had published a datatable for a flame bubble isobaric to its surroundings in nuclear statistical equilibrium(2007). I took a linear regression of the data and then a logarithmic-logarithmic regres-sion at Dr. Fisher’s urging. While I was doing this, I noticed that the Atwood numbersour models used disagreed with those computed from Calder’s data. Dr. Fisher said hehad based the original fit on data from a preprint of Calder’s paper, and that I shoulduse the updated information to find a better fit for the Atwood numbers as a functionof fuel density.

3.7 Pre-Expansion and the Virial Theorem

The virial theorem is an equilibrium equation that can be used to describe gravitationally-bound gases, such as stars. In certain conditions, the theorem may be stated as follows:

1

2

d2I

dt2= Ω + 2K

d2Idt2

is the second time derivative of the moment of inertia of the system. Ω representsgravitational binding potential energy and K is kinetic energy. If one wishes, they mayconsider the changes in each of these quantities.

d2Idt2

may also be expanded to:

2I

R[d2R

dt2+

1

R(dR

dt)2]

Where I is the moment of inertia and R is the radius of the star.Therefore:

I

R[d2R

dt2+

1

R(dR

dt)2] = Ω + 2K

I is expanded as:I = f2MR2

Where f2 is a constant that describes the distribution of mass in the star and M isthe mass of the star.

Ω becomes:

Ω = −f1GM2

R

f1 is a constant. Note that Mm may be used in the numerator instead of M2 is oneis considering the binding energy between two bodies of differing masses.

6

K is expanded as:

K =f22M(

dR

dt

2

)

Therefore:

f2MR[d2R

dt2+

1

R(dR

dt)2] = −f1GM

2

R+ f2M(

dR

dt

2

)

Initially, Dr. Fisher had me use first principles to estimate the orders-of-magnitudeof the energy terms involved so that we could gain insight into the mechanism of pre-expansion. However, we first did this for a system in which pre-expansion did not occur,so only the energies associated with the bubble were changing; the background could beignored.

After I did this, Dr. Fisher asked to me compute how much additional kinetic energywas added from nuclear burning as the bubble rose. Again, I consulted Calder’s isobarictable, and was able to calculate the thermal energy generated, which is porportional tokinetic energy by the following equation (2007):

2K = 3(γ − 1)U

U is, in this case, the thermal energy generated. γ is generally defined as follows:

γ =Cp

Cv

This equation is also valid for determining the initial internal kinetic energy of thestar.

Cp is the specific heat of the gas in isobaric conditions, while Cv is the specific heatin isochoric, or constant-volume conditions. We took γ to be 4

3 , which corresponds towhen the gas is degenerate.

After determining which terms were negligible, we set up the virial theorem for pre-expansion and began to re-arrange it in an attempt to determine a governing equation.Once we had this, we could predict the orders-of-magnitude of the energy involvedwith pre-expansion, and then refine the governing equations accordingly. We assumedhomologous expansion, in which the average density of the star would change accordingto a given function throughout the process.

4 Data and Analysis

4.1 Varying the Initial Radii

Observing Figure 1 below gave me useful insights into the general behavior of the de-flagration phase and helped highlight the limitations of bubblerise in its initial form. Anotable characteristic of the graph is the S-shape of the curves. This means that at masswas burned at an increasing rate at low offset radii, but visually the curves appear tohave inflection points at an offset of approximately 600 kilometers. After this point theburning rate decreases dramatically and starts to approach zero. bubblerise did initially

7

account for some expansion of the bubble, so this can account for the initially increasingburn rate, as more surface area is exposed to the flame. However, the star becomes lessdense with greater offset radius and the bubble is able to move faster due to accelerationfrom buoyant forces. Eventually these factors overcome the expansion of the bubble,resulting in the graph’s distinctive shape.

I also observed that while each curve had a final offset radius that was between about1500 and 1600 kilometers, each was slightly different than the those of the others andvaried with the initial offset radius. At breakout, assuming a star with a static radius:

RWD = Rb +Ro

RWD is the initial radius of the white dwarf, Rb is bubble radius, and Ro is offsetradius. In Figure 1, the initial radius of the bubble is 10 kilometers in all cases. If theinitial offset radius is increased, then the bubble will have time to expand, resulting ina lesser final bubble radius. Therefore, according to the above equation, the final offsetradius will increase. Indeed, this is the trend indicated by the data. Additionally, onecan determine that the bubble will reach a radius of between 900 - 1000 kilometers.

From these points, it follows and is supported by the graph that a bubble starting ata greater initial offset radius will tend to burn a lower fraction of the star’s mass. Thereis both less distance for the bubble to burn through and it has less surface area to burnwith.

Figure 2 shows much the same picture as Figure 1 with regard to the general shape ofthe graph, but differs in that it has a constant initial offset radius of 100 kilometers anda varying intitial bubble radius. Although it cannot be directly determined by lookingat the graph, the curves with greater fractional burnt masses have greater initial bubbleradii, following similar arguments to those made in the previous paragraphs. These samearguments explain the variations in the final offset radii.

One curve on each of Figures 1 and 2 is in common, that corresponding to an initialbubble radius of 10 kilometers and an offset of 100 km. In each case, this is the lowestcurve on the graph. The curves in Figure 2 are able to attain a greater fractional burntmass than those in Figure 1 because their greater bubble radii compensate for theirgenerally greater initial offset radii.

Given previous studies, Figure 2 perhaps provides the better description of deflagra-tion, as it seems that 100 kilometers is the most lilkely initial offset radius (Wunsch &Woosley, 2004).

8

4.1.1 Figure 1

9

4.1.2 Figure 2

Errata: In the titles of the graphs above, it should say ”Initial Bubble Radius” insteadof ”Initial White Dwarf Radius”. Additionally, the units of ”Bubble Offset Radius” arein kilometers.

4.2 Progenitor Mass Problem

I then repeated the previous two tests, except for progenitor stars with 1.370, 1.375.1.380, and 1.385 solar masses. To avoid unnecessary repetition, I shall summarize theresults below and omit graphical representations of the data. With each increase in theprogenitor’s mass, the fractional burn masses increased, but the changes were very small.To provide an example, for the curve in which the initial bubble radius was 10 kilometersand the initial offset was 100 kilometers, the fractional burn for 1.360 solar masses was4.711%, while it 4.753% for 1.385 solar masses. Likewise, when both the intitial offsetand bubble radius were 100 kilometers, the burnt mass varied from 11.29% to 11.32%,corresponding to 1.360 and 1.385 solar masses respectively.

Dr. Fisher said that studies had indicated that increasing the mass of the progenitorshould have a greater effect on the fractional mass burnt. Additionally, he said thatthe fractional burnt masses predicted by bubblerise were lower than they should be.Therefore, it was his opinion that we should include more physics to refine the predictionsmade by the program.

4.3 Turbulence

As expected, the additional of turbulence increased the fractional burnt mass of theprogenitor, as shown in Figures 3 and 4 for the case of a progenitor with 1.360 solar

10

masses. For the examples given in the previous section, ”Progenitor Mass Problem”, thefractional burnt mass increased by a factor of approximately 1.58. However, as before,only minor variation was seen between progenitors of different masses.

Dr. Fisher had initially expected turbulence to have a much greater effect, so he askedme to estimate the value of the area enhancment factor to see if this was consistentwith our result. Over the offset radius, the area enhancement factor increased at anincreasing rate, approaching factors with an order-of-magnitude of 103. However, wesoon remembered that the star’s density also decreased by several orders of magnitudebefore breakout, making the factor less effective.

4.3.1 Figure 3

Figure 3 is the turbulent counterpart to Figure 1.

11

4.3.2 Figure 4

Figure 4 is the turbulent counterpart to Figure 2.

4.4 Pressure Root-Finding

While the root finder found the correct bracketed zeros to arbitrary degrees of accuracy,this method was inefficient compared to an interpolation routine, especially as one soughtmore decimal places of accuracy. Therefore, we adopted the latter method, sacrificingsome accuracy for speed and ease of use.

4.5 Including Isobaric Expansion

Around the time that we switched to an interpolation routine, we realized that we shouldnot be modeling the flame bubble in terms of adiabatic processes. In order for a processto be considered adiabatic, it must not undergo any changes in heat. However, this ispatently untrue when one considers that nuclear burning along the surface of the bubblereleases heat energy. The situation is better described by isobaric conditions, in whichthe pressure inside and outside of the bubble remain equal to and in equilibrium witheach other.

An analysis of Calder’s isobaric NSE data revealed a very strong linear correlationbetween the logarithim of the background pressure and the logarithim of the fuel density,as shown in Figure 5 (2007). Thus, it would be viable to use it to interpolate the densitiesbetween data points in the program.

Correcting the program’s fit for the Atwood number as a function of the logarithmof fuel density was more complicated. At first I tried a linear-logarithmic fit, which wasfairly strong with a correlation coefficient of 0.94. However, when the logarithm of the

12

fuel density in cgs units had a magnitude of 10 or greater, Calder’s data lost a decimalplace of precision in order to preserve the number of significant digits. However, thisresulted in several outliers that were less reliable than the rest of the data and weakenedthe correlation. Consequently, I only considered data points when the magnitude ofthe logarithim of the fuel density was less than 10. This strengthened the correlationcoefficient of the linear-logarithmic fit to about 0.97.

However, it became apparent that a linear equation would no longer satisfactorilydescribe the entirety of the graph. Therefore, Dr. Fisher recommended that I attempt acubic fit on the linear-logarithmic axes. This time, the equation almost perfectly fit thedata. Refer to Figure 6 to see the graph and the cubic fit for the data. Furthermore, werealized that this fit with a few minor changes to the code would be sufficient to enforceisobaric expansion.

Figures 7 and 8 show the effects of the bubble’s isobaric expansion. As expected,the fractional burnt mass increased, although vastly more than we expected, with about25 − 50% of the star being burned for the conditions we tested. However, this wasconsistent with the significantly increased radii of the bubbles, which grew to about1500 - 1600 kilometers from 900 -1000 kilometers without isobaric expansion.

Another interesting feature of the graphs was that instead of the burning rate gradu-ally decreasing as the bubble approached breakout, it seemed to be increasing across thedomain of the offset radii. I hypothesize that the reason for this is that the decreasingdensity and acceleration of the bubble are no longer able to overtake the effects of thebubble’s expansion when one considers isobaric processes.

Overall, the graphs seem to obey the same trends that have been previously observedin my deflagration models, although the fractional burned masses may now be too high.

13

4.5.1 Figure 5

The horizontal axis, x, is the log of pressure. The pressure is expressed in units ofdynes per centimeter squared. The vertical axis, y, is the log of fuel density. The unitsfor density are grams per cubic centimeter. The equation of the logarithmic-logarithmicregression is y = 0.758464x - 11.246864. The correlation coefficient is 0.999935, and R2

was 0.99987. The data points were taken from Alan Calder’s isobaric NSE table (2007).

14

4.5.2 Figure 6

The horizontal axis, x, is the log of fuel density in cgs units. The vertical axis, y, isthe Atwood number. The linear-logarithmic fit for the data is described by:

y = 0.008890x3 − 0.1870507x2 + 1.110179x− 1.226925

The data points were taken from Alan Calder’s isobaric NSE table (2007).

15

4.5.3 Figure 7

The x-axis is the offset radius in kilometers, and is shown on a domain from 0 to1000. The y-axis is the fractional burnt mass of the star and is shown on a range of 0 to0.5. The initial bubble radius was held at 10 kilometers, and the progenitor had 1.360solar masses.

16

4.5.4 Figure 8

The x-axis is the offset radius in kilometers, and is shown on a domain from 0 to1000. The y-axis is the fractional burnt mass of the star and is shown on a range of 0 to0.5. The initial offset radius was held at 100 kilometers, and the progenitor had 1.360solar masses.

4.6 Pre-Expansion and the Virial Theorem

Pre-expansion could affect the total burned mass in two ways. As the radius of thestar increases, it takes more time for the bubble to reach breakout, increasing the massburned. However, as the star expands, the density would decrease, reducing the frac-tional mass burned. We would need to solve the virial theorem to determine which factorhad the greater effect.

I calculated the gravitational binding energy of the bubble at breakout to be about−1051 ergs. The kinetic energy of the bubble’s motion had a much smaller magnitude,at 1049 ergs, and nuclear burning could only contribute slightly less than 1050 ergs.

17

Therefore, gravitational binding energy seemed to be the dominant term in the virialtheorem for a type Ia progenitor, meaning that if pre-expansion were to occur, it wouldbe driven by the change in the binding energy. Likewise, we considered the kineticenergy of the bubble and nuclear burning to be negligible in our subsequent equations.However, the governing equations for pre-expansion would still need to account for thekinetic enegy of the star’s expansion, as well as homologous changes in backgroundconditions. Additionally, I needed to re-express the terms for a polytropic model, a wayof describing the hydrostatics of a star governed by the following equation:

P = kρ1+1n

P is pressure and ρ is density. k and n are both constants. The later is a particularlyimportant quantity known as the polytropic index. Depending on conditions, n mayrange from 1.5 to 3 in white dwarfs (Bowers and Deeming 1984).

In general, the binding energy for a polytropic model is:

Ω = − 3

5 − n

GM2

R

My first attempt to write a governing equation for pre-expansion was unsuccessful.When I expanded the second moment of inertia into differential expressions for gravita-tional binding energy and kinetic energy, the kinetic energy terms on both sides of thevirial theorem seemed to immediately cancel out. Rearranging this resulted in formulapredicting the gravitational collapse of the star.

However, Dr. Fisher realized that I had forgot to consider the internal kinetic energyof the star, which is associated with the microscopic movements of its atoms. Theprescence of this energy would enable the star to avoid an inevitable collapse. At first,I attempted to derive one using the virial equilibrium equation,

K = −Ω

2,

Dr. Fisher noted that the star is not strictly in equilibrium; rather the rising of thebubble and the pre-expansion each affect small changes from equilibrium. Therefore, myapproach could work only in very limited circumstances.

However, he reminded me of the correct formula for internal energy, discussed inthe Methods section, leading me to produce the following governing equation for pre-expansion:

d2R

dt2=

−3GM5−n [MR + m

r ]

f2MR+

3(γ − 1)U

f2MR

m and r are the mass of the bubble and the offset radius. U still needs to be expressedin terms of M and R.

18

5 Conclusion

bubblerise is a simple deflagration model compared to higher-dimensional simulations,but it is nonethless instructive in the basic physics of type Ia supernovae. With theadditional physics added to it during the summer, it has become capable of modeling avariety of phenomena that occur during deflagration, greatly enhancing its usefulness.While more work still needs to be done to refine the code and the governing equationfor pre-expansion, they may someday be applied to more advanced simulations, allowingnew insights into the standard candles of the universe.

6 Appendix: bubblerise Code

The following is the most recent version version of my code, virial bubble.F90, whichwas originally authored by Dr. Fisher as bubblerise.F.

program b u b b l e r i s e! /////////////////////////////////////////////////////////////////////! //! // Compute the r i s e o f a laminar bubb l e near the cen ter o f a Ia ,! // assuming background s t a r i s a n = 3 po l y t r ope .! //! // Program computes laminar and t u r b u l e n t f lame speeds us ing the! // subg r i d model p r e s c r i b e d in Khokhlov (1995) . The Atwood number! // i s de r i v ed from a l i n e a r f i t to Del ta rho / rho o f the f lame! // ashes from the en e r g e t i c s paper ( Calder , per sona l comm. ) . The! // den s i t y p r o f i l e and the g r a v i t a t i o n a l a c c e l e r a t i o n are determined! // from a f i t to an n = 3 po l y t r ope us ing the ana l y t i c approximation! // o f Liu (1996) .! //! // Two parameters comp le t e l y s p e c i f y the po l y t r ope model −− c en t r a l! // den s i t y and t o t a l mass . Both may be s e t be low .! //! /////////////////////////////////////////////////////////////////////

implicit none

#d e f i n e COMMANDLINE#d e f i n e CHARACTERLEN 3

! Def ine the c en t r a l d en s i t y and whi te dwarf mass , c e l l s i z e , in! terms o f g/cmˆ3 , s o l a r masses , and km, r e s p e c t i v e l y .

! . . . 1 .36 So lar Mass#d e f i n e RHOC 10∗∗ (9 . 34 )

19

! . . . 1 .385 So lar Mass!#de f i n e RHOC 10∗∗(9.65321251377534)

! cgs

#d e f i n e WDMASS 1.360#d e f i n e MSOLAR 1.987 e33#d e f i n e WDRADIUS 2 .5 e8#d e f i n e COURANT .001

real ∗8 L , rc , dens i ty , atwood , compute density ,& compute atwood , g , compute g , laminarspeed ,& compute laminarspeed , turbulentspeed , r i s e spe ed ,& mass , compute mass , dt , t , position , dr , dm,& pi , maxradius , bubbleradius , C,& flamespeed , compute turbulentspeed , area fac ,& chi , lambda min , area , vmax , lambda min2 ,& bubblevolume , dens i tyash , buoyancy , drag ,& compute deltaq , de ltaq , de l taqo ld , Eg , dE ,& addedmassterm , posar r (464) , densarr (464) ,& enclosed mass , check , checkrea l , f1 , f2 ,& i n e r t i a , d i n e r t i a , d2 in e r t i a , dv , d2r ,& dt2 , R, E g i n i t i a l

integer i , convertchar

#i f d e f COMMANDLINEcharacter∗CHARACTERLEN arginteger nargs , i a r g c

#endif

bubblerad ius = 16 . ∗ 1 . e5position = 40 . ∗ 1 . e5

#i f d e f COMMANDLINE! . . . Read in arguments from command l i n e .

nargs = i a r g c ( )

i f ( ( nargs . eq . 0) . or . ( nargs . eq . 1) ) then

print ∗ , ’=====================================================’

20

print ∗ , ’BUBBLERISE: ’print ∗ , ’Type Ia Flame Bubble Evolut ionary Model ’print ∗ , ’ −− wr i t t en Robert Fisher , 12/8/06 ’print ∗ , ’=====================================================’print ∗ , ’ usage : b u b b l e r i s e i n i t r a d i u s o f f s e t ’print ∗print ∗ , ’Where ’print ∗ , ’ o f f s e t = bubble i n i t i a l r a d i a l p o s i t i o n (km) ’print ∗ , ’ i n i t r a d i u s = bubble i n i t i a l r ad iu s (km) ’print ∗ , ’ and both o f f s e t and i n i t r a d i u s are 3−dig i n t e g e r s ’print ∗ , ’ −− 1 km i s entered as 001 , e t c . ’stop

else

ca l l getarg (1 , arg )bubblerad ius = convertchar ( arg ) ∗ 1 . e5

ca l l getarg (2 , arg )position = convertchar ( arg ) ∗ 1 . e5

end i f#endif

!−−−−−− INITIALIZATION −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

! . . . Compute burnt mass .! . . . Standard co l d wd model rad ius = 2195 km

pi = 4 . ∗ atan ( 1 . )t = 0 .dt = 1 . e−3

! . . = (3 . ∗ mass / (4 .∗ p i ∗ dens i t ya sh ) )∗∗ ( 1 . / 3 . )! Read in co ld whi te dwarf model .

ca l l read coldwd ( posarr , densarr )

! . . . I n i t i a l bub b l e s i z e and pos i t i on , and bubb l e p o s i t i o n v e l o c i t y .

bubblevolume = 4 . ∗ pi / 3 . ∗ bubblerad ius ∗∗3 .

21

r i s e s p e e d = 0 .f lamespeed = 0 .

maxradius = WDRADIUS

! d en s i t y = compute dens i ty ( po s i t i on , posarr , densarr )dens i ty = compute dens ity ( position )de l t aqo ld = compute deltaq ( dens i ty )dens i tyash = dens i ty ∗ ( 1 . − atwood ) / ( 1 . + atwood )mass = dens i tyash ∗ bubblevolume

! i n i t i a l g r a v i t a t i o n a l b ind ing energy

Eg = −3. / 2 . ∗ 6 .678 e−8 ∗ (WDMASS ∗ MSOLAR) /& maxradius

Eg = Eg ∗ (WDMASS ∗ MSOLAR)E g i n i t i a l = Eg

!−−−−−−−−DO LOOP −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

do i = 1 , 150000

t = t + dt

i f ( dt . l t . 0 . ) thenprint ∗ , ’ERROR : dt < 0 . ’stop

endif

! d en s i t y = compute dens i ty ( po s i t i on , posarr , densarr )dens i ty = compute dens ity ( position )enc losed mass = compute mass (position , posarr , densarr )enc losed mass = compute mass ( position )

atwood = compute atwood ( dens i ty )dens i tyash = dens i ty ∗ ( 1 . − atwood ) / ( 1 . + atwood )g = compute g (position , posarr , densarr )laminarspeed = compute laminarspeed ( dens i ty )

! t u r bu l en t s p e ed = compute turbu l en t speed ( atwood , g , L)turbu l ent speed = 0 .5 ∗ compute turbulentspeed ( atwood , g ,

& bubblerad ius )de l taq = compute deltaq ( dens i ty )

! . . . . . K95 model f o r t u r b u l e n t f lame speed , minimum uns t a b l e RT mode .

22

f lamespeed = laminarspeed! f lamespeed = max ( laminarspeed , t u r bu l en t s p e ed )

lambda min = 4 . ∗ pi ∗ f lamespeed ∗∗2 . / ( g ∗ 2 . ∗ atwood )

lambda min = lambda min

lambda min2 = 4 . ∗ pi ∗ laminarspeed ∗∗2 . / ( g ∗ 2 . ∗ atwood )

! I n t e g r a t e in mass us ing a s imple f i r s t −order i n t e g r a t o r .! I n t e g r a t e b ind ing energy us ing burnt f u e l mass ; a l s o take in t o! account a d d i t i o n a l energy r e l e a s e from o ld NSE s t a t e .

! a rea fac = 1.! area fac = ( f lamespeed / laminarspeed )! area fac = ( bub b l e r ad i u s / lambda min )∗∗ (0 .33 )! area fac = max (1 . d0 , area fac )

a r e a f a c = ( bubblerad ius / lambda min ) ∗ ∗ ( 0 . 5 )

area = 4 . ∗ pi ∗ bubblerad ius ∗∗2 .& ∗ a r e a f a c

#i f 0dr = f lamespeed ∗ dtdm = area ∗ dr ∗ dens i ty

dE = de l taq ∗ dm + ( de l taq − de l t aqo ld ) ∗ mass

mass = mass + dmEg = Eg + dE

de l t aqo ld = de l taq#endif

! . . . Increment bubb l e volume , t a k ing in t o account lower den s i t y in ash! . . . . ( d ens i t ya sh ) than f u e l ( d en s i t y ) . Update bubb l e rad ius us ing new! . . . . volume . See , eg , Dursi & Zinga le , 2006.

bubblevolume = bubblevolume + ( mass / dens i tyash )bubblerad ius = ( 3 . ∗ mass / ( 4 .∗ pi ∗ dens i tyash ) ) ∗ ∗ ( 1 . / 3 . )

23

dr = (1 + atwood ) / (1 − atwood ) ∗ f lamespeed ∗ dtbubblerad ius = bubblerad ius + dr

dm = area ∗ dr ∗ dens i ty

dE = de l taq ∗ dm + ( de l taq − de l t aqo ld ) ∗ mass

mass = mass + dmEg = Eg + dE

de l t aqo ld = de l taq

! . . . I n t e g r a t e bubb l e l o c a t i o n by computing both the buoyancy and drag! . . . . a c c e l e r a t i o n s .

C = 3.0! C = 5.

addedmassterm = 1 . /& ( 1 . + 0 .5 ∗ ( 1 . + atwood ) / ( 1 . − atwood ) )

buoyancy = addedmassterm ∗ 2 . ∗ atwood / ( 1 . − atwood ) ∗ g

drag = ( 1 . + atwood ) / ( 1 . − atwood ) ∗& ( ( addedmassterm ∗& 3 . / 8 . ∗ C ∗ r i s e s p e e d ∗∗2 . / bubblerad ius ) +& ( 3 . ∗ f lamespeed ∗ r i s e s p e e d / bubblerad ius ) )

! & ∗ area fac

r i s e s p e e d = r i s e s p e e d + ( buoyancy − drag ) ∗ dt

i f ( r i s e s p e e d . l t . 0 . ) thenprint ∗ , ’ERROR : r i s e s p e e d < 0 . ’print ∗ , ’ buoyancy = ’ , buoyancyprint ∗ , ’ addedmassterm = ’ , addedmasstermprint ∗ , ’ g = ’ , gprint ∗ , ’ atwood = ’ , atwoodprint ∗ , ’ drag = ’ , dragstop

end i f

dr = r i s e s p e e d ∗ dt

24

i f ( dr . l t . 0 . ) thenprint ∗ , ’ERROR : dr < 0 . ’stop

end i f

position = position + dr

vmax = 4 . ∗ s q r t ( atwood / ( ( 1 . + atwood ) ∗ ( 3 . ∗ C) ) ∗& addedmassterm ∗ g ∗ bubblerad ius )

! . . . Breakout c r i t e r i o n .

i f ( ( bubblerad ius + position . gt . maxradius ) ) then! p r i n t ∗ , ’ b u b b l e r ad i u s + po s i t i o n > maxr ’! p r i n t ∗ , ’ b reakout time = ’ , t! p r i n t ∗ , ’ f r a c t i o n burnt mass = ’ , mass / (1 .4 ∗ 1.987 e33 )! p r i n t ∗ , ’ b reakout v e l o c i t y = ’ , r i s e s p e ed / 1 . e5

stopendif

! . . . Pe r f e c t consumption c r i t e r i o n .

i f ( ( mass . gt . WDMASS ∗ 1 .987 e33 ) ) thenstop

endif

! . . . Adapt dt

dt = COURANT ∗ ( position /& max ( r i s e spe ed , f lamespeed ) )

dt = min ( dt , COURANT ∗ ( bubblerad ius /& max ( f lamespeed , r i s e s p e e d ) ) )

print ∗ ,! & t ,

& position / 1 . e5 ,! & ’ ’ , enc losed mass / (WDMASS ∗ 1.987 e33 ) ,! & ’ ’ , b u b b l e r ad i u s / lambda min ,

& ’ ’ , mass / (WDMASS ∗ 1 .987 e33 )! & ’ ’ , r i s e s p e ed / 1 . e5! & ’ ’ , area / (1 . e5 )∗∗2! & ’ ’ , (Eg − E g i n i t i a l ) ,

25

! & ’ ’ , ( 0 . 5 ∗ mass ∗ r i s e s p e ed ∗∗2) ,! & ’ ’ , mass ,! & ’ ’ , E g i n i t i a l ,! & ’ ’ , r i s e speed ,! & ’ ’ , bubb l e rad iu s ,! & ’ ’ , dens i tyash ,! & bubblevolume ,! & ’ ’ , buoyancy ,! & ’ ’ , drag ,! & ’ ’ , g ,! & ’ ’ , d en s i t y

end do

end

!−−−−−−−−−−−−−−−−END MAIN PROGRAM−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

subroutine read coldwd ( radius , dens )

implicit none

integer ireal ∗8 array (11600) , r ad iu s (464) , dens (464)

open (unit = 2 , f i l e = ”coldwd . dat ” )read (2 , ∗) array

do i = 1 , 464rad iu s ( i ) = array (25 ∗ i + 1)dens ( i ) = array (25 ∗ i + 8)

end do

close (2 )

end

#i f 0function compute dens ity (position , posarr , densarr )

implicit none

26

integer i , i r e adreal ∗8 densarr (464) , posar r (464) , dr , dr0 , compute density ,

& position

i r e a d = 999

do i = 463 , 0 , −1i f ( position . l t . posarr ( i + 1) ) then

i r e a d = i + 1endif

end do

dr0 = posarr (2 ) − posarr (1 )dr = dr0 − ( posarr ( i r e ad ) − position )

! . . . L inear l y i n t e r p o l a t e d en s i t y from s to r ed va l u e s

i f ( i r e a d . gt . 1) thencompute dens ity = ( densarr ( i r e ad ) − densarr ( i r e ad − 1) ) / dr0

& ∗ dr + densarr ( i r e a d )else

compute dens ity = densarr (1 )endif

return

end

function compute mass (position , posarr , densarr )

implicit none

integer i , i r e adreal ∗8 densarr (464) , posar r (464) , pi , totmass , dr , dr0 ,

& compute mass , position , rho , compute dens ity

p i = 4 . ∗ atan ( 1 . )totmass = 0 .i r e a d = 0

! . . . To avoid a l i a s i n g e r ro r s near or i g in , use a cons tant d en s i t y! . . . approximation .

27

i f ( position . l t . 40 . e5 ) thencompute mass = 4 . / 3 . ∗ pi ∗ position ∗∗3 . ∗ RHOC

else

dr0 = posarr (2 ) − posarr (1 )

do i = 1 , 463i f ( position . gt . posar r ( i ) ) then

i r e a d = iendif

end do

i f ( i r e a d . ge . 1) thendr = ( position − posarr ( i r e ad ) )

elsedr = dr0 − position

endif

i f ( ( dr . l t . 0 . ) . or . ( dr . gt . 1 . e6 ) ) thenprint ∗ , ’ compute mass ERROR : dr < 0 . ’print ∗ , ’ i r e a d = ’ , i r e a dprint ∗ , ’ dr = ’ , drprint ∗ , ’ dr0 = ’ , dr0print ∗ , ’ posarr , p o s i t i o n =’ , posarr ( i r e ad ) , ’ ’ , positionstop

endif

! . . . Sum the t o t a l mass i n t e r i o r to p o s i t i o n

i f ( i r e a d . gt . 1) thendo i = 1 , i r e a d − 1

totmass=totmass + 4 .∗ pi ∗ densarr ( i )∗ ( posarr ( i ) )∗∗2 . ∗ dr0end do

endif

totmass = totmass + 4 . ∗ pi ∗! & compute dens i ty ( po s i t i on , posarr , densarr ) ∗

& compute density ( position ) ∗& ( position )∗∗2 . ∗ dr

compute mass = totmass

i f ( compute mass . eq . 0 . ) then

28

print ∗ , ’ compute mass = 0 . ’stop

endif

endif

return

end#endif

function compute dens ity ( position )

implicit none

real ∗8 compute density , position , pi , mass , rhoc , f a c to r , rn ,& xi , xi1 , thetaprime , G, K

! . . . cons tan t s f o r Liu (1996) s p o l y t r o p i c f i t

real ∗8 alpha , C, D, F , n , A, B, beta1 , beta2 , beta , omega

p i = 4 . ∗ atan ( 1 . )G = 6.673 e−8 ! cgs

mass = WDMASS ∗ 1 .987 e33 ! Chandrasekharrhoc = RHOC ! co l d WD model

! . . . Eqn . 7.98 Hansen & Kawaler .

K = 3.841 e14 ∗ (WDMASS) ∗ 2 . / 3 .rn = s q r t (K / ( p i ∗ G) / rhoc ∗ ∗ ( 2 . / 3 . ) )x i = position / rn

! . . . Use Liu (1996) approximation to n = 3 po l y t r ope .

alpha = .53C = 5.56215d−4D = 2.745d−2F = 0 .

n = 3 .

A = (n − 1 . ) / 6 .

29

B = n ∗ (n − 1 . ) / (n + 1 . ) ∗ ∗ 2 . ∗ 6 . / 5 . ∗& ( 4 . ∗ alpha / ( 4 . + 5 . ∗ alpha ) )∗∗4 .

beta1 = 1 . / ( ( 1 . + (n − 5 . )∗∗2 ) )beta2 = 1 . / ( ( 1 . + (n − 3 . )∗∗2 ) )

beta = 6.47 − 7 .01 ∗ beta1 + 5.53 ∗ beta1 ∗∗2 .& − 25 .63 ∗ beta2 + 49.42 ∗ beta2 ∗∗2 .& − 26 .88 ∗ beta2 ∗∗2 .

! . . . omega i s the d imens ion l e s s p o t e n t i a l −− ( rho / rhoc )∗∗(1/n)

omega = −alpha ∗ ( 1 . + B ∗ x i ∗ ∗ 2 . ) ∗ ∗ ( 1 . / ( 1 . − n) ) +& ( 1 . + alpha ) ∗ ( 1 . + A ∗ x i ∗ ∗ 2 . ) ∗ ∗ ( 1 . / ( 1 . − n) )& + alpha / 6 . ∗ x i ∗∗2 . ∗ ( 1 . + A ∗ x i ∗∗2 . )∗∗ ( n / ( 1 . − n) )& + C ∗ x i ∗∗ ( 2 . ∗ beta − 1) / ( 1 . + D ∗ x i ∗∗ beta )∗∗2 .

compute dens ity = rhoc ∗ omega∗∗n

returnend

function compute mass ( position )

implicit none

real ∗8 compute mass , position , pi , mass , rhoc , f a c to r , rn ,& xi , xi1 , thetaprime , G, K

! . . . Parameters used in Liu (1996) s po l y t ope f i t

real ∗8 alpha , C, D, F , n , A, B, beta1 , beta2 , beta , omega

p i = 4 . ∗ atan ( 1 . )G = 6.673 e−8 ! cgs

mass = WDMASS ∗ 1 .987 e33 ! Chandrasekharrhoc = RHOC ! co l d WD model

! . . . Eqn . 7.98 Hansen & Kawaler .

30

K = 3.841 e14 ∗ (WDMASS) ∗ 2 . / 3 .rn = s q r t (K / ( p i ∗ G) / rhoc ∗ ∗ ( 2 . / 3 . ) )x i = position / rn

! . . . Use Liu (1996) approximation to n = 3 po l y t r ope .

alpha = 0.53C = 5.56215d−4D = 2.745d−2F = 0 .

n = 3 .

A = (n − 1 . ) / 6 .B = n ∗ (n − 1 . ) / (n + 1 . ) ∗ ∗ 2 . ∗ 6 . / 5 . ∗

& ( 4 . ∗ alpha / ( 4 . + 5 . ∗ alpha ) )∗∗4 .

beta1 = 1 . / ( ( 1 . + (n − 5 . )∗∗2 ) )beta2 = 1 . / ( ( 1 . + (n − 3 . )∗∗2 ) )

beta = 6.47 − 7 .01 ∗ beta1 + 5.53 ∗ beta1 ∗∗2 .& − 25 .63 ∗ beta2 + 49.42 ∗ beta2 ∗∗2 .& − 26 .88 ∗ beta2 ∗∗2 .

compute mass = 4 . ∗ pi ∗ rhoc ∗ position ∗∗3 . ∗& ( 2 . ∗ alpha ∗ B / ( 1 . − n) ∗ ( 1 . + B ∗ x i ∗∗2 . )∗∗ ( n / ( 1 . − n) )& + 1 . / 3 . ∗ ( 1 . + A ∗ x i ∗∗2 . )∗∗ ( n / ( 1 . − n) )& + n ∗ alpha / 18 . ∗ x i ∗∗2 . ∗ ( 1 . + A ∗ x i ∗∗2 . )& ∗∗ ( ( n / ( 1 . − n ) ) − 1)& − C ∗ x i ∗∗ ( 2 . ∗ beta − 3 . ) ∗ ( 2 . ∗ beta − 1 . − D ∗ x i ∗∗ beta )& / ( 1 . + D ∗ x i ∗∗ beta )∗∗3 .& )returnend

function compute g (position , posarr , densarr )

implicit none

real ∗8 compute g , compute mass , position , G, mass ,& posarr (464) , densarr (464)

31

G = 6.673 e−8 ! cgs

mass = compute mass (position , posarr , densarr )mass = compute mass ( position )i f ( mass . l t . 0) then

print ∗ , ’ compute g ERROR : mass < 0 . ’stop

end i fcompute g = G ∗ mass / position ∗∗2 .

returnend

function compute atwood ( dens i ty )

implicit none

real ∗8 compute atwood , dens i ty , dror

! . . . Use l i n e a r f i t to Atwood number from ACs p l o t o f d e l t a r h o /rho

! dror = −.15 ∗ l o g ( d en s i t y ) / l o g ( 10 . ) + 1.57! compute atwood = dror ∗ 1 . / ( 2 . − dror )

compute atwood = (0.00889030216837876 ∗ ( log10 ( dens i ty )∗∗3) ) −& (0.18705071631002179 ∗ ( log10 ( dens i ty )∗∗2) ) +& (1.11017911293835181 ∗ l og10 ( dens i ty ) ) −& 1.2269251387839261returnend

function compute laminarspeed ( dens i ty )

implicit none

real ∗8 compute laminarspeed , dens i ty

! . . . F i t from Timmes & Woosley

32

compute laminarspeed = 92 . e5 ∗ ( dens i ty / 2 . e9 )∗∗ ( . 8 0 5 )

returnend

function compute turbulentspeed ( atwood , g , L)

implicit none

real ∗8 compute turbulentspeed , atwood , g , L

! . . . Khokhlov (1995) subg r i d model

compute turbulentspeed = 0 .5 ∗ s q r t ( atwood ∗ g ∗ L)

returnend

function compute deltaq ( dens i ty )

! . . . Computes Del ta Q from f u e l to ash d e n s i t i e s a t NSE, f i t from t a b l e 4! . . . o f Calder e t a l , 2006. Output r e s u l t i s in erg s /gm.

implicit none

real ∗8 compute deltaq , dens i ty , de ltaq0 , de l tarho0 , sigma , A,& logrho , logrho0

A = 4.77de l taq0 = 3.05logrho0 = 6 .9logrho = log ( dens i ty ) / log ( 1 0 . )

compute deltaq = ( de l taq0 + A ∗ exp (− ( logrho − l ogrho0 )∗∗2 .& / sigma ∗∗2 . ) ) ∗ 1 . e17

returnend

!−−−−−−−−−Vi r i a l Theorem (Under Construct ion)−−−−−−−−−−−−−−−−−−−−−−−−−−−

33

! f unc t i on compu t e 2nd de r i va t i v e rad iu s (R, WDMASS, dr , d t )

!−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

! f unc t i on compute d

function convertchar ( a )

implicit none

character∗CHARACTERLEN ainteger i , j , k , val , convertchar , a s c i i

va l = 0

do i = 1 , CHARACTERLEN

a s c i i = i c ha r ( a ( i : i ) )i f ( ( a s c i i . l e . 57) . or . ( a s c i i . ge . 48) ) then

va l = va l + 10∗∗(CHARACTERLEN − i ) ∗ ( i c ha r ( a ( i : i ) ) − 48)else

print ∗ , ’ERROR : Non−Numerical Character in Conversion = ’ , a s c i istop

endifend do

convertchar = va l

return

end

7 Acknowledgements

I would like to thank Dr. Fisher for his guidance throughout the project. Additionally,I thank all CSUMS faculty and students who helped me during the summer.

8 References

Bowers, Richard, and Terry Deeming. ”Astrophysics I: Stars.” Boston: Jones andBartlett Publishers, Inc., 1984.

34

Branch, David, and G.A. Tammann. ”Type Ia Supernovae as Standard Candles.”Annual Review of Astronomy and Astrophysics. Vol. 30: 359-389. 1992.

Buchler, Jean-Robert, and T.J. Mazurek. ”On Presupernova Dynamical Mass Ejec-tion by Non-Detonated Stellar Cores.” Societe Royale des Sciences de Liege, Memoires.Vol. 8: 435-445. 1975.

Bychkov, Vitaliy V., and Michael A. Liberman. ”Self-Consistent Theory of WhiteDwarf Burning in Supenova Ia Events.” Astrophysics and Space Science. Vol. 233, Issue1-2: 287-292. Nov. 1995.

Calder, A.C., D.M. Townsley, I.R. Seitenzahl, F. Peng, O.E.B. Messer, N. Vladimirova,E.F. Brown, J.W. Truran, and D.Q. Lamb. “Capturing the Fire: Flame Energetics andNeutronization for Type Ia Supernova Simulations.” The Astrophysical Journal. Vol.656: 313-332. 10 February 2007.

Calder, A.C., E.F Brown, D.Q. Lamb, O.E. Messer, F. Peng, I.R. Seitenzahl, D.Townsley, J.W. Turan, N. Vladimirova. “Flame Energetics and the Deflagration Phaseof Type Ia Supernovae.” American Astronomical Society Meeting 208. Vol. 38: 79.June 2006.

Fisher, Robert. “bubblerise.F.” University of Massachusetts Dartmouth. Dec. 82006.

Hansen, Carl J., and Steven D. Kawaler. “Stellar Interiors: Physical Principles,Structure, and Evolution.” New York: Springer-Verlag Berlin Heidelberg, 1994.

Hillebrandt, Wolfgang, and Jens C. Niemeyer. ”Type Ia Supernova Explosion Mod-els.” Annual Review of Astronomy and Astrophysics. Vol. 38: 191-230. 2000.

Hoyle, F. and William A. Fowler. “Nucleosynthesis in Supernovae.” The Astrophys-ical Journal. Vol. 132: 565. Nov. 1960

Khokhlov, Alexei M. “Propagation of Turbulent Flames in Supernovae.” The Astro-physical Journal. Vol. 449: 695-713. Aug. 20 1995.

Liu, F.K. “Polytropic Gas Spheres: An Approximate Anayltic Solution of the Lane-Emden Equation.” Monthly Notices of the Royal Astronomical Society. Vol. 281, Issue4: 1197- 1205. Aug. 1996.

Phillips, M.M. “The Absolute Magnitudes of Type Ia Supernovae.” The Astrophys-ical Journal. Vol. 412, no. 2: L105-L108. 1993.

Plewa, T., A.C. Calder, and D.Q. Lamb. “Type Ia Supernova Explosion: Gravita-tionally Confined Detonation.” The Astrophysical Journal. Vol. 612: L37-L40. Sept. 12004.

Press, William H., Saul A. Teukolosky, William T. Vetterling, Brian P. Flannery.“Numerical Recipes in FORTRAN 77: The Art of Scientific Computing.” Second Edi-tion. Volume One. 2001. Web. June 2010. http://www.nrbook.com/a/bookfpdf.php

Rich, Paul. “Equation of State Unit.” University of Chicago. July 2 2009. Web.June 2009.

Timmes, F.X., and S.E. Woosley. “The Conductive Propagation of Nuclear Flames.I. Degenerate C + O And O +Ne + Mg White Dwarfs.” The Astrophysical Jounral.Vol. 396: 649 - 667. 2 Sept. 1992.

Timmes, F.X. ”helmholtz.f90.” Arizona State University. 8 May 2010. Web. June

35

2010. http://cococubed.asu.edu/code pages/eos.shtml.Webbink, R.F. ”Double White Dwarfs as Progenitors of R Coronae Borealis Stars

and Type Ia Supernovae.” The Astrophysical Journal. Vol. 277: 355-360. 1 Feb. 1984.Wheeler, J.C., and R.P. Harkness. ”Type I Supernovae.” Reports on Progress in

Physics. Vol. 53, Issue 12: 1467 - 1557. Dec. 1990.Whelan, John, and Icko Iben, Jr. ”Binaries and Supernovae of Type I.” The Astro-

physical Journal. Vol. 186: 1007-1014. Dec. 1973.Wunsch, Scott, and S.E. Woosley. ”Convection and Off-Center Ignition in Type Ia

Supernovae.” The American Astronomical Society. Vol. 616: 1102-1108. 1 Dec. 2004.

36