1. A Semi-Analytic Model of Type Ia Supernova Turbulent Deflagration
Kevin Jumper
Advised by Dr. Robert Fisher
May 3, 2011

2. Review of Concepts Type Ia supernovae may be “standard candles”
Progenitor is a white dwarf in a single-degenerate system
Accretion causes carbon ignition and deflagration
Fractional burnt mass is important for describing deflagration
Credit: NASA, ESA, and A. Field (STScI), from Briget Falck. “Type Ia Supernova Cosmology with ADEPT.“ John Hopkins University Web.

3. The Semi-Analytic Model
One dimensional – a single flame bubble expands and vertically rises through the star
The Morison equation governs bubble motion
t = time
R = bubble radius
ρ1 = bubble (ash) density
ρ2 = background star (fuel) density
V = bubble volume
g = gravitational acceleration
CD = coefficient of drag
Proceeds until breakout

4. The Semi-Analytic Model (Continued)
The coefficient of drag depends on the Reynolds Numbers (Re).
Coefficient of Drag vs. Reynolds Number
3.0
2.5
2.0
Coefficient of Drag
1.5
Δx is grid resolution
Higher Reynolds numbers indicate greater fluid turbulence.
1.0
0.5
0.0 20 40 60 80
100
120
140
Reynolds Number

5. The Three-Dimensional Simulation
Used by a graduate student in my research group
Considers the entire star
Proceeds past breakout
Grid resolution is limited to 8 kilometers
Longer execution time than semi-analytic model
Credit: Dr. Robert Fisher, University of Massachusetts Dartmouth

6. Project Objectives Analyze the evolution of the flame bubble.
Determine the fractional mass of the progenitor burned during deflagration.
Compare the semi-analytic model results against the 3-D simulation.
Add the physics of rotation to the semi-analytic model.

7. Comparison with 3-D Simulations (Updated)
Log Speed vs. Position
There is still good initial agreement between the model (blue) and the simulation (black).
The model’s bubble rise speed is increased due to a lower coefficient of drag. 3 2
Log [Speed (km/s)] 1
400
800
1200
1600
Position (km)

8. Comparison with 3-D Simulations (Updated)
Log Area vs. Position
The bubble’s area is decreased in the model, as it has less time to expand.
Now the model and simulation begin to diverge at about 200 km. 8 7 6
Log [Area (km^2)] 5 4 3
400
800
1200
1600
Position (km)

9. Comparison with 3-D Simulations
(Updated)
The model has greater volume until an offset of about 600 km.
The early discrepancy between the volume of the model and simulation is much smaller.
Log Volume vs. Position 12 11 10 9 8
Log [Volume (km^3)] 7 6 5 4
400
800
1200
1600
Position (km)

10. Comparison with 3-D Simulations (Updated)
Fractional Burnt Mass vs. Position
As predicted, the model’s fractional burnt mass is higher (about 3%).
The simulation predicts about 1% at breakout.
We still need to refine the model.
0.040
0.035
0.030
0.025
Fractional Burnt Mass
0.020
0.015
0.010
0.005
0.000
400
800
1200
1600
Position (km)

11. Adding Rotation to the Model Spherical Coordinates
Cartesian coordinates are inconvenient for rotation problems.
r = radius from origin
θ = inclination angle (latitude)
Φ = azimuth angle (longitude)
The above conventions may vary by discipline.
Weisstein, Eric W. "Spherical Coordinates." From MathWorld--A Wolfram Web Resource.
Image Credit: Wikipedia

12. Adding Rotation to the Model Force Equation
The rotating star is a noninertial reference frame, which causes several “forces” to act upon the bubble. F’ = Fphysical + F’Coriolis + F’transverse + F’centrifugal – mAo All forces except Fphysical depend on the motion of the bubble relative to the frame.
Credit: Fowles and Cassiday. “Analytical Mechanics.” 7th ed. Thomson: Brooks/Cole Print.

13. Adding Rotation to the Model Summary of Forces
Fphysical: forces due to matter acting on the bubble
F’Coriolis: acts perpendicular to the velocity of the bubble in the noninertial system
F’transverse: acts perpendicular to radius in the presence of angular acceleration
F’centrifugal: acts perpendicular and out from the axis of rotation
mAo: inertial force of translation
Credit: Fowles and Cassiday, page 199
Credit: Fowles and Cassiday. “Analytical Mechanics.” 7th ed. Thomson: Brooks/Cole Print.

14. Future Work
Try to narrow the discrepancy so that the model and simulation agree within a factor of two
Program the effects of rotation into the semi-analytic model

15. A Semi-Analytic Model of Type Ia Supernovae
Questions?