1. A Semi-Analytic Model of Type Ia Supernovae
Kevin Jumper
Advised by Dr. Robert Fisher
March 22, 2011

2. Review of Important Terms
White Dwarf
Chandrasekhar Limit
Type Ia Supernovae
Flame Bubble
Self-Similarity
Deflagration
Breakout
GCD Model

3. A Visualization of a GCD Type Ia Supernova
Credit: Dr. Robert Fisher

4. Project Objectives
Determine the fractional mass burned during deflagration.
Analyze the evolution of the flame bubble.
Compare the results against other models and simulations.

5. Preliminary Results For the following conditions:
Initial Mass: solar masses
Initial Radius: 2,500 kilometers
Breakout occurs in about 1.2 seconds.
Fractional burnt mass is generally too high.
Breakout velocities are around 2,600 km /s.
Maximum velocity is attained shortly before breakout.
Bubble self-similarity appears to be supported.

6. Preliminary Results: Bubble-Self Similarity
We saw the near-independence of breakout velocity from bubble conditions.
This seemed to justify our spherical assumption for the model.

7. Log Velocity vs. Log Position

8. Log Velocity vs. Log Radius

9. Comparison to 3-D Simulation Results
Dr. Fisher asked me to compare my model to David’s simulation.
Initial Conditions of Simulation
Progenitor Mass: solar masses
Progenitor Radius: ≈ 2,000 km
Initial Bubble Radius: 16 km
Initial Bubble Offset: 40 km

10. Recalibrating the Model’s Bubble Radius
Initially, the simulation had a greater fractional burnt mass.
This was a consequence of the resolution of the simulation.
To start with the same fractional burnt mass, the model’s bubble radius had to be changed to 24 km.

11. Log Speed vs. Log Position

12. Log Area vs. Position

13. Log Volume vs. Position

14. Fractional Burnt Mass vs. Position

15. Observations
There is relatively good agreement with the simulation’s velocities.
The area and volume diverge from the simulation over time.
The model area and volume obey power laws.
There is a significant discrepancy between the fractional burned mass of the model and simulation.

16. Testing the Spherical Geometry of the Simulation
The above equation is satisfied if the geometry of the bubble is spherical.
We concluded that the simulation’s bubble was non-spherical.

17. Testing the Spherical Geometry: Ratio versus Time
Credit: David Falta

18. Comparing to the Tabular Results
The model’s code also has a routine to read in data about the progenitor instead of calculating it.
However, the same general behavior as the semi-analytic model was observed.
There was also a greater discrepancy in the fractional mass burnt.

19. Varying the Coefficient of Drag
Dr. Fisher hypothesized that the coefficient of drag might be too high in the model.
I was instructed to repeatedly halve the coefficient of drag and see if we could obtain the simulation’s fractional burnt mass.
This is also affected by the Reynolds number, which describes turbulence.

20. Effect of Changing Reynolds Numbers/Drag Coefficients
Credit: Milton van Dyke

21. Fractional Burnt Mass vs. Position (C = 3/16)

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