A Semi-Analytic Model of Type Ia Supernova Turbulent Deflagration

Kevin JumperAdvised by Dr. Robert Fisher

May 3, 2011

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. 2007. Web.

The Semi-Analytic Model• One dimensional – a single flame bubble

expands and vertically rises through the star• The Morison equation governs bubble motion

t = timeR = bubble radiusρ1 = bubble (ash) densityρ2 = background star (fuel) density

• Proceeds until breakout

V = bubble volumeg = gravitational accelerationCD = coefficient of drag

The Semi-Analytic Model (Continued)

• The coefficient of drag depends on the Reynolds Numbers (Re).

Coefficient of Drag vs. Reynolds Number

• Δx is grid resolution

•Higher Reynolds numbers indicate greater fluid turbulence. Reynolds Number

Coeffi

cien

t of D

rag

0 40 12080 100 1406020

0.0

0.5

1.0

1.5

2.0

2.5

3.0

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

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.

Comparison with 3-D Simulations (Updated)

• 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.

Log Speed vs. Position

Position (km)

Log

[Spe

ed (k

m/s

)]

0 400 800 1200 1600

0

1

2

3

•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.

Log Area vs. Position

Position (km)

Log

[Are

a (k

m^2

)]

0 400 800 1200 1600

3

4

5

6

7

8

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

Position (km)

0 400 800 1200 1600

4

5

6

7

8

9

10

11

12

Log

[Vol

ume

(km

^3)]

Comparison with 3-D Simulations(Updated)

•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.

Fractional Burnt Mass vs. Position

Position (km)

0 400 800 1200 1600

Frac

tiona

l Bur

nt M

ass

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

Comparison with 3-D Simulations (Updated)

Adding Rotation to the ModelSpherical 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.

Image Credit: WikipediaWeisstein, Eric W. "Spherical Coordinates." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SphericalCoordinates.html

Adding Rotation to the ModelForce 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. 2005. Print.

Adding Rotation to the ModelSummary 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 translationCredit: Fowles and Cassiday. “Analytical Mechanics.” 7th ed. Thomson: Brooks/Cole. 2005. Print.

Credit: Fowles and Cassiday, page 199

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

Questions?

A Semi-Analytic Model of Type Ia Supernovae