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Slide 1
A Semi-Analytic Model of Type Ia
Supernova Turbulent Deflagration
Kevin Jumper
Advised by Dr. Robert Fisher
May 3, 2011
Review of Concepts
• Type Ia supernovae may
be “standard candles”
• Progenitor is a white
dwarf in a singledegenerate 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 = time
R = bubble radius
ρ1 = bubble (ash) density
ρ2 = background star (fuel) density
• Proceeds until breakout
V = bubble volume
g = gravitational acceleration
CD = coefficient of drag
The Semi-Analytic Model (Continued)
• The coefficient of
drag depends on the
Reynolds Numbers
(Re).
•Higher Reynolds
numbers indicate
greater fluid turbulence.
3.0
Coefficient of Drag
• Δx is grid resolution
Coefficient of Drag vs. Reynolds
Number
2.5
2.0
1.5
1.0
0.5
0.0
0
20 40 60 80 100 120 140
Reynolds Number
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)
Log Speed vs. Position
•The model’s bubble
rise speed is increased
due to a lower
coefficient of drag.
3
Log [Speed (km/s)]
• There is still good
initial agreement
between the model
(blue) and the
simulation (black).
2
1
0
0
400
800
1200
Position (km)
1600
Comparison with 3-D Simulations
(Updated)
•Now the model
and simulation
begin to diverge at
about 200 km.
8
Log [Area (km^2)]
•The bubble’s area
is decreased in the
model, as it has
less time to expand.
Log Area vs. Position
7
6
5
4
3
0
400
800
Position (km)
1200
1600
Comparison with 3-D Simulations
(Updated)
•The early
discrepancy
between the
volume of the
model and
simulation is much
smaller.
Log Volume vs. Position
12
11
Log [Volume (km^3)]
•The model has
greater volume until
an offset of about
600 km.
10
9
8
7
6
5
4
0
400
800
Position (km)
1200
1600
Comparison with 3-D Simulations
(Updated)
•The simulation
predicts about 1% at
breakout.
Fractional Burnt Mass vs. Position
0.040
Fractional Burnt Mass
•As predicted, the
model’s fractional
burnt mass is higher
(about 3%).
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
•We still need to
refine the model.
0
400
800
1200
Position (km)
1600
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. http://mathworld.wolfram.com/SphericalCoordinates.html
Image Credit: Wikipedia
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.
2005. Print.
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.
2005. Print.
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 semianalytic model
A Semi-Analytic Model of Type Ia
Supernovae
Questions?

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