# Slide 1

## Transcription

Slide 1
```A Semi-Analytic Model of Type Ia
Supernova Turbulent Deflagration
Kevin Jumper
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
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
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
600 km.
10
9
8
7
6
5
4
0
400
800
Position (km)
1200
1600
Comparison with 3-D Simulations
(Updated)
•The simulation
breakout.
Fractional Burnt Mass vs. Position
0.040
Fractional Burnt Mass
•As predicted, the
model’s fractional
burnt mass is higher
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
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
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.
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?
```