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 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 = 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

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

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 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)

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)

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)

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)

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 semi-analytic model

A Semi-Analytic Model of Type Ia Supernovae Questions?