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Feasibility of Core-Collapse Supernova Experiments at

the National Ignition FacilityTimothy Handy

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Physics of Dying Stars 101 Investigation of feasibility to reproduce the

standing accretion shock experimentally◦ Analytic

Can the basics of the scenario exist in the lab?◦ Semi-analytic

How do the results from the analytic work connect together?

◦ Full Simulations Are features of the supernova setting captured?

Wrap Up

What You’ve Come to See

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Hyperbolic system of conservation laws Requires an additional closure relation

(equation of state)

Euler Equations

Conserved Quantity

Multidimensional Time Dependent

One-dimensional, Steady, Arbitrary Cross-

section Area

Mass

Momentum

Energy

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Assumptions:◦ Ideal Gas◦ Isentropic (Reversible &

Adiabatic)◦ One-dimensional flow◦ Compressible

Examples:◦ Rocket Engines◦ Astrophysical Jets

de Laval Nozzle – A Basic Example

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Layers of material◦ Density gradient◦ Generated due to gravity

Steady State vs. Static Equilibrium◦ Steady State – balanced state with change

(dynamic processes)◦ Static Equilibrium – balanced state without change

Atmospheres are generally steady with dynamics◦ Pressure changes move flow◦ Heating and cooling processes trigger convection

Stratified Mediums (Atmospheres)

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Euler with SourcesGravity Gravity

+ Heating

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What’s stopping us from falling?

This pressure term comes from the interaction between atoms (well, fermions…)◦ Two atoms can’t share the same space

What happens if the pressure disappears?◦ Our businessman is in trouble!

What counters gravity?

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Core-Collapse SupernovaeIron core grows

Mass is added from silicon burning

Gravity > Degeneracy

PressureElectrons and Protons combine

to form Neutrons and Neutrinos

Sudden loss of pressure at the core

Okay BigBigge

rTOO BIG!

+ -+ = +

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Falling fluid parcels do not know new equilibrium◦ Possible overshoot of equilibrium

Compressed, high density plasma changes its properties (phase transition) and becomes nuclear matter◦ NM is much harder to compress and starts effectively

acting as a solid boundary◦ This boundary acts as a reflector for the incoming flow

Reflected flow perturbations propagate upstream and evolve into a shock

Analogy: String of springs

Bounce

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The outer stellar envelope is infalling Material passes through the shock

◦ This shock front is stationary Standing Accretion Shock Instability (SASI) In order for the supernova to continue its death, it must

revive and continue expanding The question is, how does this revival take place? What

happens to the flow field while this is happening? (Mixing?)

Finally, the shocked material is advected downstream subsonically and settles down near the surface of the reflector (proto-neutron star)

State of Affairs at this Time

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Ohnishi et al. (2008) proposed an experimental design to study the shock

Drive material toward a central reflector using lasers

The material would then strike the reflector and produce a shock

Material would continueto move through the shock

Ohnishi Design

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Loss of gravity and heating/cooling◦ Can a laboratory

shock be similar to a real shock?

Ohnishi Design

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Characterization of flows via Euler number [Ryutov et al. (1999)]◦ Essentially a material independent Mach number◦ Two shocked flows are hydrodynamically similar if the

values are equivalent.◦ Bridge between astrophysics and high energy density

physics (HEDP)◦ Same rationale as Reynolds number, Peclet number, etc.

Scaling Law (Euler number) and HEDP

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The outer stellar envelope is infalling Material passes through the shock Advected downstream subsonically and settles down near the

surface of the reflector (proto-neutron star)

The above are essential nozzle componentsHighlight difference with SN

SettlingCooling by NeutrinosGravity

ConvectionHeating by Neutrinos

The problem can now be reformulated as the composite of two problemsShock Stability ProblemSettling Flow Problem

Here our focus is on the first problem and initially without Heating

State of Affairs at this Time

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The outer stellar envelope is infalling Material passes through the shock Advected downstream subsonically and settles down near the surface of

the reflector (proto-neutron star)

The above are essential nozzle components Supernova’s additional processes

◦ Settling Cooling by Neutrinos Gravity

◦ Convection Heating by Neutrinos

The problem can now be reformulated as the composite of two problems◦ Shock Stability Problem◦ Settling Flow Problem

Our focus is on the shock stability problem (initially without heating)

State of Affairs at this Time

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Shock Stability Problem◦ Presumes the shock can be created◦ What are viable parameters?

Constraints from HEDP and supernovae◦ If existence is possible:

Is it stable “long enough”? Does it behave like the supernova phenomenon?

Convectively unstable layer? Turbulent buildup? ◦ Essentially nozzle flow (a spherical nozzle)◦ No gravity or heating/cooling

Aim of Work

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Analytic

What can we learn without a computer?

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Mathematically, it is possible to have situations with negative pressure

This is not physically motivated Should exclude situations where it occurs

Critical Mach Number (Pre-shock Pressure >0)

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Nozzle flow provides insight into how thick our domain can be

In the limit as the inner Mach number goes to 1.0 we obtain

Maximum Aspect Ratio

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Maximum Aspect Ratio

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We can also determine a relation between the Euler number and pre-shock Mach number

Euler Number vs. Pre-Shock Mach Number

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Temperatures in the lab are ≈106 Kelvin The relation between temperature and (pressure, density) is

material dependent Using ideal gas law and the above

temperature, we can derive a new “temperature” quantity (CGS)

The molar mass is expected to vary between 1 and 1000

Initial BC constraints

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HEDP provides estimates on density and pressure Says nothing about velocity Scanning a broad parameter space of inner boundary

values and limiting by subsonic flows only, it should be possible to obtain bounds on the velocity

Initial BC Constraints

Quantity Minimum Maxiumum

Density 100 104

Velocity 10-10 1010

Pressure 1012 1018

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Constraints on post-shock Mach number Constraint on domain size Constraints on gas compressibility Initial bounds on all quantities at the inner

boundary

Next:◦ See how constraints at the shock interplay with

conditions at the inner boundary

Recap

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Semi-Analytic

Will solving the ODE system connect the dots?

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Simple sampling method Combines the ability to sparsely sample

while improving coverage

Latin Hypercube Sampling

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Solutions to the ODE Euler equations combining parameters at the lower boundary with constraints imposed at the shock and domain size

After choosing all parameters, begin integrate from the lower boundary outward (aspect ratio determines how far)◦ Runge-Kutta 4/5

Apply Rankine-Hugoniot relations at the shock to obtain pre-shock values

Determine if shock constraints are satisfied

Semi-analytic Setup

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Semi-analytic Results Banding behavior of aspect ratio

Lower bound from restricting pre-shock Mach number Upper bound from restricting post-shock Mach number

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Semi-analytic Results Tight distribution of T wrt velocity

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One-D

How will steady state solutions react to perturbations?

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Solution using the FLASH Piece-wise Parabolic Method code. (Finite Volume)

Initial condition given by the ODE solution Lower boundary condition as outflow Perturbed the upstream density

Setup

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Coupling of Shock to Pert If post-shock structure begins to form, we expect the area to act as a resonator. This

should ultimately decouple from the upstream perturbation frequency and “rumble” the shock at a different mode

Points not lying on (1,1) are numerical artifacts. No important behavior is occurring there No evidence of decoupling in this manner. Problem may be solved with multidimensional

models (although 1D supernova models show this behavior).

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Stable Advective Times Post-shock structure formation should occur in a finite

number of advective crossing times (approximately 10) We see here that it is possible to maintain stationary

shocks for “long enough”

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Two-D

Will higher dimensions recreate supernova behavior?

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Select models from the one dimensional simulations were chosen to be simulated in two dimensions.

Identical boundary conditions in the radial direction

Reflecting boundary conditions in the lateral direction

Perturbations were of the form

Setup

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Qualitative Results

Richtmeyer-Meshkov instabilities Vorticity generated dissipates

◦ No large-scale structure coherence

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Flux Decomposition Decompose the energy component into different

fluxes For a convectively unstable layer, expect to see a

second kinetically-dominated region Only see one kinetic layer

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Conclusions – Parameter Ranges Constraints on post-shock Mach number

◦ Lower bound on the post-shock Mach number◦ Regulated by compressibility

Constraint on domain size◦ Maximum width regulated by compressibility◦ Generally quite narrow

Constraints on gas compressibility◦ Only certain values satisfy supernova driven

quantities◦ Values should be 5/3 or greater

Possible to create stationary shocks with HEDP conditions

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Possible to maintain a stationary shock “long enough” for flow features to develop

No coherence into a large-scale convective layer like the supernova setting◦ Flow is advected from the domain◦ No turbulent buildup causing a purely

hydrodynamic convective layer

Conclusions – SASI Recreation

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Neglected heating and cooling effects◦ While the underlying physics is different, there are

cooling effects present in HEDP experiments◦ Could help mitigate the lack of gravity

Incorporating cooling effects in an attempt to mimic low atmosphere supernova behavior would be the start to solving the “settling problem”

Future Work