1
Feasibility of Core-Collapse Supernova Experiments at
the National Ignition FacilityTimothy Handy
2
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
3
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
4
Assumptions:◦ Ideal Gas◦ Isentropic (Reversible &
Adiabatic)◦ One-dimensional flow◦ Compressible
Examples:◦ Rocket Engines◦ Astrophysical Jets
de Laval Nozzle – A Basic Example
5
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)
6
Euler with SourcesGravity Gravity
+ Heating
7
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?
8
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!
+ -+ = +
9
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
10
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
11
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
12
Loss of gravity and heating/cooling◦ Can a laboratory
shock be similar to a real shock?
Ohnishi Design
13
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
14
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
15
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
16
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
17
Analytic
What can we learn without a computer?
18
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)
19
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
20
Maximum Aspect Ratio
21
We can also determine a relation between the Euler number and pre-shock Mach number
Euler Number vs. Pre-Shock Mach Number
22
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
23
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
24
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
25
Semi-Analytic
Will solving the ODE system connect the dots?
26
Simple sampling method Combines the ability to sparsely sample
while improving coverage
Latin Hypercube Sampling
27
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
28
Semi-analytic Results Banding behavior of aspect ratio
Lower bound from restricting pre-shock Mach number Upper bound from restricting post-shock Mach number
29
Semi-analytic Results Tight distribution of T wrt velocity
30
One-D
How will steady state solutions react to perturbations?
31
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
32
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).
33
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”
34
Two-D
Will higher dimensions recreate supernova behavior?
35
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
36
Qualitative Results
Richtmeyer-Meshkov instabilities Vorticity generated dissipates
◦ No large-scale structure coherence
37
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
38
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
39
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
40
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