Feasibility of Core-Collapse Supernova Experiments at the National Ignition Facility

1. Feasibility of Core-Collapse Supernova Experiments at the National Ignition Facility Timothy Handy

2. What You’ve Come to See • 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

3. Euler Equations • Hyperbolic system of conservation laws • Requires an additional closure relation (equation of state)

4. de Laval Nozzle – A Basic Example • Assumptions: • Ideal Gas • Isentropic (Reversible & Adiabatic) • One-dimensional flow • Compressible • Examples: • Rocket Engines • Astrophysical Jets

5. Stratified Mediums (Atmospheres) • 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

6. Euler with Sources Gravity Gravity + Heating

7. What counters gravity? • 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!

8. Core-Collapse Supernovae Iron core grows Mass is added from silicon burning TOO BIG! Bigger Big Okay Gravity > Degeneracy Pressure Electrons and Protons combine to form Neutrons and Neutrinos + + = + - Sudden loss of pressure at the core

9. Bounce • 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

10. State of Affairs at this Time • 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 advecteddownstream subsonically and settles down near the surface of the reflector (proto-neutron star)

11. Ohnishi Design • 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 continue to move through the shock

12. Ohnishi Design • Loss of gravity and heating/cooling • Can a laboratory shock be similar to a real shock?

13. Scaling Law (Euler number) and HEDP • 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.

14. State of Affairs at this Time • 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 Highlight difference with SN 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 Here our focus is on the first problem and initially without Heating

15. State of Affairs at this Time • The outer stellar envelope is infalling • Material passes through the shock • Advected downstream subsonicallyand 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)

16. Aim of Work • 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

17. Analytic What can we learn without a computer?

18. Critical Mach Number (Pre-shock Pressure >0) • Mathematically, it is possible to have situations with negative pressure • This is not physically motivated • Should exclude situations where it occurs

19. Maximum Aspect Ratio • 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

20. Maximum Aspect Ratio

21. Euler Number vs. Pre-Shock Mach Number • We can also determine a relation between the Euler number and pre-shock Mach number

22. Initial BC constraints • 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

23. Initial BC Constraints • 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

24. Recap • 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

25. Semi-Analytic Will solving the ODE system connect the dots?

26. Latin Hypercube Sampling • Simple sampling method • Combines the ability to sparsely sample while improving coverage

27. Semi-analytic Setup • 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

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. Setup • 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

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. Setup • 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

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. Conclusions – SASI Recreation • 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

40. Future Work • 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”