1. Module on Computational Astrophysics Jim Stone Department of Astrophysical Sciences 125 Peyton Hall : ph. 258-3815: jmstone@princeton.edu jmstone@princeton.edu www.astro.princeton.edu/~jstone Lecture 1: Introduction to astrophysics, mathematics, and methods Lecture 2: Optimization, parallelization, modern methods Lecture 3: Particle-mesh methods Lecture 4: Particle-based hydro methods, future directions

2. Future challenges Adding more physics, stellar evolution stellar collisions

3. Fate of Massive stars, Sun-like stars, and Red Dwarfs Temperature Luminosity

4. Stellar collision J. Barnes, U. Hawaii

5. Future challenges Adding more physics, stellar evolution stellar collisions Ever larger simulations, e.g. 10 11 particles allows one to follow every star in a galaxy.

6. Is it real or a simulation?

7. The purpose of computation is understanding. A simulation that included all the physics (if possible) would be just as difficult to understand as nature. Simulations should be used to simplify physical systems so they can be understood.

8. Particle-based hydro methods. Rather than solving for the position of each particle individually, instead compute the evolution of the phase space density: f (x, v, t) evolves in time according to the Boltzmann equation: If collisions are extremely frequent, the particle distribution function (phase space density f ) will be Maxwellian. Moments of the Boltzmann equation lead to the equations of gas dynamics… For continuum approximations apply.

9. Equations of hydro express conservation of mass, momentum, and energy Conservation of mass Equation of state Conservation of momentum But how to define continuum variables (mass density  and pressure P) from discrete particles?

10. Smooth particle hydrodynamics (SPH) h As in PIC codes, average particle properties over a “smoothing length” h Then density becomes: Where W is the “smoothing kernel”, i.e. a weighting function which describes how to “smooth” the particles over h Momentum equation then becomes:

11. Strengths of SPH: 1.Method is Lagrangian; particles concentrate where  is high 2.Easy to interface to N-body codes (especially tree codes) 3.Method is simple, easy to code 4.Code always runs (robust) Weaknesses of SPH: 1.Method is Lagrangian; poor resolution in regions where  is low 2.Code always runs (sometimes gives misleading results) 3.Poor at shock capturing 4.Slow (need at least 100 particles/h ) 5.Very diffusive

12. Grid-based methods for compressible gas dynamics 1.Discretize space into zones x  x i,j,k 2.Discretize the continuous variables 3.Difference the conservation laws: as Difficulty is computing accurate and stable fluxes:

13. The two challenges of numerical MHD 1.There are 3 wave families in MHD, which are sometimes degenerate  Greatly complicates the calculation of fluxes 2.Evolved field must satisfy the divergence-free constraint  requires a conservative scheme for the magnetic flux Rewrite the induction equation using Stoke’s Law as Difference using a staggered B and EMFs located at cell edges. Still need accurate and stable EMFs (fluxes of B)… (Evans & Hawley 1988)

14. Test: Circularly Polarized Alfven Wave  = 1, P = 0.1,  = 0.1, wave amplitude = 0.1 (Toth 2000) L x = 2L y,  x =  y, wave propagates at tan -1  Exact, nonlinear solution to MHD equations - quantitative test Animation of B z

16. Test Problem: Spherical Blast Waves Not a very quantitative test, BUT check of whether blast waves remain spherical late term evolution interesting  x =  y, 400 x 600 grid, periodic boundary conditions P = 0.1  L X = 1 L Y = 1.5 P = 100 in r < 0.1 B at 45 degrees,  = 0.1 HYDRO MHD P = 0.1 

17. Hydrodynamic Blast Wave 400 x 600 grid MHD Blast Wave 400 x 600 grid

18. Successes in N-body simulation. We’ve covered the most commonly used methods for N-body simulations in astrophysics 1.Direct N-body (PP) methods 2.Tree codes 3.Particle-Mesh methods What have these methods been used for?

19. Stellar dynamics in a globular cluster (PP code)

21. Gravo-thermal oscillations: Self-gravitating systems have negative heat capacity: cool them down, they shrink, and get hotter. Result: oscillations driven by cooling from evaporation, heating by binaries Coolingheating by formation of binaries

22. Log (radius) Log(temperature) Log(density)

23. Stellar dynamics during collision of two galaxies (tree code)

24. Calculation by Chris Mihos, Vanderbilt U.

25. Evolution of the Universe is an initial value problem The past: temperature fluctuations 300,000 years after the Big Bang WMAP Formation of structure in the Universe (PM code)

28. Cosmology calculations require solving: N-body equations for collisionless dark matter Hydrodynamical equations for normal matter Radiative transfer equations for photons Microphysics: ionization/recombination, chemistry Successes: Explanation of Ly  forest Discovery that most normal matter is very hot But there are so many more problems to solve…

29. How do stars form from interstellar gas?

30. Why do massive stars explode at the end of their lives?

31. The Future of Computational Astrophysics What is certain: increases in hardware performance will enable larger problems to be tackled numerically What is needed: – More accurate algorithms – Community codes & visualization software – More realistic physics – Students trained in computation: they are the real future