Module on Computational Astrophysics

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Module on Computational Astrophysics

• Jim Stone
• Department of Astrophysical Sciences
• 125 Peyton Hall ph. 258-3815
  jmstone_at_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
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Future challenges

• Adding more physics,
• stellar evolution
• stellar collisions

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Fate of Massive stars, Sun-like stars, and Red
Dwarfs
Luminosity
Temperature
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Stellar collision
J. Barnes, U. Hawaii
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Future challenges

• Adding more physics,
• stellar evolution
• stellar collisions
• Ever larger simulations, e.g. 1011 particles
  allows one to follow every star in a galaxy.

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Is it real or a simulation?
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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.
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Particle-based hydro methods.
For continuum approximations
apply.
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
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Equations of hydro express conservation of mass,
momentum, and energy
Conservation of mass
Conservation of momentum
Equation of state
But how to define continuum variables (mass
density r and pressure P) from discrete particles?
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Smooth particle hydrodynamics (SPH)
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
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• Strengths of SPH
• Method is Lagrangian particles concentrate where
  r is high
• Easy to interface to N-body codes (especially
  tree codes)
• Method is simple, easy to code
• Code always runs (robust)
• Weaknesses of SPH
• Method is Lagrangian poor resolution in regions
  where r is low
• Code always runs (sometimes gives misleading
  results)
• Poor at shock capturing
• Slow (need at least 100 particles/h )
• Very diffusive

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Grid-based methods for compressible gas dynamics

1. Discretize space into zones x ? xi,j,k
2. Discretize the continuous variables
3. Difference the conservation laws
   as

Difficulty is computing accurate and stable
fluxes
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The two challenges of numerical MHD

• There are 3 wave families in MHD, which are
  sometimes degenerate
• ? Greatly complicates the calculation of fluxes
• Evolved field must satisfy the divergence-free
  constraint
• ? requires a conservative scheme for the
  magnetic flux

(Evans Hawley 1988)
Rewrite the induction equation
using Stokes Law as
Difference using a staggered B and EMFs located
at cell edges.
Still need accurate and stable EMFs (fluxes of
B)
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Test Circularly Polarized Alfven Wave
Exact, nonlinear solution to MHD equations -
quantitative test

• 1, P 0.1, b 0.1, wave amplitude 0.1 (Toth
  2000)
• Lx 2Ly, Dx Dy , wave propagates at tan-1 q
  1/2

Animation of Bz
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Test Problem Spherical Blast Waves
P 0.1 r 1
B at 45 degrees, b 0.1
HYDRO MHD
Dx Dy, 400 x 600 grid, periodic boundary
conditions

• Not a very quantitative test, BUT
• check of whether blast waves remain spherical
• late term evolution interesting

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Hydrodynamic Blast Wave 400 x 600 grid
MHD Blast Wave 400 x 600 grid
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Successes in N-body simulation.

• Weve covered the most commonly used methods for
  N-body simulations in astrophysics
• Direct N-body (PP) methods
• Tree codes
• Particle-Mesh methods
• What have these methods been used for?

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Stellar dynamics in a globular cluster (PP code)
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Gravo-thermal oscillations Self-gravitating
systems have negative heat capacity cool them
down, they shrink, and get hotter.
Cooling heating by formation of binaries
Result oscillations driven by cooling from
evaporation, heating by binaries
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Log(density)
Log(temperature)
Log (radius)
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Stellar dynamics during collision of two galaxies
(tree code)
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Calculation by Chris Mihos, Vanderbilt U.
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Formation of structure in the Universe (PM code)
Evolution of the Universe is an initial value
problem
The past temperature fluctuations 300,000 years
after the Big Bang
WMAP
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• 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 a forest
• Discovery that most normal matter is very hot

But there are so many more problems to solve
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How do stars form from interstellar gas?
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Why do massive stars explode at the end of their
lives?
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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