Module on Computational AstrophysicsProfessor Jim Stone

Department of Astrophysical Sciences and PACM

Module on Computational Astrophysics

Jim StoneDepartment of Astrophysical Sciences

125 Peyton Hall : ph. 258-3815: jmstone@princeton.edu

www.astro.princeton.edu/~jstone

Lecture 1: Introduction to astrophysics, mathematics, and methodsLecture 2: Optimization, parallelization, modern methodsLecture 3: Particle-mesh methodsLecture 4: Particle-based hydro methods, future directions

When is computation needed in astrophysics?

• Calculating the orbits of planets over billions of years– Requires very accurate N-body methods

• Stellar structure and evolution– Requires ODE solver, numerical linear algebra, hydrodynamics

• Formation of stars and planets– Hydrodynamic and MHD solvers, N-body methods

• Dynamical evolution of star clusters– Very fast N-body methods

• Formation of galaxies– Very fast N-body methods, hydrodynamics

• Data analysis and image processing (astronomy)– Numerical linear algebra, pattern recognition, data mining

Thus, computational methods of interest to astronomers and astrophysicists are:

• N-body methods

• ODE solvers

• Numerical linear algebra

• Hyperbolic, parabolic, elliptic PDE solvers

• Image processing, data mining and analysis

We can’t cover it all! We have to focus on one topic:

N-body algorithms

Why N-body methods?• Allow one to study interesting problems• Physics is simple• Mathematics is simple (but not too simple)• Computational methods are complex (but not

too complex)• Methods are applicable in other disciplines (e.g.

plasma physics, molecular dynamics)

Let’s look at some examples of the application of N-body methods to problems in astrophysics…

Example 1.

Orbits of solar system bodies from Doug Hamilton’s webpageSolar System Viewer

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Example 2: Stellar dynamics at the galactic center

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Example 3: Stellar dynamics in a globular cluster

Example 3. (continued)

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Movie from Frank Summers, AMNH

Example 4. Stellar dynamics during collision of two galaxies

Example 4. (continued)

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Calculation by Chris Mihos, Vanderbilt U.

Evolution of the Universe is an initial value problem

The past: temperature fluctuations 300,000 years after the Big Bang

WMAP

Example 5: Formation of structure in the Universe

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The present: distribution of galaxies near the Milky Way today (14 billion years after Big Bang)

Example 5: Formation of structure is computed by N-body simulations (particles represent dark matter)

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Movie by Andrey Kravtsov, U. Chicago

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Example 5: (continued) Zoom-in on formation of cluster of galaxies

Example 6. Dynamics of galaxies during cluster formation

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Movie by John Dubinski, U. Toronto

How are these results computed?Motion of individual particles given by Newton’s Laws

Forces computed from Newton’s Law of Gravity

“Just” have to solve two coupled ODEs, plus evaluate forces.

Introduction to mathematics of the N-body problem.

We will use numerical methods to get solutions to the equations of motion. Useful to understand mathematical properties of solutions.

For N=2, solution can be computed analytically (two-body problem).

Total momentum p = m1v1 + m2v2 Rate of change of p: dp/dt = m1dv1/dt + m2dv2/dt = F1 + F2

But F1 = -F2, --> dp/dt = 0 --> p = const.

So particles move around a common point (the Center of Mass), which moves at a constant velocity.

Two-body problem (continued)

C of M

R

Can show motion of particles follows Kepler’s Third Law

R = distance between particlesP = period of orbit

For N=3, no analytic solution possible (!!)

Example of orbits in a 3-body encounter

For N=3, no analytic solution possible (!!)

Approximate solutions are possible in certain limiting cases, e.g. when one particle is much less massive than other two.

BUT, certain constraints still apply (useful as diagnostics):Total energy conserved

Angular momentum conserved

Total momentum conserved, so center of mass moves at constant velocity (providing 6 more constraints).

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)

At any instant in time, each particle is represented by a point in the 6-dimensional space (x, v). There are so many particles, this space is filled with points. f represents the density of points in this space, and it evolves in time according to the Boltzmann equation:

Hard part is evaluating the collision term. Common to use the Fokker-Planck approximation.

Continuum methods work well for dense stellar systems, but not when single particle interactions (binaries) are important.

Timescales in N-body problems.Most fundamental is the crossing-time.

Consider a star cluster of radius R which contains N stars of mass m

By the virial theorem, for a system in equilibrium, the sum of kinetic and potential energies for a single star is:

So

For N=104, m = 0.5 solar masses, R = 4 pc, tcross = 5 million years

But age of cluster is about 10 billion years >> tcross

In addition, orbital time of binaries in cluster can be << tcross

Multi-timescale problem

Computational Tasks

1. Setup distribution of N particles

2. Compute forces between particles

3. Evolve positions using ODE solver

4. Display/analyze results

1. Setup initial distribution of particles

Can be complex for large N Need realistic model of mass distribution in cluster, galaxy, etc.

2. Computing Forces between particles.

Magnitude of force on ith particle

This diverges as distance between particles --> 0Can cause very small time steps.Solution is to use a softened potential

= softening parameter

Physically, this eliminates the formation of binaries with r <

Hard binaries (very small separation) are an important source of energy in clusters; this means should be as small as possible

Direct summation is most straightforward method of evaluating F

But, number of operations scales as N(N-1)/2

Severly limits number of particles, usually N < 104

Motivates finding more efficient techniques (more later)

3. Evolve positions using ODE solver

Could use any ODE solver (e.g. Runge-Kutta, Burlisch-Stoer, …)

Must balance accuracy versus efficiency

Need many particles to capture dynamics correctly, which can be just as important as the accuracy of any one particle orbit (except for planetary orbits, when N small).

Perhaps most popular method is leap-frog.

Leap-frog schemeDefine positions (x) and forces (F) at time level n velocities (v) at time level n+1/2Then, for ith particle

n-1 n-1/2 n n+1/2 n+1 timex, F v x, F v x, F

To start integration, need initial x and V at two separate time levels.

Specify x0 and v0 and then integrate V to t/2 using high-order scheme