Smoothed Particle Hydrodynamics Carlos Eduardo Aguiar Instituto de Física - UFRJ Outline Non-relativistic hydrodynamics. SPH equations. Applications. Relativistic hydrodynamics. Relativistic SPH. High energy nuclear physics.

Fluid Dynamics

Hydrodynamic Equations Continuity equation Ideal fluid Euler’s equation

Entropy Equation no viscosity no thermal conduction

Energy Equation

Ideal Gas

Conservation Laws

SPH Developed to study gas dynamics in astrophysical systems. Lagrangian method. No grids. Arbitrary geometries. Equally applicable in 1, 2 and 3 space dimensions. - L.Lucy, Astron.J. 82, 1013 (1977) - R.Gingold, J.Monaghan, MNRAS 181, 378 (1977) Reviews: - J. Monaghan, Annu. Rev. Astron. Astrophys. 30, 543 (1992) - L. Hernquist, N. Katz, Ap. J. Suppl. 70, 419 (1989)

Smoothing h x Error:

Particles "Monte-Carlo" sampling

Different ways of writing SP estimates (we omit the SP subscript from now on):

Derivatives No need for finite differences and grids: D D i-1 i i+1

More than one way of calculating derivatives: Exact Galilean invariance

Moving the Particles

Euler's Equation Exact momentum conservation

Entropy

Energy

Alternatively:

SPH Equations

Smoothing Kernels Gaussian: Spline:

Shock Waves shock wave x numerical calculation

Artificial Viscosity Galilean invariant. Vanishes for rotations. Conserves linear and angular momentum.

Local Resolution Length

SPH Simulation of the Hubble Volume Mass density in a thin slice (100x100x20Mpc/h) at the present epoch. This is a view one would observe if the speed of light were infinite.

SPH Simulation of Galaxy Formation Gas Dark Matter Density of gas and dark matter in a group of galaxies.

SPH Simulation of Supernova Explosion 75 ms after bounce z (km) x (km) Herant et al., Ap.J. 435, 339 (1994)

SPH Simulation of Supernova Explosion Herant et al., Ap.J. 435, 339 (1994)

SPH Simulation of Stellar Collision Disruption of a main sequence star by a close encounter with a high velocity neutron star. Colors represent log (density).

SPH Simulation of Colliding Asteroids An 8 m radius rock strikes the 1.6 km long asteroid Castalia at 5 km/s. Red is totally fractured rock, blue is intermediate fractured rock, and white particles represent the impactor.

SPH Simulation of Projectile Impact on Sand Projectile: Aluminum cylinder 30x10 cm (2d). Initial velocity: 3 km/sec Particles Temperature

Relativistic Hydrodynamics Energy-momentum conservation Baryon-number conservation

Continuity equation: Entropy equation: s = entropy density (rest frame)

Relativistic Euler equation: w = enthalpy per baryon Momentum equation:

Energy equation:

Relativistic SPH

Momentum (per baryon number) Energy (per baryon number)

Particle Velocity ?

RSPH Equations

Baryon-Free System entropy density: (Rest frame) (Lab frame)

Baryon-Free RSPH

Ultrarelativistic Pion Gas

Ultrarelativistic Pions Rarefaction Wave

SHASTA et al.

Ultrarelativistic Pions Landau Solution

Artificial Viscosity E.Chow and J.Monaghan, Journal of Computational Physics 134, 296 (1997) See also: S.Siegler and H.Riffert, astro-ph/9904070

Shock Tube Ideal nucleon gas

SHASTA et al.

Shock Tube Ideal nucleon gas

SHASTA et al.