We perform numerical simulations of a disc–planet system using various grid-based and smoothed pa... more We perform numerical simulations of a disc–planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analysed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5 per cent for the Eulerian simulations. The SPH results predict the correct shape of the gap although have less resolution in the low-density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10 per cent. The spread is larger in the tidal torques acting on the planet which agree within a factor of 2 at the end of the simulation. In the Neptune case, the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.
Basic Equations The Boltzmann Equation Conservation Laws of Hydrodynamics The Validity of the Con... more Basic Equations The Boltzmann Equation Conservation Laws of Hydrodynamics The Validity of the Continuous Medium Approximation Eulerian and Lagrangian Formulation of Hydrodynamics Viscosity and Navier-Stokes Equations Radiation Transfer Conducting and Magnetized Media Numerical Approximations to Partial Differential Equations Numerical Modeling with Finite-Difference Equations Difference Quotient Discrete Representation of Variables, Functions, and Derivatives Stability of Finite-Difference Methods Physical Meaning of Stability Criterion A Useful Implicit Scheme Diffusion, Dispersion, and Grid Resolution Limit Alternative Methods N-Body Particle Methods Introduction to the N-Body Problem Euler and Runge-Kutta Methods The Description of Orbital Motion in Terms of Orbital Elements The Few-Body Problem: Bulirsch-Stoer Integration Lyapunov Time Estimation Symplectic Integration N-Body Codes for Large N Close Encounters and Regularization Force Calculation: The Tree Method Force Calculation: Fast Fourier Transforms Smoothed Particle Hydrodynamics Rudimentary SPH Colliding Planets: An SPH Test Problem Necessary Improvements to Rudimentary SPH Summary Stellar Evolution Equations for Equilibrium of a Star Radiative, Conductive, and Convective Energy Transfer Change in Chemical Composition Boundary Conditions An Implicit Lagrangian Technique: Henyey Method Physics Packages Examples Grid-Based Hydrodynamics Flow Discontinuities and How to Handle Them A Simple Lagrangian Hydrocode Basic Eulerian Techniques Adaptive Mesh Refinement A Multidimensional Eulerian Hydrocode 2 1/2-Dimensional Simulations Examples Poisson Equation Poisson Solutions: I Poisson Solutions: II Test of the Potential Magnetohydrodynamics Basic Assumptions and Definitions MHD Source Terms Solving the Induction Equation Initial and Boundary Conditions Examples and Exercises Concluding Remarks Radiation Transport Solving the Ray Equation for the Continuum Solution for Frequency-Dependent Radiation Transfer in Spherical Symmetry Frequency-Dependent Stellar Atmospheres Technique for Flux-Limited Diffusion in Two Space Dimensions Example: Spectrum of a Rotating, Collapsing Object Example: 3-D Calculations of the Solar Photosphere Numerical Codes Radiation Transfer Stellar Evolution One-Dimensional Lagrangian Hydro ZEUS: 3-D Hydrodynamics N-Body Codes Smoothed Particle Hydrodynamics INDEX References appear in each chapter.
One of the central problems in supernova theory is the question how massive stars explode. Unders... more One of the central problems in supernova theory is the question how massive stars explode. Understanding the physical processes that drive the explosion is crucial for linking the stellar progenitors to the final rem- nants and for predicting observable properties like explosion energies, neutron star and black hole masses, nucleosynthetic yields, explosion anisotropies, and pulsar kicks. In this article we review different sugges- tions for the explosion mechanism and discuss the constraints that can or cannot be deduced from observations. The prompt hydrodynamical bounce-shock mechanism has turned out not to work for typical stellar iron cores and empirical values of the compressibility of bulk nuclear matter. Magnetohydrodynamical models on the other hand contain a number of imponderabilities and are still far behind the level of re- finement that has been achieved in nonmagnetic simulations. In view of these facts the neutrino-driven mechanism must still be considered as the sta...
This presentation discusses experiments well-scaled to the blast wave driven explosion phase of S... more This presentation discusses experiments well-scaled to the blast wave driven explosion phase of SN1987A. These experiments, performed at the Omega Laser facility, use ˜ 5kJ of laser energy to create a blast wave similar to those in supernovae. The blast wave crosses a perturbed interface with a density drop and produces Rayleigh-Taylor instability (RTI) growth. By performing experiments with more
Simulations of Type Ia supernovae are characterized by vastly disparate spatial scales, spanning ... more Simulations of Type Ia supernovae are characterized by vastly disparate spatial scales, spanning some 12 orders of magnitude. This large dynamic range cannot be captured in any modern direct numerical simulation. Therefore, a subgrid model is used to describe unresolved physical processes taking place on the smallest scales. All modern Type Ia supernova simulations currently use steady-state subgrid models. In particular, velocity perturbations are assumed to stem from either a Kolmogorov cascade from larger scales (Reinecke et al. 2002) or are generated by local Rayleigh-Taylor instabilities (Khokhlov 1995). In all of these descriptions, the effective flame speed is assumed to be a function of the local instantaneous (steady-state) velocity field. I will describe a series of studies undertaken at the Flash Center at the University of Chicago designed to confirm and extend the RT-based subgrid model of Khokhlov to account for non-steady propagation of the flame front at early times. The aim is to provide an improved implementation for large-scale simulations, as the particulars of any subgrid flame speed model can have profound effects on the total amount of kinetic energy liberated during explosion and the buoyancy of the ash bubbles formed in these calculations. Both of these factors can contribute to the explosion energy and the observed asymmetry of the event. This work was supported by the U.S. Department of Energy under grant No. BB523820.
We perform numerical simulations of a disc–planet system using various grid-based and smoothed pa... more We perform numerical simulations of a disc–planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analysed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5 per cent for the Eulerian simulations. The SPH results predict the correct shape of the gap although have less resolution in the low-density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10 per cent. The spread is larger in the tidal torques acting on the planet which agree within a factor of 2 at the end of the simulation. In the Neptune case, the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.
Basic Equations The Boltzmann Equation Conservation Laws of Hydrodynamics The Validity of the Con... more Basic Equations The Boltzmann Equation Conservation Laws of Hydrodynamics The Validity of the Continuous Medium Approximation Eulerian and Lagrangian Formulation of Hydrodynamics Viscosity and Navier-Stokes Equations Radiation Transfer Conducting and Magnetized Media Numerical Approximations to Partial Differential Equations Numerical Modeling with Finite-Difference Equations Difference Quotient Discrete Representation of Variables, Functions, and Derivatives Stability of Finite-Difference Methods Physical Meaning of Stability Criterion A Useful Implicit Scheme Diffusion, Dispersion, and Grid Resolution Limit Alternative Methods N-Body Particle Methods Introduction to the N-Body Problem Euler and Runge-Kutta Methods The Description of Orbital Motion in Terms of Orbital Elements The Few-Body Problem: Bulirsch-Stoer Integration Lyapunov Time Estimation Symplectic Integration N-Body Codes for Large N Close Encounters and Regularization Force Calculation: The Tree Method Force Calculation: Fast Fourier Transforms Smoothed Particle Hydrodynamics Rudimentary SPH Colliding Planets: An SPH Test Problem Necessary Improvements to Rudimentary SPH Summary Stellar Evolution Equations for Equilibrium of a Star Radiative, Conductive, and Convective Energy Transfer Change in Chemical Composition Boundary Conditions An Implicit Lagrangian Technique: Henyey Method Physics Packages Examples Grid-Based Hydrodynamics Flow Discontinuities and How to Handle Them A Simple Lagrangian Hydrocode Basic Eulerian Techniques Adaptive Mesh Refinement A Multidimensional Eulerian Hydrocode 2 1/2-Dimensional Simulations Examples Poisson Equation Poisson Solutions: I Poisson Solutions: II Test of the Potential Magnetohydrodynamics Basic Assumptions and Definitions MHD Source Terms Solving the Induction Equation Initial and Boundary Conditions Examples and Exercises Concluding Remarks Radiation Transport Solving the Ray Equation for the Continuum Solution for Frequency-Dependent Radiation Transfer in Spherical Symmetry Frequency-Dependent Stellar Atmospheres Technique for Flux-Limited Diffusion in Two Space Dimensions Example: Spectrum of a Rotating, Collapsing Object Example: 3-D Calculations of the Solar Photosphere Numerical Codes Radiation Transfer Stellar Evolution One-Dimensional Lagrangian Hydro ZEUS: 3-D Hydrodynamics N-Body Codes Smoothed Particle Hydrodynamics INDEX References appear in each chapter.
One of the central problems in supernova theory is the question how massive stars explode. Unders... more One of the central problems in supernova theory is the question how massive stars explode. Understanding the physical processes that drive the explosion is crucial for linking the stellar progenitors to the final rem- nants and for predicting observable properties like explosion energies, neutron star and black hole masses, nucleosynthetic yields, explosion anisotropies, and pulsar kicks. In this article we review different sugges- tions for the explosion mechanism and discuss the constraints that can or cannot be deduced from observations. The prompt hydrodynamical bounce-shock mechanism has turned out not to work for typical stellar iron cores and empirical values of the compressibility of bulk nuclear matter. Magnetohydrodynamical models on the other hand contain a number of imponderabilities and are still far behind the level of re- finement that has been achieved in nonmagnetic simulations. In view of these facts the neutrino-driven mechanism must still be considered as the sta...
This presentation discusses experiments well-scaled to the blast wave driven explosion phase of S... more This presentation discusses experiments well-scaled to the blast wave driven explosion phase of SN1987A. These experiments, performed at the Omega Laser facility, use ˜ 5kJ of laser energy to create a blast wave similar to those in supernovae. The blast wave crosses a perturbed interface with a density drop and produces Rayleigh-Taylor instability (RTI) growth. By performing experiments with more
Simulations of Type Ia supernovae are characterized by vastly disparate spatial scales, spanning ... more Simulations of Type Ia supernovae are characterized by vastly disparate spatial scales, spanning some 12 orders of magnitude. This large dynamic range cannot be captured in any modern direct numerical simulation. Therefore, a subgrid model is used to describe unresolved physical processes taking place on the smallest scales. All modern Type Ia supernova simulations currently use steady-state subgrid models. In particular, velocity perturbations are assumed to stem from either a Kolmogorov cascade from larger scales (Reinecke et al. 2002) or are generated by local Rayleigh-Taylor instabilities (Khokhlov 1995). In all of these descriptions, the effective flame speed is assumed to be a function of the local instantaneous (steady-state) velocity field. I will describe a series of studies undertaken at the Flash Center at the University of Chicago designed to confirm and extend the RT-based subgrid model of Khokhlov to account for non-steady propagation of the flame front at early times. The aim is to provide an improved implementation for large-scale simulations, as the particulars of any subgrid flame speed model can have profound effects on the total amount of kinetic energy liberated during explosion and the buoyancy of the ash bubbles formed in these calculations. Both of these factors can contribute to the explosion energy and the observed asymmetry of the event. This work was supported by the U.S. Department of Energy under grant No. BB523820.
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