An Introduction to Astrophysical Fluid Dynamics

The following sections are included:

• Preface

• Contents

The following sections are included:

• The Material Derivative

• The Continuity Equation

• The Momentum Equation

• Newtonian Gravity

• The Mechanical and Thermal Energy Equations

• A Little More Thermodynamics

• Perfect Gases

• The Virial Theorem

• Vorticity

The following sections are included:

• Hydrostatic Equilibrium for a Self-gravitating Body
  • Spherically symmetric case
  • Plane-parallel layer under constant gravity

• Equations of Stellar Structure

• Small Perturbations about Equilibrium
  • Isothermal fluctuations
  • Adiabatic fluctuations

• Lagrangian Perturbations

• Sound Waves

• Surface Gravity Waves

• Phase Speed and Group Velocity

• Order-of-magnitude Estimates for Astrophysical Fluids
  • Typical scales
  • Importance of viscosity
  • The adiabatic approximation
  • The approximation of incompressibility

The following sections are included:

• Equation of Motion in a Rotating Frame

• Equilibrium Equations for a Slowly Rotating Body

• The Roche Model

• Chandrasekhar-Milne Expansion

• Dynamics of Rotating Stellar Models

• Solar Rotation

• Binary Stars

The following sections are included:

• Convective Instability
  • The Schwarzschild criterion
  • Effects of dissipation
  • Modelling convection: the Boussinesq approximation

• The Rayleigh-Taylor Instability

• Rotational Instability

• Shear and the Kelvin-Helmholtz Instability
  • The Kelvin-Helmholtz instability
  • Critical Richardson and Reynolds numbers
  • Turbulence and the Kolmogorov spectrum

The following sections are included:

• Maxwell's Equations and the MHD Approximation

• MHD Waves

• Some MHD Applications
  • Solar prominences
  • Dynamo theory
  • Coronal heating

• MHD Instabilities

The following sections are included:

• The Formulation of Finite Differences

• The von Neumann Stability Analysis

• Various Finite-difference Schemes
  • The Lax method
  • Upwind differencing
  • The staggered leapfrog method
  • The Lax-Wendroff method
  • Implicit schemes: the Crank-Nicholson method

• Considerations for More Complex Systems

• Operator Splitting

• Examples of Implementations
  • 1-D Lagrangian scheme with artificial viscosity
  • 2-D scheme using operator splitting
  • Codes for computing astrophysical flows

• Smoothed Particle Hydrodynamics

The following sections are included:

• The Importance of Rotation: the Rossby Number

• Relative and Absolute Vorticity

• Potential Vorticity

• Baroclinicity and the Thermal Wind Equation

• Geostrophic Motion

• Some Approximate Models
  • The shallow-water model
  • f-plane and β-plane models

• Waves

• Ekman Layers

The following sections are included:

• Bernoulli's Theorem

• The de Laval Nozzle

• The Bondi Problem

• The Parker Solar-Wind Solution

• Nonlinear Acoustic Waves

• Shock Waves

• Blast Wave from a Supernova

The following sections are included:

• Role of Angular Momentum and Energetics of Accretion

• Thin Accretion Disks

• Diffusion Equation for Surface Density

• Steady Disks

• The Need for Anomalous Viscosity

The following sections are included:

• Links to Observations

• Jeans Instability

• Jeans Instability with Rotation
  • Jeans instability for a rotating system

• Ambipolar Diffusion

• Fragmentation

• Some Comments on Star Formation

The following sections are included:

• Linear Adiabatic Wave Equation for Radial Oscillations
  • Boundary conditions
  • Eigenvalue nature of the problem
  • Self-adjointness of the problem
  • A lower bound on the fundamental frequency
  • Homology scaling for the fundamental frequency of stars

• Non-adiabatic Radial Oscillations
  • Physical discussion of driving and damping

• The Quasi-adiabatic Approximation

The following sections are included:

• Nonradial Modes of Oscillation of a Star

• Mode Classification

• The Cowling Approximation

• A Simplified Discussion of Nonradial Oscillations

• A More General Asymptotic Expression

• Helioseismology: The Duvall Law

• Tassoul's Formula

• Asymptotics of g Modes

• Probing the Sun's Internal Rotation

The following sections are included:

• Appendix A Useful Constants and Quantities

• Appendix B Vector Calculus in Spherical and Cylindrical Polar Coordinates

• Appendix C Self-adjoint Eigenvalue Problems

• Appendix D The JWKB Method

• Bibliography

• Index