Prof dr A Achterberg Astronomical Dept IMAPP Radboud
Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit Gas Dynamics, Lecture 2 (Mass conservation, EOS) see: www. astro. ru. nl/~achterb/
What did we learn last time around? -Equation of motion; -Relation between pressure and thermal velocity dispersion; -Form of the pressure force
A little thermodynamics: ideal gas law Each degree of freedom carries an energy Point particles with mass m:
Alternative way to write this:
Some more thermodynamics (see Lecture Notes) Adiabatic change: no energy is irreversibly lost from the system, or gained by the system
Some more thermodynamics (see Lecture Notes) Adiabatic change: no energy is irreversibly lost from the system, or gained by the system Change in internal energy U Work done by pressure forces in volume change d. V
Gas of structure-less point particles Thermal energy density: Pressure:
Thermal equilibrium: Adiabatic change:
Thermal equilibrium: Adiabatic change: Product rule for ‘d’-operator: (just like differentiation!)
Adiabatic Gas Law: a polytropic relation Adiabatic pressure change: For small volume: mass conservation!
General case for adiabatic changes: Polytropic gas law: Ideal gas law: Thermal energy density: Polytropic index mono-atomic gas: ISOTHERMAL
Mass conservation and the volume-change law 2 D-example: A fluid filament is deformed and stretched by the flow; Its area changes, but the mass contained in the filament can NOT change So: the mass density must change in response to the flow!
Simple one-dimensional flow:
left boundary box: right boundary box:
Generalization to three dimensions:
Curves, tangent vectors and volumes carried by flow
Velocity at each point equals fluid velocity: Definition of tangent vector
Velocity at each point equals fluid velocity: Definition of tangent vector: Equation of motion of tangent vector:
Volume: definition A = X , B = Y, C = Z The vectors A, B and C are carried along by the flow!
Volume: definition A = X , B = Y, C = Z
Volume: definition A = X , B = Y, C = Z
Special choice: orthogonal triad General volume-change law
Special choice: Orthonormal triad General Volume-change law
Mass conservation and the continuity equation Volume change Mass conservation: V = constant
Mass conservation and the continuity equation Volume change Mass conservation: V = constant Comoving derivative
The continuity equation : the behaviour of the mass-density
The continuity equation : the behaviour of the mass-density Divergence product rule
The continuity equation : the behaviour of the mass-density
Summary: we are almost there! &
(Self-)gravity
Equation of motion with gravity
Self-gravity and Poisson’s equation Potential: two contributions! Poisson equation for potential associated with self-gravity: Laplace operator
Application: The Isothermal Sphere as a Globular Cluster Model
All motion is ‘thermal’ motion! Pressure force is balanced by gravity Typical stellar orbits
N-particle simulation (Simon Portugies-Zwart, Leiden)
The Isothermal Sphere: assumptions
Governing Equations: Equation of Motion: no bulk motion, only pressure! Hydrostatic Equilibrium! r
Density law and Poisson’s Equation Hydrostatic Eq. Exponential density law
‘Down to Earth’ Analogy: the Isothermal Atmosphere Low density & low pressure z Constant temperature Force balance: High density & high pressure Earth’s surface: z = 0
‘Down to Earth’ Analogy: the Isothermal Atmosphere Earth’s surface: z = 0
‘Down to Earth’ Analogy: the Isothermal Atmosphere Set to zero! Earth’s surface: z = 0
Density law and Poisson’s Equation Hydrostatic Eq. Exponential density law Poisson Eqn. Spherically symmetric Laplace Operator
Density law and Poisson’s Equation Hydrostatic Eq. Exponential density law Poisson Eqn. Spherically symmetric Laplace Operator Scale Transformation
Density law and Poisson’s Equation Hydrostatic Eq. Exponential density law Poisson Eqn. Spherically symmetric Laplace Operator Scale Transformation
WHAT HAVE WE LEARNED SO FAR…. .
Introduction dimensionless (scaled) variables Single equation describes all isothermal spheres!
Solution:
Solution:
What’s the use of scaling with r. K ? All ‘thermally relaxed’ clusters look the same!
Tidal Radius Galactic tidal force ~ self-gravity r t
- Slides: 52