Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit

-Equation of motion; -Relation between pressure and thermal velocity dispersion; -Form of the pressure force

Each degree of freedom carries an energy Point particles with mass m :

Adiabatic change: no energy is irreversibly lost from the system, or gained by the system

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

Thermal energy density: Pressure:

Thermal equilibrium: Adiabatic change:

Thermal equilibrium: Adiabatic change: Product rule for ‘d’-operator: (just like differentiation!)

Adiabatic pressure change: For small volume: mass conservation!

Polytropic gas law: Ideal gas law: Thermal energy density: Polytropic index mono-atomic gas: ISOTHERMAL

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! 2D-example:

right boundary box: left boundary box:

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

Volume change Mass conservation:   V = constant

Volume change Mass conservation:   V = constant Comoving derivative

Divergence product rule

&

(Self-)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

Typical stellar orbits All motion is ‘thermal’ motion! Pressure force is balanced by gravity

N-particle simulation (Simon Portugies-Zwart, Leiden)

The Isothermal Sphere: assumptions

Governing Equations: r Equation of Motion: no bulk motion, only pressure! Hydrostatic Equilibrium!

Density law and Poisson’s Equation Hydrostatic Eq. Exponential density law

‘Down to Earth’ Analogy: the Isothermal Atmosphere Earth’s surface: z = 0 Force balance: High density & high pressure Low density & low pressure Constant temperature z

‘Down to Earth’ Analogy: the Isothermal Atmosphere Earth’s surface: z = 0

‘Down to Earth’ Analogy: the Isothermal Atmosphere Earth’s surface: z = 0 Set to zero!

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:

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