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

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

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

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

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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

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Thermal energy density: Pressure:

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Thermal equilibrium: Adiabatic change:

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

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Adiabatic pressure change: For small volume: mass conservation!

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

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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:

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right boundary box: left boundary box:

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Velocity at each point equals fluid velocity : Definition of tangent vector

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Velocity at each point equals fluid velocity: Definition of tangent vector: Equation of motion of tangent vector:

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Volume: definition A = X, B = Y, C = Z The vectors A, B and C are carried along by the flow!

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Volume: definition A = X, B = Y, C = Z

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Volume: definition A = X, B = Y, C = Z

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Special choice: orthogonal triad General volume-change law

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Special choice: Orthonormal triad General Volume-change law

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Volume change Mass conservation: V = constant

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Volume change Mass conservation: V = constant Comoving derivative

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Divergence product rule

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&

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(Self-)gravity

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Self-gravity and Poisson’s equation Potential: two contributions! Poisson equation for potential associated with self-gravity: Laplace operator

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Application: The Isothermal Sphere as a Globular Cluster Model

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

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

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The Isothermal Sphere: assumptions

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

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

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‘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

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

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

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Density law and Poisson’s Equation Hydrostatic Eq. Exponential density law Poisson Eqn. Spherically symmetric Laplace Operator

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Density law and Poisson’s Equation Hydrostatic Eq. Exponential density law Poisson Eqn. Spherically symmetric Laplace Operator Scale Transformation

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Density law and Poisson’s Equation Hydrostatic Eq. Exponential density law Poisson Eqn. Spherically symmetric Laplace Operator Scale Transformation

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WHAT HAVE WE LEARNED SO FAR…..

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Introduction dimensionless (scaled) variables Single equation describes all isothermal spheres!

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Solution:

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What’s the use of scaling with r K ? All ‘thermally relaxed’ clusters look the same!

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Tidal Radius Galactic tidal force ~ self-gravity r t

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

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

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