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Gas Dynamics, Lecture 2 (Mass conservation, EOS) see: astro.ru.nl/~achterb/

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; R elation between pressure and thermal velocity dispersion;

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Gas Dynamics, Lecture 2 (Mass conservation, EOS) see: astro.ru.nl/~achterb/

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  1. Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, RadboudUniversiteit Gas Dynamics, Lecture 2(Mass conservation, EOS)see: www.astro.ru.nl/~achterb/

  2. What did we learn last time around? • Equation of motion; • Relation between pressure • and thermal velocity dispersion; • -Form of the pressure force

  3. A little thermodynamics: ideal gas law Each degree of freedom carries an energy Point particles with mass m:

  4. Alternative way to write this:

  5. Some more thermodynamics(see Lecture Notes) Adiabatic change: no energy is irreversibly lost from the system, or gained by the system

  6. 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 dV

  7. Gas of structure-less point particles Thermal energy density: Pressure:

  8. Thermal equilibrium: Adiabatic change:

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

  10. Adiabatic Gas Law: a polytropic relation Adiabatic pressure change: For small volume: mass conservation!

  11. General case for adiabatic changes: Polytropic gas law: Ideal gas law: Thermal energy density: Polytropic index mono-atomic gas: ISOTHERMAL

  12. Mass conservation and thevolume-change law 2D-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!

  13. Simple one-dimensional flow:

  14. right boundary box: left boundary box:

  15. Generalization to three dimensions:

  16. Curves, tangent vectors and volumes carried by flow

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

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

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

  20. Volume: definition A = X , B = Y, C = Z

  21. Volume: definition A = X , B = Y, C = Z

  22. Special choice: orthogonal triad General volume-change law

  23. Special choice: Orthonormal triad General Volume-change law

  24. Mass conservation and the continuity equation Volume change Mass conservation:  V = constant

  25. Mass conservation and the continuity equation Volume change Comoving derivative Mass conservation:  V = constant

  26. The continuity equation : the behaviour of the mass-density

  27. The continuity equation : the behaviour of the mass-density Divergence product rule

  28. The continuity equation : the behaviour of the mass-density

  29. Summary: we are almost there! &

  30. (Self-)gravity

  31. Equation of motion with gravity

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

  33. Application: The Isothermal Sphereas a Globular Cluster Model

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

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

  36. The Isothermal Sphere: assumptions

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

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

  39. ‘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

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

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

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

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

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

  45. WHAT HAVE WE LEARNED SO FAR…..

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

  47. Solution:

  48. Solution:

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