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Todor Veltchev 1,2 , Ralf S. Klessen 2 , Paul C. Clark 2

Todor Veltchev 1,2 , Ralf S. Klessen 2 , Paul C. Clark 2 1 Deparment of Astronomy, University of Sofia, BULGARIA 2 Institute of Theoretical Astrophysics, Heidelberg, GERMANY. Density distribution function in a self-gravitating, isothermal, compressible turbulent fluid.

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Todor Veltchev 1,2 , Ralf S. Klessen 2 , Paul C. Clark 2

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  1. Todor Veltchev1,2, Ralf S. Klessen2, Paul C. Clark2 1 Deparment of Astronomy, University of Sofia, BULGARIA 2 Institute of Theoretical Astrophysics, Heidelberg, GERMANY Density distribution function in a self-gravitating, isothermal, compressible turbulent fluid Sava Donkov1 Ivan Stefanov1 1 Department of Applied Physics, Technical University-Sofia, 8 Kliment Ohridski Blvd., Sofia 1000, Bulgaria

  2. Molecular Clouds – birthplaces of stars Self-gravitating compressible turbulent fluids • MC`s structure and evolution  birth of stars  IMF • MC`s structure and evolution  MC`s physics • MC`s structure  MC`s evolution • MC`s structure  PDF (Probability Distribution Function) • of mass-density

  3. PDF of mass-density Lognormal – turbulence (isothermal) PL-tail- turbulence and gravity

  4. Our model – MC ensemble • Ensemble of MCs  the same: PDF; cloud size ; • cloud edge density • Ensemble averaged representative; properties: • spherically symmetric • PDF; cloud size; cloud edge density • has statistical properties of the ensemble members

  5. Our model – the MC ensemble`s physics • Turbulence – fully saturated; all scales in the cloud belong to • the inertial range • Accretion – from the cloud`s environment • through all the cloud`s scales • Gravity • Thermodynamics – isothermal equilibrium • Magnetic fields and feedback from young stars are neglected • We suppose, also, in every shell of the spherical object: • Microscopic equilibrium (isothermal thermodynamics) • Macroscopic equilibrium – turbulence is homogeneous and isotropic

  6. Equations - 1 • The abstract scale – radius of the spherical object

  7. Equations - 2 • Navier – Stokes equation • Continuity equation

  8. Equations - 3 • Isothermal equation of state • Poisson equation

  9. The equation for p(s) – the ensemble averaged Navier-Stokes equation • Ensemble averaged Navier – Stokes equation • Because of ensemble averagingwe can replace “d” with “d/ds” • The energy per unit mass is invariant through the scales

  10. The kinetic energy per unit mass • Turbulence + accretion

  11. The thermal energy per unit mass • Our model is ensemble averaged byassumption and weused • “s” as averaged log-density, then:

  12. The gravitational energy per unit mass • According to spherical symmetry

  13. The second order nonlinear differential equation the substitution the free parameters

  14. Numerical solution

  15. Numerical solution

  16. Conclusions • Similarities with previous models (Shu 1977; Hunter 1977): • - spherical symmetry; • - isothermal flow; • - gravity and accretion; • - two slopes: (-1.5, -1.7) for low densities and -2 for high densities • Differences: there is no explicit dependence on time, we assume steady state • Contributions: • - the MCs ensemble and • the MCs ensemble averaged representative; • - including turbulence in cloud physics; • - the equation of conservation of the total energy per unit mass, • derived from the equations of the medium; • - the differential equation for p(s)

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