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Molecular Clouds: Structure, Evolution, and Birth of Stars in Turbulent Fluids

Explore the density distribution in self-gravitating, compressible turbulent fluids of molecular clouds, the birthplaces of stars. This study examines the structure, evolution, and physics of molecular clouds, focusing on the Probability Distribution Function of mass density. Our model considers turbulence, gravity, and isothermal equilibrium, neglecting magnetic fields and young star feedback. Equations based on Navier-Stokes and Poisson equations guide the analysis. The ensemble-averaged representative properties and energy conservation contribute to understanding the complex dynamics within molecular clouds.

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Molecular Clouds: Structure, Evolution, and Birth of Stars in Turbulent Fluids

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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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