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Relativistic Smoothed Particle Hydrodynamics in Astrophysical Systems

This study explores Relativistic Smoothed Particle Hydrodynamics (RSPH) for gas dynamics in astrophysical environments, emphasizing energy-momentum and baryon number conservation and entropy-based calculations. Developed to analyze systems without grids, this Lagrangian method is applicable in 1, 2, and 3 dimensions, offering insights into shocks, artificial viscosity, and thermodynamics. Drawing on seminal works, this research details the RSPH equations for studying ultrarelativistic pion gases, shock waves, and thermodynamically normal and anomalous matter. Utilizing techniques like Monte Carlo sampling and Lagrangian formulations, it presents numerical solutions for various scenarios in astrophysical settings.

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Relativistic Smoothed Particle Hydrodynamics in Astrophysical Systems

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  1. Relativistic Smoothed Particle Hydrodynamics C.E. Aguiar, T. Kodama U.F. Rio de Janeiro T. Osada,Y. Hama U. São Paulo • Outline • Relativistic hydrodynamics • Relativistic SPH • Entropy-based SPH • Shocks and artificial viscosity

  2. Relativistic Hydrodynamics Energy-momentum conservation Baryon-number conservation

  3. Baryon number conservation: comoving derivative:

  4. enthalpy per baryon: Energy-momentum conservation:

  5. Momentum equation: Energy equation:

  6. Entropy conservation: s = entropy density (rest frame)

  7. Lagrangian Equations

  8. SPH • Developed to study gas dynamics in astrophysical systems. • Lagrangian method. • No grids. • Arbitrary geometries. • Equally applicable in 1, 2 and 3 space • dimensions. - L.Lucy, Astron.J. 82, 1013 (1977) - R.Gingold, J.Monaghan, MNRAS 181, 378 (1977) Reviews: - J. Monaghan, Annu. Rev. Astron. Astrophys. 30, 543 (1992) - L. Hernquist, N. Katz, Ap. J. Suppl. 70, 419 (1989)

  9. Smoothing h x 0 Error:

  10. Particles "Monte-Carlo" sampling nb = baryon number of ''particle'' b

  11. Different ways of writing SP estimates (we omit the SP subscript from now on):

  12. Derivatives No need for finite differences and grids: D D i+1 i-1 i

  13. More than one way of calculating derivatives:

  14. Moving the Particles

  15. Momentum equation Energy equation

  16. Energy and Momentum

  17. Entropy equation

  18. ? Particle Velocity equation for g:

  19. RSPH Equations

  20. Baryon-Free Matter

  21. Lagrangian equations:

  22. Entropy-based RSPH

  23. Ultrarelativistic Pion Gas

  24. Pion Gas Rarefaction Wave

  25. Pion Gas Landau Solution

  26. numerical calculation shock wave x Shock Waves

  27. Pion Gas Shock Wave

  28. Artificial Viscosity

  29. Second Law of Thermodynamics: Thermodynamically normal matter: Thermodynamically anomalous matter:

  30. Dissipative RSPH

  31. Pion Gas Shock Wave

  32. Pion Gas Rankine - Hugoniot:

  33. QGP + Pion Gas

  34. QGP + Pions Rarefaction Shock

  35. QGP + Pions Rarefaction Shock

  36. QGP + Pions Rarefaction Shock

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