The formation of stars and planets

1. The formation of stars and planets Day 2, Topic 2: Self-gravitating hydrostatic gas spheres Lecture by: C.P. Dullemond

2. B68: a self-gravitating stable cloud Bok Globule Relatively isolated, hence not many external disturbances Though not main mode of star formation, their isolation makes them good test-laboratories for theories!

3. Equation of hydrost equilibrium: Equation of state: Equation for grav potential: Hydrostatic self-gravitating spheres • Spherical symmetry • Isothermal • Molecular From here on the material is partially based on the book by Stahler & Palla “Formation of Stars”

4. Equation of hydrostat equilibrium: Equation of state: Equation for grav potential: Hydrostatic self-gravitating spheres Spherical coordinates:

5. Hydrostatic self-gravitating spheres Spherical coordinates:

6. Hydrostatic self-gravitating spheres Numerical solutions:

7. Hydrostatic self-gravitating spheres Numerical solutions: Exercise: write a small program to integrate these equations, for a given central density

8. Hydrostatic self-gravitating spheres Numerical solutions:

9. Hydrostatic self-gravitating spheres Plotted logarithmically (which we will usually do from now on) Numerical solutions: Bonnor-Ebert Sphere

10. Hydrostatic self-gravitating spheres Different starting o: a family of solutions Numerical solutions:

11. Hydrostatic self-gravitating spheres Numerical solutions: Singular isothermal sphere (limiting solution)

12. Hydrostatic self-gravitating spheres Boundary condition: Pressure at outer edge = pressure of GMC Numerical solutions:

13. Must replace c inner BC with one of outer BCs Hydrostatic self-gravitating spheres Another boundary condition: Mass of clump is given Numerical solutions: One boundary condition too many!

14. Hydrostatic self-gravitating spheres • Summary of BC problem: • For inside-out integration the paramters are c and ro. • However, the physical parameters are M and Po • We need to reformulate the equations: • Write everything dimensionless • Consider the scaling symmetry of the solutions

15. Hydrostatic self-gravitating spheres All solutions are scaled versions of each other!

16. Hydrostatic self-gravitating spheres A dimensionless, scale-free formulation:

17. New coordinate: New dependent variable: Hydrostatic self-gravitating spheres A dimensionless, scale-free formulation: Lane-Emden equation

18. Hydrostatic self-gravitating spheres A dimensionless, scale-free formulation: Boundary conditions (both at =0): Numerically integrate inside-out

19. Remember: Hydrostatic self-gravitating spheres A dimensionless, scale-free formulation: A direct relation between o/c and o

20. Hydrostatic self-gravitating spheres • We wish to find a recipe to find, for given M and Po, the following: • c (central density of sphere) • ro (outer radius of sphere) • Hence: the full solution of the Bonnor-Ebert sphere • Plan: • Express M in a dimensionless mass ‘m’ • Solve for c/o (for given m) (since o follows from Po = ocs2 this gives us c) • Solve for o (for given c/o) (this gives us ro)

21. Mass of the sphere: Use Lane-Emden Equation to write: This gives for the mass: Hydrostatic self-gravitating spheres

22. Dimensionless mass: Hydrostatic self-gravitating spheres

23. Hydrostatic self-gravitating spheres Dimensionless mass: Recipe: Convert M in m (for given Po), find c/o from figure, obtain c, use dimless solutions to find ro, make BE sphere

24. unstable unstable Stability of BE spheres • Many modes of instability • One is if dPo/dro > 0 • Run-away collapse, or • Run-away growth, followed by collapse • Dimensionless equivalent: dm/d(c/o) < 0

25. Maximum density ratio =1 / 14.1 Stability of BE spheres

26. m1 = 1.18 Bonnor-Ebert mass • Ways to cause BE sphere to collapse: • Increase external pressure until MBE<M • Load matter onto BE sphere until M>MBE

27. Bonnor-Ebert mass Now plotting the x-axis linear (only up to c/o =14.1) and divide y-axis through BE mass: Hydrostatic clouds with large c/o must be very rare...

28. BE ‘Sphere’: Observations of B68 Alves, Lada, Lada 2001

29. Magnetic field support / ambipolar diff. As mentioned in previous chapter, magnetic fields can partly support cloud and prevent collapse. Slow ambipolar diffusion moves fields out of cloud, which could trigger collapse. • Models by Lizano & Shu (1989) show this elegantly: • Magnetic support only in x-y plane, so cloud is flattened. • Dashed vertical line is field in beginning, solid: after some time. Field moves inward geometrically, but outward w.r.t. the matter.