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

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!

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

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

Hydrostatic self-gravitating spheres Spherical coordinates:

Hydrostatic self-gravitating spheres Numerical solutions:

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

Hydrostatic self-gravitating spheres Numerical solutions:

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

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

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

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

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

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

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

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

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

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

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

Hydrostatic self-gravitating spheres We wish to find a recipe to find, for given M and P o, the following: –  c (central density of sphere) –r o (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 P o =  o c s 2 this gives us  c ) –Solve for  o (for given  c /  o ) (this gives us r o )

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

Hydrostatic self-gravitating spheres Dimensionless mass:

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

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

Stability of BE spheres Maximum density ratio =1 / 14.1

Bonnor-Ebert mass Ways to cause BE sphere to collapse: Increase external pressure until M BE <M Load matter onto BE sphere until M>M BE m 1 = 1.18

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...

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

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.