Day 2, Topic 2: Self-gravitating hydrostatic gas spheres Lecture by: C.P. Dullemond

Day 2, Topic 2: Self-gravitating hydrostatic gas spheres Lecture by: C

P. Dullemond

The formation of stars and planets

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

From here on the material is partially based on the book by Stahler & Palla “Formation of Stars”

Hydrostatic self-gravitating spheres

Spherical coordinates:

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

Different starting ?o : a family of solutions

Numerical 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:

One boundary condition too many!

Another boundary condition: Mass of clump is given

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

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:

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

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)

Hydrostatic self-gravitating spheres

Hydrostatic self-gravitating spheres

Hydrostatic self-gravitating spheres

Recipe: Convert M in m (for given Po), find ?c/?o from figure, obtain ?c, use dimless solutions to find ro, make BE sphere

Dimensionless mass:

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

Stability of BE spheres

Bonnor-Ebert mass

Ways to cause BE sphere to collapse: Increase external pressure until MBE<M Load matter onto BE sphere until M>MBE

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.