The formation of stars and planets

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The formation of stars and planets

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

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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!
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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
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Hydrostatic self-gravitating spheres
Spherical coordinates
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Hydrostatic self-gravitating spheres
Spherical coordinates
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Hydrostatic self-gravitating spheres
Numerical solutions
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Hydrostatic self-gravitating spheres
Numerical solutions
Exercise write a small program to integrate
these equations, for a given central density
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Hydrostatic self-gravitating spheres
Numerical solutions
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Hydrostatic self-gravitating spheres
Plotted logarithmically (which we will usually
do from now on)
Numerical solutions
Bonnor-Ebert Sphere
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Hydrostatic self-gravitating spheres
Different starting ?o a family of solutions
Numerical solutions
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Hydrostatic self-gravitating spheres
Numerical solutions
Singular isothermal sphere (limiting solution)
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Hydrostatic self-gravitating spheres
Boundary condition Pressure at outer edge
pressure of GMC
Numerical solutions
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Hydrostatic self-gravitating spheres
Another boundary condition Mass of clump is given
Numerical solutions
One boundary condition too many!
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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

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Hydrostatic self-gravitating spheres
All solutions are scaled versions of each other!
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Hydrostatic self-gravitating spheres
A dimensionless, scale-free formulation
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Hydrostatic self-gravitating spheres
A dimensionless, scale-free formulation
Lane-Emden equation
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Hydrostatic self-gravitating spheres
A dimensionless, scale-free formulation
Boundary conditions (both at ?0)
Numerically integrate inside-out
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Hydrostatic self-gravitating spheres
A dimensionless, scale-free formulation
A direct relation between ?o/?c and ?o
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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)

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Hydrostatic self-gravitating spheres
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Hydrostatic self-gravitating spheres
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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
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Stability of BE spheres

• Many modes of instability
• One is if dPo/dro gt 0
• Run-away collapse, or
• Run-away growth, followed by collapse
• Dimensionless equivalent dm/d(?c/?o) lt 0

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Stability of BE spheres
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Bonnor-Ebert mass

• Ways to cause BE sphere to collapse
• Increase external pressure until MBEltM
• Load matter onto BE sphere until MgtMBE

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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...
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BE Sphere Observations of B68
Alves, Lada, Lada 2001
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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.