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 of state:
Equation for grav potential:
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 hydrostat equilibrium:
Equation of state:
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 ρo :
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
Must replace
One boundary
ρc innercondition
BC with one
too many!
of outer BCs
Hydrostatic self-gravitating spheres
• Summary of BC problem:
– For inside-out integration the paramters are ρc 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:
New dependent variable:
New coordinate:
Lane-Emden equation
Hydrostatic self-gravitating spheres
A dimensionless, scale-free formulation:
Boundary conditions (both at ξ=0):
Numerically integrate inside-out
Hydrostatic self-gravitating spheres
A dimensionless, scale-free formulation:
Remember:
A direct relation between ρo/ρc and ξo
Hydrostatic self-gravitating spheres
• We wish to find a recipe to find, for given M and
Po, the following:
– ρc (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 ρc/ρo (for given m)
(since ρo follows from Po = ρocs2 this gives us ρc)
– Solve for ξo (for given ρc/ρo)
(this gives us ro)
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 Po), find ρc/ρo from figure,
obtain ρc, use dimless solutions to find ro, make BE sphere
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(ρc/ρo) < 0
unstable
unstable
Stability of BE spheres
Maximum density ratio =1 / 14.1
Bonnor-Ebert mass
m1 = 1.18
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 ρc/ρo =14.1)
and divide y-axis through BE mass:
Hydrostatic clouds with large ρc/ρo 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.