Origin of solar systems
30 June - 2 July 2009
by Klaus Jockers (email@example.com)
Max-Planck-Institut of Solar System Science
Cloud collapse (numerical calculations), initial mass function, summary
Numerical calculations of cloud collapse
Matthew R. Bate, Ian A. Bonnell and Volker Bromm, 2003
The formation of a star cluster: predicting the properties of stars and brown dwarfs
MNRAS 339, 577-599.
Matthew R. Bate, 2009
Stellar, brown dwarf and multiple star properties from hydrodynamical simulations
of star cluster formation
MNRAS 392, 590-616.
If mean atomic mass = 2.46 amu = 4.084 10-24g, then the number density of 104 cm-3
(clouds, clumps) corresponds to a mass density of 4.084 10-20g cm-3
Aim of the calculations: Not only to visualize the condensation process but also to
determine and to understand the initial mass function, i. e. the mass distribution of
the forming protostars, how many of them are multiple systems, ect.
The opacity limit for fragmentation:
As long as the gravitational energy gained by contraction can be radiated away, the
polytropic index γ = d log[p]/ d log[ρ] ≈ 1. This allows the possibility of fragmentation
because the Jeans mass decreases with increasing density if γ<4/3.
If the gravitational energy gained by contraction exceeds the rate that can be
radiated away, the gas heats up with γ > 4/3, the Jeans mass increases and the
unstable clump quickly becomes stable.
For an initial temperature T = 10 K the critical density ρcrit ≈ 10-13 g cm-3.
Minimum “stellar” mass ≈ 10 MJ and the minimum separation between stars = 10AU
(Size of pressure supported fragment).
3d Smoothed Particle Hydrodynamics (SPH), originally developed by Benz.
Parallelized using OpenMP.
Equation of state:
not modelled with
Sink particles must be introduced into the numerical code to provide a lower limit
of the scale length.
If ρ > 1000 ρcrit. a sink particle is inserted. It replaces the SPH particles contained
within racc= 5AU by a point mass with the same mass and momentum.
Sink particles interact with the gas only via gravity and accretion.
All stars and brown dwarfs start as sink particles.
Gravitation between sink particles is Newtonian but softened if the particles
approach each other by less than 4 AU. Maximum acceleration occurs when the
distance = 1AU (minimum separation of components of double stars), but part of
the calculation was redone without this softening.
Sink particles merge when they pass within 0.02 AU from each other. (23 mergers
within the whole run).
Multiple stellar systems
Multiple stellar systems are determined after the run by constructing a
Some of the binaries turn out to be very wide (several 1000 AU). They
consist of ejected objects that happen to have nearly the same velocity.
A 500 M סּmolecular cloud with radius 0.404 pc = 83300 AU.
At a temperature of 10 K the mean thermal Jeans mass is 1 M.סּ
+ supersonic turbulent velocity field:
Initially the kinetic energy of the turbulence equals the magnitude of the
gravitational potential energy of the cloud, i.e. the cloud has enough turbulent
energy to support itself against gravity.
The initial rms Mach number of the turbulence = 13.7.
At 10 K the sound speed is 184 m s-1, i.e. the mean turbulent speed = 2.52 km s-1.
This unrealistically high value is necessitated by the large size of the cloud (see
Supersonic velocity field is an essential ingredient in the stability of a molecular
from Larson, R. B.,
Turbulence and Star formation in Molecular clouds,
MNRAS 194, 809-826, 1981
Minimum Jeans mass must be resolved.
At ρcrit = 10-13 g cm-3 it is 0.0011 M.סּ
This requires 3.5 107 smoothed model particles.
Total computing time 105 CPU hours (~4000 days) on a 1.65GHz IBM p570
>459 stars and <795 brown dwarfs formed, total mass 191 M.סּ
I.e. 38% of the cloud was transformed into stars.
The movie, produced by M. Bate and coworkers, Exeter, UK, can be found at
A molecular cloud becomes unstable to
collapse simply because in a homogeneous
gas cloud with constant density and pressure
gravitational energy rises faster than volume,
while thermal energy is proportional to volume, i.e. if one increases the size of a
homogeneous cloud a point of collapse will be reached.
Instability increases with increasing mass of the cloud and with decreasing
An important issue are the “quasi-random” velocities in a molecular cloud.
Numerical models of cloud collapse assume a random velocity field of large enough
velocities to stabilize the cloud initially. As the temperature is very low these velocities
are supersonic (larger than the thermal velocity). If the cloud increases in size, these
velocities must increase to unrealistic levels (because the gravitational energy in the
cloud increases too rapidly).
Numerical models allow to calculate the initial mass function in a collapsing cloud, but
there are theoretical and observational limits to an accurate determination of this initial