Origin of solar systems 30 June - 2 July 2009 by Klaus Jockers Max-Planck-Institut of Solar System Science Katlenburg-Lindau Part 2b Cloud collapse (numerical calculations), initial mass function, summary

If mean atomic mass = 2.46 amu = g, then the number density of 10 4 cm -3 (clouds, clumps) corresponds to a mass density of g 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. 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, Matthew R. Bate, 2009 Stellar, brown dwarf and multiple star properties from hydrodynamical simulations of star cluster formation MNRAS 392,

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 ρ crit ≈ g cm -3. Minimum “stellar” mass ≈ 10 M J and the minimum separation between stars = 10AU (Size of pressure supported fragment).

Computational method: 3d Smoothed Particle Hydrodynamics (SPH), originally developed by Benz. Parallelized using OpenMP. Equation of state:

H 2 dissociation, not modelled with polytrope law

“Sink” particles Sink particles must be introduced into the numerical code to provide a lower limit of the scale length. If ρ > 1000 ρ crit. a sink particle is inserted. It replaces the SPH particles contained within r acc = 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 structure tree. 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.

+ 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 = 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 next projection). Supersonic velocity field is an essential ingredient in the stability of a molecular cloud. Initial conditions: A 500 Mסּ molecular cloud with radius pc = AU. At a temperature of 10 K the mean thermal Jeans mass is 1 Mסּ.

from Larson, R. B., Turbulence and Star formation in Molecular clouds, MNRAS 194, , 1981 Cloud velocities

Resolution: Minimum Jeans mass must be resolved. At ρ crit = g cm-3 it is Mסּ. This requires smoothed model particles. Total computing time 10 5 CPU hours (~4000 days) on a 1.65GHz IBM p570 computer node. >459 stars and <795 brown dwarfs formed, total mass 191 Mסּ. I.e. 38% of the cloud was transformed into stars. The movie, produced by M. Bate and coworkers, Exeter, UK, can be found at

Summary 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 temperature. 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 mass function.