IV From cores to stars 4.04.0 TheThe JeansJeans conditioncondition When the supporting pressure in a region is not able to hold that region up against gravitational collapse it is said to be Jeans unstable. The Jeans length (the minimum size of a region for it to collapse) is: The Jeans mass (the minimum mass of a region for it to collapse) is: 4.04.0 TheThe JeansJeans instabilityinstability If a region with a particular density and sound speed has a size greater than the Jeans length, or a mass greater than the Jeans mass it will collapse. Note that these are the initial density and sound speed, during the collapse the density will change, and the sound speed will probably change, and so collapse might not continue, and the Jeans length and mass within a collapsing region will change. 4.14.1 TheThe thermodynamicsthermodynamics ofof collapsecollapse As a prestellar core collapses it will heat-up due to the release of gravitational potential energy and cool due to radiation. Initially the core is able to radiate away all of the excess heat created by the collapse and the core stays roughly isothermal at 10K. Therefore the sound speed is constant, but the density is increasing, so the Jeans mass falls 4.14.1 TheThe thermodynamicsthermodynamics ofof collapsecollapse When the density reaches roughly 10-13 g cm-3, the core becomes optically thick to its own sub-mm radiation. At this point the core changes from isothermal to adiabatic behaviour. 4.14.1 TheThe thermodynamicsthermodynamics ofof collapsecollapse There is a minimum reached for the Jeans mass at around the critical density at which the gas becomes optically thick of ~10-2 M (roughly sun 10 Jupiter masses): the opacity limit for fragmentation 4.14.1 TheThe thermodynamicsthermodynamics ofof collapsecollapse When the Jeans mass (and length) is a minimum fragmentation is most likely to occur. Therefore the initial size of objects forming in a core during collapse are expected to be around 10-2 M , although sun they can grow very rapidly through accretion. Until it reaches the main sequence, the ©star© is a hydrostatic object ± it does not produce energy through nuclear fusion, just through the release of gravitational potential energy. Whilst the object is embedded it is referred to as a ©protostar© and afterwards as a ©pre-main sequence star©. 4.24.2 ProtostarsProtostars Protostars accrete mass rapidly from the surrounding envelope, but they are unable to radiate away their energy of contraction efficiently. Protostars initially have radii of a few au and they contract very slowly (on a Kelvin-Helmholtz timescale) until they reach a temperature of ~2000K when the temperature is high enough to dissociate molecular hydrogen. At this point the evolution becomes almost isothermal (gamma=1.1) again, as most of the kinetic energy is absorbed by dissociating H and the 2 protostar rapidly collapses to stellar densities: the second collapse.. 4.34.3 Density-temperatureDensity-temperature evolutionevolution log(T/K) 5 ©second collapse© 4 slow K-H contraction 3 2 switch to adiabatic 1 0 3 -15 -10 -5 0 Log(rho/g/cm ) 4.34.3 Density-JeansDensity-Jeans massmass evolutionevolution log(M /M ) J sun 0 opacity limit -1 secondary fragmentation? -2 -3 -15 -10 -5 0 Log(rho/g cm ) 4.44.4 ©Secondary©©Secondary© fragmentationfragmentation During the second collapse the Jeans mass falls to below even the opacity limit for fragmentation. It is possible that the collapsing protostar could fragment again in this phase to produce a very close binary system. This may solve the origin of < 1 day binaries, although simulations have found it very difficult to get secondary fragmentation to work. SummarySummary The Jeans conditions (length/mass) show if a core will collapse or not. Initially core collapse is isothermal and the Jeans mass decreases during the collapse. At a density of ~10-13 g cm-3, the core becomes optically thick to its own radiation and the collapse becomes adiabatic. -2 This sets the opacity limit for fragmentation at ~10 M . sun Collapse then proceeds on a Kelvin-Helmholtz timescale until a temperature of ~2000K when molecular hydrogen dissociates. A ©second collapse© occurs to stellar densities at this point. .