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Supercritical clouds • Rapid contraction. • Fragmentation into subregions – Also supercritical if size R ≥ clump height Z. • Compression of trapped magnetic flux. • High star-forming efficiency: – Need 20 to 50% efficiency to form bound cluster. – 20 to 50% of 270 MSun pc–2 in OB stars gives luminosity density 104 to 105 LSun pc–2. – cf. Trapezium region in Orion. AS 4002 Star Formation & Plasma Astrophysics Subcritical clouds • No way of initiating collapse if supporting magnetic flux is frozen-in. • Solution: only (molecular) ions are tied to field lines. • Ambipolar diffusion: field lines can slip through the neutrals, allowing supercritical cores to form. • Long diffusion timescale means inefficient star formation. AS 4002 Star Formation & Plasma Astrophysics Heating & cooling • Main heating agent: cosmic-ray ionization occurs at rate 1 10 17 s -1 • Main cooling agent: optically thin emission from dust, CO etc • Equilibrium temperature ~ 10 to 15 K • Cores become optically thick at n ≥ 1016 m–3 – Heat trapped => dense cores warmer – Higher T => greater critical mass => more massive stars? AS 4002 Star Formation & Plasma Astrophysics Ionization balance • Two-body recombinations of charged particles and on charged grains give ion-neutral balance: -15 -3/2 1/2 i C1/2 ; C 9.5 10 m kg n Weak function of gas temperature; value given for T ~ 10 to 30 K, metal depletion 0.1 • Typically: i 30, n 2.3 • Hence for n ~ 1010 m–3, get ionization fraction 1/2 ni C n 1.2 10 7 n i nmH AS 4002 Star Formation & Plasma Astrophysics Drag force of neutrals • Ions tied to field: Cyclotron frequency eB n mi Mean ion-neutral collision frequency = no of collisions per sec felt by ion Hence ion completes many cycles around fieldline between collisions NB: 3.5 10 m kg s 2 r i r n ud 3 10 ~ AS 4002 mi mn 3 -1 -1 m kg -1s-1 Geometric value for collisions with ud > 10 km s-1 Star Formation & Plasma Astrophysics • Collision between ions and neutrals -> exchange of momentum -> neutrals exert a frictional (or drag) force on the ions. • This force is balanced by the Lorentz force. jB 1 0 B B i n u i u n Hence the ion-neutral drift velocity ud ud ui un AS 4002 1 n i 0 B B Star Formation & Plasma Astrophysics Ambipolar diffusion • Induction equation for the ion fluid B B ui 0 t • Substitute to get nonlinear diffusion equation for neutrals: B B B un B B t 0i n AS 4002 Star Formation & Plasma Astrophysics Effective diffusion coefficient • Express RHS in terms of a diffusion coeff: D~ u2At ni , where u2A B2 0 n Alfvén speed in combined medium and t ni 1 i Collision time for neutral in sea of ions • Diffusion and dynamical timescales for field with length scale R: 2 t AD AS 4002 R R R ~ ~ and t dyn ~ D ud uA Star Formation & Plasma Astrophysics Typical timescales • Cloud core of mass 1 MSun, radius 0.1 pc and critical flux density has: 3/2 GM R R uA2 ~ tdyn ~ ~ R uA GM 1/2 • This gives a dynamical timescale of a few times 105 y, and an ambipolar diffusion time an order of magnitude or so greater. • From detailed analysis in slab geometry (see Shu 1987, Ann Rev A & A 25, 23): t AD C ~ 1/2 8 t dyn 22 G AS 4002 Star Formation & Plasma Astrophysics