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Supersonic turbulence? • If CO linewidths interpreted as turbulence, velocities approach virial values: Molecular line FWHM in km/sec 2GM 1/2 u ~ R Cloud radius • Problem: specific energy ~ u2 in macroscopic motions, length scale l gets converted to heat (cascading eddies, radiative shocks) in time ~ l/u. • Need u ~ virial speed and l << R for support. • But this means dissipation time l/u << crossing time R/u, so need constant replenishment (winds??) • Ordered magnetic field => turbulence not dominant. AS 4002 Star Formation & Plasma Astrophysics Magnetic support • A fancier form of the scalar virial theorem: 1 d2I 2 3Mc s 2 2 dt B dV 2 0 V 2 Volume integrated magnetic energy B2 1 r.dS (r.B)B.dS p 2 0 0 S S Surface term due to external gas + magnetic pressure at boundary AS 4002 Surface term due to mag. tension Star Formation & Plasma Astrophysics Contracting spherical cloud • Uniform external field B0 outside R0. • Uniformly compressed field inside R: R B 2 2 R0 B0 B0 B Total flux threading cloud R0 R • Radial field between R0 and R: R02 B(r, ) B0 2 cos r AS 4002 Star Formation & Plasma Astrophysics Magnetic volume term inside R0 B 4 3 B 20 dV 3 R 2 0 V 2 2 R0 0 R B(r, )2 20 Energy of uniform field inside inner sphere Energy of radial field between surfaces R and R0 .2r sin .r drd 2 2 3 2 2 4 1 1 B R B0 R0 3 0 3 0 R R0 AS 4002 Star Formation & Plasma Astrophysics Magnetic surface terms at R0 B2 S (r.B)B.dS r.dS 0 S 20 S 1 • Can compute surface integral over R0 directly, but beware of sign errors! • More simply, note that when R= R0 , field is uniform and so exerts no forces and makes no net contribution to Virial Theorem: 2 2 3 2 2 4 1 1 B0 R0 B0 R0 S0 3 0 30 R0 R0 • Hence deduce that volume and surface terms add to zero when R= R0 , so the sum of the two surface terms must be: 2 S AS 4002 3 0 B02 R03 Star Formation & Plasma Astrophysics Total magnetic virial contribution • Sum of volume terms: 2 2 3 2 2 4 1 1 BR B0 R0 3 0 30 R R0 • Sum of surface terms: 2 2 3 B0 R0 3 0 • Total (volume + surface) for R < R0 is: 4 2 4 1 1 B0 R0 3 0 R R0 AS 4002 Star Formation & Plasma Astrophysics Magnetic support • Add surface and volume terms, and balance against gravitational binding energy for uniform sphere: 2 4 2 4 1 1 3GM B0 R0 3 0 5R R R0 R 90 GM 2 1 R0 20 2 ie magnetic energy is sufficient to support cloud at radius R. • Set R=0 to get minimum flux/mass needed to prevent collapse: 3( 0 ) G M 20 1/2 AS 4002 1/2 Star Formation & Plasma Astrophysics Alfvén speed in critical-mass clouds • For clouds near critical state, can rearrange condition for support to get: B2 2 3 GM 3 GM Mu2A 0 5 R 5 R i.e. Alfvén speed is automatically of the magnitude needed for virial equilibrium. • May explain non-thermal velocities as Alfvénwave motions. • Nonlinear Alfvén waves may provide support against collapse along field into sheets. AS 4002 Star Formation & Plasma Astrophysics Reality check • Critical mass for clouds with typical observed R, B (Zeeman splitting of OH lines): Mcr 750 R B 2 G1 / 2 2 B R 1.66 10 MSun 3nT 2pc 3 • For constant B, get column density: Mcr R constant, 2 R u A 1/2 R1/2 AS 4002 Possible explanation of observed power-law relation between CO linewidth and clump radius Star Formation & Plasma Astrophysics Cloud contraction and fragmentation • Critical surface density and visual extinction for flattened critical-mass clouds: Mcl MSun 2 270 2 R pc B 3nT B AV (4 to 5 mag) + 2.5log 3nT (typical gas/dust ratios) • cf. Observed values: – Taurus-Auriga: AV ~ 1-2 mag in envelope, ~ 10 mag in cores with T ~ 10 to 11 K. Subcritical, low efficiency formation of low-mass stars in unbound cluster – Oph: AV ~ 6 mag in envelope, ~ 102 mag in densest cores with T ~ 30 to 35 K. Supercritical, high efficiency formation of low-mass stars + a few B stars in bound cluster AS 4002 Star Formation & Plasma Astrophysics