Astrophysical Dynamics, VT 2010 Gravitational Collapse and Star Formation Susanne Höfner [email protected] The Cosmic Matter Cycle Dense Clouds in the ISM Black Cloud Dense Clouds in the ISM Black Cloud Dense Clouds in the ISM Is a cloud stable or will it tend to collapse due to gravitational forces ? Jeans criterion: given density, temperature ⇓ critical mass M J associated with a critical length scale for a disturbance Dense Clouds: Stability or Collapse ? ∂ ∂ 1D, gravity included: u = 0 ∂t ∂ x ∂ ∂ ∂ P ∂ u uu = − − ∂t ∂ x ∂ x ∂ x ∂2 = 4G ∂ x2 isothermal gas 2 P = cs small-amplitude disturbances = in gas which initially is at rest P P0 P1 x ,t with constant pressure and density =01x ,t u= u1 x ,t =01 x ,t Dense Clouds: Stability or Collapse ? ∂ ∂ 1D, gravity included: u = 0 ∂t ∂ x ∂ ∂ ∂ P ∂ u uu = − − ∂t ∂ x ∂ x ∂ x ∂2 = 4G ∂ x2 isothermal gas 2 P = cs small-amplitude disturbances = in gas which initially is at rest P P0 P1 x ,t with constant pressure and density =01x ,t u= u1 x ,t =01 x ,t Dense Clouds: Stability or Collapse ? ∂ ∂u linearised equations: 1 1 = 0 ∂t 0 ∂ x ∂ u c2 ∂ ∂ linear homogeneous system 1 = − s 1 − 1 of differential equations ∂t 0 ∂ x ∂ x with constant coefficients ∂2 ρ Φ 1 for 1, u1 and 1 = 4G ∂ x2 1 2 isothermal perturbation P1 = cs 1 small-amplitude disturbances P=P0P1 x ,t in gas which initially is at rest = x ,t with constant pressure and density 0 1 u= u1 x ,t ρ , u , Φ ∝ exp[i(kx+ωt] 1 1 1 =01 x ,t Dense Clouds: Stability or Collapse ? ∂ ∂u linearised equations: 1 1 = 0 ∂t 0 ∂ x ∂ u c2 ∂ ∂ linear homogeneous system 1 = − s 1 − 1 of differential equations ∂t 0 ∂ x ∂ x with constant coefficients ∂2 ρ Φ 1 for 1, u1 and 1 = 4G ∂ x2 1 2 isothermal perturbation P1 = cs 1 small-amplitude disturbances P=P0P1 x ,t in gas which initially is at rest = x ,t with constant pressure and density 0 1 u= u1 x ,t ρ , u , Φ ∝ exp[ i ( kx + ωt ) ] 1 1 1 =01 x ,t Dense Clouds: Stability or Collapse ? linearised equations: i 1 0 i k u1 = 0 c2 linear homogeneous system i u = − s i k − i k of differential equations 1 1 1 with constant coefficients 0 for ρ , u and Φ 2 1 1 1 −k 1 = 4G 1 2 isothermal perturbation P1 = cs 1 small-amplitude disturbances P=P0P1 x ,t in gas which initially is at rest = x ,t with constant pressure and density 0 1 u= u1 x ,t ρ , u , Φ ∝ exp[ i ( kx + ωt ) ] 1 1 1 =01 x ,t Dense Clouds: Stability or Collapse ? linearised equations: 1 0 k u1 = 0 c2 linear homogeneous system u = − s k − k of differential equations 1 1 1 with constant coefficients 0 for ρ , u and Φ 2 1 1 1 −k 1 = 4G 1 2 isothermal perturbation P1 = cs 1 small-amplitude disturbances P=P0P1 x ,t in gas which initially is at rest = x ,t with constant pressure and density 0 1 u= u1 x ,t ρ , u , Φ ∝ exp[ i ( kx + ωt ) ] 1 1 1 =01 x ,t Dense Clouds: Stability or Collapse ? linearised equations: 1 0 k u1 = 0 c2 linear homogeneous system s k u k = 0 of differential equations 1 1 1 with constant coefficients 0 for ρ , u and Φ 2 1 1 1 4G 1 k 1 = 0 2 isothermal perturbation P1 = cs 1 small-amplitude disturbances P=P0P1 x ,t in gas which initially is at rest = x ,t with constant pressure and density 0 1 u= u1 x ,t ρ , u , Φ ∝ exp[ i ( kx + ωt ) ] 1 1 1 =01 x ,t Dense Clouds: Stability or Collapse ? 2 2 2 dispersion relation: = k cs − 4G 0 2 2 ⇒ ω lineakr chsom−og4enGeou0s sys0tem real (periodic, stable) of differential equations with c2 o2nstant coefficients ⇒ ω imaginary (growing perturb.) k cs − 4G 0 0 ρ Φ for 1, u1 and 1 2 isothermal perturbation P1 = cs 1 small-amplitude disturbances P=P0P1 x ,t in gas which initially is at rest = x ,t with constant pressure and density 0 1 u= u1 x ,t ρ , u , Φ ∝ exp[ i ( kx + ωt ) ] 1 1 1 =01 x ,t Dense Clouds: Stability or Collapse ? 2 2 2 dispersion relation: = k cs − 4G 0 2 2 ⇒ ω lineakr chsom−og4enGeou0s sys0tem real (periodic, stable) of differential equations with c2 o2nstant coefficients ⇒ ω imaginary (growing perturb.) k cs − 4G 0 0 for ρ , u and Φ 1 1 1 1/ 2 4 G critical wavenumber: k = 0 J 2 cs 1/2 or critical wavelength: = c =2/ k J s G 0 Dense Clouds: Stability or Collapse ? 1/2 1/2 P Jeans criterion: = c c = 0 J s s G 0 0 perturbations with λ > λ are unstable (growing amplitudes) J leading to collapse due to gravity 3 Jeans mass: M = J J 0 2 3/2 k T or M = 0 ∝ T 3/ 2 −1/ 2 J 0 0 0 4 mu G 0 Dense Clouds in the ISM Is a cloud stable or will it tend to collapse due to gravitational forces ? Jeans criterion: given density, temperature ⇓ critical mass M J associated with a critical length scale for a disturbance Dense Clouds in the ISM Star Formation in Dense Clouds Star Formation in Dense Clouds Black Cloud Star Formation in Dense Clouds 1 3 2 4 Dense Clouds in the ISM Is a cloud stable or will it tend to collapse due to gravitational forces ? Jeans criterion: given density, temperature ⇓ critical mass M J associated with a critical length scale for a disturbance How fast will the cloud collapse ? Typical Time Scale of the Collapse Spherical cloud collapsing under its own gravity: motion of mass shells determined by gravity (free fall), assuming that gas pressure and other forces can be neglected ⇒ free fall time 1/ 2 3 t = ff 32 G 0 ρ0 ... initial (mean) density of the cloud ρ0 = Mcloud / ( 4π Rcloud3 / 3 ) where Rcloud is the initial radius of the cloud and Mcloud its total mass Typical Time Scale of the Collapse initial cloud radius Rcloud Spherical cloud collapsing under its own gravity: motion of mass shells determined by gravity (free fall), assuming that gas pressure and other ffroerec-feasll icnagn m baess n laeygelresc itne ad homogeneous pre-stellar core ⇒ free fall time 1/ 2 3 t = ff 32 G 0 free fall time tff ρ0 ... initial (mean) density of the cloud ρ0 = Mcloud / ( 4π Rcloud3 / 3 ) where Rcloud is the initial radius of the cloud and Mcloud its total mass Protostellar Collapse Protostellar Collapse ... but reality is more complicated than that: - influence of nearby stars (radiation, winds) - rotation & turbulence on various scales - young stellar objects: disks and jets ... Protostellar Collapse simulations by Banerjee et al.( 2004): density in the disk plane and perpendicular to it Protostellar Collapse simulations by Banerjee et al.( 2004): density in the disk plane and perpendicular to it Young Stellar Objects: Disks & Jets Young Stellar Objects: Disks & Jets Young Stellar Objects: Disks & Jets.