Gravitational Collapse and Star Formation

Transcription

1 Astrophysical Dynamics, VT 010 Gravitational Collapse and Star Formation Susanne Höfner

2 The Cosmic Matter Cycle

3 Dense Clouds in the ISM Black Cloud

4 Dense Clouds in the ISM Black Cloud

5 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 MJ associated with a critical length scale for a disturbance

6 Dense Clouds: Stability or Collapse? 1D, gravity included: u = 0 t x P u uu = t x x x isothermal gas small-amplitude disturbances in gas which initially is at rest with constant pressure and density = 4 G x s P =c P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t = 0 1 x,t

7 Dense Clouds: Stability or Collapse? 1D, gravity included: u = 0 t x P u uu = t x x x isothermal gas small-amplitude disturbances in gas which initially is at rest with constant pressure and density = 4 G x s P =c P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t = 0 1 x,t

8 Dense Clouds: Stability or Collapse? linearised equations: 1 u1 0 =0 t x linear homogeneous system of differential equations with constant coefficients for ρ1, u1 and Φ1 isothermal perturbation small-amplitude disturbances in gas which initially is at rest with constant pressure and density ρ1, u1, Φ1 exp[i(kx+ωt] u1 c s 1 1 = t x 0 x 1 x = 4 G 1 s 1 P1 = c P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t = 0 1 x,t

9 Dense Clouds: Stability or Collapse? linearised equations: 1 u1 0 =0 t x linear homogeneous system of differential equations with constant coefficients for ρ1, u1 and Φ1 isothermal perturbation small-amplitude disturbances in gas which initially is at rest with constant pressure and density ρ1, u1, Φ1 exp[ i ( kx + ωt ) ] u1 c s 1 1 = t x 0 x 1 x = 4 G 1 s 1 P1 = c P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t = 0 1 x,t

10 Dense Clouds: Stability or Collapse? linearised equations: i 1 0 i k u1 = 0 linear homogeneous system of differential equations with constant coefficients for ρ1, u1 and Φ1 isothermal perturbation small-amplitude disturbances in gas which initially is at rest with constant pressure and density ρ1, u1, Φ1 exp[ i ( kx + ωt ) ] c s i u1 = i k 1 i k 1 0 k 1 = 4 G 1 s 1 P1 = c P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t = 0 1 x,t

11 Dense Clouds: Stability or Collapse? linearised equations: 1 0 k u1 = 0 linear homogeneous system of differential equations with constant coefficients for ρ1, u1 and Φ1 isothermal perturbation small-amplitude disturbances in gas which initially is at rest with constant pressure and density ρ1, u1, Φ1 exp[ i ( kx + ωt ) ] s c u1 = k 1 k 1 0 k 1 = 4 G 1 s 1 P1 = c P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t = 0 1 x,t

12 Dense Clouds: Stability or Collapse? 1 0 k u1 linearised equations: s linear homogeneous system c k 1 of differential equations 0 with constant coefficients for ρ1, u1 and Φ1 4 G 1 isothermal perturbation small-amplitude disturbances in gas which initially is at rest with constant pressure and density ρ1, u1, Φ1 exp[ i ( kx + ωt ) ] =0 u 1 k 1 = 0 k 1 = 0 s 1 P1 = c P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t = 0 1 x,t

13 Dense Clouds: Stability or Collapse? linear homogeneous s 0 ω real k c 4 G system 0 of differential equations withk constant coefficients c s 4 G 0 0 for ρ1, u1 and Φ1 s = k c 4 G 0 dispersion relation: (periodic, stable) ω imaginary (growing perturb.) isothermal perturbation small-amplitude disturbances in gas which initially is at rest with constant pressure and density ρ1, u1, Φ1 exp[ i ( kx + ωt ) ] s 1 P1 = c P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t = 0 1 x,t

14 Dense Clouds: Stability or Collapse? dispersion relation: s = k c 4 G 0 linear homogeneous s 0 ω real k c 4 G system 0 of differential equations withk constant coefficients c s 4 G 0 0 for ρ1, u1 and Φ1 critical wavenumber: or critical wavelength: (periodic, stable) ω imaginary (growing perturb.) kj = 4 G 0 c 1/ s J = c s G 0 1/ = /k

15 Dense Clouds: Stability or Collapse? Jeans criterion: J = c s G 0 1/ P0 cs = 0 1/ perturbations with λ > λj are unstable (growing amplitudes) leading to collapse due to gravity Jeans mass: or M J = 0 M J = 0 J 3 k T0 4 mu G 0 3/ T 3/ 0 1/ 0

16 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 MJ associated with a critical length scale for a disturbance

17 Dense Clouds in the ISM

18 Star Formation in Dense Clouds

19 Star Formation in Dense Clouds Black Cloud

20 Star Formation in Dense Clouds 1 3 4

21 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 MJ associated with a critical length scale for a disturbance How fast will the cloud collapse?

22 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 3 t ff = 3 G 0 1/ 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

23 Typical Time Scale of the Collapse Rcloud Spherical cloud collapsing under its own gravity: initial cloud radius motion of mass shells determined by gravity (free fall), layers in a assuming that gas pressure and other free-falling forces canmass be neglected homogeneous pre-stellar core free fall time 3 t ff = 3 G 0 1/ 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

24 Protostellar Collapse

25 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...

26 Protostellar Collapse simulations by Banerjee et al.( 004): density in the disk plane and perpendicular to it

27 Protostellar Collapse simulations by Banerjee et al.( 004): density in the disk plane and perpendicular to it

28 Young Stellar Objects: Disks & Jets

29 Young Stellar Objects: Disks & Jets

30 Young Stellar Objects: Disks & Jets