Susanne HöfnerSusanne.Hoefner@fysast.uu.se

Astrophysical Dynamics, VT 2010

Gravitational Collapse and Star Formation

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 orwill 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:

isothermal gas

small-amplitude disturbances in gas which initially is at rest with constant pressure and density

P = cs2

∂∂ t

∂∂ x u = 0

∂∂ t u ∂

∂ x uu = − ∂ P∂ x

− ∂∂ x

P=P0P1 x , t =01x , t u= u1 x , t

∂2∂ x2 = 4G

=01 x , t

Dense Clouds: Stability or Collapse ?

1D, gravity included:

isothermal gas

small-amplitude disturbances in gas which initially is at rest with constant pressure and density

P = cs2

∂∂ t

∂∂ x u = 0

∂∂ t u ∂

∂ x uu = − ∂ P∂ x

− ∂∂ x

P=P0P1 x , t =01x , t u= u1 x , t

∂2∂ x2 = 4G

=01 x , t

Dense Clouds: Stability or Collapse ?

linearised equations:

linear homogeneous systemof differential equations with constant coefficients for ρ

1, u

1 and Φ

1

isothermal perturbation

small-amplitude disturbances in gas which initially is at rest with constant pressure and density

ρ1, u

1, Φ

1 ∝ exp[i(kx+ωt]

P1 = cs21

∂1

∂ t 0∂u1

∂ x = 0∂ u1

∂ t= −

c s2

0

∂1

∂ x−

∂1

∂ x

P=P0P1 x , t =01x , t u= u1 x , t

∂21

∂ x2 = 4G1

=01 x , t

Dense Clouds: Stability or Collapse ?

linearised equations:

linear homogeneous systemof differential equations with constant coefficients for ρ

1, u

1 and Φ

1

isothermal perturbation

small-amplitude disturbances in gas which initially is at rest with constant pressure and density

ρ1, u

1, Φ

1 ∝ exp[ i ( kx + ωt ) ]

P1 = cs21

∂1

∂ t 0∂u1

∂ x = 0∂ u1

∂ t= −

c s2

0

∂1

∂ x−

∂1

∂ x

P=P0P1 x , t =01x , t u= u1 x , t

∂21

∂ x2 = 4G1

=01 x , t

Dense Clouds: Stability or Collapse ?

linearised equations:

linear homogeneous systemof differential equations with constant coefficients for ρ

1, u

1 and Φ

1

isothermal perturbation

small-amplitude disturbances in gas which initially is at rest with constant pressure and density

ρ1, u

1, Φ

1 ∝ exp[ i ( kx + ωt ) ]

P1 = cs21

i1 0 i k u1 = 0

iu1 = −cs

2

0i k 1 − i k1

P=P0P1 x , t =01x , t u= u1 x , t

−k 21 = 4G 1

=01 x , t

Dense Clouds: Stability or Collapse ?

linearised equations:

linear homogeneous systemof differential equations with constant coefficients for ρ

1, u

1 and Φ

1

isothermal perturbation

small-amplitude disturbances in gas which initially is at rest with constant pressure and density

ρ1, u

1, Φ

1 ∝ exp[ i ( kx + ωt ) ]

P1 = cs21

1 0 k u1 = 0

u1 = −c s

2

0k 1 − k 1

P=P0P1 x , t =01x , t u= u1 x , t

−k 21 = 4G 1

=01 x , t

Dense Clouds: Stability or Collapse ?

linearised equations:

linear homogeneous systemof differential equations with constant coefficients for ρ

1, u

1 and Φ

1

isothermal perturbation

small-amplitude disturbances in gas which initially is at rest with constant pressure and density

ρ1, u

1, Φ

1 ∝ exp[ i ( kx + ωt ) ]

P1 = cs21

1 0 k u1 = 0

P=P0P1 x , t =01x , t u= u1 x , t =01 x , t

cs2

0k 1 u1 k 1 = 0

4G 1 k 21 = 0

Dense Clouds: Stability or Collapse ?

dispersion relation:

linear homogeneous system ⇒ ω real (periodic, stable)of differential equations with constant coefficients ⇒ ω imaginary (growing perturb.)for ρ

1, u

1 and Φ

1

isothermal perturbation

small-amplitude disturbances in gas which initially is at rest with constant pressure and density

ρ1, u

1, Φ

1 ∝ exp[ i ( kx + ωt ) ]

P1 = cs21

2 = k 2 cs2 − 4G 0

P=P0P1 x , t =01x , t u= u1 x , t =01 x , t

k 2 cs2 − 4G 0 0

k 2 cs2 − 4G 0 0

Dense Clouds: Stability or Collapse ?

dispersion relation:

linear homogeneous system ⇒ ω real (periodic, stable)of differential equations with constant coefficients ⇒ ω imaginary (growing perturb.)for ρ

1, u

1 and Φ

1

critical wavenumber:

or critical wavelength:

k J = 4 G 0

cs2

1/2

2 = k 2 cs2 − 4G 0

k 2 cs2 − 4G 0 0

k 2 cs2 − 4G 0 0

J = c s G 0

1/2

=2/k

Dense Clouds: Stability or Collapse ?

Jeans criterion:

perturbations with λ > λJ are unstable (growing amplitudes)

leading to collapse due to gravity

Jeans mass:

or

J = c s G 0

1/2

M J = 0 J

2 3

M J = 0 k T 0

4 mu G 0

3/2

∝ T 03/2 0

−1/2

cs = P0

0

1/2

Dense Clouds in the ISM

Is a cloud stable orwill 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 orwill 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

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

t ff = 3 32 G 0

1/2

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

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

t ff = 3 32 G 0

1/2

free-falling mass layers in a homogeneous pre-stellar core

initial cloud radius Rcloud

free fall time tff

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