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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
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