Transcript Collapse

An introduction to the Physics of the
Interstellar Medium
III. Gravity in the ISM
Patrick Hennebelle
Jeans mass and length
Equilibrium solutions and stability
Collapse
Gravo-turbulent support
Jeans mass and length
Equilibrium solutions and stability
Collapse
Gravo-turbulent support
The equations
(Spitzer 1978, Shu 1992)
P  kb /m p T
Equation of state:
Heat Equation:

T  10K
Continuity Equation:
t   (v )  0
Momentum Conservation:
r
r
r r r
(t v  v v )  P  


Poisson Equation:


  4G
Thermal support
Consider a cloud of initial radius R and a constant temperature T
E therm 3 M /m p kT

R
2
E grav 2 GM /R
When R decreases, Etherm/Egrav decreases.

Thermal support
decreases as collapse proceeds.
=>Any isothermal cloud, if sufficiently squeezed, will collapse.
Gravitational Instability
(Jeans 02, Chandrasekar & Fermi 53, Ostriker 64, Spitzer 78, Larson 85, Curry 00, Nakamura et al 93,
Nakamura & Nakano 78, Nigai et al. 98 , Fiege & Pudritz 00)
Consider (Jeans Analysis, 1902) the propagation of a sonic
wave in a plan-parallel uniform medium:
  0  1 exp(it  ikx)
v  v1 exp(it  ikx)
i1  ik0v1  0
Continuity equation:

Conservation of momentum:
i0v1  ikCs 1  ik01
2
Poisson equation:
k 21  4 G1
Dispersion relation:
 2  Cs2 k 2  4G0



Dispersion relation:
 2  Cs2 k 2  4G0
k  4G 0 /Cs
if
whereas
=> sonic wave (modified by gravity)
k  4G 0 /Cs
=> there is an instability
means:

sonic propagation times smaller than the freefall time

Jeans Length:
Jeans mass:
J  Cs  /G0
both decreases with density
M J  Cs /  0
when the gas remains isothermal
3
Hoyle (1953): recursive fragmentation
as a cloud remains isothermal, it keeps fragmenting in smaller and smaller pieces
As long
Large 
wavelengths grow more rapidly than small wavelengths (problematic for fragmentation)
For:

  10 4 cm3 , T  10K , M J  1 2 M s
Jeans mass and length
Equilibrium solutions and stability
Collapse
Gravo-turbulent support

Fragmentation of sheet into filaments
Linear stability of the self-gravitating sheet (Spitzer 78)
idem:
k  kcrit   2  0
k  0   2  k 2
but for:
more unstable mode = typical width of the filaments :
J
suggest : fragmentation possible once equilibrium is reached in one direction

Fragmentation of a sheet into filaments

Filaments
Exact Equilibrium Solutions in 2D (Schmid-Burgk 1976)
Fragmentation of filament in core
Self-gravitating filaments (Ostriker 64)
(r)  0 /(1 (r /l0 )2 )2 , l0  Cs / G0
-profile in 1/r4
as for the self-gravitating sheet there is a more unstable wavelength
Suggest: the dense cores are elongated structures with a spatial period close to the
Jeans Length.

cores
Development
of the gravitational
instability in a filament:
Formation of an
elongated core
Dutrey et al 91
Fiege & Pudritz 00
Spherical equilibrium solutions
(Bonnor 56, Ebert 55, Chandrasekhar, Mouschovias 77, Tomisaka et al. 85,
Li & Shu 98, Fiege & Pudritz 00, Galli et al. 01)
Bonnor-Ebert and Singular Isothermal Sphere (SIS):
Hydrostatic Equilibrium:
Asymptotically:
also exact solutions (SIS)
Cs r (r 2r ln )/r 2  4G
2
r  ,   Cs /2Gr2
2


Non singular Solutions:
truncated at the radius
r  Cs / 4Gc
c /  14,  6.45
stable only if:


Stability using the Virial Theorem
Using the Virial theorem, it is possible to have a hint of the hydro equilibrium without
solving the problem entirely.
Consider a cloud of radius R, mass M, temperature T.
 2kb T
M2
Virial theorem: 2Utherm    3PextV  0 
V  G
 3PextV  0
m p  1
R
1  M 2kb T 1
M 2 
Pext 
 G 4 

3

4 m p  1 R
R 

leads to:

dPext
1  M 6kb T
M 2  dPext

 4G

 0  R  Rcrit


from which we get:
4 

dR
4R  m p  1
R 
dR

dPext
stability of the cloud requires:
 0  R  Rcrit
dR

There is a stable branch (weakly
condensed clouds) and an unstable one (more condensed

cloud).
Jeans mass and length
Equilibrium solutions and stability
Collapse
Gravo-turbulent support
Freefall Collapse
Consider a uniform sphere of mass M and a vanishing temperature.
Compute the acceleration of a shell (initial radius a):
2
d 2r
GM
4Ga 3
1 dr  4Ga 3 1 1 
 2 
   
  
2
2
dt
r
3r
2 dt 
3 r a 
1/ 2


r
1
8

G

 cos2 ( )    sin( 2 )  t

 3 
a
2
32G 1/ 2
r  0,    t ff  

 3 
2

All shells arrive at the same time in the centre.
The freefall time is not very different from:

t ff  G
1/ 2
Self-similar collapsing models
(Larson 69, Penston 69, Shu 77, Hunter 77, Bouquet et al. 85, Whitworth &Summer 85)
(analytical models are very important to understand the physics
and to validate the numerical methods. They present different
biaised and are complementary)
Self-similar Formalism
The fields a time t are proportional to their value at t=0.
r
x
, R(x)  (r,t)4Gt 2, u(x)  v(r,t) /Cs
Cst
Means that the initial conditions have been « forgotten ».
Spherical Collapse without rotation and magnetic field.

=>2 ordinary differential equations easy to solve !
Larson-Penston solution (69):
at t < 0 :
-the central density is rather «flat » the velocity not
far from homologous
-at infinity (supersonic part) the density is about 4
times the density of the SIS and the velocity is
supersonic (3.3 Cs).
Describes a very dynamical collapse induced by a
strong external compression.
at t > 0 :
accretion onto the singularity accretion rate:
3
30Cs /G

-make the assumption that the
prestellar phase is quasi-static
(eg slow contraction due to ambipolar
diffusion)
Density
Shu Solution (77)
-at t=0 the velocity vanishes and the
density is the SIS
Radius
The collapse starts in the centre
and propagates to the whole at the
sound speed.
Accretion rate:
Velocity
-at t>0 a rarefaction wave is launched
and propagates outwards:
inside-out collapse
Cs /G  2 106 Ms yr1
3
Radius
Shu 77

Gravitational Collapse: numerical models
Collapse of a critical Bonnor-Ebert sphere:
(Foster &Chevalier 93, Ogino et al. 99, Hennebelle et al. 03)
Velocity
In the internal region the numerical solution
Converges towards the Larson-Penston
solution.
In the external part, the collapse is well
described by the Shu solution for the density
but the velocity does not vanish.
Radius
Radius
Accretion rate
Accretion rate varies with time and reaches
about 5C 3 /G
3
Cs /G
s

Density
initial condition:
Unstable Bonnor-Ebert sphere near the
critical limit

time
time
Jeans mass and length
Equilibrium solutions and stability
Collapse
Gravo-turbulent support
Star Formation Efficiency in the Galaxy
Star formation efficiency varies enormously from place to place
(from about 0%, e.g. Maddalena's Cloud to 50%, e.g. Orion)
The star formation rate in the Galaxy is:
3 solar mass per year
However, a simple estimate fails to reproduce it.
Mass of gas in the Galaxy denser than 103 cm-3: 109 Ms
Free fall gravitational time of gas denser than 103 cm-3 is about:
From these two numbers, we can infer a Star Formation Rate of:
=> 100 times larger than the observed value
 dyn  3 /32G  2 10 6 years
=> Gas is not in freefall and is supported by some agent
Two schools of thought: magnetic field and turbulence

500 Ms/year
Turbulent Support and Gravo-turbulent Fragmentation
(Von Weizsäcker 43, 51, Bonazzola et al. 87, 92, Padoan & Nordlund 99, Mac Low 99,
Klessen & Burkert 00, Stone et al. 98, Bate et al. 02,Mac Low&Klessen 04)
turbulence observed in molecular
clouds: Mach number: 5-10
Supersonic Turbulence:
global turbulent support
If the scale of the turbulent fluctuations
is small compared to the Jeans length:
Cs,eff  Cs  Vrms /3
2
2
2
Now turbulence generates density fluctuations approximately
given by the isothermal Riemann jump conditions:

 / 0  M 2
Assuming that the sound speed which appears in the Jeans mass can be replaced by the
« effective » sound speed and since Vrms >> Cs:
Cs,eff  Cs  Vrms /3  Vrms /3
2
2
2
2
M J Cs,eff /   M J V
2
2
rms
(note that this assumes that the density fluctuation is comparable to the Jeans length which
contradicts the first assumption !)


Therefore the higher Vrms, the higher the Jeans mass.
However locally the turbulence may trigger the collapse because of converging flow that gather
material with a weak velocity dispersion.
=>a proper treatment requires a multi-scale approach similar to the Press-Schecter approach
developed in cosmology.
All numerical simulations (SPH, grid based, hydro, MHD)
show that:
Turbulence decays in 1 crossing time
MacLow & Klessen 04
Needs continuous energy injection !
External Injection: Turbulent Cascade ?
Feedback: outflows, winds... ?
Core Formation induced by Gravo-Turbulence
(Klessen & Burkert 01, Bate et al. 02, many others)
Dense cores are density fluctuations induced by the interaction between gravity and
Turbulence.
Evolution of the density field
of a molecular cloud
The calculation (SPH technique)
takes gravity into account but not
the magnetic field.
Turbulence induced the formation of
Filaments which become self-gravitating
and collapse
Klessen & Burkert 01
Without any turbulent driving:
the turbulence decays within one crossing time and the cloud
collapses within one freefall time
With a turbulent driving:
(random force is applyied in the Fourier space)
the collapse can be slown down or even suppressed
Maclow & Klessen 04
Mass accreted as a function of time:
-full line for a driving leading to a
turbulent Jeans mass of 0.6
(total mass is 1)
-dashed line for a turbulent Jeans
mass of 3
Small scale driving is more efficient
in supporting the cloud
Simulating fragmentation and accretion in a molecular clump (50 Ms)
Bate et al. 03
Collapse of a 50 solar mass cloud initially supported by turbulence.
6 millions of particules have been used and 95,000 hours of cpu have used
For a given mass, there is a maximum pressure
above which equilibrium is no more possible.
There is (often) a stable equilibrium solution and
an unstable one.
Pressure
Exact solution of the hydrostatic equilibrium
Pressure as a function of Volume
Bonnor 56
Volume
Mass
For a given pressure, there is a mass above which
equilibrium is not possible any more.
Mass as a function of radius
Chièze 87
Radius