Download turbulence - "A" Laboratory, Department of Physics/Astrophysics

An introduction to the Physics of the
Interstellar Medium
V. Magnetic field in the ISM
Patrick Hennebelle
Induction Equation:
Consider first a plasma of electrons and ions
r
r r r
t B  c E  0
r' r
B B
r' r 1r r
E  E  vi  B
c
Maxwell-Faraday Equation:
Frame of the Ions:

Neglect electrons inertia
effect), velocities small compared to the speed of
r (andr Hall
Light=> Ohm’s law j  E '
r
r r ' 1 r r  r
r
1 r r
t B  c
 B
E  v i  B 0, j 


c
4
r r
r r
r

 B   B  v  B
lead to:

t
i

0
 Ideal MHD:
Induction equation with other non-ideal effects, Ambipolar Diffusion or

Hall effect.


Induction equation can also be written as:
r r
r r r
t B   B  v  0
r r rr r rr rr r r
  t B  v .B  B.v  B.v  0


Euler equation can also be written as:
r r r r rr rr r r
t v . .v  .v  0
r
r

r
r
r
r
r r r r r r r
P 
 t v . .v  .v    
  
r
 0 if the flow is barotropic P  k 
Comparison with continuity equation is instructive:
r
r r
r
r r
t   .(v )  t   v .  .v  0
r rr
Same form except for: B.v
This term is special to vectors. It means that the field can be amplified by shear motions.

Flux freezing:
(also applies to vorticity equation)
Magnetic flux,  
along time:
r r
B.dS , across a surface S(t) is conserved

S(t )
r
r r
r r
d
  t B.dS   B. v  dl
dt S(t )
l(t )
 r r
r
r r r
r r
r r
 B. v  dl   dl . B  v   dS. B  v


l(t )


l(t )

which implies:


d

dt




S(t )
r r
 t B.dS 

S(t )
S(t )
r r
r r
dS . B  v  0


The equations (typical for molecular cloud)
(Spitzer 1978, Shu 1992)
P  kb /m p T
Equation of state:
Ionisation Equilibrium:
  i , i  c  ( 103 cm3 )

Heat Equation:

Continuity Equation:
Momentum Conservation:

Mom. Cons. for ions:

Induction Equation:

Poisson Equation:
T  10K
t   (v )  0

(t v  vv )  P     in i (vi  v )
1
i (t v i  v iv i )   in i (v  v i ) 
B  B
4
r r r r
t B  (B  v i )  0
  4G
Ambipolar diffusion
(Mestel &Spitzer 56, Mouschovias & Spitzer 76, Shu et al. 87)
-ions feel the Lorentz force
-there is a friction between neutrals and ions
but the neutrals diffuse through the ions
Approximation : Inertia of ions is negligeable.
Lorentz force ~ friction force:
1
vi  v 
B  B
4 in i
1
t B  (B  v )  
(B  (B  B))
4 in i
=>monofluide equation with a non-linear diffusion term

 ad  4 in i L /B
diffusion time:
dynamical time:
Virial equilibrium +

2
 dyn 1/ G
i  C    ad /  dyn   inC /(2 2G)
=>independant of M and L

7
for a ionisation of 10 , one obtains:


2
 ad /  dyn  8
(very) Brief description of the MHD waves
The MHD equations give rise to 3 types of waves.
Alfvén waves: transverse mode (analogous to the vibration of a string)
Slow magneto-acoustic waves (coupling between Lorentz force and thermal pressure, B
and  are anticorrelated)
Fast magneto-acoustic waves (coupling between Lorentz force and thermal pressure, B
and  are correlated)
1/ 2
r2
r2 2
2 

p  B  (p  B )  4 pBx 
Bx

ca 
,c f ,s 


2





Magnetic support
Consider a cloud of mass M, radius R, treated by B
Flux conservation:
  BR 2
magnetic / gravitational energy:
B 2R 3
2

(

/
M)
M 2 /R
of R, B dilute Gravity
Independent
Estimation of the criticalmass to flux ratio:
(Similar to Egrav=Emag but based on Virial theorem)
( / M)crit  G /0.13
mass-to-flux larger than the critical value: cloud is supercritical
mass-to-flux smaller than the critical value: cloud is subcritical
(If the cloud is subcritical, it is stable for any external
pressure !)

For a core of 1 Msol and 0.1 pc: critical B is about 20 mG
=> Ambipolar diffusion can slow down collapse by almost a factor 10
Magnetic braking
(Gillis et al. 74,79, Mouschovias & Paleologou 79,80, Basu & Mouschovias 95, Shu et al. 87)
rotation generates torsional Alfvén waves which carry angular momentum outwards

Typical time: AW propagate far enough so
that the external medium receives angular
momentum comparable to the cloud initial
angular momentum
B
cloud
B
magnetic field parallel to the rotation axis:
 para  ( core / env )  (Z core /Va )
magnetic field orthogonal to the axis:
 perp  ((1 core / env ) 1)  (Rcore /2Va )
1/ 2

cloud
Since core / env>> 1, the braking is more efficient perpendicularly to the rotation axis

Structure of the Magnetic Field
Interstellar grains are perpendicular to B, therefore the intensity is slightly polarized in this
direction.
Polarisation map for Taurus
B is organised and seems to be
roughly perpendicular to the
filaments.
Polarisation map for Orion.
The polarisation is aligned (top) or
perpendicular (bottom)
Helical structure has been proposed.
Matthews et al. 2001
Measurement of the Magnetic Field Intensity
(based on Zeeman effect)
Flux conservation:
  BR 2  cst
If the magnetic field is dynamically important, the gas
flows along the field lines

   dl  l
R 2  cst  B  
Gravitational energy:

  Gl 2  G2 /   B 2 / 
Mechanical
equilibrium:


 
2
 B  
1/ 2
An introduction to the Physics of the
Interstellar Medium
VI. Star formation:
Efficiency, IMF, disk and fragmentation
Patrick Hennebelle
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  2106 years
=> Gas is not in freefall and is supported by some agent
Two schools of thought: magnetic field and turbulence

500 Ms/year
Control of star formation by magnetic field
Density
Density
Velocity
Velocity
(Mestel &Spitzer 56, Mouschovias & Spitzer 76, Shu et al. 87)
M/ = 0.1 (M/)crit
velocity is about 0.2 Cs
(0.04 km/s)
Basu & Mouschovias 95
M/ = 1 (M/)crit
velocity is about 0.5 Cs
(0.1 km/s)
Critical
Densité
For M/ < 0.1 M/crit
the star forms after
15 freefall times.
For M/ = M/cri
the star forms after
3 freefall times.
M/ in the centre as a function of time:
Very subcritical
Time
mass/flux
Critical
M/remains lower than 2
during the collapse
Very subcritical
Basu & Mouschovias 95
Density
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.
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
Turbulent and Magnetically supported Clouds
(Li & Nakamura 04, Basu & Ciolek 04)
The clouds are magnetically critical and turbulent.
Turbulence creates supercritical regions which collapse
Li & Nakamura 04
Fraction of mass at high density
(likely to collapse) as a function of time
for different values of the normalised
mass-to-flux ratio.
Initially subcritical clouds need several
freefall times to collapse whereas
supercritical clouds collapse in a freefall time
Fraction of mass at high density as a
function of time for different value of
the initial Mach number.
The larger the turbulence, the higher the
mass fraction at high density.
This is because turbulence creates shocks
in which ambipolar diffusion occurs.
Initial mass function and Prestellar core mass function
(Motte et al. 1998, Alves et al. 2007 , Johnstone et al. 2002, Enoch et al. 2008, Simpson et al. 2008)
Motte et al. 1998
Alves et al. 2007
Initial Mass Function obtained in many different environments (Field, clusters…)
Not clear yet to which extent it is universal (Elmegreen 2008).
Theories/simulations of the IMF
-independent stochastics processes: Zinnecker 1984, Elmegreen 1997
-outflows: Adams & Fatuzzo 1996, Shu et al. 2004
-gravitation/accretion: Inutsuka 2001, Basu & Jones 2004, Bate & Bonnell 2005
-gravitation/turbulence: Padoan & Nordlund 2002, Tilley & Pudritz 2004,
Ballesteros et al. 2006, Padoan et al 2007, Hennebelle & Chabrier 2008
Many remaining questions
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
Direct numerical calculation (Bate & Bonnell 2005, Bate 2009):
Analytical calculations (Padoan & Nordlund 02, Hennebelle & Chabrier 08,09)
Statistical counting of the self-gravitating fluctuations arising in supersonic turbulence
ach 12

ach  6

Comparison between Hennebelle & Chabrier’s IMF
and numerical results from Jappsen et al. 2005
Comparison between Hennebelle & Chabrier’s IMF
and Chabrier’s IMF (compilation of observation)
Disks and Fragmentation
Grosso et al.
(2003)
Duchêne et al. 2003
Centrifugal Support and Angular Momentum Conservation
Consider a cloud of initial radius R rotating at an angular
velocity 0.
Angular Momentum Conservation:
j  R2(t)  R0 0
2
When R decreases, Erot/Egrav increases! Centrifugal support becomes dominant and
the collapse is stopped 
E rot
MR 
1


2
E grav GM /R R
2
2
Formation of a centrifugally supported disk (Larson 72)

2
v2
GM
j
 j 2 R3  2  rd 
R
R
GM
A major problem in astrophysics: Transport of Angular Momentum
Disk stability
(Toomre 64, Binney & Tremaine)
In disk, thermal and rotational supports are important. Consider a spherical piece of fluid of
radius dR of a uniformly rotating disk at the angular rotation , of surface density .
Thermal, gravitational and rotational energies are:
GM 2
E therm  MCs  C  dR , E grav 
 G2  dR 3 , E rot  M(dR   ) 2   2  dR 4
dR
Disk is stable if thermal or rotational energies are greater than the gravitational energy
(times some close to unity number).
2
2
s
2
Cs 2
E therm  E grav  dRtherm 
,
G
E rot  E grav  dRrot 
G
2
unstable
dRtherm

Stability requires:
dRtherm  dRrot  Q 
dRrot
Cs
G
1
Q is called the Toomre parameter
Rigorous (complex) linear analysis can be performed and leads to the dispersion relation:

 2  Cs2 k 2  2Gk   2
Zoom into the central part of a collapse calculation
(1 solar mass slowly rotating core)
XY
hydro
XY
MHD
m=2
B, 
XZ hydro
XZ
MHD
m=2
B, 
300 AU