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

Maxwell-Faraday Equation:

Frame of the Ions:

Neglect electrons inertia (and Hall effect), velocities small compared to the speed of Light=> Ohm’s law

lead to:

Ideal MHD:

Induction equation with other non-ideal effects, Ambipolar Diffusion orHall effect.

€

∂t

r B + c

r ∇ ×

r E =

r 0

€

rB ' =

r B

r E ' =

r E +

1

c

r v i ×

r B

€

rj = σ

r E '

€

∂t

r B + c

r ∇ ×

r E ' −

1

c

r v i ×

r B

⎛

⎝ ⎜

⎞

⎠ ⎟=

r 0 ,

r j =

1

4π

r ∇ ×

r B

€

∂t

r B +

r ∇ ×

r B ×

r v i( ) = ηΔ

r B

€

η =0

Induction equation can also be written as:

€

∂t

r B +

r ∇ ×

r B ×

r v ( ) =

r 0

=> ∂t

r B +

r v .

r ∇r B +

r B .

r ∇r v +

r B

r ∇.r v =

r 0

€

∂t

r ω +

r v .

r ∇r ω +

r ω .

r ∇r v +

r ω

r ∇.r v =

r 0

⇒ ∂t

r ω +

r v .

r ∇r ω +

r ω .

r ∇r v +

r ω

r ∇.r v = −

r ∇ ×

r ∇P

ρ

⎛

⎝ ⎜

⎞

⎠ ⎟

=r 0 if the flow is barotropic P = kρ γ

Euler equation can also be written as:

€

∂tρ +r

∇.(ρr v ) = ∂tρ +

r v .

r ∇ρ + ρ

r ∇.r v = 0

Comparison with continuity equation is instructive:

Same form except for:

This term is special to vectors. It means that the field can be amplified by shear motions.

€

rB .

r ∇r v

Flux freezing:(also applies to vorticity equation)

Magnetic flux, , across a surface S(t) is conserved along time:

which implies:

€

φ= r

B .dr S

S( t )

∫∫

€

dφ

dt= ∂t

r B .d

r S

S( t )

∫∫ +r B .

r v × d

r l ( )

l(t )

∫

€

rB .

r v × d

r l ( )

l( t )

∫ = dr l .

r B ×

r v ( )

l( t )

∫ = dr S

S(t )

∫∫ .r

∇ ×r B ×

r v ( )

€

dφ

dt= ∂t

r B .d

r S

S( t )

∫∫ + dr S

S( t )

∫∫ .r

∇ ×r B ×

r v ( ) = 0

The equations (typical for molecular cloud)(Spitzer 1978, Shu 1992)

Equation of state:

Ionisation Equilibrium:

Heat Equation:

Continuity Equation:

Momentum Conservation:

Mom. Cons. for ions:

Induction Equation:

Poisson Equation:

€

P = kb /mp ρT

€

ρ >>ρi , ρ i = c ρ (ρ >103cm−3 )

€

T =10K

€

∂tρ + ∇(ρr v ) = 0

€

ρ(∂t

r v +

r v ∇

r v ) = −

r ∇P + ρ

r ∇φ + ν inρρ i(

r v i −

r v )

€

ρi(∂t

r v i +

r v i∇

r v i) = ν inρρ i(

r v −

r v i) +

1

4π

r ∇r B ×

r B

€

∂t

r B +

r ∇(

r B ×

r v i) = 0

€

Δφ=−4πGρ

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:

=>monofluide equation with a non-linear diffusion term

diffusion time:

dynamical time:

Virial equilibrium +

=>independant of M and Lfor a ionisation of , one obtains:

€

rv i =

r v +

1

4πν inρρ i

r ∇r B ×

r B

€

∂t

r B +

r ∇(

r B ×

r v ) = −

1

4πγ inρρ i

r ∇(

r B × (

r ∇r B ×

r B ))

€

τ ad = 4πγ inρ iL2 /B2

€

τ dyn =1/ Gρ

€

ρi = C ρ ⇒ τ ad /τ dyn = γ inC /(2 2πG )

€

10−7

€

τ ad /τ 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 ρ are anticorrelated)

Fast magneto-acoustic waves (coupling between Lorentz force and thermal pressure, B and ρ are correlated)

€

ca =Bx

ρ,c f ,s =

γp +r B 2 ± (γp +

r B 2)2 − 4γpBx

2

2ρ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1/ 2

Magnetic support

Consider a cloud of mass M, radius R, treated by BFlux conservation:

magnetic / gravitational energy:

Independent of R, B dilute Gravity

Estimation of the critical mass to flux ratio:(Similar to Egrav=Emag but based on Virial theorem)

mass-to-flux larger than the critical value: cloud is supercriticalmass-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 G

=> Ambipolar diffusion can slow down collapse by almost a factor 10

€

φ∝BR2

€

B2R3

M 2 /R∝ (φ / M)2

€

(φ / M)crit = G /0.13

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 initialangular momentum

magnetic field parallel to the rotation axis:

magnetic field orthogonal to the axis:

Since ρcore

/ ρenv

>> 1, the braking is more efficient perpendicularly to the rotation axis

cloud

B

B

cloud

€

τ para ≈ (ρ core /ρ env ) × (Zcore /Va )

€

τ perp ≈ ((1+ ρ core /ρ env )1/ 2 −1) × (Rcore /2Va )

Matthews et al. 2001

Structure of the Magnetic FieldInterstellar grains are perpendicular to B, therefore the intensity is slightly polarized in this direction.

Polarisation map for TaurusB is organised and seems to beroughly perpendicular to thefilaments.

Polarisation map for Orion. The polarisation is aligned (top) or perpendicular (bottom)Helical structure has been proposed.

Flux conservation:

If the magnetic field is dynamically important, the gas flows along the field lines

Gravitational energy:

Mechanical equilibrium:

€

φ∝BR2 = cst

€

ΣR2 = cst ⇒ B ∝ Σ

€

Σ= ρdl∫ ≈ ρl

€

φ ≈Gρl2 ≈ GΣ2 /ρ ∝ B2 /ρ

€

φ ≈σ 2

€

⇒ B ≈ σρ1/ 2

Measurement of the Magnetic Field Intensity(based on Zeeman effect)

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: 500 Ms/year

=> 100 times larger than the observed value

=> Gas is not in freefall and is supported by some agent

Two schools of thought: magnetic field and turbulence

€

τ dyn = 3π /32Gρ ≈ 2 106 years

Control of star formation by magnetic field(Mestel &Spitzer 56, Mouschovias & Spitzer 76, Shu et al. 87)

M/ = 0.1 (M/)crit

M/ = 1 (M/)crit

velocity is about 0.2 Cs velocity is about 0.5 Cs (0.04 km/s) (0.1 km/s)

Basu & Mouschovias 95

Den

sity

Den

sity

Vel

ocit

y

Vel

ocit

y

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:

M/ remains lower than 2 during the collapse

Basu & Mouschovias 95

Very subcritical

Critical

Very subcritical

Criticalm

ass/

flux

Density

Time

Den

sité

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 molecularclouds: Mach number: 5-10

Supersonic Turbulence:global turbulent support

If the scale of the turbulent fluctuationsis small compared to the Jeans length:

Now turbulence generates density fluctuations approximately given by the isothermal Riemann jump conditions:

€

Cs,eff2 ≈ Cs

2 + Vrms2 /3

€

ρ /ρ 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:

(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.

€

Cs,eff2 ≈ Cs

2 + Vrms2 /3 ≈ Vrms

2 /3

€

MJ ∝ Cs,eff2 / ρ ≈ MJ ∝Vrms

2

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 notthe 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 cloudcollapses 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

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 Jeansmass of 3

Small scale driving is more efficient in supporting the cloud

Maclow & Klessen 04

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 timefor different values of the normalised mass-to-flux ratio.

Initially subcritical clouds need severalfreefall 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 themass fraction at high density. This is because turbulence creates shocks in which ambipolar diffusion occurs.

Initial mass function and Prestellar core mass function

Alves et al. 2007Motte et al. 1998

Initial Mass Function obtained in many different environments (Field, clusters…)Not clear yet to which extent it is universal (Elmegreen 2008).

(Motte et al. 1998, Alves et al. 2007 , Johnstone et al. 2002, Enoch et al. 2008, Simpson et al. 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

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

Bate et al. 03

Simulating fragmentation and accretion in a molecular clump (50 Ms)

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

€

Μach =12

€

Μach = 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:

When R decreases, Erot/Egrav increases! Centrifugal support becomes dominant and the collapse is stopped

Formation of a centrifugally supported disk (Larson 72)

A major problem in astrophysics: Transport of Angular Momentum

€

j = R2ω(t) = R02ω0

€

E rot

Egrav

=MR2ω2

GM 2 /R∝

1

R

€

v 2

R= j 2R−3 =

GM

R2⇒ rd =

j 2

GM

Disk stability(Toomre 64, Binney & Tremaine)

In disk, thermal and rotational supports are important. Consider a spherical piece of fluid of radius R of a uniformly rotating disk at the angular rotation , of surface density Σ.

Thermal, gravitational and rotational energies are:

Disk is stable if thermal or rotational energies are greater than the gravitational energy (times some close to unity number).

Stability requires: Q is called the Toomre parameter

Rigorous (complex) linear analysis can be performed and leads to the dispersion relation:

€

E therm ∝ MCs2 ∝ ΣCs

2 ×δR2, Egrav ∝GM 2

δR∝ GΣ2 ×δR3, E rot ∝ M(δR ×ω)2 ∝ Σω2 ×δR4

€

E therm > Egrav ⇒ δRtherm <Cs

2

GΣ, E rot > Egrav ⇒ δRrot >

GΣ

ω2

€

Rtherm > δRrot ⇒ Q =ωCs

GΣ>1

€

2 = Cs2k 2 − 2πGΣk + κ 2

RrotRtherm

unstable

XYhydro

XZhydro

XYMHD=2

XZMHD=2

300 AU

Zoom into the central part of a collapse calculation(1 solar mass slowly rotating core)

B,

B,