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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 cm3 ) Heat Equation: Continuity Equation: Momentum Conservation: Mom. Cons. for ions: Induction Equation: Poisson Equation: T 10K t (v ) 0 (t v vv ) P in i (vi v ) 1 i (t v i v iv i ) in i (v v i ) B B 4 r r r r t B (B v i ) 0 4G 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 2G) =>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 criticalmass 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: Gl 2 G2 / 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 R3 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 G2 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 2Gk 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