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) Equation of state: Heat Equation: Continuity Equation: Momentum Conservation: Poisson Equation:

Thermal support Consider a cloud of initial radius R and a constant temperature T 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: Continuity equation: Conservation of momentum: Poisson equation: Dispersion relation:

if => sonic wave (modified by gravity) whereas => there is an instability means: sonic propagation times smaller than the freefall time Jeans Length: both decreases with density Jeans mass: when the gas remains isothermal Hoyle (1953): recursive fragmentation As long as a cloud remains isothermal, it keeps fragmenting in smaller and smaller pieces Large wavelengths grow more rapidly than small wavelengths (problematic for fragmentation) For:

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: but for: more unstable mode = typical width of the filaments : suggest : fragmentation possible once equilibrium is reached in one direction Exact Equilibrium Solutions in 2D (Schmid-Burgk 1976) Fragmentation of a sheet into filaments Filaments

Fragmentation of filament in core Self-gravitating filaments ( Ostriker 64 ) -profile in 1/r 4 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. Dutrey et al 91 Fiege & Pudritz 00 cores Development of the gravitational instability in a filament: Formation of an elongated core

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) Non singular Solutions: truncated at the radius 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. Virial theorem: leads to: from which we get: stability of the cloud requires: 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): All shells arrive at the same time in the centre. The freefall time is not very different from:

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. 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:

Shu Solution (77) -make the assumption that the prestellar phase is quasi-static (eg slow contraction due to ambipolar diffusion) -at t=0 the velocity vanishes and the density is the SIS -at t>0 a rarefaction wave is launched and propagates outwards: inside-out collapse The collapse starts in the centre and propagates to the whole at the sound speed. Accretion rate: Velocity Density 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) initial condition: Unstable Bonnor-Ebert sphere near the critical limit 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. Accretion rate varies with time and reaches about Velocity Density Accretion rate Radius time Radius 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 10 3 cm -3 : 10 9 Ms Free fall gravitational time of gas denser than 10 3 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

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

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.

All numerical simulations (SPH, grid based, hydro, MHD) show that: Turbulence decays in 1 crossing time Needs continuous energy injection ! External Injection: Turbulent Cascade ? Feedback: outflows, winds... ? MacLow & Klessen 04

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 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 Maclow & Klessen 04

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)

Exact solution of the hydrostatic equilibrium 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 as a function of Volume Bonnor 56 For a given pressure, there is a mass above which equilibrium is not possible any more. Mass as a function of radius Chièze 87 Pressure Volume Mass Radius