1. Global Properties of Molecular Clouds Molecular clouds are some of the most massive objects in the Galaxy. mass: density: > temperature: 10-30 K -----> c s = 0.3-0.5 km/s magnetic field strength: 10-20  G lifetime < 10 8 yr --- clouds probably destroyed by UV radiation and winds (evaporation) from massive stars that form in them. Mass of molecular gas (CO) near star clusters correlate with age of clusters M o. M o. Mass distribution of molecular clouds: Clouds with masses of (Giant Molecular Clouds – GMCs) contain most of the molecular mass in the Galaxy.

2. Molecular Clouds are clumpy rather than homogeneous (Williams et al. 2000) Clumps have large (a few km/s) velocity dispersions, i.e. their individual velocities have big excursions with respect to the mean velocity) of the cloud Clumps have masses ~ 100-1000 Mo and densities ~ 500 cm -3. They comprise most of the mass of clouds but small fraction of volume N ~ M –(1.5-2) in all GMCs (universal) Clouds form from aggregation of clumps or break into clumps after they form?

3. Global equilibrium of GMCs (and smaller clouds) The Virial Theorem Idea: treat cloud globally with a general integral equation that describes the energy balance as well as the loss of energy balance. Then, if T, U, W and M are, respectively, the total kinetic energy, thermal energy, gravitational potential energy and magnetic energy, the cloud as a whole obeys: 2 tt   2T + 2U + W +  I = moment of intertia of cloud with density  In general the aim is to find out if a cloud can be in a state of energy balance (=global stability) (hence dI/dt = 0) and, if this is verified, which components of its energy budget are quantitatively most important to achieve stability Virial theorem is obtained solving integral form of Euler eq. (with Lorentz force) + Ampere’s law, continuity equation and Poisson equation discarding surface integrals (=discarding mass fluxes and external pressure as if the cloud is isolated, OK for GMCs)

4. Free-fall time Assume W cannot be balanced by any of the other forms of energies in the cloud. Then the cloud will undergo gravitational collapse and one can write: ½ d 2 I/dt 2 ~ GM 2 /R M, R = mass and radius of the cloud, assumed to be spherical for simplicity Then assuming I ~ MR 2 we can show that R wil shrink by about a factor of 2 over a (short) timescale; t ff ~ ( R 3 /GM) ½ ~ (G  ) -1/2 (use M/R 3 ~  ) ~ 7 X 10 6 yr for typical M,R for GMCs. Number smaller than observationally deduced cloud lifetimes! However no evidence that GMCs undergo global collapse, indeed internal motions of clumps are random instead of radial  means in reality one or more forces at play can balance W A more rigorous calculation (e.g. homogeneous sphere with zero internal pressure collapsing to a point) yields: t ff = (3  /G  1/2

5. Virial equilibrium: thermal, kinetic or magnetic? If cloud in equilibrium (i.e. W balanced exactly by other energies at play): 2T + 2U + W + M = 0 For typical GMCs parameters (M = 10 5 Mo, R = 25 pc, B = 20 mg, T=15 K,  V = 4 km/s) we find that: U/W ~ 3 x 10 -3, M/W ~ 0.3, T/W ~ 0.5 (use 3D  V assuming isotropy) Support comes from random kinetic energy of internal clumps,  V ~ V vir = (GM/R) 1/2 In many clouds T/W > 0.5 = clouds globally unbound (expansion instead of collapse) B not measured in typical gas of GMCs (clumps or interclump gas) yet but only in dense cores (dark clouds) or diffuse HI clouds around GMCs, so quite uncertain. If B strong and controls collapse cannot be simple geometry -- GMCs would flatten because collapse occurs perpendicular to magnetic field, but GMCs not flattened! Possibility: support does not come from ordered B but from magnetohydrodynamic waves associated with time dependent B field – MHD waves can exert isotropic “pressure” like thermal pressure (will see more on this in a few classes)

6. Three-dimensional velocity dispersion of molecular clouds decreases with decreasing molecular cloud size (observational fact, no theory): Larson’s law  V =  V 0 (L/L 0 ) n n~0.5  V 0 ~ 1 km/s L o = 1 pc Smallest clouds (below 1 pc) have three dimensional velocity dispersions comparable to gas thermal speed (~0.3-0.5 km/s). This means that for these clouds the support from internal kinetic energy is not as important as for large clouds. These small clouds are called cores and are thought to be the precursors for stars. Thermal energy alone is enough to support the cloud since: T ~ U ~ W (T ~ W for all cloud masses) The transition happens at a scale L therm = 3RTL o /  V 0 2 = 0.1 pc(T/10 K) (obtained by setting  V= v s in Larson’s law ) Virial relation for varying cloud sizes

7. Three-dimensional velocity dispersion of molecular clouds decreases with decreasing molecular cloud size (observational fact, no theory): Larson’s law  V =  V 0 (L/L 0 ) n n~0.5  V 0 ~ 1 km/s L o = 1 pc Smallest clouds (below 1 pc) have three dimensional velocity dispersions comparable to gas thermal speed (~0.3-0.5 km/s). This means that for these clouds the support from internal kinetic energy is not as important as for large clouds. These small clouds are called cores and are thought to be the precursors for stars. Thermal energy alone is enough to support the cloud since: T ~ U ~ W (T ~ W for all cloud masses) The transition happens at a scale L therm = 3RTL o /  V 0 2 = 0.1 pc(T/10 K) (obtained by setting  V= v s in Larson’s law ) Virial relation for varying cloud sizes

8. Molecular cores Cores have M of a few solar masses and L ~ 0.1 pc, so n H ~ 10 4 cm -3 = = densest component of molecular clumps Cores are found inside larger molecular cloud complexes but also in isolation (Bok Globules) -  cores are the places where stars form Internal structure of cores: (1) indication that inside cores the level of turbulence diminishes with increasing local density/smallest distance from the center (local extension of Larson’s law?). (2) there are starless cores but also cores with embedded point-like infrared sources associated with an embedded star. This is (1) proof that cores are the ultimate site of star formation and (2) shows that the star formation process is long enough that we can observe cores in different evolutionary stages (theoretical models will have to reproduce that) and (3) cores do not evaporate immediately after the star is born and heats the surrounding gas with ultraviolet and X-ray radiation.

9. Bok globules (isolated cores) 200 known Bok globules within 500 pc  proximity allows detailed study so these dense cores are ideal tool for studying the star formation process High-resolution mapping of cores. Many have embedded stars (e.g. B335 has a 1 L o star that is driving a powerful molecular outflow traced by CO) Puzzle: where are the sites of massive star formation? Known globules all contain low-mass stars (like the sun), no massive stars (> 3 Mo). Hypothesis (to be verified); Bok globules might be the only surviving part of a much larger molecular cloud complex that was destroyed by heating, radiation pressure and winds by one or more massive stars (cores in Orion, that has many massive stars, are different (mass, T, size) than cores in regions with few massive stars, e.g Taurus)