Enrique Vázquez-Semadeni Instituto de Radioastronomía y Astrofísica, UNAM, México

1. Molecular Cloud Formation and Evolution Enrique Vázquez-Semadeni Instituto de Radioastronomía y Astrofísica, UNAM, México IRyA

2. INTRODUCTION

3. The interstellar medium (ISM) is turbulent, magnetized (e.g., Heiles & Troland 2003, 2005), self-gravitating and subject to radiative heating and cooling. • These processes lead to the formation of density enhancements that constitute clouds, and clumps and cores within them (Sasao 1973; Elmegreen 1993; Ballesteros-Paredes et al. 1999). • This course: • Outline of underlying physical processes. • Discuss their interaction in the formation and evolution of molecular clouds (MCs).

4. Brief summary of ISM structure: • The ISM contains gas in a wide range of conditions: • Note these are ranges, not single values. • Possibly a continuum.

5. A hierarchical (nested) structure: Engargiola et al. 2003: Study of M33 Color image: HI distribution Circles: Giant Molecular Clouds (GMCs) GMCs seem to be the “tip of the iceberg” of the density gas distribution. They conclude thatGMCs form out of the HI.(See also Blitz et al. 2007, PPV.)

6. MCs can be observed by means of various tracers: • Molecular line emission (mm wavelengths). • Cold and warm dust emission (mm and mid-IR). • Near-IR dust extinction of background stars.

7. Image: 8-21 m emission (warm [50-100 K] dust) Contours:1.2-mm emmision (cold[10-20 K] dust) Image: I-band (8250 A) image Dark globule (BHR71) R ~ 0.4 pc M ~ 40 M n ~ 1x104 cm-3 L ~ 9 L Bourke et al. 97 Photos courtesy of D. Mardones

8. Quad1 - optical Dust absorption Visible starlight

9. Quad1 - optical CO emission

10. GMCs are extremely hierarchical as well. CO J(10) emission in Cygnus OB7 complex(Falgarone et al. 1992). “Clumps” Really, a continuum. “Cores” Typically attributed to supersonic turbulence.

11. HOW DO MCs FORM?

12. BASIC PHYSICAL PROCESSES

13. R r, P, T 1. Inestabilidad gravitacional de Jeans • Aquí: Cálculo aproximado para nube de gas esférica de radio R, y densidad r y temperatura T uniformes. • En realidad, cálculo por teoría de perturbaciones lineales. • La esfera está soportada en contra de su auto-gravedad por su presión térmica (en realidad, su gradiente) P = rkT/m = rc2, donde m= masa molecular promedio k = constante de Boltzmann c = (kT/m)1/2 = velocidad del sonido isotérmica.

14. La energía interna de la nube de gas es: • y la energía gravitacional es: • Igualando ambas encontramos el radio crítico para que la nube de densidad r y temperatura T se encuentre al borde del colapso gravitacional (que se da cuando |Eg| > Ei):

15. Longitud de Jeans • Haciendo el análisis perturbativo exacto para perturbaciones sinusoidales en un medio uniforme infinito se obtiene: • Entonces, regiones de densidad r y temperatura T=mHc2/k con tamaños mayores que la Longitud de Jeans se colapsan gravitacionalmente, si sólo están “soportadas” (“sostenidas”) por la presión térmica en contra de su autogravedad. • Se utiliza mucho también la (masa de una esfera de densidad r y radio LJ/2) Masa de Jeans

16. 2. El cociente masa/flujo magnético • Consideremos ahora el soporte proporcionado por un posible campo magnético uniforme B en la dirección x. La energía magnética es: B

17. Entonces el cociente de la energía gravitacional a la energía magnética es: donde F = pBR2 es el flujo magnético a través de la sección transversal de la nube. • En ausencia de disipación o difusión, el flujo se conserva(“congelamiento del flujo”, “flux freezing”).

18. La condición para que haya colapso, |Eg| > Em, entonces implica: • En general, el factor numérico varía dependiendo de la geometría, y de cálculos más precisos en geometría cilíndrica (Nakano & Nakamura 1978), se obtiene: • Una nube con • (M/F) > (M/F)crit se llama magnéticamente supercrítica • (M/F) < (M/F)crit se llama magnéticamente subcrítica

19. 3. Velocity convergence: A density enhancement requires an accumulation of initially distant material into a more compact region. i.e., need to have a convergence of the velocity field into the region. However, most models of clouds have relied on the notion of static equilibrium! Continuity equation (mass conservation)

20. 4. ISM thermodynamics: • (See discussion in Vázquez-Semadeni+2003, LNP, 614, 213.) • A key property of the atomic ISM is that it is thermally unstablein some regimes(Field 1965). • The internal energy equation (per unit mass) is • where n = number density, in units of cm-3, (r = mmH n) • G is the (radiative) heating function, and • L is the (radiative) cooling function. • Define thermal equilibrium by the condition • In the absence of local energetic events, the heating function to first approximation satisfies

21. The interstellar cooling function TI under the isobaric criterion. TI under the isochoric and the isobaric criteria. Dalgarno & McCray 1972; see also Wolfire+95.

22. The condition for instability is (Field 1965): At cst. P, if T , L , so T even more.  Runaway increase of T. The growth of this mode can really operate isobarically if, in a given fluid parcel, so that the flow can maintain P ~ cst by moving the gas to change r and equalize P. • The isobaric mode.

23. This mode is easiest to understand using the thermal equilibrium condition to eliminate the temperature from the ideal gas equation of state, to write

24. WNM (stable) Mean ISM thermal pressure Peq, at which heating G equals cooling nL. • Due to the forms of the cooling and heating, the behavior of Peq is: • Where dP/dr < 0, the gas is unstable under the isobaric criterion: If r , P , and the fluid parcel is even further compressed. Runaway compression until dP/dr > 0 again. • The flow segregates into a cold/dense (T~ 50 K, n ~ 50 cm-3) and a warm/diffuse (T ~ 6000 K, n ~ 0.5 cm-3) phase. CNM (stable) Wolfire et al. 1995 Thermally unstable range

25. … and, aided by gravity, an overshoot to molecular cloud conditions (Hartmann+2001; Vázquez-Semadeni+2007; Heitsch & Hartmann 2008). • The relevant scale of condensation in this case is that of the compressive wave, not the most unstable (small) scale of the linear case. • Transonic (Mach # ~ 1) compressions in the linearly stable WNM can nonlinearly trigger a transition to the CNM… (Hennebelle & Pérault 1999; Koyama & Inutsuka 2000).

26. WNM n, T, P, -v1 WNM n, T, P, v1 Can be modeled as the collision of a flow against a wall. • MCs are then probably born as thin sheets of cold atomic gas (VS+06, ApJ, 643, 245)produced by transonic converging flows in the WNM that undergo a transition to the cold phase: CNM

27. 1D numerical simulationsof thin sheet formation by transonic converging WNM streams (VS+06, ApJ, 643, 245): • Consistent with observations suggesting that CNM clouds are sheets with thicknesses ~ 0.1 pc with lengths of up to tens of parsecs (Heiles & Troland 2003, ApJ, 586, 1067).

28. 5. The atomic-to-molecular transition: • As the cloud thickens and increases its column density, it undergoes several changes (Franco & Cox 86; Hartmann+01, ApJ 562, 852; Bergin+04): • It begins to self-shield from dissociating radiation and to become molecular. • Requires column densities N ~ 1—2 x 1021 cm-2 (AV ~ 0.5—1, S ~ 10—20 Msun pc-2). • Locally, H2 molecule formation occurs on timescales t ~ 109/(n/cm-3) yr (Hollenbach+71).

29. It becomes gravitationally dominated: • The total pressure Pc inside a cloud of column density S subject to an external pressure Pe and its own weight is (Ledoux 1951; Spitzer 1978): • Thus, the cloud’s weight (assuming molecular composition, and thus a mean particle weight of ~ 2.35) exerts a pressure larger than Pe for where (Pe/k)4 is the pressure divided by Boltzmann’s constant in units of 104 K cm-3. • Thus, the column density for becoming gravitationally dominated is very similar to that for becoming molecular. • Note that P/k near the solar circle is ~ 2500 K cm-3.

30. It becomes magnetically supercritical (magnetic forces become insufficient to support its weight): • From the Virial Theorem, the condition that the magnetic energy equals the gravitational energy in a cylindrical cloud of radius R permeated by a magnetic field parallel to its axis is (Nakano & Nakamura 78; Hartmann+01; Vázquez-Semadeni+11, MNRAS, 414, 2511): • Again, very similar to the column density necessary for becoming molecular and gravitationally dominated.

31. Transalfvenic turbulence and/or gravitatonal contraction in molecular clouds. • This explains the observation that atomic clouds are observed to be magnetically subcritical, while molecular clouds and cores are observed to be supercritical (Crutcher+10; Crutcher 12, ARAA, 50, 29): Accumulation stage in CNM. Cloud evolution

32. Summarizing: • As the cloud thickens and increases its column density, it undergoes several changes for solar galactocentric distance conditions (Franco & Cox 86; Hartmann+01, ApJ 562, 852; Bergin+04): • The cloud begins to self-shield from dissociating radiation and to become molecular; • Gravity overcomes thermal pressure; • Gravity overcomes magnetic support.

33. Note that the clouds may rapidly become strongly self-gravitating : • Because, if MCs form out of a phase transition from the warm/diffuse to the cold/dense atomic phase (Gómez & VS 14, ApJ 791, 124): r 102r, T  10-2 T  Jeans mass, MJ ~ r-1/2 T3/2, decreases by ~ 104 upon warm-cold transition.

34. 6. Turbulence.

35. When a dense cloud forms out of a collision of WNM streams, it “automatically” • acquires mass (accretion of WNM from the streams); • acquires turbulence (through TI, NTSI, KHI? – Vishniac 1994; Walder & Folini 1998, 2000; Koyama & Inutsuka 2002, 2004; Audit & Hennebelle 2005; Heitsch et al. 2005, 2006; Vázquez-Semadeni et al. 2006). Hennebelle, Banerjee, Vázquez-Semadeni+08, A&A, 486, L43

36. The nonlinear thin-shell instability (NTSI; Vishniac 1994, ApJ, 428, 186) • The Kelvin-Helmholtz instability.

37. But what is turbulence? • Turbulence is a regime of flow. • Chaotic and disordered. • Involving coherent patterns of motions at many different scales. da Vinci, ca. 1500 37

38. Swift’s poem: A flea has smaller flea that on him prey; and these have smaller yet to bite’em, and so proceed at infinitum. Richardson’s adaptation to turbulence: Big whirls have little whirls which feed on their velocity, and little whirls have lesser whirls, and so on to viscosity.

39. k • La distribución de energía entre las diferentes escalas de movimientos se describe a través delespectro de energía cinéticade la turbulencia: • Se toma la Transformada de Fourier del campo de velocidades: • k es el vector de onda y k = |k| = 2p/l es el número de ondas. l es la longitud de onda. • El espectro de energía E(k) es la suma de las energías (específicas; i.e., por unidad de masa) en los modos de Fourier con números de onda k’ entre k y k+dk: kz E(k) es la energía (por unidad de masa, o “específica”) contenida |uk|2 en el cascarón esférico con k’ entre k y k+dk en el espacio de Fourier. ky kx

40. The energy spectrum E(k) is the specific (i.e., per unit mass) kinetic energy in turbulent motions of characteristic scale l ~1/k. • The most prominent theory for the functional form of the kinetic energy spectrum (i.e., for the energy in coherent motions as a function of size scale) is that ofKolmogorov(1941, a.k.a. K41). • Since turbulence is characterized by an energy spectrum, E(k), the characteristic “turbulent velocity difference” depends on scale, u = u(l). • Kolmogorov (incompressible) turbulence: u ~ l1/3 • Burgers (highly compressible) turbulence: u ~ l1/2 • (Note similarity to Larson’s scaling.)

41. If the turbulence is supersonic, it produces shocks and therefore density fluctuations, which may constitute clouds and their substructure, or seeds for subsequent gravitational collapse (von Weizsacker 1951, Sasao 1973, Elmegreen 1993, Ballesteros-Paredes+1999). Padoan et al. 2016

42. A lognormal probability distribution of density fluctuations in isothermal, supersonic turbulence: • E.g., for isothermal, hydro shocks, the density jump: r2/r1 = Ms2, where Ms= sonic Mach # in upstream gas. • Jumps are multiplicative density enhancements; additive in log r. • At a given location, a succession of compressive waves produces the instantaneous density value. Central Limit Theorem applies. In isothermal flows: lognormal distribution (Vázquez-Semadeni 94; Passot & VS 98)

43. Turbulence has been advocated as a source of pressure that may support MCs against their weight: • Why would the clouds be supported? • The Zuckerman & Palmer (1974) conundrum: • If MCs were free-falling, the SFR would be much larger than observed: • Free-fall estimate of SFR: • Observed rate is SFRobs ~ 2—3 Msun yr-1; i.e., ~100x lower. • Thus, since the 70s, MCs have been assumed to be in approximate virial equilibrium between their nonthermal motions and their self-gravity.

44. The observed nonthermal motions in MCs are strongly supersonic: 12CO • Linewidths correspond to velocity dispersions of a few km s-1. • Sound speed in MCs is cs ~ 0.2 km s-1. • Mach number • Ms ~ 10. Wilson+70

45. The “turbulent pressure”, rcs2, is then Ms2 times larger than the thermal pressure, and thus thought to provide additiional support. • The turbulent-support interpretation was reinforced by the observation of a nearly virial magnitude of the nonthermal motions:

46. Larson’s (1981) relations: Linewidth-size But see Kegel 1989, Scalo 1990 for criticisms. Density-size • S ~ nL • = cst. Larson 1981

47. Which, together, correspond to approximate virial equilibrium: Larson 1981

48. This has been interpreted in terms of a “dual role” of supersonicturbulence(Vázquez-Semadeni+00, 03; Mac Low & Klessen 04 ; Ballesteros-Paredes+07): • Provides large-scale support for cloud as a whole. • Produces small-scale local density enhancements (“cores”) that can collapse, if they exceed the local Jeans mass. • Larson’s linewidth-size relation s ~ L1/2 interpreted as the manifestation of strongly supersonic (near-Burgers) turbulence.

49. In this case, the Zuckerman-Palmer conundrum is resolved by the dual role of turbulence. • SFR given by • Mass of collapsing fragments given by integration of high-density tail of density PDF: • Timescale typically a variation of tff in high-density range (see summary by Federrath & Klessen 12). • The SFR is low because the fraction of mass undergoing collapse is small Volume fraction log n [cm-3]

50. 7. The global-collapse scenario.