Heating & Cooling in Molecular Clouds

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1 Lecture 8: Cloud Stability Heating & Cooling in Molecular Clouds Balance of heating and cooling processes helps to set the temperature in the gas. This then sets the minimum internal pressure in a core and thus the thermal energy term in the virial equation. It also sets the sound speed, the speed at which pressure waves are transmitted, and thus the strength of shocks created by supersonic motions. Overall we shall see the above features are important for setting minimum masses of gravitationally unstable cores that form stars.

2 Heating processes 1. Cosmic Rays: Relativistic particles (mostly protons). These can penetrate deeply into dense molecular cores and even protoplanetary disks. They create low levels of ionization (see Astrochemistry lecture). These ejected electrons transfer some of their kinetic energy into heating the gas.. Interstellar Radiation Field: a) Atomic Carbon (CI) Ionization by photons E>11.eV. b) Photoelectric Heating by UV photons ejecting e- from dust grains c) Irradiation of dust grains d) Stellar X-rays Cooling Processes 1. Cooling by Atoms: (consider -level system) For densities below the critical density, cooling rate per cm -3 is: Λ = nl nh!lu ΔEul n (most collisions that excite 1-> lead to radiative decay) For densities far above the critical density Λ n (most excitation from 1-> is followed by collisional deexcitation) OI and CII have fine structure energy levels with spacings that are small enough to be relevant in cool gas ~100K.

3 Cooling Processes. Cooling by Molecules: CO rotational line emission is the most important coolant in cold molecular clouds. However, in denser clouds the optical depth in the line can become large, thus trapping the photons and reducing the cooling efficiency. 3. Cooling by Dust: Dust grains emit IR and mm photons and thus cool down. If their temperature is lower than that of the gas, then they will help cool the gas down. Cloud Thermal Structure Temperature profile in outer, lowerdensity part of a molecular cloud.

4 Photo-Dissociation Region (PDR) Far UV photons destroy molecules (H, CO). These photons are absorbed by dust and by molecules as they try and penetrate into the cloud. Eventually destruction rates of the molecules become small enough that most H is in H, most C in CO and most O in O. Towards the cold center Relatively constant temperature in the center. The precise value will depend on the cosmic rate flux, radiative efficiency of the dust grains. Temperatures are ~10K at nh~ cm -3

5 Isothermal spheres and the Jeans mass S&P Ch. 9.1 Let s consider a cloud that maintains equilibrium only through the forces of selfgravity and thermal pressure: (1) () 1 ρ P φ g = 0 P = ρa T Hydrostatic equilibrium Equation of state for an ideal isothermal gas (3) φ g = 4πGρ (1) + () implies that: lnρ + φ g /a T Poisson s equation is a spatial constant! ρ(r) = ρ c exp( φ g /a T ) Substituting ρ(r) in Poisson s equation and defining: ψ = φ g /a T ( ξ 4πGρ + c * - ) a T, The isothermal Lane-Emden equation: 1 ξ " 1/ d $ dξ ξ dψ ' & ) = exp( ψ) % dξ ( r Boundary conditions: ψ(0) = 0; ψ'(0) = 0

6 By numerically integrating the LE equation, we obtain ψ(ξ): ρ(r) = ρ c exp( φ g /a T ) " ρ /ρ c = exp( ψ) 1. Density and pressure drop monotonically away from center (necessary to offset the inward pull of gravity). At large distances (ξ>>1): ρ/ρ c approaches /ξ. This implies ψ=ln(ξ /), which satisfies the LE equation, but not the boundary conditions at ξ=0 (singular isothermal sphere): ρ(r) = a T πgr In clouds, the pressure does not drop to zero, but to some value P 0 characterizing the external medium. Fixing P 0 and a T, we determine: (a) ρ 0 from the equation of state, (b) the density contrast ρ c /ρ 0, and (c) ξ 0 (and r 0 ) from the figure:! It describes an infinite sequence of models, parametrized by ρ c /ρ 0

7 For each model, we can measure the mass of pressure-bounded, isothermal spheres: r 0 M = 4π ρr dr 0 ρ /ρ c = exp( ψ) m P 1/ 0 G 3 / M 4 a T % = 4π ρ ( c ' * & ) ρ 0 1/ 1/ & ξ 4πGρ ) + c ( + r + boundary conditions ' a T * % dψ ( ' ξ * & dξ ) ξ 0 " nondimensional mass of the SIS 13 Gravitational stability All the physical characteristics of isothermal spheres follow from integration of the LE equation. However, only a limited subset of the full model sequence is gravitationally stable. In all other clouds, an arbitrary small initial perturbation in the structure grows rapidly with time, leading ultimately to collapse. For small values of ξ 0 (or ρ c /ρ 0 ) : r 0 3 3Ma T 4πP 0 stable For a stable cloud, any increase of P 0 creates both a global compression and a rise of the internal pressure, where the latter acts to re-expand the configuration. 14 Clouds of low density contrast are mainly confined by the external pressure, and not self-gravity.

8 The case of the Bok Globule B68 15 Lada et al. 003, ApJ, 586, 86; Bergin et al. 00, ApJ 570, L101 Moving to higher ρ c / ρ 0, it is more difficult for the central regions to expand after application of an enhanced P 0. All clouds with ρ c / ρ 0 > 14.1 are gravitationally unstable. The critical value of M is known as the Bonnor-Ebert mass: M BE = m a 4 1 T 1/ P 0 G 3 / unstable ρ c / ρ 0 = 14.1

9 Critical length scale In general, a certain size scale of isothermal gas is prone to collapse, regardless of the specific 3D configuration. Following Jeans classic analysis, let s consider a plane wave propagating through a uniform, isothermal gas of density ρ 0 : ρ(x,t) = ρ 0 + δρexp[i(kx wt)] where x is the direction of propagation and k π/λ is the wave number. By assumption, the small velocity induced by the perturbation is also in this direction. Substituting analogous traveling-wave forms for all variables into the: δρ δt = (ρu) P = ρa T φ g = 4πGρ ρ Du Dt = P ρ φ g " equation of mass continuity equation of state Poisson s equation momentum equation (non magnetic) Linearizing the amplitudes, canceling the exponentials, and after some algebra, we arrive at the dispersion relation, which governs the propagation of the waves: ω = k a T 4πGρ 0 ω 0 (4πGρ 0 ) 1/ For short λ (large k), ω ka T and the disturbance behaves like a sound wave, traveling at the phase velocity ω/k = a T, the isothermal sound speed. But both ω and the phase velocity pass through zero when k = k 0. The corresponding λ J π /k 0 : k 0 ω /a T ( λ J = πa + T * - ) Gρ 0, 1/ = 0.19 pc ( T + * - ) 10 K, 1/ ( n H + * - ) 10 4 cm 3, 1/ Jeans length Perturbations with λ>λ J have exponentially growing amplitudes. M J =1.0 M sun " T % $ ' # 10 K& 3 / " n H % $ # 10 4 cm 3 ' & 1/ Jeans mass Bonnor-Ebert mass

10 Considering typical parameters of clumps embedded in Giant Molecular Clouds: n(h ) = 10 3 cm -3 and T=10 K, we find M J =3 M #, two orders of magnitude below the actual masses!! Since the clumps are apparently not undergoing global collapse, an extra source of support is necessary. The most plausible source is the interstellar magnetic field.

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