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## Cosmic Rays and Thermal Instability

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**Cosmic Rays and Thermal Instability**T. W. Hartquist, A. Y. Wagner, S. A. E. G. Falle, J. M. Pittard, S. Van Loo**Outline**• The thermal instability of a non-magnetized, uniform static fluid, in the absence of cosmic rays • The thermal instability and cloud formation • The effect of cosmic rays on the thermal instability of a uniform medium • Instability of radiative, non-magnetic shocks • The effect of cosmic rays on radiative shocks**Uniform, Static, No CRs (Field 1965)**The difference between the heating rate per unit volume and the cooling rate per unit volume Thermal equilibrium**Isobaric Perturbations**Stable if**Isobaric and Isentropic**If L = Λ(T)n, then the above criterion gives stability if d ln(Λ)/d ln(T) > 1 There is also a criterion for the growth of isentropic perturbations (i.e. sound waves); for the same assumption about L, it gives stability if d ln(Λ)/d ln (T) > -3/2 for γ = 5/3**Shock-induced formation of**• Giant Molecular Clouds Sven Van Loo Collaborators: Sam Falle and Tom Hartquist**Overview**• Introduction • Thermal properties of ISM • Formation of molecular clouds • Conclusions**Introduction**• Hierarchical density structure in molecular clouds • Emission line maps of the Rosette Molecular Cloud (Blitz 1987) • MCs that do not harbour any young stars are rare • Old stellar associations (few Myr) are devoid of molecular gas • ⇒ Cloud and core formation are entangled • Not homogeneous,but highly structured • Stars embedded in dense cores**Cloud formation**• Compression + • Thermal processes in diffuse atomic gas: • Heating: photoelectric heating, • cosmic rays, soft X-rays, … • Cooling: fine-structure lines, • electron recombination, • resonance lines, … • 2 stable phases in which • heating balances cooling: • Rarefied, warm gas (w; T > 6102 K) • Dense, cold gas (c; T < 313 K) Net cooling w c Net heating (Wolfire et al. 1995; Sanchez-Salcedo et al. 2002)**Cloud formation: flow-driven**Flow-driven formation or colliding streams e.g. expanding and colliding supershells Heitsch, Stone & Hartmann (2009) Hennebelle et al. (2008) • Collision region prone to instabilities, i.e. KH, RT, NTSI • Turbulent shocked layer • Fragmentation into cold clumps • Structure depends strongly on magnetic field (both orientation and magnitude) //v ⊥v B//v**Cloud formation: shock-induced**• Shock-induced formation • e.g. shocks and winds sweeping up material • Similar processes as flow-driven • Can explain different cloud morphologies • e.g. filamentary, head-tail,… • ⇒ Shock-cloud interaction Lim,Falle & Hartquist (2005) Inutsuka & Koyama (2006) Van Loo et al. (2007) W3 GMC (Bretherton 2003) Previous work: 2D: adiabatic: MacLow et al. (1994), Nakamura et al. (2006) radiative: Fragile et al. (2005), Van Loo et al. (2007) 3D: adiabatic: Stone & Norman (1992), Shin, Stone & Snyder (2008) radiative: Leão et al. (2009) (nearly isothermal), Van Loo et al. (2010)**Numerical simulation**• Interaction of shock with initially warm, thermally stable cloud (n = 0.45 cm-3, T = 6788K, R = 200pc) which is in pressure equilibrium with hot ionised gas (n = 0.01 cm-3, T = 282500K) and β = 1. • Numerics: • Ideal MHD code with AMR (Falle 1991): • 2nd order Godunov scheme with linear Riemann solver • + divergence cleaning algorithm (Dedner et al. 2002) • Include cooling as source function • Resolution: 640/120 cells (2D/3D) across initial cloud radius • (120 cells = resolution for adiabatic convergence in 2 and 3D)**Slow-mode**shock Fast-mode shock Dynamical evolution: 2D Mach 2.5 (but similar for other values)**Dynamical evolution: 2D**From 2D simulation 12CO • Typical GMC values: n ≈ 20 cm-3 & R ≈ 50 pc • High-mass clumps in boundary and low-mass • clumps inside cloud precursors of stars • Similar to observations of e.g. W3 GMC (Bretherton 2003)**Results: 2D**Dependency on Mach number • Weak shocks (M ≤ 2): • NOT magnetically dominated • Strong shocks (M > 4): • formation time too short, because time-scale for formation of H2 is a few Myr Volume fraction of cloud for which β=Pg/Pm < 0.1 M = 2.5 M = 5 M = 1.5 • only moderate-strength shocks can produce clouds similar to GMCs(obs: β ~ 0.03-0.6)**Small (constant ram pressure)**Large (significant decrease in ram pressure ) Results: small vs. large • Pressure decrease behind shock, e.g. blast waves**Phase diagram**log(p/k) log(n) ⇒ Rapid condensation at boundary Dynamical evolution: 3D parallel Parallel shock Geometry**Dynamical evolution:3D oblique**Oblique shock ~45o Geometry Phase diagram log(p/k) log(n) ⇒ Condensation along equilibrium curve**Results: 3D**• Cloud properties: • large differences between parallel and oblique/perpendicular • Oblique/perpendicular → HI clouds; Parallel → molecular clouds • Ideal conditions (β < 1) for MHD waves to produce large density contrasts**Results: 3D**• Column density • Large column density >1021 cm-2 • Some filaments, but not much substructure Parallel Oblique**Substructure**• Effect of increasing resolution: overall the same but more detail 640 cpr 120 cpr**Future work**• Shock interacting with multiple clouds • Low resolution simulation (60 cpr) of 2 identical clouds overrun by an oblique shock (~45o) • Qualitative differences: • Shape • Density structure • Still need further study…**Conclusions**• Magnetically-dominated clouds form due to thermal instability and compression by weak or moderately-strong shocks • The time-lag between cloud and core formation is short**Uniform Media – Incoporating Cosmic Rays**• A. Y. Wagner, S. A. E. G. Falle, T. W. Hartquist, J. M. Pittard (2005) • CR Pressure Gradient Term in Momentum Density Eqn. and Corresponding Term in Energy Density Eqn.**First of Three Conditions**• Similar to condition for isobaric perturbations**Other Two Conditions**Analogous to conditions for isentropic perturbations Obviously Complicated**Limit of Large Φ and Small Diffusion Coefficient**ϕ big compared to 1 and absolute value of any other wavenumber divided by cosmic ray diffusion wavenumber (a/χ). Stable if Roughly satisfied for all values of k with small enough magnitudes**Limit of Small Φ and Diffusion Coefficient**Compared to 1; magnitudes of ratios of all other wavenumbers to the cosmic ray wavenumber are small compared to 1 (corresponds to big diffusion coefficient). Stable if**Thermal Instability**• Falle (1975); Langer, Chanmugum, and Shaviv (1981); Imamura, Wolfe, and Durisen (1984) showed that single fluid, non-magnetic, radiative shocks are unstable if the logarithmic temperature derivative, α, of the energy radiated per unit time per unit volume is less than a critical value • Pittard, Dobson, Durisen, Dyson, Hartquist, and O’Brien (2005) investigated the dependence of thermal stability on Mach number and boundary conditions**Do Magnetic Fields Affect the Themal Instability?**• Interstellar magnetic pressure is comparable to interstellar thermal pressure (about 1 eV/cc) • Immediately behind a strong shock propagating perpendicular to the magnetic field, the magnetic pressure increases by a factor of 16 • Immediately behind a strong shock the thermal pressure increases by roughly the Mach number squared**Magnetic pressure limits the ultimate compression behind a**strong radiative shock, but it does not affect the thermal instability**How About Cosmic Rays?**• In interstellar medium the pressure due to roughly GeV protons is comparable to the thermal pressure. • Krymskii (1977); Axford et al. (1977); Blandford and Ostriker (1978); Bell (1978) showed that shocks are the sites of first order Fermi acceleration of cosmic rays. • Studies were restricted to adiabatic shocks but indicated that cosmic ray pressure is great enough to modify the thermal fluid flow.**Two Fluid Model of Cosmic Ray Modified Adiabatic Shocks**• Völk, Drury, and McKenzie (1984) used such a model to study the possible cosmic ray acceleration efficiency • Thermal fluid momentum equation includes the gradient of the cosmic ray pressure • Thermal fluid equation for its entire energy includes a corresponding term containing cosmic ray pressure**Equation governing cosmic ray pressure derived from**appropriate momentum moment of cosmic ray transport equation including diffusion – diffusion coefficient is a weighted mean • Concluded that for a large range of parameter space most ram pressure is converted into cosmic ray pressure and that the compression factor is 7 rather than 4 behind a strong shock**Two Fluid Model of Cosmic Ray Modified Radiative Shocks**• Developed by Wagner, Falle, Hartquist, and Pittard (2006)**Cosmic Ray Pressure Held Constant Over Whole Grid Until t =**0**Problems**• Compression is much less than observed • Too high of a fraction of ram pressure goes into cosmic ray pressure which is inconsistent with comparable interstellar themal and cosmic ray pressures**Possible Solution**• Drury and Falle (1986) showed that if the length scale over which the cosmic ray pressure changes is too small compared to the diffusion length an acoustic instability occurs • Wagner, Falle, and Hartquist (2007, 2009) assumed that energy transfer from cosmic rays to thermal fluid then occurs**Tycho Optical FeaturesWagner, Lee, Falle, Hartquist, Raymond**(2009)