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

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ρL(ρ,T)€

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L(ρ 0,T0) = 0

Thermal Equilibrium P – n Relationship

Isobaric Perturbations

Stable if

Isentropic 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, …

Þ 2 stable phases in which Þ heating balances cooling:

1. Rarefied, warm gas (w; T > 6102 K)

2. Dense, cold gas (c; T < 313 K)

Net heating

Net cooling

wc

(Wolfire et al. 1995; Sanchez-Salcedo et al. 2002)

Cloud formation: flow-driven

Heitsch, Stone & Hartmann (2009)Hennebelle et al. (2008)

Flow-driven formation or colliding streams

e.g. expanding and colliding supershells

• 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)

B//v

//v

⊥v

Cloud formation: shock-induced

Lim,Falle & Hartquist (2005) Inutsuka & Koyama (2006)Van Loo et al. (2007)

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 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)

Dynamical evolution: 2D

Mach 2.5 (but similar for other values)

Fast-mode shock

Slow-modeshock

Dynamical evolution: 2D

• 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)

12CO

From 2D simulation

Results: 2D

• 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

Dependency on Mach number

M = 5M = 2.5

M = 1.5

Volume fraction of cloud for which β=Pg/Pm < 0.1

Þ only moderate-strength shocks can produce clouds similar to GMCs (obs: β ~ 0.03-0.6)

Results: small vs. large

• Pressure decrease behind shock, e.g. blast waves

Small (constant ram pressure) Large (significant decrease in ram pressure )

Stronger Shocks Possible

Dynamical evolution: 3D parallelParallel shock

GeometryPhase diagram

log(n)lo

g(p

/k)

⇒ Rapid condensation at boundary

Dynamical evolution:3D obliqueOblique shock ~45o

Phase diagram

log(p

/k)

log(n)

⇒ Condensation along equilibrium curve

Geometry

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

120 cpr 640 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.

Additional Parameters

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

Alpha = -1.5, M = 1.4, 2, 3, and 5

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

Including Acoustic Instability Induced Energy Transfer

Tycho Optical FeaturesWagner, Lee, Falle, Hartquist, Raymond (2009)