cosmic rays and thermal instability t. w. hartquist, a. y. wagner, s. a. e. g. falle, j. m. pittard,...
TRANSCRIPT
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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
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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
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Uniform, Static, No CRs (Field 1965)
The difference between the heating rate per unit volume and the cooling rate per unit volume
Thermal equilibrium
€
€
ρL(ρ,T)€
€
€ €
L(ρ 0,T0) = 0
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Thermal Equilibrium P – n Relationship
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Isobaric Perturbations
Stable if
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Isentropic Perturbations Stable If
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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
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• Shock-induced formation of • Giant Molecular Clouds
Sven Van Loo
Collaborators: Sam Falle and Tom Hartquist
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Overview
• Introduction• Thermal properties of ISM• Formation of molecular clouds• Conclusions
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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
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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)
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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
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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)
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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)
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Dynamical evolution: 2D
Mach 2.5 (but similar for other values)
Fast-mode shock
Slow-modeshock
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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
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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)
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Results: small vs. large
• Pressure decrease behind shock, e.g. blast waves
Small (constant ram pressure) Large (significant decrease in ram pressure )
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Stronger Shocks Possible
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Dynamical evolution: 3D parallelParallel shock
GeometryPhase diagram
log(n)lo
g(p
/k)
⇒ Rapid condensation at boundary
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Dynamical evolution:3D obliqueOblique shock ~45o
Phase diagram
log(p
/k)
log(n)
⇒ Condensation along equilibrium curve
Geometry
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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
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Results: 3D• Column density
– Large column density >1021 cm-2
– Some filaments, but not much substructure
Parallel Oblique
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Substructure• Effect of increasing resolution: overall the same but
more detail
120 cpr 640 cpr
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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…
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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
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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.
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Additional Parameters
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First of Three Conditions
• Similar to condition for isobaric perturbations
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Other Two Conditions
Analogous to conditions for isentropic perturbations
Obviously Complicated
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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
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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
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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
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Alpha = -1.5, M = 1.4, 2, 3, and 5
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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
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• Magnetic pressure limits the ultimate compression behind a strong radiative shock, but it does not affect the thermal instability
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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.
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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
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• 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
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Two Fluid Model of Cosmic Ray Modified Radiative Shocks
• Developed by Wagner, Falle, Hartquist, and Pittard (2006)
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Cosmic Ray Pressure Held Constant Over Whole Grid Until t = 0
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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
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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
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Including Acoustic Instability Induced Energy Transfer
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Tycho Optical FeaturesWagner, Lee, Falle, Hartquist, Raymond (2009)