The Interstellar Medium
IMPRS-LECTURE - Fall 2003 - Dieter Breitschwerdt Max-Planck-Institut für Extraterrestische Physik Prologue ...it would be better for the true physics if there were no mathematicians on earth. Daniel Bernoulli A THEORY that agrees with all the data at any given time is necessarily wrong, as at any given time not all the data are correct. Francis Crick TELESCOPE, n. A device, having a relation to the eye similar to that of the telephone to the ear, enabling distant objects to plague us with a multitude of needless details. Luckily it is unprovided with a bell ... Ambrose Bierce
For those who want some proof that physicists are human, the proof is in the idiocy of all the different units which they use for measuring energy. Richard P. Feynman Overview
LECTURE 1: Non-thermal ISM Components • Basic Plasma Physics (Intro) – Debye length, plasma frequency, plasma criteria • Magnetic Fields – Basic MHD –AlfvénWaves • Cosmic Rays (CRs) – CR spectrum, CR clocks, grammage – Interaction with ISM, propagation, acceleration LECTURE 2: Dynamical ISM Processes • Gas Dynamics & Applications –Shocks – HII Regions – Stellar Winds – Superbubbles • Instabilities – Kelvin-Helmholtz Instability – Parker Instability LECTURE 1 Non-Thermal ISM Components I.1 Basic Plasma Physics • Almost all baryonic matter in the universe is in the form of a plasma (> 99 %) • Earth is an exception • Terrestrial Phenomena: lightning, polar lights, neon & candle light • Properties: collective behaviour, wave propagation, dispersion, diagmagnetic behaviour, Faraday rotation • Plasma generation: – Temperature increase However: no phase transition! Continuous increase of ionization (e.g. flame) – Photoionisation (e.g. ionosphere) – Electric Field („cold“ plasma, e.g. gas discharge) • Plasma radiation: – Lines (emission + absorption) – Recombination (free-bound transition) – Bremsstrahlung (free-free transition) – Black Body radiation (thermodyn. equilibrium) – Cyclotron & Synchrotron emission 1. Definition: • System of electrically charged particles (electrons + ions) & neutrals • Collective behaviour long range Coulomb forces 2. Criteria & Parameters for Plasmas: i. Macroscopic neutrality: Net Charge Q=0 λ λ L >> D ( D ... Debye length) for collective behaviour ≈ λ3 >> ii. Many particles within Debye sphere N D ne D 1 iii. Many plasma oscillations between ion-neutral ω τ >> damping collisions en 1 Quasi-neutral Plasma • Physical volume: L • Test volume: l • Mean free path: λ
L • Requirement: l λ λ << l << L
Net Charge: Q = 0 • To violate charge neutrality within radius r requires electric potential: 4 4 q = π r 3 ()n e + n (−e) = π r 3e()n − n 3 i e 3 i e q 4 ⇒ Φ = = π r 2e()n − n r 3 i e For e = 4.803×10−10 esu, 1 esu = 1 cm3/2 g1/2 s-1 = = 11 −3 − ≈ and r 1 cm, n 10 cm , ni ne 0.01 ni we need a voltage of Φ = 6.03×103 V or T ≈ 7×107 K Debye Shielding • Violation of Q=0 only within Debye sphere
e y r e e z q ... positive test charge q + + Plasma x Equilibrium is disturbed e e e Is there a new equil. state? Look for steady state! r r • Electrostatics: E = −∇Φ(rr) • Equilibrium: Boltzmann distribution Φ r − Φ r r = e (r) r = e (r) ne (r) n0 exp , ni (r) n0 exp kBT kBT • Total charge density: ρ r = − r − r + δ r (r) e(ne (r) ni (r)) Q (r) Φ r − Φ r = − e (r) − e (r) + δ r en0 exp exp Q (r) kBT kBT • Relation between charge distribution and r r charge density (Maxwell): ∇E = 4πρ(rr)
• Disturbed potential thus given by diff.eq.: Φ r − Φ r ∇2 Φ r − π e (r) − e (r) = − π δ r (r) 4 e n0 exp exp 4 Q (r) kBT kBT • Test charge with small electrostatic potential energy: Φ r Φ r Φ r << ± (r) ≈ ± (r) e (r) kBT exp e 1 kBT kBT • Thus the transcendental equation can be approximated Φ r ∇2 Φ r − π (r) = − π δ r (r) 4 e n0 2 e 4 Q (r) kBT • Defining the Debye length:
λ = kBT D π 2 4 e n0 2 ∇2 Φ(rr) − Φ(rr) = −4π Qδ (rr) λ2 D • Electrostatic forces are central forces: Φ(rr) = Φ(r) In spherical coordinates: Coulomb and Debye Potential 1 d d 2 λ ~2.1 r 2 Φ(r) − Φ(r) = 0 D 2 λ2 r dr dr D with boundary conditions : Φ C (r) → Φ → Φ = Q r 0 : (r) C (r) r Φ D (r) r → ∞ : Φ(r) → 0 • Result: Radius r
Q − 2 Φ (r) = exp r D λ Debye-Hückel Potential r D • Is total charge neutrality fulfilled: qtot = 0 ? = ρ r 3 = − q 1 − 2 3 + δ r 3 qtot (r) d r exp rd r q (r) d r ∫V πλ2 ∫ λ ∫V 2 D V r D ∞ ∞ q 1 2 = − 4π r 2 exp− r dr + 4π r 2q δ (r) dr πλ2 ∫ λ ∫ 2 D 0 r D 0 1∞424 434 q q λ 2 = − π − D − + 2 4 r exp r Partial 2πλ 2 λ D D 0 ∞ Integration q λ 2 + 4π − D exp− r dr + q πλ2 ∫ λ 2 D 0 2 D ∞ q q λ2 2 = − []0 − 0 + 4π D exp− r + q 2πλ2 2πλ2 2 λ D D D 0 q = []0 − 2πλ2 + q = q − q = 0 q.e.d. πλ2 D 2 D Importance of Debye Shielding • Test charge q neutralized by neighbouring plasma charges within „Debye sphere“ >> λ • Charge neutrality guaranteed for r D → Φ → ∞ Φ << • For r 0 : D ( r ) bec. e k B T breaks down • Factor „2“: due to non-equil. distr. of ions
− r n = n Q λ If we neglect ion motions: i 0 : Φ (r ) = e D D r • Numbers: λ = T[K] D 6.9 -3 cm ne[cm] 6 -3 λ Ionosphere: T=1000 K, ne=10 cm , D=0.2 cm 4 -3 λ 16 ISM: T=10 K, ne~1 cm , D=6.9 m; LISM~3 10 m 4 10 -3 λ -3 Discharge: T=10 K, ne=10 cm , D=6.9 10 cm • Number of particles in a Debye sphere: 4 N = π n λ3 Plasma parameter: g = ()n λ3 -1 D 3 e D e D • Charge neutrality can only be maintained for a sufficient number of particles in Debye sphere: >> ⇔ << N D 1 g 1 λ • Collective behaviour only for r<< D for each particle: only here violation of Q=0 possible • Debye shielding is due to collective behaviour λ λ • g<<1: mfp << D Plasma frequency • Violation of Q=0: strong electrostatic restoring forces lead to Langmuir oscillations due to inertia of particles • Electrons move, ions are immobile • Averaged over a period: Q=0 • Longitudinal harmonic oscillations with plasma frequency: π 2 ω = 4 e ne p me Oscillations damped by collisions between electrons and neutral particles τ ν en ~ 1/ en ... mean collision time To restore charge neutrality we need: ω >>ν ⇔ ω τ >> en en 1 Plasma Criteria
i. L >> λ D ≈ λ3 >> ii. N D ne D 1 ω τ >>1 Exercise: iii. en
Check the validity of plasma criteria: ISM – DIG: -3 Te ~ 8000 K, ne ~ 1 cm I.2 Magnetic Fields • Magnetic Fields (MFs) are ubiquitous in universe • Observational evidence in ISM:
– Polarization of star light –> dust –> gives B⊥
– Zeeman effect –> HI –> gives B// - – Synchrotron radiation –> relativistic e –> gives B⊥ – Faraday rotation –> thermal e- –> gives B// r • Sources of MFs are electric currents j • In ISM conductivity σ is high, thus large scale r c r r electric fields are negligible and j = ()∇× B 4π Basic Magnetohydrodynamics (MHD) • Plasma is ensemble of charged (electrons + ions) and neutral particles –> characterized by distribution function ()r r in phase space f i x , p , t –> evol. by Boltzmann Eq. • MHD is macroscopic theory marrying Maxwell‘s Eqs. with fluid dynamic eqs.: „magnetic fluid dynamics“ • Moving charges produce currents which interact with MF –> backreaction on fluid motion • To see basic MHD effects, a single fluid MHD is treated (mass density in ions, high inertia compared to e-) • For simplicity gas treated as perfect fluid (eq. of state) • Neglect dissipative processes: molecular viscosity, thermal conductivity, resistivity Basic MHD assumptions ω <<ν 1. Low frequency limit: c – Consider large scale (λ big, ω small) gas motions 1 v v – Consider volume V of extension L: ω ~ ~ th <<ν ~ th τ c λ L mfp λ (hydrodynamic limit) ⇔ mfp ≡ Kn <<1 ω <<ν ≈ L – If ei then P e P i as assumed ω ω – Note that also << g (for el. + ions) holds r r r 4π r 1 ∂E – Ampére‘s law: ∇× B = j + c c ∂t r c r r r simplifies to j = () ∇ × B since 1 ∂E π 1 E 4 c ∂t c τ L v ~ = <<1 v r r 2 using (see 2.) E ~ B ∇× B B τ c c L 2. Non-relativistic limit : v/c << 1 – Electric + magnetic field in plasma rest frame are then:
r r 1 r E′ = E + ()vr × B c r r 1 r B′ = B − ()vr × E c – For v< r 1 r r r 2 r r r r ⇒ E = − []vr × B , B′ = B +1/ c []vr ×()vr × B ≈ B ⇒ j′ = j c – Current density r = r − r = − r j Zenivi eneve eneue r = r − r ρ = − ≡ with ue ve vi (drift speed), and e Zeni ene 0 Negligible drift speed between electrons and ions: j c B u ~ ~ e π ene 4 ene L µ -3 Consider ISM with B-field: B~ 3 G, L~1 pc, ne~1 cm -11 ue ~ 4.8 10 km/s as compared to cs ~ 1 km/s motion of ions and electrons coupled via collisions; small drift speed for keeping up B-field extremely low! 3. High electric conductivity σ → ∞ : ideal MHD – In principle ue is needed to calculate current density however Ohm‘s law can be used instead: r r r r r r 1 r Since B′ = B, we have j′ = j = σE′ = σ E + ()vr × B c r 1 r ⇒ E = − ()vr × B , for σ → ∞ c Thus Faraday‘s law is given by: r ∂B 1 r r 1 r r = − []∇× E = − ()∇×[]vr × B ∂t c c Magnetic pressure and tension • The equation of motion (including Lorentz force term): dvr ∂vr r r r r ρ = ρ + ρ()vr∇ vr = −∇P + F + F dt ∂t mag ext r 1 r 1 r r 1 r r r with F = ρ []vr × B = []j × B = []∇× B × B mag c c 4π 1 r r r r r Note: if [] ∇ × B × B = 0 ⇒ B = − ∇ ψ field is force free 4π Potential field! • Magnetic pressure and tension: ε ε – Aside: killing vector cross products use -tensor ijk r × r = ε and write a b a i b j ijk with summation convention ε ε = δ δ −δ δ and the identity: ijk klm ik jl il jk Thus: r r r r r r [∇× B]× B = −B×[∇× B] = − ()∂ ε ε = − (∂ )ε ε Bi j Bk jkl ilm Bi j Bk jkl ilm = ()∂ ε ε = ()∂ []δ δ −δ δ Bi j Bk jkl lim Bi j Bk ji km jm ki = ()∂ − ()∂ Bi i Bm Bi m Bi r r r 1 r = ()B∇ B − ()∇B2 2 r r r r 1 r r r (B∇)B B2 •Thus F = []∇× B × B = − mag 4π 4π 8π Magnetic tension Magnetic pressure r r r – If field lines are parallel () B ∇ B = 0 i.e. no magnetic tension, but magnetic pressure will act on fluid – If field lines are bent, magnetic tension straightens them – Magnetic tension acts along the field lines (Example: tension keeps refrigerator door closed) Ideal MHD Equations r r • Note that ∇B = 0 is included in Faraday‘s law as initial condition! ∂ρ r + ∇()ρ vr = 0 ∂t ∂vr r r 1 r r r r ρ + ρ()vr∇ vr = −∇P + ()∇× B × B + F ∂t 4π ext r ∂B r r = ∇×()vr × B ∂t d P = γ 0 dt ρ • Here we used the simple adiabatic energy equation Magnetic Viscosity and Reynolds number •For finite conductivity field lines can diffuse away • Analyze induction equation: r ∂B r r = −c()∇× E ∂t r r r r 1 r j = j′ = σ E′ = σ E + ()vr × B c r r j 1 r ⇒ E = − ()vr × B ... keep the term with σ!!! σ c r ∂B c r r r r ⇒ = − ()∇× j + ∇×(vr × B) ∂t σ c2 2 η = r r c r r r r m = ∇× ()vr × B − []∇()∇B − ∇2 B 4πσ 4πσ = ∇r × ()r × r +η ∇2 r ... magnetic viscosity v B m B • First term is the kinematic MHD term, second is diffusion term r r vB ∇×()vr × B ~ • Comparing both terms: L r B η ∇2 B ~ η m m L2 • Magnetic Reynolds number: vB vL R ≡ L = m η B η m L2 m •Rm >> 1: advection term dominates Rm << 1: diffusion term dominates cf. analogy to laminar and turbulent motions! Magnetic field diffusion r ∂B r •For R << 1 we get: =η ∇2 B m ∂t m • Deriving a magnetic diffusion time scale: B η B L2 4πσL2 ~ m ⇒τ ~ = τ 2 D η 2 D L m c Note: Form identical to heat conduction and particle diffusion • Field diffusion decreases magnetic energy: field generating currents are dissipated due to finite conductivity -> Joule heating of plasma Examples: 18 -1 τ ≈ • Block of copper: L=10 cm, σ=10 s ⇒ D 1.2 s 10 σ 16 -1 •Sun: R~ ~ L=7 10 cm, =10 s τ ≈ × 17 ≈ × 10 ⇒ D 6 10 s 2 10 yr! although conductivty is not so high, it is the large dimension L in the sun (as well as in ISM) that keep Rm high! τ ∝ σ 2 • Note that since D L turbulence decreases L and thus diffusion times thus fields in ISM have to be regenerated (dynamo?) Concept of Flux freezing • Flux freezing arises directly from the MHD kinematic (Faraday) equation (Rm >> 1): magnetic field lines are advected along with fluid, magnetic flux through any surface advected with fluid remains constant •Theorem:Magnetic flux through bounded advected surface remains constant with time •Proof:Consider flux tube Flux tube dS r = r r = r + ∆ • Consider surface S1 S(t) bounded by C1 and S2 S(t t) by C2 r B C2 vr∆t r r dS = −()vr∆t × dl r dl C1 • Surface changes postion and shape with time • Magnetic flux through surface at time t: Φ = r()r r B B r,t dS ∫S • Rate of change of flux through open surface: d r r 1 r r r r []B()rr,t dS = lim B()rr,t + ∆t dS − B(rr,t) dS ∆ → dt ∫S t 0 ∆t ∫S2 ∫S1 • Expand field in Taylor series r r r ∂B()rr,t B()rr,t + ∆t = B(rr,t)+ ∆t + ∂ K •So that t r r d r r r ∂B()r,t r 1 r r r r r r []B()r,t dS = lim dS + B()r,t dS − B()r,t dS ∫S ∆t→0 ∫ ∫S ∫S dt ∂t ∆t 2 1 • Using Gauss lawS2 r r r r ∫∫BdS = ∇ ⋅ B d 3rr = 0 V and applying to the closed surface consisting of r r r∆ S1, S2 and the cylindrical surface of length v t We obtain bottom top mantle r r r r r r r r ∫ BdS = − ∫ B()rr,t dS + ∫ B()rr,t dS − ∫ B()rr,t [(vr∆t)× dl ]= 0 S1 S2 C1 ∆ → r = r + ∆ → r = r • Noting that in the limit t 0, S2 (t) S2 (t t) S1(t) S(t) r d r r ∂B()rr,t r r r []B()rr,t dS = dS + B()rr,t ⋅ []vr × dl ∫S ∫ ∂ ∫ dt S t C r r r r and using the vector identity B()rr,t ⋅ (vr × dl )= −[vr × B()rr,t ] ⋅ dl r r r r r and Stokes‘ theorem ∫[vr × B()rr,t ] ⋅ dl = ∫∇×[vr × B()rr,t ] ⋅ dS one gets C S r r d r r r ∂B()r,t r r r r r []B()r,t dS = − ∇× []v × B()r,t ⋅ dS ∫S ∫ ∂ dt S t r • For a highly conducting fluid (σ → ∞ ) and taking v as fluid velocity, field lines are linked to fluid motion and according to ideal MHD „flux freezing“ holds d r r []B()rr,t dS = 0 dt ∫S • Note: motions parallel to field are not affected C2 C2 C 1 C1 vr • For finite conductivity field lines can diffuse out: d []r()r r =η ∇r 2 r()r r B r,t dS m B r,t dS ∫S ∫ dt S MHD waves • Perturbations are propagated with characteristic speeds • For simplicity consider linear time-dependent perturbations in a static compressible ideal background fluid •Ansatz: ρ = ρ +δρ δX 0 where <<1 = +δ P P0 P X r = r +δr v v0 v r = r +δ r B B0 B r = r +δ r E E0 E r = r +δr j j0 j r = • Assume background medium at rest: v0 0 ∂δρ r = −∇()ρ δvr = 0 Keep only ∂ 0 t first order terms! ∂δvr r 1 r r ρ = −∇δP + ()δj × B 0 ∂t c 0 r r 4π r ∇×δB = δj c r Perturbed Equations r r 1 ∂δB ∇×δE = − c ∂t r r ∇δB = 0 r 1 r δE = − ()δvr × B c 0 δ δρ P = γ ρ P0 0 • Combining equations and eliminate all variables in favour of δvr ∂P P c2 = = γ • Defining sound speed: S ∂ρ ρ S yields single perturbation equation r r ∂ 2δr B B v − 2∇r ()∇r δr − ∇r × ∇r ×δr × 0 × 0 = cs v v 0 ∂t 2 4πρ 4πρ 0 0 • Defining Alfvén speed: r B vr = 0 A πρ 4 0 ∂2δvr r r r r − c2∇()∇δvr + vr {}∇×[]∇×()δvr ×vr = 0 yields finally ∂t 2 s A A • We seek solutions for plane waves propagating parallel and perpendicular to B-field • Wave ansatz: r δvr()xr,t = A exp[i(kxr −ωt)] • Dispersion relation: − ω 2δ r + ()2 + 2 ()rδ r r + r r[](r r )δ r − ()r δ r r − ()rδ r r = v c s v A k v k v A k v A k v v A v k k v v A 0 Case Study for different type of waves r ⊥ r •Case 1:k vA dispersion relation reads then r r r r −ω 2δr + ()2 + 2 (rδr)r + r [(r )δr − ()r δr − ( δr)r ] = v cs vA k v k vAk vAk v vA v k k v vA 0 r δvr // k Longitudinal magnetosonic wave Phase velocity ω v = = c2 + v2 ph k s A r r r r •Case2:k // vA , i.e. k // B0 dispersion relation reads then c2 ()k 2v2 −ω 2 δvr + s −1 k 2 ()vr ⋅δvr vr = 0 A 2 A A vA • Two different types of waves satisfy this DR: r δr r δr – Case A: k // v ⇒ vA // v v thus ()vr ⋅δvr vr = v δv A δvr = v2δvr A A A δv A and ()2 2 −ω 2 δr = cs k v 0 the solution is just an ordinary (longitudinal) sound wave – Case B: r ⊥ δr r ⊥ δr ⇔ r ⋅δr = k v ⇒ vA v vA v 0 the DR reads in this case ()2 2 −ω 2 δr = ω vAk v 0 thus = v = v k ph A the solution is a transverse Alfvén wave (pure MHD wave) driven by magnetic tension forces ρ due to flux freezing gas (density 0 ) must be set in motion! • Magnetosonic wave (longitudinal compression wave) r r r k B δvr // k Note 1: in both cases phase • Transverse Alfvén wave velocity independent of ω and k: r dispersion free waves! B r δvr ⊥ k r Note 2: in ISM density is low, k Therefore Alfvén velocity high δvr ~ km/s I.3 Cosmic Rays Cosmic Radiation Includes - • Particles (2% electrons, 98% protons and atomic nuclei) • Photons • Large energies ( 10 9 eV ≤ E ≤ 10 20 eV ) γ-ray photons produced in collisions of high energy particles Extraterrestrial Origin • Increase of ionizing radiation with altitude • 1912 Victor Hess‘ balloon flight up to 17500 ft. (without oxygen mask!) • Used gold leaf electroscope Inelastic collision of CRs with ISM p + p → p + p +π 0 For E > 100 MeV dominant process π 0 → 2γ • Diffuse γ- emission maps the HI Galactic HI distribution • Discrete sources at high lat.: AGN (e.g. 3C279) Columbia 1.2m radio telescope •Top: CO survey of Galaxy mapping molecular gas (Dame et al. 1997) HI • Bottom: Galactic HI survey (Dickey & Lockman 1990) γ-ray luminosity of Galaxy • Probability that CR proton undergoes inelastic collision = σ σ = × −26 2 with ISM nucleus Pcoll ppnH c, pp 2.5 10 cm 0 • 1/3 of pions are π decaying with Groups of nuclei Z CR Universe Protons (H) 1 700 3000 α (He) 2 50 300 Light (Li, Be, B) 3-5 1 0.00001* Medium (C,N,O,F) 6-9 3 3 Heavy (Ne->Ca) 10-19 0.7 1 V. Heavy >20 0.3 0.06 Note: Overabundance of light elements spallation! Origin of light elements • Over-abundance of light elements caused by fragmentation of ISM particles in inelastic collision with CR primaries • Use fragmentation probabilites and calculate transfer equations by taking into account all possible channels Differential Energy Spectrum • differential ~ E-2.7 energy spectrum „knee“ is power law for 109 < E < 1015 eV ~E-3.1 1016 < E < 1015 eV „knee“ •for E > 1019 eV CRs are „ankle“ extragalactic (rg ~ 3 kpc for protons) Primary CR energy spectrum • Power law spectrum for 109 eV < E < 1015 eV: ∝ −γ γ ≈ = −γ I N (E) E with 2.70 or N(E)dE KE dE -2 -1 -1 -1 [IN] = cm s sr (GeV/nucleon) • Steepening for E > 1015 eV with γ = 3.08 („knee“) and becoming shallower for E > 1018 eV („ankle“) • Below E ~ 109 eV CR intensity drops due to solar modulation (magnetic field inhibits particle streaming) gyroradius: γ m vsinϑ pc sinϑ sinϑ Example: CR with E=1 GeV r = L 0 = = R g 12 µ ZeB Ze Bc Bc has rg ~ 10 cm! For B ~ 1 G R ... rigidity, θ ... pitch angle 15 @ 10 eV, rg ~ 0.3 pc Important CR facts: • CR Isotropy: 11 15 δI ≈ 6×10−4 – Energies 10 eV < E < 10 eV: I (anisotropy) consistent with CRs streaming away from Galaxy – Energies 1015 eV < E < 1019 eV: anisotropy increases -> particles escape more easily (energy dependent escape) 19 Note: @ 10 eV, rg ~ 3 kpc – Energies E > 1019 eV: CRs from Local Supercluster? particles cannot be confined to Galactic disk • CR clocks: – CR secondaries produced in spallation (from O and C) such 10 τ β 10 as Be have half life time H~ 3.6 Myr -> -decay into B – From amount of 10Be relative to other Be isotopes and 10B τ and H the mean CR residence time can be estimated to be τ 7 esc~ 2 10 yr for a 1 GeV nucleon CRs have to be constantly replenished! What are the sources? – Detailed quantitative analysis of amount of primaries and secondary spallation products yields a mean Galactic mass traversed („grammage“ x) as a function rigidity R: −ξ x(R) = 6.9()R g/cm2 , ξ = 0.6 20 GV for 1 GeV particle, x ~ 9 g/cm2 – Mean measured CR energy density: ε ε ε ε ≅ 3 CR ~ mag ~ th ~ turb 1 eV/cm If all CRs were extragalactic, an extremely high energy production rate would be necessary (more than AGN and radio galaxies could produce) to sustain high CR background radiation assuming energy equipartion between B-field and CRs radio continuum observations of starburst galaxy M82 ε ε give CR (M 82) ~ 100 CR (Galaxy) CR production rate proportional to star formation rate no constant high background level! CR interact strongly with B-field and thermal gas CR propagation: • High energy nucleons are ultrarelativistic -> light travel time from sources τ ~ L ≈ 3×104 yr <<τ lc c esc • CRs as charged particles strongly coupled to B-field r r r • B-field: B = B + δ B with strong fluctuation r reg component δ B -> MHD (Alfvén) waves •Cross fieldpropagation by pitch angle scattering random walk of particles! CRs DIFFUSE through Galaxy with mean speed L ≈10 kpc = -3 vdiff ~ τ 7 490 km/s ~ 10 c r 2×10 yr • Mean gas density traversed by particles ρ ≈ x × −25 3 1 ρ h τ ~ 5 10 g/cm ~ ISM c esc 4 particles spend most time outside the Galactic disk in the Galactic halo! „confinement“ volume – CR „height“ ~ 4 times hg (=250 pc) ~ 1 kpc – CR diffusion coefficient: κ × L ≈ × 28 2 ~ hCR τ 5 10 cm / s esc λ κ – Mean free path for CR propagation: CR ~ 3 / c ~ 1 pc strong scattering off magnetic irregularities! • Analysis of radioactice isotopes in meteorites: CR flux roughly constant over last 109 years CR origin: • CR electrons (~ 1% of CR paeticle density) must be of Galactic origin due to strong synchrotron losses in Galactic magnetic field and inverse Compton losses Note: radio continuum observations of edge-on galaxies show strong halo field • Estimate of total Galactic CR energy flux: ε Vconf ≈ 41 FCR ~ CR τ 10 erg/s esc Note: only ~ 1% radiated away in γ-rays! • Enormous energy requirements leave as most realistic Galactic CR source supernova remnants (SNRs) – Available hydrodynamic energy: F ~ν E ≈ 3 ⋅1051erg ≈1042 erg/s SNR SN SNR 100 yr about 10% of total SNR energy has to be converted to CRs – Promising mechanism: diffusive shock acceleration • Ultrahigh energy CRs must be extragalactic ≥ > 20 rg 100 kpc Rgal (for E ~ 10 eV) Diffusive shock acceleration: • Problem: can mechanism explain power law spectrum? • Diffusive shock acceleration (DSA) can also explain near cosmic abundances due to acceleration of ISM nuclei • Acceleration in electric fields? d ()γ r = r + 1 []r × r Lmv e E v B dt c due to high conductivity in ISM, static fields of sufficient magnitude do not exist thus strong induced E-fields from strongly time varying large scale B-fields could help -> no strong evidence! Fermi mechanism: • Fermi (1949): randomly moving clouds reflect particles in converging frozen-in B-fields („magnetic mirrors“) processes is 2nd order, because particles gain energy by head-on collisions and lose energy by following collisions (2nd order Fermi process) particles gain energy stochastically by collisions •1st order Fermi process (more efficient): – Shock wave is a converging fluid – Particles are scattered (elastically) strongly by field irregularities (MHD waves) back and forth 2nd order Fermi 1st order Fermi Shock Front Cloud • Energy gain per crossing: 2 ∆E 4 V ≈ (2nd order Fermi) E 3 c • Shock is converging fluid ∆E 4 V ≈ S (1st order Fermi) E 3 c – Energy gain per collision: ∆E/E ~ (∆v/c) – Escape probability downstream increases with energy V V P ≈ S (v ... particle speed) P =1− P ≈1− S esc v esc v – Essence of statistical process: = β let E E 0 be average energy of particle after one collision and P be probability that particles remains in acceleration process -> after k collisions: > = k = β k N( E) N0 P particles with energies E E0 ∆ ln()N N ln P E E VS ⇒ 0 = β ≡ =1+ ≈1+ ()β E0 E c ln E E0 ln β ln P ln VS V ln1+ 1+ S N E ln β c 1 ⇒ = = ≈ c ≈ − ≈ −1, (for V << v ≤ c) ln P V v V v S N0 E0 ln1− S − 1− S c v c v – Note: that N=N(>E), since a fraction of particles is accelerated to higher energies therefore differential spectrum given by N(E)dE = const.× E()ln P ln β −1dE – Note: statistical process leads naturally to power law spectrum! ln P – Detailed calculation yields: ≈ −1 e.g. Bell (1978) – Hence the spectrum is: ln β N(E)dE = const.× E −2dE – The measured spectrum E < 1015 eV gives spectral index -2.7 However: CR propagation (diffusion) is energy dependent ∝ E0.6 source spectrum ∝ E-2.1 : excellent agreement! PROBLEM: Injection of particles into acceleration mechanism Maxwell tail not sufficient „suprathemal“ particles - remains unsolved! - LECTURE 2 Dynamical ISM Processes II.1 Gas Dynamics & Applications •ISM is a compressible magnetized plasma λ << • mfp L • Pressure disturbances due to energy + momentum injection: SNe, SWs, SBs, HII regions, jets k T c = B , 0.3 ≤ c ≤120 km/s • Speed of sound: s µ s m u ISM motions are supersonic: M = >> 1 cs •Shocks (collisionless) propagate through ISM λ ∆ ( mfp >> ) II.2Shocks • Assumption: Perfect gas, B=0 ∆ ≤ λ ρ • Shock thickness mfp , u , P time independent across shock discontinuity: steady shock • Conservation laws: Rankine-Hugoniot conditions ρ = ρ 1u1 2u2 (mass) Shock Frame +ρ 2 = +ρ 2 ρ ρ P1 1u1 P2 2u2 (momentum) 1,u1, P1 2 ,u2 , P2 1 γ P 1 γ P ρu2 + 1 = ρ u2 + 2 (energy) 1 1 γ − ρ 2 2 γ − ρ 1 2 2 1 1 2 1 2 Analoga 1. TrafficTraffic JamJam:: τ „speed of sound“ cs= vehicle distance/reaction time=d/ • If car density is high and/or people are „sleeping“: cs decreases ≥ • If vcar cs then a shock wave propagates backwards due to „supersonic“ driving • Culprits for jams are people who drive too fast or too slow because they are creating constantly flow disturbances • For each traffic density there is a maximum current density jmax and hence an optimum car speed to make ρ djmax/d =0! 2. HydraulicHydraulic JumpJump:: „speed of sound“ cs= gh („shallow water“ theory) Kitchen sink experiment • total pressure: Ptot =Pram+Phyd = ρ 2 + ρ Ptot v gh • At bottom: v > cs • Therefore: abrupt jump • Behind jump: h2 > h1 V2 < V1 •V1 > c1 „ supersonic“ V2 < c2 „ subsonic“ II.3 HII Regions • Rosette Nebula (NGC2237): – Exciting star cluster NGC 2244, formed ~ 4 Myr ago – Hole in the centre: Stellar Winds creating expanding bubble Trifid Nebula: Stellar photons heat molecular cloud gas Gasdynamical expansion (e.g. jets) Facts • Lyc photon output S* of O- and B stars ionizes ambient medium to T ~ 8000 K = – In ionization equilibrium: N& I N& rec = – Energy input Q per photon: N& I Q N& recQ = – Energy loss per recombination: 3 kBTe N& rec 2 Equil. Temperature: = 2 Q Te 3 kB • Q depends on stellar radiation field and frequency dependence of ioniz. cross section • Assumption: stellar rad. field is blackbody average kinetic energy per photo-electron: ∞ ∞ Q = (hν − E ) S dν S dν , E = hν ∫ H *ν ∫ *ν H L ν ν L L For a blackbody at temperature T*: 2hν −1 hν S ν ∝ Bν (T ) = []exp()hν k T −1 * * c2 B * ν >> ≈ For h k B T * 1 Q kBTe 2 ⇒ T ≈ T e 3 * T* = 47000 K (for O5 star) Te ~ 31300 K Bad agreement with observation! Reason: forbidden line cooling of heavy elements, like [OII], [OIII], [NII] is missing! • For H the total recombination coefficient to all β (2) = × −10 −3/ 4 3 -1 excited states is: (Te ) 2 10 Te cm s = β (2) • Thus N& recQ nenH kBT* ≈ × −4 • If [OII] is the dominant ionic state: nOII 6 10 ne • Collisional excitation rate (all ions are approx. = in the ground state): Nij nenI Cij (Te ) = ()1/ 2 − Cij (Te ) Aij Te exp[ Eij kB Te ] 2 2 • Radiative energy loss by [OII] for D5/ 2 and D3/ 2 levels ≈ × −32 ( 2 1/ 2 ) − × 4 -3 -1 LOII 1.1 10 yOII n Te exp[ 3.89 10 K Te ] erg cm s ≈ taking yOII 1 = = and demanding LOII N& I Q N& recQ 1/ 4 − × 4 = × −6 Te exp[ 3.89 10 K Te ] 2.5 10 T* For T*=40000 K, we obtain: Te~8500 K, in excellent agreement with observations! • HII regions are thermostats! Dynamics of HII Regions Case A: Static HII Region • Spherical symmetric Model: – Ionization within radius r: Lyc photons = π 2 S* 4 r J RS – Recombination within r: 4 π R3 β (2)n n 3 s H e ≈ ≈ = – Ion. fract. x 1 ⇒ ne nH n – Balance of ion. + recomb. Example: 1/3 3 S* (2) −13 3 -1 R = β = 2×10 cm s (T ≈ 8000 K) S (2) 2 4πβ n = 2 -3 = 49 -1 n H 10 cm , S* 10 s (O6.5 type) ≈ ... „Stroemgren“ radius ⇒ RS 3 pc Case B: Evolving HII Region • Photon flux at IF: S 1 J = * − β (2)n2 R 4π R2 3 0 (conservation of photons) Lyc photons • Thickness of IF ~ photon mfp: R IF planar geometry; no rec. in IF R HII IF • Ambient medium at rest RS dR • IF velocity: n = J 0 dt • Define dimensionless quant. R t R −1 η = , τ = , V = S , τ = ()n β (2) R τ R τ rec 0 Equation of motion: S rec rec −τ 1/3 Solution: η = C()1− e η=()1−η3 3η2 & BC : τ → 0, η → 0 Evolution of Stroemgren sphere η •RS is reached only within 1% τ ≥ τ after 4 rec • IF velocity >> cII until R~ RS therefore n ≈ n τ II 0 then gasdynamical expansion η& • IF velocity slows down η = RS dRIF & τ rapidly rec dt dRIF << • However: cII NOT possible dt τ Case C: Gasdynamical Expansion of HII Region ≤ • When R & IF c II sound waves ck can reach IF ho S Lyc photons >> •:PII PI HII gas acts as piston RIF shock driven into HI gas • Gas pressed into thin shell: HII ≈ = RIF Rsh : R R sh • Pressure uniform in shell & τ <<τ HII region: sc dyn HI • Ionization balance in HII reg. BUT: HII grows in size + mass dM d 4 mS 1 dn II = π R3n m = − * II > 0 S II β (2) 2 dt dt 3 nII dt Simple Model: ≈ = = 2 • Psh PII 2nII kBTII nII mcII = 2 ρ 2 = ρ 2 ()γ = • For strong shock: Psh 0Vsh 0Vsh for 1 γ +1 = 4 π 2 β (2) 3 = 4 π 2 β (2) 3 • Ion. Balance: S* nII R n0 RS 3/ 2 3 3 n R II = S n0 R • Ambient medium at rest: V = R& sh 3/2 n R • Thus equation of motion: R&2 = II c2 = S c2 n II R II • Dimensionless quantities: 0 η = R = η = ηη 3/ 4 = , N cII t / RS , & R&/cII ⇒ & 1 RS BC : η →1, N → 0 4/7 −3/7 • Solution: 7 7 η =1+ N , η& =1+ N 4 4 •At = = Gasdynamical Expansion: N 0, R& cII >>τ η •Note: texp rec • Is pressure equilibrium ≈ reached: P II P I ? ≈ = PII PI ⇒ 2n f kBTII nI kBTI • Ionization balance must t [sec] 4 hold: S = π n2 β (2) R3 * 3 f f η = () & •Result: n f TI / 2TII nI = ()2/3 t [sec] ⇒ R f 2TII /TI RS M n R3 2T • Final mass: f = f f = II 3 M S n0 RS TI •Example: = = = -3 TI 100 K, TII 8000 K, n0 100 cm = × −3 ≈ ≈ ⇒ n f / n0 5 10 , R f / RS 34, M f / M S 100 • Initial mass in Stroemgren sphere is a SMALL fraction of final mass η = ≈ × 13 -2/3 For 34 ⇒ t f ~ 273 RS / cII 1.7 10 n 0 yr ≈ 7.8×107 yr • Equilibrium never reached, because star leaves main sequence before, unless density is high! ≥ × 3 -3 n0 3 10 cm II.4 Stellar Winds • Massice star BD+602522 blows bubble into ambient medium • Ionizing photons produce bright nebula NGC7635 • Diameter is about 2 pc • Part of bubble network S162 due to more OB stars Some Facts − VW ~ 2000 3000 km/s −6 M& W ~ 10 M~/yr Total outward force: σL F = W 4π r 2 In reality: σ = σ (λ) •O- and B-stars: •Strong UV radiation field –Burn Hot •Momentum transfer to gas –Live Fast •Resonance lines show P Cygni –Die young profiles (e.g. CIV, OVI) Line Driven Stellar Wind: • Assume a stationary wind flow (ignore Pth) dv GM m σL mv = − * + dr r 2 4π r 2 radiation pressure dv GM v = − * ()Γ −1 , dr r 2 σ Γ = L* = L* π 4 GM *mc Lc 4πGM mc L = * (Eddington luminosity) c σ •Stars withL*>Lc are radiatively unstable • Integration: v r GM 1 1 1 vdv = * ()Γ −1 dr ⇔ ()v2 − v2 = GM ()Γ −1 − ∫ ∫ r 2 2 0 * R r v0 R* * •Ifv(R*)=v0=0: 1/ 2 R* v(r) = v∞ 1− r 1/ 2 = 2GM * ()Γ − = L* − v∞ 1 vesc 1 R* Lc Γ This is CAK velocity profile (L* > Lc, i.e. >1 needed) σ L* v∞ = • If L >> Lc π 2 R*mc Example: 5 • For an O5 star: L*=7.9 10 L~, R*=12 R~ 1/ 2 1/ 2 1/ 2 1/ 2 σ m L R v =10 km/s p * 0 ∞ σ T m L0 R* ⇒ v∞ ≈ 2566 km/s • Mass loss rate (assume one interaction per photon): L* = M& v∞ Momentum rate c W Thus we get: ≈ × −6 M& W 6.3 10 M~/yr • Shortcomings: cross sect. λ dep., multiple scattering 1 •Note: L = M& V 2 ≈1.3×1037 erg/s << L = 3×1039 erg/s W 2 W W * Most of energy lost as radiation! Less 1 % converted into mechanical luminosity Effects of stellar winds (SWs) on ISM: • Observationally V W is better determined than M& W = −6 • We use here for estimates: M& W 10 M~/yr 1 V = 2000 km/s, L = M& V 2 =1036 erg/s W W 2 W W • Note: kinetic wind energy >> thermal energy = 49 -1 • Star also has Lyc output: S* 10 s M & W V 2000 km/s • Hypersonic wind flows into HII region: M = W ≈ = 200 cII 10 km/s • SW acts as a piston shock wave formed (once wind feels counter pressure) facing towards star • Hot bubble pushed ISM outwards facing shock • Contact surface separating shocked wind and ISM Flow Pattern: 5 4 3 • 1 Free expanding wind S+ shocked at S- IF • Energy driven shocked 2 CD 2 wind bubble bounded by contact discontinuity CD 1 S- No mass flux across C • 3 Shell of compressed HII region, bounded by IF • 4 Shell of shocked HI region (ambient gas) bounded by S+ • 5 ambient HI gas Qualitative Discussion: • Region 1 : free expanding wind has mainly ram M& pressure, P ≈ ρ V 2 , ρ = W W W W π 2 4 r VW • Region 2 : shocked wind; the post-shock temperature 3 3 is given by P = ρ V 2 = ρ V 2 = 2n k T b 4 b W 16 W W b B b = 3 m 2 ≈ × 7 ⇒ Tb VW 4 10 K 32 kB – Note: S- moves slowly with respect to wind >> – Since cb ~ 600 km/s and CD slows down ( c b R & b ) , pressure is uniform and energy is mostly thermal τ >>τ = –nb low, therefore cool dyn R b / R & b region is adiabatic – Density jump by factor 4 region extended • Region 3 : expansion of high pressure region drives outer shock S+; compresses HII gas into thin shell – Density high (at least 4 times n0) << << cooling high – Post-shock temperature T II T b since R&b VW – HII region trapped in dense outer shell, once = π β (2) 2δ 2 ≥ N& rec 4 R R nsh S* – Since wind sweeps up ambient medium into shell: = 4 πρ 3 = π 2δ M sh 0 Rsh 4 R R nshm 3 τ = δ <<τ – Pressure uniform since sc R / csh dyn 4 << • Region : between IF and S+, shock isothermal TI TII • Region 5 : ambient gas at rest at TI Simple Model for stellar wind expansion: δ << • Extension of Regions 3 + 4 is R Rb = +δ ≈ ⇒ Rsh Rb R Rb • Extension of Region 2 >> Region 1 therefore it is assumed that 2 occupies all space r • Equations: M = ρ(r′) d 3r′ (mass conservation) sh ∫ 1 r E = p(r′) d 3r′ (thermal energy) th γ −1 ∫ d (M R& ) = 4π R 2 P (momentum conservation) dt sh b b b d E th = L − 4π R 2 R& P (energy conservation) dt W b b b Similarity Solutions: • No specific length and time scales involved, i.e. r and t do not enter the equations separately = α PDE ODE, with variable Rb At = 4 π 3 3 = π 3 • Thermal energy given by Eth Rb Pb 2 Pb Rb 3 2 • Combining the conservation equations yields 3 L R4&R&& +12R3R& R&& +15R2 R& 3 = W b b b b b b b π ρ 2 0 • Substituting the similarity variable into equation α − 3L A5 []α()α −1 (α − 2)+12α 2 ()α −1 +15α 3 t 5 3 = W πρ 2 0 3 5α − 3 = 0 ⇒ α = •RHS is time-independent 5 1/5 1/5 125 L A = W π ρ 154 0 • Thus the solution reads: 1/5 1/5 125 L R = A t 3/5 = W t 3/5 b π ρ 154 0 3 R 3 − R& = V = b = A t 2/5 b sh 5 t 5 << = ρ 2 • Note: we have assumed PI Psh 0Vsh Numerical Simulation II.5 Superbubbles • NGC 3079: edge-on spiral, D~17 Mpc HST • Starburst galaxy with active nucleus • Huge nuclear bubble generated by massive stars in concert rising to z~700 pc above disk • Note: substantial fraction of energy blown into halo! More superbubbles: • Ring nebula Henize 70 (N70) in the LMC • Superbubble with ~100 pc in diameter, excited by SWs and SNe of many massive stars • Image by 8.2m VLT (+FORS) Some Facts 2 3 • Massive OB stars are born in associations (N*~10 -10 ) • If this happens approx. coeval, SWs and SN explosions are correlated in space and time! • About >50% of Galactic SNe occur in clusters τ = × 7 ()−1.6 • MS time is short: MS 3 10 m /10M0 yr stars occupy small volume during SB formation SB evolution can be described by energy injection from centre of association • Initially energy input from SWs + photon output rate of 1 O stars dominates L = M & V 2 ≈ 6 × 10 35 erg/s for O7 star W 2 W W Simple expansion model: • In early wind phase SB expansion is given by: 1/5 L 3/5 R = 269 pc 38 t cf. Cygnus supershell: R ~225 pc SB n 7 S 0 (Cash et al. 1980) L L = W , t = t 38 1038 erg/s 7 107 yr McCray & Kafatos, 1987 τ ≈ × 6 • After MS 5 10 yr last O star leaves main sequence and energy input is dominated by succesive SN explosions • Thus for 5 × 10 6 yr ≤ t ≤ 5 × 10 7 yr until last SN occurs (M ~ 7 M~) subsequent ejecta input at energy 51 E~10 E51 erg mimic a stellar wind! • If the number of OB stars is N* energy input is given by N 1051 E L = * 51 SB τ () MS M min • Here we have assumed: –A common bubble is formed – Bubble is energy driven (like SW case) – Energy input by SNe is constant with time (therefore taking MS life time of least massive SN) – Ambient density is constant • The expansion is given by: 1/5 = N*E51 3/5 RSB 97 pc t7 n0 1/5 − = N*E51 2/5 VSB 5.7 km/s t7 n0 • Most of SB size is due to SN explosions! • SB velocity larger than stellar drift explosions occur always INSIDE bubble Example: Local (Super-)Bubble • Solar system shielded Breitschwerdt et al. 2001 from ISM by heliosphere • Nearest ISM is diffuse warm HI cloud (LIC) • LIC embedded in low density soft X-ray emitting region: Local Bubble (RLB ~ 100 pc) • Origin of LB: multi- SN? NaI absorption line studies Gal. NGP • 3 sections through Plane Local Bubble GC • LB inclined ~ 20° wrt NGP LB ⊥ Gould‘s NGP Sfeir et al. (1999) Belt • LB open towards NGP? Local Chimney North Polar Spur LMC RASS Cygnus Loop Vela Crab Anticorrelation: SXRB – N(HI) HI • Anticorrel. on large angular scales for soft emission • Increase of SXR flux: disk/pole ~3 RASS • Absorption effect: ~50% local em. Local Stellar Population • Local moving groups Berghöfer and Breitschwerdt 2002 (e.g. Pleiades, subgr. B1) • 1924 B-F-MS stars (kin.): Hipparcos + photometric ages (Asiain et al. 1999) • Youngest SG B1: 27 B, τ ≈ ± ≈ 20 10 Myr, D0 120 pc • Use evol. track (Schaller 1992): det. stellar masses •IMF(Massey et al 1995): Γ −1 m = Γ = − ⇒ N(m) N 0 , 1.1 m 0 Young Stellar Content and Motion Berghöfer and Breitschwerdt 2002 • Adjusting IMF (B1): = = N(m 8M 0 ) 7 Γ−1 = m ⇒ N(m) 551.6 M 0 ≤ ≅ N(mmax ) 1⇒ mmax 20 M0 mmax ⇒ N = N(m) dm ≈ 21 SN ∫ mmin • 2 B stars still active • SNe explode in LB Formation and Evolution of the Local Bubble • Energy input by sequential SNe −α m τ = × 7 α = = τ MS 3 10 yr, 1.6 ⇒ m m( ) 10 M0 dN δ L = E SN = L t , δ = −(Γ /α +1) ≈ −0.3 SB SN dt 0 • Assumption: coeval star formation ≥ τ ≤ × 7 Star deficiency: m 10 M0 ⇒ cl 2.5 10 yr = First explosion: 15 Myr ago (mmax 20 M0 ) Superbubble Evolution r M = ρ(r′) d 3r′ (mass conservation) Analytic sh ∫ Model: r 1 E = p(r′) d 3r′ (thermal energy) th γ −1 ∫ d (M R& ) = 4π R 2 P (momentum conservation) dt sh b b b d E th = L (t) − 4π R 2 R& P (energy conservation) dt SB b b b β − r 0 54 . K 0 = β ≈ 1 − β 5 − µ 0 0 0 SNR L K r ⇒ R ) 1 7 . ~ ρ 1 + =0.6) − = µ αβ ρ = − Γ , β , 3 2 β is between 6 β . + − − − 3 µ b 1 δ ) 5 α R = 0 β 4 ( = ) K α − µ 11 β 3 , ( π + 0 3 , − 4 µ 3 = β αβ At = β − − 5 = () : dr β b β =0.4) AND SW/SB ( − − IF R r µ 0 α 7 K ( 2 r 3 π + 4 b 0 ∫ α R () = π 2 sh solution: M = ote that similarity exponent fter some tedious calculations we find: he mass of the shell is given by: A (Sedov phase: N A T • • • Similarity Local Bubble (Analytic results) • Local Bubble at present: = -3 τ = ≈ ≈ n 0 30 cm , exp 13 Myr ⇒ R b 146 pc, Vsh 5.9 km/s ≥ ≤ • Present LB mass: MLB 600 M0 , Mej 200 M0 Mass loading! (decreases radius) ≅ × 5 • Average SN rate in LB: fSN 1/(6.5 10 yr) τ ≈ ≈ × −3 −3 ≈ 6 • Cooling time: c 13 Myr (n b 5 10 cm , Tb 10 K) X-ray emission due to last SN(e)! d N E& = E • Energy input by recent SNe: SN SN dt ≈ × 36 + 0.31 ≈ × 36 E& SN 4.3 10 (1 t7 ) 5.2 10 erg/s Disturbed background ambient medium x z y Local Bubble Evolution: Realistic Ambient Medium ¾ Density ¾ Cut through Galactic plane ¾ LB originates at (x,y) = (200 pc, 400 pc) ¾ SNe Ia,b,c & II at Galactic rate ¾ 60% of SNe in OB associations ¾ 40% are random ¾ Grid resolution: 1.25 and 0.625 pc Numerical Modeling (II) ¾ Density ¾ Cut through galactic plane ¾ LB originates at (x,y) = (200 pc, 400 pc) ¾ Loop I at (x,y) = (500 pc, 400 pc) LB Loop I Results Bubbles collided ~ 3 Myr ago Interaction shell fragments in ~3Myrs Bubbles dissolve in ~ 10 Myrs II.6 Instabilities • Equilibrium configurations (especially in plasma physics) are often subject instabilities • Boundary surfaces, separating two fluids are susceptible to become unstable (e.g. contact discontinuity, stratified fluid) • Simple test: linear perturbation analysis – Assume a static background medium at rest – Subject the system to finite amplitude disturbances –Test for exponential growth – However: no guarantee, system can be 1st order stable and 2nd or higher order unstable Kelvin-Helmholtz instability • Why does the surface on a lake have ripples? Examples: Cloud – wind interface: • Shear flow generates ripples and kinks in the cloud surface due to Kelvin-Helmholtz instability • „Cat‘s eye“ pattern typical 2D Simulation of Shear Flow Vincent van Gogh knew it! U 2 U1 Normal modes analysis: r •Assume incompressible inviscid fluid at rest ( ∇ v r = 0 ) •Two stratified fluids of different densities move relative to each other in horizontal direction x at velocity U • Let disturbed density at (x,y,z) be ρ +δρ corresponding change in pressure is δP the velocity components of perturbed state are: U + u, v and w in x-, y- and z-direction thus the perturbed equations read: ρ U gr 2 Stratified fluid ρ z r 1 g can stands for any x acceleratiion! ∂u ∂u dU ∂ ρ + ρ U + ρ w = − δP ∂ ∂ ∂ = () t x dz x where U s U zs ∂v ∂v ∂ ρ + ρ U = − δP z is the surface at which ∂t ∂x ∂y s ρ changes discontinously ∂w ∂w ∂ ρ + ρ U = − δP − gδρ ∂t ∂x ∂z ∂δρ ∂δρ dρ ρ +U = −w Disturbances vary as ∂t ∂x dz [ + +σ ] ∂δz ∂δz exp i kx x k y y t s +U s = w()z ∂ s ∂ s t x ()σ < ∂u ∂v ∂w Instability if Im 0 + + = 0 ∂x ∂y ∂z • Inserting the normal modes yields a DR d dw dU gk 2w dρ ρ[]σ + k U − ρ k w − k 2 ρ[]σ + k U w = x x x []σ + dz dz dz kxU dz • At the interface U is discontinous, but the perturbation velocity w must be unique ()−ε + ε ε → • Integrate over a small „box“ z s , z s , with 0 ∆ ρ[]σ + dw − ρ dU = 2∆ ()ρ w s kxU kx w gk s BC dz dz σ + k U x s ∆ ()f = f = + − f = − where s z zs 0 z zs 0 • Since we have a discontinuity in U and ρ the DR becomes d 2 dU dρ − k 2 w = 0 since = = 0 2 dz dz dz w • Since σ + must be continous at z and w must not kxU s increase exponentially on either side, we must have = ()σ + []+ ()< w1 A kxU1 exp kz , z 0 = ()σ + []− ()> w2 A kxU 2 exp kz , z 0 • Applying the BC to solutions: ρ ()σ + 2 + ρ (σ + )2 = (ρ − ρ ) 2 kxU 2 1 kxU1 gk 1 2 • Expanding and rearranging yields the growth rate 1 ρ U + ρ U ρ − ρ ρ ρ ()U −U 2 2 σ = −k 1 1 2 2 ± gk 1 2 − k 2 1 2 1 2 x ρ + ρ ρ + ρ x ()ρ + ρ 2 1 2 1 2 1 2 r r = ϑ = ϑ where k U kU cos and kx k cos • Two solutions are possible = – If kx 0 ρ − ρ σ = ± gk 1 2 the growth rate is simply ρ + ρ R-T instability 1 2 Perturbations transverse to streaming are unaffected by it – For every other directions ofk > wave vector instability occurs if: g()ρ 2 − ρ 2 k > 1 2 ρ ρ ()− 2 2 ϑ Kelvin-Helmholtz instability 1 2 U1 U 2 cos ρ > ρ – Even for stable stratification 1 2 (against R-T) There is ALWAYS instability no matter how SMALL − U1 U 2 is! – For large velocity difference large wavelength instability Parker instability • Qualitative picture: B CRs B Gas + CRs Gas • CRs are coupled to magnetic field which is frozen into gas • Magnetic field is held down to disk by gas! • CRs exert buoyancy forces on field -> gas slides down along lines to minimize potential energy -> increases buoyancy -> generates magnetic („Parker“) loops! • Note: CRs and magnetic field have tendency to expand if unrestrained Do not despair if you did not understand everything right away! BUT Literature: • Textbooks: – Spitzer, L.jr.: „Physical Processes in the Interstellar medium“, 1978, John Wiley & Sons – Dyson, J.E.: Williams, D.A.: „Physics of the Interstellar Medium“, 1980, Manchester University Press – Longair, M.S.: „High Energy Astrophysics“, vol. 1 & 2, 1981, 1994, 1997 (eds.), Cambridge University Press – Chandrasekhar, S.: “Hydrodynamic and hydomagnetic stability“, 1961, Oxford University Press – Landau, L.D., Lifshitz, E.M.: „Fluid Mechanics“, 1959, Pergamon Press – Shu, F.H.: „The Physics of Astrophysics“, vol. 2, 1992, University Science Books – Choudhuri, A.R.: „The Physics of Fluids and Plasmas“, 1998, Cambridge University Press • Original Papers: – Dyson, J.E., de Vries, 1972, A&A 20, 223 – Weaver, R. et al., 1977, ApJ 218, 377 – McCray, R., Kafatos, M., 1987, ApJ 317, 190 – Bell, A.R., 1978, MNRAS 182, 147 – Blandford, R.D., Ostriker, J.P., ApJ 221, L29 – Drury, L.O‘C., 1983, Rep. Prog. Phys 46, 973 – Berghöfer, T.W., Breitschwerdt, D., 2002, A&A 390, 299 Bubbles everywhere!!! -- TheThe EndEnd --