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The Interstellar Medium

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The Interstellar Medium 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<<c, e- due to high mobility prevent large scale E-fields, i.e. r r ∇Φ → 0 ⇔ E′ → 0 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.
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