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Collapse of Gas Cores ISM, Star and Planet Formation Collapse of Gas Cores Robi Banerjee Hamburger Sternwarte [email protected] Molecular Clouds & Prestellar Cores Mac Low & Klessen 2004 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 2 Molecular Clouds & Prestellar Cores e.g. MC formation by colliding flows of WNM flows by: • galactic spiral arms ⟹ gravitational potential • Parker instability ⟹ flows along magnetic field lines 5000 K 20 K Vazquez-Semadeni et al. 2007 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 3 Molecular Clouds & Prestellar Cores edge-on view face-on view ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 4 Molecular Clouds & Prestellar Cores thermal instability: dP/dρ < 0 ⟹ inhomogeneous density and temperature structure ⟹ cold dense clumps and core can become gravitational unstable ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 5 Molecular Clouds & Prestellar Cores • column-density PDFs from different molecular clouds Kainulainen et al. 2009 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 6 Molecular Clouds & Prestellar Cores Bok Globule B 68: Alves et al. 2001 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 7 Bok* Globule * Bartholomeus Bok (1906-1983) • local star formation regions “hidden” protostars • radial density profile: (visible in IR) • in hydrostatic equilibrium? ⟹ Bonnor-Ebert Sphere (1956/55) Bok Globule: Barnard 68 extinction measurements by Alves, Lada & Lada 2001 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 8 Hydrostatic equilibrium • mass conservation: • momentum equation • Poisson equation (self gravity) • Equation of state: here isothermal ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 9 Hydrostatic equilibrium • hydrostatic equations (spherical symmetric): I) II) III) ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 10 Hydrostatic equilibrium with: ⟹ Lane-Emden equation (Chandrasehkar 1930): ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 11 Hydrostatic equilibrium solution of with: Φ(0) = 0, Φ’(0) = 0 (ρ(0) = ρ0) no analytic solution: ⟹ solve numerically, e.g. Runge-Kutta method )) ξ ( ρ log( ) ξ ( critical radius: ρ ξ = 6.451 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 12 Gravitational Instability Instability of a Bonnor-Ebert sphere ⟹ Rolf Ebert 1955 and W.B. Bonnor 1956 (independently) ⟹ analyse P(V) with M, T = const. : ⟹ ⟹ ; ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 13 Gravitational Instability Instability of a Bonnor-Ebert sphere ⟹ Rolf Ebert 1955 and W.B. Bonnor 1956 (independently) criterion for instability: ⟹ ξcrit = 6.451 Bonnor 1956 Ebert 1955 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 14 Gravitational Instability • critical external pressure: ; • critical Bonnor-Ebert radius: ⟹ 5 -3 −19 -3 • n ≈ 10 cm ⟹ ρ0 ~ 4×10 g cm • T ≈ 10 K ⟹ cs ≈ 0.2 km/sec 3 ⟹ rBE ≈ 2.3×10 AU ≈ 0.01 pc 3 with ξcrit = 6.451 ⟹ R ≈ 15×10 AU ≈ 0.07 pc ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 15 Gravitational Instability • critical Bonnor-Ebert mass: • • • ⟹ MBE,crit ≈ 1.65 M⨀ ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 16 Gravitational Instability: Jeans’ Mass gas sphere: • K = internal energy (thermal energy) • U = potential energy (gravitational energy) • bound: K + U < 0 • hydrostatic equilibrium: ⟹ virial theorem: 2K + U = 0 mit: MJ : Jeans mass ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 17 Jeans’ Mass • Jeans Masse vs. critical Bonnor-Ebert mass: mit ⟹ ; ⟹ MJ = 1.22 MBE,crit ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 18 Jeans’ Mass examples: • warm neutral medium (WNM): • T ≈ 5000 K ⟹ cs ≈ 5.7 km/sec • n ≈ 1 cm-3 ⟹ ρ ~ 10−24 g cm-3 7 • ⟹ MJ ≈ 10 M⨀ • ⟹ no collapse / no star formation • Bok globule: • dense, cold gas • T ≈ 10 K ⟹ cs ≈ 0.2 km/sec • n ≈ 105 cm-3 ⟹ ρ ~ 4×10−19 g cm-3 • ⟹ MJ ∼ 1 M⨀ • ⟹ regions of star formation ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 19 Gravitational Instability: Jeans analysis Jeans’ length: • from linear perturbation theory: ⟹ λJ = 2π rBE ≈ 0.97 rBE,crit 3 • diffuse gas: λJ ~ 10 pc 4 • Bok globule: λJ ~ 10 AU ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 20 Gravitational Instability: Jeans analysis • Jeans mass from Jeans length: ⟹ M’J ≈ 0.53 MJ ≈ 0.81 MBE,crit ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 21 Grav. Instability: Magnetic Fields • magnetic fields ⟹ Lorentz force: • magnetic tension: • magnetic pressure gradient: ⟹ magnetic fields can stabilise self gravitating cloud cores ⟹ critical mass-to-flux ratio: µ ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 22 Grav. Instability: Magnetic Fields • critical mass-to-flux ratio: mass M and flux Φ are conserved! from εgrav = εmag ⟹ where ρ = M/(4π/3 R3); Φ = πR2 B (magnetic flux) ⟹ (Mouschovias & Spitzer 1976) ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 23 Grav. Instability: Magnetic Fields • critical mass-to-flux ratio ⟹ numerical factor depends on geometry: ⟹ e.g. µcrit = 0.16 / √GN for flattened structures (Nakano & Nakamura 1978) ⟹ not well defined in non-closed systems, like molecular clouds ⟹ µ : volume quantity / area quantity ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 24 Grav. Instability: Magnetic Fields • with Σ = M/πR2 (Σ : column density) ⟹ mass-to-flux ratio: µ ∝ Σ/B Crutcher 2012 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 25 Collapse: free-fall time • collapse time: ⟹ e.g. from grav. acceleration of a mass element: ⟹ harmonic oscillator with frequency ω ⟹ time to reach the centre: tcoll = (2π/ω)/4 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 26 Collapse: free-fall time • collapse time (neglect pressure response): from homologous collapse solution: ⟹ all mass shells reach the centre at the same time: ⟹ free-fall time: ⟹ depends only on density • typical free-fall time: 5 Bok globules: tff ≈ 10 years ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 27 Collapse: free-fall time • free-fall time: ⟹ shorter time scales for higher density fluctuations ⟹ fragmentation: collapse of individual density fluctuations www.abenteuer-universum.de ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 28 Collapse of gas cores • gas cores (R ~ 104 to 105 AU) initial isothermal collapse 5 with times scales tff ~ 10 yr • increasing gravitational energy is released by radiation untill core becomes optically thick: isothermal collapse ⟹ gas core heats up Larson 1969 ⟹ contracts adiabatically (quasi-stationary) on Kelvin-Helmholtz time scale tKH ⟹ Hayashi track pre main sequence star, PMS ⟹ birth of a protostar ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 29 Collapse of gas cores • Shu 1977: Singular Isothermal Sphere (SIS): I) with u = vr with II) ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 30 Singular Isothermal Sphere (SIS) • self-similar solution with x = r/c t • Ansatz: ⟹ with ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 31 Singular Isothermal Sphere (SIS) solution: • static case: v = 0 ⟹ ⟹ ⟹ singular static solution? ⟹ can not be achieved in nature (e.g. Whitworth et al. 1996) ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 32 Singular Isothermal Sphere (SIS) solution general case: - - ‘wind’ solutions A= 2 Shu 1977 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 33 Singular Isothermal Sphere (SIS) solution: • general case: at time t = 0 : ⟹ mass accretion: m0(A=2) ≈ 0.95 −6 ≈ 2×10 M⨀/yr (T = 10 K ⟹ c ≈ 0.2 km/sec) 1.5 but: m0 ∝ A ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 34 Isothermal collapse 3 • SIS: accretion parameter m0: dM/dt = m0 c /G Girichidis et al. 2011 −6 −3 ⟹ dM/dt ∼ 10 − 10 M⨀/yr ! ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 35 Collapse of gas cores most important cooling processes: • radiative cooling by collisional excitations: of molecules: CO, O2, OH, NH, ..., H2O • dust cooling: gas-dust collisions + radiation of dust particles • H2-dissociation: endothermic reaction: ΔE = 4.48 eV ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 36 Collapse of gas cores • radiative cooling by collisional excitations OH, NH, CH, HCl, ... Neufeld et al. 1995 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 37 Collapse of gas cores • radiative cooling by collisional excitations ⟹ cooling power: Λ ∝ n2 temperature temperature → • comparison of tcool = (3/2n kBT) /Λ und tff ⟹ tcool/tff < 1 ⟹ collapse on free-fall time ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 38 Collapse of gas cores dust cooling : e.g. Goldsmith 2001 • energy transfer through gas-dust coupling: ΔT = Tgas−Tdust ⟹ • radiation power of dust (like a black-body radiator) ⟹ and ⟹ ⟹ efficient ‘thermostat’ ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 39 Collapse of gas cores • temperature and H2 density of the core region central temperature molecular atomic hydrogen hydrogen ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 40 Collapse of gas cores • central temperature during the collapse for different metallicities Γheat = adiabatic contraction: T ∝ n2/3 Omukai et al. 2005 • calculation from tff = tcool ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 41 Collapse of gas cores • optically thick regime: 2 ⟹ τ ≈ κ λJ < 1 ; λJ = (π c /G ρ) ⟹ slow release of compressive heating ⟹ core contracts on trajectory with MJ ≈ const. 3/2 −1/2 with MJ ∝ T ρ ⟹ T ∝ ρ1/3 ⟹ P ∝ ρ4/3 ⟹ γeff = γcrit = 4/3 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 42 Collapse of gas cores • effective EoS: P ∝ nγ during the collapse Tgas ≈ Tdust optically thick: H2 dissociation γ ≈ 4/3 efficient cooling eff ⟹ isothermal ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 43 Collapse of gas cores • Larson 1969 ⟹ first adiabatic core: optically thick regime adiabatic contraction at n ≳ 1011 cm−3 ⟹ second core: complete dissociation of H2 22 −3 at n ≳ 10 cm (cf.
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