ISM, and 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 regions “hidden” • radial density profile: (visible in IR) • in hydrostatic equilibrium? ⟹ Bonnor-Ebert Sphere (1956/55)

Bok Globule: measurements by Alves, Lada & Lada 2001

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 8 Hydrostatic equilibrium • conservation:

• momentum equation

• Poisson equation (self )

• 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

⟹ pre star, PMS ⟹ birth of a

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 :

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

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 40 Collapse of gas cores • central temperature during the collapse for different

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. mean stellar density: n ≈ 1024 cm−3)

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 44 Hayashi track Hayashi track: trajectory • adiabatic contraction of the of a protostar in the HR- optically thick core diagram • gas heats up ⟹ protostar becomes visible in the IR • large accretion :

• but not yet nuclear burning ⟹ further contraction • evolution on credits: CSIRO Australia Kelvin-Helmholtz time scales (105 − 107 years)

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 45 Kelvin-Helmholtz time scale with virial theorem:

• 2ET = − EG • for : ET = Etherm (internal energy) • 50% of EG available for Etherm ⟹ remaining energy is radiated

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 46 Pre-main-sequence stars

development of a 1 M⨀ protostar

• high opacity ⟹ energy transport by convection

• almost constant surface temperature Teff during contraction

⟹ luminosity L decreases ⟹ vertical evolution in credits: CSIRO Australia the HRD

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 47 Pre-main-sequence stars

development of a > 4 M⨀ protostar

• fast contraction ⟹ high internal temperature ⟹ large temperature gradient ⟹ energy transport by convection in the centre

• thin outer layer ⟹ energy transport by radiation credits: CSIRO Australia • horizontal development: Henyey-Track: constant luminosity L with increasing Teff during contraction phase

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 48 Protostars • protostars are embedded in dust envelopes ⟹ re-emitted radiation in the

NICMOS: near infrared camera: 0.8 − 2.5 µm

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 49 Protostars

Tobin et al., Nature 2012 • L1527 IRS: youngest observed protostar

t < 300.000 yr M ~ 0.2 M⨀ −7 dM/dt ~ 6.6×10 M⨀/yr

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 50 Protostellar Evolution • contraction + accretion ⟹ development in the HRD • also: gas outflows ⟹ expulsion of the envelope ⟹ high variable stars become visible ⟹ T-Tauri phase / T-Tauri stars

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 51 Protostellar Evolution • contraction + accretion ⟹ development in the HRD • also: gas outflows ⟹ expulsion of the envelope ⟹ high variable stars become visible ⟹ T-Tauri phase / T-Tauri stars • tenv ≤ 30 a • expansion: ~ 650 AU

• energy from: in the core region + fast rotation

XZ Tauri

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 52 T Tauri stars (TTS) • high variability of their luminosity • MTTS < 3 M⨀

• In HRD above the main sequence (larger and more luminous than MS stars)

• spectral classes: F, G, K, M (mid / late-type stars)

• visible in emission and absorption lines • emission lines mainly from their expanding envelope (up to 100 km/sec)

• mass loss up to 1 M⨀ ⟹ mass of the MS star can be much less than the initial cloud core

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 53 T Tauri stars (TTS)

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 54 Stages of protostellar evolution

ISM, Star- and Planet Formation,Bachiller, WS 16/17 RobiARAA Banerjee 199655 Stages of protostellar evolution

Bachiller, ARAA 1996

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 56 Protostellar Classes

• Class 0: compact central object, barely visible, deeply embedded in gas cores, outflows visible (disc winds)

• Class I: envelope + central object, mass ~ 0.1 M⨀

• Class II: T-Tauri stars (classical TTS, CTT), detectable in the visible spectra, disc + central object, expanding envelope

• Class III: weak-line T-Tauri stars (WTT), only weak Hα emission lines central object + remnant/debris disc

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 57 Protostellar Classes

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 58 Herbig-Haro* objects

* Georg Herbig: *1920 - 2013; Guillermo Haro: 1913-1988 further feature of young stars (YSOs) / T-Tauri stars:

• bi-polar outflows + Jets: collimated gas streams with high velocities: > 100 km/sec

• observed around many YSOs (> 400)

• interaction with the ambient gas ⟹ visible in shocks (bow shock) ⟹ knots und instabilities ⟹ Herbig-Haro objekts (since 1940) • time variations within years • life time: 104 − 105 years ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 59 Herbig-Haro Objekte

1000 AU

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 60 Herbig-Haro objects

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 61 Herbig-Haro objects

HST: HH 30 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 62 1000 AU

NGC 1333, Spitzer Telescope, IRAC, NASA, JPL ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 63 1000 AU

“Mystic Mountain” in ISM,Carina Star- and PlanetNebula, Formation, HST,WS 16/17 NASA, Robi Banerjee ESA64 Jets • jet launching?

radiation

For 391 outflows: Wu et al. (2004)

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 65 Jets

Pudritz & Norman 1986 • jet launching? ⟹ Lorentz force ⟹ magnetic fields + disc ⟹ disc-wind modell (Blandford & Payne 1982)

• magnetic field anchored in the disc • gas moves along magnetic field lines (“beads on a wire”) ⟹ gas is accelerated away from the disc surface ⟹ + magnetic fields collimate the outflow ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 66 Jets • from ideal magneto-hydrodynamic equations (MHD)

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 67 Jets • Lorentz force:

magnetic magnetic pressure tension

fmag.P f B mag.T B

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 68 Jets • Lorentz force:

Mastumoto & Shibata 1999

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 69 Jets

with Keplerian disc (stationary solution)

⟹ magneto-centrifugal acceleration

if Bp ∢ with disc axis: > 30° (Blandford & Payne 1982)

⟹ jet will be launched ⟹ and collimated

Pelletier & Pudritz, 1992 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 70 Jets

• simulation of a, rotating collapsing cloud core:

⟹ jet launching by magnetic fields

Daniel Seifried, Hamburger Sternwarte

ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 71 ISM, Star- and Planet Formation, WS 16/17 Robi Banerjee 72