Theory of star formation Monday 8th 17.15 18.00 Molecular clouds and star formation: Introduction Tuesday 9th 13.15 14.00 Molecular clouds: structure, physics, and chemistry 16.00 16.45 Cloud cores: statistics and evolution Wednesday 10th 11.15 12.00 Protostars 1
Cloud cores: statistics and evolution The goals understand factors behind core statistics understand the importance of core statistics have a general picture of the evolution from dense clouds to pre-stellar cores the connection between core/clump mass spectrum and the initial mass function stability of clouds 2
Larson relations Larson (1981) noted some fundamental correlations in the properties of interstellar clouds ( Larson laws ) 1) linewidth-size relation ~ R 0.5 2) clouds are gravitationally bound E kin vir = 1 E grav 3) clouds have similar column densities ~1.5 10 22 cm 2 N 3
R, v R typical values ~ -1.15, ~ 0.4 motions subsonic only at core level 4
at virial balance (W=-2T), the virial parameter is 1 for >1, cloud is unstable and will collapse for spherical, homogeneous cloud 2 5 R = GM note: defined so that =1 is the critical value the relations are not independent, i.e., any two imply the third one writing = pc R 0.5, one gets 5 2 vir = pc G 100 M pc = 3.7 1 km/ s 2 2 0 McKee & Ostriker (2007) 5
one can also consider non-thermal support http://cfa-www.harvard.edu/~agoodman/presentations/presentations.html 6
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http://cass.ucsd.edu/~ppadoan/talks/riverside04.pdf 8
Turbulence Mac Low (2002) 9
Turbulence if turbulence is fed at large scales, energy will be transferred to all smaller scales according to theory and numerical experiments, the result is a powerlaw power spectrum take a fourier transform of velocity or density squares of the amplitudes describe the power P(k) that exists at scales corresponding to spatial frequency k in observations, instead of density, one is forced to analyse column density or simply intensity variations alternatively: correlation functions, delta-variance 10
for incompressible gas (Kolmogorov turbulence) one has P V k ~k, P V n ~k 11/3 assuming turbulence is isotropic, statistics are described by a single number n, P(k) ~ k -n 11/ 3 n = 11/3 for 3D power spectrum for angularly averaged PS n' = n-2 = 5/3 ~ 1.67 for Burgers turbulence, n' = 2 ('shock-turbulence') one cannot assume reality is (quite) this simple ISM is not incompressible, energy dissipation is affected by complex physical processes, turbulence is driven at different scales and motions are affected by magnetic fields exponent should depend, e.g., on Mach number 11
P k k 2.70 P k k 2.25 Padoan et al. (2004): The average magnetic field strength in molecular clouds Pm va = Pd u 2 2 B = 8 u 2 P k k 2.87 12
Padoan, Juvela et al. (2006): The power spectrum of supersonic turbulence in Perseus 13
note: many methods are used to describe clouds, from two-point correlations to morphology power spectra, structure functions -variance (Stutzki et al. 1998) Principal Component Analysis (Brunt 2003) Independent Component Analysis (Maino et al. 2002) Spectral Correlation Function (Rosolowsky et al. 1999) Wavelet decomposition (Langer et al. 1993) Multifractal spectrum (Chappel & Scalo 2001) Dendrograms (Goodman, Rosolowsky et al. 2009) 14
the other prediction from numerical studies: lognormal density pdf x 2 1 log x= / exp, = 2 2 2 2 2 =2 ln 1 0.25 M 2 Vazquez-Semadeni (1994); Nordlund & Padoan (1999) 15
Nordlund & Padoan (1998) Vazquez-Semadeni et al. (2001) 16
cass.ucsd.edu/~ppadoan/talks/chicago03.pdf 17
cass.ucsd.edu/~ppadoan/talks/chicago03.pdf 18
Clump mass spectra IMF is probably a complex function of initial conditions further fragmentation of cores, multiple systems, continued accretion, competitive accretion, core collisions etc. more straightforward test of theory is provided by observed clump mass spectra simply (?), count observed clumps per mass interval clump <> core? 19
problem 1 what is a clump? gaussclumps, clumpfind 2D/3D,... results will change problem 2 how to determine the clump mass line observations: what are abundances, what are gas temperatures, are there optical depth effects,... dust continuum: are dust properties uniform, what are dust temperatures,... 20
Motte et al. (1998) Oph, 480 square arcmin, 1.3mm dust continuum (IRAM) ~60 clumps, mass spectrum dn/dm ~ m-2 21
Andre et al. (2007) 22
Enoch et al. (2008) 1.1mm Bolocam, 108 starless and 98 protostellar cores 23
... back to theory turbulence gives density structures at all scales the mass spectrum of density structures bears similarity to IMF but no stars will form unless some of these are gravitationally unstable ( >1) we will return to clump mass spectra later and now look at the stability of cores 24
Jeans condition in homogeneous medium, perturbations are unstable to collapse if the wavenumber is below critical value 'Jeans wavenumber' 2 2 k k J = 4 G 2 cs this can be converted to a limiting mass MJ 2 = = kj 3 J 3 = G 3/ 2 1/2 3 cs any perturbation more massive than Jeans mass will collapse 25
Jeans conditions alternatively, one can look at stability of a spherical object 1/ 2 M 5 4 R2 5 k T 15 k T kt G = kt = R= R 2 3R 2 G 8 G G corresponding Jeans mass 4 3 4 kt 3/ 2 1/ 2 4 3/ 2 1/ 2 3 M= R = G cs 3 3 G 3 assuming homogeneous sphere no rotation, turbulence, magnetic field, bulk motions... 26
Bonnor-Ebert sphere previous formulas give a rough stability condition... but assumed homogeneous sphere = no hydrostatic equilibrium 1 P= in reality, the sphere should fullfill gravitational potential is 2 =4 G and the equation of state gives the pressure P = c2s for 1D case this gives differential equation 1 d 2 d r r =4 G exp c 2 2 dr r dr cs r = c exp r, 2 cs c = 0 27
by using two dimensionless variables 2 s /c, = 4 G c c 2 s r one obtains the Lane-Emden equation 1 d 2 d =exp 2 d d this can be solved numerically, once boundary conditions are set =0 =0, or R= d =0 d c2s 4 G c max, R = c exp max =P ext /c2s max = max = ln Pext c c2s 28
Bonnor-Ebert sphere is stable when larger values imply collapse max 6.5, Fischera & Dopita (2007) 29
Bonnor-Ebert spheres easy to calculate numerically (even in non-isothermal case) are interesting, because many cloud cores seem to follow that density profile Alves et al. (2001) 30
Magnetic field above magnetic pressure was omitted from stability analysis Pm ~ v B2 = 8 cloud will be stable if it is below critical mass-toflux ratio M M 2 A cr or, equally, the mass is below ciritical value G c ~ 0.13 for spherical cloud M M cr = c 31
however, material is free to flow along magnetic field lines magnetic field is directly felt only by the ionized ISM component does cloud collapse preferentially along field lines? neutrals collide with ions but can still drift wrt to magnetic fields and the ionized component = ambipolar diffusion even when initial mass is small, neutral flow can increase mass-to-flux ratio above critical value in shielded regions the ionization degree is low and the time scales of ambipolar diffusion are not much longer than the free-fall time 32
Tassis & Mouschovias (2004) 33
a core cannot collapse faster than in free-fall time 3 n 6 t ff = 1.37 10 yr 32G 10 3 cm 3 fast! tff = 140 000 yr for n = 105cm-3 the timescale for ambipolar diffusion force due to friction is F i n v i v n which, if equal to Lorentz force, gives a timescale AD =3 106 yr 1/ 2 n 4 3 10 H 2 cm 3/2 B 30 G 2 L 1 pc 2 when ionization fraction is large, is much longer than free-fall time in dense regions ionized by cosmic rays only, only a few times longer than tff? 34
the importance of ambipolar diffusion can be tested by measuring the ionization fraction and the magnetic field strength new observations needed! determination of ionzation fraction may require also better chemical modelling magnetic field measurements are notoriously difficult essentially Zeeman splitting 35
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Protostars The goals know basic constituents of a protostar know the tracers available for the study of protostellar systems accretion disks and outflow phenomena extinction, molecular lines, atomic lines, radio continuum,... be aware of the basic models of low-mass/highmass star formation 37
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