Theory of star formation

Transcription

1 Theory of star formation Monday 8th Molecular clouds and star formation: Introduction Tuesday 9th Molecular clouds: structure, physics, and chemistry Cloud cores: statistics and evolution Wednesday 10th Protostars 1

2 Molecular clouds: structure, physics, and chemistry The goals understand the basic physics governing the structure and radiation of interstellar clouds understand the tools which give information on the ISM interstellar turbulence radiative transfer problem brief introduction to interstellar chemistry dust extinction, scattering, and continuum radiation atomic and molecular lines 2

3 Cloud structure clouds have structure at all scales and look similar when studied at different resolution below spiral arm scales, above Jeans length to understand global properties of star formation, one should understand statistics of these cloud structure e.g. Casuso & Beckman(2007) density and velocity fields; the forces that influence these 3

4 Cloud structure (dense) clouds are extremely inhomogeneous this can be quantified with the volume filling factor f = fraction filled by gas denser than selected threshold the observed values of f are ~ in star forming regions (e.g. Stutzki & Gusten 1990; Kramer et al 1996) ~0.1 in globules (Robert & Pagani 1993) ~0.001 in dark clouds (Falgarone & Phillips 1991) 4

5 self-similarity at scales from 0.1pc to ~100pc 5

6 Fractal clouds self-similarity means ISM is fractal the fractal dimension can be estimated e.g. from the area-perimeter relation A = K P1 / D A area of the feature e.g. with intensity above given level P length of the perimeter of the feature D is the fractal dimension D=2 log P log A log P log A 6

7 the fractal dimension has been determined e.g. from molecular line maps and IRAS data Falgarone et al. 1991; Dickman et al. 1990: IRAS maps; Stutzki et al. 1998: CO maps; Elmegreen 1997: intercloud structure the fractal dimension in 2D is within the range projection of the volume density (e.g. column density) similar values even for clouds in earths atmosphere (Lovejoy 1982; true for three orders of magnitude) D does not restrict the general structure D2D = 1 for normal geometrical structures D3D = D2D + 1 i.e. for volume structures see also Sanchez et al. (2008) 7

8 Dickman (1990) 8

9 more detailed studies have shown that interstellar clouds are not monofractals but in fact multifractals Heyer & Schloerb (1997), Chappell & Scalo (2001), Lee, Kang, Kim et al. (2008) the fractal dimension changes as the function of scale linked to properties of interstellar turbulence? fractal dimension increases with density?; lower volume filling factor lower fractal dimension? the turbulence in ISM is not of the Kolmogorov type gas is compressible large density contrasts 9

10 Structure tree / dendrograms Houlahan & Scalo (1992) described the clumpy structure using structure trees statistic cloud is divided into connected regions at different intensity levels at higher intensity levels clumps are children of the lower intensity regions to which they belong form a hierarchical tree parameters: branching ratio, size ratio between levels, density contrast 10

11 Taurus, IRAS 100µm 11

12 Goodman et al. (2009) 12

13 Turbulence fractal nature a consequence of turbulence? turbulence creates a self-similar hierarchy where energy cascades down to smaller scales Kolmogorov: for uncompressible, homogeneous, isotropic turbulence the fractal dimension is D = 4/3 Kornreich & Scalo (2000): supersonic turbulence is caused by galactic shocks and (different sources) magnetic fields, self-gravity or young stars not required; IMF result of the structure of turbulent hierarchical clouds Elmegreen 1997, 1999: fractal structure from nonlinear magnetic waves generated in the intercloud medium; no internal source of turbulent energy; source of IMF reviews: Scalo (1987); Myers (1999) 13

14 Turbulence to characterize the turbulence (or cloud structure in general), different statistical tools are used spatial power spectra intensity, density or velocity structure functions higher order correlation functions 14

15 Padoan (2004) Padoan (2002) Ossenkopf (2001) 15

16 MHD terminology some relevant parameters magnetic field strength B Reynolds number R magnetic Reynolds number Rm R M = 0 v L µ permeability, conductivity, v velocity, L structure size sound speed cs Alfvén speed va normal Mach number M=u/cs Alfvénic Mach number MA =va /cs 16

17 MHD terminology magnetic field strength B general ISM: 1-10 µg (?) e.g. Zeeman measurements Padoan et al normal Reynolds number R, magnetic Reynolds number RM in ISM ~ 109, in simulations R < 1000 RM<<1 inhomogeneities will smooth, RM >>1 => B is frozen to the plasma 17

18 MHD terminology sound speed cs (ideal gas γ =1) speed at which disturbances moves in neutral medium Alfvén speed va p C s= speed for magnetic waves 2 B V A= 8 u u M = = normal (sonic) Mach number M p Cs u is the gas velocity => rms flow speed divided by sound speed Alfvénic Mach number MA => rms speed divided by the u Alfvén speed M = VA 18

19 MHD: general results simulations show filamentary structures and dense knots where low speed shock waves meet density field is log-normal (log n has normal distribution) with gravity included, some fraction of cores become self-gravitating, some remain transient MHD simulations predict (with some extra steps) the shape of IMF Elmegreen 1997, 1999; Vazquez-Semadeni et al.1997; Padoan & Nordlund 2002, 2004; Padoan et al. 2007; Krumholz & Tan 2007 we will return to this later... 19

20 MHD-modelling usual assumptions periodic isothermal turbulence forced in a particular way modern trends higher resolution (multiresolution) realistic heating & cooling inclusion of radiative transfer coupling with chemistry feedback from star formation 20

21 Radiative transfer observations = measurement of radiation before we can make the connection to the physics of clouds, we must know how radiation is formed within the source how radiation is transported to our instruments we must solve the radiative transfer problem this can be an impossible task inversion from 2D observations to 3D cloud observers have to resort to more or less justified approximations 21

22 Radiative transfer the change of intensity over infinitesimal step dx di = I dx = absorption coefficient = emission coefficient often written using optical depth and source function S di = I S d S=, = dx 22

23 Radiative transfer for a step accross homogeneous medium d I d e I e = I e = S e d d I = I e 0 S e d = I e S 1 e easily solvable along a line-of-sight, numerically even for inhomogeneous cases angularly averaged value is often needed 1 J= I, d 4 4 for excitation, dust temperature... 23

24 Line radiation molecules / atoms / ions have a discrete set of energy levels with transitions 1. spontaneous transitions to lower energy only some transitions allowed 2. transitions stimulated by incoming radiations only some transitions allowed 3. transitions resulting from collisions with other particles only some transitions allowed (1) (2) (3) 24

25 the rates between energy levels are written with the help of Einstein coefficients, A and B, and collisional coefficients C 1. rate for spontaneous emission Rij = Aij * ni (i>j) 2. rate stimulated transitions Rij = Bij * J * ni 3. rate for collisional transitions Rij = Cij * ni * n' some relations Aul = 2h c 2 3 Bul, gu Bul =g l Blu, C ul gl = e C lu gu E l E u kt gi = statistical weights, n' = density of colliding partners at critical density, collisional and radiative rates are equal: ncrit Cul ~ Aul ( + Bul I ) 25

26 Line radiation with the radiative processes one can define absorption and emission coefficients = h nu Aul 4 = h n l B lu n u Bul 4 nuaul, nlblu-nubul = number of transitions (photons) h = energy of a single photon 4 = full solid angle (f) = frequency profile of the line 26

27 Line radiation the other side of the problem: excitation the equilibrium equation = a linear set of equations ni Aij Bij J C ij n p i=0 = n j A ji B ji J C ji n p j=0 but level populations ni depend on the radiation field and radiative transfer depends on level populations (via and )! general solution requires radiative transfer modelling = simultaneous solution for radiation and excitation 27

28 Example: a three level system C 01 n p C 02 n p B01 I A10 B10 I C 10 n p C 20 n p B 01 I C 01 n p A10 B 10 I B 12 I C 10 n p C 12 n p A21 B 21 I C 21 n p C 02 n p B 12 I C 12 n p A21 B21 I C 21 n p C 20 n p n0 n1 0 = 0 0 n2 C 01 n p C 02 n p B 01 I A10 B10 I C 10 n p C 20 n p B 01 I C 01 n p A10 B 10 I B12 I C 10 n p C 12 n p A 21 B21 I C 21 n p n2 = 0 0 n0 n1 2 nj j =0 28

29 Line radiation some shortcuts 1) optically thin medium J independent of ni and excitation temperature Tex Tbg 2) optically thick medium local thermodynamical equilibrium (LTE); Tex Tkin level populations follow Boltzmann distribution n i g i E E / kt = e nj gj i j more generally, the previous formula defines excitation temperature, Tex 29

30 Line radiation: LVG use a trick to make the problem local: large velocity gradient approximation when Doppler shifts are larger than the linewidth, regions become decoupled within the small (?) interacting region one can assume a constant source function calculate photon escape probability = 0 J=S and one has LTE =1 optically thin case, J = Ibg in general case, = 1 r, dr e d 30

31 Line radiation: LVG with escape probability fraction 1- of emitted photons remains in the medium fraction of background photons can reach a point in the medium => the mean intensity can be written as weighted average J = (1- ) S + Ibg 31

32 Example: homogeneous slab optical depth across the slab is constant source function RT problem eliminated the escape probability is t 1 1 t = d x dx e x / d t 0 0 at the centre of homogeneous slab 1 e = τ =τ z=l z z=0 Θ τ=0 32

33 Example: spherical region - towards z only points P' with vz(p')-vz(p) < vd are connected - this is the interval [z-s, z] - define logarithmic velocity interval ε r = d ln v / d ln r - the interaction length is s = r z v(r) vd / v θ v(z)=v(r) cos θ P r - calculation of J includes integration d = f v z dv z over the line profile - if s is small f v dv =dl / d, = l / s, for 0 l s z z - profile is flat within thermal width, only positions l interact with point P 33

34 the escape probability is = where = d 1 e 4 hcr 4 v = s gu B ul n l n u r in the special case when r=1 or v r, the dependence on the angle disappears and one gets 1 e =, 34

35 Line radiation: LVG once has been determined, one only has to solve modified equilibrium equation method 1: ni => S (source function) J = (1- β ) S + β Ibg (mean intensity) solve equilibrium equations as usual method 2: write equilibrium equations using β Aul instead of Aul, use induced rates β Ibg Bij instead of rates J Bij... both give the same answer, method 2 more stable? 35

36 Line radiation: exact solution include long-distance interactions RT is solved along a large number of lines-of-sight on calculates J in each position equilibrium equations are solved, resulting in a new estimate of level populations ni process iterated until convergence is reached multi-dimensional integration model volume is discretized, assume initial level populations ni Monte Carlo, long/short characteristics, finite difference methods,... see van Zadelhoff et al. (2002) 36

37 Optical/NIR recombination lines e.g., HI recombination lines H, Br, Br, Pf,.. Hummer & Storey (1987) tabulated line intensities for isothermal medium that is optically thick for Ly photons only (Menzel case B) applicable, e.g., for studies of circumstellar material and ionized winds around hot stars see Lenorzer et al. (2002), Storey & Hummer (1995), Sorochenko (1989) 37

38 strength and and line profiles are sensitive to density, velocity field, and the source geometry once lines become optically thick, scale with emitting surface area only winds optically thin, disks optically thick Lenorzer et al. (2002) 38

39 Continuum: dust no discrete levels, one calculates dust temperatures (or their probability distributions) no discrete spectra but a continuum in both absorption and emission absorption cross section abs = Qabs(f) a2, = abs(f) n emitted intensity ~ Qabs(f) a2 B(f) temperature equilibrium a b s J d = 4 a b s B, T eq d additional complication: non-isotropic scattering 39

40 Dust: extinction curve the extinction curve (e.g., A / AV ) depends on optical properties and size distribution of grains the variations are characterized a measure of the slope at visual wavelengths (e.g., Cardelli 1989) AV RV = A B AV RV=3.1 is typical for local diffuse regions value increases to RV~5 in dense clouds changes are small in the infrared part (up to 8µm) with A ~ -1.6 see, however, Fitzpatrick & Massa (2009) 40

41 intensity or colour of background source is known, the intervening dust column can be estimated optical/nir star counts (NIR) reddening of background stars comparison of a reference area or model of stellar populations NICE, NICER, NICEST, GNICER (see Lombardi 2009 and references therein) 41

42 2175Å Draine (2003) 42

43 2175Å Gr Desert et al. (1990) PAH Si 10µm 43

44 Dust: scattering function scattering mostly in forward direction, especially at short wavelengths Henyey-Greenstein approximation uses g=<cos> [ g cos = 1 g 2 [ ] 2g 1 g 2 g r ] model of Weingartner & Draine (2001) 44

45 45

46 Dust: polarization dust produces polarization by extinction of background stars longer-wavelength emission scattering the previous require asymmetric grains aligned in magnetic field tracers of magnetic field morphology e.g., outflow cavities or accretion disks, based on asymmetric illumination 46

47 NGC1333 IRS4A Goncalves (2007) Bastien et al. (1990) Girart (2006) Goodman et al. (1990) 47

48 Dust: stochastic heating to solve for emission of small grains, one needs additional knowledge of the specific heat after an absorption of a photon, grain jumps to higher temperature and gradually coold down by its own radiation transition probabilities between discrete enthalpy bins analogous to discrete transitions in the case of line radiation in full modelling, one has to consider several dust populations discretized into many size intervals, each further divided into many enthalpy bins Siebenmorgen et al. (1992), Li & Draine (2001), Juvela & Padoan (2003) 48

49 Siebenmorgen et al

50 Interstellar chemistry important because it determines abundances of our tracer elements of cooling species = power of line radiation affects mass, morphology CO isotopes, H2O, OI, CI, CII,... difficult because involves low densities, often low temperature grain surface chemistry is important corresponding laboratory measurements difficult difficult to model or to study in laboratory 50

51 51

52 Dickens et al. 2000: L183 52

53 theoretically the abundances represent the solution of the chemical reaction network the chemistry evolves with timescales up to 106 years the observed abundances are not necessarily equilibrium values one or usually many reaction paths leading to each compound; one or many destruction paths each reaction has one or more reactants and one or more reaction products, e.g. A + B C + D + E the reaction rates depend on density of the reactants kinetic temperature (= frequency of collisions) radiation field (for photoreactions and photo-ionizations) cosmic rays (for cosmic ray ionizations and cosmic-ray induced photo-reactions) carbon chains ~105 yr, sulphur compounds ~106 yr see Lee et al. (1996), Millar et al. (1996), Woodall et al. (2006) 53

54 gas phase reactions neutral-neutral reactions ion-neutral reactions often extrapolation from lab data at room temp. theoretical rates fairly reliable electron reactions (electrons are treated as another species in the chemical network) ionization by cosmic-ray photons photoreactions caused by UV field photoreactions caused by cosmic rays grain surface reactions less well understood H2 is formed on grain surfaces! R n H n HI =R[n HI 2n H 2 ] R = 3e-17 cm3/s (Jura 1974) 54

55 in practice the reaction rates k are evaluated based on the temperature, UV field, cosmic ray ionization rate etc. the reaction rate is k multiplied with the densities of the reactants leads to a set of first order ordinary differential equations dx i N = k ' r dt r=1 Mr Xp p=1 N = number of reactions, r = reaction Mr = number of reactants, Xp = density of species k' = reaction rate [s-1], <0 if Xi destroyed; includes the number of Xi created/destroyed in a single reaction 55

56 for example: A B 2C e h introduces the following terms in the set of equations dn A =... k dt dn B =... k dt dn e =... k dt dn C =... 2k dt N A N B... N A N B... N A N B... one A and one B destroyed, two C and one free electron created N A N B... electrons are counted, photons are not 56

57 Turner et al

58 Thermal balance in ISM gas heating cosmic rays cr = H 2 Q cr n H 2 UV through photoelectric effect fh n d d f y c u f df d = f0 collisions with dust grains 2 H2 formation on dust grains H =RQ H 1 f n ambipolar diffusion gravitational energy d =... n n d d T d T g / 3 4r 4/ 3 B M n H 2 X e comp... T n H 2 3/ 2 ad = cooling line radiation collisions with dust grains ( continuum radiation) line = nu A ul h 58

59 Neufeld et al. (1995) Isothermal sphere & LVG calculations Juvela et al. (2001): MHD + MC 59

60 Neufeld & Kaufman (1993) 60

61 Summary Chemistry line emission cosmic rays RT: lines X RT: continuum T(gas) T(dust) interstellar radiation field RT: continuum MHD dust emission 61

62 Bastien et al. 1990, ApJ 364, 232 Cardelli 1989, ApJ 345, 245 Casuso & Beckman 2007, 656, 897 Chappell & Scalo 2001, ApJ 551, 712 Desert et al. 1990, A&A 237, 215 Dickens et al. 2000, ApJ 542, 870 Dickman et al. 1990, ApJ 365, 586 Draine 2003, ApJ 598, 1017 Draine 2003, ARAA 41, 241 Interstellar dust grains Elmegreen 1997, ApJ 477, 196 Elmegreen 1997, ApJ 486, 944 Elmegreen 1999, ApJ 527, 266 Falgarone & Phillips 1991, IAUS 147, 119 Fitzpatrick & Massa 2009, astro-ph Girart et al. 2006, Science 313, 812: Magnetic fields in the formation on Sun-like stars Goodman et al. 2009, Nature 457, 63 Goodman et al. 1997, in Star formation near and far: Velocity coherence in dense cores Goodman et al. 1990, ApJ 359, 363 Heyer & Schloerb 1997, ApJ 475, 173 Houlahan & Scalo 1992, ApJ 393, 172 Hummer & Storey 1987, MNRAS 224, 801 Jura 1974, ApJ 191, 375 Juvela 2005, A&A 440, 531 Juvela et al. 2001, ApJ 563, 853 Juvela & Padoan 2003, A&A 397, 201 Kornreich & Scalo 2000, ApJ 531, 366 Kramer et al. 1990, A&A 307, 915 Krumholz & Tan 2007, ApJ 641, L121 Lee, Kang, Kim et al. 2008, JKAS 41, 157 Lee et al. 1996, A&AS 119, 111 Lenorzer et al. 2002, A&A 386, L5 Li & Draine 2001, ApJ 554, 778 Lombardi 2009, A&A 493, 735 Mac Low 2005, in The magnetized plasma in galaxy evolution: Magnetized turbulence McKee & Ostriker 2007, ARAA 45, 565: Theory of star formation Millar et al. 1997, A&AS 121, 139 Myers 1999, 3rd Cologne-Zermatt symp., p 227: Initial conditions and motions in star-forming dense cores Neufeld & Kaufman 1993, ApJ 418, 263 Neufeld et al. 1995, ApJS 100, 132 Ossenkopf et al. 2001, A&A 379,

63 Padoan et al. 2002, ApJ 580, L57 Padoan et al. 2004, ApJ 604, L49 Padoan & Nordlund 2002, ApJ 576, 870 Padoan & Nordlund 2004, ApJ 617, 559 Robert & Pagani 1993, A&A 271, 282 Sanchez et al. 2008, astro-ph Scalo 1987, ASSL 134, 349 Siebenmorgen et al. 1992, A&A 266, 501 Sorochenko 1989, Astron.Nachr. 310, 5, 389 Storey & Hummer 1995, MNRAS 272, 41 Stutzki & Gusten 1990, ApJ 356, 513 Turner et al. 2000, ApJS 126, 427 van Zadelhoff et al. 2002, A&A 395, 373: Numerical methods for non-lte radiative transfer Vazquez-Semadeni et al. 1997, ApJ 474, 292 Weingartner & Draine 2001, ApJ 548, 296 Woodall et al. 2007, A&A 466, 1197: The UMIST database for astrochemistry

64 Cloud cores: statistics and evolution The goals understand the importance of the study of core statistics have a general picture of the evolution from dense clouds to pre-stellar cores connection between core/clump mass spectrum and the initial mass function stability of clouds forces contributing to the collapse 64