L 3 - Stellar Evolution I: November-December, L 3: Collapse phase – theoretical models Background image: courtesy ESO - B68 with VLT ANTU and FORS 1

L 3 - Stellar Evolution I: November-December, L 3: Collapse phase – theoretical models Background image: courtesy ESO - B68 with VLT ANTU and FORS 1 The Formation of Stars Chapters: 9, 10, 12

L 3 - Stellar Evolution I: November-December, L 3: Collapse phase – theoretical models Background image: courtesy ESO - B68 with VLT ANTU and FORS 1 Barnard 68 considered a pre-collapse/collapse candidate

L 3 - Stellar Evolution I: November-December, L 3: Collapse phase – theoretical models Background image: courtesy ESO - B68 with VLT ANTU and FORS 1 If you discuss methods and techniques of collapse calculations: consider sensitivity to gridding, boundary conditions; access to a standard code? (run it)

L 3 - Stellar Evolution I: November-December, Time scales: low mass star formation

L 3 - Stellar Evolution I: November-December, Generic types of theories of collapse analytical semi-analytical numerical

L 3 - Stellar Evolution I: November-December, Jeans (1927) MNRAS 87, 720 On Liquid Stars Joel Tholine (1982) Hydrodynamic Collapse Fundamental Cosmic Physics Vol. 8, pp. 1-82

L 3 - Stellar Evolution I: November-December, Early Work Basic Insights

L 3 - Stellar Evolution I: November-December, x 2 x10 density time

L 3 - Stellar Evolution I: November-December, Penston 1969, MNRAS 144, 425 Larson 1969, MNRAS 145, 271 Shu 1977, ApJ 214, 488 Hunter 1977, ApJ 218, 834 Self-similarity solutions Isothermal spherical collapse

L 3 - Stellar Evolution I: November-December, Mass Definition Continuity Equation Momentum equation eos

L 3 - Stellar Evolution I: November-December, Similarity Variable

L 3 - Stellar Evolution I: November-December,

L 3 - Stellar Evolution I: November-December, Palla & Stahler call this Eq the isothermal Lane-Emden equation LE derived for polytropes ( P = k  n ), e.g. fully convective stars: n=3/2 (=1+1/m)

L 3 - Stellar Evolution I: November-December, LP = Larson, Penston H = Hunter EW = Expansion Wave (Shu) velocity density

L 3 - Stellar Evolution I: November-December, LP = Larson, Penston H = Hunter EW = Expansion Wave (Shu) velocity density supersonic

L 3 - Stellar Evolution I: November-December, Bonnor 1956 MNRAS 116, 351 centrally condensed flat distribution Shu 1977 extreme case

L 3 - Stellar Evolution I: November-December, Inside-out collapse (Shu 1977) Mass accretion rate a constant of the cloud Mass accretion time scale

L 3 - Stellar Evolution I: November-December, Foster & Chevalier 1993 Numerical simulations of non-singular isothermal spheres Like Hunter 1977: 1 solution has Shu’s EW as 1 limit models resemble LP with infall v ~ - 3 c s (homologous inflow) Why Shu 1977 commonly used ? (in particular, dM/dt = constant)

L 3 - Stellar Evolution I: November-December, (  = 0 at core formation;  ~ 2 t ff ) density r -2 r -3/2 Initial & boundary conditions Foster & Chevalier 1993, ApJ 416, 311

L 3 - Stellar Evolution I: November-December, compressional luminosity: pre-core formation Cloud boundary  max = Foster & Chevalier

L 3 - Stellar Evolution I: November-December, compressional luminosity: pre-core formation Foster & Chevalier Tscharnuter 1d models of 1 M o collapse: 1 st core formation 0.01 M o Cloud boundary  max = 6.541

L 3 - Stellar Evolution I: November-December, Inside-out collapse (Shu 1977) Why Shu 1977 commonly used ?...computational convenience...small number of parameters

L 3 - Stellar Evolution I: November-December, Gravitational collapse: Example inside-out (Shu 1977, ApJ 214, 488) not from Shu model p = -1.5 p = -2 R inf = c s t inf  = -0.5  = 0 adapted from Hartstein & Liseau 1998, AA 332, 703 ~ r p ~ r 

L 3 - Stellar Evolution I: November-December, predicted spectral line profiles of ground state ortho- and para-water (H 2 O) for inside-out collapse [B 335] adapted from Hartstein & Liseau 1998, AA 332, 703 Herschel HIFI S /T A ~ 500 Jy/K and o/p = 3 assumed infall region unresolved at 557 GHz

L 3 - Stellar Evolution I: November-December, Magnetised isothermal clouds Magnetic fields neglected in hydrodynamics of isothermal spheres: not important ?... Examples: Krasnopolsky & Königl 2002 Self-similar collapse of rotating magnetic molecular cloud cores, ApJ 580, 987 Allen, Shu & Li 2003 Collapse of singular isothermal toroids, I. Nonrotating ApJ 599, 351 II. Rotation & magnetic braking ApJ 599, 363 Book Chapters

L 3 - Stellar Evolution I: November-December, Allen et al: Development of pseudodisk Constant mass accretion rate

L 3 - Stellar Evolution I: November-December, Anything missing ?

L 3 - Stellar Evolution I: November-December, Isothermal eos No heating and cooling processes included Winkler & Newman 1980, ApJ 236, 201; ApJ 238, 311 Spherical, nonrotating, nonmagnetic, 1 M o momentum energy ! rad transfer ! continuity definition

L 3 - Stellar Evolution I: November-December, Pre-main-sequence evolution begins after collapse or main accretion phase Stahler, Shu & Taam 1980, ApJ 241, 637; ApJ 242, 226 protostellar evolution during main accretion phase

L 3 - Stellar Evolution I: November-December, Stahler (and Palla & Stahler ch. 11.2): stellar birthline Deuterium burning acts as a thermostat 2 H ( p,  ) 3 He Reaction rates (Harris et al. 1983, ARAA 21, 165) -> temperature sensitivity Assignment: anyone? Deuterium Burning Protostellar Pulsations

L 3 - Stellar Evolution I: November-December, Protostar evolution of a single star Fragmentation during collapse ?

L 3 - Stellar Evolution I: November-December, Analytically, Tohline (1982): fragmentation of isothermal or adiabatic spheres 1.Isothermal collapse (  = 1): Perturbation analysis of pressure-free sphere -> fragmentation during collapse No preferred wavelength -> perturbations of all sizes grow at the same rate Real clouds not pressure-free and adiabatic case more relevant...

L 3 - Stellar Evolution I: November-December, Adiabatic collapse:

L 3 - Stellar Evolution I: November-December, Numerically, General discussion: Hennebelle et al. 2004, MNRAS 348, 687 Sheets: Burkert & Hartmann 2004 ApJ 616, 288 See movie in L7 numerical simulations Rapid collapse Reid et al. 2002, ApJ 570, 231

L 3 - Stellar Evolution I: November-December, L 3: conclusions analytical collapse solutions differ in results one such solution has remained `successful´: inside-out versus outside-in collapse similarity technique applied also to magnetised and rotating clouds numerical simulations indicate otherwise, but dM/dt = constant still preferred (?) L 3: open questions how realistic are the assumptions made (resulting in e.g. supersonic/subsonic flow) ? what is the `correct eos´ ? how important is geometry ? Initial & boundary conditions ?