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

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slide1
L 3: Collapse phase – theoretical models

Background image: courtesy ESO - B68 with VLT ANTU and FORS 1

L 3 - Stellar Evolution I: November-December, 2006

slide2
L 3: Collapse phase – theoretical models

The Formation of Stars

Chapters: 9, 10, 12

Background image: courtesy ESO - B68 with VLT ANTU and FORS 1

L 3 - Stellar Evolution I: November-December, 2006

slide3
L 3: Collapse phase – theoretical models

Barnard 68 considered a pre-collapse/collapse candidate

Background image: courtesy ESO - B68 with VLT ANTU and FORS 1

L 3 - Stellar Evolution I: November-December, 2006

slide4
L 3: Collapse phase – theoretical models

If you discuss methods and techniques of collapse calculations: consider sensitivity to gridding, boundary conditions; access to a standard code? (run it)

Background image: courtesy ESO - B68 with VLT ANTU and FORS 1

L 3 - Stellar Evolution I: November-December, 2006

slide5
Time scales: low mass star formation

L 3 - Stellar Evolution I: November-December, 2006

slide6
Generic types of theories of collapse

analytical

semi-analytical

numerical

L 3 - Stellar Evolution I: November-December, 2006

slide7
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, 2006

slide8
Early Work

Basic Insights

L 3 - Stellar Evolution I: November-December, 2006

slide9
density

x10

x 2

time

L 3 - Stellar Evolution I: November-December, 2006

slide10
Self-similarity solutions

Isothermal spherical collapse

Penston 1969, MNRAS 144, 425

Larson 1969, MNRAS 145, 271

Shu 1977, ApJ 214, 488

Hunter 1977, ApJ 218, 834

L 3 - Stellar Evolution I: November-December, 2006

slide11
Mass

Definition

Continuity

Equation

Momentum

equation

eos

L 3 - Stellar Evolution I: November-December, 2006

slide12
Similarity Variable

L 3 - Stellar Evolution I: November-December, 2006

slide14
Palla & Stahler call this Eq the isothermal Lane-Emden equation

LE derived for polytropes ( P = k r n ), e.g.fully convective stars: n=3/2 (=1+1/m)

L 3 - Stellar Evolution I: November-December, 2006

slide15
density

velocity

LP = Larson, Penston

H = Hunter

EW = Expansion Wave (Shu)

L 3 - Stellar Evolution I: November-December, 2006

slide16
density

velocity

LP = Larson, Penston

H = Hunter

EW = Expansion Wave (Shu)

supersonic

L 3 - Stellar Evolution I: November-December, 2006

slide17
centrally

condensed

Bonnor 1956

MNRAS 116, 351

flat distribution

Shu 1977

extreme case

L 3 - Stellar Evolution I: November-December, 2006

slide18
Inside-out collapse (Shu 1977)

Mass accretion rate a constant of the cloud

Mass accretion time scale

L 3 - Stellar Evolution I: November-December, 2006

slide19
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 cs (homologous inflow)

Why Shu 1977 commonly used ? (in particular, dM/dt = constant)

L 3 - Stellar Evolution I: November-December, 2006

slide20
Foster & Chevalier 1993, ApJ 416, 311

r -3/2

r -2

Initial & boundary conditions

density

(t = 0 at core formation; t ~ 2 tff)

L 3 - Stellar Evolution I: November-December, 2006

slide21
Foster & Chevalier

Cloud boundary

xmax = 6.541

compressional luminosity: pre-core formation

L 3 - Stellar Evolution I: November-December, 2006

slide22
Tscharnuter 1d models of 1 Mo collapse: 1st core formation 0.01 Mo

Foster & Chevalier

Cloud boundary

xmax = 6.541

compressional luminosity: pre-core formation

L 3 - Stellar Evolution I: November-December, 2006

slide23
Inside-out collapse (Shu 1977)

Why Shu 1977 commonly used ?

...computational convenience

...small number of parameters

L 3 - Stellar Evolution I: November-December, 2006

slide24
Gravitational collapse: Example inside-out (Shu 1977, ApJ 214, 488)

~ r p

~ r a

p = -1.5

a = -0.5

p = -2

a= 0

not from

Shu model

Rinf = cstinf

adapted from Hartstein & Liseau 1998, AA 332, 703

L 3 - Stellar Evolution I: November-December, 2006

slide25
predicted spectral line profiles of

ground state ortho- and para-water (H2O)

for inside-out collapse [B 335]

infall region

unresolved

at 557 GHz

adapted from Hartstein & Liseau 1998, AA 332, 703

Herschel HIFI Sn/TA ~ 500 Jy/K and o/p = 3 assumed

L 3 - Stellar Evolution I: November-December, 2006

slide26
Magnetised isothermal clouds

Magnetic fields neglected in hydrodynamics of isothermal spheres:

not important ?...

Book

Chapters

9 + 10

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

L 3 - Stellar Evolution I: November-December, 2006

slide27
Allen et al:

Development of pseudodisk

Constant mass accretion rate

L 3 - Stellar Evolution I: November-December, 2006

slide28
Anything missing ?

L 3 - Stellar Evolution I: November-December, 2006

slide29
Isothermal eos

No heating and cooling processes included

Spherical, nonrotating, nonmagnetic, 1 Mo

definition

continuity

momentum

energy !

rad transfer !

Winkler & Newman 1980, ApJ 236, 201; ApJ 238, 311

L 3 - Stellar Evolution I: November-December, 2006

slide30
Stahler, Shu & Taam 1980, ApJ 241, 637; ApJ 242, 226

protostellar evolution during main accretion phase

Pre-main-sequence evolution begins after collapse

or main accretion phase

L 3 - Stellar Evolution I: November-December, 2006

slide31
Stahler (and Palla & Stahler ch. 11.2): stellar birthline

Deuterium burning acts as a thermostat

2H(p, g)3He

Reaction rates (Harris et al. 1983, ARAA 21, 165)

-> temperature sensitivity

Assignment: anyone?

Deuterium Burning

Protostellar Pulsations

L 3 - Stellar Evolution I: November-December, 2006

slide32
Protostar evolution of a single star

Fragmentation during collapse ?

L 3 - Stellar Evolution I: November-December, 2006

slide33
Analytically, Tohline (1982): fragmentation of isothermal or adiabatic spheres
  • Isothermal collapse (G = 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, 2006

slide34
2.Adiabatic collapse:

L 3 - Stellar Evolution I: November-December, 2006

slide35
Numerically,

Reid et al. 2002, ApJ 570, 231

See movie in

L7

numerical simulations

Rapid collapse

Sheets: Burkert & Hartmann 2004

ApJ 616, 288

General discussion:

Hennebelle et al. 2004, MNRAS 348, 687

L 3 - Stellar Evolution I: November-December, 2006

slide36
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 ?

L 3 - Stellar Evolution I: November-December, 2006

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