l 3: collapse phase – theoretical models

L 3 - Stellar Evolution I: November-December, 2006 [email protected]

L 3: Collapse phase – theoretical models

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




The Formation of StarsChapters: 9, 10, 12




Barnard 68 considered a pre-collapse/collapse candidate




Myr10Myr 1 scales time

s km 100 s km 0.1 velocities

K10K 10 estemperatur

cm g 100 cm g10 densities

pc 10 pc 10 scaleslength

star a make to

9

11

6

3321

18

--

-- -

- .

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


Time scales: low mass star formation

1968 Giuli &Cox e.g. , )(1

0

/)()(

5

3 , )(

2

pot

2

)1(3

43KH

tot

limlimff

KH

2/12/3

cloudff

2/5

oKH

o3, for 1

Myropc1.0

5.0

Myro

5.4)2(

scales timefall-free andn contractio chydrostati

-quasi const.)][ homologousfor HelmholtzKelvin

M

rdM

r

MrMrq

qrqR

GME

LR

GMqt

dt

dEL(R)

MMMMt

t

M

MRt

M

MMMt

q(x)


Generic types of theories of collapse

analytical

semi-analytical

numerical


Jeans (1927) MNRAS 87, 720 On Liquid Stars

Joel Tholine (1982)

Hydrodynamic Collapse

Fundamental Cosmic Physics Vol. 8, pp. 1-82


Early WorkBasic Insights


x 2

x10

density

time


Penston 1969, MNRAS 144, 425 Larson 1969, MNRAS 145, 271Shu 1977, ApJ 214, 488Hunter 1977, ApJ 218, 834

Self-similarity solutions

Isothermal spherical collapse


velocityradial

speed sound isothermal

inside mass

state ofequation isothermal , where

(3a) 1

(2a) 01

(1a) 4

equations fluid symmetricy Sphericall

s

2s

2

2

2

2

u

c

rM

cP

r

GM

r

P

r

uu

t

ur

ur

rt

rr

M

Mass

Definition

Continuity

Equation

Momentumequation

eos


c

cs

s

2s

c

c

c

4

4

4

density central initial where

variablesaldimensioni-non Introduce

Gt

Gc

r

c

uv

c

GGMm

D

Similarity Variable


(3b) 1

(2b) 01

(1b)

yields (3a) toeqs.(1a) intoon Substituti

2

2

2

2

mD

D

vv

v

Dv

r

D

dDm


sphere ofextent parameter h family witsimilar -self

0)0()0(

2Φ

1967)khar (Chandrase sphere isothermal of eqs.

mequilibriu chydrostati assume and

(3b) eq. into and density Let the

max

Φ2

2

)(Φ

d

d

ed

d

d

d

eD

Palla & Stahler call this Eq the isothermal Lane-Emden equationLE derived for polytropes ( P = k n ), e.g. fully convective stars: n=3/2 (=1+1/m)


2

DP

vU

LP = Larson, Penston

H = Hunter

EW = Expansion Wave (Shu)

vel

oc

ity

de

ns

ity

GcM

mx

xmG

tcM(r,t)

/975.0

and 975.0 ,0for

)(

:8 eq. 1977,Shu

3s

0

3s


2

DP

vU

LP = Larson, Penston

H = Hunter

EW = Expansion Wave (Shu)

vel

oc

ity

de

ns

ity

GcmM /3

s0

supersonic


488 214, ApJ 1977,Shu : max

451.6crit

Bonnor 1956 MNRAS 116, 351

0ext

V

P

0ext

V

P

centrally condensed

flat distribution

Shu 1977extreme case

max


Inside-out collapse (Shu 1977)

Mass accretion rate a constant of the cloud

Mass accretion time scale M

Mt acc


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)


( = 0 at core formation; ~ 2 tff)

de

ns

ity

r -2

r -3/2

Initia

l & b

ou

nd

ary

con

ditio

ns

Foster & Chevalier 1993, ApJ 416, 311


compressional luminosity: pre-core formation

Cloud boundary

max = 6.541

Foster & Chevalier


compressional luminosity: pre-core formation

Foster & Chevalier

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

o60acc ,yro510,o1.,.

acc

1-

Luminosityretion Infall/Acc

LLMMMMge

R

MMGL

Cloud boundary max = 6.541


Inside-out collapse (Shu 1977)

Why Shu 1977 commonly used ?

...computational convenience

...small number of parameters


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

not fromShu model

p = -1.5

p = -2

Rinf = cs tinf

= -0.5

= 0

adapted from Hartstein & Liseau 1998, AA 332, 703

~ r p ~ r


predicted spectral line profiles of ground state ortho- and para-water (H2O)

for inside-out collapse [B 335]

adapted from Hartstein & Liseau 1998, AA 332, 703

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

infall regionunresolvedat 557 GHz


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

BookChapters

9 + 10


Allen et al:Development of pseudodiskConstant mass accretion rate

pressure by thermal supportedy overdensit

field magneticby supportedy overdensit

/)1(

0

3s0

H

GcHM


velocityradial

speed sound isothermal

inside mass

where

(3a) 1

(2a) 01

(1a) 4

:again equations Fluid

s

2s

2

2

2

2

u

c

rM

cP

r

GM

r

P

r

uu

t

ur

ur

rr

rr

M

Anything missing ?


Isothermal eos

No heating and cooling processes included

0)(1

0)(4div

0141

t

u

0div

3

4

2

rel

2rel

rel

3

SII

rr

I

SJuP

Uut

U

uc

H

r

GMPu

uut

r

M

Q

Qr

r

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

Spherical, nonrotating, nonmagnetic, 1 Mo

momentum

energy !

rad transfer !

continuity

definition


Pre-main-sequence evolution begins after collapseor main accretion phase

Stahler, Shu & Taam 1980, ApJ 241, 637; ApJ 242, 226protostellar evolution during main accretion phase


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

Deuterium burning acts as a thermostat

2H(p, )3He

Reaction rates (Harris et al. 1983, ARAA 21, 165)-> temperature sensitivity

Assignment: anyone?Deuterium Burning

Protostellar Pulsations

9

3/19

/753.632/39

10´reverse

93/2

93/1

9/720.33/2

93

forward

99

)1(

3cm

mole1

1063.1

65.238.3112.00.11024.2

K10/ andin ratesreaction MeV; 5.494

T

T

N

eTR

TTTeTR

TTsQ


Protostar evolution of a single star

Fragmentation during collapse ?


Analytically, Tohline (1982): fragmentation of isothermal or adiabatic spheres

1. Isothermal collapse ( = 1):

Perturbation analysis of pressure-free sphere -> fragmentation during collapseNo preferred wavelength -> perturbations of all sizes grow at the same rate

Real clouds not pressure-free and adiabatic case more relevant...


2.Adiabatic collapse:

P

GM

R

R

GM

MP

0

0

2

0

2

5

5

3energy potential

2

3energy thermal

2

1

energy potential

energy thermal

R radius with cloud of balance rostaticvirial/hyd

collapse during stable moreon perturbati :4/3 (2)

collapse during unstable moreon perturbati : 4/3 (1)

/

/

/length Jeans gth toon wavelenperturbati

important more relatively pressure :4/3 (2)

decreases pressure relative : 4/3 (1)

initial,

and

along contracts sphere uniform the

1 eos adiabaticfor

2/)3/4(

J

J

2/1J

3/4

3/1

1

ii

ii

Γ

Γ

R

R

R

i

R

ρR

Γ,ρP


Numerically,

General discussion:Hennebelle et al. 2004, MNRAS 348, 687

Sheets: Burkert & Hartmann 2004 ApJ 616, 288

See movie inL7

numerical simulations

Rapid collapse 5/3 trapped,isradiation cooling :high at adiabatic

coolingdust and linemolecular :cm g10at isothermal

1 : eos baritropic

313

3/2

0

2s,0

2s

cc

P

Reid et al. 2002, ApJ 570, 231

1

d

d :eos logatropic P


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: collapse phase – theoretical models

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