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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. - PowerPoint PPT Presentation
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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
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
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
Barnard 68 considered a pre-collapse/collapse candidate
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
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)
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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)
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
Generic types of theories of collapse
analytical
semi-analytical
numerical
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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 [email protected]
Early WorkBasic Insights
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
Penston 1969, MNRAS 144, 425 Larson 1969, MNRAS 145, 271Shu 1977, ApJ 214, 488Hunter 1977, ApJ 218, 834
Self-similarity solutions
Isothermal spherical collapse
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
(3b) 1
(2b) 01
(1b)
yields (3a) toeqs.(1a) intoon Substituti
2
2
2
2
mD
D
vv
v
Dv
r
D
dDm
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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)
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
2
DP
vU
LP = Larson, Penston
H = Hunter
EW = Expansion Wave (Shu)
vel
oc
ity
de
ns
ity
GcmM /3
s0
supersonic
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
Inside-out collapse (Shu 1977)
Mass accretion rate a constant of the cloud
Mass accretion time scale M
Mt acc
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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 [email protected]
( = 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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
compressional luminosity: pre-core formation
Cloud boundary
max = 6.541
Foster & Chevalier
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
Inside-out collapse (Shu 1977)
Why Shu 1977 commonly used ?
...computational convenience
...small number of parameters
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
Allen et al:Development of pseudodiskConstant mass accretion rate
pressure by thermal supportedy overdensit
field magneticby supportedy overdensit
/)1(
0
3s0
H
GcHM
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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 ?
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
Protostar evolution of a single star
Fragmentation during collapse ?
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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...
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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
L 3 - Stellar Evolution I: November-December, 2006 [email protected]
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 - Stellar Evolution I: November-December, 2006 [email protected]
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 ?