The formation of stars and planets

Day 2, Topic 3:

Collapsing cloudsand the

formation of disks

Lecture by: C.P. Dullemond

Spherically symmetric free falling cloud

€

vff =2GM*

rIf stellar mass dominates:

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vff =2GM(r)

rFree fall velocity:

Continuity equation:

€

∂ρ∂t

+1

r2

∂(r2ρv)

∂r= 0

€

∂(r2ρvff )

∂r= 0

Stationaryfree-fallcollapse

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ρ(r)∝ r−3 / 2

Inside-out collapse of metastable sphere

r

ρ

r

ρ Suppose inner region is converted into a star:

r

ρNo support again gravity here, so the next mass shell falls toward star

ρ

r

The ‘no support’-signal travels outward with sound speed (“expansion wave”)

(warning: strongly exaggerated features)

Hydrodynamical equations

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∂ρ∂t

+1

r2

∂(r2ρv)

∂r= 0

Continuity equation:

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∂v

∂t+ v

∂v

∂r= −

1

ρ

∂P

∂r−

GM(r)

r2

Comoving frame momentum equation:

Equation of state:

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P = ρ cs2

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M(r) ≡ 4π r'2 ρ(r') dr'0

r

∫

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cs2 ≡

kT

μ mH

= const.

Inside-out collapse model of Shu (1977)

• The analytic model:– Starts from singular isothermal sphere– Models collapse from inside-out– Applies the `trick’ of self-similarity

• Major drawback:– Singular isothermal sphere is unstable and therefore

unphysical as an initial condition

• Nevertheless very popular because:– Only existing analytic model for collapse– Demonstrates much of the physics

Inside-out collapse model of Shu (1977)

Expansion wave moves outward at sound speed.So a dimensionless coordinate for self-similarity is:

€

x =r

cs tIf there exists a self-similar solution, then it must be of the form:

€

ρ(r, t) =α (x)

4π Gt 2

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M(r, t) =cs

3t

Gm(x)

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v(r, t) = csu(x)

Now solve the equations for (x), m(x) and u(x)

Inside-out collapse model of Shu (1977)

Solution requires one numerical integral. Shu gives a table.

An approximate (but very accurate) ‘solution’ is:

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g ≡1

1.43x 3 / 2

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h ≡2

x

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(x) = g(x)7 / 2 + h(x)7 / 2( )

2 / 7

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u(x) = h(x)5 / 9 − 25 / 9( )

9 /10

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m(x) =1.025 x 2 + 0.975+ 0.075 x (1− x)

For any t this can then be converted into the real solution

Inside-out collapse model of Shu (1977)

Inside-out collapse model of Shu (1977)

Inside-out collapse model of Shu (1977)

Inside-out collapse model of Shu (1977)

Singular isothermal sphere: r-2

Free-fall region: r-3/2

Transition region: matter starts to fall

Expansion wave front

Inside-out collapse model of Shu (1977)

Accretion rate is constant:

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˙ M =cs

3m0

G= 0.975

cs3

G

Stellar mass grows linear in time

Deep down in free-fall region (r << cst):

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ρ(r, t) =cs

3 / 2

17.96G

1

t

1

r3 / 2

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v(r, t) =2GM*(t)

r

A ‘simple’ numerical model

€

∂ρ∂t

+1

r2

∂(r2ρv)

∂r= 0

€

∂v

∂t+ v

∂v

∂r= −

1

ρ

∂P

∂r−

GM(r)

r2

€

P = ρ cs2

€

M(r) ≡ 4π r'2 ρ(r') dr'0

r

∫

A ‘simple’ numerical model

Temperature: 30 KOuter radius: 5000 AUInitial condition: BE sphere with ρc = 1.2x10-17 g/cm3

ρ(r)

A ‘simple’ numerical model

A more `realistic’ non-static model: Make perturbation, but keep mass the same.

ρ(r)

A ‘simple’ numerical model

Strong wobbles, but it remains stable

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benötigt.

ρ(r)

Observations of such dynamical behavior

Lada, Bergin, Alves, Huard 2003

A ‘simple’ numerical modelNow add a little bit of mass (10%) to nudge it over the BE limit:

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benötigt.

ρ(r)

Cloud collapses in a global way (not really inside-out)

Maps of pre-stellar cores

Shirley, Evans, Rawlings, Gregersen (2000)

Maps of class 0 sources

Shirley, Evans, Rawlings, Gregersen (2000)

Line profile of collapsing cloud

Flux

Blue, i.e. toward the observer

Red, i.e. away from observer

Optically thin emission is symmetric

Line profile of collapsing cloud

Flux

Blue, i.e. toward the observer

Red, i.e. away from observer

But absorption only on observer’s side (i.e. on redshifted side)

v (km/s)

T (K)

Example:Observations of B335 cloud.Zhou et al. (1993)

Collapse of rotating cloudsSolid-body rotation of cloud:

0

x

y

z

v0

r0

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v0 = ω r0 sinθ0

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j = r0v0 << GM r0

Infalling gas-parcel falls almost radially inward, but close to the star, its angular momentum starts to affect the motion.

At that radius r<<r0 the kinetic energy v2/2 vastly exceeds the initial kinetic energy. So one can say that the parcel started almost without energy.

Collapse of rotating clouds

No energy condition:

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etot ≡v 2

2−

GM

r≅ 0

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=re

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a + r = constFocal point of ellipse/parabola:

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=2rm

Equator

r rm

re

avm

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vm2 =

2GM

rm

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=GM re

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j 2 = vm2 rm

2 = 2GM rmAng. Mom. Conserv:

Radius at which parcel hits the equatorial plane:

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v0 = ω r0 sinθ0

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re =j 2

GM=

ω2 r04 sin2 θ0

GM

Collapse of rotating clouds

For larger 0: larger re

For given shell (i.e. given r0), all the matter falls within thecentrifugal radius rc onto the midplane.

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rc = re(θ0 = π /2) =ω2r0

4

GM

If rc < r*, then mass is loaded directly onto the star

If rc > r*, then a disk is formed

In Shu model, r0 ~ t, and therefore:

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rc ∝ t 4

Protostellar disks and jets

• Most of infalling matter falls on the equator and forms a disk

• Friction within the disk causes matter to accrete onto the star

• Jets are often launched from the inner regions of these disks

• A jet penetrates through the infalling cloud and opens a cavity

Spectra of collapsing cloud + star + disk

Whitney et al. 2003

Class 0

Spectra of collapsing cloud + star + disk

Whitney et al. 2003

Class I

Spectra of collapsing cloud + star + disk

Whitney et al. 2003

Class II

Spectra of collapsing cloud + star + disk

Whitney et al. 2003

Class III