the formation of stars and planets day 2, topic 3: collapsing clouds and the formation of disks...
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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:
€
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
€
∂ρ∂t
+1
r2
∂(r2ρv)
∂r= 0
Continuity equation:
€
∂v
∂t+ v
∂v
∂r= −
1
ρ
∂P
∂r−
GM(r)
r2
Comoving frame momentum equation:
Equation of state:
€
P = ρ cs2
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M(r) ≡ 4π r'2 ρ(r') dr'0
r
∫
€
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:
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ρ(r, t) =α (x)
4π Gt 2
€
M(r, t) =cs
3t
Gm(x)
€
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)
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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ρ(r)
A ‘simple’ numerical modelNow add a little bit of mass (10%) to nudge it over the BE limit:
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ρ(r)
Cloud collapses in a global way (not really inside-out)
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