Planet Formation
Topic:
Collapsing clouds
and the
formation of disks
Lecture by: C.P. Dullemond
Formation of a star from a
spherical molecular cloud core
Hydrostatic pre-stellar Cloud Core
Equation of hydrostatic equilibrium:
1 dP(r)
GM (r)
=r (r) dr
r2
r
Enclosed mass M(r):
M(r) º
ò
r
0
Equation of state:
P = rc
2
s
4p r' r(r') dr'
2
Isothermal sound speed:
kT
c º
= const.
m mH
2
s
We assume that cloud is isothermal at e.g. T = 30 K
Hydrostatic pre-stellar Cloud Core
Ansatz: Powerlaw density distribution:
q
ærö
r (r) = r0 ç ÷
è r0 ø
Put it into pressure gradient:
q
2
æ
ö
dP(r)
d r
cs r0 q q-1
2 d r (r)
2
= cs
= cs r0 ç ÷ =
r
q
dr
dr
dr è r0 ø
r0
Divide by density:
1 dP(r) c q
=
r (r) dr
r
2
s
Hydrostatic pre-stellar Cloud Core
Ansatz: Powerlaw density distribution:
q
ærö
r (r) = r0 ç ÷
è r0 ø
Put it into the enclosed mass integral:
M(r) º
ò
r
0
4p r' r(r') dr'
2
4pr0 r
2+q
= q ò (r ') dr '
0
r0
4pr0 é 3+q ùr
=
(r ') û
që
0
(3+ q)r0
4pr0 3+q
=
r
q
(3+ q)r0
(for q>-3)
Hydrostatic pre-stellar Cloud Core
1 dP(r) c q
=
r (r) dr
r
2
s
GM (r)
4p Gr0 1+q
=r
2
q
r
(3+ q)r0
Put it into the hydrostatic equil eq.:
cq
4p Gr0 1+q
=r
q
r
(3+ q)r0
2
s
Only a solution for q=-2
2c
4p G r 0 2
=r0
r
r
2
s
c
r0 =
2p Gr02
2
s
Hydrostatic pre-stellar Cloud Core
c
r0 =
2
2p Gr0
2
s
q
and
ærö
r (r) = r0 ç ÷
è r0 ø
gives:
cs2
r (r) =
2
2p Gr
Singular isothermal
sphere hydrostatic solution
Inside-out Collapse
The idea by Frank Shu (the „Shu model“)
is that a singular isothermal sphere may
start collapsing once a small disturbance
in the center makes the center lose its
pressure.
Then the next mass shell loses its support
and starts to fall.
Then the next mass shell loses its support
and starts to fall.
Etc etc.  Inside-out collapse.
Wave proceeds outward with the isothermal sound speed.
Inside-out Collapse
Once a shell at radius r starts to fall, it takes
about a free-fall time scale before it reaches
the center. This is roughly the same time
it took for the collapse wave to travel
from the center to the radius.
Let us, however, assume it falls instantly
(to make it easier, because the real solution
is quite tricky). The mass of a shell at radius r and width dr is:
2 2
2
4
p
c
r
dr
2c
2
s
s
dM (r) = 4p r (r)r dr =
=
dr
2
2p Gr
G
Inside-out Collapse
Sind the collapse wave propagates at
dr = cs dt
Meaning we get a dM(r) of
2cs2
2cs3
dM (r) =
dr =
dt
G
G
If we indeed assume that this shell falls instantly onto the
center (where the star is formed) then the mass of the star
increases as
dM 2cs3
=
dt
G
If we account for the free-fall time, we obtain roughly:
dM c
=
dt
G
3
s
The „accretion rate“ is constant!
Formation of a disk
due to angular momentum
conservation
Ref: Book by Stahler & Palla
Formation of a disk
Solid-body rotation of cloud:
z
v 0 = w r0 sinq0
0
v0
j = r0 v0 << GMr0
r0
x
y
Assume fixed M
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.
Formation of a disk
Simple estimate using angular momentum:
j = w r0 sinq 0
z
2
jz = w r0 sin q 0
2
0
v0
r0
x
Kepler orbit at r<<r0 has:
jK (r) = GMr
y
Setting
2
jK (r) = jz
yields
jz2
w 2 r04 sin 4 q 0
r=
=
GM
GM
Formation of a disk
Bit better calculation
Focal point of ellipse/parabola:
a + r = const = re = 2rm
v 2 GM
º
@0
2
r
2GM
v =
rm
No energy condition:
etot
Ang. Mom. Conserv:
j = v r = 2GM rm = GM re
2
2
m
2 2
m m
Radius at which parcel hits
the equatorial plane:
j2
w 2 r04 sin 2 q 0
re =
=
GM
GM
re
Equator
a
r
rm
vm
Formation of a disk
Since also gas packages come from the other side of the
equatorial plane, a disk is formed.
With which angular velocity will the gas enter the disk?
jz = w r02 sin 2 q 0
Kepler angular momentum at r=re:
jKe = GMre = w r02 sinq 0
Their ratio is:
jz
= sin q 0
jKe
The infalling gas rotated sub-kepler. It must
therefore slide somewhat inward
before it really enters
the disk.
Formation of a disk
For larger 0: larger re
For given shell (i.e. given r0), all the matter falls within the
centrifugal radius rc onto the midplane.
rc = re (q0 = p /2) =
w 2 r04
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 M ~ t, and therefore:
rc µ t 3
Formation of a disk
• This model has a major problem: The disk is assumed
to be infinitely thin. As we shall see later, this is not
true at all.
• Gas can therefore hit the outer part of the disk well
before it hits the equatorial plane.
Disk formation: Simulations
Yorke, Bodenheimer & Laughlin (1993)