planet formation topic: collapsing clouds and the formation of disks lecture by: c.p. dullemond

Planet Formation

Topic:

Collapsing cloudsand 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:

Equation of state:

Enclosed mass M(r):

r

Isothermal sound speed:

We assume that cloud is isothermal at e.g. T = 30 K

Hydrostatic pre-stellar Cloud CoreAnsatz: Powerlaw density distribution:

Put it into pressure gradient:

Divide by density:

Hydrostatic pre-stellar Cloud CoreAnsatz: Powerlaw density distribution:

Put it into the enclosed mass integral:

(for q>-3)

Hydrostatic pre-stellar Cloud Core

Put it into the hydrostatic equil eq.:

Only a solution for q=-2

Hydrostatic pre-stellar Cloud Core

and

gives:

Singular isothermalsphere hydrostatic solution

Inside-out Collapse

The idea by Frank Shu (the „Shu model“)is that a singular isothermal sphere maystart collapsing once a small disturbancein the center makes the center lose itspressure.

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 takesabout a free-fall time scale before it reaches the center. This is roughly the same timeit 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 solutionis quite tricky). The mass of a shell at radius r and width dr is:

Inside-out CollapseSind the collapse wave propagates at

Meaning we get a dM(r) of

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

If we account for the free-fall time, we obtain roughly:

The „accretion rate“ is constant!

Formation of a diskdue to angular momentum

conservation

Ref: Book by Stahler & Palla

Formation of a diskSolid-body rotation of cloud:

0

x

y

z

v0

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.

Assume fixed M

Formation of a diskSimple estimate using angular momentum:

0

x

y

z

v0

r0

Kepler orbit at r<<r0 has:

Setting yields

Formation of a disk

No energy condition:

Focal point of ellipse/parabola:

Equator

r rm

re

avm

Ang. Mom. Conserv:

Radius at which parcel hits the equatorial plane:

Bit better calculation

Formation of a disk

With which angular velocity will the gas enter the disk?

Since also gas packages come from the other side of the equatorial plane, a disk is formed.

Kepler angular momentum at r=re:

Their ratio is: The infalling gas rotated sub-kepler. It must therefore slide somewhat inward before it really entersthe disk.

Formation of a disk

For larger 0: larger re

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

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:

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)