1. Planet Formation Topic: Collapsing clouds and the formation of disks Lecture by: C.P. Dullemond

2. Formation of a star from a spherical molecular cloud core

3. 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

4. Hydrostatic pre-stellar Cloud Core Ansatz: Powerlaw density distribution: Put it into pressure gradient: Divide by density:

5. Hydrostatic pre-stellar Cloud Core Ansatz: Powerlaw density distribution: Put it into the enclosed mass integral: (for q>-3)

6. Hydrostatic pre-stellar Cloud Core Put it into the hydrostatic equil eq.: Only a solution for q=-2

7. Hydrostatic pre-stellar Cloud Core and gives: Singular isothermal sphere hydrostatic solution

8. 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.

9. 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:

10. Inside-out Collapse Sind 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!

11. Formation of a disk due to angular momentum conservation Ref: Book by Stahler & Palla

12. Formation of a disk Solid-body rotation of cloud: 00 x y z v0v0 r0r0 Infalling gas-parcel falls almost radially inward, but close to the star, its angular momentum starts to affect the motion. At that radius r<<r 0 the kinetic energy v 2 /2 vastly exceeds the initial kinetic energy. So one can say that the parcel started almost without energy. Assume fixed M

13. Formation of a disk Simple estimate using angular momentum: 00 x y z v0v0 r0r0 Kepler orbit at r<<r 0 has: Setting yields

14. Formation of a disk No energy condition: Focal point of ellipse/parabola: Equator r rmrm rere a vmvm Ang. Mom. Conserv: Radius at which parcel hits the equatorial plane: Bit better calculation

15. 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=r e : Their ratio is: The infalling gas rotated sub-kepler. It must therefore slide somewhat inward before it really enters the disk.

16. Formation of a disk For larger  0 : larger r e For given shell (i.e. given r 0 ), all the matter falls within the centrifugal radius r c onto the midplane. If r c < r *, then mass is loaded directly onto the star If r c > r *, then a disk is formed In Shu model, r 0 ~ t, and M ~ t, and therefore:

17. 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.

18. Disk formation: Simulations Yorke, Bodenheimer & Laughlin (1993)