1. The formation of stars and planets
Day 2, Topic 3:
Collapsing clouds
and the
formation of disks
Lecture by: C.P. Dullemond

2. Spherically symmetric free falling cloud
Free fall velocity:
If stellar mass dominates:
Continuity equation:
Stationary
free-fall
collapse

3. Inside-out collapse of metastable sphere
Suppose inner region is converted into a star: r  r 
No support again gravity here, so the next mass shell falls toward star  r
The ‘no support’-signal travels outward with sound speed (“expansion wave”)
(warning: strongly exaggerated features)

4. Hydrodynamical equations
Continuity equation:
Comoving frame momentum equation:
Equation of state:

5. 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

6. Inside-out collapse model of Shu (1977)
Expansion wave moves outward at sound speed.
So a dimensionless coordinate for self-similarity is:
If there exists a self-similar solution, then it must be of the form:
Now solve the equations for (x), m(x) and u(x)

7. Inside-out collapse model of Shu (1977)
Solution requires one numerical integral. Shu gives a table.
An approximate (but very accurate) ‘solution’ is:
For any t this can then be converted into the real solution

8. Inside-out collapse model of Shu (1977)

9. Inside-out collapse model of Shu (1977)

10. Inside-out collapse model of Shu (1977)

11. 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

12. Inside-out collapse model of Shu (1977)
Deep down in free-fall region (r << cst):
Accretion rate is constant:
Stellar mass grows linear in time

13. A ‘simple’ numerical model

14. A ‘simple’ numerical model
Temperature: K
Outer radius: AU
Initial condition: BE sphere with c = 1.2x10-17 g/cm3
(r)

15. A ‘simple’ numerical model
A more `realistic’ non-static model:
Make perturbation, but keep mass the same.
(r)

16. A ‘simple’ numerical model
Strong wobbles, but it remains stable

17. Observations of such dynamical behavior
Lada, Bergin, Alves, Huard 2003

18. A ‘simple’ numerical model
Now add a little bit of mass (10%) to nudge it over the BE limit:
(r)
Cloud collapses in a global way (not really inside-out)

19. Maps of pre-stellar cores
Shirley, Evans, Rawlings, Gregersen (2000)

20. Maps of class 0 sources
Shirley, Evans, Rawlings, Gregersen (2000)

21. Line profile of collapsing cloud
Optically thin emission is symmetric
Flux 
Blue, i.e. toward the observer
Red, i.e. away from observer

22. Line profile of collapsing cloud
But absorption only on observer’s side (i.e. on redshifted side)
Flux 
Blue, i.e. toward the observer
Red, i.e. away from observer
v (km/s)
T (K)
Example:
Observations of B335 cloud.
Zhou et al. (1993)

23. Collapse of rotating clouds
Solid-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.

24. Collapse of rotating clouds
Focal point of ellipse/parabola:
No energy condition:
Ang. Mom. Conserv:
Radius at which parcel hits the equatorial plane:
Equator r rm re a vm

25. Collapse of rotating clouds
For larger 0: larger re
For given shell (i.e. given r0), all the matter falls within the
centrifugal 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 therefore:

26. 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

27. Spectra of collapsing cloud + star + disk
Whitney et al. 2003
Class 0

28. Spectra of collapsing cloud + star + disk
Whitney et al. 2003
Class I

29. Spectra of collapsing cloud + star + disk
Whitney et al. 2003
Class II

30. Spectra of collapsing cloud + star + disk
Whitney et al. 2003
Class III