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

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

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)

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

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: 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)

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

Inside-out collapse model of Shu (1977)

Inside-out collapse model of Shu (1977)

Inside-out collapse model of Shu (1977)

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) Deep down in free-fall region (r << cst): Accretion rate is constant: Stellar mass grows linear in time

A ‘simple’ numerical model

A ‘simple’ numerical model Temperature: 30 K Outer radius: 5000 AU Initial condition: BE sphere with c = 1.2x10-17 g/cm3 (r)

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

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

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

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)

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

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

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

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)

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.

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

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:

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

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

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

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

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