Physics 778 – Star formation: Protostellar disks

Ralph Pudritz

1.2 Disk evolution – reading spectral energy distributions (SEDs)

from Hartmann 1998

d(logF) / d(log )( 1 – 10 m)

> 0 Class I< 0 y > -3 Class II

~ -3 Class III (photosphere)

I

IIIII

Annual Reviews

SEDs from Spitzer spectra:

Class 0: (bottom) L1448C

Class 1: (yellow) IRAS 04016+

Class II: (green) different small dust composition

Class III (top, blue) (spectra from FM Tau down offset by factors

50, 200, and 10,000).

Most prominent feature; ices and minerals

Class 1

from Hartmann 1998

Excess of energy above photosphereIn IR - mm

photosphere

1.3 Disk formation - gravitational collapse of rotating molecular cloud core

•Particles free-fall conserving specific angular momentum l

•l ~ ro2 sin for particle falling from ro in core with

uniform angular velocity and angle from rotation axis

•Higher l for larger separation from rotation axis

r0

•Particle from ro, shocks with particle from ro,+ on equatorial plane, vertical velocity component dissipated, particles keep rotating on equatorial plane in a disk

•Particles with ~ /2, reach the equatorial plane at the centrifugal radius Rc = ro

4 2 / GM, M central mass, Rc ~ disk radius

Collapse: streamlines and disk formation…

from Hartmann 1998

Streamlines at constant intervals of cos (dM/dt) ~ cos (dM/dt)/2=>Mass accumulates at Rc

M(core) at large radius => most of the core mass into the disk

1.4 Accretion disks: viscous evolution

• Particles at R rotating with (R) move to R+R, while particles at R+R rotating at (R+R) < (R) move to R. This motion implies a change of J in time, ie, a torque:

Tviscous ~ 2 R3 d/dR

where = surface density; = viscosity

~ v l, where v and l are characteristic velocity and length of the turbulent motions - uncertain

prescription: = cs H, where cs is the sound speed and H the scale height (Shakura & Sunnyaev 1973).

Viscous evolution of a ring: exact mathematical solution (see Pringle, ARAA, 1981)

t=0

all mass at center

all angular momentum at infinity, carried by of the mass

t >> R12/

Disk evolution: expression for viscosity

= cs H, cs sound speed, H = cs/

= cs2/ = const T R3/2

In an irradiated disk at large R, T as 1/R1/2

So, ~ const. R

Similarity solution for (R,t) (see Hartmann et al. 1998)