Physics 778 – Star formation: Protostellar disks Ralph Pudritz

Proplyds in the Orion Nebula (O’Dell & Wong)

1.2 Disk evolution – reading spectral energy distributions (SEDs) from Hartmann 1998 d(log  F ) / d(log ) ( 1 – 10  m) > 0 Class I -3 Class II ~ -3 Class III (photosphere) I II III

Annual Reviews SEDs from Spitzer spectra: Class 0: (bottom) L1448C Class 1: (yellow) IRAS 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 photosphere In IR - mm photosphere

1.3 Disk formation - gravitational collapse of rotating molecular cloud core Particles free-fall conserving specific angular momentum l l ~ r o 2  sin  for particle falling from r o in core with uniform angular velocity  and angle  from rotation axis Higher l for larger separation from rotation axis  r0r0

Particle from r o,  shocks with particle from r o,  +  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 R c = r o 4  2 / GM, M central mass, R c ~ 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 R c 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: T viscous ~ 2   R 3 d  /dR where  = surface density; = viscosity ~ v l, where v and l are characteristic velocity and length of the turbulent motions - uncertain  prescription: =  c s H, where c s 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 >> R 1 2 / 

Disk evolution: expression for viscosity =  c s H, c s sound speed, H = c s /  = c s 2 /  = const T R 3/2 In an irradiated disk at large R, T as 1/R 1/2 So,  ~ const. R Similarity solution for  (R,t) (see Hartmann et al. 1998)