Jets, Disks, and Protostars

5 May 2003

Astronomy G9001 - Spring 2003

Prof. Mordecai-Mark Mac Low

How does collapse proceed?

• Singular isothermal spheres have constant accretion rates

• Observed accretion rates appear to decline with time (older objects have lower Lbol)

• Flat inner density profiles for cores give better fit to observations.

• Collapse no longer self-similar, so shocks form.

30.975 sM c G

Accretion shocks

Yor

ke e

t al.

1993

• Infalling gas shocks when it hits the accretion disk, and again when it falls from the disk onto the star

• Stellar shock releases most of the luminosity

• Disk shock helps determine conditions in flared disk.

Accretion disks• Form by dissipation in accreting gas• Observed disks have M ~ 10-3 M << M*

• Inward transport of mass and outward transport of angular momentum energetically favored.

• How can gas on circular orbits move radially?• Either microscopic viscosity or macroscopic

instabilities must be invoked. – Balbus-Hawley instabilities can provide viscosity– gravitational instability produces spiral density

waves on macroscopic scales

• Gravitational instability will act if B-H remains ineffective while infall continues.

Disk Structure

• Nelecting pressure (Ωr >> cs) and disk self-gravity, radial force eqn:

• So long as M large, Ω ~ r -3/2 (Kepler’s law)

• Shear in Keplerian disk

• Define a shear stress tensor

• If viscosity ν 0, torque is exerted

• angular momentum transport is then

Shu, Gas Dynamics

2 2r r GM r

3

2

dr

dr

r

dr

dr π

2 rr r dz

T π

, where mass accretion 2d d r

dJM M rv

dr r

T

Alpha disk models• Viscous accretion a diffusion process, with

• molecular ν = λmfpcs; in a disk with r ~ 1014 cm,– λmfp ~ 10 cm, cs ~ 1 km s-1 => ν ~ 106 cm2 s-1

– so τacc = 1022 s ~ 3 1014 yr!

• Some anomalous viscosity must exist. Often parametrized as πrφ = – αP– based on hydro turbulent shear stress– for subsonic turbulence, δv ~ αcs

– in MHD flow, Maxwell stress

• B-H inst. numerically gives αmag ~ 10-2

– where πrφ = – αmag Pmag

2acc r

r rv v π

r rB B π

Magnetorotational instability• First noted by Chandrasekhar and Velikhov in 1950s

– ignored until Balbus & Hawley (1991) rediscovered it...

• Driven by magnetic coupling between orbits– instability criterion dΩ/dr < 0 (decreasing ang. vel.,

not ang. mntm as for hydro rotational instability)– most unstable wavelength

• so long as λc > λdiss even very weak B drives instability• if B so strong that λc >> H, instability suppressed

• Field geometry appears unimportant• May drive dynamo action in disk, increasing

field to strong-field limit

c B

MRI in protostellar disks

• MRI suppressed in partly neutral disks if every neutral not hit by ion at least once per orbit (Blaes & Balbus 1998)

• Inside a critical radius Rc ~ 0.1 AU collisional ionization maintains field coupling (Gammie 1996)

• Outside, CR ionization keeps surface layer coupled

• Accretion limited by layerGammie 1996

Simulations of MRI suppression

Hawley & Stone 1998

Sheet formation occurs in partially neutral gas

Mac Low et al. 1995

less

ion

izat

ion

tim

e

Gravitational Instability in Disks• Important for both protostellar and galactic disks

• Axisymmetric dispersion relation

– from linearization of fluid equations in rotating disk– angular momentum decreasing outwards ( )

produces hydro instability

• Differential rotation stabilizes Jeans instability– if collapsing regions shear apart in < tff then stable

2 2 2 2

223

2

where is the disk surface density, and

1the square of the epicyclic frequency

sk c G k

dr

r dr

2 0

Shu, Gas Dynamics

Toomre Criterion

• Disks with Toomre Q < 1 subject to gravitational instability at wavelengths around λT

Q

λ / λT

1

0 1/2 1

ω2 > 0 stable

ω2 < 0 unstable

Shu, Gas Dyn.

stabilized by rotationstabilized

by pressure

22 2

T 2

s

T

41 0, where

4

4 cand the Toomre parameter Q =

T T

s

Q G

c

G

• Accretion increases surface density σ, so protostellar disk Q drops

• Gravitational instability drives spiral density waves, carrying mass and angular momentum.

• Will act in absence of more efficient mechanisms• Very low Q might allow giant planet formation.

– direct gravitational condensation proposed

– may be impossible to get through intermediate Q regime though, due to efficient accretion there.

– standard giant planet formation mechanism starts with solid planetesimals building up a 10 M core followed by accretion of surrounding disk gas

• Brown dwarfs may indeed form from fragmentation during collapse (“failed binaries”).

Jets

• Where does that angular momentum go?• Surprisingly (= not predicted) much goes into jets

– lengths of 1-10 pc, inital radii < 100 AU– velocities of a few hundred km s-1 (proper motion,

radial velocities of knots)– carry as much as 40% of accreted mass– cold, overdense material

• CO outflows carry more mass – driven either by jets, or associated slower disk winds– velocities of 10-20 km s-1 – masses up to a few hundred M

Herbig-Haro objects• Jets were first detected in optical line

emission as Herbig-Haro objects• H-H objects turn out to be shocks

associated with jets– bow shocks – termination shocks– internal knots– tangential & coccoon shocks

• line spectrum can be used to diagnose velocity of shocks

Jet Observations

CO outflows

High resolution interferometric observations reveal that at least some CO outflows tightly correlated with jets. Others less collimated. Also jets?

Gueth & Guilleteau 1999

Blandford-Payne disk winds

C. Fendt

• Gas on magnetic field lines in a rotating disk acts like “beads on a wire”

• If field lines tilted less than 60o from disk, no stable equilibrium => outflow

2

0

2 20 0

0

Effective potential along a field line

1,

2

where is the footpoint of the field line

GM r rr z

r r r z

r

Jet Propagation• Collimation

– Gas dynamical jets are self-collimating – However, hydro collimation cannot occur so close to

source– Toroidal fields can collimate MHD jets quickly

• Knots in jets– knots found to move faster than surrounding jet– variability in jet luminosity seen at all scales– large pulses overtake small ones, sweeping them up

simulated IR from M.D. Smith“Hammer Jet”

Time Scales

• Free-fall time scale

• Kelvin-Helmholtz time scale (thermal relaxation: radiation of gravitational energy)

• Nuclear timescale

1 2~ 1 hr for Sunff G

27~ 3 10 yr for SunKH

GM

RL

10~ 10 yr for SunHN H He

MxE

L

Termination of Accretion

• exhaustion of dynamically collapsing reservoir?– masses determined by molecular cloud

properties?– competition with surrounding stars for a

common reservoir?

• termination of accretion?– ionization– jets and winds– disk evaporation and disruption

Protostar formation• Dynamical collapse continues until core becomes

optically thick (dust) allowing pressure to increase. n ~ 1012 cm-3, 100 AU– Jeans mass drops, hydrost. equil. reached

– radiation from dust photosphere allows quasistatic evolution

• Second dynamical collapse occurs when temperature rises sufficiently for H2 to dissociate

• Protostar forms when H- becomes optically thick. – Luminosity initially only from accretion.

– Deuterium burning, then H burning

z

C. Fendt

• deeply embedded, most mass still accreting

• disk visible in IR, still shrouded

• T-Tauri star, w/disk, star, wind

• weak-line T-Tauri star

Pre-Main Sequence Evolution• Protostar is fully convective

– fully ionized only in center– Large opacity, small adiabatic temperature gradient

• Energy lost through radiative photosphere, gained by grav. contraction until fusion begins

• Fully convective stars with given M, L have maximum stable R, minimum T– Hayashi line on HR diagram

• Pre-main sequence evolutionary calculations must include non-steady accretion to get correct starting point (Wuchterl & Klessen 2001)