# Jets, Disks, and Protostars

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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
L
bol
)

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

Collapse no longer self
-
similar, so shocks
form.

3
0.975
s
M c G

Accretion shocks

Yorke et 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 >> c
s
) and disk self
-

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 2
r r GM r
 
3
2
d
r
dr

  
r
d
r
dr



π
2
r
r r dz



T
π
, where mass accretion 2
d d r
dJ
M M rv
dr r
 

   

T
Alpha disk models

Viscous accretion a diffusion process, with

molecular
ν = λ
mfp
c
s
;

in a disk with
r ~

10
14

cm,

λ
mfp

~
10 cm,
c
s

~ 1 km s
-
1
=> ν ~ 10
6

cm
2
s
-
1

so
τ
acc

=
10
22

s ~ 3

10
14

yr!

Some anomalous viscosity must exist. Often
parametrized as
π

=

αP

based on hydro turbulent shear stress

for subsonic turbulence,
δv ~ αc
s

in MHD flow, Maxwell stress

B
-
H inst. numerically gives
α
mag

~
10
-
2

where
π

=

α
mag

P
mag

2
acc
r
 

r r
v v
 
 
 
π
r r
B 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
d
Ω/dr < 0

(decreasing ang.
vel.,
not ang. mntm as for hydro rotational instability)

most unstable wavelength

so long as
λ
c
> λ
diss

even very weak
B

drives instability

if
B
so strong that λ
c

>> 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)

R
c
~

0.1 AU collisional
ionization maintains field
coupling (
Gammie 1996)

Outside, CR ionization
keeps surface layer coupled

Accretion limited by layer

Gammie 1996

Simulations of MRI suppression

Hawley & Stone 1998

Sheet formation
occurs in partially
neutral gas

Mac Low et al. 1995

less ionization

time

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 <
t
ff

then stable

2 2 2 2
2
2
3
2
where is the disk surface density, and
1
the square of the epicyclic frequency
s
k c G k
d
r
r dr
   

  
 
 
 
 
2
0

Shu, Gas

Dynamics

Toomre Criterion

Disks with Toomre Q < 1 subject to gravitational
instability at wavelengths around
λ
T

Q

λ / λ
T

1

0

1/2

1

ω
2

> 0 stable

ω
2

< 0 unstable

Shu, Gas Dyn.

stabilized

by rotation

stabilized

by pressure

2
2 2
T
2
s
T
4
1 0, where
4
4 c
and 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,

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

If field lines tilted less than
60
o

from disk, no stable
equilibrium => outflow

2
0
2 2
0 0
0
Effective potential along a field line
1
,
2
where is the footpoint of the field line
GM r r
r 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

Nuclear timescale

1 2
~1hr for Sun
ff
G
 

2
7
~ 3 10 yr for Sun
KH
GM
RL

 
10
~10 yr for Sun
H
N H He
Mx
E
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 ~
10
12

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 H
2
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

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
-