Transcript Distribution and Properties of the ISM

Star Formation
28 April 2003
Astronomy G9001 - Spring 2003
Prof. Mordecai-Mark Mac Low
Gravitational Stability
• Criterion for gravitational stability found
λJ
by Jeans (1902).
density ρ
• Pressure opposes collapse: sound waves
must cross region to communicate pressure
changes before collapse
•
w ithout pressure, collapse occurs in a free-fall tim e
2
d r
dt
2
 
GM
r
2
 
4 G 
r
3
a harm onic oscillator w ith frequency  = 4  G  3
t ff
1 2


3



4 
 4 4 G  
2
1 2
 3  


 16 G  
1 2
Jeans instability
• Jeans swindle: in homogeneous medium
–    0   1 ; set  0  0, assum e  2  1  4  1
– not generally true, but usually justified
• Linearize equations of motion for this medium
   0   1 ( x , t ); v  v 1 ( x , t );    1 ( x , t )
1
t
  0  v 1  0
 2 1 
   cs
  1
t
0 

v1
  1  4  1
2
m ass conservation
m om entum conservation
P oisson's equat i on
T aking a tim e derivative of the first eq uation
and the divergence of the second equatio n
 1
2
t
2
  0 
v1
t
0
 2 1 
2

   cs


1

t
0 

v1
2
com bining and substituting
 1
2
t
2
 c s   1  4 G  0  1  0
2
2
T ake a trial s olution  1  C e
  c s k  4 G  0  0
2
2
2
i k x  t 
, to find
  c k  4 G  0
2
2
s
2
fo r sm all  0 o r larg e k th is red u ces to th e
d isp ersio n relatio n fo r so u n d w aves 
A s  0 in creases, 
2
d ecreases, u n til 
2
2
c k
2
s
< 0
p ro d u cin g ex p o n en tial in crease in d en sity.
T h is o ccu rs w h en
k  k 
2
2
J
4 G  0
c
2
s
Jean s w aven u m b er
2
Timescales
• What determines the rate of star formation
in galaxies?
n


• Free-fall time t  10 yr  

1 / 2
6
ff
• Galaxy lifetimes greater
3
3
 10 cm 
than 109 yr.
• Yet star formation continues today.
• How are starbursts, low surface brightness
galaxies different?
• Star formation
rate higher at
high z
• Actual value
depends on
corrections for
dust obscuration,
surfacebrightness
dimming
• Different
methods still give
drastically
different results.
Cosmic Star
Formation Rate
Lanzetta et al. (2002)
Madau et al.
(1996, 1998)
Schmidt Law
• Empirically Kennicutt
(1989, 1998) finds

 SF R   gas
  1.4  0.05
• This can be seen as a
free-fall collapse:
 SF R   SF R
  SF R


t ff

1 2
 
G  
• but what is  SFR ?
 1.5
Kennicutt 1998
dN
 m

Initial Mass Function
dm
• Salpeter (1955): α = 2.35
Galaxy
ONC
Pleiades
M35
• Describes high mass
stars well.
• Low mass stars
described by lognormal (Miller & Scalo)
or multiple power laws
(Kroupa 2002)
• Why is form universal?
• What determines peak?
Kroupa 2002
How can stars form?
• Gravity is counteracted by
– thermal pressure
– angular momentum
– magnetic pressure and tension
• Each must be overcome for collapse to occur
Isothermal Sphere Solutions
In hydrostatic equilibrium
 P    
In spherical sym m etry
cs d 
2
 dr

d
dr
   r    0 exp    c
2
s

W e can now w rite P oisson's equation as
1 d  2 d 
2
r

4

G


4

G

exp


c

0
s 


2
r dr 
dr 
This is the Lane-Emden equation, which forms the basis
2
for stellar structure. When  0   , then  
cs
2 G r
2
Bonnor-Ebert Spheres
• Ebert (1955) and Bonnor (1956) found solutions to
the modified Lane-Emden equation for finite
external pressure Pext
d  2 d 
2 



e ,


d 
d 
w h ere  
r
cs
  r 
4  G   0  ,    ln 
   0  


an d fin ite   0   B C s   0   0,
d
d
0   0
solve by change of variables
x
2
d
d
to get tw o first-order equations w ith IC s:
d
d

x

2
;
dx
d
  exp     ; x  0   0,   0   0.
2
T his has a unique solution out to radius R
 m ax 
R
cs
4 G   0 
each solution is defined by R , c s , a nd   0  .
If  m ax  6.5 
 0
 R
 14.3, then unstable equilibrium
Isothermal Collapse
• Larson (1969) and Penston (1969) found similarity
solutions for the collapse of a uniform density,
isothermal sphere
• Shu (1977) gave the collapse solution for an
initially hydrostatic isothermal sphere (but this
has a density singularity at the center).
• Whitworth & Summers (1985) showed that many
isothermal collapse solutions exist; all can be
described with two parameters measuring the
initial and final density concentration
• Isothermal sphere with central singularity is
only the most extreme case
MHD Support
• Magnetic fields can prevent collapse if
magnetic energy exceeds potential energy
• Remember, virial theorem analysis yields
M CR

0.13
G
1 2
1
 RC R
 B   M
 230 pc  
 
 3 G  1 M



• So, flux must be lost at some stage to allow
stars to form, or gas must be accumulated
along field lines over large distances.
Ambipolar Diffusion
• Neutral-ion drift (note different def’n in plasma
physics: electron-ion drift)
• Collisional drag force Fni = -Fin = γρiρn(vi - vn)
– drag coefficient γ constant for vd < 10 km s-1
• For low ionization fraction, drag balances
Lorentz
1
vd  vi  vn 
4 n  i
  B   B
• The induction equation is a non-linear diffusion
equation
B
t
   B  vn 


B

  B     B   
 4 n  i

w ith diffusion coefficient D  B
2
4  n  i  v  ni
2
A
the am bipolar diffusion tim escale is the n
t AD
L D ~  A  ni
2
2
• In molecular clouds, tAD ~ 10 tff so suggested as
solution to both flux and timescale problems.
– However, stars form within ~ 1 Myr despite varying
local ionization states
– Magnetic field measurements suggest fields already
weak when cores form
– is flux problem solved at larger scales? How?
Importance of Ambipolar Diffusion
• May be most important dissipation
mechanism in turbulence
• Mediates shock waves, reducing heating,
but causing instability
• Determines binary formation by modulating
magnetic braking in protostellar cores
• Controls viscosity in accretion disks by
suppressing magnetorotational instability
• Forms current sheets in turbulent flows,
perhaps melting meteoritic chondrules in
protoplanetary disks?
Turbulent Fragmentation
• Gravitational fragmentation in a turbulent flow
may explain some features of star formation
– collapse time depends on strength of turbulence
– slow, isolated collapse occurs in regions globally
supported against collapse by turbulence
– fast, clustered collapse occurs in unsupported
regions
– IMF appears log normal near Jeans mass
• Turbulent state of molecular clouds suggests
this mechanism indeed operates
– most observed cores magnetically unsupported
Angular Momentum
J  I ; I  MR
2
• Consider ISM with M = 1 M, n = 1 cm-3
R  M  
1 3
3 pc and 
3
10
 15
1
s , so J
10
• Angular momentum not conserved:
–
–
–
–
–
Diffuse gas > molecular clouds
molecular clouds > cloud cores
cloud cores > protostars
protostars > main sequence stars
J ~ 1048 g cm-2 s-1
• Where does it go? (Binary formation
insufficient)
56
g cm
2
s
1
Magnetic Braking
• Magnetic fields can redistribute
angular momentum away from a
collapsing region
• Outgoing helical Alfvèn waves must
couple with mass equal to mass in
collapsing region (Mouschovias &
Paleologou 1979, 1980)
b

R
 0 2vA
Binary Formation
• In absence of magnetic fields,
binary formation occurs from
the collapse of rotating regions
• Ratio of gravitational to
rotational energy determines
fragmentation
• However, magnetic braking
can effectively drain rotational
Burkert & Bodenheimer 93
energy, preventing binary
formation
• does ambipolar diffusion allow decoupling of
core from field to explain high binary rate?
Piecewise Parabolic Method
•
•
•
•
•
Third-order advection
Godunov method for flux estimation
Contact discontinuity steepeners
Small amount of linear artificial viscosity
Described by Colella & Woodward 1984, JCP,
compared to other methods by Woodward &
Colella 1984, JCP.
Parabolic Advection
• Consider the linear advection equation
a
t
u
a

a ( , 0 )  a 0  
 0;
• Zone average values must satisfy

1
n
aj 
 a  d 
j 1 2
 j
j 1 2
• A piecewise continuous function with a
parabolic profile in each zone that does so is
a    a L , j  x  a j  a 6 , j 1  x ; x 
a j  aR, j  aL, j ;
   j 1 2
 j
 n 1

a 6  6  a j  a L , j  a R , j  
2


Interpolation to zone edges
• To find the left and right values aL and aR,
compute a polynomial using nearby zone
averages. For constant zone widths Δξj
a j 1 2  a R , j  a L , j 1 
7
12
a
n
j
a
n
j 1

1
12
a
n
j2
 a j 1
n

• In some cases this is not monotonic, so add:
a
a
a
if a
 a a  a   0
a 
1

 a
a
 3a  2 a
if a  a  a  a _ a   
L, j
R, j
n
j
R, j
n
j
n
j
L, j
2
L, j
n
j
R, j
R,j
L, j

n
j
R, j
2
L, j
R, j

• And similarly for aR,j to force montonicity.
L, j
6
Conservative Form
• Euler’s equations in conservation form on a 1D
Cartesian grid
U
t

F
x

u
  




2

u

u



U 
, F 

v
 uv



 E 
  uE  up



conserved
variables
fluxes
H
x
G

0
 0 

 



 p
 g 
, H   0 , G   0 

 



0
  ug 

 


pressure
gravity or
other body
forces
Godunov method
• Solve a Riemann shock tube problem at every
zone boundary to determine fluxes
Characteristic averaging
• To find left and right states for Riemann
problem, average over regions covered by
characteristic: max(cs,u) Δt
tn+1
tn+1
or
xj-1
tn
xj
subsonic
flow
xj+1
xj-1
tn
xj
supersonic
flow
(from left)
xj+1
Characteristic speeds
• Characteristic speeds are not constant across
rarefaction or shock because of change in
pressure
Riemann problem
• A typical analytic solution for pressure (P.
Ricker) is given by the root of
f L  P ,U L   f R  P ,U R   u R  u L  0 ,
where, with I  ( L , R )

 AI 
  P  PI 

 P  BI 

 1
f I  P ,U I   


2


 2 c s , I   P 

1

   1   PI 



AI 
2

 1  I
if P  PI (shock)
if P  PI
BI 
 1
 1
PI
(rarefacti on)