Download Supersonic Turbulence and Star Formation

Supersonic Turbulence
and Star Formation
Paolo Padoan
University of California, San Diego
Alexei Kritsuk, Mike Norman (UCSD)
Liubin Pan, Rick Wagner (UCSD)
AAke Nordlund (Copenhagen)
Sergey Ustyugov (Moscow)
Copyright Paolo Padoan, 2008
Why is the gas converted into stars?
What sets the mass distribution and the formation rate of stars?
Copyright Paolo Padoan, 2008
Gravitational Instability
Instability of linear density perturbations of a uniform, isothermal, static
gas, extending to infinity (Jeans 1902):
  J =
 
2 1 /2
  th
G 0
⇒
 
J
4
M J= 
3
2
3
0 = 24 M sun

n
−3
200 cm
  
−1/2
T
10 K
3 /2
The cold interstellar medium has a complex hierarchical structure:
3
n ≈ 2×10 cm
−3
 
l
1 pc
−1
,
T ≈ 10K
So clouds of 10 pc size have n~200 cm-3, Mcl~104 Msun, and MJ~ 24 Msun.
Prediction 1:
The characteristic stellar mass in these molecular clouds is ~24 Msun
Copyright Paolo Padoan, 2008
At what rate is the gas converted into stars?
Without pressure support, a uniform sphere collapses in a free-fall time:

3
ff =
32 G 

1/ 2

n
=2.3×10 yr
−3
200 cm
6

−1/2
(roughly a sound crossing time of the Jeans length).
Prediction 2:
Molecular clouds are converted into stars in two million years.
Both predictions from the linear gravitational instability are quite wrong....
Copyright Paolo Padoan, 2008
Large range of stellar masses: 0.01 - 100 Msun
Characteristic stellar mass: 0.2 Msun
1 Msun
10-3 Msun
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100 Msun
Stellar mass distribution
.
-1
M
):
55
19
r(
te
lpe
Sa
35
Mch Hoyle (1953) MJ
1. Broad range of masses, characteristic mass Mch << MJ
2. Gas conversion into stars ~ 2% per free-fall time
Why are the predictions from the gravitational instability so wrong?
Copyright Paolo Padoan, 2008
UL
8
The cold ISM is highly turbulent, Re =
~10

The turbulence is supersonic, M s ~ 30
--> Highly non-linear velocity and density
Copyright Paolo Padoan, 2008
Turbulence Solution to Star Formation
1) Mass range of stars:
Stellar masses are set by turbulence,
not by self-gravity (M>MJ is possible).
Density peaks that become stars
are pieces of postshock gas.
Their size scales with the thickness
of the postshock gas, set by shock
jump conditions and velocity scaling.
MHD shocks: =l / M A l  , l =0 M
Velocity scaling:
M A l ~ul ~l
⇒ M max / M min =  L 0 /l 1 
M
⇒
A , 0=10
Copyright Paolo Padoan, 2008
3−2
2 / 2
= M
A
⇒ M ~l
−2 6 / 2
A, 0
M max / M min =104
⇒ M ~3 ~ l 3 0 / M
3−2
= M
~l 2
4
A ,0
2
A
2) Characteristic stellar mass:
Bonnor-Ebert mass: isothermal sphere confined by external pressure
(Ebert 1955; Bonnor 1956; McCrea 1957).
Thermal pressure:
M BE≈
 4th
G
3 /2
P th,01 /2
≈
3
 th
G 3/ 2 10/2
≈10 M sun

n
200 cm−3
  
−1 /2
T
10 K
3/ 2
≈
MJ
2.47
Dynamic pressure of turbulence (shocks --> nonlinear density jump):
M BE,t ≈
 4th
G
3/ 2
P dyn,0
1/2
≈
 3th
G 3/ 2 10/2
 Notice that M BE,t ~ n−1/ 2 T 2 −1
v 
Copyright Paolo Padoan, 2008
 
 th
v
=
M BE
M
s
≈0.4 M sun for M s =25
3) Rate of star formation:
E
Thermal energy
u 0 ≫C S
⇒
Ek
Eg
Ek,0 ≫ E th
Isothermal shocks create
a complex filamentary
density structure.
Eth
L
Gravitational energy
2
Ek
u
~
2 ,
E g  L
1/ 2
u~L
,
Ek,0
~1
Eg,0
⇒
 
E k  L L
=
E g  L
L0
−1
The turbulence can prevent the gravitational collapse.
Star formation occurs only where the density is enhanced and the
turbulence is dissipated, few % of the total mass.
Copyright Paolo Padoan, 2008
Supersonic turbulence is ubiquitous and energetically dominant
in star-forming regions.
How do we study its role in the process of star formation?
Two different numerical approaches......
Copyright Paolo Padoan, 2008
1. Brute-force approach: AMR simulations of star-formation
5 pc --> 0.5AU,
5123 --> (2x106)3
5 pc
0.02 pc
Copyright Paolo Padoan, 2008
300 AU
2. Idealized experiments of supersonic turbulence
Statistics of turbulence (universal) --> Statistical theory of star formation
Experiment setup:
Isothermal E.O.S.
Periodic B.C.
Uniform I.C. (rho, B)
Random I.C. (u)
Random acceleration (1 < k < 2)
No gravity
Up to 2,0483 (or larger with AMR)
The flow is relaxed for several tdyn before computing statistics
Euler Codes:
PPM (Colella and Woodward 1984)
PPML (Popov and Ustyugov 2007, 2008)
Stagger (Nordlund)
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10003 HD, Mach=10, Stagger Code
Copyright Paolo Padoan, 2008
10003 ideal MHD, Mach=10, Stagger Code
Copyright Paolo Padoan, 2008
Lognormal PDF of gas density
Nordlund and Padoan (1999):
  ≈ M S /2
 2ln  ≈ ln1 M S2 /4
Consistent with observations (Alyssa Goodman et al. 2008)
Copyright Paolo Padoan, 2008
Power-law velocity power spectrum:
Padoan et al. (2007):
Ek ∝ k
−1.9
Kolmogorov: k-5/3
Burgers: k-2
Is there an energy cascade in supersonic turbulence?
Supersonic turbulence as inertial motions ending into oblique shocks:
u ⊥ is dissipated by the shock
u∥ goes into postshock shear
EC
C
⇒
≈
≈ 1/2
ES
S
So there is a solenoidal cascade, but the dissipative flow geometry is
primarily
sheets (postshock regions), not filaments (vortices).
Copyright Paolo Padoan, 2008
Energy cascade in incompressible turbulence
 
u
Kolmogorov (1941):  u
=constant
l
2
3
⇒  u ∝l
⇒ u ∝l
What are the scaling exponents in supersonic turbulence?
Kritsuk et al. (2007): 1,0243 and 2,0483 PPM simulations:
Copyright Paolo Padoan, 2008
p
p/3
Energy cascade in supersonic turbulence
 
u
3


u
=constant
⇒


u
∝l
Lighthill (1955):
l
1 /3
p
1/ 3
p
p/3
v≡ u ⇒  v =  u ∝ l
2
Copyright Paolo Padoan, 2008
Copyright Paolo Padoan, 2008
Structure function exponents of
p
p
Copyright Paolo Padoan, 2008
1 /3
v ≡  u
Summary
Supersonic turbulence can explain masses and formation rate of stars.
A statistical theory of star formation can be derived from the statistics of
supersonic turbulence.
The pdf of gas density is a Lognormal and its standard deviation is a
function of the rms Mach number.
The energy spectrum is a power law, with slope ~1.9, and Es/Ec~2.
The “energy cascade” concept applies to supersonic turbulence, in the
sense that the average kinetic energy density rate does not depend on
scale.
The Log-Poisson intermittency model works well in supersonic
turbulence, and its parameters (scaling exponent and dimension of the
most dissipative structures) have the correct physical meaning.
Copyright Paolo Padoan, 2008