Phys 778 – 2009 2. Molecular Clouds

Ralph Pudritz

McMaster University

Extragalactic Molecular Clouds (Bolatto et al 2008) BIMA, OVRO, Plateau de Bue, SEST, used

Extragalactic size – linewidth

Relation:

Milky Way

Extragalactic

Dwarf galaxies

Extragalactic

Luminosity-

line width relation

Extragalactic

Luminosity – size relation

CO luminosity vs virial mass

Conclusions

Small differences between extragalactic and Milky Way GMCs down to 0.2 Zsolar

Larson type relations valid independent of environment

Departures: GMCs in dwarfs slightly larger than GMCs in galaxy

Largest departure: SMC (most metal poor) CO/H2 = xco is constant, over factor of 5 in

metallicity Photoionization regulaged star formation (McKee

1989) appears ruled out.

More Jeans masses:

Turbulence breaks up clouds into dense cores which form before big sheets are organized…

4.9

1.14

0.1

;5.27

A

J

M

n

Cluster formation in magnetized clouds (Tilley & Pudritz, MNRAS 2007)

Close-up:

spinning cores emit flux of Alfven waves extracting some angular momentum

Stage set for magnetized collapse… and formation of jets

Core, mass to flux distributions: local core Gamma is always *reduced* from initial uniform gas distribution! Ranges from supercritical to sub-critical

Arises from fragmentation of the gas..

(consider formula for Gamma)

Compare with distribution of B fields measured in cores (see Crutcher et al 2007)

Distribution of values of mass to flux values for cores is now supported by the data.

Take home: initial clumps strongly supercritical – some cores

become strongly magnetized - later, and even sometimes subcritical.

(see PN02 and earlier)

Berkeley press release, C. Heiles, ….

"You can think of this structure as a giant, magnetic Slinky wrapped around a long, finger-like interstellar cloud,'' said Timothy Robishaw, a graduate student in astronomy at the University of California, Berkeley. "The magnetic field lines are like stretched rubber bands; the tension squeezes the cloud into its filamentary shape.'' - Support for Fiege & Pudritz helical field model

II: Filamentary accretion and disk formation: FLASH – Adaptive Mesh Refinement (AMR) hydro simulation (Banerjee, Pudritz, & Anderson 2006):

- Grid adjusts dynamically to resolve local Jeans length (Truelove et al 1997); we use 12 pixels

- We added a wide variety of coolants including molecular + dust cooling, H2 formation and dissociation, heating by cosmic rays, radiative diffusion, etc.

Filamentary structure: from 0.1 pc down to

sub AU scale

- Large scale filamentary collapse onto a growing disk:

x-z plane.

y-z plane along filament: same as for x-y plane - are seeing a true filamentary collapse

Cut through disk midplane (x-y)

accretion from an off-centre sheet of material disk provides angular momentum of the disk.

highest resolution shows spiral wave structure

sheet formation – notice velocity shear..

III Turbulence: spectral energy distributions of diffuse ISM and molecular clouds

“Big power law in the sky”: radio observations - Kolomogorov- like spectrum over 11 decades:

Astrophysical power law spectra observed in range

GMCs: Size – linewidth relation (Heyer & Brunt 2004)

Armstrong et al 1995

cm 1010 scalesfor

3/5;)(

187

kkE

]6.2,5.1[

0.3 :vKolomogoro ;6.05.0

2/)1(;v

R

Is a Kolmogorov turbulent cascade picture relevant for ISM?

Many different physical processes at work over these scales Many different energy sources for “turbulence”: - galactic spiral shocks, supernovae, cosmic ray streaming,

expanding HII regions, K-H and R-T instabilities, gravitational and thermal instabilities ….

(eg. review Elmegreen & Scalo 2004) Variety of solutions of incompressible MHD turbulence weak

turbulence:

or constant flux, indices -1 to -3 (Galtier et al 2002) Damping rate of MHD turbulence very fast – eddy turnover

timeShocks: Supersonic shocks produce -2 (eg. Kritsuk et al 2007,

Vasquez-Semadeni et al 1997..) – but spectra often shallower.

2/32 or

kk

Mass spectra of cores – turbulent box simulations Gas cores Self gravitating

cores collapsed cores

Bottom 3 models driven until gravity turned on, driving scale indicated

Lognormal fits to collapsing objects

Klessen, 2001

Density structure formation Assume density changes primarily due to shock compression – after n

shock passages:

Consider shock strengths to be identically distributed random variables, in interval

Take log of both sides, apply central limit theorem. Get a log-normal distribution for density PDF:

o)(

0

)( tonormalized ;))(1( xjn

j

n

1

1ln

12

,1

1ln

2log

2

)log)(log(exp

2

1)(

2

2

2

n

n

P

)]1/(2,0[

Rapid generation of lognormal density PDFs

Convergence rapid – 3 or 4 shock passages suffices. Seen in most simulations

Mean and width grow with number of shock passages (mean RMS Mach number increasing?)

Broadest distributions for nearly isothermal gas.

In self gravitating medium, collapse sets in for dense enough fluctuations

Kevlahan & Pudritz 2009

Spherical blast wave into log-normal medium.. Most clusters impacted by spherical shock waves from

SN, massive stellar winds, HII regions. Density PDF = convolution = lognormal * PDF of spherical blast For Sedov-Taylor with sustained energy injection;

p=0 classical point SN explosion (instant shock) p=1 steady wind (injection shock) – Dokuchaev 2002 Density PDF of blast wave:

pttE )(

-4.5-9/2index 1,p

-2.8 17/6- index 0,pFor

)()()(

0

)26/()17(3

t ppdttRd

dP

Generation of power-law PDF at high densities

Initial lognormal distribution ----

Instant and injection shocks (-17/6 and -9/2)

Point: power law tail may be the result of “feedback” from massive star by blast wave

Shock generated vorticity (Kevlahan & Pudritz 2009) Curved shock waves generate vorticity on all scales – this alone

may explain structure and spectra – leading towards IMF. Vorticity jump normal to shock (Kevlahan 1997):

Term I. = 0 for spherical shocks. Shock focusing: Ms larger in concave curvature than conve

-> shock strength grows and focus at regions of minimum curvature

flow upstream ofnumber Mach turbulent

shock alongnumber Mach shock

shock acrossdirection binormal

entshock tang

normalshock

strengthshock )1)(2/1(1

1/

momentum) ang. ofon conservati (III. effects) baroclinic (II. shock) along Ms of variation(I.

2/1

1

1

1

2

2

22

2

t

s

s

s

ts

s

s

M

M

nsb

s

n

M

M

bsuS

MM

MS

Mb

Vortex sheets…

Term I produces kinks – strong jet-like vortex sheets develop *downstream* of kink .

Have index of -2 and produce K-H turbulence exponentially fast

IMPORTANT; -2 index associated with downstream flow – not associated with shock itself.

Focused shock: Solid line is shock strength, ---- shock profile. Discontinuities in strength produce vortex sheets downstream

Spectral energy distribution – kinematic approach Semi analytic approach: vorticity jump equation

used to calculate vorticity generation after each shock – velocity etc. calculated using FFT on computational domain with periodic B.C.

Computational grid for spectral code: 256^3 Ignore internal dynamic of flow – energy

redistribution due to shock Shock’s shape and strength distribution fixed Random shocks: direction and phase random

for each passage. (see example Elliott & Majda, 1995)

Multiple passages of focused shocks:energy spectra

Initial Mach No: M0=6 Top: initial uniform flow Bottom: initial

Gaussian energy spectrum

Results independent of initial condition

Energy spectrum progressively shallower with each shock passage

Re-distribution of energy to smaller scales consequence of baroclinic terms

Multiple passages of spherical shocks:energy spectra

Due to symmetry, Ms is constant

Assume downstream is irrotational with Gaussian energy spectrum

(upper) instant p=0(lower) injection p=1

A few shock passages create shallow Kolomogorov -like slopes