Lecture 25 Magnetic Fields, Stability,and Core Collapse

1. Magnetic Fields and Stability2. Singular Isothermal Sphere3. Observational Signatures of Infall

ReferencesStahler, Chs. 9 Shu ApJ 214 488 1977Evans, ARAA 37 311 1999

1. Magnetic Fields and StabilityObservations show that GMCs are large (10-100 pc), cold (~ 20 K), fairly dense (102 – 103 cm-3), and turbulent (_v ~ 1 km s-1) and that overall they are gravitationally bound. More detailed observations show that they contain small (~ 0.1 pc), cold (~ 10 K), dense (103-105 cm-3),and often turbulent regions called cores, that are also gravitationally bound.

But the cores may also contain newly formed stars and thusmanifest both stability and instability to gravitational collapse. In order to understand how cores can be both stable on onescale (their size) and unstable on another (presumably the scale that makes stars), we further examine the stability of molecular clouds and especially the role of magnetic fields.

The First Stability ResultsWithout magnetic fields, we found that the conditions for gravitational instability (collapse) can be given in terms of certain quantities like the Jeans length, time, and mass. They are determined solely by two parameters: the dispersion in the random velocity that determines the effective gas pressure and the average mass density of the cloud thatdetermines the effects of gravity:

2/1

2

2/3

sun3

JJ

2/1

2

2/1

J

2turb

2

)H(

1K1079.4

)H(

1K10pc189.0

-

-

˜̃¯

ˆÁÁË

ʘ¯

ˆÁË

Ê==

˜̃¯

ˆÁÁË

ʘ¯

ˆÁË

Ê==

+=

nTMM

nTGc

m

kTc

rl

rp

l

s

and, for no turbulence,

Stable and Unstable Virial Equilibrium

P

RRcr

A B (stable)(unstable)

For a given mass and temperature, there is a maximum pressurefor stability. Alternatively, given a value of the external pressure,there is a maximum stable mass. The solution of the Lane-Emdenequation by Bonnor (MNRAS 116 351 1956) & Ebert (Zs. f. Ap 37222 1955) gives the critical mass as

)(2.1 Jext

2/3

4

cr MOPG

cM =@

Virial Analysis for a Magnetized Cloud

MWEVPVP3

1

3

1

3

2Kext ++=Ò·-

Return to the virial equation for a static (equilibrium) cloud

Ignore rotation & turbulence (EK=0) to focus on the last(magnetic) term. In the spirit of previous approximations forthe pressure and gravitational terms, we assume that:1. the magnetization of the cloud is uniform2. outside the cloud, the field is dipolar, B(r) ~ (Z/r)-3

3. magnetic flux is conserved: _ = B !R2 ≈ const., i.e, the cloud is a good enough conductor for flux freezing

RR

GMMc

BRR

GMMcPR

2

2

22

232

23

3

1

5

33

3

1

5

334

F+-=

+-=

p

p

Virial Analysis for a Magnetized Cloud

4

2

34

2

3

2

12

1

20

3

4

3

RR

GM

R

McP

F+-=

ppp

F˜¯

ˆÁË

Ê≡F

2/1

9

51

GM

p

The magnetic and gravitational energies vary with R in thesame way. Since they stand in a constant ratio, once a cloud startscollapsing the frozen-in magnetic field cannot stop it.Magnetically dominated clouds are stable, i.e, clouds whose mass is less than the value:

More careful analysis, originally due to Mouschovias & Spitzer(ApJ 210 326 1974) leads to the slightly more accurate formula,

2

sun2/1

pc 1.0G200.113.0 ˜̃

¯

ˆÁÁË

ʘ̃¯

ˆÁÁË

Ê=Fª -

F

RBMGM

m

Magnetically Critical MassThis analysis leads to the terminology:

Magnetically Sub-Critical Mass (M < M_) – Magnetism stronger than gravity. The external pressure can maintain the cloud in equilibrium, but not collapse it

Magnetically Super-Critical Mass (M > M_) - Magnetism weaker than gravity; external pressure can collapse the cloud.

Measurements of magnetic field in cores (Crutcher et al.,(1999) indicate that they are on the margin betweensub-critical and super-critical: M ≈M_

( )2243

2

20

3

4

3F--= MM

R

G

R

McP

pp

Fate of Sub-Critical Cores

Sub-critical cores may collapse on long time scalesvia ambipolar diffusion (see Shu 2 Ch. 27).

Gravity on pulls on all of the matter, mainly neutral hydrogen & helium. The ions are tightly tied to the magnetic field and they lock the neutrals in place by microscopic ion-neutral interactions. But on long time scales, gravity will drag the neutrals through the ionsand feed the center of the core. This process is estimated to take several Myr.

The ambipolar diffusion time scale depends inversely on the electron fraction, so it occurs slowly on the outside of the core and more rapidly in the inside.

Estimating the Ambipolar Diffusion TimescaleAssuming ideal MHD, the force per unit volume is balanced by the drag force,

wnnKmm

mmBB ni

ni

ni rrr

+=¥¥— )(

4

1

p

where i and n stand for ions and neutrals, K is the rate coefficient for ion-neutral momentum-changing collisions,and w is the drift velocity of interest. One can easily check the dimensions of the right side, and that w is sensitive to: B, nH

2 and xe.

A crude estimate based on replacing the RHS by B2/4!L,B ~ 20 _G, K~ 10-9 cm3 s-1, H2 as the neutral, and typical core properties yields w << 1 km s-1.

Observations of Core Magnetic Fields

• Crutcher et al. ApJ 407 175 1993 – one 3-_ detectionout of 12 dark clouds: B1, B ≈ 19.1 +/- 3.9 _G• Crutcher ApJ 520 705 2000 – estimates of M/M_ for 27 molecular clouds (about half cores)• Crutcher & Troland (ApJ 537 L139 2000) 3-_ detectionsof two out of 8 dark clouds, including L1544, B ≈ 11_G, for which an infall signature has been observed.• Crutcher et al. ApJ 600 279 2004 non-Zeeman measure-ments of 3 pre-stellar cores: 100-200 _G

Summary: Magnetic Fields in Dense Cores

• Measurements show that magnetic fields play can play

an important role in shaping the structure and

dynamics of molecular clouds.

• Polarization maps reveal that the magnetic fields

threading molecular clouds are ordered over large

scales• The Zeeman measurements show that the mass-to-flux ratios are close to critical.

• Gravity amplifies the anisotropic structure associated

with magnetic fields, e.g., clouds collapse preferentially

along the direction of the average field.

MHD Core Formation Simulation in 3-d

Li et al. ApJ 605 800 2004• 5123 element periodic box• super-critcal field • strong supersonic turbulence• Mcloud ~ 64 MJ, L ~ 4 _J, • followed for 3 free-fall times• no microscopic physics

readily forms prolate and tri-axial super-critical cores & later centrifugally supported massive disks

column density map with disks

2. The Singular Isothermal Sphere

.411

gives1st theinto 2nd thengSubstituti . using

41

4

22

2

2

2

22

2

c

G

dr

dr

dr

d

r

cpdr

dc

dr

d

dr

d

dr

dpp

Gdr

dr

dr

d

rG

rprr

r

rffrfr

pf

pf

-=˜̃¯

ˆÁÁË

Ê

=

-=Æ-=Æ—-=—

=˜¯

ˆÁË

ÊÆ=—

We go beyond the virial analysis and try to understand how a core can appear to be both stable and collapsing bystudying the Lane-Emden equation for an isothermal sphere:

We seek equilibrium solutions satisfying:

Ú===¢ -R

rrdrrMcpR0

22ext )(4)()(0)0( rprr

The Bonnor-Ebert-McCrea Equation

01

x

1

4,

22

0

0

=+˜̃¯

ˆÁÁË

Ê

==

ydx

dy

yx

dx

d

c

Grxy

rp

rr

Use of dimensionless variables gives the BEM Equation:

schematic solution Shu ApJ 214 488 1977

The Singular Isothermal SphereShu (ApJ 214 488 1977) noticed the 1/ r2 envelope ofthe BEM solutions and showed that 1/ r2 is a particularsolution of the equation, now called the singularisothermal sphere (SIS). Substitute _ = A/r2 into

( )

rG

crM

rG

c

r

A

c

Gr

dr

d

rc

G

dr

dr

dr

d

r

2

2

2

22222

2

2)(

1

2

find to

04

21

get to0411

==

=+-=+˜̃¯

ˆÁÁË

Ê

pr

prprr

Shu rewrote the full equation as two 1st-order ODEswith no free parameters when scaled variables are used:

velocity v=u/c density _=4!G_t2

position/time x=r/ct

The Singular Isothermal Sphere

0

32/1

3

30

2/130

2

2

22

,24

1,

2: small

,1

22,)2(: large

mG

tcM

tr

am

Gr

tamux

rG

cAM

rG

cA

r

ataux

=˜̃¯

ˆÁÁË

Ê=˜̃

¯

ˆÁÁË

Ê-=

==--=

pr

pr

The solution that asymptotes to the SISis plotted on the left and has the limits:

m0 is related to A. A_ 2+ is a limiting case showing inside-outside collapse:

ctrrr

rGMu

x

ctrx

<µ=

>

- for and)(2

withcollapse fall-free: small

for (SIS)solution static: large

2/3r

For A_ 2+, m0=0.975.

Singular Isothermal Sphere

2/15

K10pc1.0yr1026.5

-

˜¯

ˆÁË

ʘ̃¯

ˆÁÁË

Ê¥=ª

Tr

c

rt

)1(2

)( :mass envelope static

)2()0( :mass infalling

)0(:mass accreted

3

0

3

0

3

-=-

-=-

=

+

+

ct

r

G

tcctMM(r)

mG

tcMM(ct)

mG

tcM

Collapse at finite r occurs after the signal that collapse started at the center reaches r, i.e., in time

It is of interest to account for the SIS mass distribution M(r):

The most remarkable result is that the accretion rate isdetermined only by the signal speed c and G:

G

cmM

3

0acc =&

Final Remarks on the SIS

• The time for the mass with r to collapse is ~ the Jeans time for the average density within that sphere.

• Rotation will cut off the r-2 envelope beyond

2/111317

C K10

s rad10cm1086.1 ˜

¯

ˆÁË

ʘ̃¯

ˆÁÁË

Ê

W¥@

W=

-- TcR

• The effects of magnetic fields have been studied in depthby Shu and collaborators, leading to singular isothermal toroids (SITs), e.g., Allen et al. ApJ 599 363 2003. Also see the work of Mouschovias and coworkers.

3. Observations of Collapse

It had been the hope of the early mm radio astronomers to discover molecular clouds in the process of collapse. They were rather surprised when, instead of collapse, outflow was discovered in the CO maps of the region around the YSO, L1551-IRS5 (next page), and later in hundreds of other sources.

Credible detections of infall were not made until the 1990s. The subject is reviewed by:• Evans NJ II, ARAA 37 311 1999• Myers, OSPS 1999• Myers, Evans, & Ohashi, PPIV 2000, p. 217• Evans 2002, astro-ph/0211526

Discovery of Bipolar Molecular OutflowsSnell, Loren & Plambeck, ApJ 239 L17 1980

CO contours on an optical photo of the L1551-IRS5 region with H-H objects.

Paradigm-setting cartoon for theoutflows from all low-mass YSOs.

Optical Jet

HH2

HH1

YSO and recent jet

Optical photo of HH1-HH2: Hz (green), S II (red) plus continuum (blue). Stapelfeldt et al,, ApJ 116 372 1998

Signature of Collapse

Evans ARAA 37 311 1999

red blue

Myers 1999 Schematic Infall Signature

blue and red shifted surfaces of constant l.o.s velocity assuming v ~ r- -1/2

Optical depth unity occurs for red and blue in regions with different excitation

Myers Two-Slab Model

Effects of varying optical depth and approach speed

B335: The Simplest Case

Choi et al. ApJ 448 742 1995H2CO & CS, with critical densities ~ 106 cm-3

dashed line is model

rounded Bok globuledistance ~ 250 pcIRAS (> 60 m): L ~ 3 Lsun

many other mm linesclumpy bipolar outflow

The Case of B335

Choi et al. (ApJ 442 742 1995) modeled the line shapes (shown on the previous page) using Shu’s SIS by varying the temperature distribution, the CS & H2CO abundances, and the beak point between the r-3/2 and r-2 density distributions. They concluded that the observations support the inside-out collapse model.

But Wilner et al. (ApJ 544 L69 2000) observed the high-frequency 245 GHz CS(5-4) transition with the PdB interferometer at much higher angular resolution (3”instead of 11”, corresponding to ~ 750 AU), and found aclumpy bipolar outflow & an inner off-center peak. The fitto the inside-out collapse model is degraded.

IRAM CS Observations of B335

Interferometer channel mapsshowing outflow using CS(5-4)

SIS modeling of line shape that includes fitting map data: dotted line

IRAM Dust Observations of B335

( )( ) ( ) AU25000for/ andfor/

AU 5000for 10KandAU 5000for AU /5000K10

02

00000

4.0

<<<=

><=--

-

rRRrRrRr

rrrTp rrr

Harvey et al. (ApJ 596 383 2003)mapped the 1.2 & 3.0 mm continuum(dust) emission at a spatial resolutionof 0.3” (10 times better than theprevious CS(5-4) maps). They detect adisk-like structure of 100 AU and modelthe dust with an inside-outside densityprofile:

They find p=1.55 +\- 0.04, R0=6500 AU, n0(H2)=3.3x104 cm-3,M(<6,500 AU) = 0.58 Msun & M(<25,000 AU) = 3.06 Msun,and argue for Shu’s 1977 model.

Starless Core L1544Tafalla et al. ApJ 504 9000 1998

HCO+

H2CO

CS

C3H2

N2H+

C34S

optically thick linesopacity increasing upwards

map of CS(2-1) profilesasymmetry extends beyond N2H+ core does not fit simple SIS models

Summary of Infall Signature Observations• There is a basic problem to start: small infall velocity requires high spatial resolution observations.• The line shape is affected by other dynamical motions, e.g., turbulence, rotation & outflow• The analysis has to use good thermal, chemical & excitation models and deal with many lines reflecting different properties; dust also plays a role.• The results depend on core shape and magnetic field. • The observations probably do show infall, but the overall result is ambiguous, e.g., low resolution surveys yield many red-peaked profiles.• ALMA is needed to go to the next level.