The collapse of dense cloud cores

•Ambipolar diffusion•Inside-out collapse•Observations of infall

Stahler & Palla, Chapters 9.3, 10.1, 10.2

Rotational Support of CoresIn Lecture 8 we considered the structure of a spherical, isothermal sphere, i.e., a Bonnor-Ebert sphere, balancing radial pressure gradient against self-gravity.

This analysis can be generalized to include rotational support (S&P, Ch. 9.2, not examined).

Observed core velocity gradients imply rotation rates that are too small to be very significant for support agains gravity.However, once collapse is underway, this initial rotation is likely important for setting the size of the disk around the protostar.

Magnetic Support of CoresSimilarly, the analysis for core support can be generalized to include magnetic support (S&P, Ch. 9.4-9.5, not examined).

Observed cores may well receive some contribution to support from magnetic fields.

In this case, there is a critical amount of magnetic flux threading the core that can keep it stable against collapse.

For cores to collapse and form stars may then require the core to lose magnetic flux... by ambipolar diffusion.

Ambipolar Diffusion

The magnetostatic equilibria resist self-gravity through both thermal pressure and magnetic forces. More precisely, electrons and ions, gyrating rapidly around B, collide with the neutrals, and the resultant drag on the latter helps to counteract gravity.

In the reference frame comoving at the local velocity of the neutral matter u, we want to express the drag force exerted on each charged particle by the sea of neutrals.

1. Electrons In any single collision with a neutral, the massive atom or molecule barely

moves. If the electron scatters in a random direction, then, on average, it transfers its full (original) momentum to the neutral, i.e., the electron gains the negative momentum -meu’e.

The number of collisions per unit volume is n<σenu’e>, where σen is the collision cross section and the angular braces denote an average over magnitude and direction in the thermal distribution of electron velocities.

€

f 'en = −nme <σ enu'e > u'edrag force oneach electron

The drag force

€

f 'in = −nmn <σ inu'i > u'i

2. Ions With their average mass of 28 mH, it is now the ions themselves that are the

heavier species in collisions with neutrals. In the ion rest frame, the momentum transfer must be the mass of a neutral, mn, times its incoming velocity. In the present frame comoving with the neutrals, each collisions thus imparts -mnu’I, on average, to an ion. The drag force is therefore:

If the cloud’s level of ionization is sufficiently low, the relative speeds between charged particles and neutrals become appreciable. The neutrals can then gradually drift across magnetic field lines in response to gravity.

This drift causes the loss of magnetic flux, so that the cloud contracts until it becomes unstable to true collapse (i.e., to the attainment of free-fall speeds). Equivalently, Mcrit decreases until it becomes lower than the cloud mass M .

€

Mcrit = MBE + MΦ

MΦ ≈ 0.12B0πR0

2

G1/ 2 = 0.12 Φcl

G1/ 2

Magnetic critical mass

Ion-neutral driftFirst note that the electrons and the ions are moving at very nearly the same velocity. This is the case if L / |ui-ue| >> tcl. Let’s see…

From Ampère Law:

€

∇ ×B =4πcj

and the expression for j in terms of the relative velocity:

€

j = nee(ui −ue )

!

€

L|ui −ue |

~ 4πneeL2

cB~ 1010 yr

For a dense core with L=0.1 pc, B=30µG, n(H2)=104 cm-3 and ne=5×10-8 n(H2) , the electrons and ions are effectively a single plasma drifting relative to the neutrals.

!

To determine vdrift ≡ ui - u we write the equations of motion for the electrons and ions, in the reference frame comoving with the neutrals. The quantity ui (and ue) includes both the tight, helical gyration about B and a much smoother drift component. The latter changes abruptly and stochastically due to collisions with neutrals, chiefly H2. Such collisions, which occur only after many helical orbits, exert the drag on ions and electrons (f’in and f’en).

An approximate equation of motion for the electrons, valid over time scales longer than that of individual collisions, and under steady state collisions:

€

0 = −ene E'+u'ec×B'

$

% &

'

( ) + nef 'en

Lorentz force

drag force

Analogously, for the ions, using ni = ne :

€

0 = +ene E'+u'ic×B'

#

$ %

&

' ( + nef 'in

Adding the two equations of motion together, and using the expression for j:

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0 = j×B /c + ne (f 'in +f 'en )

Using (1) the expressions for the drag forces, (2) u’e ~ u’I , (3) u’i ≡ ui - u = vdrift, (4) the Ampère’s law:

€

vdrift ≈(∇ ×B) ×B

4π nne[mn <σ inu'i >]

The existence of vdrift forces us to reevaluate the physical meaning of flux freezing.

In the reference frame of the neutrals (ui = u + vdrift), the ideal MHD equation becomes:

€

δBδt

=∇ × (ui ×B)

=∇ × (u×B) +∇ × (vdrift ×B)

!

Flux freezing still holds if the conductivity is large (no Ohmic dissipation), but implies that the electrons and ions are tied to B, while the neutral atoms and molecules in the cloud slip past.

How important quantitatively is this slippage?

The relevant time scale for slippage is L / |vdrift|, where L is a mean cloud diameter. Using the expression we just derived for vdrift and the relation x(e) ≈10-5 nH

-1/2

(McKee 1989, ApJ, 345, 782):

€

L| vdrift |

≈4π nnemn <σ inu'i >

| (∇ ×B) ×B |

≈ 3×106 yr n(H2)104 cm-3

'

( )

*

+ ,

3 / 2 B30 µG'

( )

*

+ ,

−2L

0.1 pc'

( )

*

+ ,

2

This time scale is comparable to the estimated life time of dense cores, thus the drift is significant. It is believed, in fact, that ambipolar diffusion is the main process setting the rate at which dense cores evolve prior to their collapse.

…but see, e.g., Ballesteros-Paredes et al. 2003, ApJ, 592, 188; Hartmann et al. 2001, ApJ, 562, 852; MacLow & Klessen 2004, RvMP, 76, 125 !!

Thus, magnetized structures are actually quasi-static. Both the gas and the magnetic field move, but so slowly that the clouds can be viewed as progressing along a sequence of equilibria, toward more centrally concentrated structures. The influence of gravity increases until the fluid velocity becomes substantial. Once |u| approaches the sound speed aT, the quasi-static description fails, and the cloud begins hydrodynamical collapse.

Ambipolar diffusion in a magnetized cloud. The drift velocity of ions relative to neutrals points away from the axis, in the same direction as the outward normal vector to the flux tube. Neutrals drift across the field in the opposite direction. MΦ decreases with time, or, equivalently, the amount of mass enclosed by a given flux tube increases with time.

One important feature of ambipolar diffusion is that it proceeds fastest toward the cloud center (the cloud’s increase in central density is an accelerating process):

Rise in central density in a contracting, magnetized cloud. The initial reference state is a uniform-cylinder, with uniform field strength of 30 µG, number density 300 cm-3, and radius and half-height both of 0.75 pc (> λJ).

When the stable equilibrium structure first forms, the density contrast is less than critical. Thus, the central density gradually increases as the result of ambipolar diffusion.

By 1.5×107 yr, the critical value for stability is reached and the condensation process picks up speed

and the contracting deep interior effectively separates from the more slowly evolving outer portion of the cloud, where ambipolar diffusion is much slower.

Inside-out collapse

Theoretical models of the formation of low-mass stars predict two important evolutionary stages:

(1) the formation of an unstable quasi-equilibrium cloud core through the slow leakage of the magnetic cloud support by ambipolar diffusion, and

(2) the formation of a central protostar and disk through a dynamical "inside-out" collapse (Shu 1977).

This inside-out collapse propagates outward as an expansion wave at the sound speed. Outside the radius of the expansion wave the cloud is static; inside this radius the flow quickly approaches free-fall velocities.

An ordered magnetic field can alter this spherical infall in the inside-out collapse.Galli & Shu (1993) have shown, that in presence of a reasonable field strength, the bulk of the infalling gas is deflected by the Lorentz force to a large (several hundred to a few thousand AU long) flattened "disk" structure oriented perpendicularly to themean magnetic field direction ("pseudodisk”).

Inside-out collapse of metastable sphere

r

ρ

r

ρ Suppose inner region is converted into a star:

r

ρNo support again gravity here, so the next mass shell falls toward star

ρ

r

The ‘no support’-signal travels outward with sound speed (“expansion wave”)

(warning: strongly exaggerated features)

Hydrodynamical equations

€

∂ρ∂t

+1r2∂(r2ρv)∂r

= 0

Continuity equation (spherical version):

€

∂v∂t

+ v ∂v∂r

= −1ρ∂P∂r

−GM(r)r2

Comoving frame momentum equation:

Equation of state:

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P = ρcs2

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M(r) ≡ 4π r'2 ρ(r')dr'0

r∫

€

cs2 ≡

kTµmH

= const.

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∂ρ∂t

= −1r2∂(r2ρu)∂r

€

∂u∂t

+ u∂u∂r

= −aT2

ρ∂ρ∂r

−GM(r)r2

€

P = ρaT2

€

aT2

Details not examined

Inside-out collapse model of Shu (1977)

Expansion wave moves outward at sound speed.So a dimensionless coordinate for self-similarity is:

€

x =raT t

€

ρ(r,t) =α(x)4πGt 2

€

M(r,t) =cs3tG

m(x)

€

v(r,t) = csu(x)

Now solve the equations for α(x), m(x) and v(x). €

M(r,t) =aT3tG

m(x)

€

u(r,t) = aTv(x)

Shu looked for similarity solutions of the form:

Details not examined

Inside-out collapse model of Shu (1977)

Singular isothermal sphere: r-2

Free-fall region: r-3/2

Transition region: matter starts to fall

Expansion wave front

Inside-out collapse model of Shu (1977)

Accretion rate is constant:

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˙ M = cs3m0

G= 0.975 cs

3

GStellar mass grows linearly in time

Deep down in free-fall region (r < aTt):

€

ρ(r,t) =cs3 / 2

17.96G1t1r3 / 2

€

v(r,t) =2GM*(t)

r

€

aT3 / 2

€

u(r,t)

€

aT3

Both the density and pressure profiles become flatter in the region of collapse

Rarefaction wave in inside-out collapse (schematic). An interior region of diminished pressure advances from radius r1 at time t1 to r2 at t2. Within this region, gas falls onto the central protostar of growing mass.

!

Observations of Infall

in Dense Cores

Line profile of collapsing cloud

Flux

λBlue, i.e. toward the observer

Red, i.e. away from observer

Optically thin emission is symmetric

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The static envelope outside rinf produces the central self-absorption dip, the blue peak comes from the back of the cloud, and the red peak from the front of the cloud. The faster collapse near the center produces line wings, but these are usually confused by outflow wings.

Evans 1999, ARA&A

The origin of various parts of the line profile for a cloud undergoing inside-out collapse

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Infall in starless cores

Lee, Myers & Tafalla 2001, ApJS, 136, 703