Lecture 7: The Collapse of Cores and...

46
Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging of Orion Protostars

Transcript of Lecture 7: The Collapse of Cores and...

Page 1: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Lecture 7: The Collapse of Cores and Infall

Herschel and Spitzer imaging of Orion Protostars

Page 2: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Review: Dense Cores in Hydrostatic Equilibrium

Page 3: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Numerical solutions: Singular isothermal sphere(limiting solution)

The Singular Isothermal SphereAs a boundary condition, we must adopt a value for ρ(0).If ρ(0) -> infinity, we get a singular isothermal sphere

Page 4: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Numerical solutions:Different starting ρo :a family of solutions

Dullemond

r-2

flat

The Bonnor-Ebert Sphere

Page 5: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Alves et al 2001, Ward Thompson et al.. 1999, Bacman et al. 2000

Page 6: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

B68: A real Bonnor-Ebert Sphere in Nature??

Page 7: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Collapse

7

Page 8: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

What Initiates Collapse?

• Formation of core (cores are never stable)• External pressure increase (shock or pressure

wave in turbulent medium)• Mass accretes onto source• Ambipolar Diffusion

8

Possibilities:

Page 9: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Magnetic Support of Cores(“Historical Diversion”)

9

IRAS 4a/4b protostar

Magnetic field lines determined through sub-millimeter polarimetry

Girart et al. 2007

Page 10: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Ambipolar Diffusion

10

Page 11: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Charges in B-Fields

11This animations illustrates charged particles frozen to the Earth‘s field

Page 12: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Collapse Initiated by Ambipolar Diffusion

• Ions & electrons are stuck to magnetic field• Neutrals are not stuck to magnetic fields• Magnetic field is strong enough to resist collapse.• Neutrals are pulled in by gravity. • Ions & electrons remain stuck to field • Relative to the contracting neutral gas, the magnetic

field, ions and electrons diffuse outward (hence ambipolar), reducing magnetic flux to mass ratio.

• Eventually cloud becomes supercritical and collapses12

Page 13: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Timescale for Ambipolar Diffusion

13Taken from Hartmann

Page 14: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

14

Page 15: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

15

Page 16: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Current Consensus: Clouds and Cores are already Supercritical.

16

Diffuse ISMSub-Critical

Molecular Clouds: Critical

Critical

Page 17: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

17

Collapse Calculations

Page 18: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

18Taken from Hartmann

Page 19: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

19

Page 20: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Spherically symmetric free falling cloud

vff =2GM*

rIf stellar mass dominates:

vff =2GM(r)

rFree fall velocity:

Continuity equation:

∂ρ∂t

+1r2∂(r2ρv)∂r

= 0

∂(r2ρvff )∂r

= 0

Stationaryfree-fallcollapse

ρ(r)∝ r−3 / 2

Slide pirated from K. Dullemond

Page 21: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

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)Slide pirated from K. Dullemond

Page 22: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Hydrodynamical equations

∂ρ∂t

+1r2∂(r2ρv)∂r

= 0

Continuity equation:

∂v∂t

+ v ∂v∂r

= −1ρ∂P∂r

−GM(r)r2

Comoving frame momentum equation:

Equation of state:

P = ρcs2

M(r) ≡ 4π r'2 ρ(r')dr'0

r∫

cs2 ≡

kTµmH

= const.

Slide pirated from K. Dullemond

Page 23: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inside-out collapse model of Shu (1977)• The analytic model:

–Starts from singular isothermal sphere–Models collapse from inside-out–Applies the `trick’ of self-similarity

• Major drawback:–Singular isothermal sphere is unstable and therefore

unphysical as an initial condition

• Nevertheless very popular because:–Only existing analytic model for collapse–Demonstrates much of the physics

Slide pirated from K. Dullemond

Page 24: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inside-out collapse model of Shu (1977)

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

x =rcs t

If there exists a self-similar solution, then it must be of the form:

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 u(x)

Slide pirated from K. Dullemond

Page 25: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inside-out collapse model of Shu (1977)Solution requires one numerical integral. Shu gives a table.

An approximate (but very accurate) ‘solution’ is:

g ≡ 11.43x 3 / 2

h ≡ 2x

α(x) = g(x)7 / 2 + h(x)7 / 2( )2 / 7

u(x) = h(x)5 / 9 − 25/ 9( )9 /10

m(x) =1.025 x 2 + 0.975+ 0.075 x (1− x)

For any t this can then be converted into the real solution

Slide pirated from K. Dullemond

Page 26: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inside-out collapse model of Shu (1977)

Slide pirated from K. Dullemond

Page 27: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inside-out collapse model of Shu (1977)

Slide pirated from K. Dullemond

Page 28: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inside-out collapse model of Shu (1977)

Slide pirated from K. Dullemond

Page 29: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

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

Slide pirated from K. Dullemond

Page 30: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inside-out collapse model of Shu (1977)

Accretion rate is constant:

˙ M = cs3m0

G= 0.975 cs

3

GStellar mass grows linear in time

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

ρ(r,t) =cs3 / 2

17.96G1t1r3 / 2

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

r

Slide pirated from K. Dullemond

Page 31: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Foster & Chevalier 1993

The Collapse of a Bonnor-Ebert Sphere

τ = free fall time for ρcν = v/cs2

Page 32: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

τ = free fall time at central density

Page 33: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Self Gravitating Sheets

Page 34: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Hartmann, Boss, Calvet & Whitney 1994 ApJ 430, L49

The Collapse of an Isotherm

al Sheet

Page 35: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

35Hartmann, Boss, Calvet & Whitney 1994 ApJ 430, L49

The Collapse of an Isotherm

al Sheet

Page 36: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Evidence for Collapse and Infall

36

Page 37: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Line profile of collapsing cloud

Flux

λBlue, i.e. toward the observer

Red, i.e. away from observer

Optically thin emission is symmetric

Slide pirated from K. Dullemond

Page 38: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Line profile of collapsing cloud

Flux

λBlue, i.e. toward the observer

Red, i.e. away from observer

But absorption only on observer’s side (i.e. on redshifted side)

v (km/s)

T (K)Example:Observations of B335 cloud.Zhou et al. (1993)

Slide pirated from K. Dullemond

Page 39: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Infall in starless cores:

Lee, Myers & Plume 2004 ApJ 153, 523

Clear infall signature for 18 of 94 starless

Page 40: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Infall in protostellar cores: Lee, Myers & Plume 2004 ApJ 153, 523

Clear infall signature in 15 of 47 sources

Page 41: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Further evidence for infall:

Optically thick lines are

blueshifted for the youngest

protostars.

δV = (Vthick - Vthin)/Vthin

Page 42: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Infall Speeds

Infall Speeds 0.04 - 0.07 kms-1

Times of a few million years

Longer than free fall time

Shorter than ambipolar diffusion time

Page 43: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

P-Cygni profiles for

winds around evolved stars

43

http://www.daviddarling.info/images/P_Cygni_profile.gif

Page 44: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inverse P-Cygni ProfilesCold infalling gas generates absorption line.

Continuum emissiom from hotter inner core.

Page 45: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Inverse P-Cygni Profiles

toward the Protostar IRAS 4A

Infall velocities of ~ 0.5 km s-1

DiFrancesco et al. ApJ 2001 562, 770

Source Center

2”

Page 46: Lecture 7: The Collapse of Cores and Infallastro1.physics.utoledo.edu/~megeath/ph6820/lecture7_ph6820.pdf · Lecture 7: The Collapse of Cores and Infall Herschel and Spitzer imaging

Summary• Mechanisms for initiating collapse in hydrostatically

supported cores–Ambipolar Diffusion–Pressure wave compresses a critical Bonnor-Ebert sphere

(increases ρc/ρo to unstable regime)–Mass accretion onto a Bonnor-Ebert sphere

• Solution for infall–Infall in a isothermal sphere–Solutions for collapse of sheet and Bonnor-Ebert sphere

• Evidence for infall–Redshifted self-absorption–Inverse P-Cygni

46