5 March 2013 Astronomy 142, Spring 2013 1

Today in Astronomy 142: star formation

Gravitational collapse, free-fall time scale The process of stellar formation

Pre-main-sequence stars and the Hayashi track

Midterm Exam #1 on Thursday

Note that there are office hours between now and then: Ask for suggestions on

your cheat sheet. Ask questions about the

online Practice Exam.

5 March 2013 Astronomy 142, Spring 2013 2

5 March 2013 Astronomy 142, Spring 2013 3

Time scale of spherical gravitational collapse

For spherically-symmetric collapse, in free fall, we’ll see that This is very fast by astrophysical standards. We should

therefore have to get very lucky to catch a star in the act of formation.

The result applies to a spherically-symmetric collapse, and we know the collapse can’t really be very spherical – as angular momentum is conserved – so this is offered only as a crude estimate.

00

12 4 19 -30

3 initial mass density32

10 sec 3 10 years for 4.7 10 gm cm .

fft nGπ ρ µρ

ρ −

= = =

= = × = ×

5 March 2013 Astronomy 142, Spring 2013 4

Spherical free-fall time: derivation

Follow a test particle that starts from rest, a distance r0 from cloud center, at t = 0. How long until it reaches the center (i.e. until the clump collapses to a point)? Multiply both sides by dr/dt, integrate over time (RHS first):

ma F

m d rdt

GMmr

=

= −2

2 2

1D, nonlinear, 2nd-order DE. M: mass interior to r. (Constant!)

0

2

2 2 20 0 0

1t t r

rdr d r dr dr GM GMdt GM dt GMdt dt r rdt r r

′′ ′= − = − = −

′ ′′ ′∫ ∫ ∫

5 March 2013 Astronomy 142, Spring 2013 5

Spherical free-fall time: derivation (continued)

On LHS, substitute v=dr/dt, dv=(d2r/dt2)dt:

Substitute back v = dr/dt:

22

20 0 0

12

t vdr d r GM GMdt v dv vdt r rdt′ ′ ′= = = −

′ ′∫ ∫

12

2 2

2

0

0

drdt

GMr

GMr

drdt

GMr

GMr

FHG

IKJ = −

= − −Choose negative root, so that the particle moves to smaller r.

5 March 2013 Astronomy 142, Spring 2013 6

Spherical free-fall time: derivation (continued)

Insert Substitute:

30 0

4 :3

M rπ ρ=

drdt

rG r

r= − −F

HGIKJ0

0 083

1π ρ

20

20 0

cos

cos 2 cos sin

r r u

dr d dur u r u udt dt dt

=

= = −

5 March 2013 Astronomy 142, Spring 2013 7

Spherical free-fall time: derivation (continued)

Then, Separate and integrate over time, RHS first:

cos2 00

00

12

83 2

83

′ ′ = =z zu duG

dtt G

u

u t ffffπ ρ π ρ

( )

02

20 02 2

81 1 12 cos sin 3 cos

8 81 11 cos3 32 cos sin 2 cos

Gdudt u u u

G Gu

u u u

π ρ

π ρ π ρ

= −

= − =

Spherical free-fall time: derivation (continued)

Now, what are u and u0? Thus

5 March 2013 Astronomy 142, Spring 2013 8

t Gudu u u

tG

ff

ff

28

3 22

4 4

332

0 20

2

0

2

0

π ρ π

πρ

π π= = +L

NMOQP =

=

z cos sin/ /

20 0 0 0 0

20

0 : , cos 0

: 0, 0 cos2ff

t r r r r u u

t t r r u u π

= = = ⇒ =

= = = ⇒ =

5 March 2013 Astronomy 142, Spring 2013 9

Star formation synopsis

Molecular clump collapses gravitationally …to a disk shape at first. Collapse along the other two dimensions happens more

slowly because of rotation (angular momentum) and, possibly and to a lesser extent, magnetic forces.

Smaller scales collapse faster (“inside-out collapse”). Central “core” gradually accretes some of the rest of the disk. To get rid of the last of the accreted material’s angular

momentum , this is accompanied by a bipolar outflow. Core becomes star; remnants of surrounding disk become planetary system. In more detail…

5 March 2013 Astronomy 142, Spring 2013 10

Star formation

1. A fragment of an interstellar molecular cloud becomes gravitationally unstable and begins to collapse… …either because the material has cooled,

or because it has been compressed (triggered) by pressure from outside.

2. Collapse proceeds from the inside out, and anisotropically. The central – denser – region collapses

faster:

The northern part of the Orion A molecular cloud in dust emission at λ = 350 µm (Lis et al. 1998).

0

332fft

Gπρ

=

5 March 2013 Astronomy 142, Spring 2013 11

Star formation (continued)

Because of conservation of angular momentum, and to a lesser degree from magnetic forces, collapse can always proceed much faster in one dimension than the other two: the fragment flattens as it collapses.

3. Soon a disk configuration is established, with a well-defined, very dense central condensation: a protostar. Contours of 13CO emission and scattered near-infrared light in IRAS 04016+2610 (Padgett et al. 1999).

1000

AU

Star formation (continued)

Henceforth the disk accretes further material from the surrounding molecular cloud, and the protostar accretes material from the disk. • Due to disk “viscosity,” mass

moves inward, angular momentum outward.

Class 0 protostar: central object still has much of its final mass to accrete; surrounding envelope still substantial;

Class I protostar: central object nearly complete, envelope settling

onto disk;

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4 1~ 10 year .dM dt M− −

( )5 7 1~ 10 year .dM dt M− − −

Wat

son

et a

l. 20

07

5 March 2013 Astronomy 142, Spring 2013 13

Star formation (continued)

4. As the disk and central object accretes gas and dust, they drive a bipolar outflow into their surroundings, which is thought to carry off the last of the accreted material’s angular momentum. In low-luminosity (low-mass)

objects the outflow takes the form of highly collimated jets. In more massive objects the outflow has a wider opening angle.

HH30 (Alan Watson et al. 2000)

5 March 2013 Astronomy 142, Spring 2013 14

B

Disk

Rotation

Ions Shock

Simplified view of magneto-centrifugal acceleration. The lines of B are stuck to the (ionized) disk and the distant ambient medium. As the disk rotates, the nearer lines of B wind up as shown, and ions in the gas above and below the disk – which are stuck to B – are driven away from the disk, along lines of B, like beads sliding on a wire. The ions can collide with neutral particles and drive them, too. The energy and angular momentum of the outflow comes from disk rotation, so the angular momentum in material at the field’s “footpoints” decreases.

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Bipolar jets and disks associated with young stars

HST

/STS

cI/N

ASA

Pat Hartigan, Rice U.

1000 AU

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Star formation (continued)

The outflows are supersonic, compared to the molecular surroundings, and thus shocks are driven into the ambient medium. The energy and momentum injected thereby contributes to the support of the molecular cloud, and can induce or hinder further star formation in the neighborhood.

L1641 VLA1/HH1-2 (Bally et al. 2002)

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Class 0/I protostar = visible wavelengths

Class II protostar: opaque disk, envelope almost gone, accretion onto star from disk at rate that decreases with age, dropping below by age 5 Myr.

Class III protostar: little dust and very little gas left in disk.

Wilking 1989

Star formation (continued) 5. The central object becomes a star, and its wind and radiation eventually stop the accretion and dissipate the gas in the disk.

10 110 yearM− −

0

0.5

1

1.5

2

2.5

3

350 450 550 650

Flux

den

sity

(arb

itrar

y un

its)

Wavelength (nm)

T TauriB. Weaver (MIRA)

G5V starPickles 1998

HαHβ

Hγ

HαHβ

Hγ

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Pre-main-sequence stars

The young stars themselves are relatively unextinguished at visible wavelengths, unless the disk is viewed edge on, and have two distinctive features: their spectra show emission

lines, notably hydrogen recom- bination lines, and UV excess emission, both produced in accretion shocks in material falling onto the star from the disk.

Spectral type G-M = T Tauri stars. Earlier types: Herbig Ae/Be stars.

Pre-main-sequence stars (continued)

It takes lower-mass young stars millions of years to settle down to their final sizes and get the fusion fires blazing. During this slow gravitational collapse, the conversion of

gravitational potential energy dominates the luminosity of young stars. For it is written (Homework #2):

The gravitational-collapse luminosity decreases with time,

and star’s effective temperature changes little while this is going on. So the young star descends almost vertically at first, through the H-R diagram. This path is called the Hayashi track.

5 March 2013 Astronomy 142, Spring 2013 19

2

235

GM dRLdtR

= −

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Pre-main-sequence stars (continued)

The Hayashi track happens to be the reverse of the path each star would follow at the end of its life, as it becomes a red giant.

For a given age the redder (lower mass, later spectral type) stars lie further above the main sequence than the bluer ones.

Kenyon & Hartmann 1995