Star Formation

28 April 2003

Astronomy G9001 - Spring 2003

Prof. Mordecai-Mark Mac Low

Gravitational Stability• Criterion for gravitational stability found

by Jeans (1902).

• Pressure opposes collapse: sound waves must cross region to communicate pressure changes before collapse

•

λJ

density ρ

2

2 2

1 2 1 22

without pressure, collapse occurs in a free-fall time

4

3

a harmonic oscillator with frequency = 4 3

1 2 3 3

4 4 4 16ff

d r GM Gr

dt r

G

tG G

Jeans instability

20 1 0 1 1; set 0, assume 4

• Jeans swindle: in homogeneous medium– – not generally true, but usually justified

• Linearize equations of motion for this medium

0 1 1

10 1

21 11

0

1

21 1

( , ); ( , );

0 mass conservation

momentum conservation

4 Poisson's equati

)

on

( ,

s

x t x t

t

ct

x t

v v

v

v

21 1

02

2 2 21 11

0

22 21

1 0 12

Taking a time derivative of the first equation

and the divergence of the second equation

0

combining and substituting

4 0

Take a trial s

s

s

t t

ct

c Gt

v

v

1

2 2 20

olution , to find

4 0

i t

s

Ce

c k G

k x

2 2 20

0

2 2 2

2 20

4

for small or large this reduces to the

dispersion relation for sound waves

As increases, decreases, until < 0

producing exponential increase in density.

This occurs w

s

s

c k G

c k

k

2 2 02

hen

4 Jeans wavenumberJ

s

Gk k

c

• What determines the rate of star formation in galaxies?

• Free-fall time

• Galaxy lifetimes greater than 109 yr.• Yet star formation continues today.

• How are starbursts, low surface brightness galaxies different?

Timescales

1/ 2

63 3

10 yr10 cmff

nt

Cosmic Star Formation Rate

• Star formation rate higher at high z

• Actual value depends on corrections for dust obscuration, surface-brightness dimming

• Different methods still give drastically different results.

Madau et al. (1996, 1998)

Lanzetta et al. (2002)

Schmidt Law• Empirically Kennicutt

(1989, 1998) finds

• This can be seen as a free-fall collapse:

• but what is

1.4 0.05

SFR gas

1 21.5

SFR SFRff

SFR

t

G

?SFR

Kennicutt 1998

Initial Mass Function

• Salpeter (1955): α = 2.35• Describes high mass

stars well.• Low mass stars

described by log-normal (Miller & Scalo)

or multiple power laws (Kroupa 2002)

• Why is form universal?• What determines peak?

Kroupa 2002

GalaxyONCPleiadesM35

dNm

dm

How can stars form?

• Gravity is counteracted by– thermal pressure– angular momentum– magnetic pressure and tension

• Each must be overcome for collapse to occur

Isothermal Sphere Solutions

This is the Lane-Emden equation, which forms the basis

for stellar structure. When

22s

0

2 202

In hydrostatic equilibrium

In spherical symmetry

cexp

We can now write Poisson's equation as

14 4 exp

s

s

P

d dr c

dr dr

d dr G G c

r dr dr

2

0 2, then

2sc

Gr

Bonnor-Ebert Spheres• Ebert (1955) and Bonnor (1956) found solutions to

the modified Lane-Emden equation for finite external pressure Pext

2 2 ,

where 4 0 , ln0

and finite 0 BCs 0 0, 0 0

s

d de

d d

rrG

c

d

d

2

22

max

solve by change of variables

to get two first-order equations with ICs:

; exp ; 0 0, 0 0.

This has a unique solution out to radius

4 0

each solution is defined by , , as

s

dx

d

d x dxx

d d

R

RG

c

R c

max

nd 0 .

0If 6.5 14.3, then unstable equilibrium

R

Isothermal Collapse• Larson (1969) and Penston (1969) found similarity

solutions for the collapse of a uniform density, isothermal sphere

• Shu (1977) gave the collapse solution for an initially hydrostatic isothermal sphere (but this has a density singularity at the center).

• Whitworth & Summers (1985) showed that many isothermal collapse solutions exist; all can be described with two parameters measuring the initial and final density concentration

• Isothermal sphere with central singularity is only the most extreme case

MHD Support• Magnetic fields can prevent collapse if

magnetic energy exceeds potential energy• Remember, virial theorem analysis yields

• So, flux must be lost at some stage to allow stars to form, or gas must be accumulated along field lines over large distances.

1 2

1

0.13

230 pc3 G 1 M

CR

CR

M

G

B MR

Ambipolar Diffusion

• Neutral-ion drift (note different def’n in plasma physics: electron-ion drift)

• Collisional drag force Fni = -Fin = γρiρn(vi - vn)

– drag coefficient γ constant for vd < 10 km s-1

• For low ionization fraction, drag balances Lorentz

• The induction equation is a non-linear diffusion equation

1

4d i nn i

v v v B B

2 2

2 2

4

with diffusion coefficient 4

the ambipolar diffusion timescale is then

~

nn i

n i A ni

AD A ni

t

D B v

t L D

B BB v B B

• In molecular clouds, tAD ~ 10 tff so suggested as solution to both flux and timescale problems.– However, stars form within ~ 1 Myr despite varying

local ionization states– Magnetic field measurements suggest fields already

weak when cores form– is flux problem solved at larger scales? How?

Importance of Ambipolar Diffusion• May be most important dissipation

mechanism in turbulence• Mediates shock waves, reducing heating,

but causing instability• Determines binary formation by modulating

magnetic braking in protostellar cores • Controls viscosity in accretion disks by

suppressing magnetorotational instability• Forms current sheets in turbulent flows,

perhaps melting meteoritic chondrules in protoplanetary disks?

Turbulent Fragmentation

• Gravitational fragmentation in a turbulent flow may explain some features of star formation– collapse time depends on strength of turbulence– slow, isolated collapse occurs in regions globally

supported against collapse by turbulence– fast, clustered collapse occurs in unsupported

regions– IMF appears log normal near Jeans mass

• Turbulent state of molecular clouds suggests this mechanism indeed operates– most observed cores magnetically unsupported

Angular Momentum

• Consider ISM with M = 1 M, n = 1 cm-3

• Angular momentum not conserved:– Diffuse gas > molecular clouds– molecular clouds > cloud cores– cloud cores > protostars– protostars > main sequence stars– J ~ 1048 g cm-2 s-1

• Where does it go? (Binary formation insufficient)

1 3 3 15 1 56 2 13 pc and 10 s , so 10 g cm sR M J

2;J I I MR

Magnetic Braking

• Magnetic fields can redistribute angular momentum away from a collapsing region

• Outgoing helical Alfvèn waves must couple with mass equal to mass in collapsing region (Mouschovias & Paleologou 1979, 1980)

0 2bA

R

v

Binary Formation• In absence of magnetic fields,

binary formation occurs from the collapse of rotating regions

• Ratio of gravitational to rotational energy determines fragmentation

• However, magnetic braking can effectively drain rotational energy, preventing binary formation

•

Burkert & Bodenheimer 93

does ambipolar diffusion allow decoupling of core from field to explain high binary rate?

Piecewise Parabolic Method

• Third-order advection

• Godunov method for flux estimation

• Contact discontinuity steepeners

• Small amount of linear artificial viscosity• Described by Colella & Woodward 1984, JCP,

compared to other methods by Woodward & Colella 1984, JCP.

Parabolic Advection• Consider the linear advection equation

• Zone average values must satisfy

• A piecewise continuous function with a parabolic profile in each zone that does so is

0)0,( ;0 aaa

ut

a

21

21

1 j

j

daaj

nj

jRjLnjjLjRj

j

jjjjL

aaaaaaa

xxaaxaa

,,6,,

21,6,

2

16 ;

;1

Interpolation to zone edges• To find the left and right values aL and aR,

compute a polynomial using nearby zone averages. For constant zone widths Δξj

• In some cases this is not monotonic, so add:

• And similarly for aR,j to force montonicity.

nj

nj

nj

njjLjRj aaaaaaa 1211,,21 12

1

12

7

6

_2

1 if 23

0 if 2

,,,,,,,

,,,,

jLjRjRjL

njjLR,jjR

njjL

jLnj

njjR

njjRjL

aaaaaaaaaa

aaaaaaa

Conservative Form

• Euler’s equations in conservation form on a 1D Cartesian grid

ug

gG

pH

upuE

uv

u

u

F

E

v

uU

Gx

H

x

F

t

U

0

0

,

0

0

0

, ,2

conservedvariables

fluxes pressuregravity orother body forces

Godunov method• Solve a Riemann shock tube problem at every

zone boundary to determine fluxes

Characteristic averaging• To find left and right states for Riemann

problem, average over regions covered by characteristic: max(cs,u) Δt

tn

tn+1

xj+1xj-1xj

or

tn

tn+1

xj+1xj-1xj

subsonicflow

supersonicflow

(from left)

Characteristic speeds• Characteristic speeds are not constant

across rarefaction or shock because of change in pressure

Riemann problem• A typical analytic solution for pressure (P.

Ricker) is given by the root of

III

I

II

Is

II

II

II

LRRRLL

PBA

PPP

Pc

PPBP

APP

UPf

RLI

uuUPfUPf

1

1

1

2

on)(rarefacti if11

2

(shock) if

,

),( with where,

,0,,

2

1

,