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The Zeldovich Box

Committees

Teaching

Research

AGE

Learning

Percent Time Spent

Fortunately, Rashid is still a young man!

The Problem of Star Formation

Stars form at a rate of about 1 Msun/yr in the Galaxy

The Problem of Star Formation

Stars form at a rate of about 1 Msun/yr in the Galaxy

Stars have masses in the range:

< 0.075 Msun: Brown dwarfs

> 100 Msun: Stars > 8 Msun explode as supernovae or collapse into black holes

How can interstellar gas with a density measured in particles cm-3

collapse into stars with densities measured in g cm-3 ?

d2R

R

GM

~

~

dt2

t2

R2

R3

1

t2

~

GM

Gr

Free-fall time: tff= (3p/32Gr)1/2

= 1.4 x 105 (105 cm-3/n)1/2 yr

Gravitational Collapse from Dimensional Analysis--1

Characteristic timescale set by self-gravity:

Bonnor-Ebert mass = maximum mass of stable isothermal sphere:

MBE = 1.18cthermal3 /(G3r)1/2

Necessary condition for star formation: M >MBE

Gravitational Collapse from Dimensional Analysis--2

Characteristic mass:

Kinetic energy/mass ~ gravitational energy/mass

cs2P/r ~GM/R M ~ Rcs2/G

Radius: R ~ cstff ~ cs/(Gr)1/2

Mass ~ Rcs2/G ~ cs3tff/G ~ cs3/(G3r)1/2

·

~ mBE / tff~ cs3/(G3r)1/2 (Gr)1/2 ~ cs3/G

·

m*

m*

For a singular isothermal sphere (Shu 1977):

=0.975cs3 / G

= 1.5 x 10-6 (T/ 10 K)3/2Msun yr-1

An isothermal gas at 10 K takes 6.5 x 105yr to form a 1 Msun star

6.5 x 107yr to form a 100 Msun star

>> age of star (~ 3 Myr)

Gravitational Collapse from Dimensional Analysis--3

Characteristic accretion rate:

need better theory for formation of massive stars

OUTLINE

Star formation problems of interest to Zeldovich:

I. Star Formation in Filaments in the Turbulent Interstellar Medium

II. Radiation Hydrodynamics of Massive Star Formation

III. The Formation of the First Stars

I. Star Formation in Filaments in the Turbulent Interstellar Medium

This paper introduced the eigenvectors of gravitational collapse

The initial collapse is into a Zeldovich pancake, but these then collapse into filaments

Dust emission from molecular filaments observed by the Herschel satellite

Light blue lines trace filaments identified by an algorithm (Andre+ 2014)

Filaments form naturally in a turbulent medium

Simulation box 4.5 pc in size with finest resolution 0.002 pc

Isothermal gas with Mach number M=10, magnetized w. Alfven Mach # MA=1

(P.-S. Li + 2014)

Temperature T=10 K, density n~ 2 x 104 cm-3

Star cluster formation in magnetized 1000 Msun clump with outflows and radiation

(A. Myers+ 14)

Initial conditions: Self-consistent MHD turbulence w. Msonic=11, MA= 0.8

Column density

Temperature

Magnetic fields reduce star formation rate and fragmentation by factor ~ 2

Strong filamentary structure in star formation

Filamentary structures observed in star-forming regions arise due to gravitational instability in sheets (Miyama+ 87), a natural extension of Zeldovich’s model. Sheets are formed by strong shocks in supersonic turbulence.

Supersonic Turbulence and the Initial Mass Function

Probability distribution function for density in isothermal turbulence is lognormal:

where x = ln ( ρ / <ρ> ),

M= Mach number, and b = 1 for compressive driving, 1/3 for solenoidal driving

Self-gravity leads to gravitational collapse of the densest structures, producing IMF

With no gravity, density PDF is log normal

After 0.42 free-fall times, self-gravity has created a high-density tail on the distribution: gas collapsing into stars

(Kritsuk+ 2011)

The initial mass function of stars (IMF) can be calculated theoretically from the distribution of masses in the log normal distribution that become unstable

(Padoan & Nordlund, Hennebelle & Chabrier, Hopkins)

Thus, a universal process—turbulence—appears to be responsible for the universal shape of the IMF (Elmegreen)

Stellar feedback greatly complicates star formation:

- Radiation pressure drives dusty gas away
- UV emission heats via photoelectric effect on dust
- Ionizing luminosity creates ionized gas (~104 K)
- Protostellar outflows carve cavities and inject kinetic energy

THE FUNDAMENTAL PROBLEM IN

MASSIVE STAR FORMATION: RADIATION PRESSURE

Eddington luminosity LE: radiative force balances gravity:

LE /4r2c = GM/r2 LE = 4GMc/(/)

(where = mass/particle)

Typical infrared cross section per unit mass for dust in massive protostellar envelopes: / 5 cm2 g-1

LE = 4GMc/(/) = 2500 (M/Msun) Lsun

Force per particle due to radiation flux F = L/4 r2:

Force = F/c = L /4r2c where here c = speed of light

= cross section

Massive stars are very luminous:

L = 10 (M/Msun)3 Lsun for M ~ (7-20) Msun

L ~ 106Lsun for M = 100 Msun

Predict growth of protostar stops when radiative force exceeds gravity:

L = 10 (M/Msun)3 Lsun> LE = 2500 (M/Msun) Lsun

Stars cannot grow past 16 Msun

But stars are observed to exist with M > 100 Msun

HOW IS THIS POSSIBLE?

THE FUNDAMENTAL PROBLEM IN

MASSIVE STAR FORMATION: RADIATION PRESSURE--II

Bipolaroutflows from protostars channel radiation away from infalling gas

(Krumholz et al. 2005; Cunningham et al 2011)

ADDRESSING THE PROBLEM OF RADIATION PRESSURE

Effect of accretion disks

Accreting gas has angular momentum and settles into a disk before accreting onto star

Previous work has shown that disk shadow reduces the radiative force on the accreting gas

(Nakano1989; Jijina & Adams 1996; Yorke & Sonnhalter 2002)

- Radiative Rayleigh-Taylor instabilities allow radiation to escape

(Krumholz et al. 2009)

where e = 0.5 v2 + u = gas energy density, E = radiation energy density;

F0 and E0in comoving frame. Accurate to lowest relevant order in v/c.

3D RADIATION HYDRODYNAMIC CALCULATIONS

Radiative transfer: gray, mixed frame, flux-limited diffusion with AMR

t + v = 0

(Mass)

(Momentum)

tv + vv = - P - + (R/c)F

te + [(e+P)v] = - v - P(4B-cE) - (R/c) vF

(Energy)

2 = 4G

(Gravity)

(Radiative energy)

tE + F= P(4B-cE) + (R/c) vF

(Flux limit)

F0 = - [c(E0) / R] E0

(Krumholz et al. 2007)

Radiation-Hydrodynamic Simulation of Massive Star Formation

Time

34000 yr

(Krumholz+ 09)

Radiation pressure created large, radiation-dominated bubble shortly after t = 25000 yr.

25000 yr

41700 yr

Radiative Rayleigh-Taylor instabilities occurred shortly after 34000 yr; at least 40% of the accretion onto the stars was due to this.

Final stellar masses in binary:

M = 42 Msun, 29 Msun

55900 yr

Ongoing research: Does RT instability occur with more accurate treatment of stellar radiation?

3000 AU

Effect of bipolar outflows on massive star formation:

Create channels for the escape of radiation

Bipolar outflows originally discovered from low-mass protostars

Herbig-Haro objects

- A clue: evidence for bipolar ejection of spinning jets.

1000 AU

C. Burrows (STScI & ESA); J. Hester (Arizona St); J. Morse (STScI); NASA

Bipolar outflows from low-mass protostars produced in rotating, magnetized disks

Observation of magnetized jet from a high-mass protostar

IRAS 18162-2048

6 cm (contours)

Synchrotron emission

L=17,000 Lsun

M 10 Msun

850 m (gray scale)

if dominated by one star

Thermal emission

(Carrasco-Gonzalez et al. 2010)

Outflow makes diskgas cooler more fragmentation (lower primary mass))

- Results for =2 g cm-2 without winds ~ same as =10 g cm-2 with winds (Trapping of radiation increases with )

0.25 pc

0.01 pc

0.01 pc

Simulations of effects of outflows on massive star formation

(Cunningham+ 2011)

= 1 g cm-2

Results on outflows at t = 0.6 tff:

mpri ~20 Msun

= 2 g cm-2

Outflow reduces radiation pressure by allowing escape

mpri ~20 Msun

= 2 g cm-2

(no wind)

mpri ~35 Msun

= 10 g cm-2

mpri~35 Msun

Column density

Temp.

Conclusion on radiation pressure in massive star formation:

Three effects—disks, radiative Rayleigh-Taylor instability and bipolar outflows—are important in overcoming radiation pressure

Outflows allow radiation to escape, reducing importance of Rayleigh-Taylor instabilities

III. The Formation of the First Stars

JETP 16, 1395 (1963)

Zeldovich’s theory before the discovery of the microwave background and inflation.

Assumed fluctuations were statistical and universe cold

Concluded galaxy formation possible only if stars created large-scale perturbations

Kindly translated by IldarKhabibullin

Three discoveries since Zeldovich’s paper:

1) The CMB => the universe was hot, not cold, when it was much denser

2) Inflation: Quantum fluctuations magnified by inflation provided initial perturbations

3) Dark matter: Perturbations in dark matter grew during the radiation- dominated era, avoiding diffusive damping (Silk damping)

A key difference between first stars and contemporary stars:

First stars formed in potential wells due to dark matter, not due to their own self gravity.

(Although baryonic gravity dominates on scales < 1 pc)

Key question: What was the mass of the first stars?

M > 0.8 Msun since no stars have been observed that have no heavy elements

Key question: What was the mass of the first stars?

M > 0.8 Msun since no stars have been observed that have no heavy elements

The mass of the star determines the nature and mass of heavy elements ejected

(Heger & Woosley 2002)

Initial conditions for gravitational collapse set by physics of H2 molecule

This physics was of interest to Zeldovich:

Density at which collisional and radiative de-excitation are in balance: ncrit ~ 104 cm-3

Minimum temperature set by spacing of energy levels, ~ 200 K

- Characteristic mass at which gravity balances thermal energy[Jeans mass ~ cthermal3 /(G3r)1/2]
- is ~ 500 Msun

(Bromm, Coppi & Larson 2002)

Mass of the first stars:

Analytic theory (McKee & Tan 08): mass set by photoevaporation of accretion disk

Isentropic collapse: entropy within factor 2 of best estimate => M* = 60 – 320 Msun

Numerical simulation (Hirano+ 14): 3D cosmological simulations

+ 2D radiation-hydrodynamic simulations of individual stars

Good general agreement between theory and simulation: First stars very massive

Challenges in the Formation of the First Stars: Magnetic Fields

High-resolution cosmological simulation of gravitational collapse

Magnetic energy increases much faster with density than expected for uniform collapse (ρ4/3): Turbulent dynamo

Initial field very weak (~10-14 G) and was not dynamically important at end of simulation

(Turk+ 12)

No stars formed in this simulation, and the effect of magnetic fields on the formation of the first stars is unknown

Challenges in the Formation of the First Stars: Magnetic Fields

But Zeldovich could have told us that magnetic fields could be important:

Title: Magnetic fields in astrophysics

Authors: Zeldovich, Ya. B.

Publication: The Fluid Mechanics of Astrophysics and Geophysics, New York: Gordon and Breach, 1983

Keywords: ASTROPHYSICS, MAGNETIC FIELDS, DYNAMO THEORY

HAPPY BIRTHDAY, YAKOV B.!

AND THANK YOU, RASHID!

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