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The Zeldovich BoxA
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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 ?
Characteristic timescale set by self-gravity:
d2Rdt2 ~
Rt2 ~
GMR2
t2 R3
GM~ 1
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 mass:
Kinetic energy/mass ~ gravitational energy/mass
cs2 P/ ~r GM/R M ~ Rcs
2/G
Radius: R ~ cstff ~ cs/(Gr)1/2
Mass ~ Rcs2/G ~ cs
3tff/G ~ cs3/(G3r)1/2
Bonnor-Ebert mass = maximum mass of stable isothermal sphere:
MBE = 1.18 cthermal3 /(G3r)1/2
Necessary condition for star formation: M > MBE
Gravitational Collapse from Dimensional Analysis--2
Characteristic accretion rate:
m*
·~ mBE / tff ~ cs
3/(G3r)1/2 (Gr)1/2 ~ cs3/G
For a singular isothermal sphere (Shu 1977):
m*
·= 0.975 cs
3 / G
= 1.5 x 10-6 (T/ 10 K)3/2 Msun yr-1
An isothermal gas at 10 K takes 6.5 x 105 yr to form a 1 Msun star
6.5 x 107 yr to form a 100 Msun star
>> age of star (~ 3 Myr)
Gravitational Collapse from Dimensional Analysis--3
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
Temperature T=10 K, density n ~ 2 x 104 cm-3 (P.-S. Li + 2014)
Star cluster formation in magnetized 1000 Msun clump with outflows and radiation(A. Myers+ 14)
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.
Initial conditions: Self-consistent MHD turbulence w. Msonic=11, MA= 0.8
Column densityTemperature
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
(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)
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
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 INMASSIVE 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
THE FUNDAMENTAL PROBLEM INMASSIVE STAR FORMATION: RADIATION PRESSURE--II
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?
L = 10 (M/Msun)3 Lsun for M ~ (7-20) Msun
Massive stars are very luminous:
L ~ 106 Lsun for M = 100 Msun
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
(Nakano 1989; Jijina & Adams 1996; Yorke & Sonnhalter 2002)
- Radiative Rayleigh-Taylor instabilities allow radiation to escape(Krumholz et al. 2009)
Bipolar outflows from protostars channel radiation away from infalling gas (Krumholz et al. 2005; Cunningham et al 2011)
3D RADIATION HYDRODYNAMIC CALCULATIONS
Radiative transfer: gray, mixed frame, flux-limited diffusion with AMR
t + v = 0
t v + vv = - P - + (R/c)F
t e + [(e+P)v] = - v - P(4B-cE) - (R/c) vF
2 = 4G
t E + F = P(4B-cE) + (R/c) vF
F0 = - [c(E0) / R] E0
where e = 0.5 v2 + u = gas energy density, E = radiation energy density;F0 and E0 in comoving frame. Accurate to lowest relevant order in v/c.
(Krumholz et al. 2007)
(Mass)
(Momentum)
(Energy)
(Gravity)
(Radiative energy)
(Flux limit)
Radiative Rayleigh-Taylor instabilities occurred shortly after 34000 yr; at least 40% of the accretion onto the stars was due to this.
Time
34000 yr
41700 yr
55900 yr
3000 AU
25000 yrRadiation pressure created large, radiation-dominated bubble shortly after t = 25000 yr.
Final stellar masses in binary:
M = 42 Msun, 29 Msun
(Krumholz+ 09)
Radiation-Hydrodynamic Simulation of Massive Star Formation
Ongoing research: Does RT instability occur with more accurate treatment of stellar radiation?
Herbig-Haro objects
• A clue: evidence for bipolar ejection of spinning jets.
Bipolar outflows from low-mass protostars produced in rotating, magnetized disks
C. Burrows (STScI & ESA); J. Hester (Arizona St); J. Morse (STScI); NASA
1000 AU
Bipolar outflows originally discovered from low-mass protostars
(Carrasco-Gonzalez et al. 2010)
6 cm (contours)
850 m (gray scale)
Observation of magnetized jet from a high-mass protostar
IRAS 18162-2048
L=17,000 Lsun
M 10 Msun
if dominated by one star
Synchrotron emission
Thermal emission
Simulations of effects of outflows on massive star formation
Results on outflows at t = 0.6 tff :
Outflow reduces radiation pressure by allowing escape
Results for =2 g cm-2 without winds ~ same as =10 g cm-2 with winds (Trapping of radiation increases with )
mpri ~20 Msun
mpri ~20 Msun
mpri ~35 Msun
mpri ~35 Msun
= 1 g cm-2
= 2 g cm-2
= 2 g cm-2
= 10 g cm-2
(no wind)
Outflow makes disk gas cooler more fragmentation (lower primary mass))
0.01 pc 0.01 pc0.25 pc
(Cunningham+ 2011)
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
Kindly translated by Ildar Khabibullin
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
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)
First stars formed in potential wells due to dark matter, not due to their own self gravity.
A key difference between first stars and contemporary stars:
(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
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 ~ cthermal
3 /(G3r)1/2] is ~ 500 Msun
(Bromm, Coppi & Larson 2002)
This physics was of interest to Zeldovich:
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
Magnetic energy increases much faster with density than expected for uniform collapse (ρ4/3): Turbulent dynamo
High-resolution cosmological simulation of gravitational collapse
(Turk+ 12)
Initial field very weak (~10-14 G) and was not dynamically important at end of simulation
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 astrophysicsAuthors: Zeldovich, Ya. B.Publication: The Fluid Mechanics of Astrophysics and Geophysics, New York: Gordon and Breach, 1983Keywords: ASTROPHYSICS, MAGNETIC FIELDS, DYNAMO THEORY
HAPPY BIRTHDAY, YAKOV B.!
AND THANK YOU, RASHID!