STAR FORMATION
In honor of
Yakov B. Zeldovich
June 20, 2014
Moscow
Chris McKee
The Zeldovich Box
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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 ?
Gravitational Collapse from Dimensional Analysis--1
Characteristic timescale set by self-gravity:
d2R
dt2

Free-fall time:
~
t2 
R
GM
~
t2
R2
R3
GM
~
1
Gr
tff = (3p/32Gr)1/2
= 1.4 x 105 (105 cm-3/n)1/2 yr
Gravitational Collapse from Dimensional Analysis--2
Characteristic mass:
Kinetic energy/mass ~ gravitational energy/mass
cs2  P/r ~ GM/R  M ~ Rcs2/G
Radius:
R ~ cstff ~ cs/(Gr)1/2
 Mass ~ Rcs2/G ~ cs3tff/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--3
Characteristic accretion rate:
·
m* ~ mBE / tff ~ cs3/(G3r)1/2  (Gr)1/2 ~ cs3/G
For a singular isothermal sphere (Shu 1977):
·
m* = 0.975 cs3 / 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)
 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)
Zoom-in shows star formation in the main filament:
T = 10 - 44 K, n ~ 104 cm-3
(P.-S. Li+ 2014)
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
Temperature
Column density
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)
II. Radiation
Hydrodynamics of
Massive Star Formation
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
Force per particle due to radiation flux F = L/4p r2:
Force = F /c = L /4pr2c
where here c = speed of light
 = cross section
Eddington luminosity LE: radiative force balances gravity:
LE /4pr2c = GM/r2  LE = 4pGMc/(/)
(where  = mass/particle)
Typical infrared cross section per unit mass for dust in massive
protostellar envelopes: /  5 cm2 g-1
 LE = 4pGMc/(/) = 2500 (M/Msun) Lsun
THE FUNDAMENTAL PROBLEM IN
MASSIVE STAR FORMATION: RADIATION PRESSURE--II
Massive stars are very luminous:
L = 10 (M/Msun)3 Lsun for M ~ (7-20) Msun
L ~ 106 Lsun 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?
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
(Krumholz et al. 2005; Cunningham et al 2011)
infalling gas
3D RADIATION HYDRODYNAMIC CALCULATIONS
Radiative transfer: gray, mixed frame, flux-limited diffusion with AMR
t r +  rv = 0
(Mass)
t rv +  rvv = - P - r + (R/c)F
(Momentum)
t re +  [(re+P)v] = - rv - P(4pB-cE) - (R/c) vF
2 = 4pGr
(Energy)
(Gravity)
t E + F = P(4pB-cE) + (R/c) vF
(Radiative energy)
F0 = - [c(E0) / R] E0
(Flux limit)
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)
RadiationHydrodynamic
Simulation of Massive
Star Formation
Time
34000 yr
(Krumholz+ 09)
Radiation pressure
created large, radiationdominated 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
L=17,000 Lsun
 M  10 Msun
if dominated by one star
6 cm (contours)
Synchrotron
emission
850 m
(gray scale)
Thermal
emission
(Carrasco-Gonzalez et al. 2010)
0.25 pc
0.01 pc
0.01 pc
Simulations of effects of outflows on
massive star formation (Cunningham+ 2011)
Results on outflows at t = 0.6 tff :
Outflow reduces radiation
pressure by allowing escape
Outflow makes disk gas 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 )
 = 1 g cm-2
mpri ~20 Msun
 = 2 g cm-2
 = 2 g cm-2
(no wind)
 = 10 g cm-2
Column density
mpri ~20 Msun
mpri ~35 Msun
mpri ~35 Msun
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 Ildar Khabibullin
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 radiationdominated 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!