The Formation of the First Star in the Universe

Abstract

We describe results from a fully self-consistent three-dimensional hydrodynamical simulation of the formation of one of the first stars in the Universe. In current models of structure formation, dark matter initially dominates, and pregalactic objects form because of gravitational instability from small initial density perturbations. As they assemble via hierarchical merging, primordial gas cools through ro-vibrational lines of hydrogen molecules and sinks to the center of the dark matter potential well. The high-redshift analog of a molecular cloud is formed. As the dense, central parts of the cold gas cloud become self-gravitating, a dense core of approximately 100 M (where M is the mass of the Sun) undergoes rapid contraction. At particle number densities greater than 10(9) per cubic centimeter, a 1 M protostellar core becomes fully molecular as a result of three-body H2 formation. Contrary to analytical expectations, this process does not lead to renewed fragmentation and only one star is formed. The calculation is stopped when optical depth effects become important, leaving the final mass of the fully formed star somewhat uncertain. At this stage the protostar is accreting material very rapidly (approximately 10(-2) M year-1). Radiative feedback from the star will not only halt its growth but also inhibit the formation of other stars in the same pregalactic object (at least until the first star ends its life, presumably as a supernova). We conclude that at most one massive (M 1 M) metal-free star forms per pregalactic halo, consistent with recent abundance measurements of metal-poor galactic halo stars.

Full text

arXiv:astro-ph/0112088v1 4 Dec 2001
Draft version February 1, 2008
Preprint typeset using L
A
T
E
X style emulateapj v. 14/09/00
THE FORMATION OF THE FIRST STAR IN THE UNIVERSE
Tom Abel
Harvard Smithsonian Center for Astrophysics, MA, US–02138 Cambridge
Institute of Astronomy, Cambridge, UK
Greg L. Bryan
Massachusetts Institute of Technology, MA, US–02139 Cambridge
Hubble Fellow
Michael L. Norman
University of California, San Diego, CA, US–92093 La Jolla
Draft version February 1, 2008
ABSTRACT
We describe results from a fully self–consistent three dimensional hydrodynamical simulation of the
formation of o ne of the first stars in the Universe. Dark matter dominated pre-galactic objects form
because of gr avitational instability from small initidal density perturbations. As they assemble via
hierarchical merging, primordial gas cools through ro-vibrational lines of hydrogen molecules and sinks
to the center of the dark matter potential well. The high redshift analog of a molecula r cloud is formed.
When the dense, central parts o f the cold gas cloud become self-gravitating, a dense core of ∼ 100 M
⊙
undergoes rapid contrac tion. At densities n > 10
9
cm
−3
a 1 M
⊙
proto-stellar core becomes fully molecular
due to three–body H
2
formation. Contrary to analytical expectations this process does not lead to
renewed fragmentation and only one star is formed. The calculation is stopped when optical depth
effects become important, leaving the final mass of the fully formed star somewhat uncertain. At this
stage the protostar is a c reting ma terial very rapidly (∼ 10
−2
M
⊙
yr
−1
). Radiative feedback from the
star will not o nly halt its growth but also inhibit the fo rmation of other stars in the same pre–galactic
object (at least until the first star ends its life, presumably as a supernova). We conclude that at most
one massive (M ≫ 1 M
⊙
) metal free star forms per pre–galactic halo, co ns istent with recent abundance
measurements of metal poor gala c tic halo stars.
1. motivation
Chemical elements heavier than Lithium are synthesized
in stars. Such “metals” are observed at times when the
Universe was only
<
∼
10% of its current age in the inter–
galactic medium (IGM) as abs orption lines in quasar spec-
tra (see Ellison et al. 2000, a nd references therein). Hence,
these heavy elements not only had to be synthesized but
also released and distributed in the IGM within the first
billion years. Only supernovae of sufficiently short lived
massive stars are known to provide such an enrichment
mechanism. This le ads to the prediction that
the first generation of cosmic structures formed massive
stars (although not necessarily only massive stars).
In the pa st 3 0 years it has been argued that the fir st
cosmologic al objects form globular cluster s (1), super–
massive black holes (2), or even low mass star s (3). This
disagreement of theoretical studies might at first seem sur-
prising. However, the first objects form via the gravita-
tional collapse of a thermally unstable reactive medium,
inhibiting co nclusive ana lytical calculations. The prob-
lem is par ticula rly a c ute because the evolution of all other
cosmologic al objects (and in particular the larger galaxies
that follow) will depend on the evolution of the first stars.
Nevertheless, in compa rison to present day star forma-
tion, the physics of the formation of the first star in the
universe is rather s imple. In particular:
• the chemical and radiative of processes in the
primordial gas are readily understood.
• strong magnetic fields are not expec ted to exist a t
early times.
• by definition no other stars exist to influence the
environment through radia tion, winds, supernovae,
etc.
• the emerging standard model for structure
formation provides appropriate initial conditions.
In previous work we have pr e sented three–dimensional
cosmologic al simulations o f the formation of the first ob-
jects in the universe (4, 5) including first applications of
adaptive mesh refinement (AMR) cosmological hydrody-
namical simulations to first s tructure formation (6, 7, ABN
hereafter) . In these s tudies we achieved a dynamic range
of up to 2 × 10
5
and could follow in detail the formatio n of
the first dense cooling region far within a pre–galactic ob-
ject that formed self–consistently from linear density fluc-
tuation in a cold dark matter cosmology. Here we report
results from simulations that extend our previous wo rk by
another 5 orders of magnitude in dynamic ra nge. For the
first time it is possible to bridge the wide range betwe en
cosmologic al and stellar scale.
2. simulation setup and numerical issues
We employ an Eulerian structured adaptive mesh re-
finement cosmological hydrodynamical code developed by
Bryan and Norman (9, 10). The hydrodynamical equa-
tions are solved with the second order accurate piecewise
1

2
parabolic method (11; 12) where a Riemann solver en-
sures accurate shock capturing with a minimum of numer-
ical viscosity. We use initial conditions appropriate for a
spatially flat Cold Dark Matter cosmology with 6% of the
matter dens ity contributed by baryons, zero cosmologic al
constant, and a Hubble constant of 50 km/s/Mpc (8). The
power spectrum of initial density fluctuations in the dark
matter and the gas are taken from the computation by the
publicly available Bo ltzma nn code CMBFAST (13) at red-
shift 100 (assuming an Harrison–Zel’dovich scale–invariant
initial spectrum).
We set up a three dimensional volume with 128 comov-
ing kpc on a side and solve the cosmological hydrodyna m-
ics equations assuming p e riodic boundary conditions. This
small volume is adequate for our purpose, because we are
interested in the evolution of the first pre–galactic object
within which a star may be formed by a redshift of z ∼ 20.
First we identify the Lagrangian volume of the first proto–
galactic halo with a mass of ∼ 10
6
M
⊙
in a low resolution
pure N–body simulation. Then we generate new initial
conditions with four initial static grids that cover this Lan-
grangian region with progressively finer resolution. With a
64
3
top grid and a refinement factor of 2 this specifies the
initial conditions in the region of interest equivalent to a
512
3
uni–grid calculation. Fo r the adopted cosmology this
gives a mass resolution of 1.1 M
⊙
for the dark matter (DM,
hereafter) and 0.07 M
⊙
for the gas. The sma ll DM masse s
ensure that the cosmological Jeans ma ss is resolved by at
least ten thousand pa rticles at all times. Smaller scale
structures in the dark matter will not be able to influence
the baryons bec ause of their shallow potential wells. The
theoretical expectation holds, because the simulations of
ABN which had 8 times p oorer DM resolution led to iden-
tical results on large scales as the simulation presented
here.
During the evolution, refined grids are introduced with
twice the spatial resolution of the parent (coarse r) grid.
These child (finer) meshes are added whenever one of three
refinement criteria are met. Two Langrangian cr iteria en-
sure that the grid is refined whenever the gas (DM) density
exceeds 4.6 (9.2) its initial density. Additionally, the local
Jeans length is always covered by at least 64 grid cells
1
(4 cells per Jeans leng th would be sufficient, 14). We have
also carried o ut the simulations with identical initial con-
ditions but varying the refinement criteria. In one series
of runs we varied the number of mesh points per Jeans
length. Runs with 4 , 16, and 64 zones per Jeans length
are indistinguishable in all mass weighted radial pro files
of physical quantities. No change in the angular momen-
tum profiles could be found, sugg e sting neg ligible numer-
ical viscosity effects on angular momentum transport. A
further refinement criter ion that ensured the local cooling
time scale to be longer than the local Courant time also
gave identical results. This latter test checked that any
thermally unstable region was identified.
The simula tion follows the non–equilibrium chemistry
of the dominant nine species species (H, H
+
, H
−
, e
−
,
He, He
+
, He
++
, H
2
, and H
+
2
) in primordial gas. Fur-
thermore, the radiative losses from atomic and molecular
line cooling, Compton cooling and heating of free elec-
trons by the cosmic background radiation are appropri-
ately treated in the optically thin limit (15, 16). To ex-
tend our previous the studies to higher densities three es-
sential modifications to the code were made . First we
implemented the three–b ody molecular hydrogen forma-
tion process in the chemical r ate equations. For tem-
peratures below 300 K we fit to the data of Orel (17)
to get k
3b
= 1.3 × 10
−32
(T/300 K)
−0.38
cm
6
s
−1
. Above
300 K we then match it continuously to a powerlaw (3)
k
3b
= 1.3×10
−32
(T/300 K)
−1
cm
6
s
−1
. Secondly, we intro-
duce a variable adiabatic index for the gas (18). The dissi-
pative component (ba ryo ns) may collapse to much higher
densities than the collisionless component (DM). The dis-
crete sampling of the DM potential by particles can then
become inadeq uate and result in artificial heating of the
baryons (cooling for the DM) once the gas density becomes
much larger than the local DM density. To avoid this, we
smooth the DM particles with a Gaussian of width 0.05 pc
for grids with cells smaller than this leng th. At this scale,
the enclosed gas ma ss substa ntially exceeds the enclosed
DM mass.
The standard messa ge passing library (MPI) was used to
implement domain decomposition on the individual levels
of the grid hierarchy as a parallelization strategy. The code
was run in parallel on 16 processors of the SGI Origin2000
supe rcomputer at the National Center for Supercomput-
ing Applications at the Unive rsity of Illinois at Urbana
Champaign.
We stop the simulation at a time when the molecula r
cooling lines r e ach an optical depth of ten at line cen-
ter because our numerical method cannot treat the diffi-
cult problem of time–dependent radiative line trans fer in
multi–dimensions. At this time the code utilizes above
5500 grids on 27 refinement levels with 1.8 × 10
7
≈ 260
3
computational grid cells. An average grid therefore con-
tains ∼ 15
3
cells.
3. results
3.1. Characteristic mass scales
Our simulations (Fig. 1, Fig. 2), identify at least four
characterisic mass scales. From the outside going in, one
observes infall and accretion onto the pre–galactic halo
with a total mass of 7 × 10
5
M
⊙
, consistent with previous
studies (4, 19, 5, ABN, and 20 for discussion and refer-
ences).
At a mass scale of about 4000 solar mass (r ∼ 10 pc)
rapid cooling and infall is observed. This is accompa nyed
by the first of three va lleys in the radial velocity distribu-
tion (Fig. 2E). The temperature drops and the molecular
hydrogen fraction increases. It is here, at number densities
of ∼ 10 cm
−3
, that the high redshift analog of a molecular
cloud is formed. Although the molecular mass fraction is
not even 0.1 % it is sufficient to cool the gas rapidly down
to ∼ 200 K. The gas cannot cool below this temperature
because of the sharp decrease in the cooling rate below
∼ 200K.
At redshift 19 (Fig. 2), there are only two mass scales;
however, as time passes the central density grows and even-
tually passe s 10
4
cm
−3
, at which point the ro- vibrational
levels of H
2
are populated at their equilibrium values and
the cooling time becomes independent of density (instead
of inversely proportional to it). This reduced cooling effi-
1
The Jeans mass whi ch is the relevant mass scale for collapse and fragmentation is thus resolved by at least 4π32
3
/3 ≈ 1.4 × 10
5
cells.

3
Fig. 1.— Overview of the evolution and collapse forming a primordial star in the universe. The top row shows projections of the gas
density of one thousands of the simulation volume approximately centered at the pre–galactic object within which the star is formed. The
four projections from left to right are taken at redshifts 100, 24, 20.4, and 18.2 respectively. The pre–galactic objects form from very small
density fluctuations, and continously merge to form larger objects. The middle and bottom row show thin slices through the gas density and
temperature at the final simulation output. The leftmost panels are on the scale of the simulation volume ∼ 6 proper kpc. T he panels to the
right zoom in towards the forming star and have s ide lengths of 600 pc, 6pc, and 0.06 pc (12000 astronomical units). The color maps (going
from black to blue, green, red, yellow) are logarithmic and the associated values were adjusted considerably to visualize the ∼ 17 orders of
magnitude in density covered by these simulations. In the left panels the larger scale structures of filaments and s heets are seen. At their
intersections a pre–galactic obj ect of ∼ 10
6
M
⊙
is formed. In the temperature slice (second panel - bottom row) one sees how the gas shock
heats as it falls into the pre–galactic object. Af ter passing the acretion shock the material forms hydrogen molecules and starts to cool. The
cooling material accumulates at the center of the object and forms the high redshift molecular cloud analog (third panel from the right) which
is dense and cold (T ∼ 200 K). Deep within the molecular cloud a few hundred Kel vin warmer core of ∼ 100 M
⊙
is formed (right panel)
within which a 1 M
⊙
proto–star is formed (yellow region in the right panel of the middle row).
ciency leads to an increase in the temperature (Fig. 2D).
As the temperature rises, the cooling rate again increases
(it is 1000 times higher at 800 K than at 200 K), and the
inflow velocities slowly climb.
In order to better understand what happens next, we
examine the stability of an isothermal gas sphere. The
critical mass for gravitational collapse given an external
pressure P
ext
(BE mass hereafter) is given by Ebert (21)
and Bonnor (22) as:
M
BE
= 1.18 M
⊙
c
4
s
G
3/2
P
−1/2
ext
; c
2
s
=
dP
dρ
=
γk
B
T
µm
H
. (1 )
Here P
ext
is the external pre ssure and G, k
B
, and c
s
the
gravitational constant, the Bo ltzma nn constant and the
sound speed, respectively. We can estimate this critical
mass locally if we set the external pressure to be the lo-
cal pressure to find M
BE
≈ 20 M
⊙
T
3/2
n
−1/2
µ
−2
γ
2
where
µ ≈ 1.22 is the mean mass per particle in units of the pro-
ton mass. Using an adiabatic index γ = 5/3, we plot the
ratio of the enclosed ga s mass to this modified BE mass in
Figure 3.
Our modeling shows (Fig. 3), that by the fourth con-
sidered output time, the central 100 M
⊙
exceeds the BE
mass at that radius, indicating unstable collaps e. This is
the third mass scale and corresponds to the second local
minimum in the r adial velocity curves (Fig. 2E). The in-
flow velocity is 1 km s
−1
is still subsonic. Although this
mass scale is unstable, it does not represent the s mallest
scale of collaps e in our simulation. This is due to the in-

4
10
−6
10
−4
10
−2
10
0
10
2
radius [pc]
10
−2
10
0
10
2
10
4
10
6
10
8
10
10
10
12
n [cm
−3
]
B
10
−4
10
−2
10
0
H
2
, HI fractions
10
−1
10
1
10
3
enclosed gas mass [M
o
]
−3
−2
−1
0
V
r,gas
, c
s
[km/s]
10
2
10
3
T [K]
r [a.u.]
10
−3
10
−1
10
1
10
3
10
5
enclosed gas mass [M
o
]
A
C
D
E
10
0
10
2
10
4
Fig. 2.— Radial mass weighted averages of various physical quantities at seven different output times. Panel A shows the evolution of the
particle number density in cm
−3
as a function of radius at redshift 19 (solid line), nine Myrs later (dotted lines with circles), 0.3 Myr l ater
(dashed line), 3 × 10
4
yr later (long dashed line), 3 × 10
3
yr later (dot–dashed line), 1.5 × 10
3
yr later (solid line) and finally 200 years later
(dotted line with circles) at z = 18.181164. The two lines b etween 10
−2
and 200 pc give the DM mass density in GeV cm
−3
at z=19 and the
final time, respectively. Panel B gives the enclosed gas mass as a function of radius. In C the mass fractions of atomic hydrogen and molecular
hydrogen are shown. Panel D and E illustrate the temperature evolution and the mass weighted radial velocity of the baryons, respectively.
The bottom line with filled symbols in panel E show s the negative value of the local speed of sound at the final time. In all panels the same
output times correspond to the same line styl es.
creasing molecular hydrogen fraction.
When the gas density becomes sufficiently large (∼
10
10
cm
−3
), three-body molecular hydrogen formation be-
comes important. This r apidly increases the molecu-
lar fraction (Fig. 2C) and hence the c ooling rate. The
increased cooling leads to lower temperatures and even
stronger inflow and. At a mass scale of ∼ 1 M
⊙
, not only
is the gas near ly completely molecula r, but the radial in-
flow has become supersonic (Fig. 2E). When the H
2
mass
fraction approa ches unity, the increase in the cooling rate
saturates, and the ga s g oes through a radiative shock. This
marks the first appearance of the proto–stellar accretion
shock at a radius of about 2 0 astronomical units from its
center.
3.2. Chemo–Thermal Instability
When the cooling time becomes independent of density
the classical criterion for fragmentation t
cool
< t
dyn
∝
n
−1/2
(23) cannot be satisfied at high densities. However,
in principal the medium may still be subject to thermal
instability. The instability criterion is
ρ

∂L
∂ρ

T =const.
− T

∂L
∂T

ρ=const.
+ L(ρ, T ) > 0, (2)
where L denotes the cooling los ses per second of a fluid
parcel and T and ρ are the gas temperature and mass
density, respectively. At densities above the critical densi-
ties of molecular hydrogen the cooling time is independent
of density, i.e. ∂L/∂ρ = Λ(T ) where Λ(T ) is the high den-
sity cooling function (e.g. 24). Fitting the cooling function
with a power-law loca lly around a temperature T
0
so that
Λ(T ) ∝ (T/T
0
)
α
one finds ∂L/∂T = ραΛ(T )/T . Hence,
under these circumstances the medium is thermally stable
if α > 2. Because, α > 4 for the densities and tempera-
tures of interest, we conclude that the medium is thermally
stable. The above analysis neglects the heating from con-
traction, but this only str e ngthens the conclusio n. If heat-
ing balances cooling one can neglect the +L(ρ, T ) term in
equation (2) and find the medium to be thermally stable
for α > 1.

5
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
M
gas
(r)
10
−2
10
−1
10
0
10
1
M
gas
(r)/M
BE
(r)
Fig. 3.— Ratio of enclosed gas mass to the locally estimated
Bonnor–Ebert mass (M
BE
≈ 61 M
⊙
T
3/2
K
n
−1/2
µ
−2
) for various out-
put times. The enclosed gas m ass exceeds the BE mass at two dif-
ferent mass scales, ∼ 1 M
⊙
and ∼ 100 M
⊙
. The line-styles in the
Figure corresp ond to the output times shown i n Fig. 2.
However, here we neglected the chemica l processes. The
detailed analysis for the case when chemical processes oc-
cur on the collapse time–scale is we ll k nown (25). This
can be applied to primor dial star formation (26) includ-
ing the three–body formation of molecular hydrogen (3)
which drives a chemo–thermal instability. Evaluating all
the terms in this modified instability criterion (26, equa-
tion 36) one finds the simple result that for molecular mass
fractions f < 6/(2α + 1) the medium is expec ted to be
chemo–thermally unstable. These lar ge molecular frac-
tions illustrate that the strong density dependence o f the
three body H
2
formation dominates the instability. Ex-
amining the three dimensional temperature and H
2
den-
sity field we clearly see this chemo–thermal instability at
work. Cooler regions have larger H
2
fractions. How-
ever, no corresponding large density inhomo geneities are
found and fragmentation does not occur. This happens
because of the short sound crossing times in the collapsing
core. When the H
2
formation time– scale b e c omes shorter
than the cooling time the instability originates. However,
as long as the sound crossing time is much shorter than
the chemical and cooling time scales the cooler parts are
efficiently mixed with the warmer material. This holds
in our simulation until the final output where for the
first time the H
2
formation time scale becomes shorter
than the sound–crossing time. However, at this point the
proto–stellar core is fully molecular and stable against the
chemo–thermal instability. Consequently no large density
contrasts are formed. Because at these high densities the
optical depth of the cooling radiation becomes larger than
unity the instability will be suppressed even further.
3.3. Angular m omentum
Interestingly, rotational support does not halt the col-
lapse. This is for two reaons . T he first is shown in panel
A of Fig. 4 , which plots the spec ific angular momentum
against enclosed mass for the same seven output times dis-
cussed earlier. Concentrating on the first output (Fig. 4),
we see that the central gas begins the collapse with a spe-
cific angular momentum only ∼ 0.1% as large as the mean
value. This type of angular momentum profile is typical
of halos produced by gravitational collapse (e.g. 27), and
means that the protostellar gas starts out without much
angular momentum to lose. As a graphic example of this,
consider the central one solar mass of the colla psing region.
It has only an order of magnitude less angular momentum
at densities n
>
∼
10
13
cm
−3
than it ha d at n
>
∼
10
6
cm
−3
although it collapsed by over a factor 100 in radius.
The remaining output times (Fig. 4) indicate that ther e
is some angular momentum tr ansport within the central
100 M
⊙
(since L plotted as a function of enclosed mass
should stay constant as long as there is no shell crossing).
In panel C, we divide L by r to get a ty pical rotational ve-
locity and in panels B and D compare this velocity to the
Keplerian rotational velocity a nd the local sound speed,
respectively.
We find that the typical rotational spee d is a factor
two to three below that required for rotational support.
Furthermore, we see that this a z imuthal speed never s ig-
nificantly exceeds the sound speed, although for most the
mass below 100 M
⊙
it is comparable in value. We interpret
this as evidence tha t it is shock waves during the turbulent
collapse that are responsible for much of the transported
angular momentum. A collapsing turbulent medium is
different from a disk in Keplarian rotation. At any radius
there will be both low and high angular momentum mate-
rial, and pressure forces or shock waves can redistribute
the angular momentum betwe en fluid elements. Lower
angular momentum material will selectively sink inwards,
displacing higher angular momentum gas. This hydrody-
namic transpo rt of angular momentum will be suppr essed
in situation where the collapse proceeds on the dynamical
time rather on the longer cooling time as in the presented
case. This difference in cooling time and the widely dif-
ferent initial conditions may explain why this mechanism
has not been observed in simulations of pre sent day star
formation (e.g. 28, and references therein). However, such
situations may also arise in the late stages o f the formation
of present day stars and in scenar ios for the formation of
supe r–massive black holes.
To ensure that the angular momentum tr ansport is not
due to numerical shear viscosity (29) we have carried out
the resolution s tudy discussed above. We have varied the
effective spatial reso lution by a factor 16 and found iden-
tical results. Furthermore, we have run the adaptive mesh
refinement code with two different implementations of the
hydrodynamics solver. The r e solution study and the re-
sults presented here were carried out with a direct piece-
wise parabolic method adopted for cosmology (11; 12).
We ran another simulation with the lower order ZEUS hy-
drodynamics (30) and still found no relevant differences.
These tests are not strict pr oof that the encountered an-
gular momentum transport is not caused by numerical ef-
fects; however, they are reassuring.
3.4. Magnetic Fields?
The strength of magnetic fields generated around the
epoch of recombination is minute. In contrast, phase
transitions at the qantum–chromo–dynamic (QCD) and
electro–weak scales may form even dynamically impor-
tant fields. While there is a plethora of such scenarios

6
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
M
gas
(r)
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
L [km/s pc]
0
0.5
1
2
L/r/V
Kepler
0
1
2
3
4
L/r [km/s]
10
−1
10
0
10
1
10
2
10
3
10
4
M
gas
(r)
10
−2
10
−1
10
0
(L/r)/c
S
B
A
C
D
Fig. 4.— Radial m ass weighted averages of various physical quantities related to the angular momentum of the gas. The seven different
output tim es correspond to the ones described in Fig. 2. Panel A shows the specific angular momentum L in km s
−1
pc as a function of
enclosed gas mass. The typical rotational speed L/r is shown in panel C and its ratio to the Keplerian velocity V
Kepler
= (GM/r)
1/2
and
the local speed of sound in panel B and D, respectively.
for primordial ma gnetic field generatio n in the early uni-
verse they are not considered to be an integr al part of our
standard picture of str uctur e formation. This is because
not even the order of these pha se transitions is known
(31), and refere nce s therein). Unfortunately, strong pri-
mordial small-scale (≪ 1 comoving Mpc) magnetic fields
are poorly constrained observatio nally (32).
The critical mag netic field for support of a cloud (33)
allows a rough estimate up to which primo rdial magnetic
field strengths we may expect our simulation results to
hold. For this we also assume a flux frozen flow with no
additional amplification of the magnetic field o ther than
the contraction (B ∝ ρ
2/3
). For a comoving B field of
>
∼
3× 10
−11
G on scales
<
∼
100 kpc the critical field needed
for support may be reached during the collapse possibly
modifying the ma ss scales found in our purely hydrody-
namic simulations. However, the ionized fraction drops
rapidly during the collapse because of the absence of cos-
mic rays ionizations. C onsequently ambipolar diffusion
should be much more effective in the formation o f the first
stars even if such strong primordial magnetic fields were
present.
4. discussion
Previously we discussed the formation of the pre–
galactic object and the primordial “molecular cloud” that
hosts the formation of the first star in the simulated patch
of the universe (7). These simulations had a dyna mic
range of ∼ 10
5
and identified a ∼ 100 M
⊙
core within the
primordial “molecular cloud” undergoing renewed gravi-
tational collapse. The fate of this core was unclear be-
cause there was the potential caveat that three body H
2
formation could have caused fr agmentatio n. Indeed this
further fragmentation had been suggest by analytic work
(26) and single zone models (3). The three dimensional
simulations described here were designed to be able to test
whether the three body process will lead to a break up of
the core. No fragmentation due to three body H
2
forma-
tion is found. This is to a large part b e c ause of the slow
quasi–hydrostatic contraction found in ABN which allows
sub–sonic damping of density perturbations and yields a
smooth distribution at the time when three body H
2
for-
mation becomes important. Instead of fragmentation a
single fully molecular proto–star of ∼ 1 M
⊙
is formed at
the center of the ∼ 100 M
⊙
core.
However, even with extrao rdinary re solution, the final

7
mass of the first star remains unclear. Whether all the
available cooled material of the surro undings will accrete
onto the proto–star or feedba ck fro m the forming star will
limit the further accretion and hence its own growth is
difficult to compute in detail.
Within 10
4
yr about 70 M
⊙
may be accreted assuming
that angular momentum will not slow the collapse (Fig. 5).
The maximum of the accretion time of ∼ 5 × 10
6
yr is
at ∼ 600 M
⊙
. However, stars large r than 100 M
⊙
will
explode within ∼ 2 Myr. Therefore, it seems unlikely
(even in the absence of angular momentum) that ther e
would be sufficient time to accrete such large masses.
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
M
gas
[M
o
]
10
1
10
2
10
3
10
4
10
5
10
6
10
7
time [years]
M/(dM/dt)
r/v
r
Fig. 5.— The accretion time as function of enclosed gas mass.
The line with symbols gives M (r)/[4πρ(r)r
2
|v
r
(r)|] . The solid line
simply shows how long it would take the mass to move to r = 0 if
it were to to keep its current radial velocity (r/v
r
(M)).
A one solar mass proto–star will evolve too slowly to
halt substantial accretion. From the accre tion time pro-
file (Fig. 5 ) one may argue that a more r e alistic minimum
mass limit of the first star should be
>
∼
30 M
⊙
because this
amount would be accreted within a few thousand years.
This is a very short time compared to expected proto–
stellar evolution times. However, some properties of the
primordial gas may make it easier to halt the accretion.
One possibility is the destruction of the cooling agent,
molecular hydrogen, without which the acreting material
may reach hydrostatic equilibrium. This may or may not
be sufficient to halt the accretion. One may also imag-
ine that the central material heats up to 10
4
K, allowing
Lyman-α cooling from neutral hydrogen. That cooling re -
gion may expand rapidly as the internal pre ssure dro ps
because of infall, possibly allowing the envelope to accr e te
even without molecular hydrogen as cooling ag e nt. Addi-
tionally, radiation pressure from ionizing pho tons as well
as atomic hydrogen Lyma n series photons may become
significant and eventually reverse the flow. The mecha-
nisms discussed by Haehnelt (1993) on galactic scales will
play an important role for the continued accretio n onto the
proto–star. This is an interplay of many complex physical
processes because one has a hot ionized Str¨omgren sphere
through which cool and dense material is trying to accrete.
In such a situatio n one expects a Raleigh–Taylor type in-
stability that is modified via the geometry of the radiation
field.
At the final output time pres e nted here there are ∼
4 × 10
57
hydrogen molecules in the entire protogalaxy.
Also the H
2
formation time scale is long because there
are no dust grains and the free electrons (needed as a
catalyst) have almost fully recombined. Hence, as soon
as the the first UV photons of Lyman Werner band fre-
quencies are produced there will be a rapidly e xpanding
photo–dissociating region (PDR) inhibiting further cool-
ing within it. This photo-dissociation will prevent fur-
ther fragmentation at the molecular c loud scale. I.e. no
other star c an be formed within the same halo before the
first star dies in a supernova. The latter, however, may
have sufficient energy to unbind the e ntire gas content of
the small pre–galactic object it formed in (35). This may
have interesting feedback consequences for the dispersal of
metals, entropy and magnetic field into the intergalactic
medium (36, 37).
Smoothed particle hydrodynamics (SPH, e.g. 38), used
extensively in cosmological hydrodynamics, has been em-
ployed (20) to follow the collapse of solid body rotating
uniform spheres . The a ssumption of coherent rotation
causes these clouds to c ollapse into a disk which devel-
opes filamentary structures which eventua lly fragment to
form dense clumps of masses between 100 and 1000 solar
masses. It has been argued that these clumps will con-
tinue to accrete and merge and eventually form very mas-
sive stars. These SPH simulation have unrealis tic initial
conditions and much less resolution then our calculations.
However, they also show that many details of the collapse
forming a primordial sta r are determined by the properties
of the hydrogen mole c ule.
We have also simulated different initial density fields
for a Lambda CDM cosmology. There we have focused
on halos with different clustering environments. Although
we have not followed the collapse in these halos to proto-
stellar densities, we have found no qualitative differences
in the “primordial molecular cloud” formation process as
discussed in ABN. Also other AMR simulations (39) give
consistent res ults on scales larger than 1 pc. In all cases
a co oling flow forms the primordial molecular cloud at
the center of the dark matter halo. We conclude that
the molecular cloud formation process seems to be inde-
pendent of the halo clustering proper ties and the adopted
CDM type cosmology. Also the mass sca le s for the core
and the proto–star are determined by the local Bonnor–
Ebert mass. Consequently, we expect the key results dis-
cussed here to be insensitive to variations in cosmology or
halo clustering.
5. conclusion
The picture arising from these numerical simulations has
some very interesting implications. It is possible that all
metal free stars are massive and form in isolation. Their
supe rnovae may provide the metals seen in even the low-
est column density quasar absorption lines (40, and ref-
erences therein). Massive primordial stars offer a natural
explanation for the absence of purely meta l free low mass
stars in the Milky Way. The consequences for the forma-
tion of galaxies may be even more profound in that the
supe rnovae provide metals, entropy, and magnetic fields
and may even alter the initial power spectrum of density

8
fluctuations of the baryons.
Interestingly, it has been re c ently argued, from abun-
dance patterns, that in low metallicity galactic halo stars
seem to have been enriched by only o ne population of mas-
sive stars (41). These results, if confirmed, would represent
strong support for the picture arising from our ab initio
simulations of fir st structure formation.
To end on a speculative note there is suggestive evi-
dence that links gamma ray bursts to sites of massive star
formation (e.g. 42). It would be very fortunate if a sig-
nificant fraction of the massive stars naturally formed in
the simulations would cause gamma ray bursts (e.g. 43).
Such high redshift bursts would open a remarkably bright
window for the study of the otherwise dark (faint) Ag e s.
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