Simulating the Formation of Primordial Proto-Stars
Department of Physics, Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan
Abstract. We study the formation of primordial proto-stars in a ACDM universe using high-resolution cosmological
simulations. Our approach includes all the relevant atomic and molecular physics necessary to follow the thermal evolution
of a prestellar gas cloud to very high densities. We describe the numerical implementation of the physics. We show the results
of a simulation of the formation of primordial stars in a reionized gas.
Keywords: Population III Stars
PHYSICS OF PRIMORDIAL STAR
The study of primordial star formation has a long his-
tory. The formation of the first cosmological objects via
gas condensation by molecular hydrogen coohng has
been studied for many years since late 1960's. One-
dimensional hydrodynamic simulations of spherical gas
collapse were performed by a number of researchers.
Omukai & Nishi  included a detailed treatment of all
the relevant chemistry and radiative processes and thus
were able to provide accurate results on the thermal evo-
lution of a collapsing primordial gas cloud up to stellar
densities. These authors found that, while the evolution
of a spherical primordial gas cloud proceeds in a roughly
self-similar maimer, there are a number of differences
in the thermal evolution from that of present-day, metal-
and dust-enriched gas clouds.
Recently, three-dimensional hydrodynamic calcula-
tions were performed by several groups. Statistical prop-
erties of primordial star-forming clouds and the over-
all effect of cosmological bias have been studied in de-
tail [2, 3]. Simulations of the formation of primordial
proto-stars have been hampered by the complexity of
the involved physics, such as radiative transfer in very
high-density primordial gases. A critical technique we
describe in this contribution is computation of molecu-
lar line opacities and continuum opacities. With the im-
plementation of optically thick coohng, the gas evolu-
tion can be accurately followed to much higher densi-
ties than was possible in the previous studies [4, 6]. We
show that the method works well in problems of collaps-
ing gas clouds, in terms of computation of radiative cool-
ing rates and resulting density and temperature structure.
Our method can be apphed to a broad range of three-
dimensional problems in non-trivial configurations. We
apply this technique to a cosmological simulation of the
formation of primordial star
OPTICALLY THICK COOLING RATE
Molecular hydrogen line cooling
When the gas density and the molecular fraction are
high, the cloud becomes opaque to molecular fines and
thenH2 line cooling becomes inefficient. The net cooling
rate can be expressed as
AH2 .thick = X ^ ^"' ftsc.u/ -4 „/ «„ ,
where «„ is the population density of hydrogen
molecules in the upper energy level u, A^i is the
Einstein coefficient for spontaneous transition, /3esc,u/ is
the probability for an emitted line photon to escape with-
out absorption, and hVui = AEui is the energy difference
between the two levels.
In order to calculate the escape probability, we first
evaluate the opacity for each molecular line as
'^/u — OI/uL,
where L is the characteristic length scale. Since the ab-
sorption coefficients «/„ are computed in a straightfor-
ward maimer, although somewhat costly, the remaining
key task is the evaluation of the length scale L. By noting
that the important quantity we need is the effective gas
cooling rate, we can formulate a reasonable and weU-
motivated approximation. To this end, we decided to use
the Sobolev method that is widely used in the study of
steUar winds and planetary nebulae.
We calculate the Sobolev length along a fine-of-sight
where vthermai = ^kT/mu
molecules, and Vr is the fluid velocity in the direction.
A suitable angle-average must be computed in order to
obtain the net escape probability. Details are found in .
is the thermal velocity of H2
CP990, First Stars m, edited by B. W. O'Shea, A. Heger, and T. Abel,
© 2008 American Institute of Physics 978-0-7354-0509-7/08/$23.00
10® 10M0^° 10" lO^MO^MO^"* 10^^
n [cm ]
The effect of molecular line opacities to the net FIGURE 2.
for a density of lO'^cm^^.
The Planck opacity as a function of temperature
We test our method by performing a three-dimensional
calculation of spherical cloud collapse. Figure 1 shows
the normalized H2 line coohng rate against local density.
We use an output at the time when the central density is
«c = 10^'*cm^^. In the figure, we compare our simulation
results with those from the full radiative transfer calcula-
tions of  (open diamonds). Clearly, our method works
very well. The steepening of the slope at« > lO^^cm^^,
owing to the velocity change where infalhng gas settles
gradually onto the center, is well-reproduced. We empha-
size that the level of agreement shown in Figure 1 can
be achieved only if all of the local densities (of chemi-
cal species), temperatures, and velocities are reproduced
Cooling by collision-induced emission
Next, we implement computation of local optical
depth to continuum radiation using the Planck opacity
table of Lenzuni et al. . The continuum opacity is
used to evaluate the net cooling rate by collision-induced
emission (CIE). The optically thin coohng rate can be
computed as in .
Figure 2 compares the opacities for a gas of primordial
composition calculated by Lenzuni et al.  and Mayer
& Duschl . We find no significant difference between
the two opacity tables, and thus we use the one by
Lenzuni et al. We calculate local optical depths in six or-
thogonal directions from a target point in a smoothed par-
ticle hydrodynamics maimer (see, e.g., ), by projecting
the SPH kernels of surrounding gas particles which have
their own density and temperature. The net energy trans-
fer rate in each direction scales as = = 1/(1 + T) when the
optical depth is small. Again, a suitable angle average
must be calculated. We simply take the mean of six di-
/r reduce ;Sr
We then use this local reduction factor to evaluate the
cooling rate ' Figure 3 shows the net reduction factor for
the CIE cooling rate against local density. We note that,
although our method is still approximate, it takes into ac-
count local density, temperature, and their structures (ge-
ometry) in a self-consistent maimer Our method can be
applied to general problems, whereas simple functional
fits that is based only on local density will fail in esti-
mating the true cooling rate when the cloud core has a
complex structure such as (weak) turbulence.
When a gas cloud core becomes opaque to continuum
radiation, the gas can still absorb thermal energy in-
put owing to gravitational contraction by dissociating
molecules. Ionization cooling is unimportant in the tem-
perature regime we consider
For densities much larger than lO^^cm^^, the reaction
time scale becomes significantly shorter than the dynam-
ical time. Then, we can use equilibrium chemistry, which
would actually simplify our handhng of chemistry evolu-
' This estimate becomes incorrect when T is large. However, the radia-
tive cooling rate is extremely small for large T and so the exact behavior
at very optically-thick regime does not matter to the net cooling rate.
number density [cm"^]
FIGURE 3. The ratio of AQIE thick/A^mthm as a function
of density. We use an output when the central density reaches
tion. The species abundances can be determined from tlie
coupled Salia-Boltzmann equations:
zn, \ h^ )
We solve these equations at a given gas density itera-
tively. The associated coohng/heating due to dissocia-
tion/formation is implemented self-consistently as
£m = Xii2^"ii2
where Xu2 = 4.48eV is the molecular binding energy.
When dissociation of hydrogen molecules is com-
pleted at T r^ 5000 K, there will be no further mech-
anisms that enable the gas to lose its thermal energy,
and then the gas temperature increases following the so-
called adiabatic track.
The equation of state in the high pressure regime must
be modified to account for various non-ideal gas effects
. Figure 4 shows the effective gas pressure for which
various non-ideal gas effects are included.
PRIMORDIAL STAR FORMATION IN A
We employ the technique described in the previous sec-
tions in a cosmological simulation. Earlier , we used
a large cosmological simulation to study the evolution of
early rehc Hii regions until second-generation gas clouds
0 5.0-10^ I.O-IO"* 1.5-10'' 2.0-10''
FIGURE 4. Pressure for a non-ideal gas, as calculated by ,
for a particle number density of lO'^cm^^.
are formed. We further explore the evolution of these
prestellargas clouds. The highest density achieved by the
simulationis'~ lO^^cm^^, at which point the central core
is optically thick even to continuum radiation. Full-scale
dissociation of hydrogen molecules is taking place in the
core, which works as an effective cooling mechanism.
We investigate in detail the structure of such gas clouds.
We then compute the gas mass accretion rate and use it
as an input to a proto-stellar calculation.
Figure 5 shows the radial temperature profile around
the proto-star and the gas mass accretion rate. The tem-
perature structure can be understood by appeahngto var-
ious atomic and molecular processes (see [4,6]). HD line
cooling brings the gas temperature below 100 Kelvin.
Note that the minimum temperature is set by the cosmic
The central proto-stellar 'seed' is accreting the sur-
rounding gas at a rate > 10^^M©/yr, and thus a star
with mass '~ lOM© will form within 10'* years. How-
ever, the final stellar mass is determined by processes
such as radiative feedback from the protostar We treat
the evolution of a protostar as a sequence of a growing
hydrostatic core with an accreting envelope. The ordi-
nary stellar structure equations are applied to the hydro-
static core. The structure of the accreting envelope is cal-
culated under the assumption that the flow is steady for a
given mass accretion rate.
Figure 6 shows the resulting evolution of the pro-
tostar. After a transient phase and an adiabatic growth
phase at M* < lOM©, the protostar enters the Kelvin-
Helmholtz phase and contracts by radiating its thermal
energy. When the central temperature reaches lO^K, hy-
drogen burning by the CNO cycle begins with a slight
amount of carbon synthesized by helium burning. This
phase is marked by a sohd circle in the figure. The en-
0 1 2
FIGURE 5. (Top) The radial temperature profile around the
proto-star as a funciton of enclosed gas mass. (Bottom) The in-
stantaneous gas mass accretion rate. We also show the accretion
rate of .
1.0 10.0 100.0
protostellar mass [M®]
FIGURE 6. Evolution of the proto-stellar radius and the
mass (solid line). The solid circle marks the time when efficient
hydrogen burning begins. The dotted line shows the mass (and
radius) growth which is calculated under the assumption that
a larger amount of gas than the parent cloud can be accreted.
For reference we also show the result from Y06 for a first
generation star that forms in an initially neutral gas cloud.
ergy generation by liydrogen burning lialts contraction
wlientlie mass is 35M© and its radius is '-^ 2.8 solar radii.
Soon after, the star reaches the zero-age main sequence
(ZAMS). The protostar relaxes to a ZAMS star within
about 10^ years from the birth of the protostellar seed.
Accretion is not halted by radiation from the protostar to
the end of our calculation.
It is important to point out that the mass of the par-
ent cloud from which the star formed is Mdoud '^ 40M©.
The final stellar mass is likely hmited by the mass of the
gravitationally unstable parent cloud. We thus argue that
primordial stars formed from an ionized gas are massive,
with a characteristic mass of several tens of solar masses,
allowing overall uncertainties in the accretion physics
and also the dependence of the minimum gas temperature
on redsluft. They are smaller than the first stars formed
from a neutral gas, but are not low-mass objects as sug-
gested by earlier studies.
The elemental abundance patterns of recently dis-
covered hyper metal-poor stars suggest that they might
have been bom from the interstellar medium that was
metal-enriched by supemovae of these massive primor-
The work is supported in part by the Grants-in-Aid for
Young Scientists 17684008 by the Ministry of Education,
Culture, Science and Technology of Japan, and by The
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