Simulation of the gravitational collapse and fragmentation of rotating molecular clouds
In this paper we study the process of the subsequent (runaway) fragmentation of a rotating isothermal Giant Molecular Cloud (GMC) complex. Our own Smoothed Particle Hydrodynamics (SPH) gas-dynamical model successfully reproduces the observed Cloud Mass-distribution Function (CMF) in our Galaxy (even...
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| Zitieren: | Simulation of the gravitational collapse and fragmentation of rotating molecular clouds / P. Berczik, M.I. Petrov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 216-223. — Бібліогр.: 27 назв. — англ. |
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irk-123456789-796462015-04-04T03:01:46Z Simulation of the gravitational collapse and fragmentation of rotating molecular clouds Berczik, P. Petrov, M.I. MS3: Physics of Stars and Galaxies In this paper we study the process of the subsequent (runaway) fragmentation of a rotating isothermal Giant Molecular Cloud (GMC) complex. Our own Smoothed Particle Hydrodynamics (SPH) gas-dynamical model successfully reproduces the observed Cloud Mass-distribution Function (CMF) in our Galaxy (even the differences between the inner and outer parts of our Galaxy). The steady state of CMF is established during the collapse within a free-fall timescale of the GMC. We show that one of the key parameters, which defines the observed slope of the present day CMF, is the initial ratio of the rotational (turbulent) and gravitational energy inside the fragmented GMC. 2005 Article Simulation of the gravitational collapse and fragmentation of rotating molecular clouds / P. Berczik, M.I. Petrov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 216-223. — Бібліогр.: 27 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/79646 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України |
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MS3: Physics of Stars and Galaxies MS3: Physics of Stars and Galaxies Berczik, P. Petrov, M.I. Simulation of the gravitational collapse and fragmentation of rotating molecular clouds Кинематика и физика небесных тел |
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In this paper we study the process of the subsequent (runaway) fragmentation of a rotating isothermal Giant Molecular Cloud (GMC) complex. Our own Smoothed Particle Hydrodynamics (SPH) gas-dynamical model successfully reproduces the observed Cloud Mass-distribution Function (CMF) in our Galaxy (even the differences between the inner and outer parts of our Galaxy). The steady state of CMF is established during the collapse within a free-fall timescale of the GMC. We show that one of the key parameters, which defines the observed slope of the present day CMF, is the initial ratio of the rotational (turbulent) and gravitational energy inside the fragmented GMC. |
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Article |
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Berczik, P. Petrov, M.I. |
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Berczik, P. Petrov, M.I. |
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Berczik, P. |
| title |
Simulation of the gravitational collapse and fragmentation of rotating molecular clouds |
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Simulation of the gravitational collapse and fragmentation of rotating molecular clouds |
| title_full |
Simulation of the gravitational collapse and fragmentation of rotating molecular clouds |
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Simulation of the gravitational collapse and fragmentation of rotating molecular clouds |
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Simulation of the gravitational collapse and fragmentation of rotating molecular clouds |
| title_sort |
simulation of the gravitational collapse and fragmentation of rotating molecular clouds |
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Головна астрономічна обсерваторія НАН України |
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2005 |
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MS3: Physics of Stars and Galaxies |
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http://dspace.nbuv.gov.ua/handle/123456789/79646 |
| citation_txt |
Simulation of the gravitational collapse and fragmentation of rotating molecular clouds / P. Berczik, M.I. Petrov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 216-223. — Бібліогр.: 27 назв. — англ. |
| series |
Кинематика и физика небесных тел |
| work_keys_str_mv |
AT berczikp simulationofthegravitationalcollapseandfragmentationofrotatingmolecularclouds AT petrovmi simulationofthegravitationalcollapseandfragmentationofrotatingmolecularclouds |
| first_indexed |
2025-07-06T03:40:20Z |
| last_indexed |
2025-07-06T03:40:20Z |
| _version_ |
1836867357273751552 |
| fulltext |
SIMULATION OF THE GRAVITATIONAL COLLAPSE AND FRAGMENTATION
OF ROTATING MOLECULAR CLOUDS
P. Berczik1,2, M. I. Petrov1
1Main Astronomical Observatory, NAS of Ukraine
27 Akademika Zabolotnoho Str., 03680 Kyiv, Ukraine
e-mail: berczik@mao.kiev.ua; petrov@mao.kiev.ua
2Astrophysics Group, Department of Physics, Rochester Institute of Technology
54 Lomb Memorial Drive, Rochester, NY 14623, USA
e-mail: berczik@cis.rit.edu
In this paper we study the process of the subsequent (runaway) fragmentation of a rotating
isothermal Giant Molecular Cloud (GMC) complex. Our own Smoothed Particle Hydrodynamics
(SPH) gas-dynamical model successfully reproduces the observed Cloud Mass-distribution Function
(CMF) in our Galaxy (even the differences between the inner and outer parts of our Galaxy).
The steady state of CMF is established during the collapse within a free-fall timescale of the GMC.
We show that one of the key parameters, which defines the observed slope of the present day CMF,
is the initial ratio of the rotational (turbulent) and gravitational energy inside the fragmented GMC.
INTRODUCTION
The SPH algorithm based on 3D hydrodynamical codes, starting with the series of pioneering works by
Monaghan and Lattanzio [8, 13, 14, 17], are always very successfully applied to the study of evolution and
fragmentation in molecular clouds and molecular cloud complexes. The early simulations have been usually
performed with a few hundred to a few thousand of SPH particles and with a fixed (few parsec) spatial resolu-
tion. The most up-to-date simulations of molecular cloud evolution (e.g., [15]) are performed using a few tens
of thousands of SPH particles with variable smoothing lengths. These simulations also include the details of
cooling and heating in the complex gas mixtures of H, H2, CO and H II species.
Our high-resolution (64 000 SPH particle) simulations, using highly flexible and adaptive smoothing lengths,
deal with the study of the runaway collapse and subsequent isothermal fragmentation of an isolated GMC
complex with different rotational (turbulent) energy parameters of the clouds. The resulting CMF is compared
with the recent observational distributions of molecular cloud complexes (dN/dM ∼ M−γ , where the slope
of the power law γ ranges from 1.4 to 1.8) derived from various CO data for different parts of our Galaxy
[6, 10, 12, 16, 21, 22].
METHOD
Continuous hydrodynamic fields in SPH are described by the interpolation functions constructed from the known
values of these functions at randomly positioned N “smooth” particles with individual masses mi [18]. To achieve
the same level of accuracy for all points in the fluid it is necessary to use a spatially variable smoothing length.
In this case each particle has an individual value of the smoothing length – hi.
A more detailed and complete description of the basic numerical equations of SPH can be found in many of
our previous publications (e.g., [1–4, 24] and the references herein). Therefore, we briefly repeat here the skeleton
SPH equations of the code. The density at the position of a particle i can be defined as:
ρi =
N∑
j=1
mj · Wij ,
c© P. Berczik, M. I. Petrov, 2004
216
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5
E
ne
rg
y
time
ETHE
EKIN
ETOT
EGRA
8000
16000
32000
64000
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.0 0.5 1.0 1.5 2.0 2.5
(
E
T
O
T
-
E
0 T
O
T
)
/
| E
0 T
O
T
|
(
in
%
)
time
8000
16000
32000
64000
GADGET - 32000
a) Evolution of the energy b) Relative total energy error
Figure 1. Time evolution of the thermal, kinetic, potential, and total energy for the adiabatic collapse of an initially
“cold” gas sphere. The different lines correspond to the different gas particle numbers. We also present the energy error
results of the GADGET code with “standard” parameters for 32 000 particles
The equations of motion for a particle i are expressed as:
dri
dt
= vi,
dvi
dt
= −
N∑
j = 1
mj
(
Pi
ρ2
i
+
Pj
ρ2
j
+ Π̃ij
)
∇i Wij −∇i Φi,
where Pi is the pressure, Φi is the self-gravitational potential, and Π̃ij is an artificial viscosity term.
The internal energy equation has the form:
dui
dt
=
1
2
N∑
j = 1
mj
(
Pi
ρ2
i
+
Pj
ρ2
j
+ Π̃ij
)
(vi − vj) ∇i Wij +
Γi − Λi
ρi
.
Here ui is the specific internal energy of the particle i. The term (Γi − Λi)/ρi accounts for non-adiabatic
processes not associated with the artificial viscosity. We present the radiative cooling in the form proposed
in [19] (see case “B”), using the MAPPINGS III software [26]:
Λ = Λ(ρ, u, Z, . . .) � Λ∗(T, [Fe/H]) · n2
i , ni = ρi/(μ · mp),
where ni is the hydrogen number density, Ti the temperature, and μ is the molecular weight.
The equation of state must be added to close the system:
Pi = (γ − 1) ρi · ui,
where γ is the adiabatic index.
One of the basic tasks in SPH is to find the nearest neighbours of each SPH particle, i.e., to construct
an interaction list for each particle. Basically, we need to find all particles with | rij |≤ 2 max(hi, hj) in order
to estimate the density and also calculate the hydrodynamical forces.
In our code we keep the number of neighbours exactly constant by defining 2hi to be the distance to the NB –
nearest particle. The value of NB is chosen such that a certain fraction of the total number of “gas” particles N
affects the local flow characteristics. From these we need to SELECT the closest NB particles. Fast algorithms
217
0
50
100
150
200
2 4 6 8
0
100
N
U
M
B
E
R
O
F
C
LU
S
T
E
R
S
ϕ
(%
)
TIME (Myr)
ω = 0.1
Nc
ϕ
0
50
100
150
200
2 4 6 8
0
100
N
U
M
B
E
R
O
F
C
LU
S
T
E
R
S
ϕ
(%
)
TIME (Myr)
ω = 0.3
Nc
ϕ
a) “Slow” initial rotation ω = 0.1 b) “Fast” initial rotation ω = 0.3
Figure 2. The time evolution of the total number of clusters Nc and the total mass fraction inside these clusters φ
during the simulations. Starting from the ∼ 5 Myr more than 80% of the total mass is already concentrated inside
the fragments. At around 6 Myr almost 95% of the total mass is already inside the clusters
for doing this exist [20]. For computational reasons, if the defined hi becomes smaller than the selected minimal
smoothing length hmin, we set the value hi = hmin.
In order to calculate the self-gravitational potential Φi and self-gravitational force −∇iΦi we use the Mitaka
Underground Vineyard (MUV) GRAPE6 computer system at the National Astronomical Observatory of Japan
[http://www.cc.nao.ac.jp/muv/]. For a more detailed description of the GRAPE6 board and for links to
publication about the GRAPE6, we refer the reader to the official homepage of Jun Makino at the Tokyo University
[http://grape.astron.s.u-tokyo.ac.jp/∼makino/grape6.html].
For the time integration of the system of hydrodynamical equations we use the second-order Runge–Kutta–
Fehlberg scheme. The time step Δti for each particle depends on the particle’s acceleration ai and velocity vi,
as well as on the sound speed ci and the heating vs. cooling balance:
Δt = Cn · min
i
[ √
2hi
| ai | ;
hi
| vi | ;
hi
ci
;
ui
u̇i
]
,
where Cn is the Courant’s number and is equal to 0.1. For computational reasons we fix the minimal integration
time step Δtmin.
The main aim of our current research is a detailed study of the isothermal fragmentation processes inside
a collapsing “cold” molecular cloud complex. For the purpose to find the fragments and its physical parame-
ters (mass and size), we use our own “cluster finding” algorithms. In these algorithms we modify the well
known and “standard“ friend-of-friend (FOF) method [11]. In addition to the use of the particle positions in
the process of “constructing” or finding the clusters (fragments) we also involve the information on the density
distribution inside each potential cluster. On the basis of the density distribution analysis we can finally select
in the more accurate way the members of our fragments (clusters). In this sense our method is more close to
the so-called SKID method, which is well described at the homepage of the Washington University “N-body
Shop” [http://www-hpcc.astro.washington.edu/tools/skid.html]. Here the reader can find a more de-
tailed description of this density basic method which is specially designed to find the gravitationally bound
groups of particles in the N-body like simulations.
One of the features of our cluster finding routine is the setting of the minimum limit of gaseous particles
to 5, in order to form a fragment. In other words, we do not count as “real” a cluster where the number of
particles is less than 5.
218
1
10
100
102 103 104 105 106 107
N
U
M
B
E
R
O
F
C
LU
S
T
E
R
S
MASS OF CLUSTERS (MSOL)
ω = 0.1
Time = 5.0 Myr
~ M -0.5
~ M -0.7
1
10
100
102 103 104 105 106 107
N
U
M
B
E
R
O
F
C
LU
S
T
E
R
S
MASS OF CLUSTERS (MSOL)
ω = 0.1
Time = 7.0 Myr
~ M -0.5
~ M -0.7
a) “Slow” initial rotation ω = 0.1, time = 5 Myr b) “Slow” initial rotation ω = 0.1, time = 7 Myr
1
10
100
102 103 104 105 106 107
N
U
M
B
E
R
O
F
C
LU
S
T
E
R
S
MASS OF CLUSTERS (MSOL)
ω = 0.3
Time = 5.0 Myr
~ M -0.5
~ M -0.7
1
10
100
102 103 104 105 106 107
N
U
M
B
E
R
O
F
C
LU
S
T
E
R
S
MASS OF CLUSTERS (MSOL)
ω = 0.3
Time = 7.0 Myr
~ M -0.5
~ M -0.7
c) “Fast” initial rotation ω = 0.3, time = 5 Myr d) “Fast” initial rotation ω = 0.3, time = 7 Myr
Figure 3. Four snapshots of the integrated cluster distribution function (ICMF)
CODE TESTING
The self-gravitating collapse of an initially isothermal “cold” gas sphere has been a common test problem for
different SPH codes [5, 7, 9, 23, 25, 27]. Following these authors, for the testing of our code, we calculate
the adiabatic evolution of the spherically symmetric gas cloud of total mass M and radius R. For the initial
internal energy per unit mass we set u = 0.05
GM
R
. The initial density profile of the cloud we obtain from
ρ(r) =
M
2 π R2
1
r
.
We distribute randomly the gas particles inside a set of spherical shells in a manner that reproduces the initial
density profile. At the start of the simulation the gas particles are at rest. For the presentation of the results
we use a system of units where G = M = R = 1.
In Fig. 1 we show the time evolution of a different types of energy and the relative total energy errors
during the calculation. For comparison of our test results we also plot the energy error results from the serial
variant of the GADGET public access SPH-TREE code [23] with “standard” parameters for the 32 000 particles
[http://www.mpa-garching.mpg.de/gadget/].
219
-1
-0.8
-0.6
-0.4
-0.2
4 4.5 5 5.5 6 6.5 7
S
LO
P
E
TIME (Myr)
ω = 0.3
ω = 0.25
ω = 0.2
ω = 0.15
ω = 0.1
Figure 4. The ICMF slope time evolution for our models with different rotation parameters
During the central bounce around t ≈ 1.1 most of the kinetic energy is converted into heat, and a strong
shock wave travels outward. For all of these runs the number of neighbours NB is assumed to be 50 and
the gravitational softening ε is assumed to be 0.01. For the integration of the system of equations we use
the second-order Runge–Kutta–Fehlberg scheme with a fixed time step Δt = 10−4.
The results presented in Fig. 1 agree very well with those of [23] and [25]. The maximum relative total energy
error is of around 0.05% even for moderate (8 000) particle numbers. The largest adiabatic test calculation (with
64 000 gas particles up to t ≈ 2.2) on an Intel Pentium 4 (3.4 GHz) host machine with the GRAPE6 board took
≈ 3.67 days of total CPU time.
INITIAL CONDITIONS
As an initial condition for our molecular cloud fragmentation study we use a model in which the parameters
are comparable with the largest GMC complexes in our Galaxy [6, 10, 16]. For the mass of the system we set
Mcloud = 107 M�. For the radius of the cloud we set Rcloud = 100 pc. For an initial density distribution we use
the previous formula where ρ(r) ∼ 1/r. For the purpose of checking the possible “resolution” effects we carry
out two sets of runs with 32 000 and 64 000 gas particles (with the corresponding indexes “low” and “high”).
The total gravitational energy of the system in such a case can be easily calculated using the simple formula:
E0
GRA = −2
3
G M2
cloud/Rcloud.
For the initial temperature we set the value, which produced the overall ratio of the thermal energy to
the gravitational energy of the system, at the fixed level: α ≡ E0
THE/ |E0
GRA| = 0.075. For the previous fixed
mass and radius of the system this condition produced an initial temperature of the cloud Tcloud ≈ 2200 K.
The corresponding sound speed was c ≈ 3.8 km/s. This is consistent with the typical measured “kinetic”
temperatures for such GMC complexes [6].
With these parameters we have an initial central concentration n0 ≈ 103 cm−3 and a free-fall time in the cloud
center τff ≈ 1 Myr. The central Jeans radius RJ is ≈ 10 pc and the corresponding Jeans mass MJ is equal
to about 105 M�. Initially, we give the whole system a rigid rotational velocity distribution with an angular
velocity value Ωcloud which we set to the unity of Ω0 = V0/ Rcloud, where
220
V0 ≡
√
G Mcloud/Rcloud.
With the above-mentioned parameters we obtain V0 ≈ 21 km/s.
The main rotational parameters (ω and β) for our two sets of models are listed in Table 1.
Table 1. The list of initial “rotational” parameters in our models
ω ≡ Ωcloud/Ω0 = 0.05 0.10 0.15 0.20 0.25 0.30
β ≡ EKIN/E0
GRA ≈ 6.2 · 10−4 2.5 · 10−3 5.6 · 10−3 1.0 · 10−2 1.5 · 10−2 2.2 · 10−2
As we can see from Table 1, the initial ratio of the rotational (or kinetic) energy of the motion of the frag-
ments (β), even in the last models with a “high” rotational parameter, does not exceed a few percents of
the gravitational bounding energy of the cloud. This value is even less for the initial ratio of the system’s ther-
mal energy to the gravitational energy (α = 7.5%). In all models, usually after the first few Myr of evolution,
these situations have changed. The ratio β is rising to an approximate value of 0.5 or even more. Its mean
is when the cloud starts the process of intensive isothermal fragmentation and the whole system of fragments
becomes almost fully “rotationally” supported.
RESULTS
In Fig. 2 we show the time evolution of the total number of clusters Nc and the total mass fraction inside
these clusters φ during the simulations. Starting from ∼ 5 Myr more than 80% of the total mass is already
concentrated inside the fragments. At around 6 Myr almost 95% of the total mass is already inside the clusters.
Figure 2a shows the results for the “slow” rotating model with ω = 0.1. In Fig. 2b we show the evolution of
the “fast” rotating model with ω = 0.3.
In Fig. 3 we show four different snapshots of the integrated cluster distribution function (ICMF) for two
selected moments of time with two different rotational parameters ω. For practical numerical reasons we use
the ICMF instead of the CMF (which is also called sometimes in the literature as the Differential Cloud Mass
Function – DCMF). Here is a simple definition of the ICMF:
ICMF =
∫ M
0
dN/dM.
Basically, it shows how many clouds we have from zero mass to any fixed mass (M). Because the CMF is
usually approximated with the power law M−γ in this case the ICMF can be simply derived as ∼ M−γ+1.
The reason for using the ICMF instead of the CMF (DCMF) is that the averaging and slope definition is
mathematically better due to the integrated CMF (which is a monotone function) and because the histograms
in this case do not have any “holes”. Of course when we compare our results with the observed (differential)
CMF slope we need to subtract one from the ICMF slope to get the corresponding CMF slope.
In Fig. 3 we can see that in most cases the ICMF slope lies between –0.5 and –0.7 (the corresponding CMF
is –1.5 and –1.7). The models with slow rotation always have a significantly lower value of the slope.
The ICMF slope time evolution for the set of our models with different rotation parameters are presented in
Fig. 4. The models with initial “slow” rotational parameters give the ICMF average slope a level of –0.8 (which
corresponds to the CMF slope of –1.8). The “fast” rotating models give the ICMF slope a level of –0.4 (CMF
slope of –1.4).
On average all models show very close values of the CMF slopes in comparison with the observed values of
γ in the different parts of our Galaxy.
The slow rotation models systematically show the slope more close to the observed values in the outer part
of our Galaxy [6, 10, 12] (γ ≈ −1.8 ± 0.03). In contrast, the fast model CMF slopes is more consistent with
the observations from the central part of our Galaxy [16, 21, 22] γ ≈ −1.5 ± 0.1.
Of course our simulations are time and also resolution limited, but even in this case we can derive a state-
ment about the two significantly different types of “population” in the molecular cloud distributions. The key
parameter which produces the different CMF slopes is the initial rotational parameter of the forming (and
subsequently fragmenting) GMC.
221
CONCLUSIONS
In this paper we present a study of the subsequent (runaway) fragmentation of a rotating isothermal GMC
complex. Our own Smoothed Particle Hydrodynamics (SPH) gas-dynamical model GRAPE’s based, successfully
reproduced the observed Cloud Mass-distribution Function (CMF) in our Galaxy. The steady state of CMF is
quickly established during the collapse approximately on a scale of a few free-fall time in the central parts of
the modelled GMC.
One of the key points in our model is that using our results we can naturally explain the source of possible
differences between the observed slope on molecular clouds mass distribution function in the Galactic center
and the outer regions of our Galaxy.
The basic idea is what if the GMC formed as a result of the galactic disc instability on the scale of the disc
height (∼100 pc). In such a case the initial angular momentum of the forming GMC can be defined by
the Coriolis force during the formation inside the differentially rotating disc. Therefore, the central GMC has
a bigger β and the external GMC has a smaller rotational parameter.
According to our models this produces the different slopes of the resulting CMF during the runaway frag-
mentation process inside the system. The observed CMF gives to the central parts of Galaxy a slope well
approximated with the value γ ≈ −1.5± 0.1 [16, 21, 22] and for the outer parts of the Galaxy the approximate
value γ ≈ −1.8 ± 0.03 [6, 10, 12].
Our results for the “slow” and “fast” rotating models give us exactly the same slopes which have a very good
agreement with the recent observations. The “slow” models corresponds to the initially more slowly rotating
GMC in the outer parts of the Galaxy. The “fast” rotating models corresponds to the GMC in the central part
of the Galaxy. The central GMC can initially get a greater angular momentum from the differential rotation of
the galactic disc during the process of GMC formation itself.
Our numerical investigation clearly shows that one of the key parameters, which determines the observed
slope of the present day molecular CMF in different parts of our Galaxy, is the initial ratio of the rotational
(turbulent) and gravitational energy inside the forming GMC.
Acknowledgements. P. B. wish to express his thanks for the support of his work to the German Science
Foundation (DFG) under the grant SFB-439 (sub-project B5). P. B.’s work was also supported by the following
grants: NNG04GJ48G from NASA, AST-0420920 from NSF and by HST-AR-09519.01-A from STScI.
He is very grateful for the hospitality to the Astronomisches Rechen-Institut (Heidelberg, Germany) where
a part of this work has been done. The research work of the authors was also supported by the Ukrainian State
Fund of Fundamental Investigations within the framework of the project 02.07.00132.
The calculation has been performed with the Mitaka Underground Vineyard (MUV) GRAPE6 system of
the National Astronomical Observatory of Japan. The authors want to express their special thanks to colleague
Naohito Nakasato (Computational Astrophysics Group, RIKEN) for his permanent help and support in handling
of the NAOJ GRAPE6 computational facilities.
The authors are also very grateful to Dan Batcheldor (Astrophysics Group, RIT) for his constructive com-
ments to the first variant of the paper.
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