1. Introduction
Protostellar cloud collisions are expected to occur in a great variety of astrophysical phenomena, for example, as a consequence of collisions between gas-rich galaxies; or due to the presence of strong tidal forces within a single giant cloud. For instance, [1] identified mechanisms they suggested could initiate the formation of agglomerations of matter within the cloud, due to the expansion of regions leading to a compression in other regions of the same cloud. He presented observational evidence of gas compression at the interface of two colliding molecular clouds, and proposed that collisions may be an alternative mechanism to explain the formation of massive stars in galaxies. Indeed, collisions between neutral molecular clouds are frequently mentioned as a possible mechanism for initiating the star formation process. For instance, based on observed carbon molecular lines, [2] suggested that the formation of the stars in the IRAS04000 + 5052 cluster was induced by the collision of two molecular clouds.
The theoretical study of collisions has a long history. It was started in the early 80 s, but even nowadays the academic interest in astrophysical collisions has never fallen off completely, see for instance [3] [4] .
In the old numerical simulations of colliding gas clouds that were done with particle based codes, the clouds partially penetrated each other; and in the case of collisions with high Mach numbers, they passed completely one to another, see [5] . These non-physical results were the consequence of the neglect of artificial viscosity that was later used by [6] .
In simulations based on grid techniques, there were also important limitations, for example, those considering only two spatial dimensions or neglecting self-gravity. In spite of this, the complexity of the phenomenon was uncovered many years ago. For instance, [7] studied head-on collisions with several two clouds combinations, in which one of them represented a giant molecular cloud, and the second one was smaller and denser than the former. They also considered several cases according to the internal density structure of the clouds: some were uniform while others clumpy. In their two dimensional hydrodynamical calculations, the effect of the self-gravity of the gas was unfortunately not taken into account.
Besides, [8] performed two dimensional collision simulations between atomic hydrogen clouds using the AMR (Adaptive Mesh Refinement) numerical technique. In their hydrodynamic equations, they included heating and cooling functions as well as a magnetic pressure term. They considered homogeneous clouds with cylindrical symmetry and surface perturbations, and showed that a filament forms at the interface of the clouds and that the gas flow outwards from the edge of the filament. They estimated that a numerical simulation based on particles with a similar resolution should include at least one million particles. Unfortunately, again their simulations did not include self gravity.
Additionally, [9] also considered collisions between giant spherical clouds within the framework of the Smoothed Particle Hydrodynamics (SPH) Lagrangian technique. The mass of each cloud was 2222M⊙ with a radius of 10 pc. Their collision models with clouds at a temperature of 20 K and with an approaching speed of 10 km∙s−1 produced fragments with a size of 0.1 pc and with a peak density number of 104 cm−3. The conclusion of their calculation was that cloud collisions certainly stimulate the formation of clumpy structures. A drawback of this study is that the authors only used 4096 particles to model such a huge cloud.
There are non-spherical collision models reported in the literature as well, for instance [10] considered two identical, parallel gas slabs, semi-infinite in extent so that each slab could be considered two dimensional, in such a way that the collision takes place in only one spatial dimension.
Although the papers mentioned above are only a representative sample of the vast amount of existing literature, in view of them, we believe it is timely to devote this paper to make new numerical simulations of the collision process, taking advantage of both new capabilities of modern computers and improved numerical techniques.
We now present numerical high resolution three-dimensional (3D) hydrodynamical simulations of spherical cloud collisions, including the self-gravity of the gas. As mentioned earlier, we restrict ourselves to considering only binary collisions between identical clouds.
Our main objective in this paper is to explore the parameter space of the collision models; we have proposed several approaching velocities, simply as sample values chosen only on the basis of the different outcomes that each simulation produces. We also emphasize that along this project we are always interested in studying the effects of the collision on a cloud which is already collapsing. Another subject deserving further study, is the case of clouds starting from hydrodynamical equilibrium.
This was already considered by [4] [11] , who simulated cloud collisions between two clouds initially in hydrodynamical equilibrium (since each cloud was modeled as Bonnor-Ebert sphere). Collisions between dissimilar clouds has also considered by [12] . An important result of our simulations, is that we have explored the possible approaching velocities of the colliding clouds (for an initial cloud with well defined physical characteristics, see Section 2). Because of this, we now know the range of approaching velocities that allow an initial binary colliding clouds system to remain as a binary system capable of undergoing further gravitational collapse and eventually produce a multiple protostellar system. It is in this sense that our results have to be compared with those of [13] , that obtained that it is more likely that a colliding system will in general result in disruption and dispersal of the cloud involved with no chance of forming proto stars.
It must be emphasized that this work hereby devoted to study collisions with clouds with a clear initial tendency to a gravitational collapse prior to collision, has enabled us to uncover subtle collision effects about the formation and fragmentation of the collapsing clumps. To have observed these effects have made an important difference in the results derived from the simulations when we compare with those results reported by [4] [11] [12] .
Besides, during the phase that comes after the collision before the density reaches the critical density ρ ~ 10−18 gr∙cm−3, shearing and Kelvin-Helmholtz instabilities are observed. Fragmentation has been observed in somefilaments, which is a direct consequence of the occurrence of the collision. The integral properties of the resulting gas clumps are calculated with the purpose of estimating if they are likely to collapse, virialize or disperse. The initial conditions for the isolated cloud are such that it will collapse in the absence of a collision as its thermal and rotational energy ratios with respect to the gravitational energy, have been chosen to have initially the numerical values α0 = 0.26 and β0 = 0.16, respectively. However, when we consider the translational kinetic energy that comes from the approaching velocity Vapp of the clouds, the collision system is unbound for Vapp greater than about Mach three. As a result of the collision process a large fraction of the translational kinetic energy is transformed into heat that is radiated away because the clouds and the shock front that forms in the interphase between the clouds are isothermal. This produces are duction of the global value of α + β for all models. For the head-on models we found that they all get a final configuration near equilibrium even for very high Vapp. This is not the case for the oblique models, which have a final value near equilibrium for Vapp less than about Mach three and all b values. The final α + β values get away from equilibrium for Vapp greater than Mach eight and high b values. Hence we only get very high dissipation, and thus the possibility of fragmentation for initially unbound systems for head-on or near head-on collisions.
The observation that collisions favor molecular cloud fragmentation is a very important result in the area of star formation, because each dense gas knot formed along the filament in the central core of the colliding cloud, could eventually form a proto-star.
The outline of the paper is as follows. In Section 2 we describe the characteristics of the particle distribution that represents the initial cloud, which will be involved in all the subsequent collisions. In Section 3 we show the geometry of the collision models and the parameter values chosen for the Gadget2 code. In Section 4, we describe the most important features of the time evolution of our simulations by means of two-dimensional (2D) isodensity plots, despite the fact that the collision events are fully 3D. In Section 5 we discuss the relevance of our results in view of those reported by previous works and finally we make some concluding remarks.
2. The Initial Cloud
We consider a spherical cloud with radius R0 = 3.0138 × 1018 cm = 0.97 pc = 2.0038 × 105 AU and mass M0 = 2.21M⊙. The average density of this cloud is ρ0 = 1.3024 × 10−21 gr/cm3, from which we can estimate the time required by a test particle to reach the center of the cloud from the outermost regions, known as the free fall time of the cloud and given by
(1)
Our initial cloud can be considered as a typical prestellar cloud, though of a very small size such that it can be called a clump according to [14] . It is relatively easy to observe these small clouds as isolated systems in low mass star forming regions such as Taurus [15] . If these clouds have also existed within more dense cloud environments such as the Orioncluster, they would have had multiple dynamical encounters. In the collision events more frequently observed, the involved clouds are very likely to be typical giant molecular clouds with radius ~ 40 pc and mass ~ 105 M⊙.
We have set five million SPH particles to generate the initial cloud by a traditional Monte Carlo scheme, in which the particles are randomly located in the volume space. Let u, v and w be random uniform variables taking real values within the interval [0,1]; then according to the fundamental probability conservation law for a system with spherical symmetry, we have that
. By means of integration we obtain that the spherical coordinates 
of the particles are related to the uniform random variables by the following equations:
(2)
where M0 is the total mass contained in the cloud of radius R0; it must be noticed that the first relation should be numerically integrated to obtain r once that u has taken an allowed uniform random value. The mathematical function f(r) in which we are interested to model the cloud’s radial distribution of matter was first introduced by [16] , and later on studied by [17] as a building block of an analytical model to study the collapse of clouds. This radial density function is given by:
, (3)
where ρc, Rc and 
are three free parameters that fix the shape of the radial density profile.
As it was mentioned by [16] , the main characteristic of this Plummer-like density function is its radial behavior: constant at first and then rapidly falling off with radius. Due to the density behavior, this mathematical model captures quite well the observed fact that prestellar clouds are mainly formed by a strongly centrally condensed low mass core, which is surrounded by a gas envelope steeply declining in density, see [15] .
The Plummer parameters are Rc = 3.0 × 1018 cm and ρc = 1.3025 × 10−21 gr∙cm−3. The initial radius R0 is slightly larger than Rc, because for this paper we are mainly interested in studying the collision of uniform density clouds, see Figure 1. The cutting density ρc is very near the average density ρ0. In an article to come, we will study collisions between centrally condensed clouds, in which Rc < R0. This future study promise to be interesting, as [18] [19] have already shown that the extension of the envelope with respect to the core in a centrally condensed cloud, is also an important dynamical factor playing a crucial role in the fate of the cloud’s collapse.
Additionally, we consider that the cloud is in counterclockwise rigid body rotation around the Z axis; therefore the initial velocity of the i-th SPH particle is given according to
, where Ω0 is the constant magnitude of the initial angular velocity. For the cloud considered in this paper, the initial sound speed and angular velocity have the following values:
(4)
As it is quite common in the literature of gravitational cloud collapse [20] [21] , the dynamical properties of the initial distribution of particles modeling the cloud are usually characterized by means of the thermal and rotational energy ratios with respect to the gravitational energy, denoted by α and β, respectively. The ρ0 and Ω0 values have been calculated for the energy ratios to initially have the following numerical values:
(5)
We recall that these selected values of α and β are known to favor the occurrence of collapse in the cloud.
As done in our previous papers, we also implement a density perturbation on the initial particle distribution, such that at the end of the evolution of the cloud as an isolated system, it might result in the formation of binary systems. This perturbation is applied to the mass of each particle mi according to:
(6)
Figure 1. Upper left panel: Radial density profile for the distribution of SPH particles modeling the initial uniform cloud, as measured from the first Gadget2 snapshot. Upper right panel: Ratio of the hydrodynamical (due to the pressure gradient only) to the gravitational acceleration for several times of the first stage of the cloud’s evolution. Lower left panel: Velocity field of an XZ view of a collision scenario where the velocity of cloud’s particles is seen to point outwards during the initial evolution stage. Lower right panel: The time evolution of the peak density obtained for the collapse of the cloud as an isolated system.
where m0 is the unperturbed mass of the simulation particle, the perturbation amplitude is set to a = 0.1 and the mode is fixed to m = 2. We have found that this mass perturbation is almost irrelevant in the scenario of the cloud collision models.
The characteristics and physical properties of the clouds expressed in Equations (5) and (6) are quite common and we refer the reader to [20] [21] for reviews of the most important results of collapse calculations which have given place to great conceptual advances in the state of the art.
Let us now say something about the way in which we account for the thermodynamics of the gas. Most authors have used an ideal equation of state. As the observed star forming regions basically consist of molecular hydrogen clouds at 10 K with an average density of 1 × 10−20 gr∙cm−3, the ideal equation of state is a good approximation. However, once that gravity has produced a substantial contraction of the cloud, the gas begins to heat. In order to take into account this increase in temperature, we use the barotropic equation of state proposed by [22] . Thus in this paper we carry out all our simulations using the following equation of state:
(7)
where for the molecular hydrogen gas the ratio of specificheats 
because we only consider translational degrees of freedom of the hydrogen molecule. For the critical density we consider only one single value of
gr∙cm−3.
However, a word of warning should be made. By comparing the results of our previous works in [23] [24] with those of [25] for the collapse of an isolated and rigidly rotating cloud with a uniform density profile, we have concluded the barotropic equation of state in general behaves quite well and that it captures all the essential thermo dynamical phases of the collapse. But this does not mean that the same comparison would always be correct for more complex initial conditions or for collision models as those we are considering in this paper.
Despite these facts and that we know that it is indispensable to include all the detailed physics of the thermal transition in order to achieve the correct results, be it in a collision or not, we carry out the present simulations with the barotropic approximation, because we know that there are other computational and physical factors that could have a stronger influence on the outcome of a simulation.
The collision clouds considered for this work have approaching velocities of up to about thirty times the sound speed, see Section 3 and Table 1. For these high velocities a shock is formed in the interface between the clouds and the artificial viscosity used for the calculations transform kinetic into thermal energy that increases the gas temperature. However, as we will now show, the cooling time 
is both much shorter than the local sound crossing time, and the free fall time, hence the isothermal condition can be used for the shock region.
For an initial density of the clouds of 
gr∙cm−3, its particle number density is 
cm−3, where mp is the proton mass and μ the mean molecular weight of the gas, which we take to be 2.4. For a strong adiabatic shock, from the Rankine-Hugoniot jump condition, the density in the thin dissipative region is about four times the pre-shock density, and so we expect the post-shock number density to be about 
cm−3.
We can calculate the cooling time from the expression [26]
(8)
Using our estimate for the post shock region we obtain that
. On the other hand, for our clouds the local sound crossing time 
and the free fall time
. We can then assume that the shock remains isothermal.
3. Collision Models and Computational Considerations
In this paper we use the Gadget 2 code which implements the SPH technique. We use five million simulation particles to model each colliding cloud, as we explain below.
3.1. The Collision Geometry
The simulations considered in this paper include both head-on and oblique collisions, see Figure 2. For the head-oncollisions, the initial position in rectangular coordinates of the center of mass (CM) of each of the colliding clouds is located at 
and
, respectively. We recall the reader that R0 is the initial radius of the cloud.
The average velocity of the center of mass of the clouds points along the Y axis and are 
and
. We define the approaching speed Vapp (with respect to the center of mass of the collid