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C o p y r i g h t 2 0 0 8 : I n s t i t u t o d e A s t r o n o m a , U n i v e r s i d a d N a c i o n a l A u t n o m a d e M x i c o

Revista Mexicana de Astronoma y Astrofsica , 44 , 259284 (2008)

HYDRODYNAMICAL SIMULATIONS OF THE NON-IDEALGRAVITATIONAL COLLAPSE OF A MOLECULAR GAS CLOUD

Guillermo Arreaga-Garca, 1 Julio Saucedo-Morales, 1 Juan Carmona-Lemus, 2 and Ricardo Duarte-Perez 2

Received 2008 January 10; accepted 2008 April 3

RESUMEN

Presentamos los resultados de un conjunto de simulaciones numericas dedi-cadas a estudiar el colapso gravitacional de una nube de gas interestelar, rgidamenterotante, aislada y esfericamente simetrica. Usamos una ecuaci on de estadobarotropica (beos por brevedad) que depende de la densidad de la nube y queincluye una densidad crtica como par ametro libre, crit . Durante el colapso tem-prano, cuando crit , la beos se comporta como una ecuaci on de estado delgas ideal. Para el colapso posterior, cuando crit , la beos incluye un terminoadicional que toma en cuenta el calentamiento del gas debido a la contracci on gravi-tacional. Investigamos la ocurrencia de fragmentaci on rapida en la nube para lo cualusamos cuatro valores diferentes de la crit . Trabajamos con dos tipos de modelosde colapso, de acuerdo con el perl radial inicial de la densidad.

ABSTRACT

In this paper we present the results of a set of numerical simulations aimed tostudy the gravitational collapse of a spherically symmetric, rigidly rotating, isolated,interstellar gas cloud. To account for the thermodynamics of the gas we use abarotropic equation of state ( beos for brevity) that depends on the density of thecloud and includes a critical density as a free parameter, crit . During the earlycollapse, when crit , the beos behaves as an ideal gas equation of state. Forthe late collapse, when crit , the beos includes an additional term that accountsfor the heating of the gas due to gravitational contraction. We investigate theoccurrence of prompt fragmentation of the cloud for which we use four differentvalues of crit . We work with two kinds of collapse models, according to the initialradial density prole: the uniform and the Gaussian clouds.

Key Words: binaries: general hydrodynamics ISM: kinematics and dynamics stars: formation

1. INTRODUCTION

The star formation process begins with a stronggravitational collapse of molecular hydrogen clouds.This process increases the cloud density by sev-eral orders of magnitude, for instance, starting from10 18 g cm 3 and ending at 10 1 g cm 3 , typical of young stellar densities (see Mathieu 1994).

Observational evidence suggests that most youngstars in the Galaxy (around 50%) form in binarysystems. Although with a lower frequency, it is also

1 Centro de Investigaci on en Fsica, Universidad de Sonora,Mexico.

2 Gerencia de Sistemas, Instituto Nacional de Investiga-ciones Nucleares, Mexico.

observed that they may group together in multiplesystems having three or more stars (see Boden 2005).

Both astronomical observations and theoreticalstudies point to the prompt fragmentation of thecloud as the leading mechanism for explaining theorigin and properties of binary stellar systems. Thereader is referred to the review of Bodenheimer etal. (2000), where several proposed theoretical mech-anisms for binary formation are discussed.

The basic idea of prompt fragmentation is thatduring the collapse a molecular cloud may sponta-neously break into two pieces in such a way that theresulting protostellar cores will orbit about one an-other. This idea is likely to be correct if the cores

259

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260 ARREAGA ET AL.

do not undergo further fragmentation upon collaps-ing to higher densities. In fact, for the Taurus darkcloud, a correlation between the masses of the newlyformed stars and the masses of the associated denseprotostellar cores (starless) in the cloud has been ob-served (Myers 1983).

Thus, it is believed that the physical characteris-tics of the protostellar cores or of the fragments arelikely to be inherited by the stars that might resultfrom them, for instance, the ability to form stable bi-nary or multiple systems with typical observed sep-arations (between 1 .0 1011 cm and 1 .0 1016 cm)and orbital periods (ranging from a couple of daysup to 10 , 000 years).

Over the last two decades, a fairly large num-ber of papers have been devoted to numerical stud-ies of the star formation process and particularly tothe collapse of an isolated uniformly rotating molec-

ular hydrogen gas cloud (see the reviews given inSigalotti & Klapp 2001, and Tohline 2002, and ref-erences therein). Earlier papers on collapse werelargely based on low spatial resolution calculations.

However, considering that hydrodynamical owscan be very sensitive to initial conditions and thatthe parameter space for the initial conditions of thecollapse of the cloud is huge, it is not as yet en-tirely clear under what conditions the fragmentationof the cloud results in a binary system of protostel-lar cores. It is for this reason that trusty modelsof the collapse process requiere a fully 3D hydrody-namical code capable of following the dynamics withan adequate resolution in order to capture the pos-sible occurrence of prompt fragmentation. In thisregard we consider that this paper may represent animprovement over earlier works in the eld because:(i) we have obtained an acceptable solution for mod-eling the collapse (Arreaga et al. 2007, where one of us has collaborated in making a convergence studyof the collapse of molecular clouds); (ii) we have fol-lowed the time evolution of the cloud beyond theoccurrence of promt fragmentation, which allows usto recognize whether there have been mergers amongthe fragments; (iii) we have measured some physical

properties of the fragments, from which we can tryto understand, for instance, how the cloud originalspin angular momentum gets transfered into orbitalangular momentum of the resulting fragments; (iv)nally, we have measured how much mass is accretedby the resulting fragments.

In addition to the resolution requirement, thereare other factors that can have a signicant inu-ence on the nature of the outcome of a given simu-

lation, for instance, the initial radial density proleand the thermodynamics of the gas. For the formerfactor, we emphasize that in this paper we considertwo cloud models with distinct initial radial densityproles: a uniform and a Gaussian cloud.

We include in this work the case of a uniform ra-

dial density cloud, because it has been studied verycarefully by several groups since the pioneering cal-culations by Boss & Bodenheimer (1979). The col-lapse of a uniform cloud has been calculated over andover again by using different numerical techniques.This calculation has therefore acquired the status of a common test calculation for convergence testingand inter-code comparison and it has been called inthe literature the standard isothermal test case sim-ulation.

There is good agreement among the differenttechniques in the outcome of the gravitational col-lapse of a uniform cloud: the cloud fragments intotwo well identied protostellar cores. This simula-tion has been so far the most illustrative example of the formation of a binary system.

We particularly refer to the paper by Truelove etal. (1997), which uncovers subtleties of the gravita-tional collapse of the uniform cloud that result frominsufficient resolution in numerical simulations. In-deed, they observed the formation of a lament (abridge connecting the protostellar cores) during anadvanced phase of the collapse only when the res-olution of the calculations was sufficiently high. Itis now agreed that simulations prior to that of Tru-

elove et al. (1997) may be numerically inaccurate(they failed to observe the lament) because theybreached the resolution requirement.

A new generation of simulations of the collapseof the uniform cloud has been carried out since thepaper by Truelove et al. (1997). We refer in partic-ular the work by Kitsionas & Whitworth (2002) andthat by Springel (2005). The resolution effect on theresults of numerical simulations has been widely ver-ied since these two groups (among others) reportedthe appearance of the lament.

We refer the reader to Bodenheimer et al. (2000)

for a review of the most important results in the re-cent history of collapse calculations which have givenplace to great conceptual advances in the state of theart; a very interesting comparison of results obtainedwith codes based on different numerical techniquesis also provided in that paper.

In this work we have not found any differencewith regard to the recent literature concerning thecollapse of the uniform density cloud. This fact al-

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HYDRODYNAMICAL SIMULATIONS OF A MOLECULAR GAS CLOUD 261

lows us to trust our simulations. It is important toemphasize this point because as far as we know, nodenitive solution for the collapse of the Gaussiancloud has so far been calculated.

Most calculations of the evolution of a Gaussiancloud have been done with mesh-based codes, such

as the Finite Differences (FD) and Adaptive MeshRenement(AMR) techniques. The AMR algorithmcreates a nested hierarchy of ever ner grids in uidregions where high resolution is needed (see Balsara2001 for a review).

In this paper we use the fully parallelizedGADGET-2 code which implements the SPH(smooth particle hydrodynamics) technique. This isa Lagrangian technique in which a nite number of particles is used to sample the continuous uid (seeMonaghan 1997, 2005 for a review).

In this work we use 1 , 200, 000 simulation par-

ticles to model the collapse of the uniform cloudand 5 , 000, 000 for the Gaussian cloud. Despite thefact that with these numbers of particles we achieveenough resolution for both cloud models, it is impor-tant to mention that for the Gaussian cloud modela comparison of our results with the literature be-comes a little more complicated.

Finally, let us say something about the thermo-dynamics of the gas. Most authors in this eld haveused an ideal equation of state to account for thethermodynamics of the gas. According to astronom-ical observations, star forming regions basically con-sist of molecular hydrogen clouds at 10 K. Therefore,the ideal equation of state is a good approximationfor this case. However, once gravity has produceda substantial contraction of the cloud, the densityreaches intermediate values, and the gas begins toheat up. In order to take this increase in tempera-ture into account in our simulations, we have madeuse of a barotropic equation of state beos , as wasproposed by Boss et al. (2000).

For instance, in the ideal gas case, the instanta-neous speed of sound c0 is constant and is equal toits average value. In the beos case, the instantaneousspeed of sound is no longer a constant but increases

with the density of the cloud. However, it is stillpossible to dene an approximate average speed of sound p/ by c20 , where is the effective adiabaticexponent of the gas (see 2.3).

It should be emphasized that in order to describecorrectly the transition from the ideal to the adi-abatic regime, one will require solving the radiativetransfer problem coupled to a fully self-consistent en-ergy equation to obtain a precise knowledge of the

dependence of temperature on density. The imple-mentation of radiative transfer has already been in-cluded in some mesh-based codes. For particle-basedcodes we are only aware of the work by Whitehouse& Bate (2006), in which they studied the collapseof molecular cloud cores with an SPH code that in-

cludes radiative transfer in the ux-limited diffusionapproximation. These authors showed that there areimportant differences in the temperature evolutionof the cloud when radiative transfer is taken into ac-count.

However, after comparing the results of our sim-ulations with those of Whitehouse & Bate (2006)for the uniform density cloud, we conclude that thebarotropic equation of state behaves in general quitewell and that we can capture the essential dynami-cal behavior of the collapse. Despite of the fact thatis indispensable to include all the detailed physicsof the thermal transition in order to achieve thecorrect results of the collapse, we carry out thesesimulations because we know that there are othercomputational and physical factors that can havea stronger inuence on the outcomes of the simu-lations, for instance, the total number of particles,the gravitational smoothing length, the initial con-ditions, among others; factors which have been morecarefully taken into account in this paper.

The outline of this paper is as follows. In 2we explain in detail all the initial characteristics of the particle distribution that we have implemented,among others, the initial mass perturbations, and

the initial energies. We also show the parameters val-ues chosen for the GADGET-2 code. In 3, we de-scribe the most important characteristic of the timeevolution of our simulations. We also report someof the physical properties of the resulting fragments.In 4 we discuss the relevance of our results in viewof those reported by previous investigations.

2. INITIAL CONDITIONS OF THE COLLAPSEMODELS

In this section we focus on the simulation pro-cess of gravitational collapse for both a uniform anda Gaussian (centrally condensed) clouds, see Table 1.In both cases we start with a rigidly rotating spher-ical cloud with radius 3 R0 = 4 .9906939 1016 cmand mass M 0 = 1.989 1033 gr. The average densityof the cloud is 0 = M 0 / (4/ 3 R 30 ) 3.82 10

18

g cm 3 . The fact that the average density is the

3 Equivalent to R 0 = 0 .016 parsecs or R 0 = 3292 AU. Themass M 0 is equal to one solar mass.

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262 ARREAGA ET AL.

TABLE 1

RESULTS OF COLLAPSE MODELS AND PROTOSTELLARCORE SYSTEMS

Model Number of Particles crit Final Outcome(millions) (g cm 3 )

Uniform clouds

UA 1.2 5.0 10 14 BinaryUB 1.2 5.0 10 13 TripleUC 1.2 5.0 10 12 BinaryUD 1.2 5.0 10 11 Binary

Gaussian clouds

GA 5.0 5.0 10 14 BinaryGB 5.0 5.0 10 13 TripleGC 5.0 5.0 10 12 BinaryGD 5.0 5.0 10 11 Binary

same for both clouds will allow us to use 0 to nor-malize the instantaneous density of the cloud duringits gravitational contraction in the plots that will bepresented.

2.1. The initial radial density prole

For the uniform cloud we follow the initial con-ditions used in Burkert & Bodenheimer (1993). Theradial density proles in which we are interested havethe following mathematical forms:

(r ) = 0 , Uniform Cloud

(r ) = c exp( r 2 /R 2c ) , Gaussian Cloud(1)

where c = 1 .7 10 17 g cm 3 is the chosen central

density; R c = 0.5777613699R0 .A very important point that must be emphasized

is the following: (i) in Boss (1991); Boss et al. (2000)made the choice of c and Rc in such a way that theradial density prole started at (r )/ 0 = 20 at r = 0and ended at 0 at r = R0 . We have abandonedthis choice in this paper; instead, our initial density

prole is almost 5 times greater at the center of thecloud than at its outermost region, see Figure 1.By means of a rectangular mesh we make the

partition of the simulation volume in small ele-ments, each with a volume x y z; at the cen-ter of each volume we place a particle the i-th ,say with a mass determined by its location ac-cording to the density prole being considered, thatis: mi = (x i , yi , zi ) x y z with i = 1 ,...,N .

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

/

0

r/R 0

U

G

Fig. 1. Radial density prole for initial distributionof particles for both models U (Uniform cloud) and G(Gaussian cloud): The continuous curve indicates the

analytical proles and the curve with crosses indicatesthe measured proles. We normalize the density withthe cloud average initial density 0 = 3 .82 10 18 gcm 3 and distance with the cloud initial radius R0 =4.9906939 1016 cm.

Next, we displace each particle from its location adistance of the order x/ 4.0 in a random spatial

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direction. This is how we have obtained a set of particles reproducing the density proles proposedby equation (1) for the initial conguration of thecloud. In fact, Figure 1 presents the initial prolesas they were numerically measured.

2.2. Initial Energies

The approximate total gravitational potential en-ergy of our initial cloud of mass M 0 and radius R0is:

E grav 35

G M 20R0

, (2)

where G is Newtons universal gravitation constant.For an ideal gas, the average speed of sound is givenby

p

c20 = k T

, (3)

where is the hydrogen molecular weight; T its equi-librium temperature and k is the Boltzmann con-stant. The average total energy E therm ( kinetic pluspotential interaction terms of the molecules) is givenby

E therm 32 N k T =

32

M 0 c20 , (4)

where N is the total number of molecules in the gas.The rotational energy of the cloud is approximatelygiven by:

E rot = 12

I 20 = 12

J 2

I

15

M 0 R20 20 , (5)

where I 25 M 0 R20 is the moment of inertia; J =

I 0 the total angular momentum of the cloud and0 its angular velocity.

According to the literature, the dynamical prop-erties of the initial distribution of particles are com-monly characterized by means of the thermal androtational energy ratios with respect to the gravita-tional energy, and , respectively. For the modelsconsidered in this paper, the values of c0 and 0 arechosen (see equation 10 and equation 14) in such away that the energy ratios are initially given the fol-lowing numerical values:

E therm

|E grav | = 0.26055

E rot|E grav |

= 0 .16134. (6)

Now, following the virial theorem, if the hydrogencloud is in thermodynamical equilibrium, then thethree previous energies satisfy the following relation:

2 (E therm + E rot ) + E grav = 0 , (7)

or, in an equivalent form,

+ = 12

, (8)

a relation that will be used in the resulting plots.

2.3. Equation of State In this paper we use the barotropic equation of

state beos proposed by Boss et al. (2000):

p = c20 1 + crit

1

, (9)

where = 5 / 3 because we only consider the transla-tional degrees of freedom of the molecular hydrogen.c0 is the initial sound speed, whose value dependsupon the initial ratio of the kinetic to the gravita-tional energy of the model under consideration. The

values for c0 appropriate for the given in equation 6of 2.2 are:

c0 = 16647 .83 cm s 1 Uniform Cloud

c0 = 19013 .16 cm s 1 Gaussian Cloud . (10)

The beos given in equation (9) depends on a sin-gle free parameter: the critical density crit . Forthe early phases of the collapse, when the maximumdensity of the cloud is much lower than the criti-cal density, max crit , the beos becomes an idealequation of state, with p and the instantaneousspeed of sound is equal to its average value according

to: dpd

= p

= c20 . (11)

For the late phases of the collapse, when max crit , there is an increase in pressure according to p 3 / 2 ; the relation given in equation (11) is nowonly approximately valid, for in this case we have:

dpd

p

c20 . (12)

We note that this thermodynamical change triesto capture the heating of the gas when the gravita-

tional contraction is signicant.In order to study the effect of the change of reg-imen from low to high pressure on the outcome of the simulation we consider four values of the criticaldensity, see Table 1.

2.4. Initial Velocities

Additionally, we consider that the cloud is incounterclockwise rigid body rotation around the z

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264 ARREAGA ET AL.

axis; therefore the initial velocity of the i-th particleis given according to the following equation:

vi = 0 ri = ( 0 yi , 0 x i , 0), (13)

where 0 , the magnitude of the angular velocity, has

been chosen according to the model under consider-ation to satisfy the value of given in equation (6),see 2.2. Thus, we have used the following two val-ues:

0 = 7.2 10 13 rad s

1 Uniform Cloud0 = 1.0 10

12 rad s 1 Gaussian Cloud . (14)

2.5. Initial Mass Perturbations

As we are interested in the formation process of binary systems of protostellar cores, we implement aspectrum of density perturbations on the initial par-ticle distribution, such that at the end of the sim-ulation it might result in the formation of binarysystems. This perturbation is applied to the massof each particle m i regardless of the cloud model ac-cording to:

m i = m0 + m0 a cos(m i ) , (15)

where m0 is the unperturbed mass of the simulationparticle, the perturbation amplitude is set to a = 0 .1and the mode is xed to m = 2.

2.6. Evolution Code

We have carried out the time evolution of theinitial distribution of particles with the parallel codeGADGET-2, which is described in detail in Springel(2005). GADGET-2 is based on the tree-PM methodfor computing the gravitational forces and on thestandard SPH method for solving the Euler equa-tions of hydrodynamics.

GADGET-2 incorporates the following standardfeatures:(i) each particle i has its own smoothinglength hi ; (ii) the particles are also allowed to haveindividual gravitational softening lengths i , whosevalues are adjusted such that for every time step i h iis of order unity.

Following Gabbasov et al. (2006), some empir-ical formulas are known to assign a value to i inorder to minimize errors in the calculation of thegravitational force on a particle i of mass m i . How-ever, GADGET-2 xes the value of i for each time-step using the minimum value of the smoothinglength of all particles, that is, if hmin = min (h i )for i = 1, 2...N , then i = hmin .

In order to move the particles forward in time acomplete time step t = tn +1 tn , GADGET-2 usesa leapfrog algorithm.

The GADGET-2 code that we have used in thispaper has implemented a Monaghan-Balasara formfor the articial viscosity (Monaghan & Gingold

1983; Balsara 1995). The strength of the viscosityis regulated by setting the parameter = 0 .75 and = 1/ 2 , see equation (14) in Springel (2005).We have xed the Courant factor to 0 .1.

2.7. Resolution

Following the work in Truelove et al. (1997), theresolution requirement for a simulation to avoid thegrowth of numerical perturbations is expressed interms of the Jeans wavelength J , which is givenby:

J =

c2

G . (16)

To obtain a form more useful for a particle basedcode, the resolution requirement length J is writtenin terms of the spherical Jeans mass M J , which isdened by

M J 43

J 2

3

=

52

6c3

G3 . (17)

Let us suppose that we are able to follow the col-lapse of the cloud until a density of (more or less)3.0 10 9 g cm 3 has been reached. Then the min-imum Jeans mass that we expect according to equa-tion (17) is approximately given by ( M J /M 0 )min =2.52 10 4 .

However, it is known that this Jeans requirementis a necessary condition to avoid the occurrence of ar-ticial fragmentation. It is less clear whether this re-quirement is also a sufficient condition to guaranteethe correctness of a given simulation. For instance,Bate & Burkert (1997) show that for particle basedcodes there is also a mass limit resolution criterion which needs to be fullled besides that of Truelove etal. (1997). They show that an SPH code producesthe correct results involving self-gravity as long as

the minimum resolvable mass is always less that aJeans mass, M J . Following Bate & Burkert (1997),the smallest mass that a SPH calculation can resolve,m r , is equal to m r 2N neigh M J where N neigh is thenumber of neighboring particles included in the SPHkernel.

Therefore, the smallest mass particle m p in oursimulations must at least be such that m p p/m r < 1,where m r M J / 80, which gives us m r /M 0 = 3 .15

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10 6 . For the uniform cloud model, all simulationparticles have the same mass m p; for the simulationwith 1 , 200, 000 particles, the ratio of the particlemass to the total mass is m p/M 0 = 8 .3 10

7 . Com-paring these mass ratios we obtain m p/m r = 0 .26.For the simulation with 5 , 000, 000 particles we ob-

tain m p/m r = 0 .063. We conclude that all the sim-ulation in this paper satisfy both resolution require-ments, that of Truelove et al. (1997) and that of Bate& Burkert (1997).

It was demonstrated by Nelson (2006) that for3D-SPH simulations to accurately resolve the cir-cumstellar discs there is a resolution criterion thatmust be fullled. This criterion establishes that forthe vertical structure of the disc to be well resolvedthe scale-height H must be greater than 4 particlesmoothing lengths h at the disc mid-plane.

As was the case for the Jeans criterion, for a

SPH simulation this criterion is better expressed interms of mass rather than in terms of length; Nel-son then denes the Toomre mass, which is givenby M T = ( T / 2)

2 = c4s /G 2 , where is thesurface density of the disc of matter; T is the wave-length which denes the stability of the disc; cs is thesound speed and G is Newtons gravitational univer-sal constant.

We can make a rough estimate of the Toomremass for our simulations by making use of the fol-lowing approximation. Let us divide the disc heightH in slices of small z. We calculate the average

volume density av for every slice and then the rela-tion between surface and volume densities for ev-ery slice is = av z. If we consider thatH for the Gaussian cloud models is approximatelygiven by H = 0.02R 0 where R0 is the initial ra-dius of the cloud, then the Toomre mass is givenby M T /M 0 = 6 .0 10

5 . Now, following Nelson(2006), the smallest mass resolution m r for a sim-ulation to prevent the appearance of numerically in-duced fragmentation of the disc must be given by6 times the average number of neighboring particlesconsidered. Therefore, for our case we have thatm r /M 0 (M T /M 0 )/ (6 40) = 2 .5 10

7 . On the

other hand, as we have seen earlier the mass of theSPH particle for the simulation of 5 , 000, 000 parti-cles is m p/M 0 = 2.0 10

7 , therefore the mass ratiowhich is directly related with the resolution criterionis given by m p/m r = 0.8. We conclude that at leastfor this rough estimate our simulations have enoughmass resolution also for the disc. Obviously, thisclaim must be taken with caution until a completestudy can be properly undertaken.

3. RESULTSIn order to study the results of the simulations

in this paper, we plot iso-density curves for a sliceof particles around the equatorial plane of the cloud.A bar located at the bottom of these plots showsthe range of values for the log of the density (t)at time t normalized to the average initial density(that is log 10 (/ 0 )) and the color allocation set bythe program pvwave version 8. According to thiscolor scale, yellow indicates the areas with higherdensities, whereas blue indicates those with lowerdensities; green and orange indicate the intermediatedensity areas.

There is a characteristic time scale relevant 4 tothe time evolution of the cloud: the free fall timetff (the time for a particle to reach the center of thecloud):

tff

3

32 G 0= 1 .07488 1012 s 3.4 104 yr

(18)We take advantage of the fact that the free fall timeis the same for both the uniform and the Gaussiancloud in order to normalize the time evolution withtff .

The mass of the initial cloud is much greater thanthe Jeans mass, M 0 M J , for the collapse to be-gin. Numerical simulations performed so far haveproved that this rotating molecular cloud contractsto an almost at conguration in less than a freefall time tff of dynamical evolution. It is less clearunder what conditions this at conguration frag-ments. Furthermore, in case this fragmentation oc-curs, how likely is it that the outcome will turn outto be a binary or a multiple system? In the caseof a multiple system, it is not known which are themost relevant dynamical properties of the resultingfragments, for instance, the mass, separation, orbitalperiods and angular momenta. These are the kindof questions we look forward to consider in the fol-lowing sections of this paper.

The main results of this paper are therefore con-tained both in the isodensity plots and in the plots of integral properties of the resulting fragments wherewe show the time evolution of some physical prop-erties of the protostellar cores. In order to calculatethe physical properties of fragments we have intro-duced a numerical denition, which is given in theAppendix A.

4 The orbital period of rotation around the axis of symme-try, t rot , is not relevant because it is very large 2 / 0 5.8tff compared with tff and t s . So, the important features of thetime evolution occur before a complete orbital revolution isreached.

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266 ARREAGA ET AL.

Fig. 2. Isodensity curves of the cloud midplane for model U A when the distribution of particles reaches a peak densityof (a)max = 5 .314 10 14 g cm 3 at time t/t ff = 1 .2626; (b) max = 7 .364 10 13 g cm 3 at time t/t ff = 1 .2897;(c) max = 3 .502 10 12 g cm 3 at time t/t ff = 1 .3212; (d) max = 5 .110 10 12 g cm 3 at time t/t ff = 1 .3437;(e) max = 7 .440 10 12 g cm 3 at time t/t ff = 1 .3549 and (f) max = 1 .361 10 11 g cm 3 at time t/t ff = 1 .3954.

3.1. The Uniform Cloud Models

In this section we show the main results for theuniform models: For model U A see Figures 2 and 3;for model UB see Figures 4 and 5; for model UC see Figures 6 and 7 and nally for model UD seeFigures 8 and 9. Below we give a more detailed ex-planation of each model.

Due to the fact that the centrifugal force alongthe equator of the cloud is grater than at the poles,the contraction of the cloud is faster along the ro-tation axis, and the cloud starts evolving through asequence of atter congurations.

When the evolution time is almost a free fall time,the cloud has lost its initial spherical symmetry, be-cause most of the particles have found a place in anarrow slice of matter around the equatorial plane.From the point of view of the rotation axis, the accre-tion of particles takes place with cylindrical symme-try. During this rst stage of gravitational contrac-tion, the cloud has reduced its dimensions to 10% of its original size.

Later on, when the attening of the cloud is ex-treme, the central part becomes slightly stretchedtaking the form of a prolate ellipsoid (Figure 6a andFigure 6b). As long as the cloud continues rotat-

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0

0.2

0.4

0.6

0.8

1

1.28 1.32 1.36 1.4

t/t ff

UA,Frag.1UA,Frag.2

0

0.2

0.4

0.6

0.8

1

1.2

1.28 1.32 1.36 1.4

t/tff

UA,Frag.1UA,Frag.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

UA,Frag.1UA,Frag.2

virial line

0

0.02

0.04

0.06

0.08

0.1

0.12

1.28 1.32 1.36 1.4

M / M

0

t/t ff

UA,Frag.1

UA,Frag.2

Fig. 3. Integral properties for the fragments found in models U A.

ing, the central part will adopt a more elongatedand slightly curved conguration. We then noticethe appearance of two small overdense cores in eachextreme of the prolate central region (Figure 2a andFigure 4a). Shortly after, we note that these smallcores get connected by a very well dened bridgeof particles. At that moment, it is adequate to de-ne these small overdensity cores as protostellar frag-ments.

Up to this point, the evolution of all uniformmodels proceeds in an identical fashion. What hap-

pens to the bridge of particles connecting the frag-ments is what makes the difference in the subsequentevolution of the uniform models; that is, the bridgewill disappear or will become a thin lament.

For instance, in the UA model the change inthe thermodynamics regime occurs earlier than inthe other models, when max > crit = 5 .0 10

14

g cm 3 , the pressure of the barotropic equation of state ( p 3 / 2max ) is greater than the pressure in theideal equation of state ( p max ) for a given density.The increase in pressure slows down the gravitational

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0.40 0.20 0.00 0.20 0.400.40

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0.00 1.19 2.38 3.57 4.76 5.95 7.14 8.09

a b c

d e f

g h i

Log / 0

Fig. 4. Isodensity curves of the cloud mid-plane for model UB when the distribution of particles reaches a peakdensity of (a) max = 4 .3640757 10 16 g cm 3 at time t/t ff = 1 .084881; (b) max = 1 .7800040 10 15 g cm 3 at timet/t ff = 1 .129897; (c) max = 6 .4807202 10 13 g cm 3 at time t/t ff = 1 .260443; (d) max = 5 .1288698 10 11 g cm 3

at time t/t ff = 1 .287452; (e) max = 4 .4799 10 10 g cm 3 at time t/t ff = 1 .318963; (f)max = 5 .7860 10 10 g cm 3

at time t/t ff = 1 .3233; (g)max = 6 .8657612 10 10 g cm 3 at time t/t ff = 1 .334179; (h) max = 7 .1663927 10 10 gcm 3 at time t/t ff = 1 .334179 and (i) max = 7 .4082296 10 10 g cm 3 at time t/t ff = 1 .340301.

collapse of both fragments. In spite of this, the frag-ments continue accreting matter from the surround-ings, although at a smaller rate.

It is noticeable that the bridge of particles be-gins to disappear because the fragments accrete theparticles of the bridge as well. Inuenced by puregravitational attraction among the fragments, theyget closer and closer together, even to the point of touching, but they do not merge, continuing their

individual trajectories until nally they enter intoorbit with one another, see Figure 2f.

We observe that the U A model results in a stablebinary system of protostellar fragments. Besides, inFigure 3 it is clearly observed that these fragmentshave a tendency to virialize. These gures suggestthat the subsequent virialization of the fragments isbasically accomplished by means of the rotationalenergy ( 1/ 2).

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t/tff

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0

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1

0 0.050.1 0.150.2 0.25 0.30.35 0.40.45 0.5

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0

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0.04

0.05

0.06

0.07

0.08

1.26 1.27 1.28 1.29 1.3 1.31 1.32

M / M

0

t/t ff

Frags

Fig. 5. Integral properties for the fragments found in models U B.

For the uniform cloud models when the change inthe equation of state occurs later, for instance, mod-els U C and U D, the nal conguration is a very thin

lament that connects two small overdensity cores,see Figures 6 and 8. For both U C and U D models,the cores also present a lengthening in such a waythat they become aligned with the lament.

To be sure that there are no articial effects pos-sibly caused by extrapolation errors when the isoden-sity curves are generated with the pvwave program,we include in Figure 10 the conguration of particles(a dot represents a single simulations particle).

For model U C , cylindrical symmetry can be ap-preciated for the cores, as well as the smooth con-nection of the cores and the lament through the

formation of small spiral arms. We never observefragmentation, neither in the cores nor in the la-ment.

However, it is possible that for model UD somebreakage is already taking place in the region be-tween the lament and the cores (see Figure 10b).

Finally, we show in Figure 11 the time evolutionof the maximum density for each uniform model. We

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0.2 0.0 0.2

0.2

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0.2 0.1 0.0 0.1 0.20.2

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0.2

0.16 0.08 0.00 0.08 0.160.16

0.08

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0.08

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0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.50

a b c

d e f

Log / 0Fig. 6. Isodensity curves of the cloud mid-plane for model U C when the distribution of particles reaches a peak densityof (a) max = 9 .80 10 16 g cm 3 at time t/t ff = 1 .1146; (b) max = 9 .37 10 15 g cm 3 at time t/t ff = 1 .2460;(c) max = 4 .19 10 13 g cm 3 at time t/t ff = 1 .2658; (d)max = 6 .93 10 12 g cm 3 at time t/t ff = 1 .2694;(e) max = 6 .74 10 10 g cm 3 at time t/t ff = 1 .2748 and (f) max = 5 .96 10 09 g cm 3 at time t/t ff = 1 .2910.

observe that the maximum density for model UAreaches its peak value and presents a tendency toremain around that maximum value; this behaviorsuggests a deceleration of the collapse of the cloud.For models U C and U D, it is evident that the cloud

keeps collapsing without decelerating even at themost advanced stage of evolution with densities ashigh as 5.0 10 7 g cm 3 .

3.2. The Gaussian Cloud Models

In this section we show the main results for theGaussian models: For model GA see Figures 12 and13; for model UB see Figures 14 and 15; for modelUC see Figures 16 and 17 and nally for model U D

see Figures 18 and 19. Below we give a more detailedexplanation of each model.

In the Gaussian models we have also introducedan initial perturbations density prole that induces

the formation of binary systems, see equation (15).However, for the Gaussian cloud this perturbationis less inuential on the subsequent development of the collapse than for the case of the uniform cloud.Due to the exponentially falling radial density proleof the Gaussian models, the density of the centralregion is almost 5 times higher than in the externalregions of the cloud; for this reason the collapse takesplace rst in the central regions and, subsequently,

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0.5

1

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UC,Frag.1UC,Frag.2

0

1

2

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4

5

1.24 1.25 1.26 1.27 1.28 1.29

t/t ff

UC,Frag.1UC,Frag.2

0

0.2

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0.8

1

0 0.5 1 1.5 2 2.5 3

UC,Frag.1UC,Frag.2

virial line

0.01

0.02

0.03

0.04

1.24 1.25 1.26 1.27 1.28 1.29

M / M

0

t/t ff

UC,Frag.1UC,Frag.2

Fig. 7. Integral properties for the fragments found in models U C .

the collapse continues by accreting particles of theoutermost regions.

It can be appreciated from the axis of rotationthat the cloud conserves a circular shape (cylindri-cal symmetry) during the rst stage of the collapse.Keeping this axial symmetry, the maximum densityin the center of the cloud can reach values that areeven higher than max 1.0 10

13 g cm 3 , whichare observed at time t/t ff 0.67, while its spatial di-mensions have already been reduced by 80% for allGaussian models.

Shortly after, the central clump of the cloud be-gins to rapidly change its geometry acquiring thecharacteristic elongated structure of bars (see for in-stance Figure 12a and Figure 14a).

For the Gaussian cloud, we must compare theoutput of our simulations with the work by Boss etal. (2000), in which they considered the collapse of a Gaussian cloud, but using an Adaptive Mesh Re-nement (AMR) technique with a very high spatialresolution.

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0.40 0.20 0.00 0.20 0.40

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0.15 0.08 0.00 0.08 0.150.15

0.08

0.00

0.08

0.15

0.00 1.37 2.74 4.11 5.48 6.84 8.21 9.31

a b c

d e f

Log / 0Fig. 8. Isodensity curves of the cloud mid-plane for model UD when the distribution of particles reaches a peakdensity of (a) max = 3 .5817632 10 16 g cm 3 at time t/t ff = 1 .08038; (b) max = 2 .9998504 10 15 g cm 3 at timet/t ff = 1 .215427; (c) max = 6 .9658080 10 11 g cm 3 at time t/t ff = 1 .260443; (d) max = 2 .2228403 10 07 g cm 3 attime t/t ff = 1 .271697; (e) max = 2 .8109875 10 07 g cm 3 at time t/t ff = 1 .273948 and (f) max = 3 .0019162 10 07 gcm 3 at time t/t ff = 1 .276198.

Additionally, long spiral arms form around thecentral bar by the effect of cloud rotation. In Fig-ure 12b we show the extension of the spiral armswith regard to the size of the bar. It is noteworthy

that Figure 12a agrees well with Figure 5a in Bosset al. (2000), even though the extension of the spiralarms in their paper cannot be clearly appreciated.

Up to this time, the evolution of all the Gaus-sian models are practically the same. The subse-quent evolution of the central bar is what makes themost signicant difference in the dynamical evolu-tion among these models.

For instance, in model GA, in which thebarotropic thermodynamics enters early in the col-lapse, we observe the notable occurrence of the decayand the subsequent fragmentation of the central bar.

In fact, the bar in this model breaks up at its centerproducing two fragments. In Figure 12 we show iso-density curves where the initial formation stage of two well dened clumps located on the extremes of the bar can be appreciated; we make this snapshotwhen the cloud has reached a maximum density of max = 4 .2 10

12 g cm 3 . These clumps will pro-voke shortly the fragmentation of the bar. Our Fig-ure 12 can indeed be compared with Figures 3, 5a

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0.18

0.185

0.19

0.195

0.2

0.205 0.21

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1.2708 1.2738 1.2768

t/t ff

Frags.

0.26

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1.2708 1.2738 1.2768

t/tff

Frags.

0.16

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0.24

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0.3

0.2 0.25 0.3 0.35 0.4

Frags.virial line

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

0.021 0.022

1.2708 1.2738 1.2768

M / M

0

t/t ff

Frags

Fig. 9. Integral properties for the fragments found in models U D.

and 5b of Boss et al. (2000), in which they used iso-density curves to illustrate the geometry of the bar inits rotational movement (see their Figures 2 and 5a).

In our case, the formation of the two clumps at theextremes of the bar can also be perfectly appreciatedand compared with their Figure 5b.

It should be noticed that the two exterior frag-ments resulting from the breakage of the spiral armsare already present at the time when the fragmen-tation of the bar occurs. Then, at this time, theresult of the simulation is four fragments. Shortly

thereafter, the occurrence of merging between twofragments leaves only two nal fragments that gointo orbit until they reach virial equilibrium (see Fig-

ure 13).We should emphasize that except for the forma-

tion of the bar, the dynamical evolution we observe isdifferent from the one shown in Figure 5d of Boss etal. (2000). These authors reported that the bar de-cayed rapidly into a single central clump surroundedby long spiral arms. They did not mention anythingabout the breakage of the long spiral arms that wehave observed.

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-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

y / R 0

x/R0

a

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

y / R 0

x/R0

b

Fig. 10. Position of particles lying within a slice of thickness z = 5 .0 10 4 about the equator of the cloud for Models(a) U C for maximum density 3 .60 10

9g cm

3at time 1 .286tff and (b) U D for maximum density 2 .8596622 10

07

g cm 3 reached at 1 .276649t ff .

Boss et al. (2000) considered the value crit =3.16 10 12 g cm 3 , which is slightly smaller thatthe value we used in the model GC and is greaterthan the value we used in model GD . They observethat the Gaussian cloud fragments into a binary pro-tostar system, but these binary clumps soon there-after evolve into a central clump surrounded by spiralarms containing two more clumps. What we observein model GC is that the central bar indeed fragments

into two clumps; but during the time that we havefollowed the collapse we do not observe the mergingof these clumps, neither in model GC nor in modelGD . It is more likely that merging will occur inmodel GD than in model GC . Had we observed themerging of these fragments, we would have claimedthat our simulation agreed with that of Boss et al.(2000) despite the fact that our initial density prolewas not the same.

As was the case for the run GA, in model GBthe central region starts deforming smoothly and itquickly adopts the elongated shape characteristic of the bars. The bar rotation provokes the appearanceof spiral arms. The arms are short-sized at the be-ginning; but they stretch out as the rotation speedof the central bar increases. As was previously thecase, these spiral arms break up, separating from thecentral fragment, and produce a couple of exteriorfragments (see Figure 14).

The bar continues to deform by the effect of tidalforces until at some point it begins to return to

0

2

4

6

8

10

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

l o g 1 0

( m a x /

0

)

t/t ff

UAUBUCUD

Fig. 11. Time evolution of the peak density for uniformmodels.

the axisymmetric (circular) conguration, presum-ably increasing its rotation speed. As a consequence,new spiral arms develop around the central fragment(see Figure 14). Again, when the difference in therotation speed is high enough, a new breakage of the younger spiral arms occurs. We have therefore amultiple system of 4-well dened fragments orbitingamong themselves. However, we never observe the

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0.10 0.05 0.00 0.05 0.100.10

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0.150 0.075 0.000 0.075 0.1500.150

0.075

0.000

0.075

0.150

0.00 0.93 1.86 2.78 3.71 4.64 5.57 6.31

a b c

d e f

Log / 0Fig. 12. Isodensity curves of the cloud mid-plane for model GA when the distribution of particles reaches a peakdensity of (a) max = 1 .7959198 10 12 g cm 3 at time t/t ff = 0 .747263; (b) max = 3 .3803476 10 12 g cm 3 at timet/t ff = 0 .765269; (c) max = 3 .6623149 10 12 g cm 3 at time t/t ff = 0 .792278; (d) max = 8 .1708079 10 12 g cm 3 attime t/t ff = 0 .819288; (e) max = 1 .6487187 10 11 g cm 3 at time t/t ff = 0 .864304 and (f) max = 2 .7573437 10 11 gcm 3 at time t/t ff = 0 .927326.

fragmentation of the central bar as was the case inmodel GA.

It is important to point out that the external

fragments rapidly accumulate mass, which mostlycomes from the remainders of the spiral arms; forinstance, Figures 13 and 15 illustrate this accretionprocess. The resulting fragments do not virializecompletely, as can be appreciated in Figures 17 and19. These gures indeed suggest that the collapseof the fragments is still in progress, but we have notfollowed the subsequent time evolution because thetime-step of the run becomes extremely small, to the

point of being almost incapable to move the simula-tion particles forward in time.

Finally, we show in Figure 20 the time evolutionof the peak density for each Gaussian model.

4. CONCLUDING REMARKS

One of the mechanisms proposed so far to ex-plain the formation of binary stars is prompt frag-mentation . The basic idea of this mechanism is thatduring the collapse of an isolated rotating moleculargas cloud, it may spontaneously break up into twofragments that enter into a stable orbit about oneanother.

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0

0.1

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t/t ff

Frag

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0.738259 0.798259 0.858259 0.918259

t/tff

Frags.

0

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0 0.2 0.4 0.6 0.8 1 1.2

Fragvirial line

0

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0.03

0.04

0.05

0.06

0.738259 0.798259 0.858259 0.918259

M / M

0

t/t ff

Frags.

Fig. 13. Integral properties for the fragments found in models GA .

In order to study theoretically the sensitivity of this mechanism to the non-ideal nature of the col-lapse, we carried out in this paper a fully three di-

mensional set of numerical hydrodynamical simula-tions at high spatial resolution aimed to model thegravitational collapse of both uniform and Gaussianclouds with a barotropic equation of state within theframework of the SPH technique.

4.1. The Uniform Cloud

We emphasize that our simulations of the uni-form cloud can follow the collapse to the maximum

density and evolution time shown in Table 2. Wehave not found any difference with regard to the lit-erature concerning the collapse of the uniform cloud,

for instance, see Kitsionas & Whitworth (2002).In all the uniform models we observe as a result

the formation of a binary system of protostellar frag-ments that are connected by a lament, irrespectiveof the value of the critical density. The propertiesof the fragments, the mass, the radius, the and ratios and the properties of the lament, stronglydepend on the critical density, as can be seen in Ta-ble 3.

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0.02 0.00 0.02

0.02

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0.02 0.00 0.02

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0.02

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0.00 1.05 2.11 3.16 4.22 5.27 6.33 7.17

a b c

d e f

g h i

Log / 0

Fig. 14. Isodensity curves of the cloud mid-plane for model GB when the distribution of particles reaches a peakdensity of (a) max = 2 .1569949 10 11 g cm 3 at time t/t ff = 0 .680639; (b) max = 3 .2251779 10 11 g cm 3 at timet/t ff = 0 .694144; (c) max = 4 .2332077 10 11 g cm 3 at time t/t ff = 0 .699546; (d) max = 5 .9035415 10 11 g cm 3 attime t/t ff = 0 .704948; (e) max = 7 .0227583 10 11 g cm 3 at time t/t ff = 0 .708549; (f)max = 7 .4701831 10 11 g cm 3

at time t/t ff = 0 .715751; (g) max = 8 .6511223 10 11 g cm 3 at time t/t ff = 0 .725655; (h) max = 1 .0928224 10 10 gcm

3 at time t/t ff = 0 .729256; and (i) max = 1 .2554817 10

10 g cm

3 at time t/t ff = 0 .737359.

Then, for the simulation U A, in which the ther-modynamical change occurs earlier, the resultingfragments are bigger and more massive than in theother uniform models runs, where the change oc-curs later. The lament entirely disappears only formodel U A, whereas it persists for the others.

The uniform models are therefore a very illustra-tive example of prompt fragmentation producing abinary protostellar system.

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0.2

0.22

0.24

0.26

0.28

0.3 0.32

0.34

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0.38

0.4

0.713051 0.721051 0.729051 0.737051

t/t ff

Frags

0

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t/tff

Frags.

0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fragsvirial line

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.713051 0.721051 0.729051 0.737051

M / M

0

t/t ff

Frags

Fig. 15. Integral properties for the fragments found in models GB .

4.2. The Gaussian Cloud

Our simulations of the Gaussian cloud can fol-low the collapse to the maximum densities and timesshown in Table 4. The properties of the fragments,the mass, the radius, the and ratios are shownin Table 5.

For Gaussian models we note that the effects of considering the non-ideal nature of the collapse aremore signicant than for the uniform cloud. Smallvalues of crit (heating of the gas is taken into ac-count earlier in the collapse) can indeed result in

more fragmentation of the cloud. For instance, inmodel GA and GB we observe the appearance of upto 4 fragments while in models GC and GD frag-

mentation is less favored. However, in the formercouple of models the occurrence of merging betweenfragments leaves us with only two nal protostellarcores.

Let us now compare our results with the litera-ture. Burkert & Bodenheimer (1996) considered theideal collapse of the Gaussian cloud using the tech-nique of nested grids. What these authors reported

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0.02 0.00 0.02

0.02

0.00

0.02

0.02 0.00 0.02

0.02

0.00

0.02

0.02 0.00 0.02

0.02

0.00

0.02

0.04 0.02 0.00 0.02 0.04

0.04

0.02

0.00

0.02

0.04

0.04 0.02 0.00 0.02 0.04

0.04

0.02

0.00

0.02

0.04

0.04 0.02 0.00 0.02 0.04

0.04

0.02

0.00

0.02

0.04

0.00 1.09 2.19 3.28 4.38 5.47 6.56 7.44

a b c

d e f

Log / 0Fig. 16. Isodensity curves of the cloud mid-plane for model GC when the distribution of particles reaches a peakdensity of (a) max = 1 .3574624 10 10 g cm 3 at time t/t ff = 0 .657231; (b) max = 4 .2892007 10 10 g cm 3 at timet/t ff = 0 .666234; (c) max = 7 .7961247 10 10 g cm 3 at time t/t ff = 0 .675237; (d) max = 3 .7783788 10 09 g cm 3 attime t/t ff = 0 .693244; (e) max = 9 .0149908 10 09 g cm 3 at time t/t ff = 0 .711250 and (f) max = 1 .3768666 10 08 gcm 3 at time t/t ff = 0 .729256.

TABLE 2

MAXIMUM EVOLUTION TIME AND DENSITY FOR UNIFORM CLOUD MODELS

Model UA UB UC UD

t max /t ff 1.526036 1.338320 1.291054 1.276649

max g cm 3 2.3075786 10 11 6.8565254 10 10 5.9576158 10 09 2.8596622 10 07

as a result of a simulation considered to be of mod-erate resolution, is a system of three fragments thatenter into orbit without merging; they distinguished

a central fragment and two additional ones whichthey refer to as the exterior binary system (see Fig-ures 1 and 2 in Burket & Bodenheimer 1996).

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0.1

0.15

0.2

0.25

0.3

0.35

0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72

t/t ff

Frags.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72

t/t ff

Frags.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Frags.virial line

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72

M / M

0

t/t ff

Frags.

Fig. 17. Integral properties for the fragments found in models GC .

In the simulation that Burkert & Bodenheimer(1996) considered of high resolution, they reported

that the fragment located at the center divides in itsturn provoking the appearance of an internal binarysystem. For this reason, they concluded that theirsimulation produced 4 fragments in orbit.

It is interesting to point out that results very sim-ilar to the ones of Burkert & Bodenheimer (1996)were observed in our model GD ; for instance, com-pare our Figure 18a with their Figure 3c and ourFigure 18d with their Figure 3f. In these simulations

we never observed the fragmentation of the centralclump.

In Boss et al. (2000) the Gaussian collapsewas calculated with the Adaptive Mesh Rene-ment(AMR), a technique in which the AMR algo-rithm creates ner grids in order to achieve higherresolution where needed.

It is noteworthy that Boss et al. (2000) also ob-served the formation of well dened dense cores onthe extremes of the central bar. They also noticed a

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0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.02 0.01 0.00 0.01 0.020.02

0.01

0.00

0.01

0.02

0.00 1.35 2.71 4.06 5.41 6.76 8.12 9.20

a b c

d e f

g h i

Log / 0

Fig. 18. Isodensity curves of the cloud mid-plane for model GD when the distribution of particles reaches a peakdensity of (a) max = 2 .4750879 10 09 g cm 3 at time t/t ff = 0 .654530; (b) max = 8 .7291514 10 09 g cm 3 at timet/t ff = 0 .656331; (c) max = 1 .1161107 10 08 g cm 3 at time t/t ff = 0 .658131; (d) max = 1 .9224208 10 08 g cm 3 attime t/t ff = 0 .660832; (e) max = 2 .5560963 10 08 g cm 3 at time t/t ff = 0 .662633; (f)max = 3 .3774650 10 08 g cm 3

at time t/t ff = 0 .664433; (g) max = 4 .4997205 10 08 g cm 3 at time t/t ff = 0 .666234; (h) max = 6 .3509960 10 08 gcm

3 at time t/t ff = 0 .668035 and (i) max = 8 .4127299 10

08 g cm

3 at time t/t ff = 0 .669835.

fragmentation of the bar but only for the model inwhich they implemented the full Eddington approxi-mation. They argued that the barotropic equation of state (which is the same equation we use in this work,

see equation 9) is an approximation to the full Ed-dington model. They reported that the bar rapidlydecayed into a single central clump surrounded byspiral arms.

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0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.663533 0.665533 0.667533 0.669533

t/t ff

Frags.

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0 .663533 0.665533 0.667533 0.669533

t/tff

Frags.

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.4 0.45 0.5 0.55 0.6 0.65 0.7

Fragsvirial line

0.004

0.0045

0.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0.663533 0.665533 0.667533 0.669533

M / M

0

t/t ff

Frags

Fig. 19. Integral properties for the fragments found in models GD .

GA would like to thank ACARUS-UNISON forthe use of their computing facilities and Ruslan Gab-basov for a very timely assistance in the making of this paper.

APPENDIX AA NUMERICAL DEFINITION OF A

FRAGMENTIn order to calculate the integral properties of

protostellar cores we need a numerical denition of a resulting fragment. In our models we proceed asfollows:

1. We locate the center of the fragment, that is,the particle with the highest density in a region(of the slice of matter surrounding the equatorof the cloud) where the fragment is located.

2. Once the location of the center has been de-termined, we nd all the particles which havedensity above or equal to some minimum den-sity value within a given maximum radius fromthe center. This set of particles dene the frag-ment and therefore its integral properties, forinstance, its mass.

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0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1

l o g 1 0

( m a x /

0

)

t/t ff

GAGBGCGD

Fig. 20. Time evolution of the peak density for Gaussian models.

TABLE 3

PHYSICAL PROPERTIES OF THE RESULTINGFRAGMENTS FOR THE UNIFORM CLOUD

Model Time ( tff ) M f /M 0 | | | |

UA 1.440506 0.094811 0.248363 0.2855930.092660 0.252887 0.262221

UB 1.319864 0.070219 0.314145 0.2835100.064592 0.258661 0.3206060.008509 0.569961 0.063677

UC 1.289253 0.047144 0.224965 0.3646210.047158 0.223713 0.382579

UD 1.277099 0.021195 0.201898 0.3264310.019875 0.219058 0.323086

TABLE 4

MAXIMUM EVOLUTION TIME AND DENSITY FOR GAUSSIAN CLOUD MODELS

Model GA GB GC GD

t max /t ff 0.972342 0.737359 0.729256 0.669835

max g cm 3

3.1654053 10 11

1.2554817 10 10

1.3768666 10 08

8.4127299 10 08

3. To calculate the ratios of energy (the and de-ned in 2.2) for a given fragment, a length h isrequired for each simulation particle in the frag-ment in order to locate its neighboring particles.We use the prescription that h R0 / (2 N f )

23

where N f is the number of particles dening thefragment. However, we cannot assure that thenumber of neighbors of each particle is always40.

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TABLE 5

PHYSICAL PROPERTIES OF THE RESULTINGFRAGMENTS FOR THE GAUSSIAN CLOUD

Model Time ( tff ) M f /M 0 | | | |

GA 0.936329 0.053970 0.280301 0.2388960.053171 0.247702 0.247702

GB 0.737359 0.012697 0.227151 0.2960110.009542 0.258132 0.3471520.007682 0.297418 0.327948

GC 0.688742 0.015067 0.155920 0.3342310.011323 0.161015 0.3538680.010044 0.153414 0.359508

GC 0.715751 0.011245 0.175698 0.2967500.006952 0.146549 0.3989850.002143 0.308841 0.221144

GD 0.669835 0.007273 0.065245 0.6281650.006951 0.065631 0.630207

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