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Mon. Not. R. Astron. Soc. 000, 000­000 (0000)

Printed 10 August 2005

A (MN L TEX style file v2.2)

The effect of a finite mass reservoir on the collapse of spherical isothermal clouds and the evolution of protostellar accretion
E. I. Vorobyov
1 2

1, 2

Department of Physics and Astronomy, University of Western Ontario, London, Ontario, N6A 3K7, Canada Institute of Physics, Stachki 194, Rostov-on-Don, Russia

and Shantanu Basu

1

Submitted November 1, 2004

Motivated by recent observations which detect an outer boundary for starless cores, and evidence for time-dependent mass accretion in the Class 0 and Class I protostellar phases, we reexamine the case of spherical isothermal collapse in the case of a finite mass reservoir. The presence of a core boundary, implemented through a constant volume approximation in our simulation, results in the generation of an inward propagating rarefaction wave. This steepens the gas density profile from r-2 (self-similar value) to r-3 or steeper. After a protostar forms, the mass accretion rate M evolves , which is a nonthrough three distinct phases: (1) an early phase of decline in M self-similar effect due to rapid and spatially nonuniform infall in the prestellar phase; (2) for large cores, an intermediate phase of near-constant M from the infall of the outer part of the self-similar density profile which has low (subsonic) infall speed in the prestellar phase; (3) a late phase of rapid decline in M when accretion occurs from the region affected by the inward propagating rarefaction wave. Our model clouds of small to intermediate size make a direct transition from phase (1) to phase (3) above. Both the first and second phase (if the latter is indeed present) are characterized by a temporally increasing bolometric luminosity Lbol , while Lbol is decreasing in the third (final) phase. We identify the period of temporally increasing Lbol with the Class 0 phase, and the later period of terminal accretion and decreasing Lbol with the Class I phase. The peak in Lbol corresponds to the evolutionary time when 50% ± 10% of the cloud mass has been accreted by the protostar. This is in agreement with the classification scheme proposed by Andr´ et al. (1993); our model adds a physical context to e their interpretation. We show how our results can be used to explain tracks of envelope mass Menv versus Lbol for protostars in Taurus and Ophiuchus. We also develop an analytic formalism which successfully reproduces the protostellar accretion rate from profiles of density and infall speed in the prestellar phase. It shows that the spatial gradient of infall speed that develops in the prestellar phase is a primary cause of the temporal decline in M during the early phase of protostellar accretion. Key words: hydrodynamics ­ ISM: clouds ­ stars: formation.

ABSTRACT

1

INTRODUCTION

Recent submillimeter and mid-infrared observations suggest that prestellar cores within a larger molecular cloud are characterized by a non-uniform radial gas density distribution (Ward-Thompson et al. 1999; Bacmann et al. 2000). Specifically, a flat density profile in the central region of size Rflat is enclosed within a region of approximately r-1

E-mail: vorobyov@astro.uwo.ca (EIV); basu@astro.uwo.ca (SB) c 0000 RAS

column density profile (and by implication an r-2 density profile) of extent Rmid . Beyond this, a region of steeper density ( r-3 or greater) is sometimes detected. Finally, at a distance Redge , the column density N seems to merge into a background, and fluctuate about a mean value that is typical for the ambient molecular cloud. The first two regions, of extent Rflat and Rmid , respectively, are consistent with models of unbounded isothermal equilibria or isothermal self-similar gravitational collapse (e.g. Chandrasekhar 1939; Larson 1969; Penston 1969). In either case, the effect


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E. I. Vorobyov and S. Basu
the assumption of constant mass and volume of a gravitationally contracting core can affect the mass accretion rate and other observable properties after the formation of the central hydrostatic stellar core. A very important question is: which of the two effects mentioned above - a gradient of infall speed in the prestellar phase, or a finite mass reservoir and associated steep outer density slope - is more relevant to explaining the observations of BATC? The evolutionary tracks of envelope mass Menv versus bolometric luminosity Lbol are another important diagnostic of protostellar evolution (Andr´ et al. 2000). BATC have fit the data using a toy e model in which M decreases with time in exact proportion to the remaining envelope mass Menv , i.e. M = Menv / , where is a characteristic time. We seek to explain the observed YSO evolutionary tracks using a physical (albeit highly simplified) model. We perform high resolution one-dimensional spherical isothermal simulations. The initial peak and decline in the mass accretion rate is modeled through numerical simulations and a simplified semi-analytic approach. A second late-time de cline in M due to a gas rarefaction wave propagating inward from the outer edge of a contracting core, is also studied in detail. Comparisons are made with the observationally inferred decrease of mass accretion rate (BATC), and evolutionary tracks of Menv versus bolometric luminosity Lbol (from Motte & Andr´ 2001). e Numerical simulations of spherical collapse of isothermal cloud cores are described in § 2. The comparison of the model with observations is given in § 3. Our main conclusions are summarized in § 4. An analytical approach for the determination of the mass accretion rate is presented in the Appendix.

of an outer boundary is considered to be infinitely far away (i.e. Rmid in our terminology). In numerical simulations of gravitational collapse in which there is a qualitative change in the physics beyond some radius (e.g. a transition from magnetically supercritical to subcritical mass-toflux ratio: Ciolek & Mouschovias 1993; Basu & Mouschovias 1994), the development of a very steep outer density profile is also seen. Finally, larger scale simulations of core formation in clouds with uniform background column density (Basu & Ciolek 2004) show an eventual merger into a near-uniform background column density, demonstrating the existence of Redge . The implication of an outer density profile steeper than r-3 is that there is a finite reservoir of mass to build the star(s), assuming that the gas beyond Redge is governed by the dynamics and gravity of the parent cloud, and thus does not accrete on to the star(s) formed within the core. An important constraint of the observations are the actual sizes of the cores. For example, in the clustered star formation regions such as Ophiuchi protocluster, Redge 5000 AU, and Redge /Rflat 5, while in the more 20000 AU, extended cores in Taurus, 5000 AU Redge 10 (see Andr´ et al. 1999; Andr´ e e, and 5 Redge /Rflat Ward-Thompson, & Barsony 2000). Clearly, only the latter case may approach self-similar conditions. Once a central hydrostatic stellar core has formed, the mass accretion rate is expected to be constant in isothermal similarity solutions (Shu 1977; Hunter 1977; Whitworth & Summers 1985). For example, for the collapse from rest of a singular isothermal sphere (SIS) with density profile SIS = c2 /(2 Gr 2 ), where cs is the isothermal sound speed, s Shu (1977) has shown that the mass accretion rate ( M ) is constant and equal to 0.975 c3 /G. However, two effects can s work against a constant M in more realistic scenarios of isothermal collapse: (1) inward speeds in the prestellar phase are not spatially uniform as in the similarity solutions, and tend to increase inward, meaning that inner mass shells fall in with a greater accretion rate; (2) the effect of a finite mass reservoir will ultimately reduce accretion. The first effect has been clearly documented in a series of papers (e.g. Hunter 1977; Foster & Chevalier 1993; Tomisaka 1996; Basu 1997; Ciolek & K¨ onigl 1998; Ogino, Tomisaka, & Nakamura 1999). It is always present since the outer boundary condition for collapse is distinct from the inner limit of self-similar supersonic infall found in the Larson-Penston solution. Rather, the outer boundary condition must represent the ambient conditions of a molecular cloud, which do not correspond to large-scale infall (Zuckerman & Evans 1974). Additionally, the finite mass reservoir and steeper than r-3 profile as a source of the declining accretion rate has been studied analytically by Henriksen, Andr´ & Bontemps (1997) and e, Whitworth & Ward-Thompson (2001), although they did not account for the physical origin of such a steep density slope. Indeed, a study of outflow activity from young stellar ob jects (YSO's) by Bontemps et al. (1996; hereafter BATC) suggests that M declines significantly with time during the accretion phase of protostellar evolution. Specifically, BATC have shown that if the CO outflow rate is proportional to M , then Class 0 ob jects (young protostars at the begin ning of the main accretion phase) have an M that is factor of 10 greater (on average) than that of the more evolved Class I ob jects. In this paper, we investigate in detail how

2 2.1

ISOTHERMAL COLLAPSE Mo del Assumptions

We consider the gravitational collapse of spherical isothermal (temperature T = 10 K) clouds composed of molecular hydrogen with a 10% admixture of atomic helium. The models actually represent cloud cores which are embedded within a larger molecular cloud. The evolution is calculated by solving the hydrodynamic equations in spherical coordinates: 1 +2 r2 vr t r r 1 r2 vr vr (vr ) + 2 t r r e 1 2 +2 (r evr ) t r r = = = 0 p GM - 2 , r r p 2 -2 ( r vr ) r r - (1) (2) (3)

where is the density, vr is the radial velocity, M is the enclosed mass, e is the internal energy density and p = ( - 1)e is the gas pressure. The ratio of specific heats is equal to = 1.001 for the gas number density n 1011 cm-3 , which implies isothermality (the value of is not exactly unity in our implementation in order to avoid a division by zero). We define the gas number density n = /m, where m = 2.33 mH is the mean molecular mass. When the gas number density in the collapsing core exceeds 1011 cm-3 , we form the central hydrostatic stellar core by imposing an adiabatic index = 5/3. This simplified treatment of the transition to an opaque
c 0000 RAS, MNRAS 000, 000­000


Finite Mass Reservoir Col lapse
protostar misses the details of the physics on small scales. Specifically, a proper treatment of the accretion shock and radiative transfer effects is required to accurately predict the properties of the stellar core (see Winkler & Newman 1980 for a detailed treatment and review of work in this area). However, our method should be adequate to study the protostellar accretion rate, and has been used successfully by e.g. Foster & Chevalier (1993) and Ogino et al. (1999) for this purpose. We use the method of finite-differences, with the time-explicit, operator split solution procedure used in the ZEUS-1D numerical hydrodynamics code; it is described in detail by Stone & Norman (1992). We have introduced the momentum density correction factor, as advocated by M¨ hmeyer & Muller (1989), to avoid the development of onc ¨ an anomalous density spike at the origin (see Vorobyov & Tarafdar 1999 for details). The numerical grid has 700 points which are initially uniformly spaced, but then move with the gas until the central stellar core is formed. This provides an adequate resolution throughout the simulations. We impose boundary conditions such that the gravitationally bound cloud core has a constant mass and volume. The assumption of a constant mass appears to be observationally justified by the sharp outer density profiles described in § 1. Physically, this assumption may be justified if the core decouples from the rest of a comparatively static, diffuse cloud due to a shorter dynamical timescale in the gravitationally contracting central condensation than in the external region. A specific example of this, due to enhanced magnetic support in the outer envelope, is found in the models of ambipolar-diffusion induced core formation (see, e.g. Basu & Mouschovias 1995). The assumption of a constant volume is mainly an assumption of a constant radius of gravitational influence of a cloud core within a larger parent diffuse cloud. The radial gas density distribution of a self-gravitating cloud with finite central density that is in hydrostatic equilibrium (e.g. Chandrasekhar 1939) can be conveniently approximated by a modified isothermal sphere, with gas density = c 1 + (r/rc )
2

3

Table 1. Model parameters Model I1 I2 n
c

r

c

r

out

c out

M

cl

T 10 10

5 .0 5 .0

0.033 0.033

0.16 0.5

24 324

5 24
-3

All number densities are in units of 104 cm masses in M , and temperatures in K.

, lengths are in pc,

sity perturbation in order to initiate evolution. Use of the modified isothermal sphere simplifies the analysis a little bit since there is a clear transition from flat central region to a power-law outer profile. The choice of central density c and outer radius rout determines the cloud mass. We study many different cloud masses - two models are presented in this section and other models are used to fit observational tracks in § 3. We also add a (small) positive density perturbation of a factor of 1.1 (i.e. the initial gas density distribution is increased by a factor of 1.1) to drive the cloud (especially the inner region which is otherwise near-equilibrium) into gravitational collapse. Table 1 shows the parameters for two model clouds presented in this section. The adopted central number density nc = 5 â 104 cm-3 is roughly an order of magnitude lower than is observed in prestellar cores (Ward-Thompson et al. 1999). Considering that these cores may be already in the process of slow gravitational contraction, our choice of nc is justified for the purpose of describing the basic features of star formation. In both models, the outer radius rout is chosen so as to form gravitationally unstable prestellar cores with central-to-surface density ratio c /out 14 (since our initial states are similar to Bonnor-Ebert spheres). In model I1, c /out 24 and by implication rout /rc 4.7, whereas in model I2 c /out 324 and rout /rc 18. Model I2 thus represents a very extended prestellar core. Models I1 and I2 have masses 5 M and 24 M respectively; the `I' stands for isothermal. 2.2 Numerical Results

(4)

(Binney & Tremaine 1987), where c is the central density is and rc a radial scale length. We choose a value rc = 1.1 cs / Gc , so that the inner profile is close to that of a Bonnor-Ebert sphere, rc is comparable to the Jeans length, and the asymptotic density profile is 2.2 times the equilibrium singular isothermal sphere value SIS = c2 /(2 Gr 2 ). s The latter is justified on the grounds that core formation should occur in a somewhat non-equilibrium manner (an extreme case is the Larson-Penston flow, in which case the asymptotic density profile is as high as 4.4 SIS ), and also by observations of protostellar envelope density profiles that are e, often overdense compared to SIS (Andr´ Motte, & Belloche 2001). For small radii (r rc ), the initial density is very close to the equilibrium solution for an isothermal sphere with a finite central density. However, at large radii it is twice the value of the equilibrium isothermal sphere, which converges to SIS . Hence, our initial conditions resemble those of other workers (Foster & Chevalier 1993; Ogino et al. 1999) who start with Bonnor-Ebert spheres and add a positive denc 0000 RAS, MNRAS 000, 000­000

Fig. 1 shows the temporal evolution of the radial gas density profiles (the upper panel) and velocity profiles (the lower panel) during the runaway collapse phase (before the formation of the central hydrostatic stellar core) in model I1. The density and velocity profiles are numbered according to evolutionary sequence, starting from the initial distributions (profile 1; note that the cloud core is initially at rest) and ending with those obtained when the central number density has almost reached 1010 cm-3 (profile 5). The dashed lines in the upper panel of Fig. 1 show the power-law index d ln /d ln r of the gas distribution for profiles 1, 2, and 3. By the time that a relatively mild center-to-boundary density contrast 150 is established (profile 2), the radial density profile starts resembling those observed in Taurus by Bacmann et al. (2000): it is flat in the central region, then gradually changes to an r-2 profile, and falls off as r-3 or steeper in the envelope at r 0.08 pc. The sharp change in slope of the density profile (e.g. at r 0.08 pc in profile 2 of Fig. 1) is due to an inward ly-propagating gas rarefaction wave caused by a finite reservoir of mass. The self-similar re-


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E. I. Vorobyov and S. Basu

Figure 1. The radial gas density (the upper panel) and velocity (the lower panel) profiles obtained in model I1 before the central hydrostatic stellar core is formed. The number 1 corresponds to the initial profiles (note that initially the cloud core is at rest) and the number 5 labels profiles when the central gas number density has almost reached 1010 cm-3 . The dashed lines in the upper panel show the power-law index d ln /d ln r of the gas distributions 1, 2, and 3. Profiles 2, 3, 4, and 5 are reached at times 0.309 Myr, 0.378 Myr, 0.392 Myr, and 0.394 Myr, respectively. For reference, the dotted line is the density profile of a singular isothermal sphere.

gion with r-2 density profile is of the Larson-Penston type, with density somewhat greater than the equilibrium singular isothermal sphere value (c2 /2 Gr 2 ). The velocity profiles s in Fig. 1 also show a distinct break at the instantaneous location of the rarefaction wave. Furthermore, the peak infall speed is clearly supersonic (since cs = 0.19 km s-1 ) by the time profile 4 is established, again consistent with LarsonPenston type flow in the inner region. Fig. 2a shows the temporal evolution of the accretion rate at a radial distance of 600 AU from the center in model I1.1 The evolution is characterized by a slow initial gravitational contraction and then a very rapid increase until about 0.4 Myr. Subsequently, a central hydrostatic stellar core forms and the mass accretion rate reaches its maximum value of 2.8 â 10-5 M yr-1 (or 17.4 c3 /G). After stellar s core formation, the evolution of the mass accretion rate has possibly three distinct phases, of which two are on display
1

Figure 2. The temporal evolution of the mass accretion rate at the radial distance r = 600 AU from the center of a cloud core obtained in a) model I1 and b) model I2. The model cloud I1 has mass 5 M and the model cloud I2 has mass 24 M . The solid lines show M during the runaway collapse phase, prior to the formation of a central stellar core. The dashed and dotted lines plot M after stellar core formation; the dashed lines show M before the gas affected by the inwardly propagating rarefaction wave has reached r = 600 AU, whereas the dotted lines show M after this gas has reached r = 600 AU. The numbers in parentheses reflect the percentage of cloud mass remaining in the envelope at the given times.

We note that the accretion rate is not expected to vary significantly in the range 0.1 AU < r < 1000 AU according to Masunaga & Inutsuka (2000).

in Fig. 2a. The early phase, plotted with the dashed line in Fig. 2a, is characterized by accretion of material that has not yet been affected by the rarefaction wave propagating inward from the outer boundary. The accretion rate is declining, even though the density profile near the stellar core was nearly self-similar at the moment of its formation. This decline is due to the gradient of infall velocity in the inner regions, an effect not predicted in the similarity solutions. However, if there is a large outer region with mass shells that are falling in at significantly subsonic speeds when the central stellar core forms (see discussion of Fig. 2b below), the accretion rate will eventually stabilize to a constant value that is consistent with the standard theory of Shu (1977). In that picture, progressively higher shells of gas lose their partial pressure support and start falling from rest on to the central stellar core almost in a free-fall manner. This would be the intermediate phase of accretion. However, the late phase of very rapid decline of the accretion rate starts at roughly 0.46 Myr (before the intermediate
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Finite Mass Reservoir Col lapse
phase can be established in the 5 M cloud), when gas affected by the inwardly propagating rarefaction wave reaches the inner 600 AU. This results in a sharp drop of M as shown in Fig. 2a by the dotted line. The existence of the (in principle) three distinct phases of mass accretion is clearly seen Fig. 2b, where M of the more extended cloud (rout /rc 18) is plotted (hereafter, model I2). The outer boundary is now at rout = 0.5 pc and it takes a time 1 Myr for the influence of the rarefaction wave to reach the inner 600 AU. As a result, the mass accretion rate has time to stabilize at a constant value of 1.34 â 10-5 M yr-1 (the dashed line in Fig. 2b), before it sharply drops at later times (the dotted line in Fig. 2b). According to Shu (1977), the collapse from rest of a power-law profile that has a density equal to twice SIS yields a mass accretion rate 5.58 c3 /G = 8.86 â 10-6 M yr-1 . Our stable s intermediate accretion rate is roughly consistent with this prediction since the density in the power-law tail is actually somewhat greater than twice SIS . It is equal to 2.42 SIS in the initial state, and grows to greater overdensities in the innermost regions. However, the bulk of the matter, which is in the outer tail, has density within 2.5SIS . Further experiments with our numerical simulations show that the intermediate phase of constant accretion rate is observed only in rather extended prestellar cores with rout /rc 15. Foster & Chevalier (1993) found an even stronger criterion 20. Since more extended cores tend to be more rout /rc massive as well, we may expect to observe the intermediate phase more frequently in the collapse of massive cores.

5

Figure 3. The temporal evolution of the mass accretion rate. The solid line shows the results of isothermal numerical simulations (model I1). The dashed line shows M obtained in a pressure-free approximation if such collapse begins from rest with the relatively mildly concentrated density profile 2 in Fig. 1. The dotted line shows the result for pressure-free collapse if a non-zero initial velocity (that of profile 2 in the bottom panel of Fig. 1) is also used. The agreement of the pressure-free model and full numerical simulation are quite good in the latter case.

2.4

Semi-analytic Mo del

2.3

Effect of Boundary Condition

Our standard simulation does not contain an external medium explicitly. In order to explore the effect of such a medium, we ran additional simulations in which the cloud core is surrounded by a spherical shell of diffuse (i.e. nongravitating) gas of constant temperature and density. The outermost layer of the cloud core and the external gas are initially in pressure balance. We found that the value of M in the late accretion phase may depend on the assumed values of the external density and temperature. For instance, if the gravitating core is nested within a larger diffuse nongravitating cloud of T = 10 K and = out , the accretion rate increases slightly as compared to that shown in Fig. 2 by the dotted line. A warmer external non-gravitating environment of T = 200 K and = out /20 shortens the duration of the late accretion phase shown in Fig. 2 by the dotted line. This phase may be virtually absent if the sound speed of the external diffuse medium is considerably higher (by a factor 1000) than that of the gravitationally bound core. This essentially corresponds to a constant outer pressure boundary condition (see Foster & Chevalier 1993). However, such a high sound speed contrast is not expected for star formation taking place in a dense (n 104 cm-3 ) environment like Ophiuchi (Johnstone et al. 2000). We believe that the constant volume boundary condition, and resulting inward propagating rarefaction wave, are best at reproducing the steep outer density profiles and the low (residual) mass accretion rate necessary to explain the Class I phase of protostellar accretion.
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Finally, we compute M of a pressure-free cloud using the analytical approach developed in the Appendix. This ap proach allows for the determination of M for a cloud with given initial radial density (r0 ) and velocity v0 (r0 ) profiles, if the subsequent collapse is pressure-free. We find that the success or failure of the analytical approach to describe the mass accretion rate of the isothermal cloud depends on the adopted (r0 ) and v0 (r0 ) profiles. For instance, if (r0 ) is determined by profile 2 (the upper panel of Fig. 1) and v0 (r0 ) = 0, the pressure-free mass accretion rate shown in Fig. 3 by the dashed line reproduces only very roughly the main features of the isothermal accretion rate (the solid line in Fig. 3). However, if we take into account the non-zero and non-uniform velocity profile v0 (r0 ) plotted in the lower panel of Fig. 1 (profile 2), then the pressure-free M shown by the dotted line in Fig. 3 reproduces that of the isothermal cloud much better. This example demonstrates the importance of the velocity field prior to stellar core formation in determining the accretion rates after its formation. The success of our analytical pressure-free approach also shows that the collapse of the isothermal cloud can be regarded as essentially pressure-free from the time of a relatively mild central concentration c /out 150, when the central number density 2 â 105 cm-3 .

3

ASTROPHYSICAL IMPLICATIONS

Class 0 ob jects represent a very early phase of protostellar evolution (see Andr´ et al. 2000), as evidenced by a relae tively high ratio of submillimeter luminosity to bolometric luminosity: Lsubmm /Lbol > 0.5%. Class 0 ob jects also drive powerful collimated CO outflows. A study of outflow ac-


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E. I. Vorobyov and S. Basu
Table 2. Model parameters for Ophiuchus M
cl

nc 1 5 2 2 â â â â 107 106 106 106

r

out

c / 15.0 14.1 18.0 18.0

out

r

out

/rc

2.0 2.0 2.4 4.0

0.17 0.23 0.55 0.9

1600 1900 4000 4000

3.7 3.6 4.1 4.1

All number densities are in cm M.

-3

, lengths in AU, and masses in

Table 3. Model parameters for Taurus M Figure 4. CO momentum flux Fco versus envelope mass Menv for 41 sources of Bontemps et al. (1996). The Class 0 and Class I ob jects are plotted with the open and filled circles, respectively. The model Fco - Menv tracks of three prestellar clouds of Mcl = 0.2 M , 0.7 M , and 2 M are shown by the dotted, dashed, and solid lines, respectively.
cl

nc 1 1 1 8 .5 â 10 â 106 â 106 â 105
6

r

out

c /out 18 26 71 73
-3

r

out

/r

c

1.9 1.8 1.5 1.8

0.46 0.65 1.0 1.5

5000 6000 10000 12000

4.1 5.0 8.4 8.5

All number densities are in cm M.

, lengths in AU, and masses in

tivity in low-mass YSO's by BATC suggests that the CO momentum flux Fco declines significantly during protostellar evolution. Specifically, Fco decreases on average by more than an order of magnitude from Class 0 to Class I ob jects. This tendency is illustrated in Fig. 4, where we plot Fco ver sus Menv for 41 sources listed in BATC. We relate Fco to M by F
co

=f

ent

â ( M w / M ) Vw â M ,

(5)

where fent is the entrainment efficiency that relates Fco to the momentum flux Mw Vw of the wind. Based on theoretical models in the literature, BATC suggested that the factor fent and the outflow driving engine efficiency (Mw /M )Vw do not vary significantly during protostellar evolution. This implies that the observed decline of Fco reflects a corre sponding decrease in M from the Class 0 to the Class I stage. Following BATC, we take fent = 1, Mw /M = 0.1, -1 and Vw = 150 km s and use equation (5) to compute Fco from our model's known mass accretion rate M (see Fig. 2). Since the sample of Class 0 and Class I ob jects listed in BATC includes sources from both the Ophiuchus and Taurus star forming regions, we develop model clouds which take into account the seemingly different initial conditions of star formation in these regions. As mentioned in § 1, the two most prominent differences between these two regions are: (1) The cores in Ophiuchus have outer radii 5000 AU) which are smaller than in Taurus, where (rout 20000 AU (Andr´ et al. 1999; Andr´ e e 5000 AU rout et al. 2000); (2) The radial column density profiles of the protostellar envelopes of Class 0 ob jects in Ophiuchus are at least 2-3 times denser than a SIS at T = 10 K, whereas in Taurus the protostellar envelopes are overdense compared to the SIS by a smaller factor 2 (Andr´ et al. 2001). e This implies that radial column density profiles of prestel lar cores in Ophiuchus and Taurus may follow the same tendency. We develop a set of Ophiuchus model cores which have rout 5000 AU, and a set of Taurus model cores which have 5000 AU rout 20000 AU. Furthermore, the factor

(by which our model density profiles are asymptotically overdense compared to SIS ) is taken to be 2.0 for Ophiuchus and < 2.0 for Taurus. Clearly, there is no unique set of model cloud parameters that would be exclusively consistent with the observational data, given the measurement uncertainties. We have chosen a set of core central densities c , radii rout , and overdensity factors so as to reasonably reproduce the observed properties of the cores in the two regions. The parameters of the model density distributions for Ophiuchus and Taurus are listed in Table 2 and Table 3, respectively. We also note that we have ensured that the cores satisfy the gravitational instability criterion rout /rc 3.6, which is similar to that for Bonnor-Ebert spheres. The Ophiuchus model cores are clustered near this limiting value of rout /rc , but the Taurus model cores are allowed to be somewhat more extended, again in keeping with observed properties. We also note that the masses of prestellar cores with the radial density profile given by equation (4) scale as 1/0.5 , c if the ratio rout /rc is fixed. The sample of 41 sources in BATC contains Class 0 and Class I ob jects from both Ophiuchus and Taurus. Hence, in Fig. 4 we take three representative prestellar clouds of Mcl = 0.23 M (Ophiuchus), 0.65 M (Taurus), and 2.0 M (Taurus), for which the Fco - Menv tracks are shown by the dotted, dashed, and solid lines, respectively. Both the data and model tracks show a near-linear correlation be tween Fco ( M ) and Menv . A slightly better fit of the model tracks to the data can be obtained by adjusting one or more of the estimated parameters fent , Mw /M , and Vw by factors of order unity. Based on the near-linear correlation of Fco and Menv , BATC developed a toy model in which M decreases with time in exact proportio