First Direct Simulation of Brown Dwarf Formation in a Compact Cloud Core
Brown dwarf formation and star formation efficiency are studied using a nested grid simulation that covers five orders of magnitude in spatial scale (– AU). Starting with a rotating magnetized compact cloud with a mass of , we follow the cloud evolution until the end of main accretion phase. Outflow of emerges yr before the protostar formation and does not disappear until the end of the calculation. The mass accretion rate declines from to – in a short time (yr) after the protostar formation. This is because (1) a large fraction of mass is ejected from the host cloud by the protostellar outflow and (2) the gas escapes from the host cloud by the thermal pressure. At the end of the calculation, () of the total mass () is outflowing from the protostar, in which () of the total mass is ejected by the protostellar outflow with supersonic velocity and () escapes with subsonic velocity. On the other hand, () is converted into the protostar and () remains as the circumstellar disk. Thus, the star formation efficiency is . The resultant protostellar mass is in the mass range of brown dwarfs. Our results indicate that brown dwarfs can be formed in compact cores in the same manner as hydrogen-burning stars, and the magnetic field and protostellar outflow are essential in determining the star formation efficiency and stellar mass.
There are several scenarios for the formation of brown dwarfs (see, Whitworth et al., 2007), though many observations indicate that brown dwarfs are formed through the gravitational collapse of a molecular cloud core (see, Luhman et al., 2007). Observations around young brown dwarfs show the existence of circumstellar disks (e.g., Pascucci et al., 2003; Luhman et al., 2005) and of gas accretion onto the young brown dwarfs (e.g., White & Basri, 2003). Recently, Whelan et al. (2005) observed an optical jet around a young brown dwarf of Oph 102. In addition, Phan-Bao et al. (2008) observed molecular outflows from the same object. To date, several outflows have been observed around young brown dwarfs (Whelan et al., 2009a, b). Since outflow is typical in the star formation process, it is considered that these flows are direct proof that brown dwarf formation occurs through the gravitational collapse of a molecular cloud core. On the other hand, observations indicate that the prestellar core mass function (CMF) resembles the initial mass function (IMF, e.g., Motte et al., 1998), suggesting that the effective reservoirs of mass required for the formation of individual stars including brown dwarfs are already determined at the prestellar core stage (André et al., 2008). The similarity of CMF and IMF implies that the prestellar cores are on a one-to-one correspndence with protostars, in which the star formation efficiency is expected to be –%.
Not all the mass of a prestellar core can convert into a star, because the outflow from a protostar can remove a significant mass from the natal cloud core (Whitworth et al., 2007). Matzner & McKee (2000) has pointed out that since the gas accretion onto a protostar is impeded by the protostellar outflow, only a fraction of the mass can convert into the protostar, with the remainder being blown out. They modeled the prestellar outflow and analytically estimated a star formation efficiency of 25–70%.
We can presume that brown dwarfs are born in more compact (or less massive) cloud cores, because they are less massive than stars. Taking a star formation efficiency of %, brown dwarfs with masses of –, where is the Jovian mass, can be formed in cores with masses of –. Cloud cores with such masses have been observed in several surveys (e.g., Motte et al., 2001; Sandell & Knee, 2001; Onishi et al., 2002). In this paper, we study the evolution of a compact molecular cloud core and the formation of a brown dwarf.
2 Model and Numerical Method
To study the evolution of star-forming cores in a large dynamic range of spatial scale, a three-dimensional nested grid method is used, in which the equations of resistive magnetohydrodynamics with a barotropic equation of state are solved (see Eq. – and  of Machida et al. 2007).
In the collapsing cloud core, we assume protostar (or proto-brown dwarf) formation occurs when the number density exceeds at the cloud center. To model the protostar, we adopt a sink around the center of the computational domain. In the region AU, gas having a number density of is removed from the computational domain and added to the protostar as a gravity in each timestep. In addition, inside the sink, the magnetic flux is removed by the Ohmic dissipation (The detailed description will appear in a subsequent paper).
As the initial state we take a spherical cloud with a Bonnor–Ebert (BE) density profile that extends up to twice the critical BE radius (). Outside the sphere (), a uniform density is adopted. For the BE density profile, we adopt a central density of and an isothermal temperature of K. For these parameters, the critical BE radius is AU. Thus, the radius of the initial sphere is AU. To promote the contraction, we increase the density by a factor of 1.68. The density contrast between the cloud center and ambient medium is about 80. The mass within () is () corresponding to (). In this paper, we call the sphere with the host cloud. The cloud rotates rigidly with s around the -axis in the region of , while a uniform magnetic field ( G) parallel to the -axis (or rotation axis) is adopted over the whole computational domain. In the region of (), the ratios of thermal , rotational , and magnetic to the gravitational energy are (0.5), (0.016), and (0.04), respectively. The gravitational force is ignored outside the host cloud () to mimic a stationary interstellar medium.
To calculate over a large spatial scale, the nested grid method is adopted (for details see Machida et al., 2005a, b). Each level of a rectangular grid has the same number of cells (). The calculation is first performed with five grid levels (–). The box size of the coarsest grid is chosen to be . Thus, a grid of has a box size of AU. A new finer grid is generated before the Jeans condition is violated. The maximum level of grids is restricted to . The grid has a box size of 40 AU and cell width of 0.63 AU.
Since we adopted an unstable core with a centrally peaked density profile as the initial state, the central part of the cloud collapses first and the gas density increases with time. In the collapsing cloud core, just after the central number density reaches , the first adiabatic core (hereafter the first core, Larson, 1969; Masunaga & Inutsuka, 2000) is formed with shock yr after the cloud collapse begins. The first core has a disk-like shape at its formation, with a size of 18 AU in the cylindrical radial direction and 3.2 AU in the vertical direction with a mass of . A low-velocity outflow (hereafter, outflow) of appears around the center of the collapsing cloud 121 yr after the first core formation. This kind of outflow in a collapsing cloud has been reported in many other studies (e.g., Tomisaka, 2002; Banerjee & Pudritz, 2006; Hennebelle & Fromang, 2008). The central density exceeds and the protostar (hereafter, the proto-brown dwarf) is formed 242 yr after the first core formation. Thus, the outflow begins to be driven yr before the proto-brown dwarf formation. We calculated the cloud evolution yr after the proto-brown dwarf formation, in which the first core increases in size keeping a disk-like structure and smoothly becomes the circumstellar disk (hereafter the circum-brown dwarf disk) with time. The outflow continues to be driven from the circum-brown dwarf disk and extends up to AU with a maximum speed of . From the start of the cloud collapse we calculated the cloud evolution for yr, corresponding to 11.7 , where is the freefall timescale at the center of the initial host cloud.
Figure 1 shows the mass accretion rate for the proto-brown dwarf, and the mass of the proto-brown dwarf, outflowing gas, and circum-brown dwarf disk against time () after the proto-brown dwarf formation. The outflowing gas is defined as gas with velocity for the whole computational domain, while the disk is defined as gas with and inside the host cloud. Figure 1 shows that the mass accretion rate remains almost constant at yr for yr, while it suddenly drops and has a very small value of – for yr. Note that it slightly decreases with time even for yr. Reflecting the accretion rate, the mass of the proto-brown dwarf continues to increase for yr, while it remains constant for yr. At the end of the calculation, the mass of the proto-brown dwarf is . Since the mass accretion rate is very small for yr, this object cannot acquire enough mass to become a hydrogen-burning star and hence evolves into a brown dwarf. On the other hand, the disk mass continues to increase for yr and saturates at . Finally, the disk mass reaches . Even if all the mass of the disk falls into the central object, the central object is within the brown-dwarf mass range ().
Figure 1 shows that the mass of outflowing gas increases over time and reaches . Although outflow weakens for yr, it continues to be driven from the disk until the end of the calculation. Figure 2 shows the configuration of the outflow. The iso-velocity of is represented by the transparent red surface, inside which the gas is outflowing from the center of the cloud with supersonic velocity. The red surface at the center indicates an iso-density of , which almost corresponds to the density at the border between the initial host cloud and the envelope (). The sphere enclosed by the white dotted circle () represents the initial host cloud. The figure shows that the outflow penetrates the host cloud to reach AU, about four times larger than the initial host cloud ( AU).
Figure 2 shows that the outflow has a wide opening angle inside the host cloud, while it has good collimation outside the host cloud. An hourglass-like configuration of field lines is realized inside the host cloud, because the field lines converge toward the center as the cloud collapses. The gas flows along the hourglass-like lines inside the host cloud and the outflow has a wide opening angle. On the other hand, gas flows along the straight field lines and has good collimation outside the host cloud. The opening angle of the outflow inside the host cloud strongly influences the mass accretion rate and star formation efficiency (Matzner & McKee, 2000).
The density and velocity distribution around and inside the host cloud are plotted in Figure 3. Figure 3a indicates that the residual matter in the host cloud is mainly distributed in a region along the rotation axis (i.e., -axis) and on the equatorial () plane. The source of the matter near the rotation axis is the outflow, while the matter on the plane corresponds to the disk which is supported by the centrifugal force. At the end of the calculation, of the residual mass remains inside the host cloud (). Since the mass of the proto-brown dwarf is and the initial host cloud mass is , the mass ejected from the host cloud is . In addition, since the outflowing mass is , the mass swept by the outflow into interstellar space is . Thus, in the host cloud, 34% () of the total mass is ejected by the outflow, while 20% () is converted into the star (or proto-brown dwarf).
In addition, at the end of the calculation, a large fraction () of the residual mass () has a positive flow velocity () and escapes from the host cloud. Note that we defined the outflowing mass as the gas with in Figure 1, while we defined the escaping mass as the gas with inside the host cloud. The rest of the mass (i.e., the gas with ) inside the host cloud is (), which is composed of two parts: the circum-brown dwarf disk and the accreting gas. As denoted above, since the disk mass is , the mass of the accreting matter is . Thus, the mass of the accreting matter is 3% () of the residual mass [or 1% () of the mass of the initial host cloud]. As a result, it is considered that the gas accretion is nearly finished. However, since the gas supply into the disk does not completely halt, (weak) outflow continues until the end of the calculation.
In Figure 3a, we divided the host cloud into three zones, according to our results. The outflowing gas has a supersonic velocity () in the outflowing zone, while it has a subsonic velocity () in the escaping zone. Note that almost all the matter in the escaping zone exceeds the escape velocity of the host cloud. A part of the matter in the escaping zone comes from outflow from the disk, while the matter near the border between the inflow and outflow (i.e., near the orange contour of ) escapes from the host cloud by the pressure gradient force. Since 34% of the total mass is already ejected from the host cloud, the gravity inside the host cloud weakens. As the initial state, we adopt a nearly equilibrium state, in which the gravity is balanced with the thermal pressure gradient force. Thus, owing to a decrease of mass (or gravity), the matter near the envelope escapes as the outflow continues. While the outflow remains inside the host cloud, the gas in the escaping zone in Figure 3a has a negative radial velocity () and falls toward the proto-brown dwarf. After the outflow penetrates the host cloud and a large fraction of mass is ejected, the radial velocity gradually decreases and finally becomes positive (), thereafter the gas escapes.
Figure 3b and 3c are close-up views of Figure 3a. As seen in Figure 3b, strong outflow appears in the region of AU. In addition, a cavity wall appears around the border between the outflow and inflow. During the calculation, strong outflow continues intermittently. Figure 3c shows the roots of the outflow located at AU, far from the proto-brown dwarf.
Recently, Phan-Bao et al. (2008) observed bipolar molecular outflow from a proto-brown dwarf ( Oph 102) with . The outflow extends up to AU with a maximum velocity of and mass of . In addition, they estimated that the circum-brown dwarf disk has a size of AU and mass of . In our calculation, the outflow driven from the proto-brown dwarf with has a typical flow speed of (Fig. 3b) and mass of , and extends up to AU. The circum-brown dwarf disk in our calculation has a mass of and size of AU. Thus, our results are quantitatively consistent with observations, except for the outflowing mass.
Phan-Bao et al. (2008) and Whelan et al. (2005) estimated the mass accretion rate to be , which is comparable to our result in the later accretion phase ( yr). However, their scenario for brown-dwarf formation is significantly different from our results. We consider that a part of a brown dwarf is formed in a (small, compact) cloud core in a similar process as low-mass star formation. However, in observational studies it seems to be considered that the brown dwarf formation is a scaled down version of that of hydrogen-burning stars, in which a very small rate of mass accretion (–) is assumed in the main accreton phase. We show that, even in the brown dwarf formation process, the mass accretion rate remains as high as for yr, and it rapidly drops to – for yr.
In the theoretical framework of star formation, the mass accretion rate can be described as , where is a constant (e.g., for Shu 1977, for Hunter 1977). Since gas clouds have temperatures of K (), the accretion rate in the main accretion phase is which is the minimum value attained in the general star formation scenario. Our results show, in the main accretion phase ( yr), the accretion rate of that corresponds to . Note that other effects such as cloud rotation, magnetic field, and turbulence only increase the mass accretion rate. Thus, the scenario in which the accretion rate of lasts for yr is not theoretically expected. On the contrary, it is reasonable that an accretion rate of lasts for yr for brown-dwarf formation. In this case, since the main accretion phase is very short, it is difficult to observe the accretion onto a proto-brown dwarf with . Thus, we may often observe low accretion rates of , because a lower accretion phase is expected to last for a longer duration ( yr). In this study, the accretion rate drops at yr because we adopted a less massive core as the initial state. However, we expect that if a massive core is adopted as the initial state, a high accretion rate of lasts for a long duration of – yr, and a stellar mass star forms. The result in this letter favors the idea that the stellar mass is determined by the initial size of the host cloud and hence brown dwarf sized objects are formed in less massive clouds.
We comment on the sudden drop of the mass accretion rate at yr, as seen in Figure 1. The epoch of the sudden drop almost corresponds to the epoch at which the outflow escapes from the host cloud. Since the outflow with a significant mass is escaped from the host cloud by this epoch, the gravitational potential of the host cloud shallows and the accretion rate weakens, which causes the sudden drop of the accretion rate. We expect that the epoch of this sudden drop depends on the size of the host cloud.
For the outflow model, Phan-Bao et al. (2008) proposed a jet-driven bow shock model (Masson & Chernin, 1993). However, our results are different from the predictions of this model. Our calculations show that the outflow is directly driven from the circum-brown dwarf disk. No appearance of the high-speed jet in our calculation is due to the fact that we adopted the region of AU as the sink, and thus, we did not calculate the region near the proto-brown dwarf. Adopting a central object with a mass of , the Kepler speed at AU is . Thus, flow of could not be resolved in our calculation. In contrast, Machida et al. (2008b) calculated the structure in the near proximity of the protostar and showed that a high-speed jet of is driven from the Jovian-mass protostar. Thus, when we resolve the structure in the near proximity to the proto-brown dwarf, a high-speed jet is expected to be driven from the proto-brown dwarf. As shown in Machida et al. (2008b), we expect that the high-speed jet hardly affects the mass accretion rate and star formation efficiency, because it has a well-collimated structure. Our calculations show that the wide-opening molecular outflow driven from the circum-brown dwarf disk strongly affects the mass accretion rate and star formation efficiency, as predicted in Matzner & McKee (2000).
- affiliation: Department of Physics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan; firstname.lastname@example.org
- affiliation: Department of Physics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan; email@example.com
- affiliation: Department of Physics Nagoya University Furo-cho, Chikusa-ku Nagoya, Aichi 464-8602; firstname.lastname@example.org
- affiliation: Faculty of Humanity and Environment, Hosei University, Fujimi, Chiyoda-ku, Tokyo 102-8160, Japan; email@example.com
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