The role of turbulent magnetic reconnection on the formation of rotationally supported protostellar disks
aa r X i v : . [ a s t r o - ph . GA ] S e p Draft version March 6, 2018
Preprint typeset using L A TEX style emulateapj v. 04/20/08
THE ROLE OF TURBULENT MAGNETIC RECONNECTION ON THE FORMATION OF ROTATIONALLYSUPPORTED PROTOSTELLAR DISKS
R. Santos-Lima , E. M. de Gouveia Dal Pino , A. Lazarian Draft version March 6, 2018
ABSTRACTThe formation of protostellar disks out of molecular cloud cores is still not fully understood. Underideal MHD conditions, the removal of angular momentum from the disk progenitor by the typicallyembedded magnetic field may prevent the formation of a rotationally supported disk during themain protostellar accretion phase of low mass stars. This has been known as the magnetic brakingproblem and the most investigated mechanism to alleviate this problem and help removing the excessof magnetic flux during the star formation process, the so called ambipolar diffusion (AD), has beenshown to be not sufficient to weaken the magnetic braking at least at this stage of the disk formation.In this work, motivated by recent progress in the understanding of magnetic reconnection in turbulentenvironments, we appeal to the diffusion of magnetic field mediated by magnetic reconnection as analternative mechanism for removing magnetic flux. We investigate numerically this mechanism duringthe later phases of the protostellar disk formation and show its high efficiency. By means of fully 3DMHD simulations, we show that the diffusivity arising from turbulent magnetic reconnection is ableto transport magnetic flux to the outskirts of the disk progenitor at time scales compatible with thecollapse, allowing the formation of a rotationally supported disk around the protostar of dimensions ∼
100 AU, with a nearly Keplerian profile in the early accretion phase. Since MHD turbulence isexpected to be present in protostellar disks, this is a natural mechanism for removing magnetic fluxexcess and allowing the formation of these disks. This mechanism dismiss the necessity of postulatinga hypothetical increase of the Ohmic resistivity as discussed in the literature. Together with ourearlier work which showed that magnetic flux removal from molecular cloud cores is very efficient, thiswork calls for reconsidering the relative role of AD for the processes of star and planet formation.
Subject headings: diffusion — ISM: magnetic fields — MHD — turbulence — star formation —accretion disks INTRODUCTION
Circumstellar disks (with typical masses ∼ . ⊙ and diameters ∼
100 AU) are known to play a funda-mental role in the late stages of star formation and alsoin planet formation. However, the mechanism that al-lows their formation and the decoupling from the sur-rounding molecular cloud core progenitor is still not fullyunderstood (see, e.g., Krasnopolsky et al. 2011 for a re-cent comprehensive review). Former studies have shownthat the observed embedded magnetic fields in molecu-lar cloud cores, which imply magnetic mass-to-flux ra-tios relative to the critical value a few times larger thanunity (Crutcher 2005; Troland & Crutcher 2008) are highenough to inhibit the formation of rationally supporteddisks during the main protostellar accretion phase of lowmass stars, provided that ideal MHD applies. This hasbeen known as the magnetic braking problem (see e.g.,Allen et al. 2003; Galli et al. 2006; Price & Bate 2007;Hennebelle & Fromang 2008; Mellon & Li 2008).Proposed mechanisms to alleviate this problem andhelp removing the excess of magnetic flux during thestar formation process include non-ideal MHD effectssuch as ambipolar diffusion (AD) and, to a smallerdegree, Ohmic dissipation effects. The AD, which Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas,Universidade de S˜ao Paulo, R. do Mat˜ao, 1226, S˜ao Paulo, SP05508-090, Brazil Department of Astronomy, University of Wisconsin, Madison,WI 53706, USA was first discussed in this context by Mestel & Spitzer(1956), has been extensively investigated since then (e.g.,Spitzer 1968; Nakano & Tademaru 1972; Mouschovias1976, 1977, 1979; Nakano & Nakamura 1978; Shu 1983;Lizano & Shu 1989; Fiedler & Mouschovias 1992, 1993;Li et al. 2008; Fatuzzo & Adams 2002; Zweibel 2002).In principle, AD allows magnetic flux to be redis-tributed during the collapse in low ionization regions asthe result of the differential motion between the ion-ized and the neutral gas. However, for realistic lev-els of core magnetization and ionization, recent workhas shown that AD does not seem to be sufficient toweaken the magnetic braking in order to allow rota-tionally supported disks to form. In some cases, themagnetic braking has been found to be even enhancedby AD (Mellon & Li 2009; Krasnopolsky & K¨onigl 2002;Basu & Mouschovias 1995; Hosking & Whitworth 2004;Duffin & Pudritz 2009; Li et al. 2011). These findingsmotivated Krasnopolsky et al. (2010) (see also Li et al.2011) to examine whether Ohmic dissipation could be ef-fective in weakening the magnetic braking. They claimedthat in order to enable the formation of persistent, ro-tationally supported disks during the protostellar massaccretion phase a highly enhanced resistivity, or “hyper-resistivity” η & cm s − of unspecified origin would See however a recent work that investigates the effects of ADin the triggering of magneto-rotational instability in more evolvedcold, proto-planetary disks where the fraction of neutral gas ismuch larger (Bai & Stone 2011).
SANTOS-LIMA, DE GOUVEIA DAL PINO & LAZARIANbe required. Although this value is somewhat depen-dent on the degree of core magnetization, it impliesthat the required resistivity is a few orders of magnitudelarger than the classic microscopic Ohmic resistivity val-ues (Krasnopolsky et al. 2010).On the other hand, Machida et al. (2010) (see alsoInutsuka et al. 2010; Machida et al. 2011) performedcore collapse three-dimensional simulations and foundthat, even with just the classical Ohmic resistivity, atinny rotationally supported disk can form at the be-ginning of the protostellar accretion phase (see alsoDapp & Basu 2010) and grow to larger, 100-AU scalesat later times. They claim that the later growth of thecircumstellar disk is caused by the depletion of the in-falling envelope. As long as this envelope remains moremassive than the circumstellar disk, the magnetic brak-ing is effective, but when the circumstellar disk becomesmore massive, then the envelope cannot brake the diskanymore. In their simulations, they assume an initiallymuch denser core than in Krasnopolsky et al. work,which helps the early formation of a tiny rotating diskfacilitated by the Ohmic diffusion in the central regions.But they have to wait for over 10 yr in order to allow alarge-scale rotationally supported, massive disk to form(see discussion in § §
2, we discuss the theoretical foundations of ourwork, in § § §
5, we discuss our results withina bigger picture of reconnection diffusion processes. Oursummary is presented in § THEORETICAL CONSIDERATIONS
The magnetic diffusion mechanism that we address isthe process deeply rooted in microscopic physics of howmagnetic fields behave in highly conductive flows. Thetextbook way to characterize these flows is to use theLundquist number S = L x V A /η , where η is Ohmic dif-fusivity and L x and V A are the scale of system at handand the Alfv´en velocity, respectively. For astrophysicalsystems L x is very large and therefore S ≫
1. The bruteforce numerical study of such systems is not feasible asthe corresponding Lundquist numbers of the numericalsimulations are much smaller. However, if the Lundquistnumber of the flow does not control magnetic reconnec-tion this opens prospects of modelling magnetic diffusionin astrophysical systems.The classical magnetic reconnection that follows thetextbook Sweet-Parker scenario (see Figure 1, upperpanel), depends on the Lundquist number and it is slowfor astrophysical systems. Indeed, in the model all thematter moving with the speed V rec over the scale L x should be ejected with the Alfv´en velocity through a thinslot ∆. The disparity of astrophysical typical scales L x and the scale ∆ determined by microphysics, i.e. resis-tivity, makes Sweet-Parker reconnection rate negligiblysmall for most of astrophysical applications, includingthe case of accretion disks. However, this scenario is notvalid in the presence of ubiquitous astrophysical turbu-lence (see Figure 1, lower panels). For the turbulencecase, LV99 showed that the reconnection becomes inde-pendent of the resistivity, i.e. becomes fast, as the out-flow region ∆ gets determined by magnetic field wander-URBULENT RECONNECTION ON PROTOSTELLAR DISKS 3 ∆∆ λλ x L Sweet−Parker modelTurbulent model blow up
Fig. 1.—
Upper plot : Sweet–Parker model of reconnection. Theoutflow is limited by a thin slot ∆ determined by Ohmic diffusivity.The other scale is an astrophysical scale L x ≫ ∆. Middle plot :reconnection of weakly stochastic magnetic field according to LV99.The outflow is limited by the magnetic field wandering.
Low plot :an individual small-scale reconnection region. From Lazarian et al.2004). ing. This challenges the well-rooted concept of magneticfield frozenness for the case of turbulent fluids (see morein Eyink et al. 2011, henceforth ELV11) and provides aninteresting way of removing magnetic flux out of, e.g.accretion disks. The LV99 model was successfully testednumerically in Kowal et al. (2009).The justification of the reconnection diffusion con-cept in L05 is based on the LV99 model (see alsoLazarian et al. 2004 for the case of magnetic reconnec-tion in partially ionized gas). Numerical effects are al-ways a concern when dealing with reconnection and mag-netic field diffusion. Indeed, unlike the numerical testsin Kowal et al. (2009), in our simulations the reconnec-tion events are happening on small scales where numer-ical effects are important. Precisely because of that,the numerical experiments with anomalous resistivity inKowal et al. (2009) are of key importance. There, usinga numerical setup with high resolution in a magnetic re-connection layer, Kowal et al. (2009) showed that in thepresence of turbulence the local non-linear enhancementsof resistivity were not important. This confirmed the cor-responding analytical prediction in LV99 (see more dis-cussion in ELV11). Appealing to that finding, we claimthat the reconnection diffusion that we observe in oursimulations is a real effect and not a numerical artefact .Analytical studies summarized in ELV11 also support thenotion that magnetic fields are generically not frozen-in As it was discussed in LV99 and shown even more explicitlyin ELV11, the plasma effects that can enhance the local reconnec-tion speed are not important in the presence of turbulence whichinduces magnetic field wandering. when conductive fluids are turbulent. In view of them,the concept of reconnection diffusion in L05 looks verynatural.As emphasized before, the concept above of magneticflux transport in turbulent flows by reconnection dif-fusion has been already successfully tested numericallyby Santos-Lima et al. (2010) for idealized models of starforming clouds. In the present paper, we study whetherthe same concept can entail substantial changes for themagnetic field removal in the formation of protostellardisks. An extension to accretion disks in general can bealso foreseen (see below). NUMERICAL SETUP AND INITIAL DISK CONDITIONS
To investigate the formation of a rotationally sup-ported disk due to turbulent reconnection magnetic fluxtransport, we have integrated numerically the followingsystem of MHD equations: ∂ρ∂t + ∇ · ( ρ u ) = 0 (1) ρ (cid:18) ∂∂t + u · ∇ (cid:19) u = − c s ∇ ρ + 14 π ( ∇ × B ) × B − ρ ∇ Ψ + f (2) ∂ A ∂t = u × ∇ × A − η Ohm ∇ × ∇ × A (3)where ρ is the density, u is the velocity, Ψ is thegravitational potential generated by the protostar, B isthe magnetic field, and A is the vector potential with B = ∇ × A + B ext (where B ext is the initial uniformmagnetic field). f is a random force term responsiblefor the injection of turbulence. An isothermal equationof state is assumed with uniform sound speed c s .We solved the MHD equations above in a three-dimensional domain using a second-order shock-capturing Godunov scheme and second-order Runge-Kutta time integration. We employed a modified versionof the code originally developed by Kowal et al. (2007),using the HLL Riemann solver to obtain the numericalfluxes in each time step.In order to compare our results with those ofKrasnopolsky et al. (2010), we have considered the sameinitial conditions as in their setup.Our code works with cartesian coordinates and vectorfield components. We started the system with a collaps-ing cloud progenitor with initial constant rotation (seebelow) and uniform magnetic field in the z direction.Given the cylindrical symmetry of the problem, weadopted circular boundary conditions. Eight rows ofghost cells were put outside a inscribed circle in the xy plane. For the four outer rows of ghost cells, we adoptedfixed boundary conditions in the radial direction in everytime step, while for the four inner ghost cells, linear in-terpolation between the initial conditions and the valuesin the interior bound of the domain were applied for thedensity, velocity, and vector potential. With this imple-mentation the vector potential A has kept its initial nullvalue. Although this produces some spurious noisy com-ponents of B in the azimuthal and vertical directions inthe boundaries, these are too far from the central regionsof the domain to affect the disk evolution. In the z direc-tion, we have applied the usual open boundary conditions SANTOS-LIMA, DE GOUVEIA DAL PINO & LAZARIAN(i.e., zero derivatives for all conservative quantities: den-sity, momentum and potential vector). We found thisimplementation far more stable than using open bound-ary conditions in the x and y directions, or even in theradial direction. Besides, adopting circular rather thansquare boundaries prevented the formation of artificialspiral arms and corners in the disk.For modelling the accretion in the central zone, thetechnique of sink particles was implemented in the codein the same way as described in Federrath et al. (2010).A central sink with accretion radius encompassing 4 cellswas introduced in the domain. The gravitational forceinside this zone has a smoothing spline function identicalto that presented in Federrath et al. (2010). We do notallow the creation of sink particles elsewhere, since weare not calculating the self-gravity of the gas. We notethat this accreting zone essentially provides a pseudo in-ner boundary for the system and for this reason the dy-namical equations are not directly solved there where ac-cretion occurs, although we assure momentum and massconservation.The physical length scales of the computational do-main are 6000 AU in the x and y directions and 4000AU in the z direction. A sink particle of mass 0 . ⊙ is put in the center of the domain. At t = 0, the gashas a uniform density ρ = 1 . × − g cm − and asound speed of c s = 2 . × cm s − (which impliesa temperature T ≈ . µ K, where µ is mean molecu-lar weight in atomic units). The initial rotation profileis v Φ = c s tanh( R/R c ) (as in Krasnopolsky et al. 2010),where R is the radial distance to the central z -axis, andthe characteristic distance R c = 200 AU.We employed a uniform resolution of 384x384x256which for the chosen set of parameters implies that eachcell has a physical size of 15 . . Although we are interested in the disk that forms in-side a radius of approximately 400 AU around the centralaxis, we have carried out the simulations in a much largerregion of 6000 AU in order to keep the dynamically im-portant central regions of the domain free from any outerboundary effects. RESULTS
We performed simulations for four models which arelisted in Table 1. Model hydro is a purely hydrody-namical rotating system. All the other models havethe same initial (vertical) magnetic field with intensity B z = 35 µ G. In order to have a benchmark, in the modelnamed resistive we included an anomalous high resis-tivity, with a magnitude about 3 orders of magnitudelarger than the Ohmic resistivity estimated for the sys-tem, i.e., η = 1 . × cm s − . According to theresults of Krasnopolsky et al. (2010), this is nearly theideal value that the magnetic resistivity should have in We note that in the two-dimensional simulations ofKrasnopolsky et al. (2010), they use a non-uniform mesh with amaximum resolution of 0 . order to remove the magnetic flux excess of a typical col-lapsing protostar disk progenitor and allow the formationof a rotationally sustained disk. We have thus includedthis anomalous resistive model in order to compare withmore realistic MHD models that do not appeal to thisresistivity excess.We have also considered an MHD model with turbu-lence injection (labeled as turbulent model in Table 1).In this case, we introduced in the cloud progenitor asolenoidal turbulent velocity field with a characteristicscale of 1600 AU and a Mach number M S ≈ − t = 0 until t = 3 × s (or ≈ resistive model: η turb ∼ V turb L inj ∼ cm s − . The induced turbulent ve-locity field has been intentionally smoothed beyond a ra-dius of 800 AU, by a factor exp {− [ R ( AU ) − / } ,in order to prevent disruption of the cloud at large radii.The injection of turbulence was stopped at t = 4 . × s ≈ .
015 Myr. From this time on, it naturally decayedwith time, as one should expect to happen in a real sys-tem when the physical agent that injects turbulent in thecloud ceases to occur.The last of the models (which is labeled as ideal MHD)has no explicit resistivity or turbulence injected so that inthis case the disk evolves under an ideal MHD condition.
TABLE 1Summary of the models η Ohm η turb Model B ( µ G) (cm s − ) (cm s − ) hydro resistive
35 1 . × turbulent
35 0 ∼ ideal MHD
35 0 0
Figure 2 shows face-on and edge-on density maps ofthe central slice of the disk for the four models at ≈ . ≈
300 AU which is typicalof a Keplerian supported disk (see also Figure 3, top-rightpanel).In the case of the ideal-MHD model (second row panelsin Figure 2), the disk core is much smaller and a thin,low density outer part extends to the outskirts of thecomputational domain. The radial velocity componentis much larger than in the pure hydrodynamical model.The bending of the disk in the core region is due to theaction of the magnetic torques. As the poloidal field linesare dragged to this region by the collapsing fluid, largemagnetic forces develop and act on the rotating flow.The resulting torque removes angular momentum fromthe inner disk and destroys its rotational support (seealso Figure 3, upper panels).The third column (from left) of panels in Figure 2shows the resistive MHD model. As in Model 1, a torus(of radius ≈
250 AU) with a rotationally dominant ve-URBULENT RECONNECTION ON PROTOSTELLAR DISKS 5locity field is formed and is surrounded by a flat, lowdensity disk up to a radius of ∼
500 AU. Compared tothe ideal MHD model (second column), the structure ofthe magnetic field is much simpler and exhibits the fa-miliar hourglass geometry.The last column (on the right of Figure 2) shows theideal MHD model with injected turbulence (labeled tur-bulent ). A high density disk arises in the central regionwithin a radius of 150 AU surrounded by turbulent de-bris. From the simple visual inspection of the velocityfield inside the disk one cannot say if it is rotationallysupported. On the other hand, the distorted structure ofthe magnetic field in this region, which is rather distinctfrom the helical structure of the ideal MHD model, isan indication that magnetic flux is being removed by theturbulence in this case. The examination of the velocityand magnetic field intensity profiles in Figure 3 are moreelucidative, as described below.Figure 3 shows radial profiles of: (i) the radial velocity v R (top left), (ii)the rotational velocity v Φ (top right);(iii) the inner disk mass (bottom left); and (iv) the ver-tical magnetic field B z (bottom right) for the models ofFigure 2. v R and v Φ were averaged inside cylinders cen-tered in the protostar with height h = 400 AU and thick-ness dr = 20 AU. Only cells with a density larger than100 times the initial density of the cloud ( ρ = 1 . × − g cm − ) were taken into account in the average evalua-tion. The internal disk mass was calculated in a similarway, but instead of averaging, we simply summed themasses of the cells in the inner region. The magneticfield profiles were also obtained from average values in-side equatorial rings centered in the protostar with radialthickness dr = 20 AU.For an ideal rotationally supported disk, the centrifu-gal barrier prevents the gas to fall into the center. Inthis ideal scenario, the radial velocity should be null (atdistances above the accretion sink zone). The top leftpanel of Figure 3 depicts the curves of the radial veloci-ties for the four models. Above the central sink accretionzone ( R > . hy-dro ) is the prototype of a rotationally supported disk,the radial velocity being smaller than the sound speed( c S = 2 × cm s − ) inside the formed disk. In theideal MHD model, the effect of the magnetic flux brakingpartially destroys the centrifugal barrier and the radial(infall) velocity becomes very large, about three timesthe sound speed. The MHD model with anomalous re-sistivity instead, shows a very similar behavior to thehydrodynamical model due to the efficient removal ofmagnetic flux from the central regions. In the case ofthe turbulent model, although it shows a persisting non-null radial (infall) speed even above R > . ≈
70 AU and ≈
150 AU).The top right panel of Figure 3 compares the rotationalvelocities v Φ of the four models with the Keplerian pro-file v K = p GM ∗ /R . All models show similar trends tothe Keplerian curve (beyond the accreting zone), exceptthe ideal MHD model. In this case, the strong suppres-sion of the rotational velocity due to removal of angularmomentum by the magnetic field to outside of the innerdisk region is clearly seen, revealing a complete failure to form a rotationally supported disk. The turbulent MHDmodel, on the other hand, shows good agreement withthe Keplerian curve at least inside the radius of ≈
120 AUwhere the disk forms. Its rotation velocity profile is alsovery similar to the one of the MHD model with constantanomalous resistivity. Both models are able to reducethe magnetic braking effects by removing magnetic fluxfrom the inner region and the resulting rotation curvesof the formed disks are nearly Keplerian. In the resistivemodel, this is provided by the hyper-resistivity, while inthe turbulent model is the turbulent reconnection thatprovides this diffusion.The bottom right panel of Figure 3 compares the pro-files of the vertical component of the magnetic field, B z ,in the equator of the four models of Figure 2. Whilein the ideal MHD model, the intensity and gradient ofthe magnetic field in the central regions are very largedue to the inward advection of magnetic flux by the col-lapsing material, in the anomalous resistive MHD model,the magnetic flux excess is completely removed from thecentral region resulting a smooth radial distribution ofthe field. In the turbulent model, the smaller intensityof the magnetic field in the inner region and smootherdistribution along the radial direction compared to theideal MHD case are clear evidences of the transport ofmagnetic flux to the outskirts of the disk due to tur-bulent reconnection (Santos-Lima et al. 2010). We notehowever that, due to the complex structure which is stillevolving, the standard deviation from the average valueis very large in the turbulent model, with a typical valueof 100 µ G (and even larger for radii smaller than 100 AU)which accounts for the turbulent component of the field.Finally, the bottom left panel of Figure 3 shows themass of the formed disks in the four models, as a func-tion of the radius. In the hydrodynamical and the MHDresistive models ( hydro and resistive ), the mass increasesuntil R ≈
250 AU and ≈
350 AU, respectively, and bothhave similar masses. The masses in the ideal MHD andthe turbulent MHD models ( ideal MHD and turbulent )increase up to R ≈
150 AU and ≈
250 AU, respectively,and are smaller than those of the other models. Nonethe-less the turbulent MHD disk has a total mass three timeslarger than that of the ideal MHD model.
Discussion
Our approach and alternative ideas
Shu et al. (2006) (and references therein) mentionedthe possibility that the ambipolar diffusion can be sub-stantially enhanced in circumstellar disks, but did notconsider this as a viable solution. The subsequent pa-per of Shu et al. (2007) refers to the anomalous resis-tivity and sketches the picture of magnetic loops be-ing reformed in the way of eventual removing magneticflux. The latter process requires fast reconnection andwe claim that in the presence of fast reconnection a morenatural process associated with turbulence, i.e. magneticreconnection diffusion can solve the problem.Krasnopolsky et al. (2010) and Li et al. (2011) showedby means of 2D simulations that an effective magneticresistivity η & cm s − is needed for neutralizingthe magnetic braking and enable the formation of a sta-ble, rotationally supported, 100 AU-scale disk around aprotostar. The origin of this enhanced resistivity is com- SANTOS-LIMA, DE GOUVEIA DAL PINO & LAZARIAN hydro ideal MHD resistive turbulent10 −2 −1 (1.4x10 −19 g cm −3 ) Fig. 2.—
Face-on (top) and edge-on (bottom) density maps of the central slices of the collapsing disk models listed in Table 1 at a time t = 9 × s ( ≈ .
03 Myr). The arrows in the top panels represent the velocity field direction an those in the bottom panels representthe magnetic field direction. From left to right rows it is depicted: (1) the pure hydrodynamic rotating system; (2) the ideal MHD model;(3) the MHD model with an anomalous resistivity 10 times larger than the Ohmic resistivity, i.e. η = 1 . × cm s − ; and (4) theturbulent MHD model with turbulence injected from t = 0 until t=0.015 Myr. All the MHD models have an initial vertical magnetic fielddistribution with intensity B z = 35 µ G. Each image has a side of 1000 AU. pletely unclear and the value above is at least two to threeorders of magnitude larger than the estimated ohmic dif-fusivity for these cores (e.g., Krasnopolsky et al. 2010).On the other hand, these same authors found that am-bipolar diffusion, the mechanism often invoked to removemagnetic flux in star forming regions, is unable to pro-vide such required levels of diffusivity (see also Li et al.2011).In this work, we have explored a different mechanism toremove the magnetic flux excess from the central regionsof a rotating magnetized collapsing core which is basedon magnetic reconnection diffusion in a turbulent flow.Unlike the Ohmic resistivity enhancement, reconnectiondiffusion does not appeal to any hypothetical processes,but to the turbulence existing in the system and fastmagnetic reconnection of turbulent magnetic fields. Oneof the consequences of fast reconnection is that, unlikeresistivity, it conserves magnetic field helicity. This maybe important for constructing self-consistent models ofdisks.In order to compare our turbulent MHD model withother rotating disk formation models, we also performed3D simulations of a pure hydrodynamical, an ideal MHDand a resistive MHD model with a hyper-resistivity co-efficient η ∼ cm s − (Figure 2). The essential fea-tures produced in these three models are in agreement with the 2D models of Krasnopolsky et al. (2010), i.e.,the ideal MHD model is unable to produce a rotationallysupported disk due to the magnetic flux excess that ac-cumulates in the central regions, while the MHD modelwith artificially enhanced resistivity produces a nearly-Keplerian disk with dimension, mass, and radial and ro-tational velocities similar to the pure hydrodynamicalmodel.The rotating disk formed out of our turbulent MHDmodel exhibits rotation velocity and vertical magneticfield distributions along the radial direction which aresimilar to the resistive MHD model (Figure 3). Thesesimilarities indicate that the turbulent magnetic recon-nection is in fact acting to remove the magnetic flux ex-cess from the central regions, just like the ordinary en-hanced resistivity does in the resistive model. We note,however, that the disk formed out of the turbulent modelis slightly smaller and less massive than the one producedin the hyper-resistive model. In our tests the later has adiameter ∼
250 AU, while the disk formed in the turbu-lent model has a diameter ∼
120 AU which is compatiblewith the observations.The effective resistivity associated to the MHD tur-bulence in the turbulent model is approximately givenby η turb ∼ V turb L inj , where V turb is the turbulent rmsvelocity, and L inj is the scale of injection of the turbu-URBULENT RECONNECTION ON PROTOSTELLAR DISKS 7 −10−8−6−4−2 0 2 0 100 200 300 400 500 V R ( c m s − ) Radius (AU) hydroideal MHDresistiveturbulent −10 0 10 20 30 40 0 100 200 300 400 500 V Φ ( c m s − ) Radius (AU) keplerian10 −4 −3 −2 d i sk m a ss ( M S UN ) Radius (AU) 10 B z ( µ G ) Radius (AU)
Fig. 3.—
Radial profiles of the: (i) radial velocity v R (top left), (ii) rotational velocity v Φ (top right); (iii) inner disk mass (bottom left);and (iv) vertical magnetic field B z , for the four models of Figure 2 at time t ≈ .
03 Myr). The velocities were averaged inside cylinderscentered in the protostar with height h = 400 AU and thickness dr = 20 AU. The magnetic field values were also averaged inside equatorialrings centered in the protostar. The standard deviation for the curves are not shown in order to make the visualization clearer, but theyhave typical values of: 2 − × cm s − (for the radial velocity), 5 − × cm s − (for the rotational velocity), and 100 µ G (for themagnetic field). The vertical lines indicate the radius of the sink accretion zone. lence . We have adjusted the values of L inj and V turb in the turbulent model employing turbulent dynamicaltimes ( L inj /V turb ) large enough to ensure that the cloudwould not be destroyed by the turbulence before form-ing the disk. We tested several values of η turb and theone employed in the model presented in Figure 2 is ofthe same order of the magnetic diffusivity of the modelwith enhanced resistivity, i.e., η turb ∼ V turb L inj ≈ cm s − . Smaller values were insufficient to produce rota-tionally supported disks. Nonetheless, further systematicparametric study should be performed in the future.As mentioned in §
1, Machida et al. (2010, 2011) havealso performed 3D MHD simulations of disk formationand obtained a rotationally supported disk solution whenincluding only Ohmic resistivity (with a dependence ondensity and temperature obtained from the fitting of theresistivities computed in Nakano et al. 2002). However,they had to evolve the system much longer, about fourtimes longer than in our turbulent simulation, in orderto obtain a rotationally supported disk of 100-AU scale. We note however, that this value may be somewhat larger inthe presence of the gravitational field (see L05, SX10)
In their simulations, a tiny rotationally supported diskforms in the beginning because the large Ohmic resistiv-ity that is present in the very high density inner regionsis able to dissipate the magnetic fields there. Later, thisdisk grows to larger scales due to the depletion of theinfalling envelope. Their initial conditions with a moremassive gas core (which has a central density nearly tentimes larger than in our models) probably helped the for-mation of the rotating massive disk (which is almost twoorders of magnitude more massive than in our turbulentmodel). The comparison of our results with theirs in-dicate that even though at late stages Ohmic, or morepossibly ambipolar diffusion, can become dominant inthe high density cold gas, the turbulent diffusion in theearly stages of accretion is able to form a light and largerotationally supported disk very quickly, in only a few10 yr.Finally, we should remark that other mechanisms to re-move or reduce the effects of the magnetic braking in theinner regions of protostellar cores have been also inves-tigated in the literature recently. Hennebelle & Ciardi(2009) verified that the magnetic braking efficiency maydecrease significantly when the rotation axis of the core SANTOS-LIMA, DE GOUVEIA DAL PINO & LAZARIANis misaligned with the direction of the regular magneticfield. They claim that even for small angles of the orderof 10 − o there are significant differences with respectto the aligned case. Also, in a concomitant work to thepresent one, Krasnopolsky et al. (2011) have examinedthe Hall effect on disk formation. They found that aHall-induced magnetic torque can diffuse magnetic fluxoutward and generate a rotationally supported disk inthe collapsing flow, even when the core is initially non-rotating, however the spun-up material remains too sub-Keplerian (Li et al. 2011).Of course, in the near future, these mechanisms mustbe tested along with the just proposed turbulent mag-netic reconnection and even with ambipolar diffusion,in order to assess the relative importance of each ef-fect on disk formation and evolution. Nonetheless, sinceMHD turbulence is expected to be present in these mag-netic cores (e.g., Ballesteros-Paredes& Mac Low 2002;Melioli et al. 2006; Le˜ao et al. 2009; Santos-Lima et al.2010, and references therein), turbulent reconnectionarises as a natural mechanism for removing magnetic fluxexcess and allowing the formation of these disks. Present work and SX10
In recent numerical study in SX10, we showed thatmagnetic reconnection in a turbulent cloud can efficientlytransport magnetic flux from the inner denser regions tothe periphery of the cloud thus enabling the cloud tocollapse to form a star.Here, also by means of fully 3D MHD simulations, wehave investigated the same mechanism acting in a rotat-ing collapsing cloud core and shown that the magneticflux excess of the inner regions of the system can be effec-tively removed allowing the formation of a rotationallysustained protostellar disk.Another empirical finding in SX10 is that the efficiencyof the magnetic field expulsion via reconnection diffusiv-ity increases with the source gravitational field. This isa natural consequence of diffusion in the presence of thegravitational field which pulls one component (gas) anddoes not act on the other weightless component (mag-netic field). In terms of the problem in hand, this impliesthat more massive protostars can induce magnetic fieldsegregation even for weaker level of turbulence. We planto explore numerically this issue in a forthcoming work.
Present result and bigger picture
In this paper we showed that the concept of reconnec-tion diffusion (L05) successfully works in the formationof protostellar disks. Together with our earlier testingof magnetic field removal through reconnection diffusionfrom collapsing clouds this paper supports a considerablechange of the paradigm of star formation. Indeed, in thepresence of reconnection diffusion, there is no necessityto appeal to ambipolar diffusion. The latter may stillbe important in low ionization, low turbulence environ- ments, but, in any case, the domain of its applicabilityis seriously challenged.The application of the reconnection diffusion conceptto protostellar disk formation and, in a more generalframework, to accretion disks in general, is natural asthe disks are expected to be turbulent, enabling our ap-peal to LV99 model of fast reconnection. An impor-tant accepted source of turbulence in accretion disksis the well known magneto-rotational instability (MRI)(Chandrasekhar 1960, Balbus & Hawley 1991) , butat earlier stages turbulence can be induced by the hy-drodynamical motions associated with the disk forma-tion. Turbulence is ubiquitous in astrophysical envi-ronments as it follows from theoretical considerationsbased on the high Reynolds numbers of astrophysi-cal flows and is strongly supported by studies of spec-tra of the interstellar electron density fluctuations (seeArmstrong et al. 1995; Chepurnov & Lazarian 2010) aswell as of HI (Lazarian 2009 for a review and refer-ences therein; Chepurnov et al. 2010) and CO lines (seePadoan et al. 2009). The application of the reconnectiondiffusion mechanism to already formed accretion diskswill be investigated in detail elsewhere. It should benoted however that former studies of the injection of tur-bulence in accretion disks have shown that at this stageturbulence may be ineffective to magnetic flux diffusionoutward (Rothstein & Lovelace 2008). SUMMARY
Appealing to the LV99 model of fast magnetic recon-nection and inspired by the successful demonstration ofremoval of magnetic field through reconnection diffusionfrom numerical models of molecular clounds in SX10 wehave performed numerical simulations and demonstratedthat:1. The concept of reconnection diffusion is applicableto the formation of protostellar disks with radius ∼ We note, however, that in the present study, we were in a highlymagnetized disk regime, where the magneto-rotational instability is ineffective.REFERENCESAllen, A., Li, Z.-Y., & Shu, F. H. 2003, ApJ, 599, 363 Armstrong, J. W., Rickett, B. J., & Spangler, S. R. 1995, ApJ,443, 209