Disk Formation Enabled by Enhanced Resistivity
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The Astrophysical Journal, 716, 1541–1550
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DISK FORMATION ENABLED BY ENHANCED RESISTIVITY
Ruben Krasnopolsky , Zhi-Yun Li , Hsien Shang
Submitted to ApJ 2009 October 21; accepted 2010 April 2
ABSTRACTDisk formation in magnetized cloud cores is hindered by magnetic braking. Previous work hasshown that for realistic levels of core magnetization, the magnetic field suppresses the formation ofrotationally supported disks during the protostellar mass accretion phase of low-mass star formationboth in the ideal MHD limit and in the presence of ambipolar diffusion for typical rates of cosmic rayionization. Additional effects, such as ohmic dissipation, the Hall effect, and protostellar outflow, areneeded to weaken the magnetic braking and enable the formation of persistent, rotationally supported,protostellar disks. In this paper, we first demonstrate that the classic microscopic resistivity is notlarge enough to enable disk formation by itself. We then experiment with a set of enhanced valuesfor the resistivity in the range η = 10 –10 cm s − . We find that a value of order 10 cm s − isneeded to enable the formation of a 10 AU-scale Keplerian disk; the value depends somewhat on thedegree of core magnetization. The required resistivity is a few orders of magnitude larger than theclassic microscopic values. Whether it can be achieved naturally during protostellar collapse remainsto be determined.
Subject headings: accretion, accretion disks — ISM: clouds — ISM: magnetic fields — magnetohydro-dynamics (MHD) INTRODUCTION
Magnetic Braking and Disk Formation
Circumstellar disks play a central role in both star andplanet formation. The disks are thought to be the con-duit through which Sun-like stars assemble the bulk oftheir mass (Shu, Adams, & Lizano 1987). They are alsothe birthplace of planets. Despite years of active research(see reviews by Bodenheimer 1995 for early work), westill do not have answers to some of the most basic ques-tions, such as “when and how does a disk appear in theprocess of star formation?”Observationally, it is difficult to separate out the pro-tostellar disk from the surrounding envelope during theearly, embedded phase of star formation with the cur-rent generation of interferometers (e.g., Jørgensen et al.2007), although the situation will improve drastically ina few years, when ALMA comes online (van Dishoeck& Jørgensen 2008). Enoch et al. (2009) inferred the ex-istence of a large (hundreds of AU in radius), massive(more than 0 . M (cid:12) ) disk around the protostar SerpensFIRS 1, based mostly on CARMA dust continuum ob-servations. Without a detailed knowledge of the centralstellar mass and gas kinematics on small (arcsec) scales,it is difficult to determine whether the inferred structureis a rotationally supported disk, a magnetically induced“pseudodisk,” or simply the inner part of an envelopethat is dynamically more complicated than the Ulrich’s(1976) rotating, collapsing sphere adopted by Enoch etal. (2009) for the protostellar envelope. Theoretically,disk formation in dense cores of molecular clouds mag-netized to the observed level (corresponding to a mean Academia Sinica, Institute of Astronomy and Astrophysics,Taipei, Taiwan Academia Sinica, Theoretical Institute for Advanced Re-search in Astrophysics, Taipei, Taiwan University of Virginia, Astronomy Department, Char-lottesville, USA dimensionless mass-to-flux ratio λ ∼ a few; Troland &Crutcher 2008) is controlled by magnetic braking, whichhas been difficult to quantify until recently.Detailed calculations have now shown that the mag-netic braking is apparently so efficient as to inhibitthe formation of rotationally supported disks during themain protostellar accretion phase for a moderate levelof core magnetization, as long as the magnetic field isfrozen in the matter (i.e., the ideal MHD limit; Allen, Li,& Shu 2003; Galli et al. 2006; Price & Bate 2007; Hen-nebelle & Fromang 2008; Mellon & Li 2008; see, however,Hennebelle & Ciardi 2009 for a different view, and alsoMachida et al. 2005 and Banerjee & Pudritz 2006 for theevolution of rotating, magnetized cores prior to the mainaccretion phase). In order for a rotationally supporteddisk to exist around a rapidly accreting protostar, themagnetic braking must be weakened one way or another.The most obvious possibility is through non-ideal MHDeffects, which are expected to be important in lightlyionized star-forming cores. However, we have recentlyshown that ambipolar diffusion, the best studied non-ideal MHD effect in star formation, does not sufficientlyweaken the braking to allow rotationally supported disksto form for realistic levels of core magnetization and ion-ization; in some cases, the magnetic braking is even en-hanced (Mellon & Li 2009, Krasnopolsky & K¨onigl 2002;see Basu & Mouschovias 1995, Hosking & Whitworth2004 and Duffin & Pudritz 2009, who considered the ef-fects of ambipolar diffusion on magnetic braking and corefragmentation during the earlier phase of prestellar coreevolution). This motivates us to examine whether two ofthe remaining non-ideal MHD effects, ohmic dissipationand the Hall effect, can weaken the coupling betweenthe magnetic field and the bulk neutral matter (and thusthe strength of magnetic braking) enough to enable theformation of rotationally supported disks. This paperconcentrates on the ohmic dissipation. a r X i v : . [ a s t r o - ph . S R ] J un Krasnopolsky, Li, and Shang
Electrical Resistivity of Cloud Cores
The resistivity η is related to the electrical conductivity σ through η = c πσ . (1)For lightly ionized molecular gas in dense cores, the elec-tric conductivity is usually dominated by the contribu-tion from electrons, although the grain contribution candominate at high densities when large amounts of smallgrains are present (e.g., Wardle & Ng 1999). To obtain arough estimate for the conductivity, we will concentrateon the contribution from the electrons, which is given by σ e = n e e m H m e ρ (cid:104) σv (cid:105) e − H = 9 . × x e ( s − ) , (2)where ρ = 2 . m H n H and x e = n e /n H . We haveadopted a value (cid:104) σv (cid:105) e − H = 2 × − cm s − for a 10 Kgas from Pinto & Galli (2008). This value is somewhatsmaller than that used in Draine et al. (1983) and War-dle & Ng (1999), which is 2 . × − cm s − , but some-what larger than that of Sano et al. (2000), which is1 . × − cm s − . It increases slowly with tempera-ture, roughly as T / , yielding a lower resistivity at ahigher temperature. Since our calculation is assumed tobe isothermal, the temperature dependence is not used.Substituting equation (2) into equation (1) yields η = 7 . × x e ( cm s − ) . (3)The fractional ionization x e can be computed from de-tailed chemical networks. At the relatively high densitiesthat are relevant for disk formation, it depends on thegrain size distribution. For the standard Mathis, Rumpl,& Nordsieck (1977) distribution, Nakano, Nishi, & Ume-bayashi (2002) computed the ionization fraction over alarge range of number density (their Figure 1). Overthe critical range of density ∼ –10 cm − where thetransition between the rapidly infalling pseudodisk anda potential Keplerian disk is expected to occur, the frac-tion can be fitted roughly by x e ≈ − (cid:18) cm − n H (cid:19) , (4)which corresponds to a resistivity of η = 1 . × (cid:18) ρ − g cm − (cid:19) ( cm s − ) . (5)The fit of fractional ionization deviates substantiallyfrom the computed values at low densities. However, thedeviation is expected to be inconsequential because thelow density envelope is sufficiently ionized to be close tothe ideal MHD limit (ignoring ambipolar diffusion, whichis investigated separately elsewhere, see e.g., Krasnopol-sky & K¨onigl 2002 and Mellon & Li 2009) accordingto either equation (4) or the computed values. If onlyrelatively large grains of size (of order 0 . µ m) exist,the ionization fraction would be higher (by as much as By “pseudodisk” we mean in this paper a structure formedby the magnetically induced flattening of infalling material due toanisotropic magnetic forces, as in Galli & Shu (1993). three orders of magnitude, see Figure 1 of Umebayashi& Nakano 1990, or compare Figures 1 and 3 of Wardle& Ng 1999), making the resistivity smaller. Equation(5) should therefore be viewed as a rough indication ofthe plausible upper range of the “classic” (as opposedto “anomalous”) resistivity, especially in view of the factthat the actual temperature in the high density regionof disk formation is expected to be higher than the 10 Kadopted in our calculation, which would lower the resis-tivity. It may be enhanced, however, through anomalousprocesses (e.g., Norman & Heyvaerts 1985).The rest of the paper is organized as follows. In § AU-scale disk is formed at a representa-tive time t = 10 s (or about 3 × yr) in the absenceof magnetic braking, and a magnetic field that is strongenough to suppress the formation of a rotationally sup-ported disk in the ideal MHD limit. We consider in § § AU-scale rotationally supporteddisk to reappear in our model problem. It turns out to beof order 10 cm s − , a few orders of magnitude largerthan the classic value expected on the 10 AU-scale, asanticipated by Shu et al. (2006) based on the concept ofthe so-called Ohm sphere, although our value is one totwo orders of magnitudes smaller than theirs, dependingon the field strength. The numerical results and theirastrophysical implications are discussed in the last sec-tion, § PROBLEM SETUP
Our primary goal is to illustrate the effects of ohmicdissipation on magnetic braking and disk formation. Forthis purpose, it is desirable to have as small a numer-ical magnetic diffusivity as possible, which demands ahigh spatial resolution. High resolution, however, puts astringent limit on the computation time step dt , whichis proportional to the square of the (smallest) grid sizefor the explicit method that we use to treat the ohmicdissipation. Although subcycling can be used to allevi-ate the time step problem substantially, a large numberof cycles are still needed to reach a reasonable protostel-lar accretion time, say, 10 s, making the treatment ofself-gravity difficult. For this initial exploration of theeffects of ohmic dissipation, we have decided to set up asimplified disk formation problem, with a rotating, non-self-gravitating envelope falling onto a central object offixed mass M ∗ . It will turn out that a large “anomalous”resistivity is needed to save the disk, an elaborate diskformation model is probably not warranted at this earlystage of the investigation.Specifically, we solve the equations of resistive MHD, ∂ρ∂t + ∇ · ( ρ v ) = 0 , (6) ∂ v ∂t + ( v · ∇ ) v = − ∇ pρ + J × B ρc − ∇ Φ g , (7) isk Formation Enabled by Enhanced Resistivity π J = c ∇ × B , (8) ∂ B ∂t = ∇ × ( v × B − η ∇ × B ) (9)under the assumption of axisymmetry, and adopt aspherical polar coordinate system ( r, θ, φ ), with a stel-lar object of M ∗ = 0 . M (cid:12) at the origin such that thegravitational potential is Φ g = − GM ∗ /r , neglecting theself-gravity of the gas. Self-gravity will be included in afollow-up work, where we will evolve the magnetized corefrom its formation to collapse in a self-consistent manner.For this initial study, we aim to capture the essence ofthe problem of magnetic suppression of disk formationin the simplest possible way, by adopting uniform dis-tributions for the initial gas density and magnetic field,and to study the effects of a new ingredient, resistivity,in this simplest problem.At time t = 0, we fill the computation domain be-tween r i = 1 . × cm and r o = 1 . × cm witha uniform density ρ = 1 . × − g cm − (correspond-ing to n (H ) = 3 . × cm − ), so that the total enve-lope mass is 1 M (cid:12) . For simplicity, we assume that thegas stays isothermal, with an isothermal sound speed a = ( p/ρ ) / = 2 × cm s − (corresponding to a tem-perature of about 10 K). For the initial rotation, weadopt the following prescription: v φ = v φ, tanh( (cid:36)/(cid:36) c ) (10)where we choose v φ, = 2 × cm s − , (cid:36) is the cylindri-cal radius, and (cid:36) c = 3 × cm. The softening of therotational profile inside the cylindrical radius (cid:36) c is toprevent the angular speed from becoming singular nearthe rotation axis. Since the inner radius of our com-putational domain is 10 AU, we are concerned with theformation or suppression of relatively large disks (of tensof AU or more) only. Our initial rotation is relativelyfast: Goodman et al. (1993) estimated observationallythat the ratio of rotational to gravitational energies fordense NH cores is typically ∼ .
02, which would imply v φ ∼ × cm s − for our setup. Our choice of a fasterrotational speed is conservative in that it is harder toremove a larger initial angular momentum through mag-netic braking (see also Mellon & Li 2008).The calculations are done with a new version of Zeus,dubbed “ZeusTW”. ZeusTW, written in idiomatic For-tran95, is based on the “Zeus36” code (Krasnopolsky,Li, & Blandford 1999, 2003), itself derived from Zeus3D(LCA version 3.4.2: Clarke, Norman, & Fiedler 1994);Zeus36 is parallel by domain decomposition, and utilizesdynamic memory for its field and grid arrays. A smallmemory pool of temporary arrays replaces the workerarrays of Zeus3D, increasing programming flexibility atessentially zero runtime cost. ZeusTW adds to Zeus36the ability to solve many non-ideal MHD problems in anexplicit form, covering the ohmic, Hall, and ambipolardiffusion terms. To treat the ohmic term relevant for thispaper, we used a resistivity algorithm based on Fleming,Stone, & Hawley (2000), which includes subcycling. Wetested the code by diffusing an initial Gaussian profile inCartesian geometry, in one, two, and three dimensions.We also tested slightly different forms of the algorithm(such as changing the operator splitting by calculatingthe current density J before or after the magnetic field B Figure 1.
Snapshot of the rotationally supported disk formed inthe absence of magnetic braking at a representative time t = 10 s.Logarithm of density (colormap); radial infall sonic transition v r = − a = − × cm s − (white line); poloidal velocity field (arrows,scale can be estimated from the radial infall component at the whiteline); and level of Keplerian support | v φ /v K | (dark gray dashes at90% and solid line at 100%), where v K = ( GM ∗ /(cid:36) ) / . is updated), and different subcycling prescriptions (suchas varying the maximum number of ohmic subcycles from1 to 100). The code passed all of these tests. Disk Formation in the Non-Magnetic Case
We adopt the standard hydro outflow boundary condi-tions at both the inner and outer (radial) boundaries ofthe computation domain (these boundary conditions im-pose continuity of material outflowing the grid, helpingto reduce artificial reflections of waves at boundaries).We use 400 non-uniformly spaced grid points in the ra-dial direction, corresponding to a smallest radial grid size dr min = 3 × cm (or 0 . . θ direction, corresponding to a minimum polar grid size of2 . × cm, comparable to the smallest radial grid size.With this setup, we obtain a rotationally supported diskof radius r d ≈
400 AU and mass m d ≈ . M (cid:12) at a time t = 10 s in the absence of magnetic braking; the timeis comparable to the duration of the deeply embeddedClass 0 phase (Andr´e et al. 2000, see, however, Evans etal. 2009 who concluded that the duration is longer). Asnapshot of the disk is shown in Figure 1. The hydro-dynamic disk is resolved in the radial direction by morethan 200 zones. We have experimented with coarser grids(e.g., a total of 200 radial zones and 90 angular zones),finding qualitatively similar results, which indicates thatthe resolution employed is sufficient. An example of thelower resolution runs is shown in § t = 10 s can be understood roughlyas follows. We first estimate the radius of the infall re- Krasnopolsky, Li, and Shang
Figure 2.
Same as Figure 1 but for the ideal MHD model. Thefield lines are drawn as thin solid lines (20 contour lines of magneticflux, with flux levels quadratically spaced from zero at the axis upto the flux value at the outer equator point of each magnetizedfigure). The field component B z is 470 µ G at the center of thefigure. No rotationally supported structure forms in this simula-tion, although the flow structure is strongly affected by numericalreconnection of field lines. gion r at time t , and the specific angular momentum (cid:96) of the material at that radius. Assuming that pressuregradients are negligible compared to the central gravity,we have the infall speed at any given radius to be nearthe free fall value v ( r ) = (2 GM ∗ /r ) / . The infall radiusis then roughly t ∼ r/v ∼ r / / (2 GM ∗ ) / , which yields r ∼ × cm at t = 10 s, much larger than (cid:36) c (thecharacteristic radius for rotation initial profile change).Therefore, the equatorial material at radius r has a spe-cific angular momentum of (cid:96) = rv = r × × cm s − .Conservation of angular momentum around a star ofmass 0 . M (cid:12) tells us that the specific angular momen-tum corresponds to a centrifugal radius of ∼ ∼
400 AU. We consider the agree-ment satisfactory, given the crudeness of the infall esti-mate, and the fact that most material on the surface ofthe infall sphere at radius r has a specific angular mo-mentum (cid:96) less than the above estimate (valid only for theequatorial region), and the centrifugal radius is sensitiveto (cid:96) (as (cid:96) ). Suppression of Disk Formation in the Ideal MHDLimit
As mentioned in §
1, previous studies have shown thata moderate magnetic field corresponding to a dimension-less mass-to-flux ratio of a few can completely suppressthe formation of rotationally supported disks in the idealMHD limit (at least when the rotation and magnetic axesare aligned; see Hennebelle & Ciardi 2009). Even though the mass-to-flux ratio is ill-defined in our current modelsetup that does not include the self-gravity of the en-velope gas, we find that the disk formation can still besuppressed if the magnetic field is strong enough. Anexample is shown in Figure 2, where the initially uni-form mass distribution in the computational domain ofthe hydro model shown in Figure 1 is threaded with auniform magnetic field B = √ π × µ G ≈ µ Galong the rotation axis. The field strength is in therange inferred by Troland & Crutcher (2008) for a sam-ple of dark cloud cores, after correcting for projection ef-fects. For comparison, we also considered a weaker fieldof B = √ π × µ G ≈ . µ G and obtained qualita-tively similar results. For MHD calculations, we adoptthe standard outflow boundary condition for the mag-netic field at the outer radial boundary, and a torque-free(i.e., B φ = 0) outflow boundary condition at the innerradial boundary, as in Mellon & Li (2008).The flow structure is strongly affected by numerical re-connection of the field lines. The reconnection is an un-avoidable consequence of the dragging of the field linesinto a highly pinched, split-monopole type configurationby gravitational collapse in the ideal MHD limit (Galliet al. 2006). The build up of magnetic flux in the centralregion as a result of continuous mass accretion forces theoppositely directed field lines above and below the equa-tor closer and closer together, eventually triggering (nu-merical) reconnection. It is present in other ideal MHDcalculations of magnetized core collapse, such as Mellon& Li (2008). Because of the (unavoidable) numerical ar-tifacts, it is difficult to quantify the exact strength of themagnetic braking. Nevertheless, it is clear from Figure2 that a coherent disk does not exist; it is replaced by aset of dense blobs, which are not rotationally supported(and may break up asymmetrically in 3D). The promi-nent dense blobs and sheets at radii 400 AU and smallerare all rotating well below their local Keplerian speed(especially the innermost ones), as can be seen from thethick dashed and solid dark gray lines at radii greaterthan (cid:36) ∼
400 AU in Figure 2, which mark the locationof | v φ | = 0 . v K and v K , respectively ( v K is the localKeplerian speed). The blobs and sheets are supportedto a large extent by the magnetic tension force from thepinched field lines. INABILITY FOR CLASSIC RESISTIVITY TO ENABLEDISK FORMATION
We now address the question of whether the illustra-tive classic resistivity given in Equation (5) can weakenthe magnetic braking enough to enable the formation ofrotationally supported disks or not. For this purpose,we have implemented into the MHD code ZeusTW anexplicit treatment of resistivity, following the work ofFleming, Stone, & Hawley (2000) and Miller & Stone(1997; see also Fendt & ˇCemelji´c 2002). In these treat-ments, the resistive term of the induction equation isincluded through operator splitting, respecting the con-strained transport condition that keeps the field diver-gence null to machine round-off. Numerical stability ofthis term limits the timestep ∆ t Ω to a value ∝ (∆ x ) , aquadratic requirement that can be much more stringentthan the time ∆ t IMHD ∝ (∆ x ) required for treating thehydrodynamics and the ideal MHD terms. Fortunately,the resistive term is computationally inexpensive com- isk Formation Enabled by Enhanced Resistivity Table 1
Model CharacteristicsModel B ( µ G) η Disk NotesHD 0 no yes Hydro simulationIMHD 35.4 no no Ideal MHDCR 35.4 yes no Classic resistivity given by Eq. (5)ER18 35.4 yes no Enhanced resistivity η = 10 ER18.5 35.4 yes no Enhanced resistivity η = 3 × ER19 35.4 yes no Enhanced resistivity η = 10 ER19.5 35.4 yes yes Enhanced resistivity η = 3 × ER20 35.4 yes yes Enhanced resistivity η = 10 IMHDw 10.6 no no Weaker field, ideal MHDER18w 10.6 yes no Weaker field, η = 10 ER18.5w 10.6 yes yes Weaker field, η = 3 × ER19w 10.6 yes yes Weaker field, η = 10 ER19.5w 10.6 yes yes Weaker field, η = 3 × ER20w 10.6 yes yes Weaker field, η = 10 Note — “Disk” in column 4 refers to “rotationally supporteddisk” at time t = 10 s. Enhanced resistivities η are given incgs units ( cm s − ). pared to the rest of the induction equation, and so it canbe efficiently subcycled. We have checked that subcy-cling speeds up the calculations typically by a substan-tial factor and does not change the results significantly.We have also tried small changes in the operator split-ting scheme, such as altering the order of the ideal MHDand the resistivity terms, or adding both terms together.These changes had no substantial effect on the results,and therefore we chose the scheme that was most con-venient for efficient subcycling, which is one identical tothat used in Fleming, Stone, & Hawley (2000).Figure 3 displays a snapshot of the simulation at therepresentative time 10 s. A dense flattened structuredevelops in the inner equatorial region. It is, however,fragmented as in the ideal MHD case, indicating that theclassic resistivity is too small to completely suppress thenumerical reconnection of field lines. The dense blob atradius ∼
400 AU, in particular, is similar to the outer-most blob in Figure 2, and is due to an early reconnectionevent. The artificial reconnection events make it difficultto carry out detailed analysis. Nevertheless, it is easyto show that the dynamics of the inner equatorial re-gion is far from that of a rotationally supported disk.First, the equatorial material inside ∼
400 AU is oftencounter-rotating, with a rotation speed that sometimesdecreases toward the center (rather than increasing aswould be expected for a Keplerian disk: see Figure 4).Second, the radial motion in the region is mostly super-sonic ( > × cm s − ). The rapid infall is an indicationthat the dense, flattened structure is a magnetically pro-duced pseudodisk (Galli & Shu 1993), rather than a ro-tationally supported Keplerian disk. There is one regionof supersonic outflow at ∼
50 AU. This region, and otheroutflow events that appeared during the simulation, arerelated to episodic reconnection events, as in the idealMHD model. In any case, classic resistivity does notappear capable of weakening magnetic braking enoughto enable the formation of a rotationally supported disk.Enhanced resistivity is needed if ohmic dissipation is tosave the disk. ENHANCED RESISTIVITY AND DISK FORMATION
It is well known that anomalous resistivity orders ofmagnitude larger than the classic value is required to
Figure 3.
Same as Figures 1 and 2 but for the MHD model withthe classic resistivity given in Equation (5). A dense, flattenedstructure exists near the equator, but it is a highly dynamic mag-netically produced pseudodisk rather than a rotationally supporteddisk. The flow structure is affected by numerical reconnection offield lines.
Figure 4.
Equatorial infall v r (solid line) and rotation v φ (dashed) for the MHD model with classic resistivity, showing thelack of a rotationally supported disk. Note the counter-rotationinside 400 AU, with rotation speed often decreasing (rather thanincreasing) toward the origin. The spikes of positive radial velocityat ∼
50 are related to episodic reconnection events. explain the 11-year sunspot cycle (Parker 2007, § Krasnopolsky, Li, and Shang
Figure 5.
Same as Figures 1 and 2 but for Model ER18 with anenhanced resistivity η = 10 cm s − . Figure 6.
Same as Figures 1 and 2 but for Model ER19 with anenhanced resistivity η = 10 cm s − . forming molecular cloud cores at high densities (Norman& Heyvaerts 1985), although its magnitude is uncertain.Here we consider the simplest case of a spatially uni-form resistivity of a range of values and for two differentmagnetic strength B = 35 . . µ G. The models
Figure 7.
Same as Figures 1 and 2 but for Model ER20 with anenhanced resistivity η = 10 cm s − . Figure 8.
Same as Figure 7, with reduced space resolution. are summarized, together with the ones discussed ear-lier, in Table 1. We will concentrate on three representa-tive, stronger field ( B = 35 . µ G) models with η = 10 (Model ER18), 10 (ER19), and 10 cm s − (ER20),respectively. The weaker field models yield qualitatively isk Formation Enabled by Enhanced Resistivity Figure 9.
Equatorial infall speed for Model ER18 ( η =10 cm s − , solid line), ER19 ( η = 10 cm s − , dashed), andER20 ( η = 10 cm s − , dash-dotted line) at the representativetime t = 10 s. The free-fall speed is plotted as the lower dottedline for comparison. similar results, and will be discussed briefly toward theend of the section.Figures 5–7 show snapshots of the three models withenhanced resistivity, and Figures 9–11 display their in-fall and rotation speeds and vertical field strength onthe equator. Additionally, Figure 8 shows the effect onModel ER20 of reducing numerical resolution by a factorof 2 in both r and θ . Compared to the models withoutresistivity (Model HD shown in Figure 1) and with theclassic resistivity (Model CR, Figure 3), the mass distri-bution and velocity field are much smoother, indicatingthat the enhanced resistivity in these models has largelyeliminated the numerical reconnection. The suppressionfacilitates quantitative analysis of the simulations.The mass distributions in the two lower resistivitymodels (Models ER18 and ER19) appear similar. Bothhave a dense equatorial region sandwiched from aboveand below by a more diffuse infalling envelope. The infallin the inner part of the envelope (within a few hundredAUs) appears to be guided by the field lines. It strikesthe surfaces of the dense equatorial structure at a super-sonic speed, forming two “accretion shocks.” These arereminiscent of the classic accretion shocks around Ke-plerian disks formed in non-magnetic simulations (e.g.,Bodenheimer et al. 1990; see Figure 1). Their produc-tion is due, however, mainly to the magnetic field ratherthan rotation. Indeed, the same shocks form even inthe absence of any rotation. For this reason, we willterm these shocks “the magnetically induced accretionshocks.” They surround a magnetically induced “pseu-dodisk” (Galli & Shu 1993) rather than a rotationallysupported Keplerian disk.Figure 9 shows that the dense equatorial structure inModels ER18 and ER19 contains a pseudodisk ratherthan a rotationally supported disk. In both cases, the Figure 10.
Same as Figure 9 but for the equatorial rotation speed.The Keplerian speed is plotted as the upper dotted line for com-parison. Radius (AU) E qua t o r i a l B z ( m i c r ogau ss ) Figure 11.
Same as Figure 9 but for the vertical magnetic fieldcomponent B z = − B θ on the equator. infall is supersonic ( > × cm s − ) within 10 AU ofthe central object, although there is significant differencein the infall speed within ∼
200 AU: the equatorial ma-terial in the less resistive model ER18 falls inward muchmore slowly than that in the more resistive model ER19.The prominent deceleration of infalling material around ∼
200 AU in Model ER18 is reminiscent of the ambipo-lar diffusion-induced C-shock (or C-transition) first dis-cussed in Li & McKee (1996; see also Ciolek & K¨onigl1998; Li 1999; Krasnopolsky & K¨onigl 2002; Tassis &Mouschovias 2007 and especially Figure 3 of Mellon &
Krasnopolsky, Li, and Shang
Li 2009). The physical origin is also similar. Just as am-bipolar diffusion, the ohmic dissipation enables the fieldlines to diffuse relative to the material that plunges intothe central object. The net effect is the formation of astrong (poloidal) field region at small radii, where theinward advection of the field lines by infalling materialis balanced by the resistivity-induced outward magneticdiffusion. A relatively small resistivity such as that inModel ER18 allows for relatively larger magnetic gradi-ents and Lorentz forces, which decelerate the infallingmaterial more strongly. When the resistivity is small,the magnetic field remains a significant barrier to proto-stellar accretion.The magnetic barrier weakens as the resistivity in-creases. In Model ER19, which is ten times more re-sistive than Model ER18, the ohmic dissipation-induceddeceleration has all but disappeared. This is becausethe higher resistivity can support only a smaller cur-rent, making it hard to effectively “trap” the magneticflux that is dragged to the small radii by the collaps-ing flow. Nevertheless, the infall speed is considerablysmaller (by about a factor of two or so) than the free-fallspeed, which is also plotted in Figure 9 for comparison.The sub-free-fall motion comes about because the equa-torial infall is retarded by a combination of centrifugaland magnetic forces, although the magnetic forces ap-pear to be the more important contributor, since therotation speed shown in Figure 10 is about 1/3 of theKeplerian speed or less (yielding a centrifugal force anorder of magnitude weaker than the gravity).Note from Figure 10, for the least resistive modelER18, the equatorial rotation speed within ∼
200 AUof the origin decreases, rather than increases, toward thecenter. Obviously, the decrease is due to magnetic brak-ing, which is apparently so efficient as to produce a regionof counter-rotation at ∼
100 AU. The counter rotationis reminiscent of the case of magnetized core collapse inthe presence of ambipolar diffusion with a relatively highcosmic ray ionization rate 10 − s − studied by Mellon &Li (2009; see the bottom panel of their Figure 4). In bothcases, the efficient braking is due to the strong poloidalfield trapped by infall at the small radii. It removes es-sentially all of the angular momentum of the equatorialmaterial that crosses the inner boundary of the compu-tation domain. In the more resistive model ER19, theequatorial rotation speed inside ∼
200 AU is initiallykept nearly constant, before increasing rapidly. The peakrotation speed remains well below the local Keplerianspeed, indicating that a rotationally supported disk is notformed. We conclude that the dense, flattened, equato-rial structure produced in both Models ER18 and ER19feature dense pseudodisks, but no rotationally supporteddense disks. These pseudodisks are thinner than the Ke-plerian disk shown in Figure 1 for the hydro case becauseof vertical compression by pinched field lines, and smallercolumn densities (due to the fast infall rate inside thepseudodisk), corresponding to reduced scale height fora given gas density. The absence of a rotationally sup-ported disk in these two cases (which have resistivitiesmuch larger than the classical value) supports our re-sults in § ∼
150 AU at the representative time t =10 s (see Figure 7), which turns out to be rotationallysupported. The strongest evidence for rotational supportcomes from Figure 10, which shows that the rotationspeed at the equator is nearly identical to the Keplerianspeed inside ∼
150 AU. The support is also evident inFigure 9, which shows that the infall has nearly stoppedat small radii (the spike near the inner edge of the com-putation grid is likely related to the outflow boundarycondition, which is not ideal for a rotationally supporteddisk; it was present in the hydro case as well). Thethick dashed and solid dark gray lines in Figure 7, whichmark the location of | v φ | = 0 . v K and v K , respectively,also make the case. The disk has a mass ∼ . M (cid:12) at t = 10 s, corresponding to an average accretion rate(from the envelope to the disk) ∼ × − M (cid:12) yr − . Asin the pure hydro case ( § B z , is plottedas a function of time for all three models with differentenhanced resistivities. The equatorial infall in the pseu-dodisk tends to compress the poloidal field lines, leadingto a sharp increase in the poloidal field strength at smallradii. Resistivity, on the other hand, tends to smooth outthe field distribution. The net effect is that the poloidalfield strength at small radii depends strongly on the mag-nitude of the resistivity, being larger for smaller resistiv-ities. Since most of the magnetic braking occurs at rela- isk Formation Enabled by Enhanced Resistivity η ∼ v K H , where H is the vertical scale lengthover which the toroidal magnetic field varies. If we take H to be the disk pressure scale height (ignoring possiblemagnetic compression of the disk), and use the standardresult for a thin-disk H/r ∼ a/v K (where r is the ra-dius and a is the isothermal sound speed), then η ∼ ra or r ∼ η/a ∼ . × AU (for η = 10 cm s − and a = 2 × cm s − ). This is about a factor of two largerthan the disk radius that we found numerically in Figure7. We should caution the reader that the above estimatedid not take into account the detailed magnetic geom-etry and braking efficiency, which depends on the fieldstrength (see below). For a weak enough field, a disk canform independent of the magnitude of the resistivity.Besides the three representative models discussedabove, we have carried out two additional simulationswith η = 3 × (Model ER18.5) and 3 × cm s − (Model ER19.5). As one would expect based on theresults for Models ER18 and ER19, no rotationallysupported disk forms in Model ER18.5. There is asmall ( ∼
40 AU in radius) rotationally supported diskin the more resistive Model ER19.5 at the fiducial time t = 10 s. The disk shrinks with time, however. By t = 2 × s, its radius decreases to ∼
25 AU; it maydisappear altogether at a later time. The trend indicatesthat η = 3 × cm s − is probably close to the criticalresistivity η c needed for disk formation.The value for the critical resistivity η c depends on theinitial field strength B . This is to be expected since, inthe limit of infinitely weak field, rotationally supporteddisks can form without any (enhanced) resistivity at all(i.e., η c = 0). For a moderately magnetized dense coreof B = 10 . µ G, we find that the transition betweenthe formation of a rotationally support disk and its sup-pression occurs between η = 10 (Model ER18w) and3 × cm s − (Model ER18.5w). This value of B isprobably close to the lower limit to the field strengthin dense cores, judging from the fact that the medianfield strength for the more diffuse, cold neutral medium(CNM) of atomic gas is ∼ µ G (Heiles & Troland 2005) and that the directly measured line-of-sight componentof the magnetic fields in a number of dense cores is ofthis order or higher (Crutcher & Troland 2008); the fullstrength of the core magnetic field is likely significantlyhigher. It is therefore reasonable to expect the criticalvalue η c for disk formation in cloud cores magnetized toa realistic level to lie somewhat between η c ∼ × and ∼ × cm s − . In what follows, we will take η c = 10 cm s − as the characteristic critical value,with the understanding that it depends somewhat on thefield strength, and can be uncertain by a factor of ∼ DISCUSSION AND CONCLUSION