MAGNETOSPHERIC ACCRETION AND EJECTION OF MATTER IN RESISTIVE MAGNETOHYDRODYNAMIC SIMULATIONS

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Draft version December 30, 2011Preprint typeset using LATEX style emulateapj v. 5/25/10

MAGNETOSPHERIC ACCRETION AND EJECTION OF MATTER IN RESISTIVEMAGNETOHYDRODYNAMIC SIMULATIONS

M. Cemeljic, H. Shang and T.-Y. ChiangAcademia Sinica, Institute of Astronomy and Astrophysics and Theoretical Institute for Advanced Research in Astrophysics, P.O. Box

23-141, Taipei 106, TaiwanDraft version December 30, 2011

ABSTRACT

We investigate the launching of outflows in the close vicinity of a young stellar object, treating theinnermost portion of an accretion disk as a gravitationally bound reservoir of matter. By solvingthe resistive MHD equations with our version of the Zeus-3D code with implemented resistivity, westudy the effect of magnetic diffusivity in the magnetospheric accretion-ejection mechanism. Physicalresistivity has been included in the whole computational region. We show, for the first time, thatquasi-stationary outflows consisting of axial and conical components can be launched from a purelyresistive magnetosphere. We identify four stages of magnetospheric interaction with distinctly differ-ent geometries of the magnetic field, and describe the effect of magnetic reconnection in re-shaping themagnetic field. The stages are the relaxation, reconnection and infall, after which two outflow compo-nents can be seen in a final flow: a fast axial component launched from above the star, dominated bymagnetic pressure, and a slow conical component, launched from the opened resistive magnetosphereof a disk gap, between the star and the disk inner radius. We show how outflows depend on thedisk to corona density ratio and on strength of the magnetic field, and compare the position of thedisk truncation radius with theoretical predictions. Results from previous investigations with resistiveMHD in the literature, which have been obtained with various setups, are recovered in our simulations.Comparisons are thus made easier for more general purposes, by identifying previous features in thesimulations within the different stages of our simulation.Subject headings: methods: numerical — processes: MHD — stars: formation

1. INTRODUCTION

Highly collimated outflows have been observed fromAGNs to Young Stellar Objects (YSOs) and young browndwarfs (Whelan et al. 2005, 2007). Accreting compactstars, like accreting white dwarfs in symbiotic binaries(Sokoloski & Kenyon 2003) and neutron stars like Cir X-1 (Heinz et al. 2007), also show similar outflowing phe-nomena. An outflow is characterized as a jet if it is super-magnetosonic, collimated into an apparent narrow open-ing, and reaches a stationary or quasi-stationary state.Such high-velocity outflowing fluxes of matter are an in-tegral part of stellar evolution. Observations in multiplewavelengths are reaching closer and closer to the objectsthat drive them.Among all the systems, models of launching outflows

in YSOs are closest to scrutiny by observations due toavailable data from star–forming regions. An accre-tion disk, through which matter accretes onto the youngstar with velocities close to a free–fall, is often associ-ated with a jet–driving YSO (Edwards et al. 1994, 2006).A correlation between the accretion rate and the high-velocity jet power was found in many Classical T-TauriStars (CTTSs) (Cabrit et al. 1990). The ratio of mass

loss in the outflow to disk accretion rate, Mw/Ma, ex-tracted from observations is hard to constrain. It isbest estimated to be approximately 0.1 through mea-surements of optical forbidden lines and veiling — seee.g. Hartigan et al. (1995) and Edwards (2008). Re-cently, He I λ10830 has offered a good probe into thehigh-velocity winds originating from the inner region

[email protected]

where the star interacts with the disk (Edwards et al.2003; Kwan et al. 2007). Despite the potential diagnos-tic power of such emission lines, the actual structure andphysical conditions of outflows can be more complex. Tofurther interpret the observed line profiles, and the ori-gins of outflows from the close vicinity of CTTSs, pre-dictions from both theoretical and numerical models arerequired.Outflows driven by energy derived from accretion are

particularly appealing in the scenarios of jet launch-ing. Many models have been proposed based onthe concept of magnetocentrifugal wind mechanisms(Blandford & Payne 1982), differing in the origins ofthe underlying magnetic fields and locations of matterlaunching. An outflow could be a disk wind driven bymagnetic fields dragged in from the envelope or gener-ated by the disk dynamo, or an inner disk wind an-chored to the narrow innermost region as in the X-wind model powered by an enhanced dynamo fromthe star-disk interaction (Shu et al. 1994, 1997), si-multaneously with an accretion funnel (Ostriker & Shu1995). It might also be a stellar wind driven along theopen field lines from the stellar surface by thermal ormagnetic pressure (von Rekowski & Brandenburg 2006;Romanova et al. 2005), or some combination of the dif-ferent possibilities. Related to the launch of winds, mag-netospheric accretion has been described in works byKonigl (1991), Ostriker & Shu (1995) and Koldoba et al.(2002) in the context of a magnetosphere interactingwith the surrounding disk, sharing some similarities withthe compact objects like neutron stars (Ghosh & Lamb1979a,b). Except for the pure disk wind models, a mag-

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2 Cemeljic, Shang & Chiang

TABLE 1List of assumptions for initial conditions in some relevant works

Paper κ star disk corona

Hayashi et al. (1996) 103 non-rotating in rotational equilibrium & adiabatic isothermal, non-rotatingHirose et al. (1997) 104 non-rotating adiabatic, Keplerian isothermal, hydrostatic

rotates different than diskMiller & Stone (1997) 102 rotating adiabatic, Keplerian isothermal, solid body

corotating with star at Rcor

Romanova et al. (2002) 102 rotating adiabatic, super-Keplerian adiabatic, corotating with starfor R≤ Rcor, else with disk

Kuker et al. (2003) 104 rotating adiabatic, Keplerian not in hydrostatic balance,non-rotating

Ustyugova et al. (2006) 103 rotating adiabatic, sub-Keplerian adiabatic, corotating with starfor R≤ Rcor, else with disk

Romanova et al. (2009) 104 rotating isothermal, sub-Keplerian isothermal, corotating with starfor R≤ Rcor, else with disk

netically connected star-disk system plays an importantrole in the making of the young stellar system and theevolution of angular momentum through the generationof strong outflows during the main phase of accretion.Numerical investigations have been followed up on the

time-dependent evolution of a system where the centralstar is magnetically connected to its accretion disk andtheir connection to jet formation and accretion. In oneof the earliest attempts by Hayashi et al. (1996), wheresimulations of only a few rotation periods were obtained,a dipole magnetosphere corotating with the central starthreaded the accretion disk that was in Keplerian rota-tion. Magnetic field lines connecting both the disk andthe star inflate outwards due to shear, and reconnectionblows out the matter along with the field, partially open-ing up the originally closed dipole loops. Gas can out-flow from those opened field lines and might form partof the X-ray jet that is often associated with flares. Re-connection as a possible origin of X-rays from such sys-tems has also been indicated in dal Pino et al. (2010).Hirose et al. (1997) investigated a magnetized star inter-acting with a truncated disk that was threaded with aninitially uniform field dragged in from the outer core,in the same direction of the magnetosphere, but sepa-rated by a neutral current sheet in the equatorial planeas a result of interaction between the fields brought to-gether. For simplicity, the star was not rotating, but thedifferentially rotating disk could anyway provide enoughshear to make the field inflate outwards, followed by areconnection event and mass transfer onto the magne-tosphere. The transferred mass diverted into two direc-tions: one that falls onto the star and the other that flowsout along the opened stellar field lines. Longer simula-tions by Goodson et al. (1997) with an aligned dipoleand a conducting accretion disk showed that differentialrotation of the disk can drive episodes of loop expan-sion. Such expansion can drive two outflow componentsof gas: one hot convergent flow along the rotation axis,and another, slower cold flow on the disk side of the ex-panding loop. Miller & Stone (1997), on the other hand,investigated interactions of magnetospheres with accre-tion disks under three different magnetic configurationsand their respective dynamical evolution. All of the men-tioned works involve resistive MHD and simple modelsof accretion disks.Numerical treatments of the physical processes and

disk structures have been improved over the years forstar-disk systems. Kuker et al. (2003) solved the disk in1D with a radiative hydrodynamic code by Kley (1989),and then extrapolated the solution to 2D as their initialcondition. For the full 2D axisymmetric MHD problem,the induction equation, Lorentz force and Ohmic dis-sipation were now included into Kley’s code, with theassumption of equal viscous and resistive dissipations.The main result was that for a smaller magnetic fieldthan 1 kG the disk is not disrupted; but for a larger fieldof the order of 1-10 kG, an outflow could be launchedfrom the disk. Without reaching a steady state, the cen-tral star was spun up by the prevalent angular momen-tum transported to it, while the magnetic field acted toslow it down. In Romanova et al. (2002) and Long et al.(2005), a star and part of the magnetosphere corotatedup to the corotation radius, and the magnetosphere coro-tated with the disk farther out. The corona was treatedin the ideal MHD regime, with effective numerical re-sistivity diffusing magnetic field in the radial direction.They found funnel flows onto the central object, spinningup or down the star, depending on the ratio of rotationrate of the star to the rotation rate of the disk inner rim.Romanova et al. (2009) (hereafter R09) later investi-

gated the effects of physical viscosity and resistivity.When the magnetic Prandtl number, the ratio betweenthe viscosity and the resistivity Pr = ν/η, is greater thanone, viscosity can strongly influence the solution. Theyfound that, in addition to the fast and light axial jetabove the star, there is another, new conical wind flow-ing up to 30 percent of the matter from the innermostportion of the disk. However, they required two differentsetups in their simulations, one for a slow and the otherfor a fast rotating star. In one of them, simulations werestarted with a slowly rotating star without any matterin the computational box, and then the stellar rotationwas gradually speeded up to its maximum value, withmatter slowly inflowing from the outer boundary. Ini-tial relaxation of the interaction between magnetic fieldand matter in such setup was different from most othersimulations. The comparison with previous results wasthen complicated even more when different initial andboundary conditions were used. Also, previous results,which did not show a stationary conical outflow in theliterature, were in the regime of Pr . 1. It was not clearif such a component emerged only for a slowly rotating

Magnetospheric Accretion and Ejection of Matter in Resistive MHD Simulations 3

star with viscosity larger than resistivity.We aim here to put previously published results in per-

spective, in systems of magnetospheric star-disk interac-tion. Table 1 lists the kinematic and thermodynamicassumptions in published works, each of them being usu-ally repeated with a variety of parameters or methods.Here, we focus on the hydro-magnetic part of the mecha-nism, assuming that thermal and radiation pressure bal-ance each other in the innermost star-disk region. Oneimportant parameter to distinguish the models is the al-ready mentioned magnetic Prandtl number. In R09 it isclaimed that if viscosity is larger than resistivity, a new,quasi-stationary conical component appears. However,we found this component even in our rendering of previ-ously studied cases, with resistivity larger than viscosity.Another parameter we study is the density ratio betweenthe disk and the corona. It is usually included as a freeparameter of the order of 102 or 103, at best 105, withoutmuch discussion, but from astronomical observations weknow that this ratio is a few orders of magnitude larger,up to 108. We investigate the influence of this ratio onthe mass and angular momentum flux in the launch ofoutflows. There are other possibilities in the setup, whichwe did not investigate here, e.g. inclusion of the stellarwind, which would probably affect the open stellar field.Most of the previously published findings in magneto-

spheric interactions could be identified in our simulationswithin four evolutionary stages of a single simulation,with both the star and the disk set in the computationalbox from the very beginning. The magnetic field hassimilar topology in the corresponding stages of differentsimulations, and the final stage differs only in the rela-tive intensity of the two outflow components. This in-tensity can be influenced by the dissipative mechanisms.Resistivity, which controls the onset of magnetic recon-nection, triggers the necessary change in the geometry ofthe magnetic field needed for the outflow launching.The organization of the paper is as follows. We first

describe our implementation of the boundary and initialconditions. In §3 we report regimes we found under abroad range of parameters. We investigated the influenceof corona to disk density ratio, strength of magnetic fieldand the physical resistivity. In §4 we address the role ofreconnection in the launching, in §5 we check a criterionfor the site of launching, and in §6 we compare position ofthe disk truncation radius in our simulations with sometheoretical predictions. Then we discuss investigated pa-rameters and the resulting outflows.

2. NUMERICAL SETUP FOR THE RESISTIVE MHDSYSTEM

We extend previous work of Cemeljic & Fendt (2004),who adopted a disk in the resistive MHD regimeand its halo in the ideal-MHD regime, followingCasse & Keppens (2002). We implement an absorbing,rotating stellar surface layer enclosing the origin, andinclude resistivity in the whole computational box. Theinitial conditions of density and magnetic field are shownin Figure 1, and the setup of the stellar surface as aboundary layer inside the computational box is shown inFigure 2.The equations of resistive MHD are solved using our

Fig. 1.— Standard initial and boundary conditions in our sim-ulations. The initial hydrostatic density distribution in the diskcorona and the disk is plotted in logarithmic color grading. Thedensity in the disk is four orders of magnitude larger than in thecorona. The dipole stellar magnetic field is plotted in white solidlines, and velocity vectors are shown in white arrows. The stellarsurface is defined as a rotating absorbing boundary layer, enclos-ing the origin — see the zoom into this region in Figure 2. Insimulations S2, the stellar absorbing layer extends into the smallportion of the disk mid-plane inside the disk gap, of radius Rg , asan outflow boundary. Along the axis of symmetry and at the mid-plane in the disk, a reflection and anti-reflection boundaries areimposed. An outflow boundary is imposed on the outer bound-aries of the computational domain, except for the disk outer rim,where a small inflow into the disk is set, to mimic the accretionflow from the portion of the disk beyond the computational box.

Fig. 2.— Zoom into the setup of the stellar surface from the Fig-ure 1. The star is set as a rotating, absorbing boundary conditioninside the computational box, enclosing the origin. Componentsof the poloidal velocity vp are copied from the layer immediatelyabove the star. The stellar rotation rate, determined by the initialtoroidal component of the velocity vt at the stellar surface, is keptconstant throughout the simulation.

version of the Zeus-3D code1, Zeus347 (Fendt & Cemeljic2002), in axisymmetry option. They are, in the cgs sys-tem of units:

∂ρ

∂t+∇ · (ρv) = 0 (1)

ρ

[

∂v

∂t+ (v · ∇)v

]

+∇p+ ρ∇Φ−j ×B

c= 0 (2)

1 For general description and numerical methods used in Zeuscode see Stone & Norman (1992a,b)


∂B

∂t−∇×

(

v ×B −4π

cηj

)

= 0 (3)

ρ

[

∂e

∂t+ (v · ∇) e

]

+ p(∇ · v) = 0 (4)

j =c

4π∇×B , (5)

where we neglected the Ohmic term in the energy equa-tion. For a complete set of equations, an ideal gas lawis assumed. The symbols in the equations of continuityof mass, momentum and induction equation have theirusual meaning: ρ and p are the matter density and ther-mal pressure, v is the velocity, Φ = −GM∗/(R

2+Z2)1/2

is the gravitational potential of the central object, andB and j are the magnetic field and the electrical current,respectively. In cgs units, magnetic diffusivity is equal toresistivity, so η stands for the electrical resistivity. Theresistive term in the induction equation (Equation 3) isincluded in the code by subtracting 4πηj/c from the elec-tromotive forces in the mocemfs procedure in Zeus347— see Appendix A in Fendt & Cemeljic (2002) for tests.In the energy equation, e = p/(γ − 1) is the internalenergy per unit volume. The density is related to thepressure by a polytropic and isentropic relation, and theinitial entropy follows S = ln(p/ργ), with an adiabaticindex γ = 5/3 in the initial conditions.We solve these MHD equations in dimensionless form.

The variables are normalized to their value measured inthe mid-plane of the disk, at a fiducial radius R0, whichwe choose to be inside the initial disk gap, R0 = 2.85R∗,where R∗ is the stellar radius. We include radial dis-tances up to 20 stellar radii in our computational box.The actual dimensional values are determined by sub-stituting the mass of the central object M∗, and thefiducial radius and density, with the mass accretionrate for the disk, in units of M0 = R2

0ρ0vK,0, where

vK,0 = (GM∗/R0)1/2 is the Keplerian velocity at R0.

The normalized coordinates are R′ = R/R0, Z′ = Z/R0,

and v′ = v/vK,0. The time in the code is measured in

units of the rotation rate timescale at R0, t0 = R0/vK,0,and the period at R0 is equal to 2π. The dimensionlessequation of motion can be written as:

∂v′

∂t′+ (v′ · ∇′)v′ =

j′ ×B′

M2A,0 ρ

′−

∇′p′

δ0 ρ′−∇′Φ′ , (6)

with ∇′ = R0∇, t′ = t/t0, ρ′ = ρ/ρ0, B

′ = B/B0 andΦ′ = −1/(R′2+Z ′2)1/2. Primes are omitted in equationsin the rest of this paper, and quantities are written incode units, unless otherwise specified.We introduced the free parameters:

M2A,0 ≡ 4πρ0v

2K,0/B

20 and δ0 ≡ ρ0v

2K,0/p0 . (7)

The Alfvenic Mach number, MA,0, at R0 and Z = 0,determines the magnetic field strength. In a typical run,MA,0 = 225 for a dipole field of the order of 100G, at thesurface of the star. The kinetic to thermal energy densityratio, δ0, is the square of the gas Mach number, whosefiducial value can be estimated from the definition of theadiabatic coefficient2. For our setup we choose δ0 = 100or 50. The typical temperatures in the corona and the

2 The sound speed is c2s = (∂p/∂ρ)|S = γℜT/m, where S denotes

disk of the YSOs are 106K and 5×103K, respectively, sothat at the inner edge of the disk it is cs,corona : cs,disk =0.8 : 0.05, in the units of vK,0.We use a uniform grid of R × Z = (90 × 90) cells,

in the axisymmetric cylindrical coordinates (R, φ, Z).The physical scale corresponds to (20× 20) stellar radii.We performed larger domain simulations in R × Z =(60 × 60)R∗, and simulations in higher resolution withR×Z = (180×180) grid cells in (20×20)R∗. These simu-lations lasted long enough for comparisons, but are moreprone to numerical problems, and tend to cease duringthe relaxation or soon afterwards, so that it is harder toperform a thorough parameter study using them. This isprobably because of numerical viscosity, which is largerwith less resolution, and helps the code to go throughproblematic events. Inclusion of physical viscosity wouldenable larger resolution, but we focus here only on effectsof resistivity.

2.1. Example of rescaling

We give an example of rescaling for a case of an YSOwithM∗ = 0.8M⊙, R∗ = 2R⊙, so that fiducial distance isR0 = 5.7R⊙ = 0.027 AU. Our computational domain isthen R×Z ≈ 0.2×0.2 AU. We can rewrite the Keplerianspeed at R0 in units of solar mass and radius as

vK,0 =

√

GM∗

R0=

√

GM⊙

R⊙

√

M∗

M⊙

/

(

R0

R⊙

)

, (8)

giving the fiducial velocity 1.64×107 cm s−1. The periodof Keplerian rotation at R0 is P0 = 2πR0/vK,0 = 1.76days. The stellar rotation rate for T-Tauri Stars isusually about 1/10 of the breakup rate, which we ob-tain from (GM∗/R

3∗)

1/2 = 2 × 10−4s−1 = 0.4 days.This means that period of rotation of a star shouldbe about 4 days. If we assume an accretion rate ofM0 = 10−8M⊙ yr−1, fiducial density and pressure areρ0 = 2.44 × 10−13 g cm−3 and p0 = 109 erg cm−3, re-spectively. With cs ∼ vK,0, the reference temperature isT0 = mv2K,0/(γℜ) ∼ 106 K. The reference value of re-

sistivity is η0 ∼ vK,0R0 = 1019cm2s−1, which is muchlarger than the classical Spitzer value.The strength of the magnetic field we obtain from the

magnetic pressure at the mid-plane of the disk, B0(Z =0) = (4πp0δ0)

1/2/MA,0 (see Equation 7), and the stellardipole field at R0 is B0 = B∗(R∗/R0)

3. A completeexpression for the fiducial magnetic field we can write inunits of solar mass and radius as:

B20 =

4π

M2A,0

Ma

M⊙/yr

M⊙

yr

√

GM⊙

R5⊙

M∗/M⊙

(R0/R⊙)5. (9)

The factor 4π is required to obtain the Gaussian cgs valuefrom the implicit normalization of the magnetic field inthe ZEUS code. When the surface strength of the dipolemagnetic field is combined with Equation 9, it gives

B∗ =667 G

MA,0

√

M8 , (10)

the constant entropy, and m the number of baryons per particle,with inclusion of free electrons. For hot, completely ionized hydro-gen in the corona, m = 0.5, but in a cold disk m = 1. ℜ standsfor the ideal gas constant ℜ = 8.31×107 erg K−1 mol−1, from theideal gas law p = ρℜT/m.


Fig. 3.— Initial angular velocity profiles along the disk mid-planein our setups. Shown are the angular velocities in simulations withthe stellar rotation rate Ω∗ =0.3, 0.15 and 0.068, long-dashed line,dashed line and solid line, respectively. For comparison, the dottedline shows the Keplerian rotation profile. The kink in the initialrotational profile is near the initial inner disk radius Ri.

where M8 is the disk mass accretion rate in units of10−8M⊙ yr−1. For M8 = 100 and MA,0 = 225, B∗ isabout 100G.

2.2. Boundary conditions

In order to mimic an absorbing stellar surface, we de-fine the “outflow” boundary condition around a centralobject. As done by Uchida & Shibata (1985), we definepart of the computational box surrounding the origin asa boundary layer, for a region further above the star. Weset up a rotating circular layer of one grid cell thicknesson top of the star, at a distance R∗ from the origin, withthe stellar rotation rate Ω∗ as a free parameter – seeFigure 2. All the other hydrodynamical quantities areabsorbed, so that the values in this layer are copied fromthe cells immediately above it. With this procedure, weneglect the stellar wind.The outer boundaries of our computational box are

open, with the flows extrapolated beyond the boundary.One exception is a small part at the disk outer boundary.There, we prescribe a small mass inflow that is consis-tent with the initial radial component of the velocity inthe disk. Reflection boundaries are imposed along theaxis of symmetry and, in simulation S1, at the disk mid-plane inside the disk, where the normal component of themagnetic field is continuous, the tangential component isreflected, and the toroidal magnetic field is anti-reflected.Under axisymmetry for the disk mid-plane and the axis,BR(R = 0, Z) = BR(R,Z = 0) = 0, and with these con-ditions ∇ · B = 0 is satisfied. We use the ConstrainedTransport (CT) method of Evans & Hawley (1988) to en-sure that it is preserved to a machine round-off precisionin computations.In simulation S2, we treat a small part of the disk mid-

plane inside the disk gap as an open boundary, effectivelyextending the stellar absorbing layer into the disk gap.Such an extension is to ensure preservation of the diskgap in the simulation, even when the magnetic field isnot strong enough to truncate the disk. This means that

some other physical effects, like physical viscosity or ra-diation transfer, which we do not include in our simu-lations, would have to act in terminating the disk. Us-ing such a setup, we can study if a weaker stellar dipolecan launch outflows from the innermost magnetosphere.Caveat is that the final disk truncation radius in simu-lations S2 is then dependent on the initial setup, and isnot self-consistently computed.

2.3. Initial conditions

We set up initial conditions for the density distribu-tion, velocity profiles, magnetic field and resistivity inthe computational domain as follows. We set up a ro-tating disk that is simply gravitationally bound furtheraway from the origin. For a self-consistent accretion disk,additional constraints such as constant fluxes throughsurfaces at different radii and stability to various modesof oscillation should be included. However, the disk sta-bility or the accretion process itself is not the subject ofstudy here, and we treat the disk only as a supply ofmatter into the stellar magnetosphere.

2.3.1. Density distribution

The initial disk density distribution is

ρd(R,Z) =R

3/2off

(R2off +R2)3/4

×

(

max10−6 ,

[

1−(γ − 1)Z2

2H2

]

)1/(γ−1)

, (11)

shown in Figure 1. The density is limited by the maxi-mum function, and ensured to be regular by a constantoffset radius Roff = 4. The disk is adiabatic with an in-dex γ = 5/3, and physically thin, with an aspect ratioof H/R = 0.1, where H is the disk height at a given ra-dius R. For the initial inner disk radius we tried variousinitial positions of the initial inner disk radius Ri in ourparameter study; here it is chosen to be at half size ofthe computational domain, Ri = 10R∗ or half of thatdistance. It is not a critical parameter, as the disk willadjust its inner rim position during the simulation, buttoo close positioned Ri can, especially in a case of strongmagnetic field, result in a too violent initial relaxation,which will stop a simulation.The corona above the star and in the disk gap coro-

tates with the central object, and further away, with theunderlying disk. The corona is in hydrostatic balance,with an initial coronal density3:

ρc(R,Z) =(R2 + Z2)−3/4

κ, (12)

which is obtained from the equality of gravitational andhydrostatic pressure. The free parameter κ = ρd/ρc de-termines density in the corona. In similar studies κ isusually in the range of 102 to 104. In simulations with-out magnetic field and S1a we used κ = 104, and in S1bκ = 105. We address the influence of this parameter onthe launching process in the resistive simulations, andinvestigate the range 102 to 106 for simulations S2.

3 For such setup it is essential to set a force-free initial magneticfield in the computational box—see e.g. Fendt & Elstner (1999,2000).


2.3.2. Velocity profiles

In our simulations, the initial stellar rotation rate is afree parameter, kept constant throughout the simulation.Since the time scale of change in stellar rotation is muchlonger than duration of simulations here, this constraintshould not influence the outcome. In the case of T-Tauritype stars, there are observational indications that stellarrotation rate is actually constant (Irwin et al. 2007), sothat for those objects it is a plausible assumption evenfor very long lasting simulations.The corotation radius, at which matter in the disk is

rotating with the angular velocity of the stellar surface,is:

Rcor =

(

GM∗

Ω2∗

)1/3

. (13)

The position of the corotation radius with respect to thedisk truncation radius Rt defines two regimes: Rcor >Rt, and Rcor ≤ Rt. Ustyugova et al. (2006) and R09 in-vestigated the latter as a “fast rotating” (or “propeller”)regime, and here we focus on the former, “slow rotating”regime. In this work we present results for a parame-ter study in a slow rotating regime, with stellar angularvelocity of 0.15, which gives the rotation period of 11.8days. The corresponding corrotation radius is 10.1R∗.For the disk, we adopt the following rotation profile:

vφ(R,Z) = (1− ǫ2)R

1/2off

(R2off +R2)1/4

exp

(

−2Z2

H2

)

(14)

The free parameter ǫ gives the departure from the Ke-plerian rotation profile, and is chosen to be 0.1 in ourtypical simulations. For ǫ = 0 the disk would go back tothe Keplerian profile. Figure 3 shows the initial angularvelocity profiles at the equatorial plane of the disk.The initial poloidal velocity profile is given by the ra-

dial inward velocity from accretion. The angular andsound speeds are both proportional to R−1/2 and, for adisk in hydrostatic equilibrium, the same is valid for theradial velocity. Both components of the poloidal velocityare given by

vR(R,Z) = −msǫR

1/2off

(R2off +R2)1/4

exp

(

−2Z2

H2

)

, (15)

vZ(R,Z) = vR(R,Z)Z

R.(16)

The constant parameter ms < 1 is used to obtain a sub-sonic inflow, and is chosen to be 0.1 in our simulationshere. We also performed runs with ms = 0.3 and 0.6,which give larger influxes of mass into the disk, withsimilar results — for more massive disk, simulations aremore prone to instabilities and tend to cease earlier thanfor lighter disk. The exponential factor in the equationeffectively confines the initial disk profile.

2.3.3. Magnetic field

The initial magnetic field is a pure stellar dipole, andwe computed it from the derivatives BR = −∂Aφ/∂Zand BZ = ∂(RAφ)/R∂R of a magnetic potential:

Aφ =µ∗R

(R2 + Z2)3/2. (17)

Fig. 4.— The mass flux ρv at T=300, in logarithmic color scalegrading, in a simulation without magnetic field. There is no sig-nificant mass or angular momentum outflow after the relaxationphase. Vectors show the velocity of matter. The disk reaches thestellar surface, where matter is accreted onto the star.

The stellar dipole magnetic moment µ∗ we set to unity.Setup with multipole expansion of magnetic field is feasi-ble in our simulations, but without stellar wind includedwe already neglect effects at the surface of the star. Weassume that dipole is leading term in the disk gap andbeyond.

2.3.4. Resistivity and artificial viscosity

The electrical resistivity η is defined through the elec-tric conductivity σ as η = c2/4πσ, where c is the speedof light. The ratio of the advection and diffusion termsin the induction equation (3) is the magnetic Reynoldsnumber, which is equal to the Lundquist number in ourproblem:

Rm =vA,0R0

η. (18)

The characteristic velocity is the Alfven speed vA,0 =

B0/(4πρ0)1/2 at R0(Z = 0). The reference time is the

period of rotation at this radius R0, t0vA,0, with vA,0 ∼vK,0.To explain the physical processes, the accretion disk re-

quires an enhanced, anomalous level of resistivity, whichis much larger than the classical value. The anomalousresistivity could be an effect of MHD turbulence or am-bipolar diffusion in a partially ionized medium4. Weset the initial constant resistivity of the disk to be ofthe same order of magnitude as the numerical resistivityη = ∆x2/∆t ∼ 10−4, which gives Rm ∼ 104, with ∆tin units of rotation at the outer disk radius Rmax, wherethe time step is smallest. We find the numerical resistiv-ity by lowering the disk constant resistivity in the codeuntil it does not affect the results.We omit the actual Ohmic term in the calculation of

the MHD equations. When the Ohmic part is included inthe internal energy equation in Zeus347, Equation 4 gainsan additional Ohmic heating term −ηj2. The inclusionof this term is expensive computationally. However, theactual difference from the solution without the Ohmicterm is negligible, as the p∇v term is much larger. Sim-ilar results have been reported in Miller & Stone (1997)and R09. It would take the Ohmic term many orders ofmagnitude larger to produce a visible effect.

4 For extensive discussion of physical conductivity in partiallyionized disks see e.g. Wardle & Ng (1999), Salmeron et al. (2007).


Fig. 5.— The mass flux ρv for the case with medium magneticfield in the simulation S1a. Mass flux is shown in a logarithmiccolor grading, vectors show the velocity of matter, and the whitesolid line show the poloidal magnetic field lines. The disk reachesthe stellar surface, where matter is accreted onto the star. A rar-efied, but fast axial outflow is launched from the magnetosphereimmediately above the star.

A resistive corona is essential for the magnetic re-connection to occur, and reconnection is crucial forre-organizing the magnetic field. The resistivity ismodeled as a function of matter density, followingFendt & Cemeljic (2002), so that η ∝ vAR ∼ ρ(γ−1)/2.For the adiabatic case

η = η0 ρ1/3 . (19)

To avoid unrealistically large η and too small Ohmictimesteps in the densest part of the domain, we limitthis value to the order of unity, with η0 = 3.0.Another diffusive process in our simulations is the nu-

merical viscosity. In our finite differencing numericalscheme, it is of the same order as numerical resistiv-ity. We do not treat the physical viscosity, only thevon Neumann-Richtmyer artificial viscosity is included,through a constant parameter that controls the numberof zones through which shocks are smoothed out. Suchviscous term is significant only in presence of shocks. Fora smooth flow, it is tiny, and for rarefactions, it is zero.The characteristic speed for viscous effects is the soundspeed cs.

3. SIMULATIONS

We started with a disk in hydrostatic balance and, asa reference, performed a hydrodynamic simulation with-out magnetic field. The stellar rotation rate in the caseshown in Figure 4 was set to Ω∗ = 0.15, but we triedother Ω∗, smaller and larger, with the similar outcomes.After relaxation, the disk remained stable for hundredsof revolutions, in a quasi-stationary state, connected tothe stellar equator, with the matter from the disk slowlyfalling onto the star. The disk got puffed up, similarto the situation in Cemeljic & Fendt (2004), where therewas no central object in the simulation, only the disk.

3.1. Simulations with truncation of the disk by a strongmagnetic field

We seek to understand the effects of magnetic diffusionon the launching of outflows from the innermost vicinityof an YSO. In our code, numerical resistivity and numer-ical viscosity are of the similar order of magnitude. Byincluding physical resistivity, but not physical viscosity,we probe the portion of parameter space with Prm < 1.As mentioned in the Introduction, an YSO magnetic field

Fig. 6.— Mass flux ρv in logarithmic color grading for the casewith strong magnetic field in the simulation S1b. The vectors showthe velocity of matter, and the white solid lines show the poloidalmagnetic field lines. Two outflows are launched from the magne-tosphere in the close vicinity of a star: a fast rarefied axial outflowimmediately above the star, and a slower and denser conical outflowabove the disk gap. The disk is truncated where its ram pressureand magnetic pressure are balanced. Poloidal magnetic field linesnearby the axis are removed from the shown sample, not to obscurethe underlying mass flux distribution.

is of the order of a few hundreds of Gauss to kG, so we setsuch magnetic field in our simulation. In the followingtext, we refer to such setup as simulations S1a. The stel-lar rotation rate remains the same as in the case withoutthe magnetic field, Ω∗ = 0.15.The disk is falling radially onto the star, but with a dif-

ference to the non-magnetic case that now a strong axialoutflow is launched above the star. One example is shownin Figure 5. The mass load in the outflow depends onthe stellar rotation rate and the magnetic field strength,and on the disk accretion rate. In our axi-symmetricsimulations it is not straightforward to conclude aboutthe nature and stability of outflows launched co-axially,and so close to the central object defined as a bound-ary condition. We leave it for non-axisymmetric, full 3Dsimulations.To investigate realistic star-disk systems, we need a

stable disk gap, which we do not obtain in any of simu-lations with magnetic field up to few hundreds of Gauss.Faster stellar rotation would help to establish it, becauseof a larger centrifugal force, but then the stellar rotationrate would become too large, compared to the range ex-pected for YSOs. We found that, with only the physicalresistivity included, the only way to realistically obtaina disk gap is to increase the magnetic field.When we increase the magnetic field to the order of few

hundred Gauss, magnetic pressure becomes sufficient totruncate the disk. When the disk gap stabilizes, in ad-dition to an axial outflow, a conical outflow is launched— see Figure 6. Such a result corroborates with simu-lations mentioned in §1, where, with an accretion rateof 10−8M⊙ yr−1, the magnetic field at which the diskbecomes truncated is of the order of kG. We assign thosesimulations as S1b in the following text.Our purely resistive simulations are, therefore, repro-

ducing the previously known results, that a sufficientlylarge magnetic field truncates the accretion disk — wediscuss the truncation radius in greater detail in §6 —and that a conical outflow is launched. With evenstronger magnetic field and varying the accretion rate,it is possible to modify the gap extension and the inten-sity of both axial and conical outflows, but it is not clearif the reason for launching a conical outflow is a large


magnetic field, or other conditions near the disk gap. Acaveat is that we neglected viscosity and radiation ef-fects, and fixed the stellar rotation rate. Therefore, it isnot unexpected that, to truncate the disk, a large mag-netic field is needed. It remained the only adjustableparameter.In a setup as described, simulations with large mag-

netic field tend to cease during, or not long after, the re-laxation, because of numerical problems. Without physi-cal viscosity, resistivity itself is not dissipative enough tostabilize the flow. To study the quasi-stationary state, weneed simulations lasting for hundreds of rotations. Also,in order to anticipate including of the stellar outflow andradiative effects, which could help in establishing a gap,we need to devise a way of producing the disk gap witha smaller magnetic field, of the order of tens to hundredsof Gauss.

3.2. Launching of a conical outflow

To probe the portion of the parameter space withsmaller stellar magnetic field, we devise a simulation inwhich the disk gap is numerically imposed, as describedin §2.2. Such simulation, dubbed S2 here, has been per-formed with a part of the disk mid-plane inside the diskgap defined as an open boundary. It means that the disktruncation radius is not determined self-consistently5.The results of our simulation S2 are shown in Figure 7.

We show the density and poloidal mass flux ρvp for thesame time step in the right and left side of the samepanel, to stress that in the density plots conical out-flows will typically not be visible even in logarithmic colorgrading. Instead, as shown in R09, outflows are well vis-ible in the poloidal mass flux plots in logarithmic colorgrading. This is probably one of the reasons why noticeof the conical outflows was not made earlier, despite ofmany numerical efforts in the portion of parameter spacewhere conical outflows should be appearing. Other fea-tures, as ejected plasmoid or accretion flow onto the starare well seen in both the density and mass flux plots, andhave been described in the literature mentioned in §1.Now we obtained long lasting simulations, which in all

respects resemble those from simulation S1b, but withthe difference that they last longer, and the magneticfield required for launching of outflows is smaller for anorder of magnitude. We can identify four evolutionarystages in progression in a system of an interacting mag-netosphere with its surrounding disk. In a case witha disk accretion rate 10−7M⊙/ year, for a rather smallstellar dipole field of 38 G, the system goes through sim-ilar relaxation and initial evolution in all cases, with thesimilar geometry of the poloidal field. The results arerobust in that they occur under a wide range of exploredparameters, although each with different details.An initially pure dipole magnetosphere has already

bulged out and brought some gas along with it at asfew rotations as T = 2. Near the axis, some gas alsoflows out at high velocity due to magnetic pressure thatis gradually building up, as shown in the large sizes of

5 For a large enough magnetic field, of the order of kG as in simu-lation S1b, such imposed disk gap is largely ignored by the disk, asmatter is lifted above the disk equatorial plane. In the next paper,Cemeljic & Shang (in preparation), we describe configurations inwhich an accretion funnel onto the star can form.

Fig. 7.— Snapshots in our simulation S2 with Rcor = 10R∗.To show the difference between density and mass flux plots, theleft half of each panel shows the poloidal mass flux ρvp, andthe right half shows the density. Both plots are in logarithmiccolor scales, shown at the bottom of the panels. The poloidalmagnetic field lines are shown in solid white lines, with linesin the axial region omitted, to show velocities in white arrows.The resolution is R × Z = (90 × 90) grid cells= (20 × 20)R∗,and the initial density contrast between the disk and the coronais κ = 104. The initial magnetic field is a pure stellar mag-netic dipole, with B∗ = (380, 38, 3.8)G for a disk accretion rate(10−6, 10−7, 10−8)M⊙/year, respectively. Each line shows, top tobottom, a characteristic stage discussed in this work: I) initial re-laxation when the magnetic field is swept in and pinched near thedisk mid-plane toward the star, II) inflation and reconnection thatend up opening the field, and strong infall of matter onto the starfrom the disk, III) retraction of the disk matter towards the coro-tation radius, with a transient inflow of matter onto the star, andthe light bullets of fast matter expelled along the axis, IV) final,quasi-stationary stage, with a light, fast axial outflow and a dense,slow conical outflow.


STAGE 1

STAGE 2

STAGE 3

STAGE 4

Fig. 8.— A schematic sketch of the evolution in star-disk interaction. Stage one: the initial stellar dipole gets pinched during therelaxation, when matter flushes in toward the central star; Stage two: the magnetic field lines are open after reconnection, and disk mattercan reach the surface of the star; Stage three: the disk matter retracts and a funnel flow forms from the disk inner radius and accretes amatter onto the star; Stage four: the system reaches a quasi-steady state, settling into a configuration consisting of an open stellar fieldand field footed in the disk. The arrows indicate the directions of the matter outflow. In the first three stages, the axial component is insome simulations strongly episodic (marked with gray shadow), on and off many times into the quasi-stationary state. In the fourth stage,a conical outflow forms, which is reaching a quasi-stationary state (marked with black shadowed arrow).

Fig. 9.— Change of the disk density profile in time, along theequatorial plane. We show densities at T=0,5,10,100,850 in thinsolid, dotted, dashed, long dashed and thick solid line, respectively.The disk density is substantially modified only during the relax-ation, afterwards it does not change much.

Fig. 10.— Velocity in the axial direction in simulation S2,throughout the simulation. We show the velocity in the axial out-flow at (R,Z) = (0.4R∗, Zmax) in a thin (red) solid line, and in theconical outflow at (R,Z) = (7.1R∗, Zmax) in a thick (black) solidline.

the arrows there. The matter has flown in at a magneticstagnation point around 8R∗, where the magnetic fielddragged in with the gas is pinched. Around T = 9, mat-ter went through a magnetic reconnection and ejectedplasmoids. The reconnected and opened field enabled

Fig. 11.— Velocity profiles for results from the bottom panel inthe Figure 7 at T=850. Profiles along the propagation directionof outflows in the Z-direction, parallel to the symmetry axis, atR = R∗, we show in the (Top panel) and at R = 5R∗ in the (Bot-tom panel). Components of the velocity in the Z, R and toroidaldirection are shown in thin solid, dashed and dotted black lines, re-spectively, and the total velocity is plotted in the thick solid (black)line. Alfven velocity is plotted in the long-dashed (blue) line, es-cape velocity in (green) dot-short-dashed line, and the sound speedin dot-long-dashed (red) line.

the disk gas to flush into the stellar surface and, at thesame time, more violent gas flows are directed outwardsboth from the axial region and from the disk. At a latertime, T = 45, after a few occurrences of the magneticreconnection events, part of the field closes back to thestellar surface, and part remains open, footed near thenew truncation radius. Matter channels through the field


Fig. 12.— Mass (Top panel) and angular momentum (Bottompanel) fluxes in the direction parallel to the axis of rotation, in aslice along the Zmax boundary, in the R-direction, in the quasi-stationary state from Figure 7. We show fluxes Fm and Fℓ atT=850 in solid line, and at T=852 in dashed line. The temporaryinfluence of the disk is visible in the rightmost portion of the massflux profile for T=850. It does not contribute to fast outflows, asthe matter in the disk rolls back towards the disk. To avoid suchapparent oscillation in the flux, we exclude fluxes from the outerportion of the box. Most of the angular momentum flux exits thesystem through the conical outflow, which also shows a periodicvariation in intensity.

lines that are footed both in the disk and the stellar sur-face. Matter along the axis is expelled in the form ofbullets or, when more stabilized, in a more continuousstream. The bottom panel in Figure 7 shows the repre-sentative snapshot in those simulations at a much latertime T = 850. The system settled into a configurationwhere the magnetic field has been opened into space witheither foot in the star or in the disk, and formed loopsthat connect to both the star and the disk. The gasflows out from the axial regions on top of the star like acoronal wind and from the boundary on top of the loopsalong the diverging field lines open to the space, formingan outflow stream of conic shape. Although not drivencompletely by centrifugal mechanisms, the matter sur-rounding the axial region comes from the disk where onefoot of the magnetic field is rooted. The disk materialstays slightly outside of the magnetic footpoint wherethe field is pinched, around the truncation radius. Whenthe disk accretion rate is well matched to the mass lossin the outflows, the simulations can last for hundreds ofrotations. The disk, after the relaxation, does not differmuch from the hydrodynamic case in Figure 4.What we observed in the simulations, and described

in snapshots in Figure 7, taken at different times, seemsto represent necessary steps in the evolution of the sys-tem consisting of a star with an accretion disk, whenprogressing towards a quasi-steady state. This processcan be generalized into four conceptual stages, when thesystem attempts to evolve from an initial condition as a

Fig. 13.— Time evolution of mass (Top panel) and angularmomentum (Bottom panel) fluxes in simulations S2, parallel to theaxis of rotation, along the Zmax boundary. We show the fluxesafter the relaxation in solid (black) line. In dashed (red) line weshow the average value, computed starting from T=100, when theflow becomes quasi-stationary. The mass flux average value is 3.0×10−5, and the angular momentum flux average value is 7.4×10−5.

Fig. 14.— Average mass and angular momentum fluxes Fm andFℓ in our simulations with increasing density contrast κ betweenthe disk and the corona. We show the mass flux in solid line,and the angular momentum flux in dashed line, in solutions withκ = (102, 103, 104, 105, 106). Results for the mass flux for κ = 102

in our simulations are always larger for a factor of 2 to 3, than forlarger, more realistic values of κ.

pure dipole threading everywhere into a disk in hydro-static balance.Stage I is the initial relaxation when the magnetic field

is swept in and pinched near the disk mid-plane towardthe star and the magnetic loops are twisted, inflatingand forming plasmoids. The gas that flows with thefield swirls in and is gradually accelerated in the axial re-gion by magnetic pressure being built up. Stage II takesthe scene after the system goes through a reconnectionthat ends up opening the field, enabling strong infall ofmatter onto the star from the disk. The axial compo-


Fig. 15.— Average mass and angular momentum fluxes Fm andFℓ in our simulations with increasing stellar magnetic field. Weshow the mass flux in solid line, and the angular momentum flux indashed line. With stellar magnetic field B∗ = (3, 10, 30, 77, 188)G,with disk accretion rate 10−6M⊙/year, both fluxes increase withincreasing magnetic field. If we assume 10−8M⊙/year, magneticfield strenghts are 1/10 of those values.

Fig. 16.— Change of resistivity with time in simulation S2. Weshow η in two positions (R,Z) in the computational box: at the exitregion of the outflow, in the position (7,Zmax), and in the middleof the computational box, in the position (10,10), in thick (black)and thin (red) solid lines, respectively. As expected in our modelwith resistivity in the disk corona dependent on the density, closerto the disk, resistivity is larger.

nent strengthens, forming a series of bullets as a resultof magnetic pressure from the twisted field. Stage IIIfollows when the disk matter retracts towards the coro-tation radius, and a time-variable inflow of matter fun-nels onto the star from the inner disk truncation radius.The system may have several passages through the firstthree stages and finally move onto the quasi-steady statewhen the magnetic field is pinched and strong enoughto balance the ram pressure of the disk gas, truncatingthe disk near the corotation radius. The magnetic fieldsettles into a geometry where the field is open into thespace both axially and conically, with some loops anchor-ing both in the star and the disk. Matter flows out alongthe open field lines, forming the axial and conical quasi-stationary outflows. Figure 8 is a schematic sketch of thestages.

3.3. Properties of the conical outflow

Here, we describe in more detail the properties of mat-ter in the quasi-stationary solutions with realistic mag-

netic field dipole strength, in our simulations S2.Figure 9 shows the change of density in time along the

disk equatorial plane. After the relaxation, our reservoirof matter in the simulation is not changing much. Thismeans that the inflow of matter into the disk from theouter boundary, which mimics the accretion of matterfrom the outer part of the accretion disk, is well chosen.In the case of a too large inflow of matter into the disk,it would pile up and the disk would become unstable.On the other side, if the disk would become drained ofmatter, it would change the conditions we want to inves-tigate.In Figure 10 we show the change of velocity in cho-

sen points in the computational domain. Oscillationsshow a sign of instability working at a short time scale,so that both outflow components are not smooth evenin the quasi-stationary phase. One possible reason isa permanent reshaping of the magnetic field by recon-nections along the flow — we discuss it in more detailin §4. We show the velocity along the outflows in thequasi-stationary state in Figure 11. The axial outflowis supersonic and sub-Alfvenic close to the stellar sur-face, but at the middle of the computational box it be-comes super-Alfvenic. Both the poloidal and total veloc-ity in conical outflows are larger than the escape velocityvesc = (2GM∗/R)1/2.We calculate the mass flux Fm and the angular mo-

mentum flux Fℓ in each half-plane, above or below thedisk equator. They are defined as:

Fm =

∫ R

0

2πρvZRdR ,

Fℓ =

∫ R

0

2π

(

ρvZvφR −BZBφR

4π

)

dR . (20)

Figure 12 shows the radial dependence of fluxes Fm andFℓ across the outer Z-boundary in the quasi-stationarystate, and Figure 13 shows the time evolution of thosefluxes. To avoid influence of the disk in those computa-tions, we integrate fluxes only to Rmax/2 in each timestep. During relaxation and stabilization of the systeminto quasi-stationary outflows, fluxes are large and vari-able, and become more steady afterwards. Reconnectionevents along the conical direction of the outflow con-tribute to the oscillation on the timescale of one rotationperiod. The contribution to the total fluxes from theaxial component is two orders of magnitude smaller.

3.4. Dependence on the density contrast and magneticfield strength

Table 1 shows that simulations in the literature wereperformed for various contrasts of the disk to corona den-sity κ. What is the influence of this parameter in oursimulations?We show results of our parameter study in Figure 14.

There are no significant differences in the average fluxesin the range 103 to 106. For κ = 102, the mass flux isalways larger, so that it is double or, in some setups,triple the value from other cases. This trend, which isillustrated here for a case of sub-Keplerian rotation ofthe disk, is true also in the case of Keplerian rotationprofile of the disk. Hence, results of simulations withκ = 102 could be unrealistic in the case of launching of


Fig. 17.— Snapshots of poloidal magnetic field lines in the simulation S2 show the reshaping of the magnetic field during simulation.The initial stellar dipole is pinched into a plasmoid and ejected during relaxation process (Left panel). Afterwards, reconnection enablesthe change of magnetic field geometry (Middle panel) into the stellar and disk components (Right panel).

Fig. 18.— Elsasser number Λ in simulation S2. Above the line inthe computational box where the Elsasser number Λ equals unity,it is Λ > 1 and launching of an outflow is possible. Below that line,Λ < 1 and the launching of matter can not occur.

the astrophysical outflows. This is not surprising, takinginto account that the realistic value of κ is of the orderof 108 for YSOs.We also check how the mass fluxes in simulation S2

depend on the magnetic field strength. Figure 15 showsthe average fluxes for magnetic fields up to the orderof 0.2 kG, for YSOs with a disk mass accretion rate of10−6M⊙ yr−1. Both the mass and angular momentumfluxes increase with increasing magnetic field strength.

4. RECONNECTION AND OPENING OF MAGNETIC FIELDLINES

Reconnection is essential for launching outflows froma star-disk magnetosphere. If, for some reason, recon-nection does not occur, a magnetic wall, or a magnetictower form, and the evolution of the system will be differ-ent. The diffusive processes, as resistivity and viscosity,facilitate the reconnection, and shape the geometry ofthe final magnetic field. Even if physical resistivity isnot included in the code, there is unavoidable numeri-cal resistivity, which can be estimated as ηnum = v∆x,where ∆x is the grid spacing. Numerical viscosity can beestimated the same way, with the coefficient νnum. We

included physical resistivity in the code. Its variationwith time in two positions in the computational box isshown in Figure 16.In the simulation with non-resistive corona magnetic

field does not relax smoothly, as it does in the simulationwith a resistive corona. In a non-resistive simulation, thecorona relaxes from the initial condition with a steepergradient between the stellar and the disk component ofthe magnetic field, because of slower reconnection pro-cess. Applied to a stage II of our typical runs, it meansthat if a current sheet is persistent, as it would be in ahigh-resolution ideal-MHD simulations, a magnetic wallcan form. The magnetosonic waves might reflect from itand destabilize the flow by strong shocks.In simulations with a disk gap, with non-negligible re-

sistivity, the stellar dipole magnetic field is reshaped intothe stellar and disk components. This occurs during theinitial stages of the simulation and is later maintained orcyclically repeated. Figure 17 shows the reshaping of themagnetic field in simulations S2. It will look similar inany configuration of the star-disk setup.

5. ELSASSER CRITERION FOR LAUNCHING RESISTIVEFLOWS

Astrophysical outflows launched by a magnetosphericaccretion-ejection are the result of the interaction of theactive magnetosphere and the innermost portion of thedisk. The Elsasser number can serve as a more generalcondition for launching resistive conical flows by mag-netic star-disk interactions. It is defined by the ratio ofthe Z component of the Alfven velocity and ηΩK (seeSalmeron et al. (2007) and references therein):

Λ ≡V 2AZ

ηΩK> 1 . (21)

In some systems, when the flow is mainly in the Z-direction, Λ can be similar or equal to the magneticReynolds number, but in general they are different.Λ > 1 is a better indicator of successful launching. InFigure 18, we plot the Elsasser number Λ > 1 in simu-lation S2. We observe that Λ ≥ 1 is valid only in themagnetosphere above the disk. For a small magneticfield of the order of 1G, there is no region in the boxthat could satisfy Λ ≥ 1, and no launching is possible.


Fig. 19.— A zoom into the closest vicinity of the star fromFigure 6 in simulation S1, to show details in the disk gap, wherethe disk is truncated. Meaning of colors and lines is the same asin Figure 7. The stellar magnetic field is about 200 G, for thedisk mass accretion rate 10−6M⊙ yr−1. Black arrow marks theposition of the truncation radius of the disk, which is in this caseRt = 3.0R∗.

6. DISK TRUNCATION RADIUS

The new truncation radius of the disk after the re-laxation is close to the line of balance of the disk rampressure and the magnetic pressure, p + ρv2 = B2/8π,located between a central object and the disk inner ra-dius — see Figure 19. In our results, it is close to theposition where the disk density drops steeply, and wherethe magnetosonic Mach number equals unity at the equa-torial plane, as matter is being launched fast along theneutral line because of reconnection. The angular veloc-ity profiles along the disk mid-plane also show a dip atthe disk truncation radii, passing from the gap into thedisk.In simulations S1, different to setup in S2, we main-

tained self-consistent boundary conditions at the diskmid-plane. Because of larger magnetic fields, such simu-lations lasted shorter than simulations S2, but enoughthat the disk would reach quasi-stationary state, andwe can estimate the disk truncation radius from ourresults. In Figure 19 we show the disk truncation ra-dius in simulation S1b. It is measured by the positionof the steep gradient in the disk density and it is atRt = 3.0R∗. The disk truncation radius can be esti-mated as Rt = αt(B

4∗R

12∗ /2GM∗M

2a )

1/7, given by or-der of magnitude as the equilibrium of the ram pres-sure of a spherical envelope in free fall and the magneticpressure of a stellar dipole (Elsner & Lamb 1977). Thenon-dimensional factor αt < 1 has been estimated tobe 0.5 in Ghosh & Lamb (1978, 1979a,b) and ∼ 1 inOstriker & Shu (1995). In our simulations the systemdeparts significantly from the spherical infall case andresults in αt for an order of magnitude too large, whencompared to the estimate above. Simulation S1b yieldsαt = 5.

Bessolaz et al. (2008) estimate αt to be equal to M2/7s ,

where Ms is the sonic Mach number measured at thedisk mid-plane, using the radial velocity of the matter inthe disk for a comparison with the sound speed. Theirderivation was for the case with the accretion columnonto the star, but conditions should be similar also in thecase of outflow launched from the inner disk radius. Forthis estimate, inserting the values from simulation S1b,

we obtain M2/7s = 0.15, which gives a too small value of

Rt = 0.3R∗. None of the two estimates is appropriatefor our result here.A further investigation of location and stability of the

disk truncation radius should be done in a simulation ofthe complete R-Z half-plane in cylindrical coordinates,to avoid effects of boundary conditions at the disk mid-plane.

7. DISCUSSION

In this work, we for the first time demonstrate launch-ing outflows with physical resistivity included in thewhole computational box (and not only in the disk),without physical viscosity. We investigate effects of re-sistivity in the innermost vicinity of the central objectsurrounded by an accretion disk, and seek a commonframework for similar works performed to date — someof them are shown in Table 1.We set up our Zeus347 code without employing any

special procedure to smooth the violent relaxation frominitial conditions. With extensive exploration of param-eter space, the results, perhaps somewhat surprisingly,showed that star-disk systems in our purely resistivesetup may undergo similar evolution to the results fromthe literature where additional free parameters or phys-ical processes, e.g. viscosity, were included.The initial evolution relies on two key ingredients: ini-

tial closed magnetic loops that connect both the stellarsurface and the innermost region of the surrounding disk,and sufficient resistivity in the system to enable the mag-netic reconnection to occur and for the field to continueits evolution to longer-lasting and more stable states.As sketched in Figure 8, common phenomena have

been identified in our simulations, which occur, some orall of them, in many simulations of interacting star-disksystems from Table 1. These phenomena can now be con-sidered robust after our extensive explorations in this pa-per. They are as follows. First, the system always relaxesas it transits from sometimes an unrealistic initial condi-tion to a more evolved configuration. The influx of mat-ter as the system relaxes pinches the magnetic field nearthe inner truncation radius. The twisting of the magneticfield lines that are connected to both the stellar surfaceand the inner parts of the disk inflates the loops. Whenenough twisting has been accumulated, magnetic recon-nection adjusts the topology of compressed field lines,and partially opens the originally closed loops. Duringthis process, strong infall floods onto the stellar surfacewhile the loops are opened up by the reconnection. Afterthe reconnection, the loops reform and the field regainsstrength, with the flooded-in disk matter retracting backto the corotation radius. Irregular, transient funnels mayoccur as the system tries to adjust itself along the evo-lution to a longer-lasting state. In some setups, thoseprocesses can repeat themselves numerous times beforea more or less quasi-equilibrium state is reached throughthe opening and closing of magnetic loops. This perhapsforms the basis of episodic accretion in the early phaseof star-disk evolution. As for the later phase of an YSOevolution, there are also many cases in our simulationswhen the system switches into the final configuration im-mediately after the relaxation, and does not change anymore.In this picture, the diffusion process, which is in our


case only the resistivity, plays an important role in ad-justing the system to a more sustainable configuration athundreds of rotations.If there is a sufficiently strong stellar magnetic field,

a two quasi-stationary outflow components form. Theirintensity depends mainly on the rotation rates of the starand the disk, and on the dissipation processes. The mag-netic field lines are opened above the star, with the moreor less episodic fast, light axial outflow, and the mag-netosphere between the star and the disk is the site oflaunching of the slower, conical component. Infall of mat-ter towards the axis might be obstructed by a magneticwall, if reconnection is suppressed (Lynden-Bell 1996,2003), because resistivity can only moderately modifythe flow shape on a slower time scale than the forma-tion of a magnetic wall. Such a wall could mount to a“magnetic tower”, reported in simulations by Kato et al.(2004). More details on longer-lasting and stable funnelsin our setup are being investigated in Cemeljic & Shang(in preparation).This axial, fast outflow component stabilizes in a last

stage. High velocities along the axial region have oftenbeen obtained in simulations of astrophysical jets, butit was often considered a numerical artifact, and not apart of the magnetospheric accretion-ejection of matterin the star-disk system. However, it was present in mod-els based on observations (Kwan & Tademaru 1988) andin models of evolution of an axial jet with reconnection ofmagnetic field, related to disk evolution (Goodson et al.1997, 1999). R09 also obtained a jet as a contributionto the outflow from the star-disk magnetosphere. Wechecked that our axial flow is dependent on conditions atthe base of the flow (atop the star) and not formed justby magnetic collimation. The flow is highly variable atthe beginning, as it builds up during the relaxation fromthe initial conditions, and during the intermittent phasesof reconnection and inflation of the magnetic field, andremains variable in intensity throughout the simulation.A numerically produced flow would tend to be less depen-dent on actual physical parameters. However, to studythe behavior of this outflow in greater detail, it would beneeded to include stellar wind in the simulation, and toconsider a larger physical domain in the computationalbox. Such a study would be more convincing, too, if per-formed in the full 3D simulations, as it could then answerquestions about stability of the axial jet. We leave it forfurther study.In the first three transition stages, our results can

be related to various stages obtained in many othersimulations, with some of them listed in Table 1, andpresented also in Ostriker & Shu (1995), Lovelace et al.(1995). The highly time-dependent radial outflows inGoodson et al. (1997); Goodson & Winglee (1999) alsoresemble those we obtained here. Similar conclusions toours can be found about the position of corotation ra-dius and to the occurrence of reconnection. For longerduration of simulations, which Kuker et al. (2003) ob-tained for magnetic Prandtl number about unity, resultsabout the disk truncation remain valid. The final stagein our simulations can be compared also with the resultsin R09, of which our simulations are necessarily a sub-set. Because of including physical viscosity, simulationsof R09 have one degree of freedom more. A caveat is

that their initial conditions differ from setups in othersimulations in the literature, as well as ours, and the re-sults might be affected by this. We confirm that R09results are still valid, in the part concerning the resis-tive dissipation, with a more conventional initial setup.A quasi-stationary solution for the resistive simulationwith substantial magnetic field in the star-disk magneto-sphere will consist of two outflow components, even whenthe viscosity is much smaller then resistivity.The strength of the stellar dipole, which we vary

in the range of 0.1–200G for the disk accretion rate10−6M⊙/year, affects the actual fluxes of mass and an-gular momentum. A very strong initial field may causestrong shear motion at the beginning, and can largelydisturb the relaxation process, especially in the caseof a non-resistive corona, because pinching and recon-nection of the magnetic field depend critically on theconditions in the magnetosphere (Lovelace et al. 1995;Goodson et al. 1997)6. In ideal MHD simulations per-formed with a coarser grid — see e.g. Romanova et al.(2002) — numerical resistivity can mimic resistive effects,and such results could be more realistic than those inthe high resolution ideal-MHD simulations (Yang et al.1986). In high resolution simulations, physical resistivityshould be included, or results could depend on numericalresistivity, which is not related to physical quantities ofthe setup. Any study of results is then necessarily flawedby purely numerical effects. However, a model for physi-cal resistivity itself can affect the results, as was pointedout in Goodson et al. (1997).In the accretion process, the angular momentum is

partly transported through the system by means of hy-drodynamical viscosity. To some extent, viscosity isincluded in the effective resistivity when the magneticPrandtl number is assumed to be of order unity — seeFerreira & Pelletier (1995) and Casse & Ferreira (2000).However, as shown in Meliani et al. (2006), the viscoustorque extracts angular momentum from the disk lessefficiently than the magnetic torque for turbulent disksand can be neglected in simulations, as we have done inour simulations.To identify locations in which the launching is possi-

ble in our computational box, we compute the Elsassernumber Λ. For a sufficiently large magnetic field andresistivity, it shows that launching occurs from the mag-netosphere above the innermost disk region and the star.There is no disk component of outflow in our results. Toself-consistently determine the contribution from the in-ner disk itself to the overall outflow from the star-disksystem, larger portion of the disk and proper boundaryconditions should be included in the simulation. Here wesimulate only the innermost magnetosphere of the star-disk system, and outflow components which we obtainare then necessarily limited to the contribution from thedisk gap. Essential role of this innermost part of the sys-tem is the re-shaping of the geometry of magnetic field

6 It is worth noticing that the opening of the field lines doesnot happen only through reconnection. Another process which fa-cilitates it and can act in combination with reconnection is theinflation of the magnetic field lines. It occurs because differentialrotation at the footpoints of the magnetic field loops which threadthe star and the disk, tends to open the field lines. It has been de-scribed in e.g. Gold & Hoyle (1960), Aly (1980) and Lovelace et al.(1995).


through the reconnection, which is enabled by resistivity.In one of our setups, with large resistivity of the or-

der of η = 0.5 in the disk, which is a thousand timeslarger than the numerical resistivity, matter is drainedfrom the disk across the magnetic field lines. Such mech-anism could not lead to long-lasting outflows, except foran unusually large accretion rate, for example in closebinaries. However, it is worth mentioning, as this couldbe one mechanism for disk disappearance in setups asours. In astronomical objects, this could happen whenthe disk matter becomes stripped of some ingredients byplanet formation or interaction with the environment,e.g. a neighbor star in close binary system. The resis-tivity in the vicinity of the central object could change,leading to an abrupt change in the accretion process,with the innermost part of the disk disrupted in time ofthe order of ten rotation periods at the inner disk ra-dius, much less than one rotation at the outer radius ofthe disk. Consequences could be observed in YSOs or atthe end of the active accretion phase in planetary disks.

8. SUMMARY

We report results of our numerical simulations in theresistive MHD regime of magnetospheric accretion andejection in the closest vicinity of a central object. Forthe first time, we show the launching of two long-lastingoutflow components in the case of purely resistive mag-netosphere of a slowly rotating star. We also find thatresults, which are usually scattered in various setups andnumerical methods in the literature, can be shown as fourstages in the time-evolution of one setup for a star-disksystem. This simplifies comparison of results from dif-ferent researchers.There are four common stages in our simulations: the

flow relaxation with pinching of the magnetic field, theinflation with reconnection and partial opening of thestellar dipole field, the retraction of the disk towards thecorrotation radius with transient funnel flow onto thestellar surface, and the equilibrium of magnetic and hy-drostatic pressure, with two outflow components fromthe innermost magnetosphere. The axial outflow is fast

and of low-density, expelled by the magnetic pressureabove the stellar surface. The conical outflow is slowerand denser, and is launched by the magnetic force in themagnetosphere.In our simulations, only the stellar dipole is set as an

initial condition for the magnetic field. The inclusionof a large scale interstellar or disk field does not changethe outcome of simulations, as the stellar dipole is theleading term in the disk gap. The stellar dipole fieldalso determines the disk inner radius, as it is positionedwhere the magnetic and the disk ram pressure balance.We identify possible location where the reconnection out-flow originates from the innermost magnetosphere by theElsasser number Λ. The outflow components which weobtain in this work could just be part of the overall out-flow phenomena, or transient, when the inner portionof the disk actually participates dynamically and mag-netically. Current investigated solutions play a role inre-distribution of the initial stellar dipole magnetic fieldinto the stellar and disk fields, enabled by resistivity.Our setup shares one caveat with most of the present

works for a star-disk problem: we do not include the stel-lar wind in simulations. It would influence the solutionsnearby the axis of symmetry and could even affect thevery existence of the axial outflow component. Togetherwith investigation of stability of axial outflows in full 3Dtreatment, we leave it for future work.

This work was supported by funding to Theoretical In-stitute for Advanced Research in Astrophysics (TIARA)in the Academia Sinica and National Tsing Hua Univer-sity through the Excellence Program of the NSC, Tai-wan. MC performed part of work when resident in theEuropean Community’s Marie Curie Actions - HumanResource and Mobility within the JETSET network un-der contract MRTN-CT-2004005592 in Athens, Greece.The authors thank Ruben Krasnopolsky, Oscar Morata,Jose Gracia and Nektarios Vlahakis, for their very help-ful discussions throughout the project. We thank theLCA team and M. Norman for the possibility to use theZEUS-3D code.

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MAGNETOSPHERIC ACCRETION AND EJECTION OF MATTER IN RESISTIVE MAGNETOHYDRODYNAMIC SIMULATIONS

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