by about a factor of 10 in the innermost box to a factor of 104 in the outermost box. It is shown below that the magnetic diffusivity determines the nature of the interaction between the stellar magnetosphere and the accretion disk. The Astrophysical Journal, 524:159-168, 1999 October 10
© 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.
Jets from Accreting Magnetic Young Stellar Objects. II. Mechanism Physics
Anthony P. Goodson
Department of Physics, University of Washington, Seattle, WA 98195; anthony@geophys.washington.edu
and
Robert M. Winglee
Geophysics Program, University of Washington, Seattle, WA 98195; winglee@geophys.washington.edu
Received 1998 May 27; accepted 1999 June 5
ABSTRACT
This paper addresses the physical principles underpinning a new jet-launching mechanism described in a companion paper. In this new jet formation model, magnetic loops that connect the star to the disk become twisted and expand via helicity injection. This expansion drives an outflow, with the axial symmetry of the disk leading to a concentration of outflowing plasma along the rotation axis, forming the jet. In the companion paper, it is found that the radial location of the inner edge of the disk undergoes oscillations. In this paper, the physical causes of the disk oscillations are investigated. This investigation leads to the conclusion that there are three classes of flows that can arise, depending on the role of diffusive instabilities. The most diffusive flows allow the stellar magnetic field to slip through the accretion disk and yield steady accretion flows. Such configurations are unlikely to produce outflows. The flows with intermediate diffusivity have been described by Lovelace, Romanova, & Bisnovatyi-Kogan and represent conditions in which the field is effectively frozen into the accretion disk azimuthally but slips radially. In the absence of magnetic reconnection, such configurations are predicted to produce steady flows with logarithmically collimated disk winds. The least diffusive flows, in which the bulk radial disk velocities are greater at times than the speeds with which magnetic field lines can diffuse into the disks, lead to the formation of the collimated unsteady jets described in the companion paper and are the primary interest of this paper. The jet velocity is also addressed.
Subject headings: accretion, accretion disks; ISM: jets and outflows; methods: numerical; MHD; stars: magnetic fields; stars: pre-main sequence1. INTRODUCTION
In a companion paper (Goodson, Böhm, & Winglee 1999, hereafter Paper I), high-resolution low-diffusivity numerical magnetohydrodynamic (MHD) simulations of accretion flows around strongly magnetic aligned rotators are presented and compared with observations. These simulations corroborate a conceptual model laid out in Goodson, Winglee, & Böhm (1997), in which differential rotation between a central star and the surrounding accretion disk twists the stellar magnetic field. The twisted magnetic field expands because of helicity injection (in the presence of a thin conducting plasma). Plasma attached to field lines flows outward because of the frozen-in flux approximation. Axial symmetry of the system leads to the concentration of some of the outflowing plasma along the rotation axis, which is possibly associated with jets observed from young stellar objects (YSOs). The symmetry also concentrates some of the outflowing plasma along the surface of the accretion disk. This component of the outflow is possibly associated with the observed "disk wind" from T Tauri outflows.
In the simulated jet formation process of Paper I, the inner edge of the accretion disk undergoes periodic oscillations. Furthermore, each of these oscillations is associated with three observed phenomena: a "blob" of overdense plasma in the jet (as in the blobs of the HH 30 jet), an extremely large magnetic reconnection event that should be associated with a large X-ray flare and a rapid increase in the accretion rate, observable as an increase in UV excess. While the timing of the simulated oscillations (and blob productions) differed from the observed timing in the jet source HH 30 by about an order of magnitude (the simulated periods were too short), significant insights can be gained by examining the physics underpinning these oscillations.
The nested grid scheme used for the simulation is described in some detail in Paper I. In this scheme, the simulation is conducted in a series of nested boxes, each with twice the resolution of the box exterior to it. This allows the region near the star to be modeled with high resolution and allows flows at great distances from the star to be captured at lower resolution.
A consequence of using the nested box scheme is that the numerical diffusivity of the code depends on location. The diffusivity has been experimentally measured to be approximately 5 × 1016 cm2 s-1 in the innermost box and increases to 5 × 1019 cm2 s-1 in the outermost (11th) box, whose outer boundary is 50 AU from the modeled star. These values for magnetic diffusivity are greater than the modeled Shakura-Sunyaev viscosity (which sets the disk description) evaluated at 10 R by about a factor of 10 in the innermost box to a factor of 104 in the outermost box. It is shown below that the magnetic diffusivity determines the nature of the interaction between the stellar magnetosphere and the accretion disk.
This paper is primarily concerned with understanding the physics of the oscillations of the inner edge of the disk. Section 2 addresses this issue. Section 3 goes on to derive an estimate of the jet velocity based on physical parameters.
2. MAGNETOSPHERIC OSCILLATIONS
In this section, the physics underpinning the radial oscillations of the inner edge of the accretion disk (seen in the simulations presented in Paper I) are investigated. First a broad overview of the oscillation mechanism is given, then this overview is qualitatively corroborated with simulation results. The details of the oscillation mechanism are then derived, and the conclusion is reached that there are three possible classes of magnetically dominated axially symmetric accretion flow, with the average value of the magnetic diffusion in the disk making the physical distinction. The most diffusive flows have been described in detail by Ghosh & Lamb (1978, 1979a, 1979b, hereafter collectively GL), the intermediate-diffusivity flows by Lovelace, Romanova, & Bisnovatyi-Kogan (1995, hereafter LRBK), and the least diffusive flows by Goodson et al. (1997) and in Paper I.
As in Miller & Stone (1997), the magnetic field in these simulations tends to strip off a ring of plasma from the disk. It is this ring that is accreted by the star. Figure 1 shows the location of the magnetospheric boundary as a function of time for the baseline simulation case of Paper I. Here the magnetosphere is defined as the region of closed magnetic flux. The magnetospheric oscillations are clearly periodic. At the minimum radial position of each oscillation, the magnetosphere is unloaded via field-aligned accretion. The first two oscillations have reduced periods (and a higher accretion rate) than later oscillations, presumably because of the choice of initial conditions. By the end of the simulation, the oscillations appear to have reached a fixed period. There is no indication that the system is approaching a stationary state.
 | Fig. 1 Radial oscillations of the magnetospheric boundary in the baseline case. |
Examination of the local disk/magnetosphere interaction produces the picture outlined in Figure 2. The upper left panel shows the situation after the magnetosphere has expanded back outward. There is a clear boundary between the stellar magnetosphere and the inner edge of the disk, although the expanding magnetosphere has begun to be pinched by the disk. The magnetic field lines contained in the initial magnetosphere are drawn in black for all panels. The open stellar field lines and open disk field lines are drawn in gray. In the upper middle panel of Figure 2, the outer boundary of the stellar magnetosphere has begun to diffuse into the disk. For a brief time, the diffusion velocity (vdiff, which is a measure of the speed with which magnetic field lines will diffuse into the disk) is greater than the radial bulk velocity of the disk plasma and cross field diffusion can be important. Once the stellar flux threads the disk, differential rotation between the star and disk will inflate and open the field, producing the LRBK configuration with the magnetospheric flux. The LRBK field configuration produces two main magnetic stresses on the plasma at the inner edge of the disk. The pinched field produces an outward radial force on the disk (which gravity must eventually overcome as the amount of plasma on the field lines increases through continued diffusion).
 | Fig. 2 Illustration of the magnetospheric oscillation mechanism. Upper left panel: The expanding magnetosphere. Magnetospheric flux is mixed diffusively into the disk in the upper middle panel. Spin down torques on the inner edge of the disk lead to inward displacement in the upper right panel. Lower left panel: The reconnection event, which produces a magnetic topology, which favors field-aligned accretion. The magnetospheric boundary is unloaded in the lower middle panel, and the outward radial expansion commences again in the lower right panel. Black lines are used to denote magnetic field lines that constitute the original magnetosphere. |
The second major stress on the disk is the spin-down torque associated with flinging the plasma attached to the open field lines that thread the inner edge of the disk. If the torque (Tmag) is small (in particular, if Tmag 
L/
orbit, where L is the angular momentum of the ring at the inner edge of the disk and orbit is the orbital time), then the ring at the inner edge of the disk will spiral in over several orbital periods. The radial location of the inner edge of the disk can then be calculated assuming quasi-static equilibrium (these calculations are carried out below). The upper right panel of Figure 2 shows the inner edge of the disk spiraling inward. The inward radial velocity vr becomes greater than the cross-field diffusion velocity vdiff, and diffusive effects become unimportant. By the lower left panel of Figure 2, the disk has spiraled in sufficiently far to cause reconnection at the enhanced current sheet that separates the open disk field from the open stellar field.
Field-aligned accretion, which unloads the stellar magnetosphere, occurs in the lower middle panel of Figure 2. By the lower right panel, the unloaded magnetosphere is expanding outward again. By highlighting the initial stellar magnetospheric field lines in black, the flux exchange mechanism can be seen. As stellar flux diffuses into the disk, the opening mechanism takes hold and (by the upper right panel of Fig. 2) that portion of the stellar magnetosphere becomes open. Reconnection (lower left panel) returns the flux to the magnetosphere. This flux exchange process repeats. The simulation results shown in Paper I qualitatively reproduce this picture, but there are some quantitative results that play important roles as well.
In the low-diffusivity case shown in Paper I, and in the cartoon description of Figure 2, angular momentum is removed from both the star and the disk, including those regions of the disk beyond the Keplerian corotation point rco. This is in contradiction to steady state (diffusive) model of GL, in which the stellar magnetic field threads the disk, but slippage is provided by strong diffusive processes. In steady state models, the magnetic interaction between the star and the region of the disk beyond rco to which it is coupled works to transfer angular momentum from the star to the disk. In the low-diffusivity model, those regions of the disk beyond rco are initially spun up by the magnetic field that connects them to the star, but the field rapidly becomes open. In the "open" field configuration, magnetic flux that threads the disk is not connected to the star. A prior field-aligned accretion event has left these field lines loaded with plasma that is ejected as the disk wind and the jet. The disk wind is centrifugally flung by the field, transferring angular momentum from disk plasma to wind plasma as in Blandford & Payne (1982). Jet plasma is centrifugally flung by the open stellar field, transferring angular momentum from the star to the jet. An important difference between this mechanism and that of Blandford & Payne (1982) is that the field lines are loaded with plasma a priori; there is no need to extract substantial mass from the disk after the expansion has occurred. The opening of the magnetic field determines the magnetic field contribution to the angular momentum evolution of the accretion disk.
The opening of the magnetic field through helicity injection is well studied. Aly & Kuijpers (1990) predict that magnetic arcades that are subjected to differential rotation (helicity injection) will evolve quasi-statically through a series of force-free magnetic configurations until a critical shear is reached. Beyond this critical shear, no equilibrium configuration is calculable. Miki
& Linker (1994) use numerical simulation to show that quasi-equilibrium solutions are possible past the critical shear and also find that the magnetic field "opens" at the critical shear point. As the field opens, a strong current sheet develops between the regions of oppositely directed field. This produces the field configuration of LRBK.
While the simulation results described here are forced to evolve more quickly (because of the high rotation rate relative to the Alfvén speed of the flow) than those examined by Miki & Linker (1994), and the simulated flows are by definition not force free (they drive an outflow), the field configuration does rapidly transition to the open state. The time required for the field to open is typically less than 1 differential rotation between the star and the disk.
If the basic conceptual picture of Figure 2 holds, then significant angular momentum should be removed from the inner edge of the disk by the magnetic field. This angular momentum is placed into the disk wind. Once the magnetic field has been opened by differential rotation, the disk can centrifugally fling the disk wind. The preceding accretion event has preloaded the disk field with plasma, and the expansion event has given the disk wind positive radial velocity. Figure 3 shows the time-averaged angular momentum flux through a spherical surface at R = 5.8 AU for the baseline run of Paper I. The angular momentum flux is averaged over a period of 100 days. The disk wind removes a substantial amount of angular momentum from the accretion disk, even those regions beyond rco. In Figure 3, the minimum in angular momentum flux at 
= 0.7 is due to the current sheet being located at essentially the same polar angle for each plasmoid. The post reconnective flows in the plasmoid can actually have negative angular velocity.
 | Fig. 3 Angular momentum flux through a spherical surface at 5.8 AU. Note that the disk wind carries away substantial angular momentum. |
The outflow through opening angles less than 
0.7 removes angular momentum from the star. The plasma associated with this outflow forms the jet itself and the low-density hot plasma around the core of the jet (see Paper I). This outflow is initially driven along open stellar field lines that connect to the star at high latitudes, and the resulting back torque on the star is shown below to produce spin-down times of a few 105 yr.
To summarize the angular momentum evolution for the disk, the opening of the magnetic field rapidly leads to a magnetic configuration that produces spin-down torques on the disk plasma. For regions of the disk inside the Keplerian corotation radius, magnetic torques will be negative even prior to the field opening. For regions beyond the corotation radius, the torque is initially positive while the disk is magnetically connected to the star, then transitions to a negative torque as the magnetic field opens and plasma on disk field lines is centrifugally flung.
As soon as the torque on the inner edge of the disk transitions to a spin-down torque, the inner edge of the disk will move inward and will eventually be accreted by the star.
This conceptual view implies a time-dependent relation between the magnetic torque on the star and the accretion rate onto the star itself. Figure 4 shows the time-dependent mass accretion rate, magnetic torque, and spin-down time for a 40 day period of the baseline simulation run of Paper I. In the top panel, the time-dependent mass accretion rate is shown. Two accretion events are visible, one at t = 30 days and one at t = 55 days. The average accretion rate over this time interval is about 5 × 10-8 M yr-1.
 | Fig. 4 (Top panel) Accretion rate, (center panel) magnetic torque, and (bottom panel) spin-down time. Immediately prior to the decrease in the magnitude of the spin-down torque, a reconnection event occurs. |
Prior to each accretion event, the inner edge of the disk has been threaded by the magnetospheric flux, the field has opened because of differential rotation, and the inner edge of the disk has been spun down. As the ring of disk plasma spirals inward, dragging the frozen-in flux with it, the current sheet that separates the open stellar flux from the open disk flux is enhanced and eventually disrupted via magnetic reconnection. The altered field topology favors field-aligned accretion. The two accretion events shown in the top panel of Figure 4 are such events.
The magnetic reconnection events between the open stellar flux and the open disk flux, which precede the accretion events, dramatically decrease the magnetic torque on the star. The middle panel of Figure 4 shows the time dependent magnetic torque, which is the integral of
(
)/c over the surface of the star, where is the surface current at the stellar surface.
Prior to the first reconnection event, the magnetic torque on the star is negative, with a peak magnitude of 1.8 × 1037 dyn cm. At this point in time, the field is open. Thus the negative torque on the star is due primarily to the stellar field driving an outflow of attached plasma, part of which is the collimated jet seen in Paper I. In Paper I it is demonstrated that the outflow being driven by the open stellar field is a portion of a past accretion stream and therefore consists of plasma that originated in the disk.
A reconnection event between the open stellar field and the open disk field occurs at t = 28 days. The magnetic torque on the star dramatically decreases in magnitude, and an accretion event immediately follows. At this point, the star has been reconnected with the inner regions of the disk. Furthermore, the inner edge of the disk has spun in radially very close to the star. Some of the stellar flux remains open (and is imposing a spin-down torque on the star), so the total magnetic torque remains negative. After the accretion event, the magnetic torque on the star begins to increase in magnitude again. During this time, an expansion of the magnetic field is occurring. There is some residual accretion during this phase, but in general the residual accretion stream is being pushed outward by the expanding loops (see Paper I). By t = 39 days, the magnitude of the spin-down torque has reached a maximum. It is at this point that the magnetic field has become completely open and an Alfvén surface forms on the open stellar field. At the Alfvén surface, the flow transitions from sub-Alfvénic to super-Alfvénic. A steady spin-down torque continues for about 15 days, until t = 53 days, when a second reconnection event occurs, and the process repeats.
The spin-down time for the star is shown in the bottom panel. Here the angular momentum of the modeled T Tauri star is taken to scale from a solar interior model (Stix 1989) and is assumed to have a value of 6.4 × 1049 g cm2 s-1. The average spin-down time for the star is 2.5 × 105 yr.
The spin down time of a few 105 yr is consistent with the deduced upper limit for the lifetime of the jet producing phase of YSOs. Reipurth, Bally, & Devine (1997) have examined parsec scale outflows from YSOs (HH 111 is one such outflow) and estimated the lifetime of the jet producing phase to be on the order of 105 yr.
In summary, the mechanism for jet formation outlined in Paper I works to remove angular momentum from the both the disk and the star. Angular momentum is removed from the disk after the magnetic field opens. At this point, the disk drives a wind as in Blandford & Payne (1982), although the field has been preloaded with plasma (it is in the residual accretion stream). Angular momentum is removed from the star by the open stellar field. The angular momentum removed from the star is placed into the jet and into the diffuse plasma between the star and the disk wind. The torques on the star are episodic but are on average significant and can produce spin down times for YSOs of a few times 105 yr.
In following subsections, the details of the disk oscillation mechanism are derived. The radial trajectory of the inner edge of the disk is derived assuming quasi-equilibrium. It is found that the resulting spin-down time for the inner edge of the disk is in good agreement with the simulation results of Paper I. Then it is shown that there are three fundamental classes of magnetically dominated accretion, with the distinction being the rate at which magnetic flux can diffuse into the disk.
2.1. Mixing Magnetospheric Flux into the Disk
To gain an understanding of the magnetospheric oscillations, we consider first the plasma entry into the stellar magnetospheric boundary. Figure 5 is a schematic of the stellar field diffusing into the disk plasma. The disk has a scale height h and the region that the magnetospheric flux ultimately threads is denoted by ldiff. The diffusion velocity of field lines into to the disk is found by neglecting the convective term in the induction equation. Thus
where 
is the diffusivity (assumed to be spatially uniform) at the inner edge of the disk.
 | Fig. 5 Illustration of the diffusion terms used in eq. (2) |
The value of must necessarily reflect the role of any diffusive instabilities that work to mix the stellar magnetospheric field into the disk. Solving equation (1) for the radial component of v (to first order in h) yields the diffusion velocity (vdiff):
where it is assumed that in general Br Bz. The justification for this assumption is the enhancement of Bz on the inside face of the ring because of the radial compression of the magnetosphere.
Within the context of the jet formation model presented in Paper I and Goodson et al. (1997), the relative value of vdiff compared to the bulk radial velocity of the disk vr determines the relevance of cross-field diffusion for the open field configuration predicted by LRBK. When vdiff > vr, the field will diffuse into the disk, loading the magnetospheric field lines with disk plasma. The field will open and the plasma will spiral in toward the star because of the magnetic torques alluded to above. As it does so, the radial velocity will increase. For low diffusivity, the radial velocity can become greater than the cross-field diffusive velocity (vr > vdiff). When this occurs, diffusion across field lines becomes less important than the bulk motion of the disk plasma toward the star.
Intuitively, the relationship between vdiff and vr controls the amount of plasma loaded onto the magnetospheric boundary. If the view of Figure 5 is taken to be just at the point when vr = vdiff, then the diffusive length is given by ldiff = vdiffdiff, where diff is the time for which vdiff > vr. After this time, vr > vdiff and cross-field diffusion becomes unimportant. Thus the mass of the ring that will ultimately be accreted by the star is
where 
(R) is the surface density of the disk, Rd is the inner radius of the disk from which rings of plasma are stripped and accreted by the star, and it is assumed that ldiff is much less than Rd. Mring is the amount of plasma that will be accreted with each disk oscillation, so that the stellar accretion rate is Mring/osc, with osc the oscillation period of the magnetosphere.
2.2. Estimating vr
The dynamic behavior of the ring of disk plasma that is diffusively threaded by magnetospheric flux controls the timing of the transition point where diffusion ceases to be important and so effectively controls the accretion rate. To understand the dynamics of this ring of plasma, we first address the effect of the two main magnetic stresses (the radial and azimuthal stresses) on the inner edge of the disk. The discussion below closely follows those of Aly (1980), Aly & Kuijpers (1990), and Kuijpers & Kuperus (1994).
For the derivation below, we assume that the inflated magnetic field is described by
which is an assumption supported by the simulation results of Paper I.
Integrating the radial (
×)/c force on the ring of plasma at the inner edge of the disk yields the radial magnetic force on that ring:
The total radial force is thus related to the angular momentum via the force balance equation,
where L = Mringv
r is the angular momentum of the ring of plasma at the inner edge of the disk. Assuming that the azimuthal ( )/c term is comparable to the radial term, the magnetic torque on the inner edge of the disk is
Solving equation (6) for L and setting the time derivative of L equal to equation (7) allows expressions for the bulk radial velocity and infall times to be calculated. The bulk radial velocity as a function of r is
Substituting the definition for Mring from equation (3) into equation (8) yields
Note that equation (9) is independent of ldiff. Under the assumptions stated above, the radial trajectory of the ring of plasma is determined strictly by the surface density at the inner edge of the disk (Rd) from which the accreting rings of plasma are stripped. Figure 6 plots log10 of the magnitude of the radial velocity as a function of r for Rd = 10, 20, and 30 R. The value of (Rd) is obtained by assuming a standard 
-model accretion disk (Shakura & Sunyaev 1973) with
= 10-7 M yr-1 and = or-1.5, with = 0.2 at r = 1 AU. This is the model disk used in Paper 1.
 | Fig. 6 Predicted radial velocity of rings of plasma that are stripped from the inner edge of the disk. Trajectories are shown for various inner disk radii, which sample different surface densities |
The Keplerian azimuthal velocity is shown in Figure 6 for reference. Figure 6 shows that the radial infall velocity associated with magnetic torques is typically about 10-2 times the Keplerian velocity, except in the region close to the star. Thus the typical orbital decay time is on the order of 102/(2
) orbits.
The middle case (Rd = 20 R) is consistent with the results of the simulation in Paper 1. In those results, a ring of plasma is stripped from the disk at a radius of about 20 R and spirals in toward the star. The ring of plasma is threaded by significantly more flux than the main body of the disk from which it came. The main body of the disk is exposed to less magnetic torque than the ring, leading to radial separation between the two as in the cartoon sequence shown in Figure 2 and in some cases of Miller & Stone (1997).
Figure 6 shows that the "steady" location of the inner edge of the disk (the radius from which rings of plasma are stripped) controls the infall velocity of the ring. Plasma accreting from Rd = 30 R has a lower infall velocity at all radii than plasma originating in a disk at R = 10 R. This is because of the radial behavior of the linear mass density 2Rd(Rd). If the disk linear mass density increases with increasing R, then the infall velocity will decrease. This is because an annulus of plasma at higher Rd will have higher mass and higher angular momentum than an annulus from smaller radii and will spin down more slowly under the same magnetic torque.
The data in Figure 6 are best interpreted in the following sense. If the cross field diffusion velocity (see eq. [2]) is less than the largest velocity shown on a trajectory in Figure 6, then the bulk velocity exceeds the diffusion velocity and the flow will become unsteady. Unsteady flows of this nature will produce the outflows simulated in Paper I. If the cross-field diffusion velocity is greater than all expected dynamic velocities, radial diffusion dominates and the flow is likely steady. Steady flow may occur in the form laid out by LRBK (if vdiff < r
K) or in the form laid out by GL (if vdiff 
rK).
The baseline case of Paper I (where Rd 20 R) is an illustration of the low-diffusivity flow. From Figure 6, it can be seen that unsteady flow requires that the diffusion velocity be less than about 25 km s-1, which is the maximum radial disk velocity near the star. The maximum value of the effective diffusivity corresponding to this diffusion velocity is max = hvdiff 2 × 1017 cm2 s-1 (about 100 times the Shakura-Sunyaev viscosity at 10 R). As stated earlier, the of the simulation is about an order of magnitude less than max. Diffusivities of this magnitude require the action of diffusive instabilities.
It should be noted that the overall strength of the magnetic field plays an important role. If the stellar field is weak enough, or if magnetic reconnection does not produce field configurations favorable to field-aligned accretion, the magnetosphere can be swept completely onto the star at the equator. This result (produced by some cases of Miller & Stone 1997) allows for steady accretion, primarily by viscous stresses. Viscous stresses can dominate accretion onto the star in weak-field (but low-diffusivity) cases as the magnetic field is swept out of the disk and pinned to the star.
2.3. Mass Accretion Rates
Figure 6 shows that the baseline case of Paper I is accreting rings of plasma that originate at Rd = 20 R. This particular configuration is predetermined by the global accretion rate in the disk and the cross field diffusion velocity.
For secular stability, the "average" magnetospheric mass accretion rate (roughly Mring/infall) must be equal to the global mass accretion rate through the disk. Within the model of Shakura & Sunyaev (1973), the global mass accretion rate and the prescription determine the mass surface density of the disk. For any given diffusivity and mass surface density description (R), the system will self adjust until the global mass accretion rate is equal to Mring/infall.
Any particular disk model will completely prescribe (R). From equation (9), it is seen that this (Rd), in conjunction with the magnetic field strength and stellar mass and radius, determine the ring's radial velocity vr(Rd). Integrating equation (9) with respect to time yields infall(Rd). The total mass of the ring is required to estimate the mass accretion rate (Mring/infall). From equation (3), the mass of the ring is given by vdiff and diff, where diff is the time for which vdiff > vr. Picking any value of vdiff allows Mring to be estimated for any given ring trajectory (where a ring trajectory is completely defined by the disk model, the stellar field, mass, and radius, and the disk truncation radius Rd). Thus the mass accretion rate and infall period can be calculated for any given value of , , and Rd. (Note that this discussion assumes that the diffusion time is small compared to the entire infall time.)
Figure 7 shows the inner radius of the disk Rd (upper panel) and the oscillation period osc (which is assumed to be equal to infall, the spin down time for the ring; lower panel) as a function of the effective diffusion , for = 10-7 to 10-9 M yr-1 with the prescription fixed as described above. The data show that increasing leads to shorter oscillation periods and smaller inner disk radii. Increasing leads to smaller gaps between the star and the disk for all values of . In general, increasing leads to shorter infall periods infall, although the 10-7 M yr-1 case approaches limiting behavior.
 | Fig. 7 Upper panel: Radial location of the inner edge of the disk from which rings of accreting plasma are stripped, as a function of effective diffusivity and mass accretion rate. Lower panel: Spin-down time for rings of plasma as a function of effective diffusivity and mass accretion rate. Spin-down time is approximately equal to the magnetospheric oscillation period. The effective magnetic diffusivity is expressed both in cgs units and as a ratio to the modeled Shakura-Sunyaev viscosity at 10 R |
2.4. Summary: Three Classes of Magnetically Dominated Accretion Flows
The particular details shown in Figures 6 and 7 are determined by the model accretion disk. We have chosen a disk with an with a relatively strong radial dependence to reflect the role of local magnetic "viscous" effects such as the Balbus-Hawley instability (Balbus & Hawley 1991; Hawley & Balbus 1991, 1992). Other models would produce different periods and disk inner radii than those seen in the simulations of Paper I, but the fundamental physical conclusion would remain the same: there are three possible classes of magnetically dominated accretion flows. If the cross-field diffusion velocity becomes less than the dynamic velocity of plasma at the inner edge of the disk, and the magnetic field is strong enough to induce field-aligned accretion (rather than equatorial accretion), the flow will almost certainly be unsteady. The attendant oscillations of the magnetospheric boundary can explain many of the observational characteristics of jet producing YSOs, including the formation of optical knots on the jet axis and very large X-ray flares.
In this subsection, we classify the three types of magnetically dominated accretion flows. The classification scheme presented here implicitly assumes the magnetic field is sufficiently strong to allow field-aligned accretion near the star and does not include weak-field cases with equatorial accretion.
Figure 8 shows the distinctions between the three classes of flows. The figure shows the magnitudes of the Keplerian velocity and the radial velocity resulting from the magnetic torques described above. There are three velocity regions highlighted. The top region shows where the model of GL is applicable. In this case, the field can continuously slip through the disk and is not wrapped up. The physics of the model of GL are well founded and the model has been employed to address the spin-up and spin-down of accreting pulsars.
 | Fig. 8 Summary of three classes of magnetically dominated accretion flows. For high diffusivity, the model of GL applies, as the field can continuously slip through the disk and not be constantly wrapped up. For intermediate diffusivity, the model of LRBK applies as the field is wrapped up until it opens, but steady solutions are possible if the diffusion velocity remains greater than the peak expected radial velocity of the inner edge of the disk. For low diffusivity, the model of Goodson et al. (1997) applies, as the radial velocity exceeds the diffusion velocity and the flow must (for strong field cases) be unsteady. |
The second major region of Figure 8 shows the region over which the steady model of LRBK is applicable. In their model, the diffusion velocity is sufficiently low so that the field wraps up and opens via the mechanism described in § 1. Once the field is open, the flow field can approach a steady state. If the diffusion velocity is greater than the expected bulk radial velocity, then the disk plasma can move radially inward until it reaches the point where field-aligned accretion can occur. Once the field is open and the disk is not magnetically connected to the star, the disk field can remain (effectively) frozen into the disk plasma with respect to azimuthal motion and spiral in very slowly because of its cross-field radial motion.
Unsteady flows are a possibility in the intermediate diffusivity case of LRBK. The current sheet that separates the open stellar flux from open disk flux in the model of LRBK is tearing mode unstable. Reconnection between these two regions will cause time-dependent torques that can lead to time-dependent accretion. These time-dependent effects may be small, and it is unlikely that the intermediate diffusivity case of LRBK would produce the isolated accretion events produced by the low-diffusivity case.
Should steady accretion flows arise from the intermediate diffusivity case, a disk wind can arise, roughly as described by Blandford & Payne (1982) and similar in many respects to the outflow predicted by Shu et al. (1994). This outflow would be expected to collimate logarithmically and would be hollow, barring an outflow from the star itself.
The final region of Figure 8 represents the cases studied in Goodson et al. (1997), Paper I, and some of the cases of Miller & Stone (1997). The diffusion velocity in these cases is less than the expected peak bulk velocity. There are thus times when the magnetospheric boundary field lines are not being loaded with plasma. If a reconnection event alters the field and allows field-aligned accretion, the magnetosphere will become unloaded and will expand back outward to mix into the disk again. Thus the last case (in the presence of strong fields that are allowed to reconnect) represents an unsteady configuration.
It is this low-diffusivity case that is shown in Paper I to produce collimated jets with disk winds that are in qualitative agreement with observations of outflows from YSOs. The jets have knots along the jet axis, the velocity spectra show a two-component outflow, and a means for producing very large X-ray flares is inherent in the jet-launching mechanism itself. Within the context of the jet-launching mechanism presented in Paper I, the period of the magnetospheric oscillations sets the time-scale for the formation of the optical condensations in systems such as HH 30 and also sets the time scale for the very large X-ray outbursts observed from YSOs.
While Figure 8 shows only three regimes, we note that there is possibly a fourth that applies to diamagnetic disks. If the stellar magnetic field is completely excluded from the disk, and if the total magnetic diffusivity is significantly lower than the turbulent viscosity of the disk (i.e., if the viscous spin-down time is less than the timescale for mixing magnetic flux into the disk), then the magnetic field will be pinched onto the star as disk plasma slowly spirals inward. These types of configurations have been examined to some extent by Aly (1980) and Aly & Kuijpers (1990).
3. JET VELOCITY
In the jet-launching mechanism presented in Paper I, the velocity of the core of the jet is set when the plasma attached to field lines is accelerated to the Alfvén velocity. Within the Alfvén radius, magnetic effects dominate, and plasma attached to the open stellar field will move outward as a bead on a wire. For the derivation below, we assume that the expansion of the poloidal field due to differential rotation has placed the jet plasma beyond the centrifugal barrier. The plasma then moves "downhill" in the effective potential of the rotating star.
At the Alfvén radius, the mass outflow rate of the jet jet is given by
where
jet is the geometric factor of the jet in the stellar vicinity. Referring to Paper I, the half-angle jet, which defines the initial opening of the jet field lines, can be quite large, with a typical value being /4. The collimation of the jet occurs at larger radii than rA.
For plasma attached to open stellar field lines, the velocity of the plasma will be vjet vA *rA sin jet. As in § 3, we assume that the open magnetic field behaves as B(r) = B*(r*/r)2. Substitution into the Alfvén velocity definition and simplifying,
The jet velocity can be solved for and expressed in units appropriate to jets from YSOs:
where Pd is the stellar rotation period in days, BkG is the stellar surface field in kilogauss, r1 is the stellar radius in solar radii and 8 is the jet mass flux in units of 10-8 M yr-1. For the baseline case simulated, with Pd = 1.8, BkG = 1, r1 = 1.5, 8 = 5, and jet = /4, the predicted jet velocity is 250 km s-1, in good agreement with the velocity spectra presented in Paper I. Those spectra show a high-velocity component of the outflow with a velocity of 140
200 km s-1.
4. SUMMARY AND DISCUSSION
The basic physics underpinning the jet-launching mechanism of Paper I are not complex and have been explored in some depth by other researchers. What is new about this mechanism how the features of helicity injection and low-diffusivity magnetic accretion combine to produce an unsteady jet with episodic accretion by the star. In this paper, we have shown that the jet-launching mechanism outlined in Paper I and in Goodson et al. (1997) can work to remove angular momentum from both the star and the disk, to include those regions of the disk beyond the Keplerian corotation radius rco. The removal of angular momentum from the disk serves to load the stellar magnetosphere with plasma, which in turn can lead to magnetospheric oscillations.
There appear to be three classes of magnetospheric flows around accreting magnetic stars, with the distinction between the three being determined by the effective diffusivity in the disk. For high diffusivity, the field can slip through the disk azimuthally and avoid wrapping up. The resulting flows and star/disk interactions in this case are described by GL. No outflow is expected in this case. For intermediate diffusivity, the field can slip radially through the disk, but not azimuthally. The steady state consequences of this flow are described by LRBK. For low diffusivity, the field slips only radially for a time, threading only the inner edge of the disk. The resulting magnetic spin-down torques in the disk produce radial motions that exceed the diffusion velocity, diffusion becomes unimportant, and the flow becomes unsteady. This flow, the subject of Paper I and Goodson et al. (1997), produces collimated outflows with knots, in good qualitative agreement with observations.
The simulated evolution of Paper I agrees with the conceptual model for the new jet-launching mechanism, and the simulation results are consistent (or at least not inconsistent) with observations. The successful generation of stellar jets seems to require only that the star have a sufficiently strong magnetic field, that the star and the regions of the disk to which it is coupled rotate at different angular velocities, and that the diffusion velocity be fast enough to allow new plasma to be loaded onto magnetospheric field lines during each oscillation but slower than both the expected peak radial velocity and the Keplerian velocity of the disk. The role of axial symmetry seems to be important but is unquantifiable to date.
It should be noted that this work is subject to many of the criticisms offered by Safier (1998). As stated, axial symmetry is assumed. Proper investigations into tilted dipole rotators will require a full three-dimensional treatment. Furthermore, a stellar magnetic dipole field is assumed. While Safier (1998) bases his argument against dominant dipolar fields on the existence of very large X-Ray flares, and we show here (as have Hayashi, Shibata, & Matsumoto 1996) that large X-ray flares can be a natural consequence of a dipole field coupled to a conducting disk, the effect of higher order multipole terms in the magnetic field may be important, especially close to the star where the higher order terms may dominate the field. Finally, assuming that the magnetic field initially threads the (near-perfectly) conducting disk everywhere is questionable: the field is unlikely to penetrate the disk everywhere if the conductivity is high.
It is perhaps best to summarize the results of this work by paraphrasing GL: "Plasma entry into the magnetosphere is a physically interesting process." As advancements are made in understanding magnetic systems ranging from the earth's magnetosphere to accretion by YSOs, magnetic cataclysmic variables, and neutron stars, it is becoming clear that the mechanism by which plasma enters the magnetosphere sets many of the characteristics of the magnetosphere itself, as well as determining many characteristics of the system's environment. In the case of accreting compact objects, the work presented here implies that the mechanism of plasma entry into the magnetosphere ultimately produces an outflow.
This work was supported by the University of Washington Royalty Research grant 65-2597 and NSF grant AST 97-29096.
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