Mem. S.A.It. Vol. 82, 112c© SAIt 2011 Memorie della

Formation of magnetohydrodynamic jets: ares as triggers of internal shocks

Christian Fendt

Max Planck Institute for Astronomy, Konigstuhl 17, D-69117 Heidelberg, Germanye-mail: fendt@mpia.de

Abstract. We investigate how the overall jet formation process is affected by a variationin the accretion disk magnetic flux profile and/or the existence of a central stellar magneto-sphere using axisymmetric magnetohydrodynamic (MHD) simulations. These simulationsevolve from an initial, hydrostatic equilibrium state in a force-free magnetic field config-uration. Two different simulation setups will be considered. In the first approach the roleof the disk magnetic flux profile and disk mass loss profile is investigated concerning thejet collimation degree. Our results suggest (and quantify) that in general outflows launchedfrom a very concentrated region close to the inner disk radius tend to be un-collimated. Inthe second approach, jet formation is numerically investigated from a magnetic field con-figuration consisting of a stellar dipole superposed by a disk field. The central dipole isfound to de-collimate the disk wind considerably. Reconnecting flares are launched by theinteraction of the disk and stellar magnetic field and may change the overall mass flux in theoutflow by a factor of two. We apply the energetics and time scales of our numerical flaremodel to XRB sources.

Key words. accretion – accretion disks – magnetohydrodynamics: MHD methods: numer-ical – stars: formation – stars: magnetic fields – stars: mass loss – stars: winds, outflows –ISM: jets and outflows

1. Introduction

Astrophysical jets are highly collimated beamsof high velocity material, observed in a va-riety of astronomical sources - among themyoung stellar objects (YSO), micro-quasars(MQs, XRBs), or active galactic nuclei (AGN).Somewhat less collimated beams of compara-tively lower speed are usually called outflows.The current understanding of jet formation isthat these outflows are launched by magneto-hydrodynamic (MHD) processes in the closevicinity of the central object – an accretion disk

surrounding a protostar or a compact object(Blandford & Payne 1982; Pudritz et al. 2007).

The geometrical setup of a stellar dipolarmagnetic field surrounded by an accretion diskcarrying its own magnetic flux is a frequentastrophysical scenario which seem to be re-alized in young stars, cataclysmic variables,high-mass and low-mass X-ray binaries, andother micro-quasar systems.

Numerical simulations of MHD jet forma-tion are an inevitable tool to understand the un-derlying physical processes. Furthermore, thejet-launching region can be resolved numer-ically, but not observationally. These simula-

Fendt: Jets and flares 113

tions can be distinguished in those taking intoaccount the evolution of the disk structure andothers considering the disk surface as a fixed-in-time boundary condition for the jet. Both ap-proaches are somewhat complementary, eachof them having their pros and cons.

The first approach allows to directly inves-tigate the mechanism lifting matter from thedisk into the outflow (Uchida & Shibata 1984;Miller & Stone 1997; Goodson et al. 1999;Casse & Keppens 2002; Romanova et al. 2002;Meliani et al. 2006). This approach is compu-tationally expensive and still somewhat limitedby spatial and time resolution. Also, the diskmodel underlying the jet formation simulationsis usually rather basic. Studying the accelera-tion and collimation of a disk/stellar wind re-quires essentially to follow the jet dynamicalevolution for i) very long time ii) on a suffi-ciently large grid with iii) appropriate resolu-tion. For such a goal, the second approach isbetter suited (Ustyugova et al. 1995; Ouyed &Pudritz 1997; Krasnopolsky et al. 1999; Fendt& Cemeljic 2002; Kigure & Shibata 2005;Fendt 2006, 2009; Porth & Fendt 2010). Ofcourse, the mass flux ratio of jet and disk can-not be determined by such an approach. Thecase of superposed stellar/disk magnetic fieldis rarely treated in simulations, although thefirst models were discussed already in Uchida& Low (1981). Simulations of a dipole withaligned vertical disk field are presented byMiller & Stone (1997); Matt et al. (2002). Thestellar field has important impact on the jet for-mation process as enhancing the magnetic flux,adding a central pressure, and providing excessangular momentum for the launching region.

2. Disk jets and stellarmagnetospheres

In the following we discuss several aspectswhich consider the jet formation process inpresence of a central stellar magnetic field.

Additional magnetic flux. In comparisonto the situation of a pure disk magnetic field,the stellar magnetic field adds substantial mag-netic flux to the system. For a polar fieldstrength B0 and a stellar radius RTTS resp. RNS,

the large-scale stellar dipolar field

Bp,?(r) ' 40 G( B0

1 kG

) ( r3 RTTS

)−3

(1)

= 40 kG( B0

1 MG

) ( r3 RNS

)−3

is to be compared to the disk poloidal magneticfield which could be provided either by dy-namo action or by advecting the ambient inter-stellar field. Equipartition arguments suggest amaximum disk magnetic field of

Bp,disk < Beq(r) = 20 G1√α

(Ma

10−6 M/yr

) 12

(2)

×(

M?

M

) 14(

H/r0.1

)− 12(

r10 R

)− 54

= 2 MG1√α

(Ma

10−8 M/yr

) 12

×(

M?

2M

) 14(

H/r0.1

)− 12(

r5 RNS

)− 54

,

where the first part refers to the protostellarcase and the second one for a compact star. Thestellar magnetic dipole will not remain closed,but will partly inflate and open up due to shearbetween the foot points of the poloidal mag-netic field lines on star and disk (e.g. Uchida& Shibata 1984; Fendt & Elstner 2000; Matt& Pudritz 2005). The additional Poynting fluxthat threads the disk may support jet launch-ing by MHD forces. The stellar field may alsoserve as an additional energy source for the jetkinetic energy, thus implying a greater asymp-totic jet speed (Michel scaling; Michel 1969;Fendt & Camenzind 1996).

Additional magnetic pressure. The stellarmagnetic field also provides an additional cen-tral magnetic pressure which may result in ade-collimation of the overall outflow. The cen-tral stellar magnetic field may launch a strongstellar wind which will remove stellar angu-lar momentum. Such an outflow will interactwith the surrounding disk wind. If true, ob-served jets and outflows from stellar sourcesmay consist of two components – the stellarwind and the disk wind. Note that so far this

114 Fendt: Jets and flares

argument is ”ad-hoc” and numerical simula-tions are needed to figure out the actual dynam-ical evolution (see below). Simulations of stel-lar MHD winds have been provided by Matt &Pudritz (2005, 2008).

Angular momentum exchange by thestellar field. In the scenario of magnetic “disklocking”, the stellar field which threads thedisk will re-arrange the global angular momen-tum budget. If the star looses angular momen-tum to the disk, both disk accretion and out-flow formation is affected. In this case the an-gular momentum is transferred by the dipolarfield. It is deposited close to the inner disk ra-dius, not farther out than the last closed fieldline. Therefore, the matter in this region maybe accelerated to slightly super-Keplerian rota-tion which has two interesting aspects. (i) Dueto the super-Keplerian speed this disk mate-rial could be easily expelled into the coronaby magneto-centrifugal launching (Blandford& Payne 1982; Ferreira 1997) and form adisk wind. (ii) The excess angular momen-tum will stop accretion unless it is removed bysome further (unknown) process. A disk out-flow launched from the very inner part of thedisk can be an efficient way to do this.

The torque on the star by the accretion ofdisk matter is τacc = Macc (GM?rin)1/2 (e.g.Matt & Pudritz 2005; Pudritz et al. 2007),withthe disk accretion rate Macc, the stellar massM? and the disk inner radius rin inside theco-rotation radius. For “disk locking”, the starmay be braked-down by the magnetic torquedue to stellar field lines connecting the starwith the accretion disk outside the co-rotationradius. The differential magnetic torque act-ing on a disk annulus of dr width is dτmag =

r2BφBzdr. However, while Bz may be derivedby assuming a central dipolar field, the induc-tion of toroidal magnetic fields (electric cur-rents) is model dependent.

Non-axisymmetric effects from a tippedmagnetic dipole. A central dipolar field in-clined to the rotation axis of star and disk maystrongly disturb the axisymmetry of the sys-tem. In extreme cases this may hinder jet for-mation at all, while weaker non-axisymmetricperturbation may lead to warping of the in-ner disk, and thus a precession of the outflow

launched from this area. A rotating inclineddipole also implies a time-variation of the mag-netic field which may lead to a time-variationin the mass flow rates for both the accretiondisk and the outflow.

Investigations of the warping process byPfeiffer & Lai (2004) using numerical simu-lations show that the warp could evolve intoa steady state precessing rigidly. Disks can bewarped by the magnetic torque that arises fromthe a slight misalignment between the disk andstar’s rotation axis (Lai 1999). This disk warp-ing mechanism may also operate in the absenceof a stellar magnetosphere as purely inducedby the interaction between a large-scale mag-netic field and the disk electric current and,thus, may lead to the precession of magneticjets/outflows (Lai 2003).

3. MHD simulations: disk jets with ofdifferent magnetic flux profiles

Here we discuss simulations of jet formationwhere jets are form from pure disk winds (fordetails see Fendt 2006; Pudritz et al. 2006).The physical grid size corresponds to (r × z) =(150 × 300) rin.

We start from a force-free initial field dis-tribution in hydrostatic equilibrium. The sim-ulation evolves under the boundary conditionof a fixed mass inflow from the accretiondisk into the outflow. However, we run var-ious models, covering a wide range of diskmagnetic field profiles and disk wind massflux profiles, parameterized by a power law,Bp,wind(r) ∼ r−µ, ρwind(r) ∼ r−µρ . Both quan-tities can be combined in the disk wind mag-netization parameter (Michel 1969), σwind ∼B2

pr4Ω2F/Mwind ∼ rµσ . We quantify the colli-

mation degree by comparing the axial and lat-eral mass fluxes (see Fendt & Cemeljic 2002;Fendt 2006). Figure 1) shows the degree ofcollimation measured by the parameter ζ isplotted against the power law exponent of thedisk wind magnetization µσ. The main result isthat steep magnetization profiles, resp. the diskmagnetic field profiles, are unlikely to generatehighly collimated outflows. However, flat pro-files which generally lead to a higher jet colli-

Fendt: Jets and flares 115

Fig. 1. Time evolution of the axial mass flux close to the upper boundary. The mass flux changes duringthe initial evolution (sweep-out of the initial corona), but also during the flaring events.

Fig. 2. Initial magnetic field distribution for star-disk jet formation simulations, shown are poloidal mag-netic field lines. Arrows indicate the magnetic field direction. Note the different location of the X-points. Different strength and orientation of the superposed stellar and disk magnetic field component,Adisk = 0.0, 0.01,−0.01,−0.1, resp. Astar = 1.0, 5.0, 1.0, 3.0 (from left to right). Note that here we show theleft hemisphere (rotation axis directs upwards).

mation degree, tend to be unstable, i.e. do notestablish a steady state.

Thus, result may indicate that transient jetfeatures arise from accretion disks with flatmagnetic flux profiles. transient jet features.

4. MHD simulations: outflows fromdisk-star magnetospheres

Here we present results of MHD simulationsconsidering the co-evolution of a stellar mag-netosphere and a disk magnetic field where

both field components are fed by a mass fluxfrom the underlying boundary condition - rep-resenting the stellar surface and the accretiondisk. The field direction of both componentscan be aligned or anti-aligned. Similar config-urations were considered by Uchida & Low(1981) and were recently reconsidered in theform of reconnection X-winds Ferreira et al.(2006).

In our model we apply cylindrical coordi-nates (r, φ, z), and divide the equatorial in threeparts - the stellar surface with r < r? = 0.5rin,

116 Fendt: Jets and flares

Fig. 3. Time evolution of a star-disk magnetosphere from initial state of Fig. reffig:star-disk-ini, middle.Time step is 50, 400, 2606, 2700 rotations of the inner disk (from top to bottom). Colors show logarithmicdensity contours, black lines are poloidal field lines (magnetic flux contours).Note that here we show both upper hemispheres (rotation axis directs upwards).

the accretion disk at radii r > rin = 1.0, andalso the gap between star and disk. The out-

flow mass flux consists of a stellar wind contri-bution and a disk wind contribution. The cen-

Fendt: Jets and flares 117

tral star is rotating with a magnetospheric co-rotation radius equal to the disk inner radius.The grid size is (r × z) = (80 × 80) inner diskradii which refers to different physical scaleswhen applied to e.g. protostars or XRBs. Theinitial magnetic field distribution is taken as asuperposition of the stellar (dipolar) field andthe disk field (force-free potential field),

Ψtotal(r, z) = Adisk fdisk(r, z) + Astar fstar(r, z) (3)

(see Fig. 2), where Ψ0,disk and Ψ0,star mea-sure the strength of both components and thefunctions f (r, z) describe the initial (force-free)magnetic field distribution of both components(Fendt 2009).

Figure 3) shows how the coronal fieldstructure evolves in time for the example sim-ulation with Ψ0,disk = −0.1 and Ψ0,star = 3.0.In this case, disk magnetic field and stellardipolar field (along the equatorial plane) arealigned. We evolve the simulations for 2800 ro-tations at the inner disk radius correspondingto 4 rotations at the outer disk radius. At inter-mediate time scales (about 700 inner disk ro-tations) a quasi-stationary state emerges. Oneclearly sees the de-collimating effect of thecentral stellar wind component. Note, how-ever, that at this time the outer disk has rotatedonly about 0.15 times and the coronal struc-ture above the outer disk will further evolve intime and disturb the quasi-steady state. Overthe long run such quasi-stationary states maybe approached again, what we observed is acyclic behavior of the opening angle with a pe-riodicity of about 500 (inner disk) rotation pe-riods.

Independent of the alignment, the centraldipole does not survive on the large scale.A two-component outflow emerges as stellarwind surrounded by a disk wind. For a reason-ably strong disk magnetic flux a collimated jetemerges. If the overall outflow is dominatedby a strong stellar outflow the low mass fluxdisk wind remains un-collimated. The favor-able setup to launch a collimated jet from astar-disk magnetosphere is that of a relativelyheavy disk wind and high disk magnetic flux.

We also observe that reconnection pro-cesses close to the remaining inner dipoleleads to sudden flares (see also Goodson et

al. 1999)which seem to trigger the large-scalecyclic behavior. The propagation of these flaresis very fast, reconnection islands propagateacross the jet magnetosphere within a few ro-tation time steps. In our case the reconnec-tion/flares seem to be triggered by the evolu-tion of the outer disk wind. Even for our verylong time-scales the outer disk outflow is stilldynamically evolving, thus changing the cross-jet force equilibrium and forcing the innerstructure to adjust accordingly. The flare eventsare accompanied by a temporal change in theoutflow mass flux and momentum. Figure 5shows the mass loss rate in axial direction in-tegrated across the jet. We see two flares witha 10%-increase in the mass flux followed bya sudden decrease of mass flux by a factor oftwo. This behavior is also seen in the poloidalvelocity profile.

Considering the ejection of large-scaleflares and the follow-up re-configuration ofoutflow dynamics, we hypothesize that the ori-gin of jet knots is triggered by such flaringevents. Our time-scale for flare generation isof 1000 rotational periods and longer than thetypical dynamical time at the jet base, butsimilar to the observed knots. The flare itselffor about 30-40 inner disk rotation times (seeFendt (2009) for a comparison to the Sweet-Parker reconnection time scale which turns outto be of the same order for the simulation pa-rameters applied).

5. Reconnection flares triggeringunsteady jets?

Massi & Kaufman Bernado (2008) provide asummary on magnetic field strengths, accre-tion rate, and jet parameters in micro quasars(MQs) and X-ray binaries (XRBs). For thesources carrying a neutron star in the center(i.e. not a black hole) the flaring model due todisk - star magnetic field interaction discussedabove may be applied.

We first need an energy estimate. The max-imum magnetic energy density available is ∼B2/8π. As typical field strength we take thestellar dipolar field at the inner disk radius (seeeq. 1). The volume of the flaring region we esti-mate as a cylinder with radius of the inner disk

118 Fendt: Jets and flares

Fig. 4. Poloidal magnetic field evolution during one example flare around t = 1800. Solid and dashed linesindicate the direction of total magnetic flux of the superposed dipolar and disk magnetic field components.Shown are time steps: 1760, 1790, 1810 (from left to right). Note that here we show the right hemisphere(rotation axis directs upwards).

Fig. 5. Time evolution of the axial mass flux close to the upper boundary. The mass flux changes duringthe initial evolution (sweep-out of the initial corona), but also during the flaring events.

radius rin, a height of five disk scale heights,∆z = 5h = 2.5rin (assuming a thick disk modelh ∼ 0.5r), a gauge of ∆r = ∆z, thus a volumeof ∆V ' 2.5πr3

in. With that the magnetic energyavailable is about

EB = 1.5 × 1034erg( B0

109G

)2 (rin

5 RNS

)−3

(4)

This corresponds to an upper limit for the flareenergy, assuming that a global 3D structure (acylinder ring is flaring at the same time and thatall magnetic energy is transferred. Applyingthe flare time scale derived from our numer-ical simulations above the flare luminosity isabout LB ' EB/τflare ' 2 × 1028erg/s. Sincethe dipolar stellar field decays rapidly with ra-

Fendt: Jets and flares 119

dius, reconnection events from radii ∼ 50RNSwould provide much less (factor 1000) energy.

This should be compared to typical kineticjet energies,

Lkin = 1.5 × 1033ergs−1( vjet

0.5 c

)2(5)

(Mjet

10−3Macc

) (Macc

10−10Myr−1

).

This value is clearly beyond the capabilities ofa flaring event discussed above. However, onemay understand the flare energy transferred toa much smaller mass load, creating a wave rid-ing on the jet stream which higher velocityand triggering a shock wave visible as tran-sient flow. Still the energy provided by the flareseems to be too low considering even a tran-sient flow of, say 10−4 of the bulk jet kineticluminosity. Only if we consider extremely lowmass fluxes for the bulk jet, the large-scale re-connection may cause a significant contribu-tion to the kinetic energy propagating along thejet. On the other side, the jet mass flux may infact be the essential parameter dividing tran-sient from steady jet flows.

Acknowledgements. I like to thank the organizers ofthe workshop for a lively and efficient meeting. Partsof this paper benefitted from discussions with MariaMassi and Sergei Komissarov.

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