Mem. S.A.It. Vol. 76, 372c© SAIt 2005 Memorie della

Launching jets from resistive accretion disks

C. Zanni1, A. Ferrari1, S. Massaglia1, G. Bodo2 and P. Rossi2

1 Dipartimento di Fisica Generale dell’Universita, Via Pietro Giuria 1, I - 10125 Torinoemail: zanni@ph.unito.it

2 INAF – Osservatorio Astronomico di Torino, Strada dell’Osservatorio 20, I-10025 PinoTorinese email: bodo@to.astro.it

Abstract. We present numerical time-dependent MHD simulations of the acceleration andcollimation process of astrophysical jets from accretion disks. We begin to perform a pa-rameter space analysis by varying the amount of resistivity introduced in the disk to allowa steady accretion motion. The aim of this work is to study the effect of the resistivity indetermining the properties of both accretion and ejection flows.

Key words. Accretion disks – Jets and outflows – Magnetohydrodynamics

1. Introduction

Jets and outflows are observed in very differ-ent astrophysical systems, ranging from youngstars (YSO) where Herbig-Haro jets take theirorigin, to X-Ray binaries, up to extragalacticradio jets which are accelerated in the coresof Active Galactic Nuclei (AGN). Nowadaysthere is a general agreement on the fact that asimilar physical scenario can explain the originof outflows in the most diverse systems: thiswidely accepted model is based on the interac-tion of an open large-scale magnetic field withan accretion disk around a central compact ob-ject. The magnetic field extracts energy and an-gular momentum from the accretion disk andtransfer them to the outflow which is magneto-centrifugally accelerated.

In the ideal MHD regime this mechanismhas been studied for the first time by Blandford& Payne (1982) by means of self-similarsteady solutions in cylindrical geometry. In

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that work and in the following self-similarstudies that extended the model (Sakurai 1985;Contoupolos 1994; Sauty et al.; 2002) the diskis treated as a rotating boundary condition. Tostudy the dynamics of the accretion disk andthe back reaction of the jet on the disk, resistiveeffects must be introduced at least in the accre-tion disk: in fact it is necessary to allow theaccretion flow to cross the poloidal magneticfield lines to obtain a steady solution (Wardle& Koenigl 1993; Casse & Ferreira 2000).

Besides of the cited analytical self-similarstudies, another approach is based on numer-ical time-dependent simulations where, on theother hand, the problem is to deal with the largerange of spatial and temporal scales involved.Most of the simulations performed were lim-ited to the study of the outflow dynamics,treating the disk as a fixed boundary condi-tion (Ustyugova et al. 1995; Ouyed & Pudritz1997). Due mostly to the high computationalcost, simulations which include the disk in thecomputational domain are performed on shorttime-scales (Uchida & Shibata 1985; Kato et

C. Zanni et al.: Launching jets from resistive disks 373

al. 2002) or are limited to the study of onesample case, without exploring the parame-ter space which defines the initial conditions(Casse & Keppens 2004; Zanni et al. 2004).

In the present work we begin to explorethe parameter space by studying the effectsof resistivity in determining the properties ofthe accretion disk and of the accelerated out-flow. In Section 2 we describe the initial setupand the set of simulations that we performed,in Section 3 we analyze the acceleration andcollimation mechanisms, in Section 4 we be-gin to study the influence of the resistivity onthe asymptotic behavior of both the accretionand ejection flows, in Section 3 we analyze theoutflow energetics in the different simulationswhile in Section 6 we give a brief summary ofour results.

2. Numerical setup and simulations

To study the mechanisms which can lead tothe launch of a jet from a magnetized accretiondisk we perform a set of time-dependent MHDnumerical simulations in axial symmetry. Weuse a modified 2.5D version of the MHD mod-ule provided with the FLASH multi-purposehydrocode (http://flash.uchicago.edu).In the initial setup we model a Keplerian diskin equilibrium with the gravitational potentialof a point-like central object. An equilibriumsolution can be obtained assuming a polytropicrelation between pressure and density in thedisk (P ∝ ργ). The disk density ρdisk is givenby

ρdisk = ρd0

[γ − 1γαd

( r0

R− ε2 r0

r

)] 1γ−1

(1)

where γ = 5/3 is the adiabatic index of theplasma, R = (r2 + z2)1/2 is the spherical ra-dius, r0 is the normalization length and ρd0 isthe disk density evaluated at (r = r0, z = 0).αd = c2

d/v2k = 0.01 is the constant ratio be-

tween the squares of the disk isothermal soundspeed cd and the Keplerian speed vk evaluatedat the disk midplane. The initial rotation speedis given by vφ = ε(GM/r)1/2. The parameterε2 = 1 − αdγ/(γ − 1) . 1 gives the devia-tion of the toroidal speed from the Keplerian

value due to the gradient of the disk pressurewhich is given by Pdisk = Kργdisk, with K =

v2k0ρ

1−γd0 αd (vk0 is the Keplerian speed evaluated

at the cylindrical radius r = r0). The disk issurrounded by a rarefied atmosphere in hydro-static equilibrium. Assuming a polytropic re-lation between pressure and density, the atmo-sphere density ρatmo is defined as

ρatmo = ρa0

(γ − 1γαa

r0

R

) 1γ−1

(2)

where ρa0 = 10−4ρd0 is the corona densityevaluated at R = r0 and αa = c2

a/v2k = (γ −

1)/γ = 0.4 is the ratio between the squaresof the atmosphere isothermal sound speed andthe Keplerian speed evaluated at (r = r0,z = 0). The equilibrium pressure is Patmo =

Kργatmo with K = v2k0ρ

1−γa0 αa. The surface at

which Pdisk = Patmo marks the boundary be-tween the disk and the atmosphere solution.The initial magnetic field is a bent poloidalfield whose magnetic energy at the disk mid-plane is proportional to the disk pressure, thatis β = 2Pdisk/B2

z |z=0 = 3.33. The precise topol-ogy of the initial magnetic field is shown in thefirst panel of Fig. 1 and 2. As pointed out inthe Introduction, a certain amount of resistiv-ity should be introduced in the disk to avoidthe accumulation of magnetic flux and allowa steady accretion motion. In our simulationsthe resistivity νm is parametrized following theShakura & Sunyaev (1973) prescription for theturbulent viscosity:

νm = αmVAHexp[−(z/H)4

](3)

where VA = B/√ρ is the Alfven speed and

H =√αdr = 0.1r is the height scale of the

disk. The exponential function cancels the re-sistive effects outside the disk. αm defines theamount of resistivity which we introduce. Inthe present work we study the effects of resis-tivity by performing three simulations varyingthe value of αm = 0.1, 0.5, 1. The compu-tational domain of the simulations has a sizerd × zd = 20r0 × 60r0. The equivalent resolu-tion of the Adaptive Mesh is 256 × 768 points.Suitable symmetry conditions are imposed onthe axis of symmetry (r = 0) and on the disk

374 C. Zanni et al.: Launching jets from resistive disks

Fig. 1. Time evolution of density maps of the simulation characterized by αm = 1. Density isplotted in logarithmic scale and normalized on ρd0. Sample magnetic field lines are overplotted.

midplane (z = 0) while outflow conditions areimposed on the outer boundaries. To avoid thesingularity at the origin a sink condition is im-posed inside the R < 0.5r0 region. In the fol-lowing lengths will be given in in units of r0,speeds in units of vk0 and the densities in unitsof ρd0. Expressed in these units the period ofrotation of the disk at r = r0 is equal to 2π.

3. Acceleration and collimation of theoutflow

In Fig. 1 and 2 we show the time evolutionof density maps of the cases characterized byαm = 1 and αm = 0.1 respectively. Followingthe simulation up to a time t = 120, which cor-responds approximatively to 20 turns of the in-ner radius of the disk r0, a strong outflow isproduced in both simulations, although withdifferent characteristics. First of all we can no-tice that due to the lower resistivity of theαm = 0.1 case, more magnetic flux is accu-mulated toward the center, causing the out-flow to be hollow and unsteady. Neverthelessthe mechanism which starts the launching ofthe jet is the same. A detailed balance of theforces which drive the dynamics of the systemin the αm = 1 case is shown in Fig. 3: in thefirst panel on the left we plot cuts of the in-tervening forces along the z axis; we see that

only the thermal pressure gradient can counter-act the pinch due to both the gravitational ac-celeration and Lorentz force. Nevertheless theLorentz force changes sign at the disk surface(z ' 1.5) thus driving the acceleration of theoutflow. Analogously, also the toroidal compo-nent of the Lorentz force changes sign (secondpanel in Fig. 3): therefore the magnetic fieldbrakes the disk, which is then losing angularmomentum and accretes, while it centrifugallyaccelerates the outflow. The collimation of theoutflow against the centrifugal pull is ensuredby the radial Lorentz force (right panel in Fig.3) which is mainly given by the tension of thetoroidal field −B2

φ/r (magnetic “hoop stress”).Even if this discussion refers specifically to theαm = 1 case, it is quite general and is valid forthe lower resistivity cases too. The main differ-ence in the αm = 0.1 case is that, due to thelow resistivity, the field lines at the surface ofthe disk are much more bent: the greater radialcomponent of the field generates an accelerat-ing force in the outflow which is perpendicularto the field lines which are then continuouslydistorted and inverted (right panels in Fig. 2).

4. Steady accretion and ejection

Since the timescales which we are able to sim-ulate are much shorter than the lifetime of the

C. Zanni et al.: Launching jets from resistive disks 375

Fig. 2. Same plot as Fig. 1 for the αm = 0.1 simulation.

Fig. 3. Plots of the forces driving the dynamics of the αm = 1 simulation. Left panel. Cuts ofthe forces directed along the z direction at r = 4: Lorentz force (solid line), thermal pressuregradient (dashed line) and gravitational acceleration (dotted line). Central panel. Cut along the zaxis of the toroidal Lorentz force. Right panel. Cuts of the radial forces at z = 20: Lorentz force(solid line), centrifugal force (dot-dashed line), pressure gradient (dashed line) and gravitationalacceleration (dotted line).

Fig. 4. Left panel. Angle between the poloidal speed and the poloidal magnetic field along aselected magnetic field line at t = 120. Central panel. Time evolution of the mass ejection ratecalculated a the disk surface between r = r0 and r = 10r0. Right panel. Time evolution of theratio between the ejection rate and the accretion rate on the central object. In the three panelssolid lines refer to the αm = 1 simulation, dotted lines to the αm = 0.5 case while the dashedlines to the αm = 0.1 simulation.

376 C. Zanni et al.: Launching jets from resistive disks

sources that we intend to model, it is importantto understand if our simulations are evolvingtoward a steady state which can be sustainedon longer timescales. In a steady state solu-tion in the idel MHD regime the poloidal ve-locity should be parallel to the poloidal mag-netic field lines. In the first panel on the left ofFig. 4 we plot the angle between the poloidalspeed and the poloidal magnetic field along aselected magnetic field line at t = 120: we seethat, while in the resistive disk the two vectorsare almost perpendicular due to the accretionmotion, in the outflow, in which the resistivityis negligible, the two vectors tend to be par-allel. To evaluate the steadiness of our solu-tions we also plot in the second panel the massejection rate measured at the disk surface be-tween r = r0 and r = 10r0 as a function oftime: we can see that in all the three simula-tions, after a sudden increase, the ejection ratetends to assume an asymptotic value. We canalso see that the higher is the resistivity, thelower is the rate: the lower resistivity cases de-velop stronger gradients of the magnetic fieldat the disk surface; moreover they have moreopened field lines at the disk surface, thus re-ducing the centrifugal-gravitational potentialbarrier which must be overtaken to acceleratethe outflow (Ogilvie & Livio 1998). In the thirdpanel we finally plot the ratio between the massejection rate and the inner accretion rate on thecentral objects as a function of time: we obtainasymptotic ratios around ' 1, 0.5 and 0.4 forαm = 0.1, 0.5, 1 respectively. A ratio equal to1 means that an equal amount of mass comingfrom the outer accretion goes into the outflowand onto the central object.

5. Energy balance of the outflow

We finally study which are the main differ-ences in the energetics of the outflows pro-duced by disks characterized by a different re-sistivity. In the first panel in Fig. 5 we plot theenergy flux integrated at the disk surface be-tween r = r0 and r = 10r0 as a function oftime; the energy flux is given by the sum of thePoynting flux (v × B) × B and the kinetic flux= 1/2ρv2v. We see that asymptotically the out-flows generated in the three simulations are ex-

tracting about the same amount of energy fromthe disk. Differences can be found when weplot the ratio between the Poynting flux andthe kinetic flux as a function of the z coordi-nate along the outflow (second panel of Fig.5): outflows with a lower ejection rate, whichare then produced by a disk with a higher re-sistivity, show a higher ratio between the twofluxes at the disk surface. On the disk scalethe Poynting flux mainly determines the ex-traction of specific angular momentum fromthe disk (see Zanni et al. 2004): our resultsshow then that the lower is the mass loading ofthe outflow, the more efficient is the magneto-centrifugal process which accelerates the out-flow. Since the Poynting flux which is domi-nant on the disk height scale is then transferredto the kinetic flux along the outflow (secondpanel in Fig. 5), the asymptotic speed of theoutflow is therefore higher for jets with a lowermass flux (third panel in Fig. 5). The asympo-totic speeds are around the escape velocity andsuper-fastmagnetosonic.

6. Summary

The simulations presented in this paper al-lowed us to study in detail the mechanismswhich are responsible for the acceleration andcollimation of astrophysical jets. Our resultsshowed how a poloidal magnetic field thread-ing a Keplerian disk can extract angular mo-mentum from the disk itself and transfer it to anoutflow which is therefore centrifugally accel-erated. Moreover the extraction of angular mo-mentum allows the disk to accrete on the cen-tral object. We performed a series of three sim-ulations varying the resistivity of the disk char-acterized by the adimensional parameter αm =0.1, 0.5, 1. We therefore studied how the ac-cretion and ejection processes are affected bya different resistivity showing that less resis-tive disks produce higher ejection rates. Dueto the different ejection rates also the out-flow showed different properties: the magneto-centrifugal mechanism driving the accelerationis more efficient in the jets with a lower massejection rate thus producing a higher terminalspeed. Nevertheless the total energy flux ex-tracted by the outflow through the disk surface

C. Zanni et al.: Launching jets from resistive disks 377

Fig. 5. Left panel. Time evolution of the energy flux through the disk surface between r = r0 andr = 10r0. The flux is given by the sum of the kinetic and the Poynting flux. Central panel. Plotof the average ratio between Poynting and kinetic fluxes along the outflow at the final simulatedtime t = 120. Right panel. Plot of the average z component of the jet speed along the outflow att = 120. In the three panels solid lines refer to the αm = 1 simulation, dotted lines to the αm = 0.5case while the dashed lines to the αm = 0.1 simulation.

was similar at the final simulated time for allthe three cases studied.

Acknowledgements. The software used in thiswork was in part developed by the DOE–supported ASCI/Alliance Center for AstrophysicalThermonuclear Flashes at the University ofChicago. This work has been co–funded by MIURunder the grant 2002028843. The numericalcalculations have been performed at CINECA inBologna, Italy, thanks to the support of INAF.

References

Blandford, R. D., & Payne, D. G. 1982,MNRAS, 199, 883

Casse, F. & Ferreira, J. 2000, A&A, 353, 1115Casse, F. & Keppens, R. 2004, ApJ, 601, 90

Contopoulos, J. 1994, ApJ, 432, 508Kato, S., Kudoh, T. & Shibata, K. 2002, ApJ,

565, 1053Ogilvie, G. I. & Livio, M. 1998, ApJ, 499, 329Ouyed, R. & Pudritz, R. 1997, ApJ, 482, 712Sakurai, T. 1985, A&A, 39, 821Sauty, C., Trussoni, E. & Tsinganos, K. 2002,

A&A, 389, 1068Shakura, N. I. & Sunyaev, R. A., 1973, A&A

24, 337Uchida, Y., & Shibata, K. 1985, PASJ, 37, 515Ustyugova, G. V., Koldoba, A. V., Romanova,

M. N., Chetchetkin, V. M. & Lovelace, R. V.E. 1995, ApJ, 439, L39

Wardle, M. & Koenigl, A. 1993, ApJ, 410, 218Zanni, C., Ferrari, A., Massaglia, S., Bodo, G.,

& Rossi, P. 2004 Ap&SS, 293, 99