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Relativistic Jets from Accretion Disks

Relativistic Jets from Accretion Disks

arXiv:astro-ph/0409441v2 17 Sep 2004

R.V.E. Lovelace, P.R. Gandhi, & M.M. Romanova Cornell University February 2, 2008
Abstract The jets observed to emanate from many compact accreting objects may arise from the twisting of a magnetic ?eld threading a di?erentially rotating accretion disk which acts to magnetically extract angular momentum and energy from the disk. Two main regimes have been discussed, hydromagnetic jets, which have a signi?cant mass ?ux and have energy and angular momentum carried by both matter and electromagnetic ?eld and, Poynting jets, where the mass ?ux is small and energy and angular momentum are carried predominantly by the electromagnetic ?eld. Here, we describe recent theoretical work on the formation of relativistic Poynting jets from magnetized accretion disks. Further, we describe new relativistic, fully-electromagnetic, particle-in-cell simulations of the formation of jets from accretion disks. Analog Z-pinch experiments may help to understand the origin of astrophysical jets.

1. INTRODUCTION Powerful, highly-collimated, oppositely directed jets are observed in active galaxies and quasars (see for example Bridle & Eilek 1984), and in old compact stars in binaries - the “microquasars” (Mirabel & Rodriguez 1994; Eikenberry et al. 1998). Further, highly collimated emission line jets are seen in young stellar objects (B¨ hrke, Mundt, & Ray 1988). Di?erent models have been put foru ward to explain astrophysical jets (Bisnovatyi-Kogan & Lovelace 2001). Recent observational and theoretical work favors models where twisting of an ordered magnetic ?eld threading an accretion disk acts to magnetically accelerate the jets. Here, we discuss the origin of the relativistic jets observed in active galaxies and quasars and in microquasars. We ?rst discuss a theoretical model (§1), and then new results from relativistic particle-in-cell (PIC) simulations (§2). 2. POYNTING JETS The powerful jets observed from active galaxies and quasars are probably not hydromagnetic out?ows but rather Poynting ?ux dominated jets. The motions of these jets measured by very long baseline interferometry correspond to 1

bulk Lorentz factors of Γ = O(10) which is much larger than the Lorentz factor of the Keplerian disk velocity predicted for hydromagnetic out?ows. Furthermore, the low Faraday rotation measures observed for these jets at distances < kpc from the central object implies a very low plasma densities. Similar arguments indicate that the jets of microquasars are not hydromagnetic out?ows but rather Poynting jets. Poynting jets have also been proposed to be the driving mechanism for gamma ray burst sources (Katz 1997). Theoretical studies have developed models for Poynting jets from accretion disks (Lovelace, Wang, & Sulkanen 1987; Lynden-Bell 1996; Romanova & Lovelace 1997; Levinson 1998; Li et al. 2001; Lovelace et al. 2002; and Lovelace & Romanova 2003). Stationary non-relativistic Poynting ?ux dominated out?ows were found by Romanova et al. (1998) and Ustyugova et al. (2000) in axisymmetric MHD simulations of the opening of magnetic loops threading a Keplerian disk. Here, we summmarize a model for the formation of relativistic Poynting jets from a disk (Lovelace & Romanova 2003). Consider a dipole-like coronal magnetic ?eld - such as that shown in the lower part of Figure 1a - threading a di?erentially rotating Keplerian accretion disk. The disk is perfectly conducting, high-density, and has a small accretion speed (? Keplerian speed). The ?eld may be generated in the disk by a dynamo action and released. Outside of the disk there is assumed to be a “coronal” or “force-free” plasma (ρe E+J × B/c ≈ 0, Gold & Hoyle 1960). We use cylindrical (r, φ, z) coordinates and consider axisymmetric ?eld con?gurations. Thus the ? ? r magnetic ?eld has the form B = Bp + Bφ φ, with Bp = Br ? + Bz z. We have Br = ?(1/r)?Ψ/?z and Bz = (1/r)?Ψ/?r. where Ψ(r, z) ≡ rAφ (r, z) is the ?ux function. Most of the azimuthal twist ?φ of a ?eld line of the Poynting jet occurs along the jet from z = 0 to Z(t) as sketched in Figure 1a, where Z(t) is the axial location of the “head” of the jet. Along most of the distance z = 0 to Z, the radius of the jet is a constant and Ψ = Ψ(r) for Z >> r0 , where r0 is the radius of the O-point of the magnetic ?eld in the disk. Note that the function Ψ(r) is di?erent from Ψ(r, 0) which is the ?ux function pro?le on the disk surface. Hence r2 dφ/dz = rBφ (r, z)/Bz (r, z). We take for simplicity Vz = dZ/dt = const. We determine Vz subsequently. In this case H(Ψ) = [r2 ?(Ψ)/Vz ]Bz can be written as a function of Ψ and dΨ/dr. With H known, the relativistic Grad-Shafranov equation, 1? r? c

?? Ψ ?

?Ψ ·? 2r2

r 4 ?2 c2

= ?H(Ψ)

dH(Ψ) , dΨ


can be solved (Lovelace & Romanova 2003). The quantity not determined by equation (1) is the velocity Vz , or Lorentz factor Γ = 1/(1 ? Vz2 /c2 )1/2 . This is determined by taking into account the balance of axial forces at the head of the jet: the electromagnetic pressure within the jet is balanced against the dynamic pressure of the external medium which 2

2 is assumed uniform with density ρext . This gives (Γ2 ? 1)3 = B0 /(8πR2 ρext c2 ), or for Γ ? 1, 1/6 1/3 1/3 10 B0 1/cm3 Γ≈8 , (2) R 103 G next

where R = r0 /rg ? 1 and rg ≡ GM/c2 , and B0 the magnetic ?eld strength at the center of the disk. A necessary condition for the validity of this equation is that the axial speed of the counter-propagating fast magnetosonic wave (in the lab frame) be larger than Vz so that the jet is e?ectively ‘subsonic.’ This value of Γ is of the order of the Lorentz factors of the expansion of parsecscale extragalactic radio jets observed with very-long-baseline-interferometry (see, e.g., Zensus et al. 1998). This interpretation assumes that the radiating electrons (and/or positrons) are accelerated to high Lorentz factors (γ ? 103 ) at the jet front and move with a bulk Lorentz factor Γ relative to the observer. The r 2 2 ˙ luminosity of the +z Poynting jet is Ej = c 0 0 rdrEr Bφ /2 = cB0 R3/2 rg /3 ? 2.1 × 1046 (B0 /103 G)2 (R/10)3/2 (M/109 M⊙ )2 erg/s, where M is the mass of the black hole. For long time-scales, the Poynting jet is time-dependent due to the angular momentum it extracts from the inner disk (r < r0 ) which in turn causes r0 to decrease with time (Lovelace et al. 2002). This loss of angular momentum leads to a “global magnetic instability” and collapse of the inner disk (Lovelace et al. 1994, 1997, 2002) and a corresponding outburst of energy in the jets from the two sides of the disk. Such outbursts may explain the ?ares of active galactic nuclei blazar sources (Romanova & Lovelace 1997; Levinson 1998) and the one-time outbursts of gamma ray burst sources (Katz 1997). 3. RELATIVISTIC ELECTROMAGNETIC PIC SIMULATIONS We performed relativistic, fully electromagnetic, particle-in-cell simulations of the formation of jets from an accretion disk initially threaded by a dipolelike magnetic ?eld. This was done using the code XOOPIC developed by Verboncoeur, Langdon, and Gladd (1995). Earlier, Gisler, Lovelace, and Norman (1989) studied jet formation for a monopole type ?eld using the relativistic PIC code ISIS. The geometry of the initial con?guration is shown in Figure 1b. The computational region is a cylindrical “can,” r = 0 ? Rm and z = 0 ? Zm , with out?ow boundary conditions on the outer boundaries, and the potential and particle emission speci?ed on the disk surface r = 0 ? Rm , z = 0. Equal ?uxes of electrons and positrons are emitted so that the net emission is e?ectively space-charge-limited. About 105 particles were used in the simulations reported here. The behavior of the lower half-space (z < 0) is expected to be a mirror image of the upper half-space. Figure 2 shows the formation of a relativistic jet. The gray scale indicates the logarithm of the density of electrons or positrons with 20 levels between the lightest (1012 ) and darkest (4 × 1015 /m3 ). The lines are poloidal magnetic ?eld lines Bp . The total B??eld is shown in Figure 3. The computational region 3

has (Rm , Zm ) = (50, 100) m, the initial B??eld is dipole-like with Bz (0, 0) ≡ B0 = 28.3 G and an O-point at (r, z) = (10, 0) m, and the electric potential at the center of the disk is Φ0 = ?107 V relative to the outer region of the disk. Initially, the computational region was ?lled with a distribution of equal densities of electrons and positrons with n± (0, 0) = 3 × 1013 /m3 . Electrons and positrons are emitted with equal currents I± = 3 × 105 A from both the inner and the outer portions of the disk as indicated in Figure 1b with an axial speed much less than c. For a Keplerian disk with r0 ? rg , the scalings are Φ0 ? 2 3/2 1/2 B0 (r0 rg )1/2 , I ? cB0 r0 and the jet power is ? cB0 r0 rg . The calculations were done on a 64 × 128 grid stretched in both the r and z directions so as to give much higher spatial resolution at small r and small z. These simulations show the formation of a quasi-stationary, collimated current-carrying jet. The ˙ Poynting ?ux power of the jet is Ej ≈ 7 × 1011 W and the particle kinetic energy power is ≈ 4.7 × 1010 W. The charge density of the electron ?ow is partially neutralized by the positron ?ow. Simlations are planned with the positrons replaced by ions. Scaled Z-pinch experiments con?gured as shown in Figure 1b can allow further study of astrophysical jet formation. We thank C. Birdsall, S. Colgate, H. Li, J. Verboncoeur, I. Wasserman, J. Wick, and T. Womack for valuable assistance and discussions. This work was supported in part by NASA grants NAG5-13060 and NAG5-13220, by NSF grant AST-0307817, and by DOE cooperative agreement DE-FC03 02NA00057. REFERENCES Bisnovatyi-Kogan, G.S. & Lovelace, R.V.E. 2001, New Astron. Rev., 45, 663 Bridle, A.H., & Eilek, J.A. (eds.) 1984, in Physics of Energy Transport in Extragalactic Radio Sources, (Greenbank:- NRAO) B¨ hrke, T., Mundt, R., & Ray, T.P. 1988, A&A, 200, 99 u Eikenberry, S., Matthews, K., Morgan, E.H., Remillard, R.A., & Nelson, R.W. 1998, ApJ, 494, L61 Gisler,G., Lovelace, R.V.E., & Norman,M.L. 1989, ApJ, 342, 135 Gold, T., & Hoyle, F. 1960, MNRAS, 120, 89 Katz, J.1. 1997, ApJ, 490, 633 Levinson, A. 1998, ApJ, S07, 145 Li, H., Lovelace, R.V.E., Finn, J.M., & Colgate, S.A. 2001, ApJ, 561, 915 Lovelace, R.V.E., Li, H., Koldoba, A.V, Ustyugova, G.V, & Romanova, M.M. 2002, ApJ,572,445 Lovelace, R.V.E., Newman, W.I., & Romanova, M.M. 1997, ApJ, 484, 628 Lovelace, R.V.E., Romanova, M.M., & Newman, W.I. 1994, ApJ, 437, 136 Lovelace, R.V.E., Wang, J.C.L., & Sulkanen, M.E. 1987, ApJ, 315, 504 Lovelace, R.V.E., & Romanova, M.M. 2003, ApJ, 596, L159 Lynden-Bell, D. 1996, MNRAS, 279, 389 4

Figure 1: (a) Sketch of the magnetic ?eld con?guration of a Poynting jet from Lovelace and Romanova (2003). The bottom part of (a) shows the initial dipolelike magnetic ?eld threading the disk which rotates at the angular rate ?(r). The top part shows the jet at some time later when the head of the jet is at a distance Z(t). At the head of the jet there is force balance between electromagnetic stress of the jet and the ram pressure of the ambient medium of density ρext . (b) Sketch of the initial conditions for the relativistic PIC simulations of jet formation from an accretion disk. Mirabel, I.F., & Rodriguez, L.F. 1994 Nature, 371, 46 Romanova M.M., & Lovelace R.V.E. 1997, ApJ, 475, 97 Romanova, M.M., Ustyugova, G.V, Koldoba, A.V, Chechetkin, VM., & Lovelace, R.V.E. 1998, ApJ, 500, 703 Ustyugova, G.V, Lovelace, R.V.E., Romanova, M.M., Li, H., & Colgate, S.A. 2000 ApJ, 541, L21 Verboncoeur, J.P., Langdon, A.B., & Gladd, N.T. 1995, Comp. Phys. Comm., 87, 199 Zensus, J.A., Taylor, G.B., & Wrobel, J.M. (eds.) 1998, Radio Emission from Galactic and Extragalactic Compact Sources, IAU Colloquium 164, (Ast. Soc. of the Paci?c)


Figure 2: Relativistic PIC simulations of the formation of a jet from a rotating disk. (a) -(c) give snapshots at times (1, 2, 3) × 10?7 s, and (d) is at t = 10?6 s.


Figure 3: Three dimensional magnetic ?eld lines originating from the disk at r = 1, 2 m for the same case as Figure 2.



ρ ext




outflow boundary conditions

initial plasma


outflow boundary







Bp (t=0)






(equal e+/-)

Fig. 1

rotating conducting disk N E r = - vφ Bz /c

(equal e+/-)

Fig. 2

Fig. 3



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