The Jets and Disc of SS 433 at Super-Eddington Luminosities
aa r X i v : . [ a s t r o - ph . H E ] J un Mon. Not. R. Astron. Soc. , 1–11 (2004) Printed 7 June 2018 (MN L A TEX style file v2.2)
The Jets and Disc of SS 433 at Super-EddingtonLuminosities
T. Okuda ⋆ , G. V. Lipunova , D. Molteni Hakodate Campus, Hokkaido University of Education, Hachiman-cho 1-2, Hakodate 040-8567, Japan Sternberg Astronomical Institute, Universitetskiy pr. 13, Moscow, 119992, Russia Dipartimento di Fisica e Tecnologie Relative, Universita di Palermo, Viale delle Scienza, Palermo, 90128, Italy
Accepted
ABSTRACT
We examine the jets and the disc of SS 433 at super-Eddington luminosities with˙ M ∼
600 ˙ M c by time-dependent two-dimensional radiation hydrodynamical calcula-tions, assuming α -model for the viscosity. One-dimensional supercritical accretion discmodels with mass loss or advection are used as the initial configurations of the disc.As a result, from the initial advective disc models with α =0.001 and 0.1, we obtainthe total luminosities ∼ . × and 2 . × erg s − . The total mass-outflow ratesare ∼ × − and 10 − M ⊙ yr − and the rates of the relativistic axial outflows in asmall half opening angle of ∼ ◦ are about 10 − M ⊙ yr − : the values generally con-sistent with the corresponding observed rates of the wind and the jets, respectively.From the initial models with mass loss but without advection, we obtain the totalmass-outflow and axial outflow rates smaller than or comparable to the observed ratesof the wind and the jets respectively, depending on α . In the advective disc modelwith α = 0 .
1, the initially radiation-pressure dominant, optically thick disc evolves tothe gas-pressure dominated, optically thin state in the inner region of the disc, andthe inner disc is unstable. Consequently, we find remarkable modulations of the discluminosity and the accretion rate through the inner edge. These modulations manifestthemselves as the recurrent hot blobs with high temperatures and low densities at thedisc plane, which develop outward and upward and produce the QPOs-like variabilityof the total luminosity with an amplitude of a factor of ∼ ∼
10– 25 s. This may explain the massive jet ejection and the QPOs phenomena observedin SS 433.
Key words: accretion, accretion discs – black hole physics – hydrodynamics – radi-ation mechanism: thermal– X-rays:individual: SS 433.
Disc accretion is an essential process for such phenomenaas energetic X-ray sources, active galactic nuclei, and pro-tostars. Since the early works by Pringle & Rees (1972) andShakura & Sunyaev (1973), a great number of papers havebeen devoted to studies of the disc accretion onto gravi-tating objects. When the accretion rate is not too high,the accretion disc luminosity is directly in proportion tothe accretion rate and can be successfully described by theShakura-Sunyaev (S-S) model. However, for the supercriti-cal accretion rate, matter flows out of the disc, and the rateof accretion onto the central black hole is reduced, regu-lating the luminosity to the Eddington limit. The galacticmicroquasar SS 433 is a promising super-critically accret- ⋆ E-mail:[email protected] ing sterllar-mass black hole candidate and has stimulatednumerous studies, because it displays remarkable observa-tional features, such as its extremely high energy, two op-positely directed relativistic jets, and the precessing mo-tion of the jets. Although a number of observational andtheoretical studies on SS 433 have been published (for de-tailed reviews see Margon 1984; Fabrika 2004), there arestill many problems of the jets and the disc to be solved.The super-Eddington accretion discs are generally expectedto possess vortex funnels and radiation-pressure drivenjets from geometrically thick discs (Shakura & Sunyaev1973; Lynden-Bell 1978; Fukue 1982; Calvani & Nobili1983). Such supercritical disc models have been numericallyexamined by two-dimensional radiation hydrodynamicalcalculations (Eggum, Coroniti & Katz 1985, 1988; Okuda2002; Okuda et al. 2005; Ohsuga et al. 2005; Ohsuga 2007;Ohsuga & Mineshige 2007), especially focussing on the puz- c (cid:13) T. Okuda, G. V. Lipunova, and D. Molteni r/Rg m ( r ) H / R . h/r m. Figure 1.
Solid lines: accretion rate ˙ m ( r ) normalized to the inputaccretion rate and relative thickness h/r of the supercritical discwith mass loss for the case of the input accretion rate ˙ m ∼
600 and the viscosity parameter α = 0 .
1. Dashed lines show thesolution for S-S model without mass loss. r/Rg l ogT l og ρ T ρ Figure 2.
Same as Fig.1 but for the profiles of temperature T (K) and density ρ (g cm − ). The temperature profile is the samefor the both models, as far as the radiation-pressure dominantdisc is concerned. zling X-ray source SS 433 (Eggum, Coroniti & Katz 1985,1988; Okuda 2002; Okuda et al. 2005). However, these worksleave something to be desired as far as SS 433 is concerned,because the accretion rates adopted in these studies are verysmall, compared with those estimated for SS 433. In this pa-per, we examine the properties of the jets and the disc ofSS 433 with the plausible supercritical accretion rate. Weperform time-dependent two-dimensional radiation hydro-dynamical calculations, using 1D models of the supercriticalaccretion disc models with mass loss or advection (Lipunova1999) as initial disc configurations. r/Rg l og l ogT H / R -10-8-6-4-20 56789 00.40.81.21.62 ρ h/r T ρ Figure 3.
Disc thickness h/r , temperature T (K), and density ρ (g cm − ) of the supercritical disc with advection for the case of˙ m ∼
600 and α = 0 . Our previous studies of SS 433 (Okuda 2002; Okuda et al.2005) are based on the initial discs by S-S model with˙ m
20. Here, ˙ m is the input accretion rate normalizedto the Eddington critical accretion rate ˙ M c (= L E /ηc =4 πGM/ηcκ ), where L E is the Eddington luminosity, G thegravitational constant, M the black hole mass, κ the Tho-moson scattering opacity, c the velocity of light, and η theconversion efficiency of the energy of the accreting matterinto radiation. Adopting η = 1 /
12, we have ˙ M c = 2 . × g s − and L E = 1 . × erg s − for M = 10M ⊙ . Inthis paper we consider ˙ m ∼ M ∼ . × − M ⊙ yr − and is a more plausible valuefor SS 433 (Fabrika 2004). The S-S disc with such a highaccretion rate is geometrically too thick and becomes in-valid against assumptions used. When the accretion rate isvery high, the disc luminosity exceeds the Eddington lu-minosity and matter flows out of the disc surface, causingthe accretion rate onto the central object to decrease. Theoutflow takes place inside the radius at which the disc thick-ness becomes comparable to the disc radius. This radius iscalled a spherization radius R sp (Shakura & Sunyaev 1973).Lipunova (1999) proposed supercritical disc models, takingaccount of mass loss and advection through the accretiondiscs, which were applied to SS 433 and the ultraluminousX-ray sources (ULXs) by Poutanen et al. (2007). In theirmodels with mass loss, it is assumed that a fraction ǫ W ofthe radiation energy flux is spent on the production of theoutflow within the spherization radius. For ǫ W = 1, if advec-tion is neglected, we obtain analytical solutions (Lipunova1999) for the disc variables at the disc plane, and the ra-tio R sp / R g is estimated to be ∼ ˙ m , where R g is theSchwarzschild radius. The accretion rate in the disc withinthe spherization radius is given by˙ M ( r ) = ˙ M (cid:18) rR sp (cid:19) ( r R g ) − / ( R sp R g ) − / for r R sp . (1)This law has the asymptotic form c (cid:13) , 1–11 he Jets and Disc of SS 433 Table 1.
Results and comparison with observationsModel α ˙ M L L d ˙ M out ˙ M ◦ ˙ M edge t ev (M ⊙ yr − ) (erg s − ) (erg s − ) (M ⊙ yr − ) (M ⊙ yr − ) (M ⊙ yr − ) ( R g /c )ML-1 10 − . × − . × . × . × − . × − . × − . × ML-2 0.1 1 . × − . × . × . × − . × − . × − . × AD-1 10 − . × − . × . × . × − . × − . × − . × AD-2 0.1 1 . × − . × . × . × − . × − . × − . × Observation – 10 − – 10 − – 10 – 10 − – 10 − − – 10 − – – Time (Rg/c) l ogL ( e r g / s ) l og M ( g / s ) . LL d Input accretion rateM M. out M edge. Figure 4.
Time evolution of total luminosity L , disc luminosity L d , total mass-outflow rate ˙ M out ejected from the entire outer boundarysurface, mass-outflow rate ˙ M ◦ of the axial outflow with a half opening angle 1 . ◦ along the rotational axis, and mass-inflow rate ˙ M edge into the black hole through the inner boundary for model ML-2, where time is shown in units of R g /c . ˙ M ( r ) ≈ ˙ M rR sp for 3 R g ≪ r R sp . (2)The temperature T and the density ρ in the central planeof the disc are approximately given in the region of 3 R g ≪ r R sp by T ( r ) = 6 . × (cid:18) M ⊙ (cid:19) − / (cid:16) α . (cid:17) − / (cid:18) r R g (cid:19) − / K , (3) ρ ( r ) = 1 . × − (cid:18) M ⊙ (cid:19) − (cid:16) α . (cid:17) − (cid:18) r R g (cid:19) − / g cm − , (4)where α is the viscosity parameter.Figs 1 and 2 show the accretion rate ˙ m ( r ), the relativedisc thickness h/r , the central plane temperature T , and thedensity ρ for the 1D model with mass loss and the viscosityparameter α = 0 .
1, along with the solution for S-S modelwithout mass loss. Outside the spherization radius the solu-tions are identical.The disc at the supercritical accretion rates be-comes geometrically thick. The emission from the sur-face of a thick disc is not an efficient cooling mecha-nism, and the advective transport of the viscously gener-ated heat becomes important in the energy balance equa-tion (Paczy´nski & Bisnovatyi-Kogan 1981). An advection-dominated disc model at moderately super-Eddington accre-tion rates ˙ m
70 was first proposed by Abramowicz et al.(1988). A supercritical disc with even higher accretion rateswas approached by Lipunova (1999), where mass loss and advection were included, and the discs were geometricallythick ( h/r ∼ . h/r > m ∼ α = 0 .
1. This solution and the analytic solution for thedisc with mass loss, described above, are used in the currentwork as initial configurations of the disc in the 2D numericalmodel.
A set of relevant equations for the numerical calculation con-sists of six partial differential equations for the density, themomentum, and the thermal and radiation energy. Theseequations include the full viscous stress tensor, the heat-ing and cooling of the gas, and the radiation transport.The pseudo-Newtonian potential (Paczy´nski & Wiita 1980)is adopted in the momentum equation. The radiation trans-port is treated in the gray, flux-limited diffusion approxi-mation (Levermore & Pomraning 1981). We use sphericalpolar coordinates ( r , ζ , ϕ ), where r is the radial distance, ζ is the angular distance measured from the equatorial planeof the disc, and ϕ is the azimuthal angle. The above setof differential equations is numerically integrated in timeusing a finite-difference method, which is an improved ver-sion of that described in Kley (1989). The method is basedon an explicit-implicit finite difference scheme, whose de-tails are described by Okuda, Fujita & Sakashita (1997) and c (cid:13) , 1–11 T. Okuda, G. V. Lipunova, and D. Molteni
Okuda et al. (2005). The computational domain is dividedinto N r × N ζ grid cells, where N r grid points (=100) in theradial direction are spaced logarithmically as ∆ r/r = 0 . N ζ grid points (=150) in the angular direction areequally spaced, but more refined near the equatorial plane,typically with ∆ ζ = π/
150 for π/ > ζ > π/ ζ = π/
300 for π/ > ζ >
0, in order to resolve the struc-ture of the accretion disc. Although the radial mesh-sizes donot have a fine resolution to examine detailed disc structure,we consider the mesh-sizes to be sufficient for examinationof the global behavior of the disc, the jets, and the windmass-outflow rates.
We consider a Schwarzschild black hole with mass M =10M ⊙ and take the inner-boundary radius R in = 2 R g andthe outer boundary radius of the spherical computationaldomain R out = 4 . × R g ( ∼ . × cm). For the kine-matic viscosity, we adopt the usual α - model. In Table 1,we give the parameters of the discs with mass loss (ML)and advection (AD), where ˙ M is the input accretion rate,and α is the viscosity parameter. Starting with 1D solutionsdescribed in section 2, we perform time integration of theequations until a quasi-steady solution is obtained. The initial conditions consist of a dense, optically thick ac-cretion disc and a rarefied optically thin atmosphere aroundthe disc. Physical variables ρ and T at the equatorial planefor r > R g are given by the 1D solutions for the super-critical discs with mass loss or advection. We construct thevertical structure of the disc in the approximation of thehydrostatic and radiative equilibrium by integrating the rel-evant equations with given boundary values at the equato-rial plane. As for the initial atmosphere around the disc,assuming that the gas is in the optically thin limit and inthe radially hydrostatic equilibrium, we have F r = cE r = − λcρκ e ∂E r ∂r , (5) ∂P∂r = − ρ GMr , (6)where F r is the radial component of the radiative flux F , E r the radiation energy density, λ the flux-limiter, κ e theelectron-scattering opacity, and P the total pressure. Fur-thermore, if it is assumed that the flux-limiter λ is constantand that the radiation pressure is dominant throughout thegas, we have for the initial radiation energy density E r andthe density ρ , from the above equations, E r = 1 κ e GMr , (7) ρ = 2 λκ e r . (8)Actually, the flux-limiter λ is taken to be ∼ − . How-ever, we note that a particular initial distribution of the gasaround the disc does not influence the results at the suffi-ciently late simulation times. r/Rg l og l ogT -8-6-4 68101214 10 -3 -2 -1 ρ TT r βρ l og β Figure 5.
Radial profiles of density ρ (g cm − ), temperature T (K), radiation temperature T r (K) (dashed line), @ and ratio β of the gas pressure to the total pressure on the equatorial planeat t = 5 . × R g /c for model ML-2. Physical variables at the inner boundary, except for the ve-locities, are calculated by extrapolation of the variables nearthe boundary. We impose limiting conditions that the radialvelocities at the inner boundary are given by a fixed free-fallvelocity and the angular velocities are zero. On the rota-tional axis and the equatorial plane, the meridional tangen-tial velocity w is zero and all scalar variables must be sym-metric relative to the axis and the plane. The outer bound-ary at r = R out is divided into two parts. One is the discboundary through which matter is entering from the outerdisc. At this outer-disc boundary we assume a continuousinflow of matter with a constant accretion rate ˙ M . Theother part is the outer boundary region above the accretiondisc. We impose free-floating conditions on this outer bound-ary and allow for outflow of matter, whereas any inflow isprohibited here. We also assume the outer boundary regionabove the disc to be in the optically-thin limit, | F | → cE r . In order to obtain a reliable configuration of the jet, the sim-ulation time should be longer than the jet arrival time t jet atthe outer boundary. On the other hand, to get a steady stateof a viscous disc in a region of radius r , we need the sim-ulation time longer than the characteristic dynamical time t dyn and the viscous time-scale t vis . These times are givenas follows: t jet ∼ R out V jet = 2 . × (cid:16) V jet . c (cid:17) − R g c , (9) t dyn ∼ Ω − = (cid:18) rR g (cid:19) / R g c = 3 × (cid:18) r R g (cid:19) / R g c , (10)and t vis ∼ t dyn α (cid:16) hr (cid:17) − = 3 × (cid:16) α . (cid:17) − (cid:16) hr (cid:17) − (cid:18) r R g (cid:19) / R g c , (11) c (cid:13) , 1–11 he Jets and Disc of SS 433 where Ω and V jet are the Keplerian angular velocity and thetypical jet velocity. For the supercritical disc of SS 433, wehave h/r ∼ V jet ∼ . c . Large-scale mass loss sets innear the spherization radius of the disc at r ∼ R g . Forthis radius and α = 0 .
1, the maximum time of t jet , t dyn , and t vis is ∼ × R g /c . We set this time as an approximatecomputational goal, exceeding it in the cases of models ML-2and AD-2 in order to examine the disc instability. Whereasthe quasi-steady values of the luminosities are almost at-tained at the final phases, the simulation times are still notsufficient for the disc and the outflow to settle into a com-pletely steady state, because the input accretion rate is notequal to the total mass-outflow rate plus the mass-flux rateof the falling gas into the black hole. In spite of the lim-ited computational time, we are able to derive the generalproperties of the disc and the outflow.The luminosity curve is a good measure to check if asteady state of the disk and the outflow is attained. Thetotal luminosity L and the disc luminosity L d are calculatedas R F d S , where the surface integral is taken over the outerboundary surface and the disc surface, respectively. The discsurface is placed at the height where the density drops to atenth part of the value at the equatorial plane. This lead tosome uncertainty in L d because the vertical structure of ageometrically thick disc is treated rather approximately.To compare with the observational data for SS 433,we calculte the total mass-outflow rate ˙ M out and the “ax-ial outflow” rate ˙ M ◦ . The value ˙ M out is calculated for theentire outer-boundary surface and corresponds to the ob-served mass-outflow in the wind of SS 433. Observationsindicate that the opening angle of the X-ray and opticaljets is ∼ . ◦ (Marshall, Canizares & Schulz 2002; Fabrika2004). Throughout the paper, we call the relativistic out-flow through the outer base of the cone directed along therotational axis with the half opening angle of 1 . ◦ “the ax-ial outflow”. The mass-flux rate ˙ M ◦ of the axial outflowshould correspond to the mass-outflow rate in the jets of SS433, provided the size of the present computational domainis comparable to the distance of the observed jets from thecentral source. In order to check the total mass-flux rate,we also calculate the mass-flux rate ˙ M edge of the gas fallingthrough the inner boundary into the black hole.In Table 1 we give L , L d , ˙ M out , ˙ M ◦ , and ˙ M edge atthe final simulation time t ev for each of the models. Val-ues L , L d , and ˙ M edge show QPOs-like features in modelAD-2; Table 1 shows their averaged values around the fi-nal phase. In the columns for ˙ M out and ˙ M ◦ , one can seethe observational mass-outflow rate in the wind and the jetsof SS 433 (Dopita & Cherepashchuk 1981; van den Heuvel1981; Kotani 1998; Marshall, Canizares & Schulz 2002;Cherepashchuk et al. 2003; Fabrika 2004). In the model ML-1, the initial disc is radiation-pressuredominant and optically thick. The perturbed density andtemperature waves are generated initially near the inneredge, and then heating and cooling waves move up and downbetween the inner edge and the radius of ∼ R g , raisingthe disc temperature to ∼ K. As a result, the gas-pressure dominat, optically thin region is formed in the in-nermost region. The process is repeated for a while, then - - - - - - - - - - - - - - - - r/Rg z / R g -12 -9 -6 -4 log ρ c Figure 6.
Velocity vectors and density contours near the spher-ization radius on the meridional plane at t = 3 . × R g /c formodel ML-2. The reference vector of light speed is shown by along arrow. The axial outflow propagate with relativistic veloci-ties ∼ . c along the Z-axis. r/Rg z / R g log T Figure 7.
Same as Fig.6 but for temperature contours. Thevelocity vectors are indicated by unit vectors. stops after t ∼ × R g /c . Finally, we get stable, opti-cally thick, and radiation-pressure dominant disc. The re-sultant disc temperatures at the equatorial plane are notdifferent significantly from the initial temperatures of thedisc, while the densities are enhanced by a factor of 10 –30 in the region of r R g . The final luminosities L and L d ∼ × erg s − are one order of magnitude largerthan the Eddington luminosity. The total mass-outflow rate˙ M out ∼ . × − M ⊙ yr − is less by an order than the mass-outflow rate of the wind observed in SS 433, and the rate ofthe axial outflow ˙ M ◦ ∼ . × − M ⊙ yr − is marginallyin the range of the observed mass-outflow rate of the jets.Fig. 4 shows the time evolutions of L , L d , ˙ M out , ˙ M ◦ ,and ˙ M edge for model ML-2 with the viscosity parameter α =0.1. The total luminosity L becomes comparable to thedisc luminosity L d at the time t ∼ R out /c = 4 . × R g /c ,which is the photon transit time from the inner edge to theouter boundary. The total luminosity L ∼ erg s − ob- c (cid:13) , 1–11 T. Okuda, G. V. Lipunova, and D. Molteni
Time (Rg/c) l ogL ( e r g / s ) l og M ( g / s ) Input accretion rate MM out . . M edge . . L d L Figure 8.
Time evolutions of L , L d , ˙ M out , ˙ M ◦ , and ˙ M edge formodel AD-1. tained finally is greater by a factor of 6 than the Eddingtonluminosity.The initial disc of ML-2 is radiation-pressure dominantand optically thick throughout the whole disc region. In thesame way as in the case of ML-1, the perturbations of den-sity and temperature, generated initially near the inner edge,propagate outward and inward as heating and cooling waves,and the high temperature, gas-pressure dominant, opticallythin region is formed in the inner disc. The instability inthe optically thin region is never damped, as indicated by˙ M edge in Fig. 4, and appears as an oscillating hot blob witha variable size of 10 – 100 R g at the equatorial plane. Aftera large-scale hot blob at t ∼ . × R g /c , the instabil-ity persists but the absolute variation amplitude of ˙ M edge becomes smaller. In spite of the considerable variability of˙ M edge , the modulations of L and L d are weak, except forthe phase of the large-scale blob, and become negligible atthe later phase. At the final phase, the first outward heat-ing wave reaches the spherization radius and merges intothe outer Shakura-Sunyaev disc, and the whole disc tendsto settle gradually into a steady state. Eventually, the discevolves into two zones: the gas-pressure dominant and op-tically thin inner disc and the radiation-pressure dominantand optically thick outer disc.Fig. 5 shows the radial profiles of density ρ (g cm − ),gas temperature T (K), radiation temperature T r (K), andratio β of the gas pressure to the total pressure on the equa-torial plane at t = 5 . × R g /c for model ML-2. The ra-diation temperature T r is defined as T r = ( E r /a ) / , where a is the radiation density constant. When the gas is fullyoptically thick, the gas temperature T should be equal tothe radiation temperature T r . When the gas becomes opti-cally thin, T r is much lower than T as it is found in Fig. 5.Initially, the radiation temperature T r and the gas temper-ature T are ∼ K near the inner edge of the opticallythick disc. However, after some time, the gas temperaturenear the inner edge goes up to 10 – 10 K and is variableby a factor of 10. Similarly, the radiation temperature is alsomodulated between 10 – 10 K in the region of r R g .The gas-pressure dominant, hot, optically thin inner disc isseparated from the radiation-pressure dominant outer discby the transition region at r R g . The gas variables T and ρ vary sharply across the transition region, but theradiation temperature T r is smooth everywhere.Figs 6 and 7 show the contours of density ρ (g cm − )and temperature T (K) with the velocity vectors of the gasnear the spherization radius at t = 3 . × R g /c for modelML-2. Here we see a rarefied, hot, and optically thin high-velocity outflow region and a dense, cold, and geometricallythick disc. The outflow propagates with relativistic velocitiesof 0.08 – 0.2 c along the Z-axis. The disc is geometrically thickwith h/r ∼ R g r × R g and the large-scale outflow sets in the region. In the high-velocity region along the rotational axis, the temperature isas high as ∼ K and the density is as low as ∼ − –10 − g cm − . Fig. 8 shows the time evolutions of L , L d , ˙ M out , ˙ M ◦ , and˙ M edge for model AD-1. The evolutionary features of the discis very similar to the case of ML-1. Finally, we get stable,optically thick, and radiation-pressure dominant disc withthe maximum disc temperature ∼ K in the innermostregion. The total luminosities L ∼ . × and 2 . × erg s − in models AD-1 and AD-2 are in the same rangeas those in models ML-1 and ML-2. The total mass-outflowrates are ∼ . × − and 1 . × − M ⊙ yr − and the rates ofthe relativistic axial outflow are 9 . × − and 1 . × − M ⊙ yr − in models AD-1 and AD-2 respectively. These valuesof ˙ M out and ˙ M ◦ are consistent with the corresponding ob-served rates of the wind and the jets.Fig. 9 shows the time evolutions of L , L d , ˙ M out , ˙ M ◦ ,and ˙ M edge for model AD-2. In this case, we find the remark-able variabilities of ˙ M edge , L d , and L and, to examine theirproperties, we took a long simulation time t ∼ R g /c ( ∼
100 s). The high-velocity jets propagate vertically to the discplane and expand gradually from the rotational axis with in-creasing time. After the time R out / . c , the jets arrive at theouter boundary in the polar direction, and the mass outflowbegins and gradually approaches the steady state. In modelAD-2, most of the accreting matter is flown out as wind andonly 10 percent of the input matter is swallowed into theblack hole.The modulations of ˙ M edge in AD-2 show two types ofvariability: (1) the small amplitude variations with a shorttime-scale, (2) the large amplitude variations with a longtime-scale. The disc luminosity is also modulated by a fac-tor of a few to ten, synchronously with ˙ M edge . The varia-tions with the small amplitude and short periods stronglyinfluence the disc luminosity but not the total luminositymeasured at the distant outer boundary. If the atmospherebetween the disc surface and the outer boundary was fullyoptically thin, the total luminosity would suffer from thesame modulations as the disc luminosity. However, in the su-percritical accretion flow with high input density, only thelarge modulations with the long periods contribute to thetotal luminosity, due to the atmospheric absorption aroundthe disc. These modulations behave as QPOs-like variabili-ties. In Fig. 10, we plot the power density spectra of ˙ M edge , L d , and L and recognize the QPOs-like features of thesespectra with characteristic signals of ν ∼ × − – 10 − and 0.5 – 2 Hz for ˙ M edge and L d . Only longer-period vari-ations are obtained for L ; therefore, we expect modulations c (cid:13) , 1–11 he Jets and Disc of SS 433 Time (Rg/c) l ogL ( e r g / s ) l og M ( g / s ) . Input accretion rateM M M
L L d Figure 9.
Time evolutions of L , L d , ˙ M out , ˙ M ◦ , and ˙ M edge for model AD-2. The disc luminosity and the mass accretion rate at theinner edge show two types of variability: one with the small amplitude and short periods and another with large amplitude and longperiods. Only the longer-term variations influence the total luminosity in the form of small modulations measured at the distant outerboundary. ν P o w e r o f L Ld P o w e r o f M -2 -1 -100001000 01500 015003000 LL d M edge. e dg e . Figure 10.
Power spectra of time variations of the total lumi-nosity L , the disc luminosity L d , and the mass-inflow rate ˙ M edge at the inner edge of the computational domain for model AD-2. of the observed luminosity with the quasi-periods between ∼
10 and 25 s.Fig. 11 shows the radial profiles of density ρ (g cm − )and temperature T (K) at the equatorial plane for t =1 . × (dashed lines) and 1 . × R g /c (solid lines)for model AD-2. Similarly to the case of model ML-2, thedisc obtained in model AD-2 consists of three regions: (1)the gas-pressure dominant, optically thin disc in the in-ner region, (2) the advection-dominated, radiation-pressuredominant, optically thick disc in the intermediate region,(3) the outer radiation-pressure dominant, optically thickShakura-Sunyaev disc at r > R g . The regions (1) and(2) are sharply separated by the critical radius r c . The criti-cal radius moves randomly up and down, and usually reaches ∼ R g with a heating wave, advancing sometimes to ∼ R g and rarely to the maximum radius ∼ R g , thenrecedes to the inner edge with a cooling wave. These insta-bilities in the inner disc lead to recurrent hot blobs with high r/Rg ρ l ogT -7-6-5-4 68101214 l og T ρ Figure 11.
Radial profiles of density ρ (g cm − ) and tempera-ture T (K) at the equatorial plane for AD-2, where dashed andsolid lines show the profiles at the phases of t = 1 . × and1 . × R g /c . The gas-pressure dominant, optically thin discin the inner region is separated from the advection-dominated,radiation-pressure dominant, optically thick disc by the criticalradius r c , which moves randomly up and down, usually reaches ∼ R g , and sometimes advances to ∼
100 – 200 R g . temperatures and low densities, like bubbles in the boillingwater. The hot blobs typically grow to the size of ∼ R g inthe disc plane, go up with increasing size, and finally decayat z ∼ R g . The time evolution of the hot blobs is shownin Fig. 12.Figs 13 and 14 show the contours of gas temperature T (K) and radiation temperature T r (K) with velocity vectorsin the whole computational domain at t = 1 . × R g /c formodel AD-2. The gas temperature T in the high-velocityjets region is as high as ∼ K at r > cm and T ≫ T r because the jet region is mildly optically thin, but T ∼ T r in the outer optically thick disc. From Fig. 14, we findthat the contours of T r , as well as the contours of radiationenergy density E r , show an anisotropic distribution of radial c (cid:13) , 1–11 T. Okuda, G. V. Lipunova, and D. Molteni r/Rg Y r/Rg Y r/Rg z / R g r/Rg z / R g TIME= 1.74E5 log T
Figure 12.
Time evolution of hot blobs in model AD-2, where the temperature contours of the blobs are shown at t = 1 . × ,1 . × , 1 . × , and 1 . × R g /c . The hot blobs typically grow to the size of ∼ R g in the disc plane, go up with increasingsize, and finally decay at z ∼ R g . component F r of the radiative flux F , where F ∝ − grad E r .Actually, the radial components F r in the direction of ζ > ◦ exceed those in the direction of ζ ∼ ◦ by a factor of 5– 7. Fig. 15 shows the contours of density ρ (g cm − ) withmagnified velocity vectors over the whole computational do-main, where the velocity vector of 0.2 c is denoted in thelegend. From detailed analyses of the density contours andthe velocity vectors, we recognize three characteristic out-flows originating in the different regions of the disc shown inFig.11: (1) the most relativistic axial outflow with velocities ∼ c ejected perpendicularly to the innermost hot,optically thin disc, (2) the high-velocity (0.1 – 0.05 c ) out-flow within a half opening angle ∼ ◦ ejected from theadvection-dominated, optically thick disk in the interme-diate region, (3) the slow outflow with velocities ∼ . c flowing from the disc region near the spherization radius.The slow outflow (3) from the outer disc, interacting withthe high-velocity outflow (2), is blown obliquely beneath thehigh-velocity flow and is accelerated up to the velocities of0.002 – 0.05 c for 10 ◦ ζ ◦ at the outer boundary.The present result for the supercritical disc with ˙ m ∼ ∼ ◦ in the previous study with˙ m ∼
20 (Okuda et al. 2005). This is due to the reason thatthe supercritical disc with a very high accretion rate has avery large spherization radius, where a massive outflow (3)is going out distortedly from the disc.
The initial discs in all cases considered here are radiation-pressure dominant and optically thick throughout the whole disc region. Therefore, their stability may be supposedly in-terpreted in terms of the slim disc model at highly super-Eddington luminosity; that is, these discs should be sta-ble against the local and global perturbations. Actually, thetime evolution of the disc in models ML-1 and AD-1 withsmall α = 0 .
001 eventually shows stable features. On theother hand, just at the beginning of the simulations in ML-2 and AD-2, the unstable behaviors of the accretion rateat the inner edge and the disc luminosity are revealed. Thiscan be attributed to the fact that the gas-pressure dominant,optically thin, high temperature state is triggered in the in-nermost region of the disc due to initial perturbations of thedisc variables. The optically thin disc region, as that devel-oping repeatedly in model AD-2, is not attained finally inmodels ML-1 and AD-1, because the disc densities in thesemodels are much higher than those in ML-2 and AD-2 dueto ρ ∝ α − in the initial disc.The thermal instability of advection-dominated one-temperature discs was examined byKato, Abramowicz & Chen (1996). It was shown that,in the case of α - viscosity, the optically thin advectivedisc is unstable against local perturbations if the viscosityparameter α is small, but that two-dimensional analysis isnecessary to investigate stability if α is large. The stabilityof an optically thin, advection-dominated accretion discwith large α = 0 . c (cid:13) , 1–11 he Jets and Disc of SS 433 r/Rg z / R g log T c Figure 13.
Velocity vectors and contours of gas temperature T (K) in logarithmic scale on the meridional plane at t = 1 . × R g /c for model AD-2. . . . . . . r/Rg z / R g log Trc Figure 14.
Same as Fig.13 but for radiation temperature T r (K).If the gas is fully optically thick, T r is equal to the gas temperature T (K). significantly. Thus, our two-dimensional calculations of theadvection-dominated discs confirm the above theoreticaland 1D numerical results. On the other hand, for modelML-2 with large α , we find that ˙ M edge , T , and T r aremodulated by a factor of 3 – 10 in the innermost region ofthe optically thin disc, that is, the disc is locally unstable.However, no significant modulation of the disc luminosityis found especially at the later phases. This is attributedto the fact that, in model ML-2, the absolute value of thelocal accretion rate in the inner region and its variationamplitude are small compared with the input accretionrate. Therefore, the modulations do not influence the lu-minosities of L and L d , because the resultant gravitationalenergy release due to the accreting gas could not contributelargely to the total radiation. r/Rg z / R g ρ Figure 15.
Same as Fig.13 but for density contours ρ (g cm − )with magnified velocity vectors. SS 433 is a typical stellar-mass black-hole candidate with ahighly super-Eddington luminosity. The present results canbe compared with the observations of SS 433, because themodel input accretion rate ˙ M ( ∼ . × − M ⊙ yr − ) isin the observed range of SS 433 mass transfer rate (Fabrika2004).The unique jet velocity V jet (=0.26 c ) of SS 433 is ex-plained in terms of the relativistic velocity of plasma accel-erated by the radiation-pressure force in the inner disc. TheX-ray spectral lines from the gas moving with relativisticvelocities are emitted in the very hot, optically thin regionalong the rotational axis. The temperature T ∼ – 10 Kand the density ρ ∼ − – 7 × − g cm − obtained in thepresent simulations at distances 6 × cm r . × cm in the conical axial outflow are generally consistent withthe values T ∼ . × – 6 × K and ρ ∼ × − –7 × − g cm − , obtained by fitting the model of X-rayjet emission lines produced at distances up to ∼ × cmfrom the jet base (Marshall, Canizares & Schulz 2002).In models AD-1 and AD-2, we obtain the total mass-outflow rates ˙ M out (3 . × − and 1 . × − M ⊙ yr − )and the axial mass-outflow rates ˙ M ◦ (9 . × − and1 . × − M ⊙ yr − ) comparable with the observed wind( ∼ − – 10 − M ⊙ yr − ) and jet ( ∼ − – 10 − M ⊙ yr − ) mass-ouflow rates, respectively. On the other hand,the mass-outflow rate ˙ M out in ML-1 and the axial outflowrate ˙ M ◦ in ML-2 are by more than an order of magnitudelower compared with the corresponding observed values ofthe wind and jets, respectively.The observed half opening angle of SS 433 jets is foundto be very small, about ∼ ◦ , in the distant region of ∼ – 10 cm for the X-ray jets (Marshall, Canizares & Schulz2002) and 10 – 10 cm for optical jets (Fabrika 2004). Ourcomputational domain ( r ∼ . × cm ) is located roughlyat the base of the observed X-ray jet, and the obtained halfopening angle θ c for the high-velocity outflow with radialvelocities of V r > . c is rather large, θ c > ◦ . The rela-tively large half opening angles of high-velocity flows, ∼ ◦ c (cid:13) , 1–11 T. Okuda, G. V. Lipunova, and D. Molteni – 60 ◦ , have been also found in other numerical studies of thesupercritical accretion discs (Eggum, Coroniti & Katz 1985,1988; Okuda 2002; Okuda et al. 2005; Ohsuga et al. 2005;Ohsuga 2007), and in SS 433 such large opening angles ofthe high-velocity flow were at odds with observations. Thequestion in the previous studies was how to collimate thehigh-velocity flow to a smaller angle. In the present study,this is not a problem. Indeed, the mass-outflow rates ob-tained for the relativistic axial outflow in the half openingangle of ∼ ◦ are sufficiently high to explain the observedmass-outflow rate of SS 433 jets, as far as the results of AD-1and AD-2 are considered. However, we have another ques-tion, why the relativistic axial outflow in our simulations israther undistingishable from the broader high-velocity out-flow, while X-ray and optical jets of SS 433 are so brilliantlyobserved as the hot relativistic streams with a small open-ing angle in a region far beyond the present computationaldomain.We interpret the small opening angles of the jets interms of the proposal (Marshall, Canizares & Schulz 2002;Fabrika 2004) that the observed expansion velocity in thetransverse direction of the jets coincides with the soundvelocity c s in the region with a temperature ∼ K(Marshall, Canizares & Schulz 2002) and that the half open-ing angle θ c observed in the distant region at r ≫ cm should be equal to ∼ c s /V jet , where V jet = 0 . c .From our hydrodynamical results, in the axial flow we have c s ∼ . × − c and T ∼ K, which agrees very wellwith the idea above. We suggest that, if future observationsresolve the jets deeper to the central source, the jets (high-velocity outflows) will be viewed with large opening angles, ≫ ◦ .Studying H and He absorption lines during the precess-ing motion of SS 433 disc, Fabrika (2004) derives the windvelocity V w of SS 433 as a function of the angle ζ from thedisc plane: V w ≈ (8000 ± · sin ζ + 150 km s − for 0 ζ ◦ . This gives velocities V w = 400 – 2000 km s − (0.001– 0.007 c ) for 10 ◦ ζ ◦ , which are smaller by a factorof 2 – 6 than the radial velocities V r ∼ c at theouter boundary in our simulations. The high wind velocity ∼ − (=0.005 c ) was suggested from analysis of HeIIemission near angles ζ ∼
70 – 80 ◦ (Fabrika 2004). However,we obtain higher outflow velocities in these directions, 0.08 –0.1 c . Therefore, we have to presume some mechanism of de-celerating the high-velocity flow in the distant region, suchas interactions with the walls of gas cocoon surrounding theaxial funnels where the jets expand.Although the absolute luminosity of SS 433 is not di-rectly observed, the kinematic luminosity L k is generallyestimated to be > erg s − (Kotani 1998), which is con-sistent with ∼ × erg s − obtained in all cases.The absolute luminosity of SS 433 is interesting becauseit is the maximum luminosity among the accreting stellar-mass black holes. The compact objects in the recently ob-served ULXs are considered as stellar-mass black hole orintermediate mass black hole, depending on whether theirluminosity far exceeds the Eddington luminosity L E or liesat sub-Eddington values (Mushotzky 2004). The total lumi-nosities L obtained here are higher by a factor of 3 thanthat given by the estimate L/L E = 0 . . m ob-tained numerically in 1D model (Lipunova 1999) and arehigher by more than an order than the Eddington luminos- ity 1 . × erg s − . The collimated outflows in our sim-ulations also lead to the outgoing radiation directed alongthe rotational axis. Actually, from the anisotropic radiationfield obtained at the outer boundary for each model, we havethe radial component F r ∼ × – 5 × erg s − cm − of the radiative flux F in the cone with a half opening an-gle 30 ◦ . If an observer within this cone assume the radiativeflux to be isotropic over the entire surface, the apparent lu-minosity is 7 × – 10 erg s − . Thus, SS 433 can bea representative of the supercritically accreting stellar-massblack hole candidate, observed as the ULXs in nearby galax-ies (King et al. 2001; Fabrika, Abolmasov & Karpov 2006;Begelman, King & Pringle 2006).We should pay attention to the remarkable modulationsof ˙ M edge and L d in model AD-2. Their power spectra showQPOs-like signals at ν ∼ × − – 10 − and 0.5 – 2 Hz,but only the larger amplitude modulations survive as smallmodulations of the total luminosity with a quasi-period of ∼
10 – 25 s, as mentioned in subsection 4.2. These instabilitiesdevelop as the recurrent hot blobs with variable size risingthrough the accretion disc and the polar funnel region. Suchinhomogeneties could cause the luminosity modulations. Al-though occasional massive jet ejections, which are recog-nized as a clustering of flare events in radio light curves,were exhibited by Microquasars, such as SS 433 (Fiedler1987) and GRS 1915+105 (Foster 1996), X-ray observationsof these events in SS 433 have been hardly performed so far,because the massive jet ejections are rare, short, and ape-riodic. Nevertheless, Kotani et al. (2006) reported recentlya variety of new phenomena, including a QPO-like featurenear 0.1 Hz, rapid time variability, @and shot-like activities,and suggested that an irregular massive jet ejection mightbe caused by the formation of small plasma bullets or knotsin the continuously emanating flow. We propose that the ob-served QPO-like phenomena in SS 433 can be explained interms of the recurrent hot blob phenomena found in modelAD-2.
We examined the jets and the disc of SS 433 with ˙ M ∼
600 ˙ M c by time-dependent two-dimensional radiation hydro-dynamical calculations, assuming α -model for the viscosity.The initial discs are given by 1D supercritical disc modelswith mass loss or advection. As the result, the total lumi-nosities obtained are 1 – 2.5 × erg s − , which are 6 to 10times higher than the Eddington luminosity for M = 10M ⊙ .From the initial models with advection, we obtain the totalmass-outflow rates ˙ M out ∼ × − and 10 − M ⊙ yr − , andthe relativistic axial outflow rates ˙ M ◦ ∼ − M ⊙ yr − .These outflow rates agree well with the observed mass-outflow rates of the wind and the jets in SS 433. On theother hand, from the initial models with mass loss but with-out advection, we obtain the total mass-outflow rates andthe axial outflow rates smaller than or comparable to the ob-served rates of the wind and the jets respectively, dependingon α . Still, while the mass-flow rate ˙ M ◦ of the axial outflowagrees well with the observed mass-outflow rate of SS 433jets covering the same half opening angle 1 ◦ , a problem isto be solved, why in the simulations the axial outflow is notdistinguishable from the other high-velocity flow. c (cid:13) , 1–11 he Jets and Disc of SS 433 The calculated radial velocities of the high-velocity out-flow at the outer boundary around the disc are larger by afactor of 2 – 6 than the observed wind velocities (Fabrika2004), which are given as a function of the elevation anglefrom the disc plane of SS 433. Thus, for the radial veloci-ties to be consistent with the observed wind velocities, theoutflow gas except the relativistic axial outflow must be de-celerated in the distant region beyond the present computa-tional domain by some mechanism, such as interaction withthe inhomogeneous matter or the gas cocoon around the SS433 disc.The initial advective disc with large α evolves to thegas-pressure dominant, optically thin state in the inner re-gion. The inner disc generates instabilities in agreement withthe previous stability analyses of the advection-dominatedoptically thin discs. As a result, we find remarkable mod-ulations of the mass-inflow rate at the inner edge and thedisc luminosity. The modulations of ˙ M edge present two typesof variability; (1) the small amplitude variations with shorttime-scales of 0.5 – 2 s, (2) the large amplitude variationswith long time-scales of ∼
10 – 25 s. Only the variability (2)survives in the total luminosity because of the atmosphericabsorption around the accretion disc. The disc instabilityresults in the recurrent hot blobs, which develop outwardand upward and produce QPOs-like phenomena in the totalluminosity with the quasi-periods of ∼
10 – 25 s. The QPOs-like behavior of the luminosity and the hot blobs phenomenafound here may explain the recent observations of a varietyof new phenomena in SS 433, such as a QPO-like featurenear 0.1 Hz, rapid time variability, and a shot-like activityascribed to the formation of small plasma bullets.
ACKNOWLEDGMENTS
G. V. Lipunova has been supported by the Russian Foun-dation for Basic Research (project 09-02-00032). G. V.Lipunova is grateful to the Offene Ganztagesschule of thePaul-Klee-Grundschule (Bonn, Germany) and Stadt Bonnfor providing a possibility for her full-day scientific activity.
REFERENCES
Abramowicz M. A., Czerny, B., Lasota, J. P., Szuszkiewicz,E., 1988, ApJ, 332, 646Begelman M. C, King A. R., Pringle J. E., 2006, MNRAS,370, 399Calvani M., Nobili L., 1983, in Astrophysical jets, ed. A.Ferrari & A. G. Pacholczyk (Dordrecht: Reidel), 189Cherepashchuk A. M., Sunyaev R. A., Seifina E. V.,Panchenko I. E., Molkov S. V., Postnov K. A., 2003, A&A,411, L441Dopita M. A., Cherepashchuk A. M., 1981, Vistas Astron.,25, 51Eggum G. E., Coroniti F. V., Katz J. I., 1985, ApJ, 298,L41Eggum G. E., Coroniti F. V., Katz J. I., 1988, ApJ, 330,142Fabrika S., 2004, Astrophysics and Space Physics Reviews,12, 1 Fabrika S., Abolmasov P., Karpov S., 2006, in Proceed-ings IAU Symposium No. 238, Black Holes : from Starsto Galaxies - across the Range of Masses.Fiedler R. L., Johnston K. J., Spencer J. H., Waltman E.B., Florkowski D. R., Matsakis D. N., Josties F. J., Anger-hofer P. E. Klepczynski W. J., McCarthy D. D., 1987, AJ,94, 1244Foster R. S., Waltman E. B., Tavani M., Harmon B. A.,Zhang S. N., Paciesas W. S., Ghigo F. D., 1996, ApJ,467, L81Fukue J., 1982, PASJ, 34, 163Kato S., Abramowicz M. A., Chen X., 1996, PASJ, 48, 67King A. R., Davies M. B., Ward M. J., Fabbiano G., ElvisM., 2001, ApJ, 552, L109Kley W., 1989, A&A, 208, 98Kotani T., 1989, PhD. The Institute of Space and Astro-nautical Sciences. JapanKotani T., Trushkin S. A., Valiullin R., Kinugasa K., Safi-Harb S., Kawai N., Namiki M., 2006, ApJ, 637, 486Levermore C. D., Pomraning G. C., 1981, ApJ, 248, 321Lynden-Bell D., 1978, Phys.Scr., 17, 185Lipunova G. V., 1999, Astron. Lett., 25, 508Manmoto T., Takeuchi M., Mineshige S., Matsumoto R.,Negoro H.,, 1996, ApJ., 464, L135Margon B., 1984, ARA&A, 22, 507Marshall H. L., Canizares C. R., Schulz N. S., 2002, ApJ,564, 941Mushotzky R., 2004, Prog. Theor. Phys. Suppl., No.155,27Ohsuga K., 2007, PASJ, 59, 1033Ohsuga K., Mineshige S., 2007, ApJ, 670, 1283Ohsuga K., Mori M., Nakamoto T., Mineshige S., 2005,ApJ, 628,268Okuda T., 2002, PASJ, 54, 253Okuda T., Fujita M., Sakashita S., 1997, PASJ, 49, 679Okuda T., Teresi V., Toscano E., Molteni D., 2005, MN-RAS, 357, 295Paczy´nski B., Bisnovatyi-Kogan G., 1981, Acta & Astro-nomica, 31, 283.Paczy´nski B., Wiita P. J., 1980, A&A, 88, 23.Poutanen J., Lipunova G., Fabrika S., Butkevich A. G.,Abolmasov P., 2007, MNRAS 377, 1187Pringle J. E., Rees M. J., 1972, A&A, 21, 1Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337van den Hueuvel E. P. J., 1981, Vistas Astron., 25, 95 c (cid:13)000