Magnetized 1.5-dimensional advective accretion flows around black holes
aa r X i v : . [ a s t r o - ph . H E ] A ug August 28, 2019 0:50 WSPC Proceedings - 9.75in x 6.5in main page 1 Magnetized 1 . Tushar Mondal ∗ and Banibrata Mukhopadhyay † Department of Physics, Indian Institute of Science, Bengaluru 560012, India ∗ E-mail: [email protected] † E-mail: [email protected]
We address the role of large scale magnetic stress in the angular momentum transport,as well as the formation of different kinds of magnetic barrier in geometrically thick,optically thin, vertically averaged 1.5-dimensional advective accretion flows around blackholes. The externally generated magnetic fields are captured by the accretion processfrom the environment, say, companion stars or interstellar medium. This field becomesdynamically dominant near the event horizon of a black hole due to continuous advectionof the magnetic flux. In such magnetically dominated accretion flows, the accretingmatter either decelerates or faces magnetic barrier in vicinity of the black hole dependingon the magnetic field geometry. We find that the accumulated strong poloidal fields alongwith certain toroidal field geometry help in the formation of magnetic barrier which mayknock the matter to infinity. When matter is trying to go back to infinity after gettingknocked out by the barrier, in some cases it is prevented being escaped due to cumulativeaction of strong magnetic tension and gravity, and hence another magnetic barrier. Wesuggest, this kind of flow may be responsible for the formation of episodic jets in whichmagnetic field can lock the matter in between these two barriers. We also find thatfor the toroidally dominated disc, the accreting matter rotates very fast and deceleratestowards the central black hole.
Keywords : accretion, accretion discs – black hole physics – MHD (magnetohydro-dynamics) – gravitation – X-rays: binaries – galaxies: jets
1. Introduction
In 1972, Bekenstein proposed an idealized engine, namely Geroch-Bekenstein en-gine , that makes use of the extreme gravitational potential of a black hole (BH) toconvert mass to energy with almost perfect efficiency. However, the practical realiza-tion of this engine is very difficult in astrophysical systems. Generally astrophysicalBHs do convert mass to energy via accretion process with modest efficiencies. Laterthe idea came to address the importance of large scale dipole magnetic field in anaccretion flow . Following this idea, it was suggested and also verified numeri-cally that this efficient conversion is practically possible in the presence of largescale poloidal magnetic field. Such an efficient accretion phenomenon is named asMagnetically Arrested Disc (MAD) .The origin of strong magnetic field near the event horizon of a BH is as fol-lows. For the case of advective accretion flow, the magnetic fields are capturedfrom accreting medium or companion star. These fields are dragged inward withcontinuous accretion process and become dynamically dominant through flux freez-ing in vicinity of a BH. Theoretical models suggest the importance of large-scalemagnetic field in accretion and in the formation of strong outflows/jets, as well ashigh-energy radiation . Observations of all well-known jetted sources also indi- ugust 28, 2019 0:50 WSPC Proceedings - 9.75in x 6.5in main page 2 cate the presence of dynamically important magnetic field at the jet-footprint . Inthis proceeding, we explore the underlying role of such large-scale strong magneticfields on the disc flow behaviours for optically thin, geometrically thick, advectiveaccretion flows around BHs. Unlike MAD, here the advection of both poloidal andtoroidal magnetic fields are considered. We address the possible origin of differentkinds of magnetic barriers depending on field geometry.
2. Model equations
We address magnetized, optically thin, advective, axisymmetric, vertically aver-aged, steady-state accretion flow around BHs in the pseudo-Newtonian frameworkwith Mukhopadhyay potential . The flow parameters, namely, radial velocity ( v ),specific angular momentum ( λ ), fluid pressure ( p ), mass density ( ρ ), the radial ( B r )and toroidal ( B φ ) components of magnetic fields, are solved simultaneously as func-tions of radial coordinate r . Throughout in our computation, we express the radialcoordinate in units of r g = GM BH /c , where G is Newton’s gravitational constant, M BH is the mass of the black hole, and c is speed of light. We also express λ in unitsof GM BH /c and velocity in units of c , and the other variables accordingly to makeall the variables dimensionless. Hence, the continuity equation, the components formomentum balance equation, and the energy equation can read as, respectively, ddr ( rρhv ) = 0 , (1) v dvdr − λ r + ρh d ( hp ) dr + F = − √ hB φ πρhr ddr (cid:16) r √ hB φ (cid:17) , (2) v dλdr = rρh ddr (cid:0) r W rφ h (cid:1) + √ hB r πρh ddr (cid:16) r √ hB φ (cid:17) , (3) hv Γ − (cid:16) dpdr − Γ pρ dρdr (cid:17) = Q + − Q − = f m Q + = f m ( Q + vis + Q + mag ) , (4)where F is the magnitude of the gravitational force corresponding to the pseudo-Newtonian potential. W rφ is the viscous shearing stress, which can be expressedusing Shakura-Sunayev α -viscosity prescription with appropriate modifications due to advection as given by W rφ = α ( p + ρv ). Following the vertical momentumbalance equation, the disc half-thickness can be written as h = r / F − / s(cid:18) p + B π (cid:19) (cid:14) ρ. (5)The energy equation is written by taking care of the proper balance of heating,cooling and advection. Q + is the energy generated per unit area due to magnetic( Q + mag ) and viscous dissipation ( Q + vis ), whereas Q − infers the energy radiated outper unit area through different cooling mechanisms . However for the present pur-pose, we do not incorporate cooling processes explicitly. The factor f m measuresthe degree of cooling and it varies from 0 to 1 for two extreme cases of efficientcooling and no cooling respectively. The details of dissipation terms are followed ugust 28, 2019 0:50 WSPC Proceedings - 9.75in x 6.5in main page 3 from the Ref. 12. Note that in the energy equation, we do not include the heatgenerated and absorbed due to nuclear reactions .The presence of magnetic field provides two other fundamental equations,namely, the equation for no magnetic monopole and the induction equation. Theseare respectively ∇ . B = 0 , and ∇ × ( v × B ) + ν m ∇ B = 0 , (6)where v and B are the velocity and magnetic field vectors respectively and ν m isthe magnetic diffusivity. For this accretion disc solution, we consider the inductionequation in the limit of very large magnetic Reynolds number ( ∝ /ν m ).
3. Solution procedure
The set of six coupled differential equations (1) − (6) for six flow variables v , λ , p , ρ , B r , and B φ are solved simultaneously to obtain the solutions as functions of theindependent variable r . The appropriate boundary conditions are as follows. Veryfar away from the BH, matter is sub-sonic and the transition radius between theKeplerian to sub-Keplerian flow is the outer boundary of our solutions. Very nearthe BH, matter is super-sonic and the event horizon where matter velocity reachesspeed of light is the inner boundary. In between these two boundaries, matterbecomes tran-sonic where matter velocity is equal to (or similar to) medium soundspeed and this location is called as sonic/critical point. We use this point as one ofthe boundary. Different types of sonic/critical points, say, saddle, nodal, and spiral,are described in the Ref. 12.
4. Results
The angular momentum transport in accretion physics had been a long standingissue until Balbuas & Hawley described the importance of magnetorotational in-stability (MRI), particularly in an ionized medium in the presence of weak magneticfields. Later Mukhopadhyay & Chatterjee showed that the efficient transport ofangular momentum is also possible by large-scale magnetic field in geometricallythick, advective accretion flow, even in the complete absence of α -viscosity. Here,we address the importance of large-scale strong magnetic field in accretion geome-try. When the magnetic field is strong enough, presumably that corresponds to theupper limit to the amount of magnetic flux which a disc around a BH can sustain,it disrupts the accretion flow. Then the force associated with the magnetic stressbecomes comparable to the strong gravitational force of the BH. As a consequence,the magnetic field arrests the infalling matter in vicinity of the BH. Here we plan tounderstand such magnetic activity depending on different kinds of field geometry.In Figure 1 we show three different natures of accretion depending on differentmagnetic field geometries. The left column indicates the Mach number ( M ), definedas the ratio of the radial velocity to the medium sound speed, whereas the right ugust 28, 2019 0:50 WSPC Proceedings - 9.75in x 6.5in main page 4 M ) and the corresponding magnetic field components ( B i )for different relative field strengths in the disc. The model parameters are M BH = 10 M ⊙ , ˙ M =0 .
01 ˙ M Edd and f m = 0 . column represents the corresponding magnetic field components ( B i ). In Figure 1( a ), the Mach number increases monotonically towards the central BH until it facesmagnetic barrier at r ≃
12. After facing such a barrier, matter goes back to infinity.To explain the origin of such barrier, we focus on the underlying role of magneticforce along with strong gravity, centrifugal force, and force due to gas pressure.When matter falls towards central BH, the forces supporting gravity may suppressby the forces acting outward. In such circumstances, following radial momentum ugust 28, 2019 0:50 WSPC Proceedings - 9.75in x 6.5in main page 5 balance equation, the condition for which barrier appears as − πρh √ hB φ d ( √ hB φ ) dr ! − ρh d ( hp ) dr + λ r > F + 14 πρh ( √ hB φ ) r , (7)where the first term appears from magnetic pressure and the last term from mag-netic tension. Simply it suggests the essential condition to appear magnetic barrieris ddr ( √ hB φ ) ≪ − √ hB φ r . On the other hand, the equation for no magnetic monopoleleads to ddr ( √ hB r ) = − √ hB r r . These combine conditions suggest that the matterfaces magnetic barrier only when the field strength is strong enough to satisfy equa-tion (7) and the disc is poloidal magnetic field dominated. Figure 1( b ) infers suchfield conditions.In Figure 1( c ), the Mach number initially increases monotonically and finallyslows down to the central BH. In this case, the disc is toroidal magnetic field dom-inated as shown in Figure 1( d ). In this situation matter is rotating very fast com-pared to the inward dragging. In Figure 1( e ), the Mach number increases up to r ≃ .
5, where the magnetic barrier appears due to dominant poloidal magneticfield. After knocking out by such a barrier, matter tries to go away from the BH,but faces another barrier at r ≃ .
6. This is because of the cumulative actionof the inward strong gravity force and the inward force due to magnetic tensionsalong the field lines. In between these two barriers matter faces spiral-type criticalpoint and again falls back to the central BH. The corresponding field componentsare shown in Figure 1( f ). Note that the magnetic field strength within plungingregion is few factor times 10 G for 10 M ⊙ BH for all the cases mentioned above.
5. Discussions
Accretion disc can carry small as well as large-scale strong magnetic fields. Small-scale field may generate, locally, through some physical scenario, Biermann baterymechanism, dynamo process etc. However, the origin of large-scale strong magneticfields is still not well understood. It was suggested that the externally generatedmagnetic field either from companion star or from interstellar medium may captureand drag inward through continuous accretion process. This field becomes dynami-cally dominant in vicinity of a BH due to flux freezing. In such circumstances, strongmagnetic fields arrest the infalling matter and disrupt the conventional accretionflow properties.In this proceeding, we address three different types of accretion flow propertiesdepending on field geometry in the presence of large scale strong magnetic fields.First, when the disc is poloidal field dominated, the infalling matter faces magneticbarrier and tries to go back to infinity. On its way away from the BH, matter maycompletely lose its angular momentum and at the same time the combined effectsof inward strong gravity and the magnetic tension along field lines can preventthe matter being escaped. In this particular case, matter again falls back to theBH by facing second magnetic barrier. The important application of such a flow ugust 28, 2019 0:50 WSPC Proceedings - 9.75in x 6.5in main page 6 behaviour is the formation of episodic jets in which magnetic field can lock theinfalling matter in between these two magnetic barriers. Second, for such a poloidalfield dominated case, matter can completely go back to infinity after knocked outby the first magnetic barrier. Magnetic tension is not strong enough to preventthe matter being escaped in this case. This type of flows is the building block toproduce unbound matter and hence strong continuous outflows/jets. Third, whenthe disc is toroidal field dominated, the infalling matter rotates very fast ratherthan being dragged inward in vicinity of the BH. In such case, matter falls slowlyinto the BH. For all these scenarios the maximum magnetic field strength nearthe stellar mass BH is few factor times 10 G, which is well below the Eddingtonmagnetic field limit and hence perfectly viable. The other important aspect of thisquasi-spherical advective flow is the angular momentum transport. The outwardtransport of angular momentum occurs through large-scale magnetic stress. Thespecific angular momentum is quite below the local Keplerian value.In the other proceedings of this volume, we discuss more general disc-outflowsymbiotic model and its observational implication to ultra-luminous X-raysources . Acknowledgments
TM acknowledges P. V. Lakshminarayana travel grant ODAA/INT/18/11, Officeof Development and Alumni Affairs, Indian Institute of Science, Bengaluru, India.
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