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The nature of Jet in AGN: the BlandfordZnajek effect 23 PhD cycle: General Relativity rd Elisa Prandini 29 January 2009 Contents ● Active Galactic Nuclei ● Kerr black hole ● The BlandfordZnajek mechanism ● the mathematical framework ● physical hypotesis and consequences ● – on the accretion disk – on the hole surface Power output 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 2 Active Galactic Nuclei Supermassive black hole (1061010 Mʘ) Accretion disk (keplerian motion) Relativistic jets perpendicular to the disk plane (in Radio Loud Objects) 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 3 Active Galactic Nuclei Supermassive black hole (1061010 Mʘ) Accretion disk (keplerian motion) Relativistic jets perpendicular to the disk plane (in Radio Loud Objects) Unified model: the observed features depend on the line of sight of the observer Radio Galaxies/ Seyfert 2 RL Quasars/ Seyfert 1 Blazars/ RQQ The overall emission from an AGN covers the entire electromagnetic spectrum 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 4 Active Galactic Nuclei Supermassive black hole (1061010 Mʘ) Accretion disk (keplerian motion) Space around the central object: GENERAL RELATIVITY Relativistic jets perpendicular to the disk plane (in Radio Loud Objects) Unified model: the observed features depend on the line of sight of the observer Radio Galaxies/ Seyfert 2 RL Quasars/ Seyfert 1 Blazars/ RQQ The overall emission from an AGN covers the entire electromagnetic spectrum 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 5 Active Galactic Nuclei Supermassive black hole (1061010 Mʘ) Rotating object: Kerr metric Accretion disk (keplerian motion) Relativistic jets perpendicular to the disk plane (in Radio Loud Objects) Unified model: the observed features depend on the line of sight of the observer Radio Galaxies/ Seyfert 2 RL Quasars/ Seyfert 1 Blazars/ RQQ The overall emission from an AGN covers the entire electromagnetic spectrum 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 6 Kerr black holes Kerr metric in BoyerLindquist coordinates: ds2 =− 1− 2mr 4Mar A dt 2 − sen2 dt d dr 2 d 2 sen2 d 2 =r 2 a2 cos2 =r 2−2Mra2 A= r a 2 29 Gennaio 2009 2 2 2 a= 2 J M specific angular moment −a sen E.Prandini The BlandfordZnajek Effect 7 Kerr black holes Kerr metric in BoyerLindquist coordinates: ds2 =− 1− 2mr 4Mar A dt 2 − sen2 dt d dr 2 d 2 sen2 d 2 =r 2 a2 cos2 =r 2−2Mra2 A= r a 2 2 2 2 a= 2 J M specific angular moment −a sen singularities: =0 coordinate singularity r ±. = M ± M 2−a2 =0 curvature singularity r =0 ∧ =/2 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 8 Kerr black holes Kerr metric in BoyerLindquist coordinates: ds2 =− 1− 2mr 4Mar A dt 2 − sen2 dt d dr 2 d 2 sen2 d 2 =r 2 a2 cos2 =r 2−2Mra2 A= r a 2 2 2 2 a= 2 J M specific angular moment −a sen singularities: =0 coordinate singularity r ±. = M ± M 2−a2 =0 curvature singularity r =0 ∧ =/2 Key properties: ● ● existence of the ergoregion gravitational dragging 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 9 The ergoregion g 00=0 ergosurface 1− 2Mr =0 r 0.= M M 2−a2 cos2 the ergosurface is also called “stability limit” : inside it everything rotate w.r.t. an observer located at infinity (the metric itself rotate) 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 10 The ergoregion g 00=0 ergosurface 1− 2Mr =0 r 0.= M M 2−a2 cos2 the ergosurface is also called “stability limit” : inside it everything rotate w.r.t. an observer located at infinity (the metric itself rotate) ergoregion: region between the ergosurface and the coordinate singularity (r+). Inside it: everything rotate w.r.t. infinity ● a particle may have negative energy w.r.t. infinity > Penrose process ● it is possible, in principle, to exctract rotational energy from a Kerr BH! ● 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 11 Gravitational dragging In the gravitational field of rotating sources the gravitational dragging effect takes place. This effect induce a rotation of the metric surrounding the black −g hole, with a velocity: 2Mar = t = g A 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 12 Gravitational dragging In the gravitational field of rotating sources the gravitational dragging effect takes place. This effect induce a rotation of the metric surrounding the black −g hole, with a velocity: 2Mar = t = g A The following ZAMO tetrad (in which the dragging effect is subctracted), is useful for physical measurement: = 29 Gennaio 2009 1 2 A 0 = ∂ 0 ∂ 2 sen A 1 2 ∂ 1 2 ∂ r = = E.Prandini The BlandfordZnajek Effect − 1 2 r ∂ 13 Gravitational dragging In the gravitational field of rotating sources the gravitational dragging effect takes place. This effect induce a rotation of the metric surrounding the black −g hole, with a velocity: 2Mar = t = g A The following ZAMO tetrad (in which the dragging effect is subctracted), is useful for physical measurement: Ω = r+ 29 Gennaio 2009 r0+ 1 2 A 0 = ∂ 0 ∂ 2 sen A r 1 2 ∂ 1 2 ∂ r = = − 1 2 r ∂ Allowed angular velocities of a test particle in a Kerr black hole (as measured from infinity): inside the ergoregion only corotating orbits are allowed E.Prandini The BlandfordZnajek Effect 14 Note The tetrad: 1 2 A 0 = = ∂ 0 ∂ A sen2 1 2 1 2 ∂ r = ∂ = 1 − 2 r with ∂ can be obtained after the “diagonalization” of the metric: ds 2 =− 1 2 1 2 ∂ r = ∂ 0' '= A sen2 29 Gennaio 2009 d '= d − dt dt '= dt 2 2 A 2 2 dt ' dr d sen d ' A A 0' = g t 2Mar = = g A − 1 2 ∂ ' = − 1 2 r ∂ E.Prandini The BlandfordZnajek Effect 15 The origin of jet energy Large amount of energy emitted by jet energetic particles in AGN the key question is: where does this energy comes from?? 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 16 The origin of jet energy Large amount of energy emitted by jet energetic particles in AGN the key question is: where does this energy comes from?? ● Accepted: rotational energy of the black hole and surrounding accretion disk 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 17 The origin of jet energy Large amount of energy emitted by jet energetic particles in AGN the key question is: where does this energy comes from?? ● ● Accepted: rotational energy of the black hole and surrounding accretion disk How? Several models have been proposed: ● BlandfordZnajek mechanism (1977): extraction of an energy and angular ● ● momentum outflow though the black hole magnetosphere, then converted into Poynting flux and finally transferred to charged particles. The BlandfordPayne model (1982) magnetocentrifugal acceleration Punsly & Coroniti model (1990): similar to BZ model but with different boundary conditions 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 18 The origin of jet energy Large amount of energy emitted by jet energetic particles in AGN the key question is: where does this energy comes from?? ● ● Accepted: rotational energy of the black hole and surrounding accretion disk How? Several models have been proposed: ● BlandfordZnajek mechanism (1977): extraction of an energy and angular ● ● momentum outflow though the black hole magnetosphere, then converted into Poynting flux and finally transferred to charged particles. The BlandfordPayne model (1982) magnetocentrifugal acceleration Punsly & Coroniti model (1990): similar to BZ model but with different boundary conditions ● Jet emission hypotesys: ● Kerr black hole surrounded by an accretion disk ● A magnetosphere filled with plasma extends far away from the hole Controversial: boundary conditions (on the hole surface...) and properties of the magnetosphere 29 Gennaio 2009 19 ● E.Prandini The BlandfordZnajek Effect BlandfordZnajek mechanism Is the most popular mechanism for jet formation ● The mechanism itself is rather complicated, and has controversial interpretations. ● 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 20 BlandfordZnajek mechanism Is the most popular mechanism for jet formation ● The mechanism itself is rather complicated, and has controversial interpretations ● Original paper: R.D.Blandford and R.L.Znajek, “Electromagnetic extraction of ● energy from Kerr black hole”, Mon.Not.R.astr.soc (1977), 179, 433456 ● that is a generalization in a Kerr metric of the Newtonian: R.D.Blanford, “Accretion disc electrodynamics”, Mon. Not. R. astr. Soc. (1976) 176, 465481; 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 21 BlandfordZnajek mechanism Is the most popular mechanism for jet formation ● The mechanism itself is rather complicated, and has controversial interpretations ● Original paper: R.D.Blandford and R.L.Znajek, “Electromagnetic extraction of ● energy from Kerr black hole”, Mon.Not.R.astr.soc (1977), 179, 433456 ● ● that is a generalization in a Kerr metric of the Newtonian: R.D.Blanford, “Accretion disc electrodynamics”, Mon. Not. R. astr. Soc. (1976) 176, 465481; I will also follow: ● D. Macdonald and S.Thorne, “Black hole electrodynamics: an absolute space/universaltime formulation”, Mon. Not. R. astr. Soc. (1982) 198, 345382; ● ● S.S.Kommissarov, “BlandfordZnajek mechanism versus Penrose process”, arXiv: 0804.1912v1, 2008. Frank, King, Raine, “Accretion Power in Astrophysics”, Cambridge Univ. Press. 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 22 Macdonald & Thorne: 3+1 formulation Explain the BlandfordZnajek effect in a 3+1 absolutespace/universaltime formulation 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 23 Macdonald & Thorne: 3+1 formulation Explain the BlandfordZnajek effect in a 3+1 absolutespace/universaltime formulation MATHEMATICS spacetime geometry as a stationary axisymmetric black hole: 2 2 2 2 d − dt exp 2 dr exp 2 d ds2=−2 c 2 dt 2 1 2 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 24 Macdonald & Thorne: 3+1 formulation Explain the BlandfordZnajek effect in a 3+1 absolutespace/universaltime formulation MATHEMATICS spacetime geometry as a stationary axisymmetric black hole: 2 2 2 2 d − dt exp 2 dr exp 2 d ds2=−2 c 2 dt 2 1 2 “absolutespace” formalism: 3dimentional space: 2 2 2 2 d exp 2 dr exp 2 d ds2= jk dx j dx k= 1 2 axial simmetry : ,1, 2 are independent of 2 this means that m= ∇ is a Killing vector field : m j∣k mk∣ j =0, moreover : ≡∣∣ : 2 is the crf r , =const , m tangent we will call these circles m−loop , the cylindrical radius of an m−loop 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 25 Macdonald & Thorne: 3+1 formulation Explain the BlandfordZnajek effect in a 3+1 absolutespace/universaltime formulation MATHEMATICS spacetime geometry as a stationary axisymmetric black hole: 2 2 2 2 d − dt exp 2 dr exp 2 d ds2=−2 c 2 dt 2 1 2 “absolutespace” formalism: 3dimentional space: 2 2 2 2 d exp 2 dr exp 2 d ds2= jk dx j dx k= 1 2 axial simmetry : ,1, 2 are independent of 2 this means that m= ∇ is a Killing vector field : m j∣k mk∣ j =0, moreover : ≡∣∣ : 2 is the crf r , =const , m tangent we will call these circles m−loop , the cylindrical radius of an m−loop “absolutetime”: is the global time t, equal to the time coordinate t in 4dim. metric: Ȧ = 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect ∂A ∂t 26 ZAMO and absolute space observers In the absolutespace formalism, the observables are measured by ZAMOs Rotation of the hole drags the ZAMO into toroidal motion along mloops: d / dt of ZAMO rest frame= 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 27 ZAMO and absolute space observers In the absolutespace formalism, the observables are measured by ZAMOs Rotation of the hole drags the ZAMO into toroidal motion along mloops: d / dt of ZAMO rest frame= despite this motion, the ZAMO world lines are orthogonal to the 3dimentional hypersurfaces of constant time t: our absolute space is regarded by the ZAMO as a space of constant time in its local Lorentz frames. 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 28 ZAMO and absolute space observers In the absolutespace formalism, the observables are measured by ZAMOs Rotation of the hole drags the ZAMO into toroidal motion along mloops: d / dt of ZAMO rest frame= despite this motion, the ZAMO world lines are orthogonal to the 3dimentional hypersurfaces of constant time t: our absolute space is regarded by the ZAMO as a space of constant time in its local Lorentz frames. Gravitational redshift of ZAMO clock (dτ proper time, dt global time): d / dt of ZAMO clock = where α is the lapse function , , Axial symmetry: constant on mloops (dX/dφ=0) Horizon H of the black hole (α=0): ZAMO moves with ΩH 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 29 Spatial coordinate system In BoyerLindquist coordinates: 2 A sen ds = / dr d 2 2 2 2 2 =r 2a2 cos2 , =r 2−2 GMr /c 2a2 , A= r 2a2 −a2 sin2 1 /2 = A 29 Gennaio 2009 1 /2 2aGMr A , = , = cA sin E.Prandini The BlandfordZnajek Effect 30 Spatial coordinate system In BoyerLindquist coordinates: 2 A sen ds = / dr d 2 2 2 2 2 =r 2a2 cos2 , =r 2−2 GMr /c 2a2 , A= r 2a2 −a2 sin2 1 /2 = A lapse function 29 Gennaio 2009 1 /2 2aGMr A , = , = cA sin cylindrical radius of an mloop E.Prandini The BlandfordZnajek Effect 31 Electrodynamics in absolute space The fields, currents and charge densities are measured by the ZAMOs. MAXWELL EQUATIONS: ∇⋅E=4 e ∇⋅B=0 ∇ × B =4 j /c 1/c [ Ė Lm E− E⋅∇ m ] , ∇ × E =−1/c [ Ḃ Lm B− B⋅∇ m ] ● LmE is the Lie derivative of E along the toroidal Killing vector m 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 32 Electrodynamics in absolute space The fields, currents and charge densities are measured by the ZAMOs. MAXWELL EQUATIONS: ∇⋅E=4 e ∇⋅B=0 ∇ × B =4 j /c 1/c [ Ė Lm E− E⋅∇ m ] , ∇ × E =−1/c [ Ḃ Lm B− B⋅∇ m ] ● in the case electric and magnetic field stationary and axisymmetric LmE is the Lie derivative of E along the toroidal Killing vector m 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 33 Electrodynamics in absolute space The fields, currents and charge densities are measured by the ZAMOs. MAXWELL EQUATIONS: ∇⋅E=4 e ∇⋅B=0 ∇ × B =4 j /c 1/c [ Ė Lm E− E⋅∇ m ] , ∇ × E =−1/c [ Ḃ Lm B− B⋅∇ m ] ● LmE is the Lie derivative of E along the toroidal Killing vector m αE and αB are the electric and magnetic field measured by ZAMO, if he uses t rather than τ in computing d p / dt = d p/ d =q Ev× B ● ● Moreover: v= [ d proper distance d 29 Gennaio 2009 ] [ vm= ZAMO d proper distance dt E.Prandini The BlandfordZnajek Effect ] abs.space 34 Electrodynamics in absolute space − Charge conservation : where ∂ e m⋅∇ e ∇⋅ j =0 ∂t ∂ m⋅∇ is the global time derivative along ZAMO trajectories ∂t 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 35 Electrodynamics in absolute space − Charge conservation : where ∂ e m⋅∇ e ∇⋅ j =0 ∂t ∂ m⋅∇ is the global time derivative along ZAMO trajectories ∂t Electric and magnetic field can be derived from a scalar potential A0 and a vector potential A: 29 Gennaio 2009 E = 1 [ ∇ A0 / c ∇ A ]− c 1 Ȧ Lm A with A= A⋅m B= ∇ × A − − E.Prandini The BlandfordZnajek Effect 36 Electrodynamics in absolute space − Charge conservation : where ∂ e m⋅∇ e ∇⋅ j =0 ∂t ∂ m⋅∇ is the global time derivative along ZAMO trajectories ∂t Electric and magnetic field can be derived from a scalar potential A0 and a vector potential A: E = 1 [ ∇ A0 / c ∇ A ]− c 1 Ȧ Lm A with A= A⋅m B= ∇ × A − = ZAMO mass − energy density S = ZAMO flux of energy W = ZAMO stress tensor 29 Gennaio 2009 − densities and fluxes of redshiftedenergy and of angular momentum: e= S⋅m / c 2, S E = S W ⋅m L= S⋅m / c 2 , S L =W ⋅m E.Prandini The BlandfordZnajek Effect 37 Electrodynamics in absolute space 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 38 Electrodynamics in absolute space Energy −momentum conservation : T ; =0 becames in our 3+1 formalism: ● ● energy conservation: 1 [ ∂ / ∂ t m⋅∇ ] 2 ∇ ⋅ 2 S 1 m⋅W ⋅∇ =0 =− j⋅E matter not −included − − − momentum conservation: −1 [ ∂ / ∂ t Lm ] −1 S⋅m ∇ − gc 2 −1 ∇⋅ W =0 2 =−c [ e E j /c× B ] matter not−included 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 39 Electrodynamics in absolute space Energy −momentum conservation : T ; =0 becames in our 3+1 formalism: ● energy conservation: ● momentum conservation: [ ∂ / ∂ t m⋅∇ ] ∇ ⋅ S m⋅W ⋅∇ =0 =− j⋅E matter not −included − 1 − 2 2 − 1 −1 [ ∂ / ∂ t Lm ] −1 S⋅m ∇ − gc 2 −1 ∇⋅ W =0 ● ● 2 =−c [ e E j /c× B ] matter not−included from these laws it is possible to derive the conservation laws for redshifted energy and for angular momentum the momentum conservation law define the condition of forcefreeness 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 40 The physics case We can divide the space in 3 regions: Region D (the disk and the BH surface) Region FF: the magnetosphere outside the equatorial plane re e ph s o t ne g ma Region A: the magnetosphere far out from the BH region from D.Macdonald, S.Thorne Mon. Not. R. astr. Soc. (1982) 198, 345382 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 41 The physics case We can divide the space in 3 regions: Region D (the disk and the BH surface) Region FF: the magnetosphere outside the equatorial plane re e ph s to e gn a m Region A: the magnetosphere far out from the BH region from D.Macdonald, S.Thorne Mon. Not. R. astr. Soc. (1982) 198, 345382 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 42 Hypotesys Stationary axisymmetric system + magnetic field ● The disk and the magnetosphere are degenerate ● The magnetosphere is forcefree: ● Far out the magnetosphere is no longer forcefree nor degenerate ● 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 43 Hypotesys Stationary axisymmetric system + magnetic field ● The disk and the magnetosphere are degenerate ● The magnetosphere is forcefree: ● Far out the magnetosphere is no longer forcefree nor degenerate ● 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 44 Consequences I Stationary axisymmetric system + magnetic field ● ● This means that: ∂ F = Ḟ = 0 , ∂t ∂ f = f˙ =0 , ∂t ● Lm F =0 for all vector fields m⋅∇ f =0 for all scalar fields In this case B,E, ρe and j can be derived from three scalar potentials 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 45 Consequences I Stationary axisymmetric system + magnetic field ● ● This means that: ∂ F = Ḟ = 0 , ∂t ∂ f = f˙ =0 , ∂t ● Lm F =0 for all vector fields m⋅∇ f =0 for all scalar fields In this case B,E, ρe and j can be derived from three scalar potentials: ● ● A0: electric potential I: total current passing downwards through an mloop I x =−∫A j⋅d ● ψ: total magnetic flux passing upward through an mloop x =∫A B⋅d 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 46 Consequences I Stationary axisymmetric system + magnetic field ● ● z the poloidal magnetic field lines are parabolic (Blandford 1976) R 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 47 Consequences I Stationary axisymmetric system + magnetic field ● ● ● ● z the poloidal magnetic field lines are parabolic (Blandford 1976) The magnetic flux ψ is simply related to the component Aφ of the vector potential: ψ(x)= 2πAφ R the fields E and B can be expressed in their toroidal and poloidal part 2 E⋅m m E ≡ E T E P , E T =w 2 B⋅m m B ≡ B T B P , B T =w − − ● ψ is constant through an mloop (stationarity): ET=0 ● Axysimmetry of B (LmB=0): ∇ ⋅BT =0 = ∇ ⋅B P 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 48 Consequences I Stationary axisymmetric system + magnetic field ● ● Hence: 1 E= ∇ A0 ∇ 2 c 2I T B =− 2 m c m× ∇ P B =− 2 2 P P m× ∇ I ∇ I=−2 m× j j = 2 2 ∇ =2 m×B ● P the flow of electromagnetic angular momentum and redshifted energy in the magnetosphere is described by the poloidal parts of the flux vectors: I P B 2 c I E×m S PE = BP − 2 2 c P SL = 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 49 Hypotesys Stationary axisymmetric system + magnetic field ● The disk and the magnetosphere are degenerate ● The magnetosphere is forcefree: ● Far out the magnetosphere is no longer forcefree nor degenerate ● 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 50 Consequences II Degeneracy: (E · B = 0) ● ● the magnetic field lines are frozen into the disk plasma, and rotate at the plasma local angular velocity (ΩF) 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 51 Consequences II Degeneracy: (E · B = 0) ● ● ● the magnetic field lines are frozen into the disk plasma, and rotate at the plasma local angular velocity (ΩF) An observer locally at rest w.r.t the field lines, does not experience any F − F P electric field; the ZAMO experiences the field: E =− v / c × B =− ∇ 2c F (v is the field velocity w.r.t. ZAMO) 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 52 Consequences II Degeneracy: (E · B = 0) ● ● ● ● the magnetic field lines are frozen into the disk plasma, and rotate at the plasma local angular velocity (ΩF) An observer locally at rest w.r.t the field lines, does not experience any F − F P electric field; the ZAMO experiences the field: E =− v / c × B =− ∇ 2c F (v is the field velocity w.r.t. ZAMO) Each field line rotate with constant angular velocity ΩF, hence ΩF is constant along field lines: B P⋅∇ F =0 (FERRARO'S LAW OF ISOROTATION, 1937) 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 53 Consequences II Degeneracy: (E · B = 0) ● ● ● ● ● the magnetic field lines are frozen into the disk plasma, and rotate at the plasma local angular velocity (ΩF) An observer locally at rest w.r.t the field lines, does not experience any F − F P electric field; the ZAMO experiences the field: E =− v / c × B =− ∇ 2c F (v is the field velocity w.r.t. ZAMO) Each field line rotate with constant angular velocity ΩF, hence ΩF is constant along field lines: B P⋅∇ F =0 (FERRARO'S LAW OF ISOROTATION, 1937) the poloidal redshifted energy is in this case: P E F P L S = S = 29 Gennaio 2009 F I BP 2 c angular momentum and redshifted energy both flow along poloidal magnetic field lines and dE=Ω FdL E.Prandini The BlandfordZnajek Effect 54 Hypotesys Stationary axisymmetric system + magnetic field ● The disk and the magnetosphere are degenerate ● The magnetosphere is force free ● ● there are free charges: ● from the disk (GoldreichJulian mechanism) ● in the BH region: pairs production from vacuum Far out the magnetosphere is no longer forcefree nor degenerate ● 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 55 Consequences III Forcefree condition: (E · B = 0 and ρeE+(j/c)xB=0) ● ● current flows on poloidal magnetic field surfaces: I(ψ) ● ● hence: poloidal magnetic field sourfaces rotates with constant angular velocity, and are surfaces of constant I and ψ jP everywhere parallel to BP 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 56 Consequences III Forcefree condition: (E · B = 0 and ρeE+(j/c)xB=0) ● ● current flows on poloidal magnetic field surfaces: I(ψ) ● hence: poloidal magnetic field sourfaces rotates with constant angular velocity, and are surfaces of constant I and ψ ● jP everywhere parallel to BP ● fluxes of angular momentum and redshifted energy flow without loss along the poloidal magnetic field lines 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 57 Hypotesys Stationary axisymmetric system + magnetic field ● The disk and the magnetosphere are degenerate ● The magnetosphere is forcefree: ● Far out the magnetosphere is no longer forcefree nor degenerate ● 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 58 Consequences IV Far out the magnetosphere is no longer forcefree nor degenerate ● (related to the fact that particles cannot travel at speed higher then c) ● ● In the A region (magnetosphere not forcefree, nor degenerate) particles are forced to move at speed lower then the field lines: DRAG EFFECT that slows down the field line (in radial motion) This cause a difference between the angular velocity of the disk ΩD and that of poloidal field lines ΩF 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 59 Consequences IV Far out the magnetosphere is no longer forcefree nor degenerate ● (related to the fact that particles cannot travel at speed higher then c) ● ● In the A region (magnetosphere not forcefree, nor degenerate) particles are forced to move at speed lower then the field lines: DRAG EFFECT that slows down the field line (in radial motion) This cause a difference between the angular velocity of the disk Ω D and that of poloidal field lines Ω F > angular momentum extraction from the disk (BZ effect) 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 60 Blandford Znajek effect Velocity difference between the flux tube and the disk induces in the rest frame of the disk plasma > radial electric field (Ω ∝ DΩF)BP from D.Macdonald, S.Thorne Mon. Not. R. astr. Soc. (1982) 198, 345382 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 61 Blandford Znajek effect Velocity difference between the flux tube and the disk induces in the rest frame of the disk plasma > radial electric field (Ω ∝ DΩF)BP radial current: j ∝ D − F B P /resist. that interacts with BP producing a net torque on the flux tube d L D D − F − = 2 2 D dt 4 c Z from D.Macdonald, S.Thorne Mon. Not. R. astr. Soc. (1982) 198, 345382 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 62 Blandford Znajek effect Velocity difference between the flux tube and the disk induces in the rest frame of the disk plasma > radial electric field (Ω ∝ DΩF)BP radial current: j ∝ D − F B P /resist. that interacts with BP producing a net torque on the flux tube d L D D − F − = 2 2 D dt 4 c Z that will be transmitted, loss free, through the FF region: d LD I − dt = 2 c and into the A region, where it will act on charged particles. from D.Macdonald, S.Thorne Mon. Not. R. astr. Soc. (1982) 198, 345382 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 63 Blandford Znajek effect Velocity difference between the flux tube and the disk induces in the rest frame of the disk plasma > radial electric field (Ω ∝ DΩF)BP radial current: j ∝ D − F B P /resist. that interacts with BP producing a net torque on the flux tube d L D D − F − = 2 2 D dt 4 c Z that will be transmitted, loss free, through the FF region: d LD I − dt = 2 c and into the A region, where it will act on charged particles. Redshifted power transferred to charged particles: from D.Macdonald, S.Thorne Mon. Not. R. astr. Soc. (1982) 198, 345382 29 Gennaio 2009 A I F d L P = = 2 c dt F E.Prandini The BlandfordZnajek Effect 64 The horizon 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 65 The horizon: boundary conditions Can be studied by assuming a surface charge and current densities lying in the horizon (Znajek 1978) ● ● In our formalism, the horizon is a two dimensional surface with α = 0 ● The poloidal magnetic field is perpendicular to the horizon 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 66 The horizon: boundary conditions Can be studied by assuming a surface charge and current densities lying in the horizon (Znajek 1978) ● ● In our formalism, the horizon is a two dimensional surface with α = 0 ● The poloidal magnetic field is perpendicular to the horizon ● Hypotesis: the horizon fields are degenerate: EH·BH=0: ● the electric field EH is the field induced by the relative, to the tetrad, motion between the horizon and the poloidal magnetic field lines: F − H H F P E =− v /c ×B =− ∇ 2 c ● There is an induced current on the horizon surface, and the em field exerts a torque: dL H F − H = B⊥ H 4 c d dt 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 67 The horizon the net torque exerted by the horizon on the annular tube of magnetic flux ∆ψ and the transmitted redshifted power are: d L H H − F 2 − = B ⊥ dt 4 c optimal condition: 29 Gennaio 2009 F d LH , P = − dt F H ~ /2 E.Prandini The BlandfordZnajek Effect 68 Power output In the optimal conditions described before, if: black hole mass: M~108M. angular velocity ΩH ~1 rad/1000s magnetic field strenght 104 G Power output from the disk ~ Power output from the hole~1044 erg s1 29 Gennaio 2009 E.Prandini The BlandfordZnajek Effect 69
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