General relativistic computation of shocks in accretion disc and bipolar jets

Size: px

Start display at page:

Download "General relativistic computation of shocks in accretion disc and bipolar jets"

Easter Ray
5 years ago
Views:

1 General relativistic computation of shocks in accretion disc and bipolar jets Rajiv Kumar Co-author Dr. Indranil Chattopadhyay Aryabhatta Research Institute of observational sciences (ARIES) January 21, 2016 Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

2 Overview 1 Introduction Relativistic disc-jet sources 2 Relativistic fluid equations Accretion and jet equations, EoS and assumptions. 3 Results Accretion and ejection solutions. 4 Conclusions Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

3 Introduction: Jets are found in AGNs & microquasars. Since black holes don t have intrinsic atmosphere, so jets have to originate from accretion disc. Figure: Artistic picture of accretion disc and jets. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

4 General Relativistic Fluid Equations Geometrical unit: G = M = c = 1. Metric used: ds 2 = g ttdt 2 + 2g tφ dtdφ + g rrdr 2 + g φφ dφ 2 + g θθ dθ 2. where, g tt = (1 2r/Σ), g tφ = 2arsin 2 θ/σ, g rr = Σ/, g φφ = Asin 2 θ/σ and g θθ = Σ. = r 2 2r + a 2, A = (r 2 + a 2 ) 2 a 2 sin 2 θ, Σ = r 2 + a 2 cos 2 θ. The governing equations of the relativistic fluid is, T δβ ;β = 0, (ρuβ ) ;β = 0, where T δβ = (e + p)u δ u β + pg δβ }{{} +tδβ viscous stress tensor shear tensor(σ δβ ) Ideal fluid part Projecting it onto a spatial coordinate gives R. Navier-Stokes eqn. If along the 4-velocity gives us first law of thermodynamics. The general form of viscous shear tensor: σ δβ = 1 2 (u δ;β + u β;δ + a δ u β + a β u δ ) 1 3 Θ exph δβ, where, a δ 4-acceleration, u β;γ co-variant derivative of 4-velocity, Θ exp = u γ ;γ expansion of fluid world line, h δβ projection tensor. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

5 General Relativistic Fluid Equations We assumed r φ component of viscous stress tensor is dominant and disc is in steady state, axis-symmetric and in vertical hydrostatic equilibrium. The accretion equations are Energy generation equation, u δ T δβ ;β = 0 [( ) ] uδ e+p n n,δ e,δ = Q +, (1) where, Q + = t δβ σ δβ is the heating term and for the time being we are ignoring cooling term. The radial component of relativistic Navier-Stokes equation, du u r r dr + γ2 v A r 4 + A r 2 [ A r 4 r2 a2 (r A r 2 )]u φ u φ 2a r 2 [ A r 3 + ]u t u φ 2a r 3 ut u φ + γ2 v dp = 0, (2) e+p dr the integrated version of the azimuthal component equation (h φ δ T δβ ;β = 0), ρu r (L L 0 ) = 2µσφ r, where, µ = ρν, ν = αarfc, fc = (1 v2 ) 2, (3) Now, integral form of mass-conservation equation, Ṁ = 4πρHur r. (4) H = 2Θr 3, where γ φ is angular Lorentz factor. tγ 2 φ Integrating equations (1-4) and re-arranging, we get E = hγvr /A exp(x f ), X f = [( A r 4 r2 A (r a2 r 2 )) l2 (1 ωλ) r 2 γ 2 v + ur g φφ (L L 0) 2 2νhg rr ]dr, (5) E is constant of motion in presence of viscous dissipation and is called as general relativistic Bernoulli parameter. If α = 0, exp(x f ) = (1 ωλ)/γ φ, E = hu t, where, ω = g tφ g φφ. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

6 Relativistic Equation of State (EoS) The relativistic EoS for multispecies flow is (Chattopadhyay, 2008 and Chattopadhyay & Ryu, 2009), e = n e m e c 2 f = ρ e c 2 ρc f = 2 f, (6) [2 ξ(1 1/η)] where, f = (2 ξ) ˆ1 + Θ ` h i 9Θ+3 3Θ+2 + ξ 1 + Θ 9Θ+3/η. η 3Θ+2/η This is relativistic equation of state for multispecies gas flow. It is an approximation but is very accurate to the actual EoS for relativistic Maxwell-Boltzmann gas (see Chandrasekhar 1939 and Ryu et al. 2006). Here, adiabatic index is defined as Γ = 1 + 1/N, where, N = (1/2)df /dθ is polytropic index. Θ = kt/(m e c 2 ) is non-dimensional temperature, ξ = n p +/n e and η = m e /m p +. we have integrated first law of thermodynamics with equation (6) for adiabatic gas, we obtained (Kumar et al. 2013), ρ = K exp(k 3) Θ 3/2 (3Θ + 2) k 1 (3Θ + 2/η) k 2 where, k 1 = 3(2 ξ)/4, k 2 = 3ξ/4, and k 3 = (f t)/(2θ). This is the Relativistic adiabatic equation of state for multispecies flows, where K is constant of entropy. This eqn is analogous to Newtonian adiabatic EoS, p = Kρ Γ. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

7 Jet equations We assumed rotating and inviscid outflow from the disc.the eqn of stream-line or von-zeipel parameter is defined as (Abramowicz 1974, Chakrabarti 1985), «1/2 ϑφ Z = ϑ φ = g! φφ + g tφ /ϑ φ 1/2 g tt + ϑ φ, ϑ φ = u φ, ϑ φ = uφ g tφ u t u t. (7) It was proposed that the angular momentum of jets would be related to the von Zeipel parameter, ϑ φ = c φ Z n, (8) where, c φ and n are some constant parameters. Constant of motion of the jet R j = h j u tj [1 c 2 φ Z(2n 2) ] β, (9) Figure: von Zeipel surfaces. q q where, u tj = (1 2/r j ) 1/2 γ j, γ j = γ vj γ φj, γ vj = 1/ (1 vj 2), γ φj = 1/ (1 cφ 2Z(2n 2) ), v j = γ φ v p and β = n/(2n 2). where, all the symbols have usual means but subscript j represents flow variable for jet. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

8 Calculation of critical point In order to find the accretion solutions, so, we have to find critical point by solving (dv/dr = f (N)/f (D) = 0/0 = ) critical point conditions. Let us look into two properties. 1- Matter velocity on the horizon is free fall velocity i.e., v h = p 2/r s. Asymptotically close to horizon i.e. r 2 (Spin, a s = 0), we assumed v in = δ p 2/r in, where, δ < Since viscous stress tensor becomes negligible very close to the horizon, so, we assumed at very close to the horizon E = E = hu t, then the equation (5) must satisfy the relation, γ φ exp(x f ) = 1. This condition written as quadratic in L 0. Which is b 2L b 1L 0 + b 0 = 0, Now, we have v in, Θ in, L in at r in = and L 0 or λ 0 at horizon by using four flow parameters, E, ξ, α and λ in, so we can integrate equations (1-3) simultaneously outward from r in. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

9 Typical GR accretion solution with various flow quantities M = v/a Mach number v Bulk velocity Θ Non-dimensional temperature E Viscous relativistic Bernoulli parameter R j Relativistic Bernoulli parameter for the jet M Entropy accretion rate Γ Adiabatic index λ Specific angular momentum L Bulk angular momentum Figure: This ploted for the flow parameters, E = , ξ = 1.0, α = 0.01, λ in = 2.99 and a s = 0. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

10 All possible accretion solutions & MCP region Figure: Simulation result shows accretion disc and outflows. Lee et al... Figure: Multiple critical point (MCP) parameter space for ξ = 1.0, α = 0.01 and a s = 0. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

11 Jet solution & jet variables Figure: In this figure we have represented a typical accretion - jet solutions together. Here solid (red) curve represents disc-half height for accretion parameters, E = , L 0 = 2.92, α = 0.01, a s = 0 and ξ = 1. Dotted -dashed (blue) line is jet stream line for von Zeipal parameter, Z φ = Arrows represent direction of bulk motion and a quadratic part of circle (solid black thick line) represents black hole surface. Figure: In plot (a) We presented together variation of accretion Mach number (M) with radial distance, log(r) and corresponding outflow solutions Mach number (M j ) with axial distance, log(z j ). Variation jet flow variables shown in plots (b) outflow velocity (v j ), (c) jet constant of motion (R j ), (d) jet dimensionless temperature (Θ j ), (e) jet adiabatic index (Γ j ) and (f) jet entropy (Ṁ j ) with jet axis, log(z j ). Flow parameters are used L 0 = 2.906, α = 0.01, E = 1.001, a s = 0, ξ = 1.0 and relative mass outflow rate is Rṁ = Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

12 Multiple critical points (MCP) in the jet Figure: In panel (a) and (c), we have drawn von Zeipel surfaces (VZS) with various von Zeipel parameters and Kerr spin parameter, respectively. In panel (b) and (d), we have shown variations of the jet relativistic Bernoulli parameter (R jc ) with jet critical points. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

13 Possiple jet solutions Figure: Depending upon number and nature of critical points, we have represented variation of of Mach number (M j ) with jet spherical radius (r j ) for different von zeipel parameters. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

14 Jet solution for rotating BH Figure: Jet shock solution for the disc parameters E = 1.001, λ = 2.1, ξ = 1.0, a s = 0.95 and corresponding shock location in the disc, r s = (In panel (a) accretion solution, blue solid line). Shock locations in the jet are at r js = 13.55, and Z = Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

15 Variation of outflow rate Figure: Variation of plot (a) shock location (r s) with viscosity parameter (α), plot (b) mass outflow rate (Rṁ) with r s, plot (c) compression ratio (R) with r s and plot (d) Rṁ with R. Plots are plotted with E = , ξ = 1.0, a s = 0 and keeping outer boundary fixed at r inj = and corresponding Keplerian angular-momentum, λ K = λ inj = Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

16 Conclusions Pure relativistic accretion solutions are nicely satisfying the GR inner boundary condition. Which we don t see in pseudo-potential because it blows up very close to the horizon. We have investigated self-consistently jet solutions from shock accretion solutions and also estimated mass outflow rate. We have found MCP and therefore, the shocks in the jet. According to Nakayama (1992) paper, in case of wind flows, the inner shock is stable and the outer one may be unstable. If these shocks are unstable then they can generate QPOs and variability in the jets as observed in the Blazars. GR accretion-jet solutions some features are qualitatively same as in pnp but quantitatively they differ. The real difference appears close to the horizon, where radiative properties around the horizon will be different in GR than in pnp. Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

17 Thanks Rajiv Kumar (ARIES, Nainital) Accretion-Ejection around black holes January 21, / 17

Charged particle motion around magnetized black hole

Charged particle motion around magnetized black hole Martin Kološ, Arman Tursunov, Zdeněk Stuchĺık Silesian University in Opava, Czech Republic RAGtime workshop #19, 23-26 October, Opava 2017 Black hole