Modeling Relativistic Magnetized Plasma

1. MODELING RELATIVISTIC MAGNETIZED PLASMA Komissarov Serguei University of Leeds UK

2. RELATIVISTIC IDEAL MHD Conditions: Equations: - perfect conductivity • stress-energy-momentum of • electromagnetic field -stress-energy-momentum of matter

3. RELATIVISTIC IDEAL MHD Advantages: • Allows adiabatic transfer of energy and • momentum between the electromagnetic • field and particles; • Allows dissipation at shocks; • All wave speeds below c. Disadvantages: • Complexity; • Difficult to implement if .

4. RELATIVISTIC IDEAL MHD Godunov-type schemes: • Robust and simple Lax-Friedrichs-type Riemann solvers; • More accurate and complex linear Riemann solver • (contacts, shears; Komissarov 1999); • No exact Riemann solvers so far - too expensive; • A number of ways to handle the “divB-problem”; • (i) constrained transport (Evans & Hawley 1988); • (ii) generalised Lagrange multiplier (Dedner et al. 2002); • (iii) smoothing operator (Toth 2000) .

5. RELATIVISTIC IDEAL MHD Example: - X-ray image of the Inner Crab Nebula based on 2D relativistic MHD simulations (Komissarov & Lyubarsky 2003) Chandra image of the real Crab Nebula

6. MAGNETODYNAMICS (MD) Condition: Equations: (Komissarov 2002) Perfect conductivity: or MAGNETODYNAMICS is MAGNETOHYDRODYNAMICS without the HYDRO part

7. MAGNETODYNAMICS Advantages: • Simple hyperbolic system of conservation laws • (linearly degenerate fast and Alfven modes); • Perfectly describes force-free magnetospheres • of black holes and neutron stars; Disadvantages: • Does not allow adiabatic transfer of energy and momentum • transfer between the electromagnetic field and particles; • 2) Does not allow dissipation; • Fast wavespeed equals to c; • Often breaks down;

8. MAGNETODYNAMICS Example: Stability of the Blandford-Znajek solution. H f Analitical and numerical solutions for a black hole with a=0.1, 0.5, and 0.9 at r =10 and t=120. (Komissarov 2001)

9. MAGNETODYNAMICS Breakdowns of the MD approximation. 1D example: Initial solution Time evolution 2 2 B - E B x B y E=0 x x A need for finite conductivity in order to keep E down !

10. RESISTIVE ELECTRODYNAMICS Covariant 3+1 form Equations: Constitutive a, b - lapse function and shift vector (space-time metric)

11. RESISTIVE ELECTRODYNAMICS Ohm’s Law: (no particle inertia) - drift current - anisotropic conductivity Typical conditions of BH and pulsar magnetospheres: In current sheets: or even

12. RESISTIVE ELECTRODYNAMICS Advantages: • Simplicity; • Drives solutions towards the force-free state; • 3) Allows dissipation in current sheets (transfer of • energy between the electromagnetic field and radiation); Disadvantages: • Does not allow adiabatic transfer of energy • between the electromagnetic field and particles; • Fast wave speed equals to c.

13. RESISTIVE ELECTRODYNAMICS Example: Ergospheric current sheet. W 2 2 Kerr black hole in uniform at infinity magnetic field; plasma version. (Komissarov, 2004) B - E

14. RELATIVISTIC IDEAL MHD with a density floor Prescription: do not let the particle energy density to slip below a curtain small fraction of the electromagnetic energy density. Advantages: • All the advantages of Ideal MHD; • Allows to get quite close to the MD limit; Disadvantages: • All the disadvantages of Ideal MHD; • How to handle current sheets ?

15. RELATIVISTIC IDEAL MHD with a density floor Example: Inertial effects in the Blandford-Znajek problem. Analytical and numerical solutions for a Kerr black hole with a=0.1. (Komissarov, 2001) Lorentz factor (colour) and the critical surfaces at t=170; a=0.9.

16. CONSLUSIONS • These have been few first steps in exploring ways of • modelling relativistic magnetized plasma; • A number of important results have been obtained; • There is still a long way to go and a promise of new • important results in near future.