Nick Camus Niccolo Bucciantini Philip Hughes Maxim Lyutikov

1. RMHD simulations of the Crab Nebula Serguei Komissarov (University of Leeds) Nick Camus Niccolo Bucciantini Philip Hughes Maxim Lyutikov

2. Plan of the talk 1. Crab Nebula and its wisps; 2. Theoretical MHD models; 3. High-resolution 2D simulations – strong variability; 4. Modelling synchrotron emission – moving wisps; 5. Statistical analysis of the variability; 6. Gamma-ray emission; 7. Summary.

3. I. The Crab Nebula and its wisps Thermal filaments (supernova remnant) Non-thermal diffuse emission (plerion) Optical image The total mass, MN + MNS ~ 6M3 < 9M3, too low for core-collapse supernova ?

4. X-rays “torus” pulsar X-ray map of of the inner Crab Nebula Visible light jet Chandra image ( Weisskopf et al. 2000)

5. Wisps of The Crab Nebula HST movie Hester et al.(1995) Quasi-periodic (?) emission of wisps. Knot 1 is the most compact permanent feature on the map, 0.5 arcsec or 6 light days

6. II. MHD model Termination of spherical wind 1D relativistic MHD model; Particle dominated relativistic pulsar wind with purely azimuthal magnetic filed; s-problem: conversion of the magnetic energy into the kinetic energy of the wind. Dissipation of magnetic energy. v B B v

7. Termination of equatorial wind Magnetic hoop stress redirects the flow towards the poles Lyubarsky (2002); Bogovalov & Khangoulian (2002) “jet” B v v “torus” v “torus” termination shock B “jet” shocked pulsar wind unshocked pulsar wind with ram pressure Michel (1973)

8. III. High-resolution 2D simulations The same setup as in Komissarov & Lyubarsky (2004) but 1) higher resolution; 2) improved model of synchrotron emission; 3) no equatorial symmetry imposed. RMHD equations: Numerical scheme: improved version of Komissarov (1999); conservative upwind scheme; second order in time and third order in space; spherical coordinates; hierarchical time stepping; dynamical grid size (not AMR).

9. Simulations setup: Initial solution and boundary conditions outer boundary: supersonic outflow 2D; axisymmetry. inner boundary: supersonic inflow (stationary pulsar wind) Unshocked pulsar wind Number of grid points in q: Supernova ejecta (cold Hubble flow) Duration of runs ~ the Crab’s age (1000 yr)

10. Supernova remnant (shell): scaled to fit the observed expansion rate of the Crab Nebula Model of pulsar wind: Total energy flux: (scaled to fit the spindown power of the Crab pulsar) Kinetic energy flux: Michel (1973) (?) s = (Poynting flux)/(kinetic flux) ~ 0.01 (too low?) g =10 - bulk Lorentz factor;

11. Azimuthal magnetic field: x = 0.16 - magnetization parameter (particle dominated flow). Dissipative current sheet (magnetic dissipation, flow acceleration) m W 3 3 7 Coroniti (1990), Michel(1994), Lyubarsky & Kirk (1991) etc.

12. RESULTS: Strong variability of the plerion flow Animation 1: Total pressure ( CGS units ) near the end of the run with the highest resolution. Only the inner part of the computational domain is shown.

13. RESULTS: Strong variability of the plerion flow Animation 2: Magnetic field ( Gauss ) near the end of the run with the highest resolution

14. IV. Modelling synchrotron emission Initial electron spectrum at the termination shock: Downstream spectrum (synchrotron + adiabatic losses): n – density of advective tracer; n0 ~ r -2 – its value at the shock; - cut-off energy. Synchrotron emissivity: - Doppler factor, - normal to the line of sight component of comoving magnetic field, - radiation frequency.

15. Evolution equations for the spectrum parameters: - suspension equation for n ; - advective scalar equation for n0 ; - reaction-advection eq. for evolves due to synchrotron and adiabatic losses. These equations are integrated simultaneously with the main system (of RMHD).

16. Synthetic optical image. Role of the Doppler beaming in the appearance of the nebula. OFF Doppler beaming switched off

17. Synthetic optical image. Role of the Doppler beaming in the appearance of the nebula. ON Doppler beaming switched on HST knot 1

18. observer Lorentz factor of the post-shock flow Emissivity in the frame of the observer

19. Origin of the knot 1 pulsar knot 1 HST

20. Geometry of the knot 1 - oblique shock equations - Doppler beam angle - flow towards the observer Transverse size of the knot:

21. Animation 3: Synchrotron emission near the end of the run with the highest resolution • Initial wisp speed ~ 0.5c; • wisps slow down and • pile-up further out; • sometimes wisps contract; • proper motion in the jet.

22. V. Statistical analysis of data Time series to study numerical convergence and to quantify variability Measure the magnetic field at a point near equator downstream of the termination shock. Four runs with increasing resolution.

23. Auto-correlation function, Convergence not reached. Getting close? The characteristic time scale decreases with resolution; around 1 year for . turbulence numerical noise characteristic time scale of variability

24. Search for quasi-periods (Wavelet transform). Morlet Two quasi-periods: ~1.5yr and ~3yr. Observations suggest: ~ few months. In simulations a) the shock radius is ~3 times higher, b) the period decreases with resolution.

25. Unexplained feature – the “inner ring” ( its bright knots ).

26. VI. Gamma-rays from the termination shock ? Termination shock size ~ 120 light days. Gamma-rays come from the very vicinity of the termination shock and hence must be subject to strong Doppler beaming. wisps knot 1 Extrapolation of the knot 1 optical emission using the power law (Tziamtzis et al., 2009) gives the observed total flux at 100 MeV !

27. - maximum energy of electrons accelerated by the electric field Evidence of Doppler boosting ? ( Vittorini et al. 2011 )

28. Variability of the knot 1 at 100 MeV time in years ( No data for shorter time scales )

29. Mechanism of the gamma-ray variability ? - Doppler boosting Variable Doppler boosting ? When the viewing angle decreases from 1/g to 0 the Doppler factor increases from g to 2g. For a = 3 this yields 30-fold increase of the observed emissivity. What can result is such a variability of the flow pattern on the timescale of gamma-ray flares ???

30. VII. Summary • New high-resolution axisymmetric MHD numerical simulations • reveal highly unsteady flow dynamics in pulsar wind nebulae; • The most dynamic region is inside the TS cusp – the jet base. • Statistical analysis indicates turbulent cascade in the main body of • the nebula. • Synthetic synchrotron maps are remarkably similar to the HST and • Chandra maps of the Crab Nebula: Jet, torus, knots, wisps, fine fibrous • structure of emission. The inner ring is still a puzzle; • Wisps move with relativistic speeds similar to the observed; • The predominant motion is expansion, though contraction is also seen • from time to time;

31. For the highest resolution, the characteristic time-scale of the flow • variability and wisp production is around 1 year; • Wavelet transform reveals quasi-periods of ~1.5yr and ~3.0 yr • (only slightly longer compared to the observations). • Inner knot (knot 1) is a highly Doppler-boosted patch of the termination • shock. It could be the main contributor to the observed gamma-ray • emission from the Crab nebula. A correlation of the optical emission • from the knot and the gamma-ray emission is expected. • How different is the 3D dynamics ??? The End

32. Crab movie Hester et al.(2002)

33. snapshot at ~ the Crab’s age bright wisps HST knot 1 fine wisps other knots Fine fibrous structure of the synchrotron emission, similar to that of the Crab Nebula (Scargle1969, Hester et al. 1995) north-south asymmetry (Doppler beaming)

34. Wavelet transform (Morlet) Search for quasi-periods. There is a quasi-periodic behaviour (more than one period?) ! The period decreases with resolution … .