Polarisation of high-energy emission in a pulsar striped wind

Transcription

1 Polarisation of high-energy emission in a pulsar striped wind Jérôme Pétri Observatoire Astronomique de Strasbourg, Université de Strasbourg, France. X-ray polarimetry, November 2017, Strasbourg Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 1/ 16 X-ray polarimetry, 14th November / 16

2 Outline 1 Motivation 2 Striped wind model 3 Phase-resolved synchrotron polarization 4 GR & QED corrections to pulsar magnetospheres 5 Conclusions & Perspectives Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 2/ 16 X-ray polarimetry, 14th November / 16

3 1 Motivation 2 Striped wind model 3 Phase-resolved synchrotron polarization 4 GR & QED corrections to pulsar magnetospheres 5 Conclusions & Perspectives Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 2/ 16 X-ray polarimetry, 14th November / 16

4 Motivation Several models of pulsed high-energy emission polar cap (Harding, 1981). outer gap (Cheng et al., 2000). two-pole caustic (Dyks & Rudak, 2003). striped wind (Pétri, 2011). How to discriminate between them? fit a large band of the pulsar spectrum. use predictions about polarization of pulsed emission. done for the Crab pulsar in optical (Dyks et al., 2004; Pétri & Kirk, 2005) still waiting for X/γ-rays data. Figure: Emission models (Aliu et al., 2008) Figure: Optical polarization (Pétri & Kirk, 2005). Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 3/ 16 X-ray polarimetry, 14th November / 16

5 Polarisation of the Crab pulsar Figure: Comparison of several high-energy models (S lowikowska et al., 2009). Energy band observations PD PA references optical phase-resolved 9.8% 110 o (S lowikowska et al., 2009) X/γ-rays off-pulse 72% 120 o (Forot et al., 2008) X/γ-rays off-pulse 48% 123 o (Dean et al., 2008) Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 4/ 16 X-ray polarimetry, 14th November / 16

6 1 Motivation 2 Striped wind model 3 Phase-resolved synchrotron polarization 4 GR & QED corrections to pulsar magnetospheres 5 Conclusions & Perspectives Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 4/ 16 X-ray polarimetry, 14th November / 16

7 The structure of the striped wind Figure: Near the star a magnetic dipole. Figure: At large distances the striped wind (Bogovalov, 1999). Important parameters: Ω: rotation axis χ: magnetic axis inclination with respect to Ω ζ: line of sight inclination with respect to Ω - hot and magnetized plasma - relativistic beaming Γ v 1 } pulsed emission Figure: Striped wind emission (Kirk et al., 2002). Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 5/ 16 X-ray polarimetry, 14th November / 16

8 y z Striped wind model: magnetic field structure Exact analytical solution for the electromagnetic field in a realistic striped wind finite thickness for the current sheet. relativistic magnetized flow wind. radial expansion. constant speed V = β vc with β v 1. (Pétri, 2013) Χ Ζ r Figure: Meridional section. B r = β 2 vb L r 2 L r 2 tanh(ψs/ ) B ϕ = β vb L r L r sinϑ tanh(ψ s/ ) E ϑ = βvcb 2 r L L sinϑ tanh(ψ s/ ) r [ Ψ s = cosϑ cosχ+sinϑ sinχ cos ϕ Ω(t r ] β ) vc satisfies the homogeneous Maxwell equations x Figure: Equatorial section. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 6/ 16 X-ray polarimetry, 14th November / 16

9 Striped wind model: particle density number Two component emission model including (Pétri, 2013) a cold plasma in the well organized magnetic field outside the stripe n cold (r,t) = Nctanh2 (ψ/ ) r 2 a hot almost unmagnetized plasma inside the stripe n hot (r,t) = N h[1 tanh 2 (ψ/ )] r 2 Parameters of the model starting emission region r 0 r L. Lorentz factor of the wind Γ v. ratio of hot to cold particles N h /N c. power law index p. obliquity χ. inclination of line of sight ζ. (Lyubarsky & Kirk, 2001) (Bühler & Blandford, 2014) How do these parameters influence the phase-resolved synchrotron polarization? Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 7/ 16 X-ray polarimetry, 14th November / 16

10 1 Motivation 2 Striped wind model 3 Phase-resolved synchrotron polarization 4 GR & QED corrections to pulsar magnetospheres 5 Conclusions & Perspectives Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 7/ 16 X-ray polarimetry, 14th November / 16

11 I Polarization with respect to the Lorentz factor Γ v Χ 60,Ζ 60, Nh Nc 10., r0 10, p phase asymmetric light-curves for modest Γ v. falling time longer than rising time. shape of pulses = sheet thickness + relativistic beaming. correct as long as the opening angle of the beaming is larger than the thickness of the stripe. Π opening angle decreases like 1/Γ v sharpening of the pulses. pulse width tends to a minimum independent of Γ v for Γ v 1. Ψ degrees low Γ v correspond to high polarization degree. relativistic flow depolarizes the synchrotron emission significantly in the striped wind. polarization angle not significantly affected. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 8/ 16 X-ray polarimetry, 14th November / 16

12 I Polarization with respect to the power-law index p v 10., Nh Nc 10.,Χ 60,Ζ 60, r p intervenes through the power in Doppler beaming factor D v = 1/Γ v(1 β n). relativistic beaming efficiency depending on p. p = 3 reinforce the variation with D v. Π efficient beaming implies less emission in the off-pulse phase. 0 polarization degree follows a similar trend. Ψ degrees increasing p also increases Π substantially from 11% to 19%. polarization angle variation similar for every p. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 9/ 16 X-ray polarimetry, 14th November / 16

13 1 Motivation 2 Striped wind model 3 Phase-resolved synchrotron polarization 4 GR & QED corrections to pulsar magnetospheres 5 Conclusions & Perspectives Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 9/ 16 X-ray polarimetry, 14th November / 16

14 Corrections to Maxwell equations An accurate and quantitative analysis of phenomena at the neutron star surface must take into account an important space time curvature. a magnetic field strength of the order or larger than B qed. a relativistic pair plasma e ± (classical/quantum?). high-energy radiation processes (curvature, synchrotron, inverse Compton). QED processes. gravitational redshift. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 10/ 16 X-ray polarimetry, 14th November / 16

15 Space-time in 3+1 formalism A first approach starts with an effective theory for the electromagnetic field including quantum and gravitational corrections to Maxwell equations formulation in space and time based on general relativity (3+1). Lagrangian description of the electromagnetic field. The metric is divided in space Σ t and a time coordinate ds 2 = α 2 c 2 dt 2 γ ab (dx a +β a c dt)(dx b +β b c dt) with lapse function α. shift vector β. spatial metric γ ab. Figure: Space-time split in 3+1. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 11/ 16 X-ray polarimetry, 14th November / 16

16 The QED Lagrangian X classical Lagrangian of electromagnetic field. + QED corrections à la Euler-Heisenberg. I i, A i : 4-current and 4-potential. F ik : electromagnetic tensor. I 1 = F ik F ik, I 2 = F ik F ik field invariants. L QED = QED Lagrangian Classical terms QED corrections. η 1 Euler-Heisenberg Born-Infeld α sf 180π 1 2µ 0 B qed 2 7 η 2 η1 η µ 0 b 2 with α sf the fine structure constant and b = T. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 12/ 16 X-ray polarimetry, 14th November / 16

17 The QED Lagrangian X classical Lagrangian of electromagnetic field. + QED corrections à la Euler-Heisenberg. I i, A i : 4-current and 4-potential. F ik : electromagnetic tensor. I 1 = F ik F ik, I 2 = F ik F ik field invariants. L QED = 1 4µ 0 F ik F ik I i A i + QED Lagrangian Classical terms QED corrections. η 1 Euler-Heisenberg Born-Infeld α sf 180π 1 2µ 0 B qed 2 7 η 2 η1 η µ 0 b 2 with α sf the fine structure constant and b = T. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 12/ 16 X-ray polarimetry, 14th November / 16

18 The QED Lagrangian X classical Lagrangian of electromagnetic field. + QED corrections à la Euler-Heisenberg. I i, A i : 4-current and 4-potential. F ik : electromagnetic tensor. I 1 = F ik F ik, I 2 = F ik F ik field invariants. L QED = 1 4µ 0 F ik F ik I i A i + η 1I 2 1 +η 2I 2 2 QED Lagrangian Classical terms QED corrections. η 1 Euler-Heisenberg Born-Infeld α sf 180π 1 2µ 0 B qed 2 7 η 2 η1 η µ 0 b 2 with α sf the fine structure constant and b = T. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 12/ 16 X-ray polarimetry, 14th November / 16

19 Maxwell equations in GRQED Constitutive relations in general relativity (Pétri, 2013) ε 0E = αd+ε 0c β B µ 0H = αb β D ε 0c Curvature of absolute space taken into account by lapse function α in first term. frame dragging in second term, cross-product between the shift β and fields. constitutive relations for QED (Pétri, 2015) (ξ 1 = 1 K η 1 and ξ 2 η 2) Homogeneous Maxwell equations F = ρ G = J+ 1 γ t( γf) F = ξ 1D+ ξ2 c B G = ξ 1H ξ2 c E Inhomogeneous Maxwell equations E = 1 γ t( γb) B = 0 GR and QED seen as material media in standard classical Newtonian theory. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 13/ 16 X-ray polarimetry, 14th November / 16

20 Simple approximation for plasma: GRFFQED For a pair plasma e ± infinite conductivity σ = +. zero temperature T = 0 then zero pressure. negligible mass m e = 0. In the force-free approximation from which we deduce the electric current J = ρ E B B 2 J E = 0 ρe+j B = 0 + B G F E B 2 B. The system of Maxwell equations is complete, the charge density ρ adapting to the field configuration. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 14/ 16 X-ray polarimetry, 14th November / 16

21 Vacuum perpendicular solutions Spin parameter a = 2 5 R R r L. Spindown luminosity Spindown (L/L vac dip) L flat dip = 8π 3µ 0c 3 Ω4 B 2 R Vacuum orthogonal dipole (N, -3) (GR, -3) (N, 0) (GR, 0) R/r L Figure: Spindown luminosity for different rotation rates, magnetic field strengths given by log(b/b q) and gravitational field (Newtonian or GR). (Pétri, 2014), (Pétri, 2016) QED has no impact on spindown luminosity. Also true for plasma filled magnetospheres (FFQED). Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 15/ 16 X-ray polarimetry, 14th November / 16

22 1 Motivation 2 Striped wind model 3 Phase-resolved synchrotron polarization 4 GR & QED corrections to pulsar magnetospheres 5 Conclusions & Perspectives Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 15/ 16 X-ray polarimetry, 14th November / 16

23 Conclusions & Perspectives Conclusions constraints on the arbitrary parameters concerning the configuration of the emission region. the distribution function of the emitting particles. future observations of polarization properties in the high-energy band (IXPE) will be able to disentangle between several models. solve the decade long standing problem about the emission mechanism and location. Perspectives deeper knowledge about the particle distribution functions. joining the plasma kinetic to the fluid regime from surface up to 100r L. local study including possibly magnetic reconnection and turbulence within the sheet. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 16/ 16 X-ray polarimetry, 14th November / 16

24 References I Aliu E. et al., 2008, Science, 322, 1221 Bogovalov S. V., 1999, A&A, 349, 1017 Boyarsky A., Fröhlich J., Ruchayskiy O., 2015, Physical Review D, 92, Bühler R., Blandford R., 2014, Reports on Progress in Physics, 77, Cheng K. S., Ruderman M., Zhang L., 2000, \apj, 537, 964 Dean A. J. et al., 2008, Science, 321, 1183 Dyks J., Harding A. K., Rudak B., 2004, ApJ, 606, 1125 Dyks J., Rudak B., 2003, ApJ, 598, 1201 Forot M., Laurent P., Grenier I. A., Gouiffès C., Lebrun F., 2008, ApJ, 688, L29 Haas F., 2005, Physics of Plasmas, 12, Harding A. K., 1981, \apj, 245, 267 Kirk J. G., Skjæraasen O., Gallant Y. A., 2002, A&A, 388, L29 Lyubarsky Y., Kirk J. G., 2001, ApJ, 547, 437 Mignani R. P., Testa V., González Caniulef D., Taverna R., Turolla R., Zane S., Wu K., 2017, Monthly Notices of the Royal Astronomical Society, 465, 492 Pétri J., 2011, MNRAS, 412, 1870 Pétri J., 2013, Monthly Notices of the Royal Astronomical Society, 433, 986 Pétri J., 2013, MNRAS, 434, 2636 Pétri J., 2014, Monthly Notices of the Royal Astronomical Society, 439, 1071 Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 1/ 7 X-ray polarimetry, 14th November / 7

25 References II Pétri J., 2015, Monthly Notices of the Royal Astronomical Society, 451, 3581 Pétri J., 2016, Astronomy and Astrophysics, 594, A112 Pétri J., Kirk J. G., 2005, ApJ, 627, L37 Shabad A. E., Usov V. V., 1984, \apss, 102, 327 Shaviv N. J., Heyl J. S., Lithwick Y., 1999, \mnras, 306, 333 S lowikowska A., Kanbach G., Kramer M., Stefanescu A., 2009, MNRAS, 397, 103 Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 2/ 7 X-ray polarimetry, 14th November / 7

26 I Polarization with respect to starting emission radius r 0 v 10., Nh Nc 10.,Χ 60,Ζ 60, p 2. Ψ degrees Π phase lag increasing with r 0. explained by retardation effect. weak influence on light-curves. pulses not symmetric with respect to rising/falling time. increase in polarization degree Π for large r 0. no significant Π for r 0 r L. Π up to 28% for r 0 > 10r L. off-pulse in cold plasma and well-ordered magnetic field high Π. hot plasma depolarizes synchrotron emission low Π. polarization angle suffers sharp gradient switching from -60 o /-80 o to +60 o /+80 o within each pulse. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 3/ 7 X-ray polarimetry, 14th November / 7

27 I Polarization with respect to the particle density ratio N h /N c v 10.,Χ 60,Ζ 60, r0 10, p ratio between peak intensities proportional to N h /N c hot plasma associated with pulses. cold plasma associated with off-pulse emission. Π phase light-curves do all overlap, all peaks phase-aligned. polarization degree similar for each curve. 5 0 highest polarization degree decreases for increasing density contrast. Ψ degrees hot and unpolarized component reduces Π. no distinction for polarization angle. density can not influence the polarization angle. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 4/ 7 X-ray polarimetry, 14th November / 7

28 I Polarization with respect to the inclination of the line of sight ζ v 10., Nh Nc 10.,Χ 60, r0 10, p as in the previous plots, two pulses are detected only if χ > π/2 ζ. the polarization degree depends strongly on the inclination angle. too a small ζ produces weakly polarized emission, for instance with ζ = 30 o wet get a maximum of 9%. Π 10 5 for maximal inclination with ζ = 90 o the polarized component is high, up to 20%. 0 close to the pole the striped wind is circularly polarized. Ψ degrees close to the equatorial plane it is almost exclusively linearly polarized. amplitude of the polarization angle decreases significantly with increasing ζ. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 5/ 7 X-ray polarimetry, 14th November / 7

29 I Polarization with respect to obliquity χ v 10., Nh Nc 10.,Ζ 60, r0 10, p for small obliquity, χ < π/2 ζ, two peaks not well separated. single pulse. Ψ degrees Π for χ = 30 o and ζ = 60 o, line of sight not significantly crossing the stripe. for χ = {60 o,90 o }, looking through the stripe and two pulses per period. separation being maximal and equal to half a period for the orthogonal rotator χ = 90 o. polarization degree and angle evolve according to the shift between both pulses. In any case, Π decreases sharply within the pulse(s). angle shows sharp gradient. off-pulse, Π constant and maximal. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 6/ 7 X-ray polarimetry, 14th November / 7

30 Possibles applications of QED wave propagation quantum/relativistic plasmas vacuum birefringence, signature from optical? (Mignani et al., 2017) do photons follow curve field lines (magnetic lensing)? (Shabad & Usov, 1984), (Shaviv et al., 1999) do we need QMHD and chiral MHD? (Haas, 2005), (Boyarsky et al., 2015) as a long term task, possibility to test QED in strong magnetic AND gravitational fields. Figure: The IXPE satellite. Jérôme Pétri (Observatoire Astronomique) Polarization in pulsar striped wind 7/ 7 X-ray polarimetry, 14th November / 7