Electrodynamics of neutron star magnetospheres Theory Electrosphere - PowerPoint PPT Presentation

Electro- dynamics of pulsars Jérôme Pétri INTRO Observations Electrodynamics of neutron star magnetospheres Theory Electrosphere An example of non-neutral plasma in astrophysics How to do ? Geometry Stability Diocotron Jérôme Pétri Magnetron Non-linear evolution Quasi-linear Centre d’Étude des Environnements Terrestre et Planétaires - Vélizy, FRANCE model Laboratoire de Radio Astronomie, École Normale Supérieure - Paris, FRANCE Conclusions Non Neutral Plasma Workshop, New York - 20/6/2008

Outline Electro- dynamics of pulsars Jérôme What is a pulsar ? Pétri 1 Observations INTRO Theory Observations Theory Electrosphere Electrosphere 2 How to do ? How to do ? Geometry Geometry Stability Diocotron Magnetron Non-linear Stability properties evolution 3 Quasi-linear Diocotron instability model Magnetron instability Conclusions Non-linear evolution Quasi-linear model Conclusions 4

Electro- dynamics of pulsars Jérôme Pétri INTRO Observations Theory Electrosphere How to do ? Geometry INTRODUCTION Stability Diocotron Magnetron Non-linear evolution Quasi-linear model Conclusions

Discovery Electro- The first pulsar dynamics of pulsars discovered fortuitously at Cambridge Observatory (UK) in 1967 at radio-frequencies Jérôme signal made of a series of pulses separated by a period P = 1 . 337 s Pétri pulse profile changes randomly but arrival time stable INTRO Observations duration of a pulse ∆ t ≈ 16 ms Theory ⇒ size of the emitting region : L ≤ c ∆ t ≈ 4800 km Electrosphere ⇒ evidence for a compact object How to do ? Geometry Stability Radio signal measured from PSR1919+21 Diocotron Magnetron (Bell & Hewish, 1968) (already 40 years ago) Non-linear evolution Quasi-linear model Conclusions Basic assumption Pulsar = strongly magnetised rotating neutron star

Typical neutron star parameters Electro- Orders of magnitude dynamics of pulsars mass M ∗ = 1 . 4 M ⊙ Jérôme Pétri radius R ∗ = 10 km ( R ⊙ = 700 . 000 km ) mean density ρ ∗ = 10 17 kg / m 3 ( ρ ⊙ = 1 . 410 kg / m 3 ) INTRO Observations crust temperature T ∗ = 10 6 K Theory moment of inertia I ∗ = 10 38 kg m 2 Electrosphere How to do ? magnetic field strength at the stellar surface B ∗ = 10 5 ... 8 T ( B ⊙ = 5 × 10 − 3 T) Geometry induced electric field E ∗ = 10 10 ... 13 V / m ⇒ particles extracted from the surface Stability Diocotron particle density in the magnetosphere n = 10 17 m − 3 Magnetron Non-linear evolution Quasi-linear model Magnetic field strength estimation Conclusions Magnetic field intensity at the stellar crust estimated from dipolar magnetic radiation in vacuum, assuming a dipolar magnetic field (period P slowly increases ˙ P = dP / dt > 0) p B ∝ P ˙ P Period P P B ∗ Pulsar Period derivative ˙ 10 8 T 10 − 15 radio 1 s 10 5 T 10 − 18 millisecond 1 ms

The “standard cartoon” (not a physical model) Electro- dynamics of pulsars Jérôme Pétri INTRO Observations Theory Electrosphere How to do ? Geometry Stability Diocotron Magnetron Non-linear evolution Quasi-linear model Conclusions Fundamental problem in astrophysics no measurement/experiment in situ possible only observations coming from the electromagnetic radiation emitted ⇒ underlying plasma processes must be studied indirectly

The space charge distribution in the magnetosphere Electro- dynamics of pulsars Basic physics Assumptions Jérôme (Goldreich-Julian, 1969) Pétri aligned rotator ( � µ ) Ω ∗ � � closed magnetosphere entirely filled INTRO Observations with the corotating plasma Theory electrostatic equilibrium : Electrosphere � E + � v ∧ � B = 0 How to do ? Geometry the corotating charge density at Stability Ω ∗ · � B equilibrium ρ = − 2 ε 0 � Diocotron Magnetron the null surface : region where the Non-linear evolution B = 0) charge density vanishes ( � Ω ∗ · � Quasi-linear model particles follow the electric drift motion Conclusions Extracting charges from the stellar surface E ∧ � B direction in the � ⇒ non vacuum solution open field lines sustain a wind Does it work ? Is this picture self consistent and stable ? does not pulse (because aligned rotator) ! simulations have shown that it is NOT stable ! (Smith, Michel and Thacker, MNRAS, 2001)

Charged wind : source of particles Electro- dynamics of pulsars Pair creation cascades (Sturrock 1970, Assumptions Jérôme Pétri Ruderman & Sutherland 1975) corotation impossible outside the light INTRO cylinder R L = c / Ω ∗ Observations Theory charged wind emanating from the polar caps Electrosphere How to do ? charged particles ( e − e + ) are Geometry produced by γ + B → e + + e − in the Stability polar caps Diocotron Magnetron open field lines sustain a wind made Non-linear evolution of particles of both signs Quasi-linear ⇒ increase or decrease of the total model charge of the system Conclusions (star+magnetosphere) ⇒ no constraint to force charge conservation Inconsistent global picture ⇒ problem of the current closure

Electro- dynamics of pulsars Jérôme Pétri INTRO Observations Theory ELECTROSPHERE Electrosphere How to do ? Geometry = Stability Diocotron PART OF THE MAGNETOSPHERE Magnetron Non-linear evolution Quasi-linear FILLED WITH PLASMA model Conclusions

Magnetospheric model Electro- What is the structure of the magnetosphere ? dynamics of pulsars How does a stable plasma distribution looks like for a pulsar ? Jérôme All models proposed so far are electrodynamically unstable and not self-consistent ! ! ! Pétri INTRO Assumptions Observations Theory the neutron star = perfect spherical conductor of radius R ∗ , generating a dipolar Electrosphere magnetic field of strength B ∗ and in solid body rotation with speed Ω ∗ How to do ? Geometry an aligned rotator, i.e., magnetic moment and spin axis are parallel Stability charges extracted freely from the stellar crust whatever their nature Diocotron Magnetron magnetic field induced by the magnetospheric currents are neglected Non-linear evolution ⇒ constant and dipolar Quasi-linear model any force other than electromagnetic is neglected (even the gravitational attraction, F g / F em = 10 − 9 ! ! !) Conclusions E ∧ � � B v = electric drift approximation � B 2 Numerical simulations extract particles from the neutron star crust and let them fill the magnetosphere until they reach equilibrium stop when no more charges are left on the stellar surface

Plasma configuration Electro- dynamics of pulsars An example (Pétri, Heyvaerts & Bonazzola, A&A 2002) Jérôme Pétri 3D structure of the electrosphere 3 Rotation rate 3 INTRO 2.5 Observations 2.5 Theory Polar axis 2 2 � � r � Electrosphere 1.5 1.5 How to do ? 1 Geometry 1 0.5 Stability 1.5 2 2.5 3 3.5 4 Diocotron 0.5 1 1.5 2 2.5 3 3.5 4 Radius r Magnetron Equatorial plane Non-linear evolution Surface charge Charge density Quasi-linear model 1 1.2 Conclusions 1 0.8 0.8 Σ s � Θ � 0.6 Ρ � r � 0.6 0.4 0.4 0.2 0.2 0 20 40 60 80 1.5 2 2.5 3 3.5 4 Colatitude Θ en degree Radius r Total charge of the system Q tot = only free parameter

Results : magnetospheric structure Electro- dynamics of pulsars Jérôme Main features Pétri magnetosphere of both sign of charge INTRO Observations finite in extent Theory large gaps appear between the equatorial belt and the polar domes Electrosphere no electric current circulation in the gaps How to do ? Geometry differential rotation of the disk, overrotation Stability ⇒ shearing between magnetic surfaces responsible for instabilities Diocotron Magnetron same qualitative conclusions apply whatever the total charge Q tot of the neutron Non-linear evolution star+magnetosphere Quasi-linear model Conclusions Neutron stars : a natural trap for non-neutral e ± plasma ? The electromagnetic field acts as a trap, confinement of the non-neutral e ± plasma : in the “radial” direction by the dipolar magnetic field in the “axial” direction by the quadrupolar electric field

Electro- dynamics of pulsars Jérôme Pétri INTRO Observations Theory Electrosphere How to do ? Geometry STABILITY PROPERTIES Stability Diocotron Magnetron Non-linear evolution Quasi-linear model Conclusions

The question of the stability of the electrosphere Electro- Equilibrium & stability dynamics of pulsars Equilibrium does not imply stability. Jérôme The previous model can be unstable to non-neutral plasma instabilities Pétri INTRO Different approximations Observations Theory Studying the whole 3D structure to complicated ⇒ simplifications Electrosphere How to do ? We restrict to a 2D configuration of the equatorial disk only Geometry Stability a cylinder of infinite axial extent Diocotron an infinitely thin disk Magnetron Non-linear evolution Quasi-linear Different analysis model Conclusions linear stability ⇒ growth rate obtained from an eigenvalue problem non-linear simulations quasi-linear model Different regimes diocotron/magnetron non relativistic/relativistic