S. Shibata 柴田　晋平Yamagata Univ. 山形大学

Department of Phys 物理学科

The Structure of the Pulsar Magnetosphere

via Particle Simulation

＠ FAN3 22 June 2012

Aim of our numerical study

1. To give the theoretical proof of the outer gap (the mechanism has been proposed)

2. To look for other particle acceleration sites; polar cap, & slot gap to be confirmed, and look for other possibilities?

3. To study the pulsar wind formation

via particle simulation, the main problems to be challenged are

9%5%

65%

11%2%7% 1%

11'2M2'34

黄色系Ｐ１　メインピーク一つ

緑系Ｐ１＋Ｐ２　２ピーク＋ブリッジＭ型

青系Ｐ１＋Ｐ２＋Ｐ３以上

４４サンプルFrom “The First Fermi Large Area Telescope Catalog of Gamma-ray Pulsars,Abdo et al., astro-ph/0910.168v2

Although pulse profiles have some varieties,

Two-pole caustic (TPC) geometry(Dyks & Rudak, 2003)

Double peak and bridge emission can be understood if emission is along the last open field lines; due to multi-ply superposition of photon arrival times: caustic.

Favors Slot gap / Outer gap

Closed field(dead zone)

Open field region

Observation (Vela)

model

Caustic in human life

Double pulses and spectrum support the outer gap model to give the proof of the outer gap.

After Ray & Parkinson 2010Astro-ph 1007.2183

Third peak, may not be caustic

3rd peak, 4th,,,,,many components are observed and shall be observed in the future. look for other acceleration sites by simulation.

The rotation power is mostly carried off by the pulsar wind. formation of the pulsar wind.

Q1 how much fraction of the wind energy is carried by electromagnetic flux (Poynting flux), and how much by kinetic energy of particles; acceleration efficiency

Q2 where the wind acceleration takes place; near the l.c. or gradually in a long distance to the nebula.

X-ray image of Crab Nebula, which is exited by the pulsar wind, but the contents of the wind is not certain

Basic Physics I

Model:NS magnetized electrically conducting sphere; B=10^12 G, rotating wiht P= 0.1sec

v

BV

+-

Electric potential difference appears between the poles and the equator.Rotating magnet has emf

charge separation

＋＋

ーー

ー

＋

B

V

F

Release of rotational energy (in reality ]: how?

Note ∂B /∂t =0

Unipolar inductor: rotating magnet produces emf

φ=(Ωμ/cR^3) [(x^2+y^2)cos χ – zx sin χ]

v

BV

+-

Electric potential difference appears between the poles and the equator.

For oblique rotator, potential difference becomes

current

magnetic field

By forming a closed current circuit, one can extract energy: one can light the lamp.

As a back reation, electromagnetic breaking on the magnet causes spin-down: Thus rotational energy of the neutron star

is extracted.

The rate of energy loss L_rot is (by dimensional analysis within the classical physics)

One can estimate μ 　 by this relation.

currentmagnetic field

hν/c or γmcAngular momentum is lost by photons back reaction on the emitting material make mutual mo. extraction possible

One must notice that the circuit experiences a force in such a direction that the circuit is force to corotate with the star. Thus the circuit and the plasma around the neutron star have strong tendency of corotation.

If the ciruit corotates with the star, no mutual motion, no extraction of emf.

Co-rotation speed becomes c

If the lever-arm distance is c/Ω, angular momentum is properly lost.

Summary:The extraction of rotation energy can be kept continuously by emission of photons and particles near and beyond the light cylinder with levar-arm distance of c/Ω. As a corollaly, the polar cap and any other radiations well within LC cannot be a dominant emission site. I think that most of the rotation power is emitted tangentially from the light cylinder by a mixture of photons and relativistic particles, called the pulsar wind.

Breaking index

Younger pulsars may have worse efficiency (?)

n < 3 ?

Basic Physics II

We are working on higher order approximation!

If there is enough plasma, it corotates with the star. However, because of relativity principle, the corotation is limited within the LC.In 1973, Holloway thought that if some plasma is leaked from the LC , there appears a gap in the corotation region. (Holloway N.J.,1973 Nat. Phys. Sci 246, 6)

Eq. of motion:

Co-rotation electric fieldBC; rigid rotation of the star; star is a good conductor

Drift is corotation; for oblique rotator

The corotation electric field is supported by the “corotation charge density”, GJ density

In number density,

“quasi-neutral plasma?”

This charge density is difficult to realize in the pulsar magnetosphere because of large gravity and because source of plasma is limited. We do not have quasi-neutral plasma in the pulsar magnetosphere as far as electron-positron pair creation, which we shall discuss later, takes place.

Let us suppose somewhat idealized situation where the electric field is the corotaional one (no field-aligned comp.) and therefore the charge density is GJ. The plasma is supported against gravitational force from the star so that the plasma is completely charge separated. Let us add a perturbation that some plasma is leaked from LC by say centrifugal force.

Holloway 1973Deficiency of plasma within the LC manifests itself as a gap round the boundary between the negative and positive GJ density.

strong charge separation causesgaps

Positive GJ

GJ negative

GJ negative

electrons

positrons

star

Light cylinder

For the first time, the particle simulation is applied for this situation by Krause-Polstorf and Michel (1985).

Gap

Field-aligned Electric field

Gap

These pictures are reproduced by our code.

In the gap, GJ density is not filled, and the electric field deviates from the co-rotation filed.

Inertial force, non-EM forces must balance EM force; Particle acceleration

Evacuated space (Gap) is unstable against pair creation. As a result, Holloway gap creates pairs continuously or quasi-periodically. This is the outer gap.

By many authors, the gap structure is calculated but is affected by many factors: e.g., geometry of back ground magnetic field, soft-photon field, radiation process taken into account, boundary conditions for the electric field.

(Hirotani & Shibata 1999, Takata Shibata & Hirotani 2004, Hirotani 2008, Takata Wang & Cheng 2010,,,,)

One can in principle calculate the structure of the outer gap if sufficient conditions are given.

However, we do not think the outer gap structure can be determined as far as it is local model. We always have free parameter, which must be free to adjust with the surroundings, especially the pulsar wind. More specifically the total current and current distribution in the gap is controlled by the pulsar wind. Thus the model must be global , and the outer gap must be solved together with the wind.

Particle simulation.

(Wada & Shibata 2007, Wada & Shibata 2011, Yuki & Shibata 2012)

Method of Particle Simulation

For the case of axisymmetry,

GRAPE-6@nao.jpSpecial purpose computer forAstrononomical N-body Problem

Plasmas are represented by several tens of thousands of super-particles.(number of particles are not fixed, but particles are added suitably from the stellar surface or because of pair creation.)

Cal .Domain is 3D (60RL)^360RL

60RL

60RL

We iteratively solve the equation of mo. and EM field

GRAPE-6@nao.jpSpecial purpose computer forAstrononomical N-body Problem

With this method, the corotation boundary (i.e. emf of the star) is strictly satisfied,

For the magnetic field, we use

In general, surface charge on the star exists, but it is replaced by simulation particles (free emission) until no surface charge is emitted from the stellar surface.

with

Treatment for pair creation:Gap is unstable against e+e- pair creation

A simple treatment

Bγ ＋ B e+ ＋ e-

γ ＋ X-ray e+ ＋ e-Two photon pair creation

Unscreened Gap Electric field

If E > Ec, n=f * int(E/Ec) pairs are created (on the spot approx.)

NS

1. Particles flows

2. (m, q) are chosen so that any plasma drift motions are tracked correctly.

3. Radiation drag force is taken into account in the lowest order approximation.

Curvatureradiation

Results of Particle Simulation

＊ Outer gap is formed with continuous pair creation & E// just aboe Ec

Paris are separated in the gap by E//＊ outgoing flow LC＊ ingoing flow Star

The outer gap is proven by the particle simulation. Ingredients are eq. of mo., Maxwell’s eq., BC rotating NS and pair creatoin if E//>Ec

The outgoing flow rotate with high Lorentz factor, so that curvature radiation carries off the angular momentum. Back reaction of the curvature radiation is in –φ direction cross drift motion.

Rotating plasmas makes toroidal current to open some magnetic flux.Particles on the open magnetic flux flows out along the magnetic field.

In going flow from the outer gap fall on to the star. If this continues, the star is charged up and up. So steady state is reached by emission of pariticles from the polar cap. There must be accelerating field to extract particles from the polar cap.

E// map

E// is just above Ec: necessary minimum for pair creation.

Map of Φ(Non-corotational electric potential)

Dead zone

Curren

t neu

tral d

ead

zone

If Φ=0, the potential is as the same as the star,Co-rotation and E//=0

E//= -B ・∇ Φ

The non-corotaion potential Φ= φ- Ωxr ・Ａ　is constant on the star, it is set to be zero.

Map of Φ(Non-corotational electric potential)

Dead zone

Curren

t neu

tral d

ead

zone

We find another dead zone in the middle latitudes.This dead zone locates on the field lines which separates the outgoing current and ingoing current.

Let us call this dead zone the current neutral dead zone.

Light

cyl

inde

r

Super- rotatoin

Discussion

Discussion The ploidal current loop is such that it start from the star, go through the outer gap, goes to the nebula, and go back to the star in higher latitudes.We have two dead zones:One is the dead zone in the equatorial closed magnetic flux.The other extends along the separatrix of the oppositely directed currents.

Discussion

Φ = 0 in the dead zone.The outer gap is sandwiched by two dead zone. Therefore, previous boundary conditions used for the outer gap is correct.

Discussion The polar current is surrounded by the current neutral zone;Inside of hollow cone of the current neutral zone. If there is slot-gap, it resides above the current neutral dead zone.

Discussion

Some particles leak due to radiation drag drift.It is noticeable that E is larger than B in this region.

Map o f E/B

（ Uzdensky, 2003 ）

Udzdenski の force free 解

light

cyl

inde

r

PIC simulation for an axisymetric Y-point初期の磁力線

（ Uzdensky,

2003 ）

Magnetic reconnection takes place

t ～ 4 x 10^5 ⊿ t

Z / RLC

0.99 1.00

での磁力線

赤道面

0.00

0.01

0.00

0.01

1.01

0.020.02

R / RLC

Z / RLC

⇒ heating and acceleretion

Discussion

Flow out of magnetic islands with pairs.Thin current sheet with outgoing positrons and backward electron beam along last open field line another component

Future work

Number of particles must be still larger

(now Ndot = 0.7 GJElectric field in the polar flow has not been resolved.)

Gamma-ray coming into the screened region also make pairs. These pair must be included.

Oblique rotator (not Grape, PIC)

Realistic pair creation process (include photons)

GR effect near the star.

Thank you!