Theory of pulsar`s electrosphere

Transcription

Theory of pulsar’s electrosphere
10 mai 2005 - Cargèse
Jérôme Pétri
[email protected]
Max Planck Institut für Kernphysik - Heidelberg
Cargèse, 9-13/5/2005 – p. 1/??
Outline
1. What is a pulsar ?
Observations
Theory
2. Structure of the magnetosphere
Polar cap
Outer gap
3. Electrosphere of a pulsar
Equilibrium state
Stability analysis
Pair creation
4. On the pulsar wind
5. Conclusions & Perspectives
Cargèse, 9-13/5/2005 – p. 2/??
Discovery
First pulsar discovered fortuitously at Cambridge Observatory in 1967:
Radio signal measured from PSR1919+21
(Bell & Hewish, 1968)
* signal made of a series of pulses separated by a period P = 1.337 s ;
* pulse profile changes randomly but arrival time stable in time ;
* duration of a pulse ∆t ≈ 16 ms
⇒ size of the emitting region: L ≤ c ∆t ≈ 4800 km
⇒ compact object.
Pulsar = strongly magnetised rotating neutron star.
Cargèse, 9-13/5/2005 – p. 3/??
Observations: general aspects
to date about 2000 pulsars are known ;
identified as galactic objects concentrated in the equatorial plane of the Milky Way ;
rotation period P between 1.5 ms and a few seconds (8 s) ;
pulse arrival time extremely stable but increases slowly (Ṗ > 0)
⇒ interpreted as a rotational braking of the neutron star.
P − Ṗ diagram for 539 pulsars
-12
Log10 P’
-14
-16
-18
-20
0.001
0.01
0.1
Période P Hen sL
1
10
Cargèse, 9-13/5/2005 – p. 4/??
Observations: radio emission
Mean profile of PSR 1133+16
and a sample of 100 individual pulses
the structure of the pulse change
randomly but the mean profile remains
extremely stable ;
the spectral flux density for radio pulsars
decreases like a power law
⇒ non thermal emission ;
the radio luminosity is three to five orders
of magnitude lower than the total rotational
energy losses (mostly released in particle acceleration forming a wind illuminating the supernova remnant by synchrotron
radiation) ;
the brightness temperature for the radio emission is T=1023−25 K
⇒ coherent emission mechanism ;
* for incoherent emission : intensity ∝ N ;
* for coherent emission when size of emitting region L ≪ λ : intensity ∝ N 2 .
Radio emission = non thermal coherent radiation mechanism.
Cargèse, 9-13/5/2005 – p. 5/??
Variety of mean profiles
Classification due to Rankin (1983)
120
PSR1642-03 à 1.42 Ghz
100
PSR0809+ 74 à 4.85 Ghz
1
St
80
0.8
60
0.6
40
0.4
20
0.2
22.5 25 27.5 30 32.5 35 37.5 40
Phase du pulse Hen degréL
PSR0329+ 54 à 2.25 Ghz
300
100
80
100 120 140 160 180 200 220
PSR525+ 21 à 1.61 Ghz
120
250
D
200
60
150
40
100
20
50
25 30 35 40 45 50 55 60
120
PSR1237+ 25 à 1.4 Ghz
100
80
Sd
T
230 240 250 260 270 280
120
PSR0355+ 54 à 1.4 Ghz
100
M
80
60
60
40
40
20
20
45
55
40
50
60
65
T12
80 90 100 110 120 130 140
Cargèse, 9-13/5/2005 – p. 6/??
Some useful definitions
An empty magnetosphere
Ω : rotation axis of the pulsar ;
µ : magnetic moment ;
α : angle between the magnetic moment
and the rotation axis ;
the light cylinder : surface where the
speed of a particle corotating with the star
at a rate Ω∗ reaches the speed of light c ;
⇒ radius of the light cylinder RL = c/Ω∗ .
energy losses by dipolar magnetic radiation
Cargèse, 9-13/5/2005 – p. 7/??
Typical neutron star parameters
mass M∗ = 1.4 M⊙ ;
radius R∗ = 10 km (R⊙ = 700.000 km) ;
mean density ρ∗ = 1017 kg/m3 (ρ⊙ = 1.410 kg/m3 ) ;
crust temperature T∗ = 106 K ;
moment of inertia I∗ = 1038 kg m2 ;
magnetic field strength at the stellar surface B∗ = 105−8 T (B⊙ = 5 × 10−3 T).
Magnetic field intensity at the stellar crust estimated from dipolar magnetic radiation in
vacuum, assuming a dipolar magnetic field :
p
B ∝ P Ṗ
For typical values of P and Ṗ :
P = 1 s and Ṗ = 10−15 implying B∗ = 108 T for s pulsars ;
P = 1 ms and Ṗ = 10−18 implying B∗ = 105 T for ms pulsars.
Cargèse, 9-13/5/2005 – p. 8/??
Phenomenological model
The hollow cone model (Radhakrishnan & Cooke, 1969)
What does the magnetospheric structure look like to explain this model ?
Cargèse, 9-13/5/2005 – p. 9/??
Polar cap model
The magnetosphere
(Goldreich-Julian, 1969)
Assumptions :
~∗ k µ
an aligned rotator (Ω
~) ;
a closed magnetosphere entirely filled
with the corotating plasma up to the light
cylinder ;
in electrostatic equilibrium :
~ + ~v ∧ B
~ =0;
E
the Goldreich-Julian charge density at
equilibrium ;
the null surface: region where the charge
~ ·B
~ = 0) ;
density vanishes (Ω
Extracting charges from the
stellar surface
⇒ non vacuum solution
particles follow a electric drift motion in the
~ ∧B
~ direction ;
E
Cargèse, 9-13/5/2005 – p. 10/??
Charged wind
Pair creation cascades
(Sturrock 1970, Ruderman &
Sutherland 1975)
the corotation impossible outside the light
cylinder RL ;
a charged wind emanating from the polar
caps
the charged particles (e+ e− ) are
produced by γ + B → e+ + e− in the
polar caps.
the open field lines sustain a wind made of
particles of both signs ⇒ increase or decrease of the total charge of the system
(star+magnetosphere) ;
⇒ problem of the current closure.
Cargèse, 9-13/5/2005 – p. 11/??
Solution for the axisymmetric rotator
(Contopoulos et al.,1999)
Closure of the electric circuit :
by following the last open magnetic field
line : current sheet ;
by violation of the electric drift approximation. Particles follow the surface of the light
cylinder (Beskin, Mestel).
Cargèse, 9-13/5/2005 – p. 12/??
Outer gap model
Aim: to explain the high energy component of the pulsar’s spectrum (gamma emission)
(Cheng, Ho & Ruderman 1986)
Assumptions :
the outer gaps are located between the
light cylinder and the null surface ;
the photon disintegration is impossible
because B too weak ;
the pair cascade initiated by
photon-photon interaction in the outer
gaps, γ + γ → e+ + e− ;
the curvature photons emitted tangentially
to the local magnetic field lines.
An alternative model:
the two pole caustic = slot gap from the light cylinder down to the stellar surface (Dyks &
Rudak 2003).
Cargèse, 9-13/5/2005 – p. 13/??
Summary of the “standard” model
Cargèse, 9-13/5/2005 – p. 14/??
Collaboration with Silvano and Jean
Electrosphere with gaps :
what is a gap ?
Construction of an electrosphere
finite in extend and in electrostatic equilibrium
Charge transport and stability
diffusion of particles by turbulence
Stability study :
by a linear analysis ;
by non linear numerical simulations ;
by quasilinear modeling.
Associated emission
pair cascade in the gaps
Cascades pairs in the gaps
Cargèse, 9-13/5/2005 – p. 15/??
Electrospheric model
Aim
Construction of a self-consistent electrosphere for a charged and aligned pulsar.
Assumptions :
the neutron star = perfect spherical conductor of radius R∗ , generating a dipolar
magnetic field of strength B∗ and in solid body rotation with speed Ω∗ ;
an aligned rotator, i.e., magnetic moment and spin axis are parallel ;
particles located well within the light cylinder ;
charges extracted freely from the stellar crust whatever their nature ;
magnetic field induced by the electrospheric currents are neglected ⇒ constant and
dipolar ;
any force other that electromagnetic are neglected (even the gravitational attraction !!) ;
the electric drift approximation ;
the spacetime curvature neglected (frame dragging effect).
Cargèse, 9-13/5/2005 – p. 16/??
Iterative scheme
Qualitative picture
Algorithm :
1. Transfer a fraction p of the stellar surface
charge ;
2. Evaluate the total electric potential φ
(star+electrosphere)
3. Find the differential rotation speed Ω din
the non corotating region ;
4. Deduce the associated differential charge
density ρ6= ;
5. Find the new volume occupied by the
plasma ;
6. Return to step 2 until self-consistency is
achieved between φ, Ω et ρ6= ;
Total charge Qtot
= only parameter of the model
7. Return to step 1 until total vanishing of the
stellar charge.
Cargèse, 9-13/5/2005 – p. 17/??
Results: electrospheric structure
(Pétri, Heyvaerts & Bonazzola, A&A 2002a)
Structure 3D de l’électrosphère
3
Vitesse de rotation
3
2.5
2
WHrL
Axe polaire
2.5
1.5
2
1.5
1
1
0.5
1.5
0.5
1
1.5
2 2.5 3
Plan équatorial
3.5
2
4
Charge stellaire
2.5
Rayon r
3
3.5
4
3.5
4
Densité de charge
1.2
0.8
1
0.6
0.8
rHrL
ss HqL
1
0.6
0.4
0.4
0.2
0.2
0
20
40
60
Colatitude q en degré
80
1.5
2
2.5
Rayon r
3
Cargèse, 9-13/5/2005 – p. 18/??
Results: electrospheric structure
Main features :
electrosphere of both sign ;
is finite in extent for Qtot ≤ 3 Qc . Moreover, if Qtot ≪ 3 Qc , plasma is confined well
inside the light cylinder ;
has large gaps appear between the equatorial belt and the polar domes ;
there is no electric current circulation in the gaps ;
a differential rotation of the disk, overrotation
⇒ induce a shearing between magnetic surfaces responsible for the growing of an
instability ;
the same qualitative conclusions whatever apply the total charge Qtot .
The electromagnetic field acts as a Penning trap, confinement of the non-neutral plasma:
in the radial direction by the magnetic field :
in the “axial” direction by the electric field.
Cargèse, 9-13/5/2005 – p. 19/??
Results: electrosphere with charge loss
4
3.5
Vitesse de rotation
3
3
WHrL
Axe polaire
2.5
2.5
2
2
1.5
1.5
1
1
1.5
2
0.5
0.5
1
1.5
2
2.5
3
Plan équatorial
3.5
2.5
Rayon r
3
3.5
4
3.5
4
4
Charge de surface
Densité de charge
1.2
0.8
1
0.6
0.8
rHrL
ss HqL
1
0.6
0.4
0.4
0.2
0.2
0
20
40
60
Colatitude en degré
80
1.5
2
2.5
Rayon r
3
Cargèse, 9-13/5/2005 – p. 20/??
Results: electrosphere with charge loss
Main features :
the global geometric and cinematic structure unchanged ;
the differential rotation of the disk is preserved ;
there is no static equilibrium state due to the stellar charge loss ;
only negative charges are expulsed from the polar caps
⇒ increase of the total charge of the system ;
it prevents an ongoing escape of other particles when the system has reached a
sufficiently high charge.
Cargèse, 9-13/5/2005 – p. 21/??
Diocotron instability: assumptions
Goal
Study the electrodynamic stability of the differentially rotation disk.
Simplifications :
the thin disk approximation, differential rotation ;
we lift the domes, replaced by a charge density on the stellar crust ;
equilibrium rotation rate electrodynamically self-consistent ;
the dipolar magnetic field is produced by the neutron star ;
motion in the plasma only allowed in the equatorial plane, non warping ;
cold plasma, charge separated, and non relativistic ;
no relevant effect due to the current in the plasma on the external magnetic field ⇒
perturbation essentially electrostatic ;
in the electric drift regime, (RLarmor ≪ RL ) ;
the diocotron instability ≈ Kelvin-Helmholtz for a non neutral plasma.
Cargèse, 9-13/5/2005 – p. 22/??
Diocotron instability: linear analysis
Equations governing the motion in the disk :
8
∂σ
>
div
(σ
~
v
)
+
>
>
>
>
σ ∂t
>
>
< ∆Φ +
δ(z)
ε0
>
~
>
E
>
>
>
>
>
:
~v
=
0
=
0
=
~ Φ
−∇
~ ∧B
~
E
B2
=
(1)
Fourier analysis :
χ(r, ϕ, t) = χ̃(r) ei (m ϕ−ω t)
(2)
Fredholm integral equation for the Fourier components of the perturbation σ̃1m :
ωm σ̃1m (r) = m Ω0 (r) σ̃1m (r) +
Z
r2
Km (r|r′ ) σ̃1m (r′ ) dr′
(3)
r1
It is eigenvalue problem with the unknown eigenfunctions σ̃1m and eigenvalues ωm .
Cargèse, 9-13/5/2005 – p. 23/??
Linear analysis: results
Main features :
for a given charge, the growth rate
increases from m = 2 to a maximum at
m ≈ 18 and then decreases slowly
when m increases ;
Eigenvalues for the diocotron
instability
(Pétri, Heyvaerts & Bonazzola,
A&A 2002b)
Carte des valeurs propres
7
Charge totale
18
16
1420
6
12
0
Q = 0.5 Qc
10
5
9
8
7
g=ImHwL
4
6
3
Q = Qc
25
30
Q = 1.5 Qc
35
5
Q = 2 Qc
4
2
3
instability disappear for large values of m
(>40), γm = 0 (collective behavior lost) ;
for a given value of the azimuthal
mode m, the growth rate decreases for an
increasing total charge Qtot of the
system ;
m=2
1
1
1.5
Re HwL
Wp =

m
2
2.5
the growth rate is of the same order of
magnitude than the pattern speed of the
perturbation ⇒ instability develops on a
very short timescale, much less than any
other instability.
Need to study the saturation of the instability due to non linear effects in order to obtain
correct picture of the development on a long timescale ⇒ numerical simulations.
Cargèse, 9-13/5/2005 – p. 24/??
Diocotron instability: non linear evolution
Goal
Determine the regime reached due to the diocotron instability
two problems :
isolated disk ;
disk feeded with charges by pair creation.
Solving the equations by means of numerical simulations:
the continuity equation in cylindrical coordinates with a source term s :
∂σ
1 ∂
1 ∂
+
(r σ vr ) +
(σ vϕ ) = s
∂t
r ∂r
r ∂ϕ
(4)
Integration done by a fluid dynamics code ClawPACK available on the net.
the Poisson equation with help on Green functions G :
ΦD (~
r) =
ZZ
G(~
r|~
r ′ ) σD (~
r ′ )dS
(5)
Disk
Cargèse, 9-13/5/2005 – p. 25/??
Non linear evolution: results (1)
Disk without any source Bidimensional structure
Cargèse, 9-13/5/2005 – p. 26/??
Diocotron instability: quasilinear model
Model applies only when a few modes m are excited and incoherent.
Description of the mean density profile < σ > (t) (1D problem).
Idea
Decompose all physical quantities ψ(r, ϕ, t) in a mean value < ψ > (r, t) and a
fluctuation δψ(r, ϕ, t) around this average as follows :
ψ(r, ϕ, t) =< ψ > (r, t) + δψ(r, ϕ, t)
non linearities kept in the evolution of < σ > ;
non linearities ignored for the perturbations.
Cargèse, 9-13/5/2005 – p. 27/??
Quasilinear model: results
Disk with an external source of charges
(Pétri, Heyvaerts & Bonazzola, A&A 2003)
Vitesse de rotation moyenne
Densité moyenne
10
4
0
<s>
<W>
3
-10
2
-20
1
-30
0
4
2
6
8
10
4
2
6
r
Courant radial moyen
10
coefficient de diffusion
40
0
20
-2.5
log10 HDL
<i>HrL
8
r
0
-20
-5
-7.5
-10
-12.5
-40
-15
4
2
6
8
10
4
2
r
6
8
10
r
Taux de croissance
Charge du disque
0.2
100
0.175
Qd Hen Qc L
0.15
gm HtL
0.125
0.1
0.075
0.05
80
60
40
20
0.025
20
40
60
Temps t
80
100
0
20
40
60
Temps t
80
100
Cargèse, 9-13/5/2005 – p. 28/??
Conclusions of the simulations
Non linear simulations:
without charge feeding, particles tend to accumulate to form supercharges of high
density and rotating at a high speed around the neutron star ;
with an external source of charges, the initial extension of the disk grows slowly in
time ;
there is no significant current sustained at this stage.
⇒ simulations on a longer timescale are necessary
⇒ quasi linear model.
Quasi linear model:
model in agreement with the full non linear evolution ;
there exists a situation for which the diocotron instability generates a net flux of
charges radially outwards ;
A positive equatorial current exists
It is an alternative solution to the current closure problem.
Cargèse, 9-13/5/2005 – p. 29/??
Cylindre lumière
e−
Vent polaire négatif
Ligne polaire
Current circuit within the closed magnetosphere
Dôme
e−
Zone de formation
de paires e+ e−
e+
Étoile
Disque
Plan équatorial
e+
Courant équatorial positif
RL
Observational signature resulting from the pair creation activity in the gaps
⇒ high energy component of the spectrum.
Cargèse, 9-13/5/2005 – p. 30/??
Pair creation e+ e− (1)
a quantum electrodynamics process in a strong magnetic field of the order of critical
magnetic field Bc ≈ 4.4 · 109 T : γ + B → e+ + e−
R⊥
L
k′
k
ξ
~
B
~
B
x
y
In the frame R⊥ , the threshhold energy is given by k′ = k sin ξ ≥ 2 me c2 .
(Eγ = k me c2 )
for a given photon energy, the efficiency increases with the magnetic field intensity
⇒ occurs solely in the innermost regions of the electrosphere ;
the photons are produced by :
* curvature and synchrotron radiation
⇒ constraint on the maximum energy reached by the charges, γ < 107 ;
* inverse Compton diffusion ;
Cargèse, 9-13/5/2005 – p. 31/??
Optical depth for γ
+ B → e+ + e−
For the s pulsars
B=108 T
1. ´ 10
Profondeur optique t
12
Energie Eg
1. ´ 108
100 MeV
1 GeV
10000
10 GeV
100 GeV
1
1 TeV
10 TeV
0.0001
100 TeV
1 PeV
1
5
10
50 100
Position d’émission R0 Hen R*L
500
For the ms pulsars
B=105 T
1. ´ 107
Energie Eg
10000
100 GeV
1 TeV
10
10 TeV
0.01
100 TeV
0.00001
1 PeV
1
1.5
2
3
5
Position d’émission R0 Hen R*L
Cargèse, 9-13/5/2005 – p. 32/??
Pair creation e+ e− (2)
photon-photon interaction via the process :
γ + γ → e+ + e−
interaction between a gamma photon emanating from the outer gaps and a thermal
photon created by the black body radiation from the hot stellar crust ;
Optical depth for the process γ + γ → e+ + e−
10000
Interaction g-g
Température T
100
106 K
1
107 K
0.01
108 K
0.0001
109 K
1. ´ 10-6
100
10000 1. ´ 106 1. ´ 108
Enérgie Eg du photon g Hen me c2 L
Cargèse, 9-13/5/2005 – p. 33/??
Consequences on the electrospheric structure
Electrosphere with a charge source
WHrL
Axe polaire
4
3
Vitesse de rotation
2.5
2.25
2
1.75
1.5
1.25
1
5
2
1
2
4
6
8
Plan équatorial
10
2.5 5 7.5 10 12.5 15 17.5 20
Rayon r
Charge de surface
ss HqL
SrcHr,WHrLL
1
0.8
Fonction source Src
0.6
0.4
0.2
0
20
40
60
80
Colatitude q en degrés
0.175
0.15
0.125
0.1
0.075
0.05
0.025
0
2.5 5 7.5 10 12.5 15 17.5 20
Rayon r
Cargèse, 9-13/5/2005 – p. 34/??
Cylindre lumière
e−
Vent polaire négatif
Ligne polaire
Conclusion
Dôme
e−
Zone de formation
de paires e+ e−
e+
Étoile
Disque
Plan équatorial
e+
Courant équatorial positif
RL
in the aligned rotator, electrosphere do exist, finite in extension and in electrostatic
equilibrium ;
a instability of electrostatic nature (diocotron) develops in the equatorial belt allowing
particles to diffuse across the magnetic field lines ;
numerical simulations have shown the formation of an equatorial current carrying a net
flux of charges towards the light cylinder ;
a pair cascade can only take place in the innermost regions of the magnetosphere,
very close to the neutron star surface. It is a source feeding the disk with charged
particles.
the striped wind model in very good agreement with recent optical polarisation
properties of the Crab pulsar
Cargèse, 9-13/5/2005 – p. 35/??
Perspectives
study in detail the pair creation process with Monte Carlo simulations ;
close the electric circuit via a negatively charged polar wind ;
generalise the diocotron instability by taking into account the perturbation in the
magnetic filed
⇒ magnetron instability ;
study the oblique rotator ;
theory of charged wind with gaps : quantitative theory of Cheng, Ho et Ruderman.
Cargèse, 9-13/5/2005 – p. 36/??
Physics of pulsars
Apart from the obvious need of :
the general relativity (neutron star equation of state, gravitational waves,
electromagnetic field enhancement by frame dragging effect) ;
the quantum electrodynamics (pair creation),
the physics of pulsar needs two essential ingredients :
1. non neutral plasma physics ;
trapping of particles in Penning like traps ;
stability properties of this equilibrium ;
diocotron and magnetron instability.
2. plasma distribution does not overlap with the magnetosphere
⇒ large vacuum gaps and thus “electrospheric solution” and NOT magnetospheric
one.
1967-2005 = 38 years with little progress !!
Cargèse, 9-13/5/2005 – p. 37/??