How the inclination angleaffects the electrodynamics

and statistics of radio pulsars

V.S.BeskinLebedev Physical Institute

• Pulsar MagnetosphereAxisymmetric caseOrthogonal case Inclined rotator

• Statistics of Extinct Neutron StarsEvolution of the inclination angleExtinct radio pulsarsSupersonic propeller

Pulsar Magnetosphere

1. Axisymmetric magnetosphere• Explosion in the pulsar magnetosphere science

(Contopoulos, Kazanas & Fendt, Gruzinov, Uzdensky, Mestel et al, Ogura & Kojima, Harding & Muslimov, Spitkovsky, Timokhin, Komissarov, McKinney, Arons et al, Istomin, Beskin & Nokhrina)

• In all the works an implicit assumption was done that the pair creation mechanism in polar region can support any current which is necessary to have MHD solution up to infinity.

1. Axisymmetric magnetosphere• Explosion in the pulsar magnetosphere science

(Contopoulos, Kazanas & Fendt, Gruzinov, Uzdensky, Mestel et al, Ogura & Kojima, Harding & Muslimov, Spitkovsky, Timokhin, Komissarov, McKinney, Arons et al, Istomin, Beskin & Nokhrina)

• In all the works an implicit assumption was done that the pair creation mechanism in polar region can support any current which is necessary to have MHD solution up to infinity.

Contopoulos, Kazanas & Fendt, ApJ, 1999

I(Ψ) Michel, 1973

3

6420

tot ~c

RΩBW

Contopoulos, Kazanas & Fendt, ApJ, 1999

I(Ψ) Arons

3

6420

tot ~c

RΩBW

Beskin, Gurevich & Istomin, JETP, 1983Beskin & Malyshkin, MNRAS, 1998

ΩΩ ≠F

const || =j

2. Orthogonal rotator

• Exact solution for zero longitudinal electric current

• Energy loss for local GJ current• Position of the light surface• PSR B1931+24• Inclined split monopole

Orthogonal rotator, I = 0

Beskin, Gurevich & Istomin (1983)

Orthogonal rotator, I = 0

Beskin, Gurevich Mestel, Panagi & Istomin (1983) & Shibata (1999)

Orthogonal rotator, I = 0

• For I = 0 the toroidal magnetic field vanishes at the light cylinder.

• It means that for I = 0 there is NO ENERGY LOSS even for inclined rotator.

• All the energy loss is to be connected with the longitudinal current.

Orthogonal rotator - energy lossBeskin, Gurevich & Istomin, JETP,1983,

Beskin & Nokhrina, Astr. Lett, 20042/1

22⎟⎠⎞

⎜⎝⎛−≈−=

cΩR

cΩB

cGJ ππρ ΩB

dS ][[1tot ∫ ××= − BJr SΩcW

Orthogonal rotator - energy lossBeskin, Gurevich & Istomin, JETP,1983,

Beskin & Nokhrina, Astr. Lett, 20042/1

22⎟⎠⎞

⎜⎝⎛−≈−=

cΩR

cΩB

cGJ ππρ ΩB

cj GJ||A ρ/i =

A3

6420

3*

tot i64

⎟⎠⎞

⎜⎝⎛=

cΩR

cRΩBfW

dS ][[1tot ∫ ××= − BJr SΩcW

Orthogonal rotator - energy lossMestel, Panagi & Shibata, MNRAS,1999

cj

RERB (0)GJ

||LL ρ)()( ≈ϕ

1/2

3

6420

tot ⎟⎠⎞

⎜⎝⎛<

cΩR

cRΩBW

2LLL

Rtot )()(

4dS ][

4L

RRBREccW ϕππ≈×= ∫ BE

Orthogonal rotator - energy lossSpitkovsky, ApJ Lett., 2006

( )θsin 141 2

3

6420

tot +≈c

RΩBW

It means that iA >>1

Orthogonal rotator - energy loss

• If iA ~ 1, then Bϕ << Βp on the light cylinder.• It means that the lightsurface |E| = |B| is to Locate in the very vicinity of the light cylinder.

Inclined rotator – PSR B1931+24• Switch-off – magnetodipole loss only

• Switch-on – current loss only (f* = 1.592 –1.96)

• The ratio

χsin61 2

3

6420

tot cRΩBW =

χcosi8 03

6420

2*

tot cRΩBfW =

5.1χsinχcosi

43

20

2*

off

on

off

on ≈==f

PP

WW

Inclined split monopole (Bogovalov, A&A,1999)

For arbitrary inclinationangle χ

along the axis, so that

Hence, incline split monopolecannot describe real dipolefield of radio pulsars.

cj (0)GJ|| ρ =

)()( LL RERB ≈ϕ

Statistics of Extinct Neutron Stars

Energy loss for χ< π/2Magnetodipole loss

3

6420

tot 61

cRΩBIW =ΩΩ=

t

P

P0

P/Pdot

t1/2

Energy loss for χ< π/2Magnetodipole loss

χsin61 2

3

6420

tot cRΩBIW =ΩΩ=

const χcos =Ω

t

P

P0

P/Pdot

t1/2

P/cosχ0

Energy loss for χ< π/2Magnetodipole loss

Current loss

χcosi8 03

6420

2*

tot cRΩBfW =

χsin61 2

3

6420

tot cRΩBW =

const χcos =Ω

const χsin =Ω

t

P

P0

P/Pdot

t1/2

P/sinχ0

• Time scale of the inclination angle evolution _______is the same as .

• Energy loss decreases mainly by the evolution of the inclination angle.

• Death line depends on the inclination angle.

χχ/τχ = PP/τD =

Death line

Death line depends on the inclination angle

Death line

Tauris & Manchester, MNRAS, 298, 625, 1998

Gould Rankin

χ χ

P P

New point• Extinct radio pulsars – supersonic propeller

transition line depends on the inclination angle as well.

• E.g., for magnetodipole energy loss

χsin 21

Epr PP =

( )sP

Bvcμ 1002

1

6ext

21

72

17

21

30E

−

≈(Lipunov et al, 1996)

Extinct radio pulsars (RS)

Radio pulsars Supersonic Propeller

Beskin & Eliseeva, 2005

Extinct radio pulsars

s

Extinct radio pulsars (Arons)

Radio pulsars Supersonic Propeller

Beskin & Eliseeva, 2005

Extinct radio pulsars

s

Supersonic propeller (RS)

21

cPR2πcosχ ⎟

⎠⎞

⎜⎝⎛=

Supersonic propeller (Arons)

Бескин, Елисеева, 2005b

( ) ( )

0.93.1

1.7

12

P

0.185P 1427

3.7

12

PA69

P105

2.74

12

2

fNE

5

3

Py0.1B1

dPP

y

B1BPNkP101.2PN

0.8E

25-2

674

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞⎜

⎝⎛+

∫×

×+∫⋅= −⋅

⋅

−−

−

We predict for supersonic propellers

• P ~ 6−12 s• Pdot ~ 10−13

• N ~ 10−4 Νtot• Aligned or orthogonal

Radio transient sources?

ConclusionDo not remember that radio pulsarsare nonaxisymmetric objects.

For discussion• Can the electric current be much

larger/smaller that local GJ one?• Whether the last observations of B1931+24

mean the hard evidence against the free particle escape from the NS surface.

• Is it possible to determine the evolution of the inclination angle?