spindown m2 o4 K128 N8 ro100 c90 - Unical · 1 A brief overview 2 Vacuum electrodynamics 3 Plasma...

Pulsar magnetospheres and winds

A challenge for plasma physicists and astrophysicists

Jérôme Pétri

Observatoire astronomique de Strasbourg, Université de Strasbourg, France.

Copanello

Jérôme Pétri (Observatoire astronomique) Pulsar magnetospheres 1 / 19 VLASOVIA, May 30 - June 2 1 / 19

Summary

1 A brief overview

2 Vacuum electrodynamics

3 Plasma magnetosphere

4 Conclusions


1 A brief overview



4 Conclusions


What is a pulsar ?

1 neutron starcompact object ⇒ strong gravity effects

ξ ≡G M

R c2≈ 0.35

2 strongly magnetized⇒ plasmas, QED effects (pair creation)

Bq ≡m2 c2

e ~≈ 4.4 × 109 T

3 rotating⇒ huge electric fields

Eschw ≡ c Bq ≈ 1.3 × 1018V/m


What is a pulsar ?

1 neutron starcompact object ⇒ strong gravity effects

ξ ≡G M

R c2≈ 0.35

2 strongly magnetized⇒ plasmas, QED effects (pair creation)

Bq ≡m2 c2

e ~≈ 4.4 × 109 T

3 rotating⇒ huge electric fields

Eschw ≡ c Bq ≈ 1.3 × 1018V/m

Some useful definitions

obliquity χ : angle between magnetic moment ~µ and rotation axis ~Ω

aligned/perpendicular/oblique rotator : χ = 0/90o/any value

light cylinder radius : surface on which a particle corotating with the neutron star reaches thespeed of light c : rL = c/Ω⇒ transition from quasi-static to wave zone (⇒ very different plasma regimes)


Neutron star magnetospheres : orders of magnitude

period P ∈ [1 ms, 1 s]

period derivative P ∈ [10−18, 10−15]

⇒ spin-down losses well constrained

Lsp = 4π2I P P

−3 ≈ 1024−31

W

very different from black holes or accreting

neutron stars

inferred magnetic field estimate by dipole

radiation

B = 3.2 × 1015√

P P = 105−8

T

⇒ consistent with magnetic flux

conservation during gravitational collapse

but no constrain on the geometry

(obliquity χ)

probably not a good guess if multipoles

present.FIGURE : P − P diagramm.


Pulsar magnetosphere : orders of magnitude

Electromagnetic and gravitational field characteristics

electric field induced at the stellar crust

E = ΩB R = 1013

V/m

⇒ instantaneous acceleration at ultra-relativistic speeds, Lorentz factor γ ≫ 1

(τacc < 10−20 s)

negligible gravitational force for protons ! ! !

Fgrav

Fem

≈ G M mp/R2

eΩB R≈ 10

−12 ≪ 1

even smaller for electrons/positrons (me/mp).

⇒ dynamic of the magnetosphere dominated by the electromagnetic field

Neutron star characteristics

masse M ≈ 1.4 M⊙.

radius R ≈ 10 km.

centrale density ρc ≈ 1017 kg/m3.


Pulsar magnetosphere : the challenges

Quantity Estimation Second Millisecond

Rotation frequency (Hz) ν∗ = 1P

1 1.000

Luminosity (W) L = 4π2 I P P−3 6.3 × 1024 6.3 × 1030

Magnetic field (T) B =√

3 µ0 c3

32 π3

√I P PR3 7.4 × 107 7.4 × 104

Electric field (V/m) E = ΩB R 7.5 × 1012 7.5 × 1012

Gavitational/electric force G M me

R2 e E9.7 × 10−12 9.7 × 10−12

Light cylinder radius (km) rL = cΩ

47 700 47.7

Particle number density at R (m−3) n = 2 ε0Ω B

e6.9 × 1016 6.9 × 1016

Particle number density at rL (m−3) 1.1 × 106 1.1 × 1015

Particle flux (s−1) F =4 π ε0

eΩ2 B R3 7.5 × 1029 7.5 × 1032

Plasma frequency at R (Hz) νp = 12 π

√

n e2

ε0 me2.3 × 109 2.3 × 109

Plasma frequency at rL (Hz) 9.4 × 103 2.9 × 108

Cyclotron frequency at R (Hz) νB = e B2 π me

2.8 × 1018 2.8 × 1015

Cyclotron frequency at rL (Hz) 4.5 × 107 4.5 × 1013

Characteristic age (years) τ = P

2 P1.6 × 107 1.6 × 107

Gravitational potential energy (J) Eg = 35

G M2

R2.6 × 1046 2.6 × 1046

Rotational kinetic energy (J) Ek = 12

I Ω2 3.2 × 1039 3.2 × 1045

Magnetic energy (J) EB = 4 π3

B2 R3

2 µ01.62 × 1034 1.62 × 1028

Thermal energy (J) Eth = 32

N k T 3.4 × 1040 3.4 × 1040

TABLE : The fundamental parameters of a normal and a millisecond pulsar.


1 A brief overview



4 Conclusions


Exact solutions

exact analytical solution for a rotating dipole in vacuum (Deutsch, 1955)

spindown power due to magnetodipole losses. For a oblique rotator

Lvac⊥ =

8π B2 Ω4 R6

3µ0 c3sin

2 χ

torque exerted on the surface by charges and currents

(Michel & Goldwire, 1970; Davis & Goldstein, 1970)

⇒ secular evolution of the inclination angle

two singular open field lines leading to a two armed archimedean spiral

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

y/r L

x/rL

Deutsch orthogonal dipole

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

y/r L

x/rL

Deutsch orthogonal dipoley/r L

x/rL

y/r L

x/rL

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0.01 0.02 0.05 0.1 0.2 0.5

Lu

min

osity

(L/L

vac

dip

)

Spin rate (R/rL)

Vacuum orthogonal dipole

n=3

Newtonian

GR

exact analytical solutions for multipoles also exist

(Bonazzola et al., 2015; Pétri, 2015)

⇒ useful to enhance the pair production rate at the polar caps

(Harding & Muslimov, 2011)


1 A brief overview



4 Conclusions


The role of the plasma

plasma required observationally ⇒ broadband radiation detected on Earth.

particles needed to furnish charges and currents in the magnetosphere.

Analytical study intractable, recent progress via numerical simulations of which most

extensively studied

force-free electrodynamics (FFE or magnetodynamics) : zero mass limit. No

energy dissipation.

resistive magnetodynamics : transfer of energy from field to particles. Prescription

not unique. Plasma motion not solved.

magnetohydrodynamics (MHD) : particle inertia taken into account and the full

stress-energy tensor, matter and field, is solved. Ideal and resistive MHD regimes.

multi-fluids : evolve each species independently, coupling through electromagnetic

interactions.

fully kinetic treatment : individual particle acceleration that are out of thermal

equilibrium. Needs to solve the full Vlasov-Maxwell equations.

radiation reaction limit : acceleration compensated by radiation reaction. Particle

motion solved analytically in terms of the external electromagnetic field.


The “standard model” of an ideal pulsar

The full system to solve :

∇µ(Tµνem + T

µνmat ) = 0

∇µ∗F

µν= 0

∇µ(ρm uµ) = 0

Fµν

uν = 0

Some simplification : force-free magnetosphere (Fµν Jν = 0)

ρe~E +~j ∧ ~B = ~0

magnetic energy density B2

2 µ0≫ any other energy densities.

particle inertia neglected : zero mass limit.

no dissipation : ideal MHD

~E + ~v ∧ ~B = ~0

no pressure : cold plasma.


Force-free magnetospheres

Simplest approach to pulsar electrodynamics

ideal MHD without particle inertia and without radiation

Maxwell equations

∇ · B = 0

∇× E = − 1√γ∂t(

√γ B)

∇ · D = ρe

∇× H = J +1√γ∂t(

√γ D)

FFE current prescription (constraints E · B = 0 and E < c B)

J = ρe

E ∧ B

B2+

B · ∇ × B/µ0 − ε0 E · ∇ × E

B2B

ρe = ε0 ∇ · E

No fluid quantity enters into the system to be solved. (Spitkovsky, 2006; Komissarov,

2006; McKinney, 2006; Pétri, 2012; Paschalidis & Shapiro, 2013; Cao et al., 2016)


Force-free magnetospheres

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

x

Equatorial magnetic field lines for the orthogonal rotator

FIGURE : Magnetic field of the perpendicular rotator χ = 90o .

0.5

1

1.5

2

2.5

0 15 30 45 60 75 90

Lum

inosity

(L/L

FF

Edip

)

Obliquity (χ)

FFE oblique dipole

N 0.1N 0.2N 0.5

FIGURE : Spin-down luminosity vs χ from simulations in red and fit inblue.

Plasma filled magnetosphere spindown

LFFEsp ≈ 3

2L

vac⊥ (1 + sin

2 χ)

to be compared with vacuum

Lvacsp ≈ L

vac⊥ sin

2 χ

(Spitkovsky, 2006; Pétri, 2012)


MHD magnetospheres

Includes particle inertia but not particle acceleration

FIGURE : Perpendicular rotator χ = 90o

FIGURE : Spin-down luminosity vs obliquity χ.

(Tchekhovskoy et al., 2013)


PIC magnetospheres

Includes particle inertia AND particle acceleration self-consistently

Equation of motion for a particle (Lorentz force)

mduα

dτ= q F

αµuµ

FIGURE : Plasma properties

FIGURE : Particle and Poynting energy flux

(Belyaev, 2015)


PIC magnetospheres with radiation

Equation of motion for a particle (Lorentz force + radiation reaction)

mduα

dτ= q F

αµuµ + g

α

FIGURE : Lorentz factor and characteristics synchrophotonfrequency.

FIGURE : Electron and positron trajectories.

(Cerutti et al., 2016)Jérôme Pétri (Observatoire astronomique) Pulsar magnetospheres 14 / 19 VLASOVIA, May 30 - June 2 14 / 19

Radiation reaction limit

Radiation back reacts on particle motion, a friction appears and at equilibrium

q (E + v ∧ B) = K v

where K represents the intensity of emission.

For ultra relativistic motion, analytical solutions exist

K2 ≈ q2

2 c2

[

E2 − c

2B

2 ±√

(E2 − c2 B2)2 + 4 c2 (E · B)2

]

The equation of motion becomes an algebraic equation for v depending solely on the

electromagnetic field E,B

(K 2 + q2

B2) v = q

2E ∧ B + q K E + q

3 E · B

KB

FIGURE : With (right) and without (left) radiation reaction (Contopoulos, 2016)


Do we need general relativity ?

-2

-1

0

1

2

-2 -1 0 1 2

y/r L

x/rL


Newtonian

GR

FIGURE : Magnetic field lines in the equatorial planefor R/rL/R = 0.2.

1.41.61.8

22.22.42.62.8

33.23.43.6

0.01 0.02 0.05 0.1 0.2 0.5

Lu

min

osity

(L/L

FF

Edip

)

Spin rate (R/rL)

FFE orthogonal dipole

n=3

Newtonian

GR

FIGURE : Spindown luminosity.

(Pétri, 2016)

YES we need GR for a good quantitative analysis of energy budget an electromagnetic

field topology.


Do we need QED on a global scale ?

On small scales obviously yes for pair

creation.

The critical magnetic field is

Bqed =m2

e c2

e ~≈ 4.4 × 10

9T

Maxwell equations become non-linear for

B & Bqed.

⇒ Corrections to lowest order by

expansion of Euler-Heisenberg Lagrangian

post-Mawxellian parameters like

post-Newtonian gravity

⇒ Quantum vacuum equivalent to a

medium : D(E,B),H(E,B)

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0.01 0.02 0.05 0.1 0.2 0.5

Spin

dow

n(L/L

vac

dip

)

R/rL


(N, -3)

(GR, -3)

(N, 0)

(GR, 0)

FIGURE : Spindown luminosity for different rotation rates,magnetic field strengths given by log(B/Bqed) and gravitationalfield (Newtonian or GR).

(Pétri, submitted)

QED does not influence neutron star global vacuum electrodynamics.

Same conclusions for FFE magnetospheres.


Towards general agreements about dense pulsar magnetospheres

filled with electron/positron pairs

almost everywhere

FFE approximation satisfactory on a

global scale

formation of an equatorial current

sheet

efficient particle acceleration and

emission in this sheet

Y-point of great importance for the

dynamics/spindown losses

EY ≈(

rL

RY

)2

EL > EL

maybe solution for braking index ?

breakdown of ideal MHD/FFE in some

small regions

magnetic reconnection invoked in

these regions and in the sheet

(Contopoulos, 2016)

(Timokhin, 2006)


1 A brief overview



4 Conclusions


Conclusions

o Pulsar magnetosphere and wind

global structure well constrained

global FFE picture satisfactory

good agreement between FFE/MHD and PIC simulations

magnetosphere naturally linked to its striped wind

! Caveats

some dissipation regions required for emission

self-consistent acceleration of particles only through PIC/Vlasov simulations

particle injection rate unknown

a lot of microphysics still missing

realistic fully kinetic simulations impossible because

ωB

Ω& 10

12 − 1018

If you have good ideas to deal numerically with such strong electromagnetic fields (and

may be also with radiation) you are welcome to help.


References I

Belyaev M. A., 2015, MNRAS, 449, 2759

Bonazzola S., Mottez F., Heyvaerts J., 2015, A&A, 573, A51

Cao G., Zhang L., Sun S., 2016, MNRAS, 455, 4267

Cerutti B., Philippov A. A., Spitkovsky A., 2016, MNRAS, 457, 2401

Contopoulos I., 2016, ArXiv e-prints

Davis L., Goldstein M., 1970, ApJL, 159, L81

Deutsch A. J., 1955, Annales d’Astrophysique, 18, 1

Harding A. K., Muslimov A. G., 2011, ApJL, 726, L10+

Komissarov S. S., 2006, MNRAS, 367, 19

McKinney J. C., 2006, MNRAS, 368, L30

Michel F. C., Goldwire, Jr. H. C., 1970, Astrophysical Letters, 5, 21

Paschalidis V., Shapiro S. L., 2013, Physical Review D, 88, 104031

Pétri J., 2012, MNRAS, 424, 605

Pétri J., 2015, MNRAS, 450, 714

Pétri J., 2016, MNRAS, 455, 3779

Pétri J., submitted, A&A, submitted

Spitkovsky A., 2006, ApJL, 648, L51

Tchekhovskoy A., Spitkovsky A., Li J. G., 2013, MNRAS, 435, L1

Timokhin A. N., 2006, MNRAS, 368, 1055


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