The dissipation of Poynting-flux in pulsar winds John Kirk - - PowerPoint PPT Presentation

The dissipation of Poynting-flux in pulsar winds

John Kirk Max-Planck-Institut f¨ ur Kernphysik Heidelberg, Germany Collaborators: O. Skjæraasen, Yuri Lyubarsky, Yves Gallant

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Preliminaries

Poynting dominated:

σ =

Poynting flux particle-born flux ≫ 1

Wind:

γ > γfms ≈ √σ

Possible examples: GRB, Jets from AGN. . . Best example: pulsar wind

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Pulsar winds

The Crab Nebula Central star is source of particles and magnetic field (Piddington 1957) and waves (Rees & Gunn 1974).

• Few particles: magnetic dipole radiation?

Damping ⇒ propagation only for ωpe < Ω. For Crab: r > 108rL (e.g., Melatos & Melrose 1996)

• Many particles, MHD wind + shock

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The σ problem

Michel’s parameter µ = γσ:

µ =

wind luminosity mass-loss rate × c2

“Standard” estimate for Crab: µ ≈ 106. But, including radio electrons:

µ = 5 × 1038 erg/s 1040 pairs/s ≈ 104

At fast magnetosonic point γ = µ1/3 ≈ 20, σ ≈ 400

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The σ problem II

• Force-free axisymmetric fl
• ws accelerate: Γ ∝ r

(Buckley 1977, Contopoulos & Kazanas 2002) in subsonic region (γ < √σ)

• Steady, relativistic, axisymmetric winds are

quasi-spherical, e.g., cold spherical (monopole) wind:

n ∝ 1/r2 Bφ ∝ 1/r

No acceleration Γ =constant

⇒ σ =constant

But, at inner edge of Nebula σ ≈ 10−3.

⇒ dissipation required

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Striped wind

Oblique, split-monopole solution, Bogovalov A&A 349, 1017 (1999) Meridional plane Equatorial plane The striped wind (Coroniti 1990)

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Current sheets

• ✁

Governing equations: continuity, energy, entropy. Key question: What controls the thickness of the sheets?

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Slow dissipation

Ampère’s law in comoving frame:

c 4π

• 2B′ℓ

= I′ < 2en′(ℓλ′)c βd

Minimum sheet thickness for βd = 1.

a = λD βd

Coroniti (1980) (sheet thickness>gyro radius= T/eB′), Michel (1994), Lyubarsky & Kirk (2001).

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Slow dissipation, II

p ∝ r−3 T ∝ r−1/2 ∆ ∝ r1/2 Γ ∝ r1/2

• Hot plasma performs work in accelerating the fl
• w
• Dissipation timescale dilated

rmax rL = ˆ L1/2 ˆ L = L(π2e2/m2c5),

(= 1.5 × 1022 for Crab)

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Tearing-mode limited dissipation

• Current sheet unstable — e.g., tearing mode at

rate γt: thinner sheet ⇒ faster growth

• Complex fl
• w with reconnection sites inside

annihilation region

• Overall expansion speed cβexp = aγt

βexp = λD a 3/2

(Lyubarsky 1996)

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Tearing-mode limited dissipation II

p ∝ r−17/6 T ∝ r−5/12 ∆ ∝ r5/12 Γ ∝ r5/12 rmax rL = µ4/5 ˆ L3/10

Faster than “slow dissipation” for µ < ˆ

L1/4

(= 3.5 × 105 for Crab)

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Fast dissipation

How rapidly can the dissipation zone expand?

• Fast m.s. speed in

external medium? Drenkhahn (2002)

• Total fl

ux conserved

⇒ causal connection

required

⇒ βexp < βs βexp = Min

• 1

√ 3,

• T

mc2

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Fast dissipation, II

Relativistic T:

p ∝ r−8/3 T ∝ r−1/3 ∆ ∝ r1/3 Γ ∝ r1/3 rmax rL = 0.1µ2

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Fast dissipation, III

• Faster than “slow dissipation” for µ < 10ˆ

L1/4 no

consistent dissipative solution otherwise.

• For Crab, ≈ 10% of Poynting fl

ux dissipated in relativistic regime

• Remaining 90% dissipated inside termination

shock for µ = 2 × 104

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Conclusions

• Dissipation in a wind is slowed by free expansion
• Minimum mass-loading required for complete

dissipation

• Provides acceleration to γ = µ, with γ ∝ rq, q = 1/3

to 1/2.

• “Conventional” µ estimates for Crab too high for

complete dissipation

• Low µ and maximum rate permits ∼ 10%

dissipation by r = 106rL, complete dissipation by

2 × 109rL — inside the termination shock

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