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 – p.1/15
Preliminaries Poynting dominated: Poynting flux σ = particle-born flux ≫ 1 Wind: γ fms ≈ √ σ γ > Possible examples: GRB, Jets from AGN. . . Best example: pulsar wind – p.2/15
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 ω pe < Ω . For Crab: r > 10 8 r L (e.g., Melatos & Melrose 1996) • Many particles, MHD wind + shock – p.3/15
The σ problem Michel’s parameter µ = γσ : wind luminosity µ = mass-loss rate × c 2 “Standard” estimate for Crab: µ ≈ 10 6 . But, including radio electrons: 5 × 10 38 erg/s µ = 10 40 pairs/s 10 4 ≈ At fast magnetosonic point γ = µ 1 / 3 ≈ 20 , σ ≈ 400 – p.4/15
The σ problem II • Force-free axisymmetric fl ows 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 /r 2 B φ ∝ 1 /r No acceleration Γ = constant ⇒ σ = constant But, at inner edge of Nebula σ ≈ 10 − 3 . ⇒ dissipation required – p.5/15
Striped wind Oblique, split-monopole solution, Bogovalov A&A 349, 1017 (1999) Meridional plane Equatorial plane The striped wind (Coroniti 1990) – p.6/15
✁ ✡ � ✁ ✂ ✄ ✂ ☎ ✆ ☎ ✝ ✞ � ✂ ✟ ✠ Current sheets Governing equations: continuity, energy, entropy. Key question: What controls the thickness of the sheets? – p.7/15
Slow dissipation Ampère’s law in comoving frame: � c � 2 B ′ ℓ I ′ = 4 π 2 en ′ ( ℓλ ′ ) c � β d � < Minimum sheet thickness for � β d � = 1 . a = λ D β d Coroniti (1980) (sheet thickness > gyro radius = T/eB ′ ), Michel (1994), Lyubarsky & Kirk (2001). – p.8/15
Slow dissipation, II r − 3 p ∝ r − 1 / 2 T ∝ r 1 / 2 ∆ ∝ r 1 / 2 Γ ∝ • Hot plasma performs work in accelerating the fl ow • Dissipation timescale dilated r max ˆ L = L ( π 2 e 2 /m 2 c 5 ) , = ˆ L 1 / 2 ( = 1 . 5 × 10 22 for Crab) r L – p.9/15
Tearing-mode limited dissipation • Current sheet unstable — e.g., tearing mode at rate γ t : thinner sheet ⇒ faster growth • Complex fl ow with reconnection sites inside annihilation region • Overall expansion speed cβ exp = aγ t � 3 / 2 � λ D β exp = a (Lyubarsky 1996) – p.10/15
Tearing-mode limited dissipation II r − 17 / 6 p ∝ r − 5 / 12 T ∝ r 5 / 12 ∆ ∝ r 5 / 12 Γ ∝ r max = µ 4 / 5 ˆ L 3 / 10 r L Faster than “slow dissipation” for µ < ˆ L 1 / 4 ( = 3 . 5 × 10 5 for Crab) – p.11/15
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 ⇒ β exp < β s � � � 1 T β exp = Min √ 3 , mc 2 – p.12/15
Fast dissipation, II Relativistic T : r − 8 / 3 p ∝ r − 1 / 3 T ∝ r 1 / 3 ∆ ∝ r 1 / 3 Γ ∝ r max = 0 . 1 µ 2 r L – p.13/15
Fast dissipation, III L 1 / 4 no • Faster than “slow dissipation” for µ < 10ˆ consistent dissipative solution otherwise. • For Crab, ≈ 10% of Poynting fl ux dissipated in relativistic regime • Remaining 90% dissipated inside termination shock for µ = 2 × 10 4 – p.14/15
Conclusions • Dissipation in a wind is slowed by free expansion • Minimum mass-loading required for complete dissipation • Provides acceleration to γ = µ , with γ ∝ r q , 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 = 10 6 r L , complete dissipation by 2 × 10 9 r L — inside the termination shock – p.15/15
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