Superluminal waves in pulsar winds The striped wind and its - - PowerPoint PPT Presentation

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Superluminal waves in pulsar winds

Ioanna Arka

in collaboration with John Kirk Max Planck Institut f¨ ur Kernphysik Heidelberg, Germany

HEPRO III, Barcelona, 28/6/2011

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Outline

The striped wind and its termination shock Brief review Poynting flux reflection and propagation in the upstream Superluminal waves Model and calculations Results Conclusions

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

The striped wind

◮ Rotating neutron star, frequency Ω, magnetic moment µ ◮ Misaligned rotator, Bφ ∝ r −1 dominant at r ≫ rLC = c Ω ◮ Striped wind: entropy wave with Γ ≫ 1

Magnetization σ =

B2 4πΓnmc2 ≫ 1

Strength parameter a =

eB mcΩ ≫ 1

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Magnetic flux dissipation

◮ In the nebula σ < 1 (Kennel & Coroniti 1984)

Need for field dissipation in the wind or at the shock

◮ reconnection at the current sheets (Coroniti 1990) ◮ however, reconnection slows down as flow accelerates

(Lyubarsky & Kirk 2001)

◮ reconnection at the termination shock (Lyubarsky

2003, P´ etri & Lyubarsky 2007, Lyubarsky & Liverts 2008)

σ ≫ 1 a ≫ 1 δl ≪ λ

Michel 1982: square wave

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Interaction of wind with termination shock

◮ ”instantaneous” current reversal at shock ◮ alternating current emits electromagnetic waves ◮ in vacuum this wave would have an amplitude

Bref = ρ − 1 4 B0

ρ: shock compression ratio, B0: wind field amplitude at shock

◮ However perpendicular shock with σ ≫ 1 →

ρ ≃ 1 + 1

2σ (Kennel & Coroniti 1984) ◮ only a small fraction of the flux is reflected ◮ reconnection at shock: reflection of strong wave:

Bref = B0 4 → possibility of a precursor to the shock

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Wave propagation in plasma

Assume that there is a reflected wave that propagates in the upstream cold plasma. Wave propagation condition in plasma:

◮ linear waves ω > ωp, ωp: plasma frequency ◮ strong waves ω > ωp √a ⇐

⇒ Γ4 > a

σ

(e.g. Max 1973)

Reconnection at shock (P´

etri & Lyubarsky 2007):

◮ no dissipation: Γ <

a 4σ3/2

◮ partial dissipation:

a 4σ3/2 ≤ Γ ≤ a σ

◮ full dissipation: Γ > a σ

for full dissipation the reflected wave always propagates

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Strong waves in magnetized plasma

Self-consistent approach needed:

◮ two-fluid approach, cold e+ − e− plasma ◮ wave propagating transversely to the magnetic field:

X-mode

◮ linear X-mode in e+ − e− plasma is purely transverse

(Iwamoto 1993)

Non-linear X-mode already studied in the past analytically in some limiting cases (Kennel & Pellat 1976, Clemmow 1974,

Asseo et al. 1978 and others)

◮ waves with βφ > 1 can have arbitrarily large amplitudes

and can propagate in thin plasmas

◮ full treatment is possible numerically

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Wind conversion

Assumptions:

◮ conversion happens in δR ≪ R → plane wave

approximation

◮ wave’s frequency imposed by central rotator (pulsar)

Conditions: conservation of phase-averaged

• 1. particle flux
• 2. energy flux
• 3. momentum flux
• 4. magnetic flux

at conversion → ”jump conditions” for the transition

(see also Kirk 2010)

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Computation

• 1. Choose pulsar parameters:

◮ Luminosity L ◮ Magnetization σ ◮ Lorentz factor of outflow Γ

• 2. Solve equations for cold e− − e+ plasma:

◮ Maxwell’s equations ◮ Equations of motion + continuity equations

• 3. Apply ”jump conditions” to choose from above

solutions. Solutions dependend on radius and the parameter χ where χ = B

• B2

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Superluminal waves in the upstream

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1/βφ log(R) χ=0.65 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1/βφ log(R) χ=0.95

(plots for σ = 100, Γ = 100)

◮ χ ranges from 0 to 1 from equator to end of striped

wind zone

◮ minimum radius for conversion ◮ upstream propagating modes possible (propagating

inwards from the shock)

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

New ”magnetization parameter”

◮ Energy transferred from fields to particles ◮ introduce wave parameter σw:

Field energy flux Particle energy flux

◮ σw/σ plotted, where σ: magnetization in striped wind

(plots for σ = 100, Γ = 100)

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 log(σw/σ) log(R) χ=0.65

• 2.5
• 2
• 1.5
• 1
• 0.5

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 log(σw/σ) log(R) χ=0.95

Superluminal waves in pulsar winds Ioanna Arka The striped wind and its termination shock

Brief review Poynting flux reflection and propagation in the upstream

Superluminal waves

Model and calculations Results

Conclusions

Conclusions

◮ Striped wind can be converted to a strong, superluminal

wave through interaction with the termination shock

◮ During conversion, energy gets transferred from fields to

particles: → efficient particle acceleration → magnetic field dissipation

◮ Implication for particle acceleration at the shock: σ < 1

shocks are efficient particle accelerators

◮ Possibility for recovery of Fermi I at shock front