Ideal Magnetic Acceleration of Relativistic Flows Long history: Camenzind, Chiueh, Li, Begelman, Heyvaerts, Norman, Beskin, Bogovalov, Begelman, Tomimatsu, Tsinganos, Contopolous, Konigl, Vlahakis, Spruit, Lovelace, Romanova, Lynden-Bell, Michel, Okamoto, Eichler, Narayan, Phinney and others. Recent burst: Beskin & Nokhrina (2006); Komissarov et al. (2007,2009); Zakamska et al.(2008), Tchekhovskoy et al. (2008,2009), Lyubarsky (2009). The nature of ideal MHD acceleration has become clearer.

Quick overview: Ideal magnetic acceleration of relativistic flows is a slow process; It requires continuous collimation (bunching of magnetic field lines) ; This requires causal connectivity across the flow; This is achieved only for narrow externally confined flows; For the effective energy conversion ( kinetic energy flux ~ Poynting flux ) this translates into the half-opening angle. Too narrow for GRBs! Even if realised this still implies too week MHD internal and reverse shocks in the context of GRBs. Alternatives: 1) The acceleration is magnetic and effective but non-ideal? 2) The acceleration is effective but not magnetic? 3) The acceleration is magnetic but not effective – GRB jets remain Poynting-dominated up to the zone of prompt emission? Ideal MHD acceleration:

z Basics of axisymmetric steady-state ideal MHD cold flows Faraday eq.: Perfect conductivity: along magnetic field lines; “angular velocity” of the lines. Poloidal energy fluxes: rest mass: matter: Poynting:

Basics of axisymmetric steady-state ideal MHD cold flows zz Conservation laws: Additional constants along the magnetic field lines: is the upper limit on Michel’s sigma,, magnetization parameter. Low initial gamma:

Beyond the light cylinder (Alfven surface) Light cylinder definition: azimuthal component dominates These allow to writeas (Why the last result is important is explained on the next slide)

Smaller is, higher is the Lorentz factor of the flow. When decreases by 2 compared to the initial value Consider then. Smaller is, stronger is the flow collimation (bunching). B r where along the magnetic field line; is the total magnetic flux enclosed by the magnetic surface; is the collimation (bunching) function which governs the growth of r Collimation and acceleration

  

, where is the fluid frame field. For unconfined flows the connectivity across flow is partially lost when ( ), greatly reducing the efficiency of collimation/acceleration. Causal connectivity Relativistic Mach number: - fast magnetosonic speed as measured in the fluid frame; Mach cone (cone of influence): v cfcf or. For initially Pointing-dominated flows we must have and, thus, the kinetic energy flux still much lower than the Poynting flux at the fast point, and

(In fact, only are reached on reasonable length scales. So at most equipartition flows with ) For flows confined within the half-opening angle the connectivity across flow is partially lost when. For we obtain For example if the acceleration becomes inefficient after the Lorentz factor reaches ! In fact, we can do better: For full conversion of the Poynting flux into kinetic energy,, this gives us the upper limit on the opening angle

  

Implications for GRB jets. Such jets are too narrow to fit the statistics of GRBs and SNe. The observed rate of GRBs: After correction for beaming: The observed rate of core-collapse SNe: The observed rate of type Ib/c SNe: Moreover, the “equipartition jets” give too weak shocks to fit the inner shocks model for prompt gamma emission and weak reverse shock as well (Mimica et al., 2008);

Alternative acceleration models: 1) Acceleration of GRB jets is non-ideal: conversion of Poynting flux first into heat and then into kinetic energy ? 2) GRB jets lose causal connectivity, become ballistic and remain Poynting dominated all the way (Lyutikov & Blandford, 2003) ? 3) Jets are never Poynting dominated (the fireball model) ? The end