tokens: m e / m i so that the ion distribution is approximately a delta-function around p ∥ i = γ CC m i c carrying current at speed c . Details of two-stream instability are discussed in Appendix B.2 . 3.3. Ion Acceleration in the Nose We now consider the particle acceleration at the end of jets (i.e., the nose region where the jet magnetic field and current turn to the radial direction). Internal kink instability due to j z occurring in the Central Column, as analyzed in Colgate et al. ( 2015 ) and used in Section 3.2 , should persist for some distance into the nose driven by j r as the current turns radially in the nose, now less affected by curvature radiation as B φ falls like 1/ r giving curvature parameter α C ∝ 1/ r . But a new acceleration mechanism arises due to conditions in the nose region different from those in the cylindrical Central Column. One new condition concerns the sensitivity of the curvature radiation, Equation 11 (a), to R C that allows ions to be accelerated to the full voltage in the nose region, where R C = r becomes large. Another new condition concerns the escape of ions from the nose, escape itself turning out to be a necessary ingredient for the powerful acceleration mechanism not present in the Central Column. There are two ways that ions can escape from the nose. First, while ion orbits are magnetically confined in the Central Column, in the absence of an electric field ion orbits in the nose are not confined due to drifts of guiding centers in z caused by the curvature of field lines and gradients in the magnetic field intensity. Opposite drifts by electrons establish an electric charge giving E ∝ T e , the electron temperature, which can cause ions to escape by E × B motion. In addition, ions can also escape by hyper-resistive diffusion, which is what we will mainly discuss. 3.3.1. Hyper-resistive Diffusion in the Nose As in Section 3.2.1 , we represent hyper-resistivity in the nose as diffusion, noting that an electric field larger than that produced by charge separation is created if turbulence in the nose causes ions to escape by diffusion. To see this, consider the radial component of the relativistic Ohm’s law in Equations 5 (a). In the nose, with unit vector , and electron flow v e is either that due to drifts in the z -direction or that parallel to B giving so that we can drop the relativistic correction to E r . Applying Equation 5 (a) in the r -direction (perpendicular to B ) gives Equation 13 (b) giving the cosmic-ray accelerator is obtained as follows. Early in jet propagation when in the jet and nose, all of the voltage drop in the jet (giving net zero voltage around the loop) can be approximated as , which appears as an inductive electric field in our disk-centered reference frame. It is this inductive field that serves to buildup B φ behind the nose advancing at velocity . Even as the jet slows down, it continues to be true that it is that builds up B φ , giving Equation 13 (b) with in our reference
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