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tokens: velocity. A transient current system is generated during a time corresponding to at least the travel time at the Alfvén velocity between the interaction site (e.g., the equator in the case of a satellite–magnetosphere interaction) and the planetary ionosphere. Because of the large size of the current system (several planetary radii in this example), this transient phase can last for a long time and may even be longer than the interaction time itself (see e.g. Neubauer 1980 ; Gurnett Goertz 1981 ; Saur et al. 2004 , for the Io-Jupiter case). In this case, electron acceleration is due to the parallel electric field associated to kinetic Alfvén waves above the ionosphere of the planet. This electric field is modulated at the Alfvén wave frequency and does not form electric potential drops, and hence does not directly form auroral cavities either (although cavities may slowly build up due to the excitation of ion acoustic waves, Hess et al. 2010a ; Matsuda et al. 2012 ). Thus, the electron density in the current system is the same as outside of it, with the cold component of the plasma remaining present, and the electron distribution is either a ring or a Kappa-like distribution ( Swift 2007 ; Hess et al. 2007b , 2010b ). 3.3. Unstable electron distributions Wave-particle resonance is reached when the Doppler-shifted angular frequency of the wave ω = 2 πf in the frame of the electron ( ω − k ∥ v r ∥ ) is equal to that of the gyration motion of resonant electrons (). Components k ∥ and v r ∥ are the parallel components (to the magnetic field lines) of the wave vector and of the velocity of the resonant particle, respectively. The electron cyclotron angular frequency is ω c = 2 πf c = eB / m e , with B being the magnetic field amplitude and e and m e the charge and mass of the electron. Finally, Γ r is the Lorentz factor associated with the resonant electron motion. The resonance condition therefore writes with In the weakly relativistic case ( v r c ), the resonance equation can be rewritten as: The resonance equation therefore forms a circle in the [ v ∥ , v ⊥ ] velocity space, whose centre v 0 , located on the v ∥ axis, is given by: which can be rewritten as where θ is the direction of the emission ( k cos θ = k ⋅ b = k ∥ ) and N = ck / ω is the refractive index value (which must be taken into account to reproduce the observations, as shown by Ray Hess 2008 ). The resonance condition can therefore be rewritten as Therefore, the resonance condition does not only define the angular frequency of the amplified wave ω = 2 πf , but also the direction of the emission θ . The resonance equation is under-constrained as there are two variables ( ω and θ ) to determine for a single equation. To solve it, one must consider another constraint brought by the CMI amplification equation, which states that the amplification occurs for positive gradients of the electron perpendicular velocity distribution around the resonant velocities v r ( Wu 1985 ). Given an