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tokens: factor is retained explicitly in order to track the resultant change if the ion wind does not completely fill the polar cap. The initial electron density will also be given by Equation ( 44 ). For ions, we scale , and we will use the midrange value of to make estimates. The primary beam electrons accelerate to with , curvature-radiate, and produce secondary pairs with the multiplicity ; thus, the initial Lorentz factor of the dense secondary pair plasma is of order . 6.2.2. Magnetic Field <inline-formula> <tex-math> </tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjaa984cieqn219.gif"></inline-graphic> </inline-formula> For the polar cap flux tube, in dipole (primed and starred) coordinates , , and, above the acceleration sheath, the essentially field-aligned current is . The perturbed magnetic field () can be obtained from Ampere’s law for the vector potential ,where H ( x ) is the Heaviside step function. Matching solutions inside and outside of the polar cap, we obtainThe perturbed magnetic fields within the polar cap are Since the wind equations are best formulated in the inertial coordinate system, in order to obtain the inertial perturbed azimuthal field , we must project onto the inertial -axis, , using the coordinate transformation relations To order , we obtainAveraging over the polar cap yieldsNote that in the inertial coordinate system, the dipole field does have a φ -component (), but that . 6.2.3. Polar Cap Poynting and Particle Energy Fluxes In the inertial coordinate system, the polar cap electric field , where is the electric field in the co-rotating frame. Since smaller than , we have . Integrating the outgoing Poynting flux () over the north and south polar caps yields the total radiated electromagnetic luminositywhere is the vacuum dipole radiated power. Integrating the ion energy flux over both polar caps yields the total ion luminosityThe pair plasma particle luminosity from the electron outflow half of the polar cap is much smaller than L i . Combining Equations ( 51 ) and ( 52 ) and taking , we obtain the initial global value of the ratio of electromagnetic to particle powerwhich is independent of . 6.2.4. Initial Wind Parameters We evaluate the wind-conserved energy parameter E by using the initial wind conditions to obtainFor the fiducial values of f and g , . The modified Kennel et al. ( 1983 ) conserved -parameter becomes 6.3. The Wind Z-parameter In their discussion of hot relativistic winds, Kennel et al. ( 1983 ) introduce the wind parameter that differentiates the types of wind solutions. Winds with are basically cold and centrifugally slung, corresponding to the classic pulsar wind. Winds with start with either a relativistic high temperature or high initial Lorentz factor. Clearly, for the above value of σ and , . Kennel et al. ( 1983 ) numerically show that high- winds experience very little radial acceleration with increasing distance. 6.4. The Alfvén Critical Point