We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci & Sokal. In addition, we consider the relationship of functions of W to the class of completely monotonic functions and show that W is a complete Bernstein function.
Magnetotail dipolarizations, often associated with substorms, produce significant energetic particle enhancements in the nighttime magnetosphere. In this paper, we apply our recently developed magnetotail dipolarization model to the problem of energizing electrons and ions. Our model is two‐dimensional in the meridional plane and is characterized by the ability to precisely control the location of the transition from the dipole‐like to tail‐like magnetic fields. Both magnetic and electric fields are calculated, self‐consistently, as the transition zone retreats farther into the tail and the area around the Earth occupied by dipole‐like lines increases in size. These fields are used to calculate the motion of electrons and ions and changes in their energies. We consider the energizing effects of the fields restricted to ±15° and ±30° sectors around the midnight meridian, as well the axisymmetric case. Energies of some electrons increase by a factor of 25, which is more than enough to produce observable ionospheric signatures. Electrons are treated using the Guiding Center approximation, while protons and heavier particles generally require description based on the Lorentz equations.
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