## 1. Introduction

[2] One of the important components of space weather is the relativistic electron population of the radiation belts. During periods of magnetic disturbance the enhanced flux of these particles can seriously threaten the growing number of civilian and military assets in space. The vulnerability of these assets continually increases as the trend toward smaller spacecraft results in less radiation protection and the trend toward smaller chips results in less effective radiation hardening.

[3] In a recent paper [*Inan et al.*, 2003], it was proposed that in situ injection of VLF whistler mode waves from electric dipole antennas on spacecraft within the radiation belts would dramatically increase the pitch angle scattering of the relativistic electrons and cause these particles to be rapidly lost from the belts, thereby mitigating the flux enhancement. In order to assess the number of spacecraft-based VLF transmitters necessary to achieve the proposed mitigation it is necessary to determine the maximum VLF electromagnetic power that can be radiated by a typical spacecraft system using electric dipole antennas. One of the most important factors in this determination is the current distribution along the length of the dipole, since it is this current which ultimately determines the amount of VLF power which can be radiated from the antenna into the plasma.

[4] At the present time there is very little known in general about the current distribution of dipole antennas at VLF frequencies in a magnetoplasma such as the magnetosphere. Much of the past work concerning the characteristics of dipole antennas in a magnetoplasma has proceeded by first assuming a current distribution, usually triangular, and then determining, for example, the input impedance, radiation resistance, and radiation pattern of the antenna. However, the accuracy of the triangular current assumption has never been established. Numerous references concerning early work on this topic can be found in work by *Balmain* [1972, 1979], *Wang and Bell* [1969], and *Wang* [1970].

[5] In a uniform dielectric medium, one can reasonably approximate the current distribution along a center driven, thin dipole antenna through the relation [*King et al.*, 2002, chapter 1]

where *h* is the antenna half-length, β_{d} = ω/*c* = 2π/λ is the wave number for waves of frequency ω which propagate in the medium, λ is the wavelength of the waves, ε_{d} is the relative dielectric constant, *c* is the velocity of light in free space, and *s* is the distance along the antenna measured from the current input terminals at *s* = 0. If the antenna length is small compared to the wavelength of the radiated waves, then (β_{d}*h*)^{2} ≪ 1 and the sinusoidal terms can be approximated by their arguments and (1a) becomes the triangular current distribution

[6] If *h* is fixed in (1a), *I*(*s*) depends only upon the unique wavelength λ = 2π/β_{d} which is possessed by all propagating waves of frequency ω in the medium, independent of their direction of propagation. This circumstance is quite different in a magnetized plasma such as the magnetosphere, since this medium is anisotropic for electromagnetic waves, and the wavelength of these waves depends strongly upon their direction of propagation. Since there is no unique wavelength in this medium, it is not clear that the current distribution along an in situ dipole antenna can be reasonably described by relations such as (1a) or (1b). Furthermore, if it is possible, what is the value for the wavelength that should be used in (1a)?

[7] The variation of λ at VLF frequencies in the plasmasphere results from the variation of the refractive index *n*(ψ) as the angle ψ between the propagation vector *k* and the Earth's magnetic field *B*_{o} is changed. In Figure 1 we plot *n*(ψ) for a typical VLF whistler mode wave (assuming a cold plasma) as a function of ψ. The plot shows a cross section of the refractive index surface, which is a surface of revolution generated by rotating the refractive index plot around the *B*_{o} direction. The refractive index *n* increases monotonically as a function of ψ and extends asymptotically toward infinity along the resonance cone surface at the resonance cone angle ψ = ψ_{r}.

[8] Figure 2 shows λ(ψ) for whistler mode waves of 5 kHz frequency whose refractive index is similar to that shown in Figure 1. For waves propagating directly along *B*_{o}, λ ≃ 3 km, while for waves propagating at angles ψ close to ψ_{r}, λ ∼ 0. The lower limit is an artifact of the assumption that the plasma is cold. Finite temperature effects generally prevent the wavelength from approaching zero. Extensive spacecraft observations of these very short wavelength whistler mode waves, also commonly known as quasi-electrostatic lower hybrid waves, indicate that their wavelengths are seldom smaller than ≃2 m [*Bell et al.*, 1983; *Bell and Ngo*, 1988; *James and Bell*, 1987; *Bell and Ngo*, 1990; *Bell et al.*, 1991a, 1991b, 1994]. Since the dipole antenna length proposed by *Inan et al.* [2003] was roughly 100 m, we have a situation in which some of the waves of frequency ω radiated by the antenna will possess wavelengths much longer than the antenna while other radiated waves of frequency ω will possess wavelengths much shorter than the antenna. In view of the complexity of this system, some method of determining *I*(*s*) from first principles is clearly required.

[9] Our primary goal in the present paper is to construct from first principles integral equations of the Hallén type [*Hallén*, 1938] to determine *I*(*s*) for two orientations of the dipole antenna, one parallel to *B*_{o} and one perpendicular to *B*_{o}. These equations are developed in the limit of short wavelengths, ie, for λ^{2} ≪ λ_{o}^{2}, where λ_{o} is the wavelength of whistler mode waves propagating parallel to *B*_{o}. This approach is suggested by the results of *Wang and Bell* [1969], who assumed a current distribution similar to (1b) and found that the major portion of the radiated power was carried by quasi-electrostatic whistler mode waves for which λ ≃ *h*. This result suggests that if the antenna length is restricted to 100 m or less, one can reasonably neglect the effects of the waves with λ ≥ 1 km and consider only the shorter wavelength quasi-electrostatic waves. Our second goal is to determine the conditions under which *I*(*s*) is approximately triangular in order to judge the applicability of past work [e.g., *Balmain*, 1964; *Wang and Bell*, 1969; *Wang*, 1970] in which such distributions were assumed to apply.

[10] Our third goal is to compare the results of our model with the results of past workers who used both electromagnetic and quasi-static models to describe the fields generated by a dipole antenna. The quasi-static model is based on a modified form of Poisson's equation, and *Balmain* [1964] was one of the first to apply the model to a thin electric dipole in a magnetized plasma. He assumed that the antenna possessed a triangular current distribution and was short compared to the characteristic wavelength of the medium. This characteristic wavelength is not defined in the model, but must be determined through other means [*Chugunov*, 1968]. With the aid of the quasi-static model, the input impedance of the antenna was defined through concise analytic formulas, and the predictions of the model were partially tested in laboratory experiments [*Balmain*, 1964].

[11] Subsequently, *Wang and Bell* [1969] and *Wang* [1970] studied the same problem with the aim of establishing the general characteristics of dipole antennas in the plasmasphere. In contrast to the approximate quasi-static model of *Balmain* [1964], Wang's model relied on a numerical solution of the full set of Maxwell's equations to find the antenna input impedance, assuming a triangular current distribution. Comparison of the full wave model with the quasi-static model generally showed good agreement when the antenna length was of the order of 10–100 m and the driving frequency lay in the range: *f*_{lhr} ≪ *f* ≪ *f*_{ce}, where *f*_{lhr} is the lower hybrid resonance frequency, defined below, and *f*_{ce} is the electron gyrofrequency.

[12] Notable advances in the development of the quasi-static model were achieved by *Chugunov* [1968], who considered a wider variety of antenna forms and developed an integral equation relating the integral of the unknown antenna surface charge density to known functions. More recently, *Mareev and Chugunov* [1987] have extended the model further by incorporated spatial dispersion and particle collisions.

[13] It is important to note here that we do not use the quasi-static model in the present paper. In contrast to the quasi-static model which makes use of a modified form of Poisson's equation, our own model makes use of the full set of Maxwell's equations to construct integral equations for the dipole current distribution. One of the additional outcomes of our development is that we can determine the conditions for the applicability of the quasi-static model in the two cases in which the antenna is either parallel or perpendicular to the ambient magnetic field.