## 1. Introduction

[2] Each of the four satellites of the Cluster space fleet carries a pair of mutually perpendicular long-boom antennas with spherical probes at their tips, each about 88 m tip to tip, located in the spin plane [*Gustafsson et al.*, 1997]. Several wave experiments use these antennas at different frequencies, ranging from direct current up to about 580 kHz in passive as well as active modes of operation, within a coordinated consortium [*Pedersen et al.*, 1997]. Since the behavior of such antennas is known to depend strongly on the frequency, their design is quite sophisticated in order to comply with the constraints of each experiment. For the modeling of the antenna in terms of finite elements as presented here, we tried to follow the real structure as closely as possible. This study applies to a so-called “high-frequency” (HF) range, from a few kilohertz up to about 100 kHz, a wide band which includes the local plasma frequency along the Cluster orbit. We will ignore the influence of the ions on the waves in this range, and we will neglect the anisotropy due to the steady magnetic field whenever the electron cyclotron frequency lies well below the plasma frequency. This contrasts with the quite different antenna behavior in the near-zero and ULF ranges, which have their own constraints mainly dominated by the magnetic field and the conductivity of the ambient plasma, including the current of photoelectrons escaping from the Sun-exposed spacecraft (S/C) surfaces [*Pedersen et al.*, 1998].

[3] In the above defined HF domain the ambient medium must be considered as a dielectric with a frequency-dependent permittivity, which includes the losses due to the radiation of kinetic waves from every conducting surface in contact with the plasma [*Béghin and Kolesnikova*, 1998]. The usual zero-order approximation to the basic theory for antennas shorter than the electromagnetic (EM) wavelengths consists of attributing to the ambient medium the scalar permittivity ɛ_{0} of free space, or possibly the tensor permittivity of a cold plasma. In these conditions, most experimenters assume that the effective length of a dipole antenna is its tip-to-tip length in the case of a double sphere (or double probe) and half this length for a double wire; here the term “probe” signifies a sphere together with its connecting wire if this wire is not shielded in any way. In some cases, a more or less arbitrary correction factor is introduced to take account of the disturbances in the electrostatic field near to the spacecraft (Figure 1) and perhaps also of the noninfinite input impedance of the receiver.

[4] The above approximations can lead to significant errors in the interpretation of received signals, especially at frequencies close to plasma resonances whenever critical parameters, such as the ratio between the local Debye length and the system size, are underestimated (see, e.g., review by *Béghin* [2002]). This will be shown in more detail in section 4.3 in the specific case of Cluster for typical plasma conditions encountered along the orbit. First, we summarize the general context in which our modeling method is applicable, then we give the results of the calculation for the effective antenna length and of the simulation for various active modes of operation that the present hardware design unfortunately does not allow to be put into practice. As shown by our study, the complex behavior of such antennas could indeed be understood more easily by using a combination of active modes such as self-impedance and mutual impedance measurements. Finally, we give the results of a preliminary test of the mutual impedance measurement which has been done recently between the long-boom dipole and the double-probe antenna by coordinated simultaneous operations of the relaxation sounder Wave of High frequency and Sounder for Probing of Electron density by Relaxation (WHISPER) [*Décréau et al.*, 1997] and the waveform receiver Wide Band Data instrument(WBD) [*Gurnett et al.*, 1997]. This test validated our modeling, though the results are somewhat more complicated than we predicted initially with a single Maxwellian electron distribution. In our view, such a measurement can be regarded as an in-flight calibration of the Cluster antennas and also as a promising complementary method of plasma diagnostics.