Capacitance of DSX VLF Antenna in Plasma

The DSX transmitting antenna consists of two 40 m booms extending from the satellite along the Y-axis. The conductive elements of each boom are three 0.4 mm diameter wires arranged on a 20 cm diameter circle. The wires are supported by a structure that is tenuous and largely (or entirely) dielectric. In this paper we calculate the capacitance of this structure (1) using various analytical methods; (2) using PIC simulations with the computer code Nascap-2k; and (3) by comparison with ground-based and space-based measurements.

The textbook method for calculating the capacitance between two concentric cylinders can be extended to calculate the capacitance of a wire array enclosed in a large outer cylinder. The diameter of the equivalent single wire depends only on the number and size of the array wires. For the DSX configuration, the equivalent single wire has a diameter of 3.64 cm.

Before flight, one 40 m boom was stretched out 1 meter above the floor and three meters below the ceiling. Its capacitance was measured to be "800+" pF. This implies an equivalent diameter of nearly 20 cm. For Nascap-2k modeling we chose a compromise, modeling the equivalent boom as 10 cm in diameter.

The literature contains good approximations for the capacitance of a vertical mast over a ground plane. We take the ground plane as the symmetry plane between the two masts. For the 3.64 and 10 cm diameters, the single mast capacitance is 330 and 389 pF respectively. As the DSX configuration is two such masts in series those values are divided by two to obtain 165 and 195 pF.

The "ion matrix" approximation for a cylindrical boom in plasma gives the capacitance and sheath radius as functions of voltage and plasma density. For the 3.64 cm boom in plasma of density 10⁹ m^-3, we calculate a capacitance of 11 pF/m and sheath radius of about 5 m.

We use Nascap-2k with PIC ions and fluid electrons to simulate the DSX sheath dynamics, with the capacitance as one result. We obtain a vacuum capacitance of 197 pF, in excellent agreement with the analytic result of 195 pF. For 1 kV bias the capacitance increases to about 220 pF in 10⁸ m^-3 plasma, and about 250 pF in 10⁹ m^-3 plasma. DSX measurements in the plasmasphere give about 280 to 325 pF.