The Voyager spacecraft have a 3.66 m diameter parabolic high-gain antenna. I've had a nose around NASA's archives but I can't find the exact geometry of the parabola; specifically the distance from the vertex to the focus, the latus rectum or alternatively the quadratic equation of the parabola.
Any suggestions where to look, or does anyone happen to know the shape of the antenna's parabola?
tl;dr: From Fig. 3.14 below:
And:
Further reading:
While the Voyagers' primary reflector is very close to a parabola, they've slightly tweaked the shape as part of the design of the secondary reflector (it's a Cassegrain-like antenna for the X-band system.)
And the secondary reflector is optimized to minimize spillover and fill the aperture as uniformly as possible. I think what those mean exactly and why they were important will make excellent followup questions!
Here are some drawings and text from Chapter 3 of the Descanso volume 8 publication Spaceborne Antennas for Planetary Exploration, William A. Imbriale, Editor
Also see Sections 1.2.2 through 1.2.4 for some more mathematical background.
3.2.2 Requirements
The HGA consists of a paraboloidal reflector with a 3.66-m (12-ft.) diameter circular aperture and suitable S- and X-band feeds. The X-band feed utilizes dual shaped Cassegrain optics, and the S-band feed utilizes a prime focus feed. A frequency selective subreflector (FSS) reflects the X-band signal and passes the S-band signal. The HGA has a focal length to diameter (F/D) ratio of 0.338. The HGA is RHCP and operates over the frequency ranges of 2115 ±5 and 2295 ±5 MHz. It also operates over the frequency range of 8422 ±20 MHz, with a dual polarized feed that yields a right-hand or left-hand circularly polarized wave from the HGA depending on which of its input ports is excited by the RFS. S-band signals are received by the HGA at 2115 ±5 MHz and routed to the RFS receiver. S-band signals at 2295 ±5 MHz from the RFS S-band transmitter are radiated via the HGA. X-band signals at 8422 ±20 MHz from the RFS X-band transmitter are radiated via the HGA. The LGA radiates a circularly polarized, broadbeam pattern directly to Earth. The LGA requirements are summarized in Table 3-1.
3.2.3 Voyager High-Gain Antenna
Since a 3.66-m reflector was the largest solid reflector diameter that could fit into the nose cone fairing without deployment, it was desirable to have the highest aperture efficiency possible. A high-efficiency dual-reflector system generally requires that (1) most of the feed energy be intercepted by the reflectors (i.e., low spillover), and (2) the field in the aperture of the main reflector be distributed as uniformly as possible. Ordinarily, reduction of spillover requires tapering the aperture distribution, and a uniform aperture distribution generally involves substantial spillover. Consequently, optimum performance traditionally involves a compromise that has limited efficiencies of conventional systems to about 55–60 percent. The shaped dual-reflector concept permits the apparent contradiction between the two requirements for high efficiency to be overcome with the following rationale: a feed is selected with a high taper at the edge of the subreflector to minimize forward spillover; the subreflector profile is designed to distribute the highly tapered energy uniformly over the aperture of the main reflector. By designing for constant aperture illumination (see Section 1.2.4), the classical hyperboloid subreflector is transformed into an empirical contour with a smaller radius of curvature than a hyperboloid in the central section to deflect more of the rays to the outer part of the main reflector. Thus, there is little spillover and, at the same time, a nearly uniform aperture distribution. The main reflector must then be slightly reshaped from its original paraboloidal contour to produce a constant-phase distribution
The HGA was dual-shaped for optimum efficiency at X-band (see Fig. 3-14). The dichroic subreflector is transparent to radiation from an S-band prime-focus horn nestled behind it. At S-band, the main reflector differs little from a paraboloid. The focus of the resultant best-fit paraboloid was chosen as the prime focus for the S-band feed.
The most detailed Voyager general arrangement drawing we have is this one. (huge drawing which I won't add as an image here)
and we know the gain, from Descanso 4:
S-band gain is approximately 36 dBi; X-band gain is approximately 48 dBi.
