A plagiary

AVERTISSEMENT: What follows is not my work, except for the numbers I plugged in. Call it a plagiary if you like. In fact, please do call it a plagiary. Plagiary is what this is. All I wanted to do was to understand what it would take to drive a regular, simple Aldrinesque train from Earth to Venus and back ("EVE") on a Free Return.

Don't credit me for this - except, if you like, for the numbers. Here are the people who wrote the outline, and did well-enough at it that I could follow their argument well-enough: Alan L. Friedlander, John C. Niehoff, Dennis V. Byrnes, James M. Longuski. And here is the pdf they wrote: "Circulating Transportation Orbits between Earth and Mars", 1986.

That settled, here follows what their project looks like when the orbits go between Earth and Venus - or rather, from Venus to Earth (it's an outbound perspective). At least, here follows the section I figured out before I decided I was done, for my side of the project. I underlined where I plugged in my own numbers or expanded on the parts of their arguments which I, your humble plagiarising moron, had to look up for myself.

II. Prelude to Circulating Orbit Design

Earth and Venus revolve about the Sun with orbit periods of 1.0 year and 0.615198 years, respectively. Earth travels in the ecliptic plane with an orbit eccentricity of 0.0168 while Venus travels in a plane inclined by 3.394° to the ecliptic with an eccentricity of 0.006772. As a help in understanding the properties of circulating orbits, ignore for the moment the non-circular, non-coplanar motion of the real world. Additionally, approximate the period of Venus (PV) by 0.615 years, so that every eight years Earth makes eight revolutions about the Sun while Venus makes exactly 13 revolutions. Since the relative geometry between Earth and Venus repeats with the synodic period of 1/((13-8)/8) = 8/5 = 1.6 years, trajectories connecting them will re-occur with this period also. The inertial geometry repeats every eight years. These relationships have important implications in the design and repeatability of circulating orbits between these planets.

The VISIT type of circulating orbit, first proposed by Niehoff evolved from consideration of a class of trajectories which might display low relative velocities at both Earth and Mars terminals. In order for this to occur it is necessary that the connecting heliocentric orbit be nearly tangent to the planetary orbits. For Venus-Earth the perihelion of the transfer orbit must be near Venerean semimajor axis 0.723 AU and the aphelion near 1.0 AU. The orbit which satisfies this exactly in the circular coplanar model has a semi-major axis of 0.86 AU and - by Kepler - a period (P) of root-0.636 = 0.7975 years. If, in addition, it is required that the transfer orbit be resonant with both Venus and Earth, that is: nP=m
and
jP=kPV
where n, m, j, and k are small integers, then I see two possibilities. For Venus-Earth we are limited by the lack of factors for 13.

The first possibility (denoted VISIT-1) has a period of 0.308 (4/13) years and a semi-major axis of cuberoot-0.0946745 = 0.455 AU. This orbit completes 13 revolutions about the Sun while Earth completes four. It also completes two revolutions about the Sun while Venus completes one. Hence, the 13:4 resonance with Earth and 2:1 resonance with Venus means that Earth encounters will occur once every four Earth years and Venus encounters every Venerean year, 225 Earth days.

The second possibility (denoted VISIT-2) has a period of half a year and a semi-major axis of 0.794 AU. This orbit completes one revolution about the Sun while Earth completes two. It also completes four revolutions about the Sun while Venus completes 13. The 1:2 resonance with Earth and 4:13 resonance with Venus means that Earth encounters occur once every two Earth years; while Venus encounters occur once every 13 Venerean years, eight of our years.

In the circular coplanar world both VISIT orbits must have perihelia of less than Venus' 0.723 AU since their periods are less than the value of 0.7975 computed above. In the real world, the eccentricity of Earth's orbit makes the VISIT-1 orbit ideal for encountering Earth near its perihelion and the VISIT-2 orbit ideal for encountering Earth near its aphelion.

...

Since encounter locations on a 8/5 synodic period orbit rotate 3/5 of a circle per revolution, 216°; there exists a synodic REcession of the line of apsides equal to 144° per orbit which must be accommodated by either propulsive maneuvers (which are not desirable) or, to the extent possible, by gravity-assist swingbys of the planets (which have about the same mass). Given this effective trajectory shaping, both Earth and Venus will be encountered sequentially every 1-3/5 years. The encounter speeds at both planets, however, will be significantly higher than those of VISIT orbits because the 1-3/5 year period transfer orbit has an aphelion distance of about [don't know, or much care] AU, thereby crossing the orbit of Earth at steeper (non-tangential) angles.