Relative movement of the planets
As the planets move in their orbits, their distance and relative velocity toward each other are continuously changing. Because the velocity between them is changing, they have a relative acceleration. This acceleration between the two planets is not due to gravitational interaction between them but rather because they are moving in changing directions with varying speeds. The planetary explorer graphs distance and relative speed and acceleration for any pair of planets.
The links in the narrative below create a graph of the mentioned planetary configuration. Any graph can be captured with he "Capture URL" button that creates a link to the currently display information. The motions of any pair of planets can be explored.
Orbital events that are searchable by the explorer include identifying each planet's perihelion and aphelion. (when the planet is the closest or furthest from the sun). In addition, the explorer will report the details of when when a pair of planet's distance, relative speed or acceleration are at a maximum or minimum. The algorithm searches local maxima and minima, where the movement changes direction. Because orbits are not circular there are situations where there are more than one maximum and minimum in a given cycle.
Opposition and conjunctions
There are also events when a pair of planets line up related to the sun. These events have different names depending on which planet is closer to the sun.
For a planet closer to the sun, inferior conjunction occurs when the planet is between the outer planet and the sun. For example, an inferior conjunction of Venus and the Earth. When an outer planet is aligned so that the outer planet is close to 180° from the sun, it is called opposition.
When a planet is on the opposite side of the sun from the analyzed planet, it is called superior conjunction.
For example, with the Earth and Venus, inferior conjunction occurs when Venus is between the Earth and the sun. This inferior conjuction usually doesn't have Venus make a transit in front of the sun because the plane of motion of Earth and Venus are different. Instead, the triangle between the Sun, Earth and Venus is the most acute.
One thing that is notable is that the time of opposition and minimum distance are not identical. Also, the time of superior conjunction and maximum distance are generally not the same.
Maximum elongation and quadrature
This website also calculates other interesting arrangements. One event is the moment of the maximum elongation of an inner planet. When Venus or Mercury are at maximum elongation, the angle in the sky between the planet and sun is the greatest. This is the time that it is easiest to see the inner planets in the sky.
The concept of maximum elongation is relevant for any planet for a planet closer to it than the sun. For example, one can select Jupiter as the outer planet and Earth as the inner planet. The explorer reports the maximum elongation of the Earth as an inner planet. These elongations are called western and eastern depending on whether the planet appears in the morning or evening. When the planet is at maximum eastern elongation, it is east of the sun in the sky so that it is visible after sunset.
Another interesting event is the time of quadrature. Quadrature happens when an outer planet makes a right angle in the heavens. Thus Mars can be in quadrature related to the Earth. Just as with maximum elongation, a quadrature can happen the the planet is to the east or west of the sun in the sky.
Because the tables I am using calculate the position of the Earth rather than the center of mass of the Earth-Moon system, the moon has an effect on the earth's motion. This shows up especially with the relative acceleration between the Earth and Mars. The effect is present with other planets, but it is most obvious with Mars.
Definition of opposition and conjunction
My calculation of the oppositions and conjunctions have non-standard definitions. I'm basing my interactions on the angles between the planets rather than their astronomical [longitude] in the sky relative to the sun. These two definitions differ only slightly. The formulas for the most extreme triangles shapes are easier to compute with the current tables.
The information that this website presents are truncated in the precision I present. The accuracy of the formulas that I use can identify events with inner planets more precisely, but knowing the exact second that an event happens is excessive. Similarly, I present the distant, velocity and acceleration in less precision than the formulas provide.
Although more modern analyses are more accurate, the VSOP87 analysis is good enough. I'm not trying to present information accurately enough to pilot interplanetary space probes, but rather to present information in an interesting form.
Hopefully astronomers could use the website to identify when certain events happen. Currently I do not calculate the positions of the planets in the sky.