Notes on Ecliptical Coordinates and Kepler's 3rd Law

Inferior Planets

Mercury
Venus

Superior Planets

Mars
Jupiter
Saturn
Uranus
Neptune

Ecliptical Coordinates

Geometric Geocentric Coordinated are typically used.
We will assume Geometric Geocentric unless stated otherwise
There is an alternative set that are Heliocentric (sun-centered)

The Ecliptic is defined as zero latitude with North being positive and South negative. Thus, b = 0 degrees at the ecliptic.
The Vernal Equinox is used to define the zero of longitude. The ecliptical longitude is an Eastward angle. Thus, the Vernal Equinox has l = 0 degrees.
Note: ecliptical longitude is always measured in degrees (not hours of angle).
Distance (r) is measured in Astronomical Units (AU), the average Earth-Sun distance.

Conjunctions and Elongations

Object A is in conjunction with Object B if
l_A - l_B = 0 degrees.

If only one object is specified, then the other is the Sun by default.
Object A is in conjunction, if
l_A - l_sun = 0 degrees.

Inferior planets have two conjuctions with the Sun.
Inferior conjunction occurs when r < 1 AU
Superior Conjunction occurs when r > 1 AU

Superior planets have only one conjunction with Sun (superior).

Inferior planets have two greatest elongations:
Greatest Eastern Elongation occurs when l_A - l_sun is at maximum (evening star longest after sunset).
Greatest Western Elongation occurs when l_sun - l_A is at maximum (morning star longest before sunrise).


Oppositions and Quadratures

Object A is in opposition with Object B if
|l_A - l_B| = 180 degrees.

If only one object is specified, then the other is the Sun by default.
Object A is in opposition, if
|l_A - l_sun| = 180 degrees.

Only superior planets have oppositions with the Sun. At opposition, a superior planet will be near its closest distance to Earth (i.e., r close to a minimum).

Superior planets can have quadratures with the Sun
Eastern Quadrature occurs when
l_A - l_sun = 90 degrees.

Western Quadrature occurs when
l_sun - l_A = 90 degrees.

Sidereal Orbital Period and Synodic Period

On Earth, we observe a repeaing cycle of conjunctions, oppositions and phases for planets. This period is called a synodic period (just like for the Moon). The planetary synodic periods are related to the sidereal orbital periods of the planets. If P is the sidereal orbital period of a planet and S is its synodic period observed from Earth, then
S = P_Earth * P / |P_Earth - P|

Kepler's Law

Here is Newton's Form of Kepler's 3rd Law where P is the sidereal orbital period measured in years, a is the semi-major axis of the orbit in AU, and M in solar masses:
P² = a³ / (M₁ + M₂)

In the case of something orbiting the Sun, the mass of the Sun is 1 solar mass and the other object is small in comparison. So that M₁ + M₂ = 1. This case reduces to Kepler's original form:
P² = a³