Planetary Opposition

 

Consider two planets whose orbits about the sun are circular and in the same plane called the “ecliptic”.  These planets are said to be in opposition when they are the closest together.   From the perspective of the inner planet, this will happen when the outer planet is exactly opposite the sun.   This is illustrated below for the Earth and Mars.

At opposition the outer planet will appear to slow down and briefly stop in the sky.  Immediately afterwards the outer planet will apparently reverse direction before again resuming its normal path.   This is the phenomenon of “retrograde motion” and happens once at each opposition.

 

TIME BETWEEN OPPOSITIONS

 

The angle “ of any planet relative to the sun changes a rate determined by its period, T, which is the time required to make one revolution.

 

 

We will denote the time in days it takes for the Earth and Mars to revolve once around the Sun as TEarth and TMars respectively. 

Opposition will occur at intervals of t = TOpposition when their relative angular rates are equal to multiples of 360° or 2π radians.  The is expressed as

 

 

 

We can then use this relation to calculate either the length of the Martian year in the heliocentric model, Tmars, or the observed time between oppositions, TOpposition, as

 

or

 

From long multi-generational study and by as early as 3000 B.C., the length of the Earth year or TEarth , (about 365.25 days or more precisely 365.2422 days) was known from measurement of the equinoxes.  We also knew the time between oppositions or TMars (about 780 days or more precisely 779.94 days).  Assuming a heliocentric model, this allows us to calculate the length of the Martian year TMars (about 687 days or more precisely 686.98 days).  As for instance