**Planetary
Opposition**

Consider two planets orbiting the sun in perfectly circular orbits. These planets are said to be in opposition when the distance between them is the smallest possible value. From the perspective of the inner planet, this will happen when the outer planet is exactly opposite the sun in the sky. This is illustrated below for the Earth and Mars.

From the perspective of the inner planet, the outer one will appear to move along the same circular path in the sky called the elliptic. As the two planets approach opposition the outer planet will slow down and stop in the sky at opposition. Immediately after opposition the outer planet will seem to briefly reverse direction before again resuming its normal path. And this phenomenon of “retrograde motion” will happen only once at each opposition.

**TIME BETWEEN OPPOSITIONS**

We will denote the time (in days) it takes for the Earth and
Mars to revolve once around the Sun as T_{earth} and T_{mars} respectively.
From long term observations of the motion of the Sun and stars in our sky even
in ancient time (by 3000 B.C.) is was possible to accurately determine the
length of the Earth year, or T_{earth }. This is approximately 365.25
days (or more precisely 365.2422 days).

In any event, the angle of the planet relative to the sun is or

At opposition, this angle, , of each of the planets will be equal. But this angle must be less than or radians in order to compare the angle of Earth with that of Mars. And since it takes more than two years for the Earth to catch up with Mars we need to subtract out whole numbers of revolutions. So the formula is

where n = 2 (Earth years) and m = 1 (Mars year). We can
then use this relationship to calculate either the time between oppositions or
the length of the year on Mars, T_{mars}.

or

For the time between oppositions, t, where T_{mars}
= 687 days (or more precisely 686.98 days) we have