Interior planets are the planets whose orbits are inside the orbits of the earth and superior planets are those planets whose orbits are outside the orbits of the earth.

Synodic period (S) is the former time interval of planet and Sidereal period (P) is the latter time interval of planet.

Conclusion:

The required diagram is shown in Figure 1.

Figure 1

The above figure shows the orbital motion of the earth and mars.

The Earth makes SP⊕ orbits during the time required for another planet to make SP orbits. If the other planet is superior, then to overtake the planet earth must make one extra orbit around the sun and it is expressed as,

    SP⊕=SP+1

By rearranging the above equation.

    SP⊕−SP=1S(1P⊕−1P)=11S=1P⊕−1P

In the case of inferior planet, the planet has to make one extra trip to overtake the earth.

    SP=SP⊕+1

By rearranging the above equation

    SP−SP⊕=1S(1P−1P⊕)=11S=1P−1P⊕

Thus, the relationship between a planet’s synodic period and its sidereal period is 1S=1P⊕−1P for superior planet and 1S=1P−1P⊕ for inferior planet.