how do i work out synodic period?

Question

if i know a planet's sidereal period and i know how far that planet is from the sun, assuming a circular orbit, how do i find out its synodic period?

i have uranus' radius of orbit = 19.23 AU
and its sidereal period = 84.33 earth years
how do i get its synodic period?
is there some simple formula?

cheers.

Answer 1

In 84.33 Earth years, Uranus will have orbited the Sun once, and Earth will have orbited the Sun 84.33 times. During those 84.33 orbits, Earth will have passed Uranus 83.33 times. The difference of exactly 1 is because of Uranus's exactly 1 orbit during that time.

Therefore, Earth has an opposition to Uranus every 84.33 / 83.33 years, which is Uranus's synodic period, or 1.012 years.

Answer 2

This wikipedia page has the formula.