planetary orbit times?

Question

if i know uranus' orbital period, i know its distance from the sun and it moves from opposition to quadrature with respect to earth, how do i work out how long it takes to move between these 2 points. a circular orbit is assumed.

radius = 19.23 AU
sidereal period = 84.33 years
synodic period = 1.012 years

i feel like this is such a simple thing to do but ive ran out of steam tonight.

cheers.

Answer 1

From a heliocentric point of view, the angle q at quadrature must be sin(q)=1/19.23, so q = 2.98 degrees. The angle at opposition is 90, so Uranus must move 90-2.98 = 87.02 degrees from opposition to quadrature. This is equal to 87.02 / 360 = .2417 of a circle. Since the synodic period is 1.012 years, the time from opposition to quadrature must be 1.012 * .2417 = .2446 years.

Answer 2

I am not sure if moving from opposition to quadrature takes exactly a quarter of a synodic period, but in this case (such a distant planet and the synodic period so close to 1 year) it must be very close to that. So I say a quarter of 1.012, or 0.253 years.