Question

What is the distance between Uranus and Saturn in km?

Answer 1

Overlooking orbital-plane small inclinations `2.49^o and 0.77^o` to the ecliptic, the average distance in-between Saturn and Uranus is the semi-major axis of the Uranus orbit = 2872.5 million km

Explanation:

The orbit of Uranus is outer to the orbit of Saturn.

The distance between any two orbiters is maximum, when they are farthest from the Sun and also in opposition. This distance is the sum of the aphelions.

The distance is least when they are on the same side of the Sun, the outer planet being at perihelion and the inner planet is at aphelion.

So, the average is easily seen as

( (sum of aphelions) + (outer planet perihelion-inner planet aphelion))/2.

= (aphelion + perihelion, of the outer planet)/2

= the semi-major axis of the elliptic orbit of the outer plant,
using aphelion = a ( 1 + e ) and perihelion = a ( 1 - e )..

Here, it is Uranus a = 2872.5 Million km.

Rigorous formula for distance at any time is

Saturn-Uranus distance

`r_(su)=sqrt(r_s^2+r_u^2-2 r_s r_u cos theta_(su))`,

where`r_s`=Sun-Saturn distance, `r_u`=Sun-Uranus distance and `theta_(su)` is the angle between the radii to the planets from the Sun.