Checking Kepler's third law for the inner planets

Updated 31 May 26

(You'll need to view this in landscape mode if using a small screen).

It's easy to see from the diagram in the previous post that the distance from the Sun of an inner planet can be found by measuring the angle of greatest elongation (East or West) from the Sun and then taking the sine of that angle.

There will be some variation, so we'll need to take an average over many measurements. We will consider orbits to be more-or-less circular and hence will loosely refer to this distance as the orbital radius.

For Earth and the inner planets:

Planet	Orbit radius (r) in AUs	r^3	Period (t) in years	t^2	r^3/t^2
Mercury	sin 22.5° * = 0.38	~0.06	7/29	0.06	0.96
Venus	sin 46° *  = 0.72	~0.37	8/13	0.38	0.98
Earth	1	1	1	1	1

(* from approximate average greatest elongation data here)

So far, so good, but to extend this to the outer planets, I'll need a method of determining the orbital radii for those. 🤔