Tropical timings – the orbit of Uranus

Uranus [image credit: NASA]

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The idea here is to demonstrate that the Uranus orbit period on the solar simulator is in tropical years, using the same process as in our Neptune orbit post (here).

The alternative would be a sidereal year period based on the so-called ‘fixed stars’ as a reference frame. As Wikipedia points out, ‘The term fixed stars is a misnomer because those celestial objects are not actually fixed with respect to one another or to Earth’, but their movement from an Earth perspective is extremely slow due to their vast distances away from us.

From the NASA planetary factsheet for Uranus we find:
Sidereal orbit period (days) — 30,685.40 (U) — 365.256 (E) — 84.011
Tropical orbit period (days) — 30,588.74 (U) — 365.242 (E) — 83.749

The last figure in each row is the time of one Uranus orbit in Earth years. The data can be used on Arnholm’s solar simulator to test planetary motions.

To improve the accuracy we use the tropical year of Earth as published – 365.24219 days, so the NASA orbit period is:
30588.74 / 365.24219 = 83.7492 tropical years (TY)
The period of a Uranus-Earth conjunction using this result is:
83.7492 TY * 1 / (83.7492 – 1) = 83.7492/82.7492 = 1.0120847 TY.

Since 83.7492 * 4 = 99.999% of 335 years, a simple formula to test on the solar simulator is:
4 Uranus orbits = 331 Uranus-Earth conjunctions = 335 TY.

On the solar simulator, the note at the top says: ‘Looking down on Sun’s north pole’. This defines its frame of reference. For a first check we compare two Uranus-Earth conjunctions 335 tropical years apart.

That shows 4 Uranus and 335 Earth orbits (= tropical years) is a period when their conjunctions occur in the same orientation to the Sun.

For longer checks, the 3000-year window of the simulator allows time for 8 sets (of 4 each) test Uranus orbits = 32*83.75 = 2680 TY. The screenshots will show two comparisons of the orientation of Uranus and Earth, one from an initial conjunction in year zero, and another from initial year 310. The comparison years are 2680 TY (335*8) ahead, i.e. year 2680 itself and year 2990. (The first comparison ends exactly 8 Uranus orbits after the 2010 graphic shown above.)

These results confirm that the simulator represents the planetary movement of Uranus using the tropical orbit data, completing 32 orbits within a day or two of 83.75 TY * 32. The test orbit period of 83.75 TY = 30589.033 days, only 0.293 days more than the NASA published tropical orbit period (30588.74 days).

Note that using the NASA sidereal orbit number the whole test period would be over 8 years greater, at about 2688.35 years, and would not return to the same orientation, due to the 0.35 years ‘overlap’.
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Image: Uranus [credit: NASA]
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For other ‘tropical timings’ posts see here:
https://tallbloke.wordpress.com/?s=tropical+timings