Weird render settings!
To get an extremely clean looking function render in the very small or sharp details I've found it best to use 'Ray trace everything' in the renderer, even an AA of 2 will give cleaner results than a render detail of '1' without RTE. All the flat 2D images have AA=2
The problem using RTE is that once you introduce displacement, the surface of the planet, even at render detail '1' is terribly optimized.
RTE, detail=1, AA=4...
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To get around it and have a lovely smooth RTE render with displacement you'll want to first disable GI relative detail, create a fill lighting instead(because of the next step) and then raise the render detail to way past '1', until you find a good balance of displacement and subdivision.
The 1280px image on the 2nd page was RTE, detail=8, AA=4. The 700px image was RTE, detail=10, AA=4.
You'd be alarmed using detail '10' in most cases but here it's the cleanest option and it's still fast to render since there is no GI used, the detail slider is really only being used to smooth the planet surface and do the shadows.