Recently, I've watched this video by Veritasium describing Poisson's Spot, or the Arago Spot. It is explained in the video that near circular (or spherical) objects can produce this optical effect, given that there aren't many obstructions about the circumference of the object to diffuse and scatter the light and the light producing the shadow is more or less in phase.
My question is, Are Arago Spots possible/Have Arago Spots been observed in the solar shadows of any celestial bodies?
The Arago spot phenomenon requires the light source to be point-like and very bright. The sun is not a point-like light source, as it has an apparent diameter of about 0.5 degrees
The only body large enough to cast a full solar shadow is the moon. Arago's spot has not been observed during a total solar eclipse.
@JamesK's answer is basically correct. I'd change "be point-like" to "be parallel" or "be collimated", but then I'd change that to "have a high degree of partial coherence" or maybe "have high etendue because a slightly converging or diverging wave front could produce a similar spot-like structure.
The important thing is that the spread in the directions of rays from the source does not blur the spot out so much that it becomes so weak and spread out that it is not noticeable, or isn't really a spot anymore.
There are several other problems as well, mostly related to scale.
For visible light, a planet's roughness is so huge compared to a wavelength that it will not produce a small well-defined spot. See how Deviation from circularity points out that the spot's existence and size can be though of in terms of a point-spread function of the source.
You wouldn't compare the planet's roughness with a wavelength directly, but perhaps to something like $\sqrt{\lambda \ R_{planet}}$. For the Earth that's a few meters, so the Earth and other planets are too rough. For the roughness example in Wikipedia using a 4 mm sphere, that's 45 microns, and that nicely explains why the simulated edge corrugations of 10 microns do not remove the central peak, but they are weaker at 50 microns and nearly gone at 100 microns!
And while much of the contribution comes from rays that pass fairly near the obstruction, a planet's atmosphere's refraction will muddle that and have more refractive than diffractive effect.
This is why most demonstrations use the collimated light from a laser. It's not for the wavelength purity (the detailed fine structure depends on wavelength, but the spot is an enhancement extended along the axis for any wavelength) but for the extremely high partial coherence.
But that shouldn't completely stop you from looking for an Arago spot in the radio wavelengths, especially at low frequency where all the problems related to scale are less severe. Long wavelength signals from the Earth (e.g. AM radio) are blocked from reaching space by the ionosphere unfortunately, so you can't sneak around behind the Moon to look for your favorite radio station's Arago spot. That's too bad because the Moon would likely be smooth enough for those half-km wavelengths, and it's quite round (rather than oblate) as well. The nice thing about that would be the narrow frequency of the carrier in standard AM transmissions; detection of that would be far simpler than a broadband radio source.
