We are enjoying a planetary parade here on Earth, but my alien friend from the distant planet Pyrknot isn’t too impressed. Pyrknot is a perfect sphere of radius R. Its single-star system has six planets other than Pyrknot with such chaotic orbits that they appear in uniformly random locations in the Pyrknothian sky.
Precisely: the star and the six planets are all so far away that each is effectively a point at infinity, i.e. just a direction in the sky. At the moment in question, the seven directions (the star's and the six planets') are independent and uniformly distributed over the whole sphere of directions. A body is above the horizon at a surface point exactly when it is strictly above that point's local horizon (the plane tangent to the surface there). The star is visible from a point whenever it is above the horizon there. A planet is visible from a point exactly when the planet is above the horizon there and the star is not (planets aren’t visible in daylight). There is no weather or other obstruction.
My friend stands at a point chosen uniformly at random on the surface, independently of the star and planets. Suppose it is known that there exists some point on Pyrknot’s surface from which all six planets are visible. Given this, let α be the probability that all six planets are also visible from my friend’s position.
My friend is considering building a tower at their position to improve their chances of seeing these planetary parades. From the top of the tower, a celestial body (planet or star) is above the horizon exactly when it is above the horizon at at least one surface point less than a distance r (measured along the surface) from the base of the tower; visibility from the top then follows the same rules as above. In particular, the top of the tower is in daylight, and sees no planets at all, whenever the star is above the horizon at any surface point within distance r of the base. For r small compared with R, the probability (given the same fact) that all six planets are visible from the top of the tower is
α + β·(r/R) + O((r/R)²).
Find α and β.
Answer format: two exact fractions, in this order: α, β.