We are enjoying a planetary parade during the beginning of this month here on Earth, but my alien friend from the distant planet Pyrknot (well modeled as a perfect sphere) isn’t too impressed. Their single-star system has six planets other than Pyrknot with such chaotic orbits that each night these planets independently appear in uniformly random locations in the Pyrknothian sky (the planets aren’t visible in daylight). If at a given moment there exists somewhere on Pyrknot that all six planets are visible, then from my friend’s position on the surface they have an α probability of also seeing all of the planets.

My friend is considering building a tower to improve their chances of seeing these planetary parades. Assume the tower allows the viewing of every celestial body that could be seen from at least one location on the surface of Pyrknot less than a distance r from the base of the tower. In the limit of r being small compared to the radius R of Pyrknot, the new probability of seeing the planetary parade from the top of the tower is linearly approximated by α + β·(r/R). Find α and β in exact terms.