{"knowledge_schema": {"broad_category": "Astronomy \u2192 Stellar Distribution", "refinement": "This problem examines the relationship between the distance from an observer and the number of stars observed within that distance, considering a uniform distribution of stars in three-dimensional space.", "specific_scope": "The focus is on how the density of stars changes as one moves away from a point in space, specifically looking at the volume of space that increases with distance.", "goal": "To understand and determine how the number of stars per unit range of distance (r) varies as a function of r."}, "summary": "In this problem, we are exploring how the number of stars changes with distance in a uniformly distributed stellar environment. As we move away from a point in space, the volume of the sphere increases with the square of the radius (r^2), leading to an increase in the number of stars observed. Therefore, the number of stars per unit range of distance (r) is proportional to r^2, as the surface area of a sphere increases with the square of the radius."}