{"knowledge_schema": {"broad_category": "Astronomy \u2192 Stellar Observations \u2192 Parallax and Distance", "refinement": "This problem examines the relationship between the observed number of stars and their distance from the observer, specifically how parallax affects star visibility.", "specific_scope": "The focus is on understanding how the number of stars, which varies with parallax as 1/plx^5, translates to a change in the number of stars with distance (r).", "goal": "Determine the mathematical relationship between the number of stars and distance based on the given parallax dependency."}, "summary": "In this problem, we explore how the number of stars in a specific region of the sky changes with distance, given that the number of stars varies with parallax as 1/plx^5. To find the relationship with distance (r), we need to understand that parallax (plx) is inversely related to distance. As distance increases, parallax decreases, leading to a change in the number of stars observed. The goal is to derive the relationship between the number of stars and distance, which results in a dependence of r^3, indicating that the number of stars increases with the cube of the distance."}