{"knowledge_schema": {"broad_category": "Astronomy \u2192 Stellar Properties \u2192 Brightness and Distance", "refinement": "This problem examines the relationship between a star's apparent brightness and its distance from the observer, as measured by parallax.", "specific_scope": "The focus is on understanding how the apparent brightness of a star changes as its parallax angle varies, specifically in the context of uniform distribution of stars in space.", "goal": "Identify the mathematical relationship that describes how apparent brightness decreases with increasing distance (or decreasing parallax) of a star."}, "summary": "In astronomy, the apparent brightness of a star is inversely proportional to the square of its distance from the observer. This relationship is expressed mathematically as brightness \u221d 1/distance\u00b2. Since parallax (plx) is a measure of distance, the apparent brightness of a star changes with its parallax according to the formula 1/plx\u00b2. Thus, as the parallax decreases (indicating the star is farther away), the apparent brightness diminishes significantly."}