{"knowledge_schema": {"broad_category": "Astronomy \u2192 Stellar Properties \u2192 Luminosity", "refinement": "This problem examines the relationship between a star's luminosity, temperature, and radius, applying the Stefan-Boltzmann Law.", "specific_scope": "The focus is on comparing the luminosity of two stars with different temperatures and radii but the same mass.", "goal": "Calculate the ratio of the luminosity of Star_A to Star_B using their respective temperatures and radii."}, "summary": "In this problem, we analyze the luminosity of two stars, Star_A and Star_B, using the Stefan-Boltzmann Law, which states that luminosity is proportional to the square of the radius and the fourth power of the temperature. Given that Star_A has a surface temperature twice that of Star_B and a radius half that of Star_B, we can derive the luminosity ratio. Despite the differences in temperature and radius, the calculations show that the luminosity ratio of Star_A to Star_B is 1:1, indicating that both stars emit the same amount of energy."}