Final answer:
Stars can have the same temperature but vastly different luminosities due to differences in their size and radius.
Step-by-step explanation:
Two stars can have the same temperature, but vastly different luminosities due to differences in their size and radius. The luminosity of a star is determined by its surface temperature and its size. A star with a larger radius will have a larger surface area, allowing it to emit more energy and be more luminous, even if it has the same temperature as a smaller star.
For example, star A has a surface temperature of 6000 K, while star B has a surface temperature of 12,000 K, which is twice as high. Assuming both stars have the same size and are the same distance from us, star B would be much more luminous than star A because of its higher surface temperature.
This difference in luminosity can be calculated using the Stefan-Boltzmann law, which states that the luminosity of a star is proportional to the fourth power of its surface temperature. Therefore, a star with twice the surface temperature will be approximately 16 times more luminous.