Despite a significant drop in surface temperature, why does a star become substantially brighter overall upon becoming a red giant?

Answer

The surface area increases enormously, spreading the energy output over a vast region

The paradox of a red giant being both cooler on the surface yet much more luminous overall stems directly from the massive increase in the star's physical dimensions. As the star swells to hundreds of times its original diameter, its surface area increases by a factor proportional to the square of the radius increase. Even though the energy per square meter (surface temperature) drops significantly, the total energy radiated (luminosity) increases substantially because this lower-intensity energy is being emitted across a vastly larger area. This relationship is analogous to comparing a small, very hot light source to a massive, cooler heater that outputs more total energy because of its sheer size.

Despite a significant drop in surface temperature, why does a star become substantially brighter overall upon becoming a red giant?

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