How does the energy output of a K-type star with $0.7 M_ ext{sun}$ compare to a G-type star with $1.0 M_ ext{sun}$?
Answer
The $0.7 M_ ext{sun}$ star emits significantly less energy because it has less material to fuse and lower core pressure.
Stellar energy output scales extremely steeply with mass, often cited as being proportional to the mass raised to the fourth or fifth power ($L ext{ scales as } M^4 ext{ or } M^5$). Because the $0.7 M_ ext{sun}$ star possesses substantially less mass than the $1.0 M_ ext{sun}$ star, the gravitational forces compressing its core are weaker, leading to lower core pressure and a slower rate of nuclear reactions. This results in a vastly reduced total energy output compared to the solar-type star, which is the fundamental reason for its longer lifespan.
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