If a star has twice the mass of the Sun ($2 M_\odot$), what is its approximate luminosity relative to the Sun using $\alpha=3.5$?
Answer
About 11.3 times brighter ($2^{3.5}$)
When applying the mass-luminosity relation using the common approximation $\alpha=3.5$, the luminosity scales by the mass raised to that power. For a star with twice the solar mass ($2 M_\odot$), its luminosity ($L$) is calculated as $2^{3.5}$. Performing this calculation yields a result of approximately $11.3$. This demonstrates the non-linear effect: doubling the mass does not double the brightness; instead, it results in over eleven times the luminosity because the core fusion process becomes dramatically more vigorous under the increased gravitational compression.
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