During contraction, what is the approximate luminosity scaling relationship for high-mass stars relative to their mass ($M$)?
Answer
Luminosity is often proportional to $M^{3.5}$
The text establishes a very steep relationship between mass and luminosity for high-mass protostars as they undergo gravitational contraction. Because these objects are dominated by radiation-driven energy loss due to their high internal temperatures and transparency, their luminosity ($L$) scales very steeply with their mass ($M$). The approximate relationship provided for this scaling is $L$ proportional to $M^{3.5}$. This extreme dependence on mass means that even small increases in mass yield disproportionately large increases in energy output, which directly impacts how quickly they shed gravitational energy and contract.
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