What is the inverse scaling relationship governing the contraction timescale ($ au$) for high-mass stars based on mass ($M$)?
Answer
The timescale scales roughly as $M^{-1.5}$
The contraction timescale ($ au$) is highly sensitive to mass, governed by the interplay between the energy generated and the efficiency of its release. For high-mass stars, where luminosity ($L$) scales steeply as $M^{3.5}$, the resultant contraction timescale exhibits a steep inverse relationship with mass. Specifically, $ au$ scales roughly as $M^{-1.5}$. This inverse relationship implies that as mass increases, the time required to reach the Zero-Age Main Sequence (ZAMS) decreases significantly, explaining why massive stars finish forming so much faster than low-mass stars.
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