How much brighter is a 10 M☉ star expected to be using the M^3.5 relation?
Answer
Roughly 3,162 times more luminous
The Mass-Luminosity Relation, approximated by $L ext{proportional to } M^{3.5}$, dictates the luminosity increase relative to the Sun ($1 M_{ ext{Sun}}$). To calculate the expected luminosity for a star ten times the mass of the Sun ($10 M_{ ext{Sun}}$), one must raise the mass multiple to the power of 3.5. Calculating $10^{3.5}$ yields a factor of approximately 3,162. This calculation demonstrates that a star only ten times heavier than the Sun is over three thousand times brighter, highlighting the dramatic output difference driven by the exponential relationship between mass and luminosity.
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