= Solution
Integrating $dP/dr=-GM_r\rho/r^2$ in order of magnitude, with $\rho\sim M_*/R_*^3$, gives
$$
P_c\sim\frac{GM_*^2}{R_*^4}.
$$
The <ideal gas law> independently gives $P_c\sim M_*k_BT_c/(\mu m_HR_*^3)$. Equating the two estimates yields
$$
\boxed{R_*\sim\frac{G\mu m_H}{k_BT_c}M_*.}
$$
For similar composition and an approximately fixed hydrogen-burning <central temperature>, $R_*\propto M_*$. Calibrating this rough relation on the Sun and using part <S2/1/a> gives a nominal <mass> range of about $0.05$ to $5\,M_\odot$.
The endpoint <luminosity> ratio is $10^8$, whereas the inferred <mass> ratio is about $10^2$, giving $L\propto M^4$ across that range. The \i[exponent] is therefore consistent at this schematic level. The solar <normalization> and individual endpoint <masses> are much less reliable: using the <luminosity> relation itself gives approximately $0.1$ to $10\,M_\odot$. <Central temperatures> are not truly constant, low-mass stellar structure differs, and the uncorrected $V$ <luminosities> in the diagram are not bolometric. These limitations prevent treating the rough radius-based <masses> as a contradictory precision measurement.
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