Main Sequence Scaling Laws

Main Sequence Scaling Laws

On the main sequence, stars fuse hydrogen in their cores. Their bulk properties — luminosity, radius, and temperature — scale predictably with mass via empirical power laws.

Luminosity

For solar-type main-sequence stars, luminosity scales steeply with mass:

$\frac{L}{L_\odot} = \left(\frac{M}{M_\odot}\right)^4$

This strong dependence explains why massive stars are so short-lived: they burn fuel far faster.

Radius

Stellar radius scales more gently with mass:

$\frac{R}{R_\odot} = \left(\frac{M}{M_\odot}\right)^{0.8}$

Effective Temperature

The effective temperature follows from the Stefan–Boltzmann law ($L = 4\pi R^2 \sigma T_{\rm eff}^4$). Combining the luminosity and radius scaling laws:

$T_{\rm eff} \propto \frac{L^{0.25}}{R^{0.5}} \propto M^{4 \times 0.25 - 0.8 \times 0.5} = M^{0.6}$

In absolute terms:

$T_{\rm eff} = T_\odot \cdot \left(\frac{M}{M_\odot}\right)^{0.6}$

where $T_\odot = 5778$ K is the solar effective temperature.

These approximations are valid for main-sequence stars in the range $0.5$–$50\ M_\odot$.

Your Task

Implement three functions. All constants must be defined inside each function.

• main_sequence_luminosity(M_solar) — returns $L/L_\odot = (M/M_\odot)^4$
• main_sequence_radius(M_solar) — returns $R/R_\odot = (M/M_\odot)^{0.8}$
• main_sequence_temperature(M_solar) — returns $T_{\rm eff}$ in K, using $T_\odot = 5778$ K