Stellar Luminosity

Stellar Luminosity

A star radiates as a near-perfect blackbody. Its total luminosity is governed by the Stefan-Boltzmann law:

$L = 4\pi R^2 \sigma T_{\text{eff}}^4$

where:

• $R$ is the stellar radius in metres
• $T_{\text{eff}}$ is the effective surface temperature in Kelvin
• $\sigma = 5.6704 \times 10^{-8}$ W m⁻² K⁻⁴ is the Stefan-Boltzmann constant

Solar Units

Astronomers often express luminosity relative to the Sun:

$\frac{L}{L_\odot} = \left(\frac{R}{R_\odot}\right)^2 \left(\frac{T}{T_\odot}\right)^4$

The Sun's parameters: $R_\odot = 6.957 \times 10^8$ m, $T_\odot = 5778$ K, $L_\odot = 3.828 \times 10^{26}$ W.

A star with twice the radius and the same temperature has four times the luminosity. A star twice as hot (same radius) has sixteen times the luminosity.

Inverse: Effective Temperature

Given the luminosity and radius, you can recover the effective temperature:

$T_{\text{eff}} = \left(\frac{L}{4\pi R^2 \sigma}\right)^{1/4}$

This is used to determine stellar surface temperatures from measured luminosities and radii.

Your Task

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

• stefan_boltzmann_luminosity(R_m, T_K) — returns $L$ in watts
• luminosity_solar(R_m, T_K) — returns $L / L_\odot$ (dimensionless)
• effective_temperature(L_W, R_m) — returns $T_{\text{eff}}$ in Kelvin

Use $\sigma = 5.6704 \times 10^{-8}$ W m⁻² K⁻⁴ and $L_\odot = 3.828 \times 10^{26}$ W.