PX282 - A9 - connection to luminosity

luminosity is the total energy emitted:

\begin{aligned} L & = \int d Ω \int_{s t a r} d A \int_{0}^{\infty} B_{λ} d λ \\ = \int_{0}^{2 π} \int_{0}^{π} \cos θ \sin θ d θ d ϕ (4 π R_{*}^{2}) (σ \frac{T^{4}}{π}) \end{aligned}

where,
$d Ω = \sin θ d θ d ϕ$ ,
$\int_{s t a r} d A = 4 π R_{*}^{2}$ ,
$\int_{0}^{\infty} B_{λ} d λ = (σ \frac{T^{4}}{π})$
- $\cos θ$ arises from the angular profile of the emitting area

only looking at light coming out of the star limits in $θ$ half:

\begin{aligned} L & = 2 π {[- \frac{1}{2} \cos^{2} θ]}_{0}^{\frac{π}{2}} 4 R_{*}^{2} σ T^{4} \\ ∴ L & = 4 π R_{*}^{2} σ T_{e f f}^{4} \end{aligned}

this is the luminosity-radius-temperature relation of a star
the flux:

f = σ T_{e f f}^{4}

notes:
- $T^{4}$ makes errors inflate rapidly
- real spectra are not perfect blackbodies
- $T_{e f f}$ is very effective for classifying stars