= Limb-darkened disk flux
{title2=$\overline I=I_c(1-u/3)$}
For a circular stellar disk whose <specific intensity> is independent of <azimuth>, its mean <specific intensity> over <projected area> is
$$
\overline I=\frac{1}{\pi R_\star^2}\int_{\rm disk} I\,dA_{\rm proj}
=2\int_0^1 I(\mu)\mu\,d\mu.
$$
The <polar coordinates> element is $dA_{\rm proj}=2\pi r\,dr$ after angular integration, and $r\,dr=-R_\star^2\mu\,d\mu$. Substitution of the <linear limb-darkening law> gives
$$
\overline I=2I_c\int_0^1(1-u+u\mu)\mu\,d\mu=I_c(1-u/3).
$$
For a distant observer at distance $D\gg R_\star$ receiving <light> normally to the detecting plane, the <observed radiative flux> is
$$
f_\nu\simeq\frac{\pi R_\star^2}{D^2}\overline I_\nu
=\pi\left(\frac{R_\star}{D}\right)^2I_{\nu,c}(1-u_\nu/3).
$$
It is spectral <irradiance>, with units $\mathrm{W\,m^{-2}\,Hz^{-1}}$ when $I_\nu$ is per hertz. The local outward <radiative flux> per actual surface <area> is instead $F_\nu=2\pi\int_0^1 I_\nu(\mu)\mu\,d\mu=\pi\overline I_\nu$, a spectral <radiant exitance>. In a spherical model with the same <stellar atmosphere> everywhere, the <luminosity> is $L_\nu=4\pi R_\star^2F_\nu$ and $f_\nu=L_\nu/(4\pi D^2)$, consistent with the <inverse-square law>. Compare disks at fixed $I_c$ or fixed total <observed radiative flux> explicitly: increasing $u$ lowers the total in the first comparison, while the second requires a larger $I_c$.
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