= Limb darkening
{wiki}
Limb darkening is the decrease of emergent <specific intensity> toward the apparent edge of a stellar disc. A transit depth then depends on the intensity sampled by the planet as well as its projected area; the uniform-disc area ratio is only an approximation.
For a spherical <star> of <radius> $R_\star$, let $r$ be distance from the centre of its projected disk and let $\theta$ be the <angle> between the outward <unit normal> and the <line of sight>. The angular coordinate is
$$
\mu=\cos\theta=\sqrt{1-(r/R_\star)^2}.
$$
Disk centre has $\mu=1$; the apparent edge approaches $\mu=0$. A resolved image measures <specific intensity> in each direction. <Limb darkening> means that this <specific intensity> itself falls toward the edge; the <projected area> factor used when collecting <radiant flux> is a separate geometric effect.
A simple explanation assumes a horizontally uniform <plane-parallel atmosphere>, a sufficiently deep <optically thick medium>, negligible <scattering of light>, and thermal absorption and emission in <local thermodynamic equilibrium>. Let $\tau_\nu$ be inward vertical <optical depth> at fixed <frequency> $\nu$. An oblique <light ray> has <line of sight> <optical depth> $\tau_\nu/\mu$, so it encounters the same attenuation at a smaller vertical <optical depth>. The <formal solution of the radiative transfer equation> gives
$$
I_\nu(0,\mu)=\int_0^\infty S_\nu(\tau_\nu)e^{-\tau_\nu/\mu}\frac{d\tau_\nu}{\mu},\qquad 0<\mu\le1.
$$
Here $S_\nu$ is the <source function>; under these assumptions it is the <Planck function> $B_\nu[T(\tau_\nu)]$. If <temperature> increases inward, an oblique view preferentially samples cooler layers and hence lower <specific intensity>. <Radiative Transfer in Stellar Atmospheres by Robert J. Rutten (8th edition)>[§2.2.2, pp. 17–19, equations (2.37)–(2.44) and Figure 2.4]{ref}
The weight $e^{-\tau_\nu/\mu}/\mu$ is an <exponential distribution> in vertical <optical depth>, with <expected value> $\mu$ and <standard deviation> $\mu$. Thus “seeing down to <optical depth> one” refers to the <line of sight>, and represents a broad sampling of layers rather than a sharp emitting surface. For a <source function> that is linear throughout the atmosphere, the <Eddington-Barbier relation> is exact:
$$
S_\nu(\tau_\nu)=a_\nu+b_\nu\tau_\nu
\quad\Longrightarrow\quad
I_\nu(0,\mu)=a_\nu+b_\nu\mu=S_\nu(\mu).
$$
This follows by integrating the constant and linear terms separately. A constant <source function> instead gives constant <specific intensity>, as in the <blackbody limit of an isothermal atmosphere>. An outward-rising <temperature> or important <scattering of light> changes the argument. Close enough to the actual edge, a <plane-parallel atmosphere> also loses validity because of curvature. Neither one <limb-darkening coefficient> nor a universal dependence on <wavelength> follows from this simple model.
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