= Linear limb-darkening law
{title2=$I(\mu)=I_c[1-u(1-\mu)]$}
The <linear limb-darkening law> parametrizes angular <specific intensity> by a stipulated <limb-darkening coefficient> $u$:
$$
I(\mu)=I_c[1-u(1-\mu)],\qquad I_c=I(1),\qquad 0\le u\le1.
$$
Here $I$ denotes <specific intensity> at one <frequency>, or <radiance> integrated over a specified interval of <frequency> or <wavelength>; it does not denote <radiant intensity>. Within this stipulated range, $u=0$ gives a uniform disk and $I(0)/I_c=1-u$. The <linear limb-darkening law> is linear in $\mu$, while its radial profile $I(r)=I_c[1-u+u\sqrt{1-(r/R_\star)^2}]$ is curved. In the linear <source function> example above, $u=b_\nu/(a_\nu+b_\nu)$ when $a_\nu,b_\nu\ge0$ and $a_\nu+b_\nu>0$.
For the illustrative choice $u=0.6$, the <specific intensity> at $r/R_\star=0$, $0.8$, and the limiting edge is respectively $I_c$, $0.76I_c$, and $0.4I_c$. These are synthetic values, not measurements of the <Sun>. Real <optical spectra> and <stellar atmospheres> need not obey a <linear limb-darkening law>; more flexible profiles can be necessary for accurate <exoplanet transit photometry>. <Analytic Light Curves for Planetary Transit Searches by Kaisey Mandel and Eric Agol>[§1, p. L171: the linear law is insufficient for the cited high-quality transit observations; §4 defines a quadratic law]{ref}
\Image[/stellar-limb-darkening-disk-profiles.png]
{title=Synthetic stellar disk and angular and radial <limb darkening> profiles}
{description=The disk uses the stipulated <linear limb-darkening law> with $u=0.6$. The profiles compare four stipulated <limb-darkening coefficients> at equal central <specific intensity>. Shading represents relative brightness, without prescribing a <wavelength> or copying a solar measurement.}
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