= Small-planet transit weighting
{title2=$\delta\simeq k^2I(\mu_p)/\overline I$}
A dark, opaque circular <planet> of <radius> $R_p$ blocks <specific intensity> over its overlap with the projected <star>. With $k=R_p/R_\star$ and no <transit dilution>, the instantaneous <exoplanet transit depth> is exactly
$$
\delta=\frac{\displaystyle\int_{\rm overlap}I\,dA_{\rm proj}}{\pi R_\star^2\overline I}.
$$
When $k\ll1$, the entire planetary disk is inside the stellar disk, and <specific intensity> varies little across that disk, <small-planet transit weighting> reduces to
$$
\delta\simeq k^2\frac{I(\mu_p)}{\overline I}
=k^2\frac{1-u+u\mu_p}{1-u/3},\qquad
\mu_p=\sqrt{1-z^2},\qquad z<1-k.
$$
Here $z$ is the projected centre separation divided by $R_\star$. The numerator samples the local <specific intensity>; the denominator normalizes by the whole <star>'s mean <specific intensity>. The small-size argument and the distinction between full overlap and ingress are explicit in the original analytic treatment. <Analytic Light Curves for Planetary Transit Searches by Kaisey Mandel and Eric Agol>[§2, p. L171, geometry; §5, p. L174, small-planet approximation restricted to the interior $z<1-p$]{ref}
For a short approximately straight crossing at constant transverse <velocity> $v_\perp$, write $x=v_\perp t/R_\star$ with $t=0$ at mid-transit, and let $b$ be the <transit impact parameter>. Then $z^2=x^2+b^2$ and $\mu_p=\sqrt{1-b^2-x^2}$. With stipulated $u=0.6$ and $k=0.05$, a central crossing has mid-transit <exoplanet transit depth> approximately $0.003125$, whereas $b=0.8$ gives $0.002375$. A uniform disk would give $k^2=0.0025$ for either fully overlapping crossing. The brighter central chord produces a deeper dip and the off-centre chord a shorter crossing at the same $v_\perp$.
\Image[/limb-darkened-transit-chord-weighting.png]
{title=Two transit chords across the same synthetic <limb-darkened disk flux> distribution}
{description=Stipulated $u=0.6$ and $k=0.05$; the coloured curves integrate the occulted <specific intensity> over the finite planetary disk, including ingress and egress. Dashed curves show <small-planet transit weighting> only where the whole planetary disk is inside the <star>. Horizontal dotted segments give the uniform-disk full-overlap depth $k^2$. The chord panel shows mid-transit positions; the horizontal coordinate in the <light curve> is distance travelled divided by stellar <radius>, equivalent to scaled <time> at constant transverse <velocity>.}
At fixed $z<1$ away from the edge, a two-dimensional <Taylor's theorem> expansion over the symmetric planetary disk cancels the first <derivatives>; the first correction to the blocked <radiant flux> is of order $k^4$ if the second <derivatives> remain bounded. This error estimate is not uniform at the edge, where the radial <derivative> of the <linear limb-darkening law> diverges. Ingress, egress, a grazing <exoplanet transit>, and a crossing of a <starspot> require the overlap <integral> or a suitable more complete model. Inferring $k=\sqrt\delta$ without the local-to-mean <specific intensity> ratio can bias the estimated <planet> <radius>.
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