= Solution
A <exoplanet transit> occurs when a planet crosses the projected stellar disk and blocks some stellar <flux>. For a circular <orbit> with isotropically distributed angular-momentum directions, $\cos i$ is uniform on $[0,1]$. A transit requires $a|\cos i|<R_\star+R_p$, so
$$
\boxed{p_{\rm tr}=\frac{R_\star+R_p}{a}\simeq\frac{R_\star}{a}.}
$$
For a dark planet, a uniformly bright stellar disk and negligible dilution, the transit <exoplanet transit depth> is $\boxed{\delta\simeq(R_p/R_\star)^2}$. For $R_\star/a\ll1$, a small planet and <impact parameter> $b=a\cos i/R_\star$,
$$
\boxed{t_{\rm tr}\simeq\frac{P}{\pi}\frac{R_\star}{a}\sqrt{1-b^2}.}
$$
The central-transit result sets $b=0$. A first-to-fourth-contact duration retaining planet size replaces $\sqrt{1-b^2}$ by $\sqrt{(1+R_p/R_\star)^2-b^2}$ and includes the exact projected-speed geometry when needed. <Eccentricity>, <limb darkening> and grazing geometry modify these expressions.
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