Total duration of a circular planetary transit ($T_{14}=\frac{P}{\pi }\arcsin \sqrt{\frac{(1+k{)}^2-b^2}{(a/R_{\star }{)}^2-b^2}}$)

The first-to-fourth-contact duration of an exoplanet transit by a spherical opaque planet of radius ratio $k=R_p/R_{\star }$ on a circular orbit. The transit impact parameter is $b=(a/R_{\star })\cos i$. The formula includes finite planet size and orbital curvature; it reduces to the transit duration of a small planet when $k\to 0$ and $R_{\star }/a\ll 1$. It describes contact geometry, not a uniquely specified light-curve threshold in the presence of limb darkening.

The total duration $T_{14}$ runs from first to fourth contact; the fully superimposed duration $T_{23}$ runs from second to third contact. Circular-orbit formulas contain $1+k$ for the former and $1-k$ for the latter. Confusing the two changes inferred impact parameter, even if the period and transit depth are unchanged.

For a circular orbit, define $k=R_p/R_{\star }$, $A=a/R_{\star }$ and $s=\sin (\pi T_{14}/P)$. Inverting contact geometry gives $b^2=[(1+k{)}^2-A^2{s}^2]/(1-s^2)$. The inferred value is conditional on the stellar scale, the total-duration convention and the orbital model; it must satisfy $0\le b<1+k$. A non-grazing complete crossing additionally has $b<1-k$.

The orbital speed near transit differs from the circular value by approximately $(1+e\sin \omega )/\sqrt{1-e^2}$. At otherwise fixed geometry, duration is shortened by the inverse of this factor. Without eccentricity and periapsis constraints, a measured duration need not select one unique impact parameter.

Assume Kepler's third law, a negligible planet mass, and $M_{\star }/M_{\odot }=R_{\star }/R_{\odot }$. Define $A_{\odot }=[(GM{)}_{\odot }{P}^2/(4{\pi }^2){]}^{1/3}/R_{\odot }$. The total duration of a circular planetary transit determines $a/R_{\star }$, giving the displayed stellar radius. Unknown orbital eccentricity, transit dilution or a different stellar mass-radius calibration introduces additional uncertainty.