Exoplanet transit

An exoplanet passes in front of its host star as seen by an observer and blocks part of the stellar flux. Exoplanet transit photometry infers geometry and radii from the resulting light curve. A true alignment involves the true anomaly, not generally the mean anomaly.

The probability that an isotropically oriented circular orbit produces an exoplanet transit, assuming $a>R_{\star }+R_p$. Uniformity of $|\cos i|$ gives the displayed result. Orbital eccentricity, a restriction to full rather than grazing crossings, or a non-isotropic orientation distribution changes the expression.

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.

The minimum projected planet-star separation during an exoplanet transit, divided by the stellar radius. It determines the stellar chord length and hence the transit duration of a small planet.

For two exactly coplanar circular orbits around the same star, transit impact parameters scale as semi-major axis, hence as $P^{2/3}$. Any overlap by the outer planet requires $b_2<1+k_2$. In the point-planet limit the period ratio is bounded by $b_1^{-3/2}$. Tangency has zero duration, so this is a geometric supremum; practical detection also requires sufficient signal-to-noise ratio and survey coverage.

For a small planet on a nearly circular Kepler orbit, a stellar chord at normalized transit impact parameter $b$ has length $2R_{\star }\sqrt{1-b^2}$. Dividing by transverse orbital speed $2\pi a/P$ gives the duration. A central transit has the longest duration within this approximation. Eliminating $a$ with Kepler's third law yields $P/D^3\ge {\pi }^{2}G{\rho }_{\star }/3$ for known mean stellar mass density. Finite planet radius, appreciable orbital eccentricity, and $R_{\star }/a$ corrections change the bound.

For an opaque or detectably extinguishing spherical debris cloud of radius $R_c=\eta R_H$ much larger than the stellar radius, a central exoplanet transit lasts approximately $2R_c/(na)$. Combining this with the Hill radius gives the displayed mass estimate. For cloud impact distance $s$, replace $R_c$ by $\sqrt{R_c^2-s^2}$: the central formula then gives a lower mass estimate unless the chord geometry is known.