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For two uniform angular motions with $P_b<P_c$, an inner exoplanet transit at zero and next outer exoplanet transit at $\tau$, the next conjunction solves $(n_b-n_c)t+n_c\tau =2\pi$. Hence its longitude modulo $2\pi$ isFor an eccentric outer body this is only an approximation. Let $f_L,M_L$ be its line-of-sight anomalies and ${\Delta }_e(t)=f(t)-M(t)-(f_L-M_L)$. Exact true alignment instead requires $(n_b-n_c)t+n_c\tau -{\Delta }_e(t)=2\pi$. The correction follows directly from uniform mean anomaly and nonuniform true anomaly.

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.

Write $R_p$ for the physical planet radius and $R_c=\eta R_{H1}$ for the debris-cloud radius; the source uses $R_1$ for both even though their stated size orderings are opposite. A central transit duration of a small planet on a nearly circular Kepler orbit is $D_p\simeq 2R_{\star }/(n_1{a}_1)=P_1{R}_{\star }/(\pi a_1)$. Kepler's third law, with $M_{\star }=4\pi {\rho }_{\star }{R}_{\star }^3/3$, givesA noncentral planetary exoplanet transit is shorter, so within the small-radius and nearly circular approximations, a duration requiringis too long for the planet alone and favors extended debris. For a central cloud transit with $R_c\gg R_{\star }$, the path length is approximately $2R_c$. Using the Hill radius $R_{H1}=a_1(M_1/(3M_{\star }){)}^{1/3}$ gives $D_1/P_1\simeq R_c/(\pi a_1)$, hence the Hill-cloud transit mass estimateThis equality assumes a central cloud chord. An orbit described merely as nearly edge-on need not have central cloud crossings. If its projected cloud impact distance is $s$, the measured half-chord is $\sqrt{R_c^2-s^2}$, so the boxed mass is multiplied by $[1-(s/R_c{)}^2{]}^{-3/2}$. It is a lower estimate when that geometry is unknown.

For the dynamical calculation use ${\mu }_2=M_2/M_{\star }$, $A=G{M}_2/a_2$, and $\mathcal{R}=Af(\alpha )e_1\cos {\varphi }_1$. The derivatives in the supplied Lagrange planetary equations hold the other osculating orbital elements fixed:Since $A/(n_1{a}_1^2)=n_1{\mu }_2\alpha$, the physical mean-longitude equation for a first-order resonant term isThe source's epoch notation needs care when $n_1$ varies. Literally differentiating ${\lambda }_1=n_1(t)t+{ϵ}_1(t)$ adds $t{\dot{n}}_1$; one cannot retain that definition and also identify the printed ${\dot{ϵ}}_1$ with ${\dot{\lambda }}_1-n_1$. A consistent version writes ${\lambda }_1={\int }_0^t{n}_1(s)ds+{ϵ}_{\mathrm{int}}$ and assigns the displayed correction to ${\dot{ϵ}}_{\mathrm{int}}$. Equivalently, the canonical mean-longitude equation for a first-order resonant term follows from the disturbing function directly. The result for $g$ is the intended physical equation under that convention.

Away from resonant-argument libration, neglect changes in $e_1,\alpha ,{\varpi }_1$ on one slow cycle to this order. With $n_0=2\pi /P_1$ the reference mean motion, the signed slow frequency isThis is the near-resonant transit-timing superperiod. Choose ${\varphi }_1(0)=0$. Differentiating $n_1\propto a_1^{-3/2}$ with the semi-major axis equation givesLet $C=3j{\mu }_2{n}_0^2\alpha f{e}_1$ and $n_s=n_1(0)$. Direct integration giveswith zero initial mean longitude. The constant and linear terms are absorbed into the fitted transit ephemeris: the measured mean frequency is $\bar{n}=n_s-C/\omega$. This distinction avoids mistaking an arbitrary initial frequency for the long-term fitted period.

For $|\Delta |\ll 1$, the double-integrated frequency term dominates the direct $g$ term provided $g/f$ stays finite. Relative to the fitted linear ephemeris, ${\lambda }_1(t_k)=2\pi k$ givesHere $P_1$ denotes the fitted mean period at the retained order. The signed-frequency convention fixes the phase sign. Replacing $\omega$ by $2\pi /P_t>0$ changes the sign of the signed $A_t$ when needed; a nonnegative amplitude hasUsing a signed $P_t=2\pi /\omega$ instead gives the source's signed proportionality $A_t/P_t\simeq 3{\mu }_2\alpha f{e}_1/(2\pi \Delta )$. The leading implicit transit equation can equally use $P_{1}k$ inside the sine, since the difference is higher order in the perturbation. This is the eccentricity-enhanced near-resonant transit-timing variation; its derivation assumes small longitude oscillations, ${\mu }_2{e}_1\ll {\Delta }^2$ at fixed resonance coefficients, and the effectively fixed eccentricity used in this truncated model. Dominance over forced-eccentricity timing terms typically also requires $e_1\gg |\Delta |$ at fixed $j$. A full transit calculation also includes forced orbital eccentricity and the conversion from mean longitude to the actual sky-crossing angle. Those are lower order in this eccentricity-enhanced regime, rather than identically absent; broader near-resonant formulas are derived by Lithwick, Xie and Wu.

Inside a mean-motion resonance, ${\varphi }_1$ undergoes resonant-argument libration and its evolution must be solved together with $a_1,e_1,{\varpi }_1$. The constant circulation frequency $\omega$ and the associated ${\Delta }^{-2}$ expansion then fail. At approximately fixed eccentricity, the resonant resonant pendulum approximation has libration frequency of order $j{n}_0\sqrt{{\mu }_2\alpha |f|e_1}$, up to numerical factors. The transit-timing variation follows that libration period and depends on its libration amplitude of a resonant argument. Exactly at a resonant equilibrium there need be no libration signal; near a separatrix the period grows, and eccentricity dynamics can modify the simple estimate. Resonant amplitudes remain controlled by the bounded resonant motion rather than diverging as the circulating detuning tends to zero.

For constant mean motions, consecutive mean conjunctions are separated by $P_{\mathrm{syn}}=P_b{P}_c/(P_c-P_b)$ and advance in longitude by $\Delta \Lambda =2\pi /(P_c/P_b-1)$. To follow one of the $q$ branches, compare conjunctions $q$ steps apart. At exact resonance that advance is $2\pi p$; subtract it to measure the slow drift. Hence conjunction-pattern precession near a mean-motion resonance isThe printed expression is its leading small-offset term:Positive $\delta$ gives retrograde drift and negative $\delta$ prograde drift. The tracked-branch rotation time is about $q{P}_b/(p|\delta |)$; the full set of $q$ interchangeable directions repeats after a rotation $2\pi /q$, giving the usual mean-pattern super-period $P_b(1+\delta )/(p|\delta |)$. The slow, continuously tracked branch interpretation assumes the resonance neighbourhood $|\delta |(p+q)\ll q$; merely $|\delta |\ll 1$ need not give a slow pattern for an arbitrarily large integer ratio $p/q$. At $\delta =0$ there is no offset-driven drift.

The allowed directions can consequently move through the observer's line of sight. Triple exoplanet transits require the $bc$ and $cd$ patterns, together with their timing phases, to overlap that line of sight within the finite exoplanet transit windows; differing drift rates can open and close such windows. A small offset neither guarantees a triple event nor prohibits it permanently.

Moreover, in a true mean-motion resonance, bounded ${\varphi }_{bc}$ can coexist with a period offset because $(p+q)n_c-p{n}_b=q{\dot{\varpi }}_c+{\dot{\varphi }}_{bc}$. The mean conjunction pattern then follows apsidal precession plus the resonant-argument libration correction. For appreciable orbital eccentricity, the true-conjunction geometry has the periodic corrections identified above; the drift calculation is a mean-phase result.

First adopt the uniform-angular-motion approximation implicit in the requested formula. Set the line of sight to longitude zero, the $b$ exoplanet transit to $t=0$, and the next $c$ exoplanet transit to $t=\tau$, with $0\le \tau <P_c$. Write $n_b=2\pi /P_b$, $n_c=2\pi /P_c$, with $P_c>P_b$. The next conjunction satisfiesThe common longitude is $n_{b}t$ modulo $2\pi$. Subtracting one $2\pi$ gives the conjunction longitude from a transit time lagSubstitution of the period ratio yieldsIt is exact in this uniform-angle model; no first-order expansion in $\delta$ is necessary here. For other choices of which exoplanet transit is used, choose the appropriate whole-turn branch.

For the finite-eccentricity orbit specified earlier, this is an approximation. Let $f_L$ and $M_L$ be the true anomaly and mean anomaly at the line of sight. The eccentric planet has $M_c(t)=M_L+n_c(t-\tau )$ and true angular advance $n_c(t-\tau )+{\Delta }_e(t)$, whereTrue conjunction therefore requires $(n_b-n_c)t+n_c\tau -{\Delta }_e(t)=2\pi$. The PDF omits this term, which is generally $O(e_c)$.

For a concrete counterexample take $p=q=1$, $\delta =0$, $e_c=1/5$, $\tau =P_b/4$, and a line of sight along $c$'s periapsis. The uniform model predicts $t=7P_b/4$ and longitude $3\pi /2$. At that time $M_c=3\pi /2$, but Kepler's equation gives $f_c\ne 3\pi /2$, so the planets are not truly aligned. Accurate finite-eccentricity conjunctions must be found from the corrected equation.

A simultaneous transit in a resonant chain requires both conjunction patterns to admit the observer's longitude at the same time. In the leading mean-conjunction geometry put $x=\Lambda -{\varpi }_c$. The necessary phase compatibility at symmetric centres isLet $d=\gcd (q,n)$. Eliminating $x$, and using that $q\ell -nk$ ranges over multiples of $d$, gives the exact modular compatibility criterionHere $\gcd$ denotes the greatest common divisor. For the orders $q,n\in \{1,2,3\}$ above, the two phase sets intersect only whenThe common directions are the two quadratures relative to periapsis. In reduced resonances this requires $p$ and $m$ odd; otherwise the apparent second-order ratios reduce to different resonance orders. There is no additional restriction on those odd values from angular compatibility alone. A shared direction and suitable temporal phase can be chosen, and the rational period ratios allow its recurrence.

This is a potential low-amplitude configuration, not a guarantee of an observed triple exoplanet transit. The observer must lie in the common orbital plane and near an allowed direction. Finite stellar radii, nonzero libration amplitudes of a resonant argument, apsidal motion and asymmetric centres broaden or change the possibilities. At finite $e_c$, true simultaneous alignment obeys the more precise equations $(p+q)y-px={\varphi }_{bc}$ and $(m+n)x-my={\varphi }_{cd}$, with $y=M_c(x)$. Thus the order-only conclusion is not a universal necessary condition for every eccentric resonant chain.

In leading mean-conjunction geometry a common observer phase $x$ for two resonant arguments requires $qx={\varphi }_1$ and $nx={\varphi }_2$ modulo $2\pi$. With $d=\gcd (q,n)$, the exact modular criterion isNecessity follows by subtracting the two congruences after multiplication; sufficiency follows from the integer linear combinations of $q$ and $n$, whose set is $d\mathbb{Z}$. For exterior centres $\pi$ and interior centres $0,\pi ,0$ at orders one to three, only $q=n=2$ satisfies it. This is phase compatibility, not a guarantee of an observed exoplanet transit: timing, observer orientation, resonant-argument libration widths and true-versus-mean longitude must also be considered.

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.

Transit spectroscopy infers the properties of an exoplanet atmosphere from the wavelength dependence of an exoplanet transit. Its measured product is an exoplanet transmission spectrum, while an annulus model for transmission spectroscopy connects the spectrum to atmospheric optical depth. The observing technique is described by NASA.