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For an eccentric inner body and circular outer perturber near the reduced $(m+n):m$ period ratio, the leading resonant argument is $\varphi =(m+n){\lambda }_{\mathrm{out}}-m{\lambda }_{\mathrm{in}}-n{\varpi }_{\mathrm{in}}$. Its disturbing function amplitude is of order $e_{\mathrm{in}}^n$. At a mean conjunction, $\varphi =n(\Lambda -{\varpi }_{\mathrm{in}})$; at actual alignment the exact expression is $(m+n)x-my$ with $x=f_{\mathrm{in}}$ and $y=M_{\mathrm{in}}(x)$.

The Lagrange planetary equations express changes of osculating orbital elements in terms of derivatives of a disturbing function. Their small-eccentricity secular limit becomes a linear equation for complex eccentricity.

For disturbing function $\mathcal{R}=(G{M}_2/a_2)f(\alpha )e\cos \varphi$, $\alpha =a/a_2$, the leading planar Lagrange planetary equations give $\dot{\lambda }=n+n(M_2/M_{\star })eg(\alpha )\cos \varphi$. The derivative with respect to $a$ holds the other osculating orbital elements fixed. If one writes $\lambda =n(t)t+ϵ(t)$, its literal derivative includes $t\dot{n}$; the physical mean-longitude equation cannot be obtained by silently dropping that term. A consistent accumulated-phase parametrization uses $\lambda ={\int }_0^{t}n(s)ds+{ϵ}_{\mathrm{int}}$.

For a circular orbit the origin of mean anomaly is arbitrary because there is no distinguished pericentre. Choose $M=0$ at the ascending node, so $M$ is also the argument of latitude. If a different periapsis convention is used, replace $M$ below by $M+\omega$. The orbital-frame rotation from inclination and node givesIn particular the crossing at $M=0$ has increasing $z$ for prograde $n>0$ and $0<I<\pi$, confirming the ascending node convention.

Put $ϵ=a/a_{\mathrm{pl}}$ and $c=\cos \psi =\hat{\mathbf{r}}\cdot {\hat{\mathbf{r}}}_{\star }$. The binomial series expansion of the direct disturbing function isThe indirect disturbing function subtracts $G{M}_{\star }ac/a_{\mathrm{pl}}^2$, cancelling the dipole term. The constant $G{M}_{\star }/a_{\mathrm{pl}}$ has no spatial derivative and does not perturb the particle. After omitting that constant, the leading stellar quadrupole disturbing function isThus the PDF's equality is understood modulo a dynamically irrelevant additive constant and higher multipoles.

Define $\ell =M+\Omega$ and let ${\ell }_{\star }$ be the star's orbital longitude as seen from the planet. Its orbital plane has $I_{\mathrm{pl}},{\Omega }_{\mathrm{pl}}$. The vector from planet to star is the negative of the planet's heliocentric vector: if ${\ell }_{\mathrm{pl}}$ describes that latter vector, use ${\ell }_{\star }={\ell }_{\mathrm{pl}}+\pi$. This sign matters in $\cos \psi$ but drops out of ${\cos }^2\psi$. Expanding the two rotated unit vectors givesFor example, the particle's horizontal correction is $-(I^2/2)\sin (\ell -\Omega )(-\sin \Omega ,\cos \Omega )$, while its vertical component is $I\sin (\ell -\Omega )$; their dot products with the corresponding stellar components produce these terms. No cubic terms occur because simultaneous reversal of both small orbital inclinations leaves the dot product unchanged.

The two mean longitudes are uniform for the circular Kepler orbits, so orbit averaging is uniform in both $\ell$ and ${\ell }_{\star }$. Set $c_0=\cos (\ell -{\ell }_{\star })$ and $c_2=\cos \psi -c_0$ through quadratic order. The needed averages follow by integrating sine and cosine products over $[0,2\pi {]}^2$:For the last identity, expand $c_0=\cos \ell \cos {\ell }_{\star }+\sin \ell \sin {\ell }_{\star }$ and factor the two integrations, each using $⟨{\sin }^2\ell ⟩=⟨{\cos }^2\ell ⟩=1/2$. Squaring $c_0+c_2$ now yieldswhere the error includes all fourth-order combinations of the two orbital inclinations. ThereforeMultiplying this by $G{M}_{\star }{a}^2/a_{\mathrm{pl}}^3$ gives the requested secular disturbing function. Equivalently, the exact angular factor for two circular rings is $(3{\cos }^2{I}_m-1)/8$, where $I_m$ is their mutual inclination; its small-angle expansion gives the same result. This is the double-averaged circular tidal potential.

For aligned equatorial and planetary orbital planes, set $I_{\mathrm{pl}}=0$. The secular stellar quadrupole disturbing function then has derivative ${\partial }_I{\overline{\mathcal{R}}}_{\star }=-3G{M}_{\star }{a}^{2}I/(4a_{\mathrm{pl}}^3)$. The supplied Lagrange planetary equations for nodal precession givesThe planetary quadrupole coefficient $J_2>0$ contributes ${\partial }_I{\overline{\mathcal{R}}}_{J_2}=-3G{M}_{\mathrm{pl}}{J}_2{R}_{\mathrm{pl}}^{2}I/(2a^3)$, henceBoth are retrograde nodal precession rates. Equating them cancels $n$ and gives the Laplace radiusThe planetary quadrupole dominates inside $a_L$ and the stellar tide outside it. These are small-inclination rates for planet-bound Kepler orbits in the hierarchical regime $a\ll a_{\mathrm{pl}}$. At exactly $I=0$, the node itself is undefined; the rates are the limiting precession frequencies of an infinitesimal tilt.

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 brevity write $b_1=b_{3/2}^{(1)}(\alpha )$, $b_2=b_{3/2}^{(2)}(\alpha )$, and $K=G{M}_p\alpha /(4a_p)$. The quadratic disturbing function is $\mathcal{R}=K[\frac{1}{2}{b}_1{e}^2-b_2{e}_{p}e\cos ({\varpi }_p-\varpi )]$. Differentiation in Lagrange planetary equations givesThe separate longitude of pericentre equation is singular at $e=0$, because a circular orbit has no distinguished periapsis. The complex eccentricity, $z=e{e}^{i\varpi }$, is regular there. Using $z_p=e_p{e}^{i{\varpi }_p}$ and $\dot{z}=e^{i\varpi }(\dot{e}+ie\dot{\varpi })$ givesSince $n{a}^2=\sqrt{G{M}_{\star }a}$ and $n_p=\sqrt{G{M}_{\star }/a_p^3}$,The approximations are planar, small-orbital eccentricity, first order in the planet-to-star mass ratio, and secular averaging away from important mean-motion resonances. The semimajor axis values remain constant at this order.

For a planet with fixed orbit, set $g=A{b}_1$ and $z_f=(b_2/b_1)z_p$. Then $\dot{z}=ig(z-z_f)$, so $z-z_f=(z(0)-z_f)e^{igt}$. With $z(0)=0$,On the Argand diagram, the complex eccentricity traces a circle centred on the forced eccentricity $z_f$, of radius $|z_f|$. For an interior test particle, $0<\alpha <1$ and $g>0$, so the motion is counterclockwise, starting at the origin. The rotating radius $z-z_f$ is the free eccentricity. The orbital eccentricity ranges from zero to $2|z_f|$, attaining its maximum at half a secular period. If $z_p$ is placed on the positive real axis, the initial tangent points downward, which is consistent with counterclockwise motion from the leftmost point of this circle.

Now let the planetary mass grow as $M_p(t)=M_{p}t/t_p$ for $0\le t\le t_p$, while the planet's orbital elements and the test particle's semimajor axis remain fixed. Both the precession and forcing coefficients acquire the same factor $t/t_p$. Their ratio, and hence $z_f$, remain constant. ThusandThis is the secular phase integral for proportional mass growth. The Argand diagram circle and the free eccentricity amplitude are unchanged; only the rate of traversal changes. The angular speed about $z_f$ starts at zero and rises linearly to $g$. The early displacement is quadratic in time rather than linear. Since $t_p=2\pi /g$, $\Phi (t_p)=\pi$, soA planet already at its final mass would instead return the particle to $z=0$ at $t_p$. After growth stops, the circle continues at rate $g$, with $z(t)=z_f[1+e^{ig(t-t_p)}]$ for $t\ge t_p$. Slow mass growth does not damp the free eccentricity here: the fixed centre of the circle and the conservative equation preserve its amplitude.

For several planets with fixed masses, Laplace-Lagrange secular theory gives $\dot{\mathbf{z}}=i\mathbf{Az}$ for the planets andfor a massless particle, with real secular coupling coefficients ${\nu }_j$. Resolve planetary complex eccentricities into secular eigenmodes, $z_j(t)={\sum }_k{E}_{jk}{e}^{i{g}_{k}t}$. Away from secular resonances, the test-particle solution has the formIts Argand diagram trajectory is a superposition of rotations: generally a quasiperiodic curve with beats, rather than a circle with one fixed centre. Even if the planetary orbits are artificially held fixed, their combined forcing shifts the centre to the sum of the individual forced eccentricities. At $g_k=g_0$, the resonant contribution is proportional to $t{e}^{i{g}_{0}t}$ in linear theory; its growing amplitude eventually invalidates the small-orbital eccentricity approximation.

If one planetary mass changes, the Laplace-Lagrange secular matrix, its eigenvalues, its eigenvectors, and the particle's forced response generally change with time. Planetary modes need not remain independent. Frequencies drift and a secular resonance can sweep across the particle, changing or exciting its free eccentricity. Very slow evolution away from resonances permits an approximate adiabatic mode description; rapid evolution or a resonance passage can mix modes. The simple substitution of a phase integral works only in the special proportional-growth case above, where the entire particle equation is multiplied by one scalar and its forced centre stays fixed. It does not apply to a general many-planet system.

In the weak, non-crossing, leading-eccentricity model the exterior resonant coefficient has positive sign: write its disturbing function as $R_{\mathrm{res}}=B_q{e}_c^q\cos {\varphi }_{bc}$, with $B_q>0$ for the orders considered. The physical feedback can be seen for first order. Before an outer conjunction near apoapsis, the faster inner planet pulls the outer one backwards; after passing it pulls forwards. On the outward side, the nearer, stronger earlier pull wins, reducing the outer semi-major axis and increasing its mean motion. Thus ${\varphi }_{bc}<\pi$ is driven upward. On the inward side, the later forward pull wins and drives ${\varphi }_{bc}>\pi$ downward. The paired impulses cancel at the symmetric centre.

For higher order, sum the feedback over all $q$ conjunction phases. The first surviving angular Fourier harmonic is $\sin {\varphi }_{bc}$; the lower Fourier harmonics cancel over the equally spaced directions. A local Hamiltonian reduction with canonical momentum $J$ hasThe negative curvature of the Kepler orbit Hamiltonian gives the displayed sign. Linear stability analysis at $\pi$ gives a restoring acceleration. The symmetric libration centres of an eccentricity resonance are thereforeThe corresponding conjunction phases relative to periapsis are $\{\pi \}$, $\{\pi /2,3\pi /2\}$ and $\{\pi /3,\pi ,5\pi /3\}$. They suppress repeatedly close passages and provide resonance protection.

These are expected centres of the specified leading-harmonic model, not centres fixed solely by the integer order for arbitrary orbital eccentricity and masses. Asymmetric resonant-argument libration can occur when higher Fourier harmonics matter. For example, if $R=A\cos \varphi +B\cos 2\varphi$, $A>0$, $B>A/4$, the symmetric $\pi$ equilibrium loses stability and stable centres satisfy $\cos \varphi =-A/(4B)$. Thus further dynamical information is needed beyond the PDF's unrestricted eccentric orbit to assert unique centres. Exterior asymmetric branches are documented by Winter and Murray, “Resonance and chaos. II. Exterior resonances and asymmetric libration”.

Use conventional mean longitudes $\lambda$ and take the positive integers $p,q$ in lowest terms. Rotational invariance requires the angular coefficients of a resonant argument to sum to zero. Since the inner planet has zero orbital eccentricity, the eccentricity-type term involving its undefined apsidal angle has zero amplitude. The surviving outer-eccentricity argument isIts leading disturbing function term is proportional to $e_c^q\cos {\varphi }_{bc}$. Near the period commensurability, $(p+q)n_c-p{n}_b$ is small, so the resonant argument is slow and repeated perturbations can add coherently. Trapping produces resonant-argument libration, rather than circulation, with $(p+q)n_c-p{n}_b-q{\dot{\varpi }}_c\simeq 0$.

A near-integer period ratio alone does not prove mean-motion resonance: the diagnostic is bounded resonant-argument libration. At $e_c=0$ this eccentric resonant term vanishes and ${\varpi }_c$ is undefined. If $p,q$ have a common factor, reduce the ratio first; the unreduced $q$ is not the actual lowest resonance order.

Differentiating the disturbing function in Lagrange planetary equations givesFor the complex eccentricity $z=e{e}^{i\varpi }$,Use $\sin \delta +i\cos \delta =i{e}^{-i\delta }$. Each planetary term becomes $i{A}_j{e}_j{e}^{i{\varpi }_j}=i{A}_j{z}_j$, soUnlike the polar orbital element equations, this linear equation is nonsingular at $e=0$.

Expanding the disturbing function in orbital elements gives a Fourier series of the formwhereThe radial coefficients are combinations of Laplace coefficients. Rotational and reflection symmetry impose the D'Alembert characteristic: the integer coefficients sum to zero, nodal coefficients obey a parity rule, and each harmonic begins at the corresponding order in eccentricities and inclinations.

Subtract the star's acceleration from that of $M_1$ in an inertial frame. With ${\mathbf{r}}_i$ measured from the star,This has the requested form ${\ddot{\mathbf{r}}}_1={\nabla }_1(U_1+{\mathcal{R}}_1)$ withand disturbing functionThe first term is the direct disturbing function, the attraction of $M_2$ on $M_1$. The second is the indirect disturbing function, which subtracts the acceleration of the star-centered origin by $M_2$.

At the exterior $(p+q):p$ mean-motion resonance,Using the Kepler third law for $M_{\ast }\gg M_1$ givesThe disturbing function is a Fourier series in integer combinations of the orbital angles. The D'Alembert characteristic permits the eccentric termAway from resonance, terms with rapidly circulating angles average away. Here, however,so ${\varphi }_1$ is a slow resonant argument. Successive astronomical conjunctions then act coherently, making this term dominate the long-period resonant dynamics even though a $q$th-order resonance has coefficient proportional to $e^q$ at small eccentricity.

The disturbing function may be expanded in harmonics of the planets' orbital angles. Its terms fall into three useful classes:

Well-separated planets far from resonance are governed on long timescales mainly by the secular terms.

The dimensionless coefficient $J_2$ describes the leading axisymmetric departure of a planet's external Newtonian gravitational potential from a monopole: $\Phi =-G{M}_p/r+[G{M}_p{J}_2{R}_p^2/r^3]P_2(\cos \vartheta )$, where $\vartheta$ is colatitude from the spin axis. An oblate mass distribution has positive $J_2$. The disturbing function is minus the nonspherical potential term. On a circular inclined Kepler orbit, $\cos \vartheta =\sin I\sin M$, so orbit averaging gives ${\overline{\mathcal{R}}}_{J_2}=G{M}_p{J}_2{R}_p^2{a}^{-3}(1/2-3{\sin }^{2}I/4)$. The exact circular nodal rate is $-3n{J}_2(R_p/a{)}^2\cos I/2$; its small-tilt limit is $-3n{J}_2(R_p/a{)}^2/2$.

For a particle far inside a planet's stellar orbit, the planet-centred disturbing function is $G{M}_{\star }/|{\mathbf{r}}_{\star }-\mathbf{r}|-G{M}_{\star }\mathbf{r}\cdot {\mathbf{r}}_{\star }/r_{\star }^3$. Expanding the first term cancels its dipole against the indirect disturbing function. Dropping the spatially constant monopole leaves $G{M}_{\star }{r}^2(3{\cos }^2\psi -1)/(2r_{\star }^3)$, where $\psi$ is the angle between the two offset vectors. The omitted instantaneous terms are $O(G{M}_{\star }{r}^3/r_{\star }^4)$. The gradient of the retained expression is the leading differential stellar tidal acceleration.

In the weak, non-crossing, leading-eccentricity planar model, exterior orders one to three have positive resonant disturbing function coefficients $B_q$. Interior eccentricity-type coefficients of these orders have sign $(-1{)}^q$. A local resonant Hamiltonian $H=-A{J}^2/2-B{e}^q\cos \varphi$, with $A>0$, gives $\ddot{\varphi }=AB{e}^q\sin \varphi$. Linear stability analysis therefore selects $\varphi =\pi$ for $B>0$ and $\varphi =0$ for $B<0$. Exterior symmetric centres are $\pi$; interior centres are $0,\pi ,0$ for orders one, two and three. These are conditional leading-harmonic predictions, not universal centres at arbitrary orbital eccentricity; asymmetric resonant-argument libration changes the picture.