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The signed instantaneous contribution of apsidal precession to azimuthal motion during isotropic stellar mass loss is $\dot{\varpi }/\dot{\theta }=-(\dot{\mu }/\mu n)(1-e^2{)}^{3/2}\sin f/[e(1+e\cos f{)}^2]$. Its maximum absolute magnitude is small when $n|\mu /\dot{\mu }|\gg F(e)$, where $F(e)\sim 1/e$ near a circular orbit and $F(e)\to 3\sqrt{3}/4$ as $e\to 1$. The low-orbital eccentricity coordinate singularity can be avoided by evolving the eccentricity vector.

Time averaging over a Kepler orbit gives $⟨\cos f⟩=-e$, as follows from $dt=(1-e\cos E)dE/n$ and $\cos f=(\cos E-e)/(1-e\cos E)$. Consequently slow isotropic stellar mass loss preserves the secular orbital eccentricity. Conserved specific angular momentum then gives $\mu a=\mathrm{constant}$, so pericentre, apocentre and semimajor axis expand inversely with the remaining gravitational parameter.

The difference between the angular momentum of circular coplanar orbits at fixed semi-major axes and the system's actual angular momentum along the reference normal. With circular momenta $L_j=m_j\sqrt{G{M}_{\star }{a}_j}$ it is ${\sum }_j{L}_j(1-\sqrt{1-e_j^2}\cos I_j)$. For small planar orbital eccentricities its quadratic part is $\frac{1}{2}{\sum }_j{L}_j|z_j{|}^2$. Its conservation in $\dot{z}=iAz$ requires $L_j{A}_{jk}=L_k{A}_{kj}$.

A phase-mixed population secularly forced by a distant eccentric planet has typical orbital eccentricity proportional to $e_{p}a/a_p$. Consequently a relative speed of order orbital eccentricity times Keplerian speed gives $v_{\mathrm{col}}^2\sim e_p^2(a/a_p)GM/a_p$. Comparing specific impact energy with a catastrophic disruption threshold gives a lower disruption radius $a/a_p\gtrsim Q_D^{\ast }{a}_p/(GM{e}_p^2)$, up to an order-one coefficient depending on velocity statistics and impact-energy convention. The planet mass sets the secular stirring time, but cancels from this energy scaling.

With constant mean motions and $P_c/P_b=(1+\delta )(p+q)/p$, mean conjunctions advance by $\Delta \Lambda =2\pi /(P_c/P_b-1)$ each synodic period. Tracking one branch every $q$ conjunctions and subtracting $2\pi p$ givesThe usual formula $-2\pi p\delta /(q{P}_b)$ keeps only first order in $\delta$. A labelled branch takes $q{P}_b(1+\delta )/(p|\delta |)$ to turn once, whereas the full set of $q$ interchangeable directions repeats after $P_b(1+\delta )/(p|\delta |)$. Finite orbital eccentricity introduces true-conjunction corrections.

For resonant trapping of dust in an exterior $(p+q):p$ mean-motion resonance, let Poynting–Robertson drag have parameter $A$. Holding semi-major axis $a$ fixed and retaining secular rates through second order in orbital eccentricity givesFor $5p>q$, squared orbital eccentricity approaches this equilibrium exponentially in the truncated model. If the predicted value is not small, higher-order terms are needed for quantitative dynamics. Finite planetary torque also imposes a minimum eccentricity for balance, which alone does not guarantee capture during migration.

If retained Lagrange planetary equations have $\dot{a}=-2(p+q)a{e}^q{C}_r\sin \varphi$ and $\dot{e}=-q{e}^{q-1}{C}_r\sin \varphi$, then $d(e^2)/d\log a=q/(p+q)$. Thus $e^2-[q/(p+q)]\log (a/a_{\mathrm{ref}})$ is constant. A resonantly oscillating orbital eccentricity–semi-major axis trajectory retraces one curve rather than enclosing area in that projection.

A first-order resonant orbital perturbation changes the semi-major axis and therefore the mean motion. Integrating the sinusoidal mean-motion variation over the near-resonant transit-timing superperiod gives a mean-longitude amplitude proportional to ${\mu }_{2}e/{\Delta }^2$. Converting phase to transit time gives the displayed amplitude ratio. It assumes small perturbations, circulating resonant argument, effectively fixed free orbital eccentricity, and dominance of this enhanced term over direct phase and forced-eccentricity contributions. The formal divergence as $\Delta \to 0$ signals failure of this expansion, not infinitely large transit times.

For an attractive Kepler orbit with gravitational parameter $GM$, velocity $\mathbf{v}$ and specific angular momentum $\mathbf{h}=\mathbf{r}\times \mathbf{v}$, the eccentricity vector isIts magnitude is the orbital eccentricity, and it points toward periapsis for a noncircular orbit. It is a normalized attractive Laplace-Runge-Lenz vector; it remains constant under the pure inverse-square force.

With a fixed planetary pericentre, the linear secular equation $\dot{z}=iA(z-Z_f)$ has solution $z=Z_f+(z(0)-Z_f)e^{iAt}$. An initially circular orbit follows $z=Z_f(1-e^{iAt})$, a counterclockwise circle in the complex plane centered at the forced eccentricity $Z_f$, with free eccentricity amplitude $Z_f$. Its maximum orbital eccentricity is $2Z_f$ and its phase-averaged mean orbital eccentricity is $4Z_f/\pi$.

Uniform planar kick angle $\theta$ gives $c=\cos \theta$ with probability density $1/(\pi \sqrt{1-c^2})$. This maps onto a one-dimensional curve in orbital eccentricity–semi-major axis space. The two radial-kick signs share that curve but have opposite tangential components of the eccentricity vector. The resulting density is enhanced near tangential-kick endpoints.

Uniform mean anomaly implies a line probability inversely proportional to the orbital speed. For small orbital eccentricity and true anomaly $f$, it is proportional to $1-e\cos f$.

For a local two-body hyperbolic encounter with central parameter $\mu$ and allowed centre separation $r_{\min }\ge r_0$, conservation of energy gives equal asymptotic speeds. The gravitational scattering angle therefore impliesPut $z=v_{\infty }\sqrt{r_{\min }/\mu }$ and use $2z/(1+z^2)\le 1$. The optimizing grazing orbit has $z=1$, orbital eccentricity two and deflection $\pi /3$. For finite spherical bodies $r_0$ is the sum of their radii; excluding actual contact makes the upper bound a supremum. Without a finite exclusion radius the ideal point-mass problem has no finite bound.

The oscillatory integration constants determine the orbital eccentricity vector, the constant radial offset determines the semi-major axis, and the constant along-track offset fixes the mean longitude. The radial and tangential epicycle amplitudes have ratio $1:2$.