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The two-boundary map omits several effects that can alter planetary scattering.

The initial semi-major axis, orbital eccentricity, orbital inclination and orbital phase determine whether encounters occur and their relative velocities. A strongly bound comet needs more energy to escape; a nearly parabolic one needs less. The planetary radius and mass density, the finite comet radius, and gravitational focusing determine collision probabilities. Tidal disruption, atmospheric gas drag, sublimation and physical fragmentation can destroy a body before a nominal point-particle scattering sequence is completed.

Other planets can hand a comet from one scatterer to another, eject it, or lift its periapsis clear of the original scatterer's orbit. Mean-motion resonances and secular perturbations can protect objects from encounters or correlate kicks, contradicting the independent random walk assumption. Planetary migration changes the encounter geometry over time.

Stellar flybys and a galactic tide can change distant comet periapses, allowing new encounters or detaching an object from the planetary region. Stellar mass evolution changes both binding and planetary orbits. Finally, the age and the supply rate of new comets determine whether the observed population is a residual one or continuously replenished. Thus the mass-radius map is a useful conditional classification, not a complete survival law.

The planetary expansion is a decomposition into secular eigenmodes of conservative linear Laplace-Lagrange secular theory. It assumes small orbital eccentricities, coplanarity, approximately constant semi-major axis values, and averaging over fast orbital phases away from important mean-motion resonances and close encounters. Dissipation, migration, and nonlinear secular effects are absent from this constant-coefficient representation.

The $g_i$ are real eigenfrequencies of the Laplace-Lagrange secular matrix, describing apsidal precession. For mode $i$, the signed real coefficients $e_{ji}$ give its amplitude in planet $j$; their relative magnitudes are fixed by an eigenvector, and their signs distinguish aligned and anti-aligned apsides. The common phase ${\beta }_i$ specifies that mode's orientation at the chosen time origin. A planet's actual orbital eccentricity is $|z_j(t)|$, which generally varies through interference of modes; it is not a sum of the positive mode amplitudes.

To first order in orbital eccentricity, the semi-minor axis is $a+O(e^2)$, while the ellipse's center is displaced from the star by $-a\mathbf{e}$. Thus each orbit is a circle of radius $a$ about $-az$, in complex spatial coordinates.

Write $z=z_f+e_p{e}^{i{\varpi }_p}$ at one secular epoch. The centers of the individual circles lie on a circle of radius $a{e}_p$ about $C_f=-a{z}_f$. Their union is the proper-eccentricity annulusIts full radial width is $2a{e}_p$ and its center is displaced by $a{e}_f$ toward the forced apoapsis direction. Relative to the star, if the forced longitude of periapsis is zero,The middle panel of the preceding figure shows this construction. For fully sampled orbital phases and proper phases, the annulus is occupied throughout, but its density need not be uniform; turning points in the proper radial excursion give integrable edge enhancements. The construction requires $e_f,e_p\ll 1$ and is an instantaneous secular geometry.

Coplanarity requires both $I=I_d$ and $\Omega ={\varphi }_d$, up to the orbital-plane orientation degeneracy. Under that hypothesis the measured planet angle must obeyFor a nearly edge-on belt this offset is small except close to the projected crossings where $\cos f=0$. Measurements at several orbital phases can therefore test the entire projected ellipse: a systematic offset from the predicted relation implies a nonzero mutual inclination. A single projected location generally cannot establish coplanarity because $f$, the front-back branch, and the nodal orientation can be degenerate.