= Solution
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 $z=z_f+e_pe^{i\varpi_p}$ at one secular epoch. The centers of the individual circles lie on a circle of radius $ae_p$ about $C_f=-az_f$. Their union is the <proper-eccentricity annulus>
$$
\boxed{a(1-e_p)\leq|\mathbf r-C_f|\leq a(1+e_p),\qquad C_f=-a\mathbf e_f.}
$$
Its full radial width is $2ae_p$ and its center is displaced by $ae_f$ toward the forced <apoapsis> direction. Relative to the <star>, if the forced <longitude of periapsis> is zero,
$$
r_{\rm inner,outer}(\theta)\simeq a[1-e_f\cos\theta\mp e_p].
$$
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 $e_f,e_p\ll1$ and is an instantaneous secular geometry.
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