= Solution
If the <proper eccentricity> is negligible and the <semi-major axis> ranges over $a\pm\Delta a$, particles lie on nearly aligned forced <Kepler orbits>. For a locally constant forced vector, these are geometrically similar <ellipses>, represented to first order by circles of radius $a'$ centered at $-a'\mathbf e_f$.
The boundaries of the <aligned eccentric ring> are therefore
$$
r_\pm(\theta)\simeq(a\pm\Delta a)[1-e_f\cos\theta],\qquad
\boxed{W(\theta)\simeq2\Delta a(1-e_f\cos\theta).}
$$
It is narrower at forced <periapsis> and wider at forced <apoapsis>, unlike the constant-width <proper-eccentricity annulus>. Its centers shift slightly between the two edges, as shown in the right panel of the figure. The condition $e_p/e_f\ll\Delta a/a$ makes the proper radial excursion negligible compared with the variation in forced center location.
The forced vector need not actually be constant across the interval. More generally
$$
W(\theta)\simeq2\Delta a\left|1-\operatorname{Re}\left[(z_f+a\,dz_f/da)e^{-i\theta}\right]\right|.
$$
For aligned apsides this becomes $2\Delta a[1-(e_f+a e_f')\cos\theta]$, provided the orbits remain nested. The radial gradient of <forced eccentricity> matters for the density comparison even in a narrow ring.
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