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tokens: in this limit. is given bywhere n , a , e , i are a resonant eKBO’s mean motion, semimajor axis, eccentricity, and inclination (relative to HP’s orbit plane), is the planet’s semi-minor axis, and c is a numerical coefficient whose asymptotic value is ∼5.8 for a ≪ a ′ ( c will be larger for a approaching a ′). For numerical estimates we adopt e ′ = 0.2 and c = 8. We find that for m ′ = 10 M ⊕ , the small amplitude oscillation periods for the four eKBOs are 0.6–0.94 Gyr, a factor of a few smaller than the age of the solar system. We conclude that 10 M ⊕ is approximately the minimum mass necessary to maintain periodic orbits of the third kind for the four eKBOs. 2.3. Planet’s Current Location in Its Orbit Each resonant eKBO constrains the longitude of HP because the critical resonant angle must librate about either 0° or 180°. For illustration, consider the libration of about 180° for Sedna’s 3/2 resonant orbit. Given that Sedna is presently near its perihelion, λ ≈ π , the planet can be in proximity to one of three longitudes: 60°, 180°, or 300° from Sedna’s longitude of perihelion, π . The trace of the resonant orbit in the rotating frame (Figure 2 ) shows that libration of φ would allow the planet to be located as far as ∼54° from these longitudes. This means that only three small zones of ±6° around each of three locations, π , and are excluded for the planet’s present location. For e ′ = 0.2, we find that the excluded zone is a little larger, ±75 around each of the same three locations. However, such geometric estimates of exclusion zones do not account for the planet’s gravity. To account for this, we computed surfaces of section for the circular planar restricted three body problem for several values of the planet mass in the range of 10 −6 –10 −3 M ⊙ , for the neighborhood of the 3/2, 5/2, 3/1, and 4/1 MMRs and the extreme eccentricities of the four eKBOs (0.85, 0.87, 0.85 and 0.7, respectively). Figure 4 shows a sample of these for planet mass 3 × 10 −5 M ⊙ (approximately 10 M ⊕ ). From these surfaces of section, we determine the range of excluded longitudes of the planet by measuring the complement of the large libration zones of the 3/2 and 5/2 MMRs. We also conservatively exclude a small range, ±5°, around the homoclinic points of the 3/1 and 4/1 resonant eKBOs. Figure 5 shows our estimates of the excluded zones, projected on the sky, for a planet of mass 3 × 10 −5 M ⊙ ; we plot a representative case of the planet’s low inclination orbit (in green) and also a representative case of its high inclination orbit (in red). (Note that a range of orbital planes near each of these representative cases is allowed by our calculations.) We find that approximately half of the ecliptic longitudes are excluded in both cases; the excluded zone is not contiguous. Figure 4. Surfaces of section for the circular planar restricted three body problem, for mass ratio , and Jacobi constant values as indicated. These are phase space portraits in the