  We propose a closed-form (i.e. without expansion in the orbital
eccentricities) scheme for computations in perturbation theory in the
restricted three-body problem (R3BP) when the massless particle is in an orbit
exterior to the one of the primary perturber. Starting with a multipole
expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a
sequence of normalizations in Delaunay variables by Lie series, leading to a
secular Hamiltonian model without use of relegation. To this end, we introduce
a book-keeping analogous to the one proposed in Cavallari and Efthymiopoulos
(2022) for test particle orbits interior to the one of the primary perturber,
but here adapted, instead, to the case of exterior orbits. We give numerical
examples of the performance of the method in both the planar circular and the
spatial elliptic restricted three-body problem, for parameters pertinent to the
Sun-Jupiter system. In particular, we demonstrate the method's accuracy in
terms of reproducibility of the orbital elements' variations far from
mean-motion resonances. As a basic outcome of the method, we show how, using as
criterion the size of the series' remainder, we reach to obtain an accurate
semi-analytical estimate of the boundary (in the space of orbital elements)
where the secular Hamiltonian model arrived at after eliminating the particle's
fast degree of freedom provides a valid approximation of the true dynamics.
