Copyright ©2017 by Paul Niquette. All rights reserved. 

A visualization
of the seven planets orbiting the star
TRAPPIST1.
NASA/JPL/Caltech Twentyfive
years ago, all the known planets
were
in our own Solar
System. Until 1992, exoplanets were the
stuff of speculation. Thanks
to advances in
detection
methods, as of this writing
(May 2017), more than 3,600 exoplanets
have been confirmed in more
than 2,700 planetary systems,
and there
are billions
more to be
found
in just the Milky Way Galaxy. TRAPPIST (Transit Planets and Planetesimals Small Telescope) is the name for a pair of robotic telescopes that made the initial discovery in 2015 of... TRAPPIST1 is an exceptional planetary system surrounding an ultracool dwarf star, which is about the size of our own planet Jupiter but situated 40 lightyears away in the constellation of Aquarius .Exceptional indeed, inasmuch as TRAPPIST1 comprises a total of seven temperate terrestrial exoplanets  the most found in any exoplanet system discovered so far. They are tightly packed, with all seven planets whirling through space well within the orbital dimensions of our planet Mercury. For the Trappist Orbits puzzle, let us name the planets p1, p2, p3, ... p7, and observe...
Close as they are to each other, the TRAPPIST1 planets have not collided for the last few million years. The seven orbits appear to be 'stable'  even harmonious... The linked reference postulates that orbital resonance plays a key rôle in assuring 'harmony' among the seven planets in the Trappist Orbits. It is easy to show that adjacent Trappist Orbits have periods that are ratios of small whole numbers and thus meet the criterion for orbital resonance... Consider the first three Trappist Orbits: τ1 = 1.511, τ2 = 2.422, τ3 = 4.050 earth days... τ2 / τ1 = 1.603 versus 8/5 = 1.600  99.8% agreement...therefore, while p2 completes 5 orbits, p1 completes 8 orbits and p3 completes 3 orbits. Sure enough, using gazillions of simulations on a supercomputer, astrophysicists have tested the 'harmony' concept and confirmed that the present configuration of Trappist Orbits will be sustainable for billions of years. This puzzle will use a simple spreadsheet to do the same... Here is a snapshot of the first three planets in circular counterclockwise orbits around TRAPPIST1. We observe that p1 has recently passed p2 where r12 = r2  r1, and p2 has recently passed p3, where r23 = r3  r2. Solvers can use this sketch to derive the distances r12 and r23, which by Newton's Law determine the instantaneous gravitational forces acting between the planets... Four forces are shown acting on p2: Gravitational force f2(r) imposed on p2 by the sun and varies inversely with the square of the orbital radius r2^{2}. Centrifugal force f2(ω,r) varies directly with the orbital radius r2 and with the square of the angular velocity ω2^{2}^{}^{}. For a circular orbit f2(r2) = f2(ω2,r2), thus ω2 = (GM)^{1/2} r2^{ 3/2}. Planets p1 and p3 in adjacent orbits impose gravitational forces f12 and f23 on p2, which vary inversely with r12^{2}^{ } and r23^{2}. As planet p1 passes below p2, ϕ1 = 0, r12 decreases to its minimum, r12 = r2  r1, and f12 reaches its maximum, f12 = m1 m2 G / (r2  r1)^{2}. Likewise, as planet p2 passes below p3, ϕ3 = 0, r23 decreases to its minimum, r23 = r3  r2, and f23 reaches its maximum, f23 = m2 m3 G / (r3  r2)^{2}. This
sketch shows f12
and f23 each
decomposed
into their respective Tangential
and Radial
components. We
observe that... Planet p1 has recently passed below p2, and its gravitational force f12 is now acting to accelerate p2 tangentially along its orbital path while simultaneously pulling p2 radially toward the sun.Newton's Third Law of Motion, of course, assures that f12 and f23 apply to p1 and p3, respectively, in equal magnitudes and opposite directions. Before taking on the challenge in the Trappist Orbits puzzle, some solvers may need to be advised that in orbital mechanics, the motions of celestial bodies resulting from applied forces are counterintuitive. Accordingly, we need to indulge in some orbital mechanics here... We might acquire an understanding of the subject at Kerbal Space Program, which provides a practical treatment of maneuver nodes in rocketry. Maneuver nodes are defined as planned orbital locations where thrust vectors are prescribed to accomplish various objectives: transfer orbit, circularization, rendezvous, stationkeeping. Solvers may regard the 'points of conjunction' between adjacent planets as 'spontaneous maneuver nodes' and treat the components of gravitational attraction between planets as if they were orthogonal thrust vectors as produced in command modules and artificial satellites... From a circular orbit, a brief Tangential force applied in the same direction as a satellite's motion ('prograde') changes the orbit to elliptical; the satellite will ascend and reach the highest orbital point (the apoapse) at 180 degrees away from the point of application ('firing point' in rocketry); then it will descend back, returning to the firing point. So too, thrust applied in the opposite direction of the satellite's motion ('retrograde') changes the orbit to elliptical with the lowest orbital point (the periapse) at 180 degrees away from the point of application; then it will ascend back, returning to the firing point. From any location in an elliptical orbit, a brief Radial force applied toward the focus of the orbit  the sun  rotates the whole orbit in its plane through an angle in the direction of the satellite's motion, thereby moving both the apoapse and the periapse to new locations still opposite to each other. The effect is generally described as orbital 'precession'. A brief force applied away from the focus of the orbit 