Copyright ©2017 by Paul Niquette. All rights reserved.
A visualization of the seven planets
orbiting the star TRAPPIST-1.
Twenty-five years ago, all the known
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
Exceptional indeed, inasmuch as TRAPPIST-1
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.
The linked reference postulates that
resonance plays a key rôle in assuring 'harmony' among the seven
planets in the Trappist Orbits.
Consider the first three Trappist
Orbits: τ1 =
1.511, τ2 = 2.422, τ3 =
4.050 earth days...
...therefore, while p2
completes 5 orbits, p1 completes 8
orbits and p3 completes 3 orbits.
This sketch shows f12
and f23 each decomposed into their
respective Tangential and Radial
components. We observe that...
Third Law of Motion, of course, assures that
f12 and f23 apply to
p1 and p3,
respectively, in equal magnitudes and opposite
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
counter-intuitive. Accordingly, we
need to indulge in some orbital mechanics here...
In the sketch above, we have coined the terms
'apotrap' and 'peritrap' for the Trappist Orbits puzzle.
Whenever the elliptical orbits of p1
and p2 happen to become aligned,
such that p1 passes its p1
apotrap just as p2
simultaneously passes its p2 peritrap
directly above, then r12 reaches
its absolute minimum, which results in
radial force f12 reaching its absolute
maximum. The p2 orbit
rotates in its plane through some angle γ2 in the direction of the planet
motion, as indicated by p2 peritrap rotated
ccw. Meanwhile, being acted on
equally in the opposite direction, the p1 orbit
rotates in its plane through some angle γ2 in angle γ1 in the direction opposite to the
planet motion, as indicated by p1 peritrap