Editing Orbital resonance (section)

=== Laplace resonance ===
[[File:TheLaplaceResonance2.png|thumb|300px|Illustration of Io–Europa–Ganymede resonance. From the centre outwards: Io (yellow), Europa (gray), and Ganymede (dark)]]
The Laplace resonance involving Io–Europa–Ganymede includes the following relation locking the ''orbital phase'' of the moons:

:<math>\Phi_L=\lambda_{\rm Io} - 3\cdot\lambda_{\rm Eu} + 2\cdot\lambda_{\rm Ga}=180^\circ</math>

where <math>\lambda</math> are [[mean longitude]]s of the moons (the second equals sign ignores libration).

This relation makes a triple conjunction impossible. (A Laplace resonance in the [[Gliese 876]] system, in contrast, is associated with one triple conjunction per orbit of the outermost planet, ignoring libration.) The graph illustrates the positions of the moons after 1, 2, and 3 Io periods. <math>\Phi_L</math> librates about 180° with an amplitude of 0.03°.<ref name="Sinclair1975">{{cite journal |last1=Sinclair |first1=A. T. |year=1975 |title=The Orbital Resonance Amongst the Galilean Satellites of Jupiter |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=171 |issue=1 |pages=59–72 |bibcode=1975MNRAS.171...59S |doi=10.1093/mnras/171.1.59|doi-access=free }}</ref>

Another "Laplace-like" resonance involves the [[Moons of Pluto|moons]] [[Styx (moon)|Styx]], [[Nix (moon)|Nix]], and [[Hydra (moon)|Hydra]] of Pluto:<ref name="ShowalterHamilton2015">{{cite journal |last1=Showalter |first1=M. R. |author1-link=Mark R. Showalter |last2=Hamilton |first2=D. P. |year=2015 |title=Resonant interactions and chaotic rotation of Pluto's small moons |journal=[[Nature (journal)|Nature]] |volume=522 |issue=7554 |pages=45–49 |bibcode=2015Natur.522...45S |doi=10.1038/nature14469 |pmid=26040889|s2cid=205243819 }}</ref>

:<math>\Phi=3\cdot\lambda_{\rm S} - 5\cdot\lambda_{\rm N} + 2\cdot\lambda_{\rm H}=180^\circ</math>

This reflects orbital periods for Styx, Nix, and Hydra, respectively, that are close to a ratio of 18:22:33 (or, in terms of the near resonances with Charon's period, 3+3/11:4:6; see [[Orbital resonance#Coincidental 'near' ratios of mean motion|below]]); the respective ratio of orbits is 11:9:6. Based on the ratios of [[synodic period]]s, there are 5 conjunctions of Styx and Hydra and 3 conjunctions of Nix and Hydra for every 2 conjunctions of Styx and Nix.<ref name="ShowalterHamilton2015" /><ref name="Witze2015">{{cite journal |last1=Witze |first1=A. |date=3 June 2015 |title=Pluto's moons move in synchrony |journal=[[Nature News]] |doi=10.1038/nature.2015.17681|s2cid=134519717 }}</ref> As with the Galilean satellite resonance, triple conjunctions are forbidden. <math>\Phi</math> librates about 180° with an amplitude of at least 10°.<ref name="ShowalterHamilton2015" />
{{center|
{{Annotated image
 |image=Hydra, Nix, Styx conjunctions cycle.png
 |image-width=625
 |width=625
 |height=124<!-- to crop the lower part of the image -->
 |float=center
 |annotations=<!-- this parameter must be there, empty or not! -->
 |caption=Sequence of conjunctions of Hydra (blue), Nix (red), and Styx (black) over one third of their resonance cycle. Movements are counterclockwise and orbits completed are tallied at upper right of diagrams (click on image to see the whole cycle).}}
}}