This shows the evolution of the orbits of three planets at Gliese 876. From closest to the star to the farthest, the planets are d, c, and b. Since the creation of this video, a fourth planet, e, has been discovered further out. We see the planets b and c in dynamic secular interaction where the orbits of both planets are precessing. The point of a planet’s orbit where it is closest to the star is the longitude of perihelion, ω. If ω changes, than the orbit rotates around its focus. This change, Δω, occurs because the two planets are influencing each other strongly enough to cause their orbits to change. A line connects the star with each the perihelions of b and c to show another intriguing feature of this 2:1 resonance. The angle between ω_b and ω_c goes up and down like a pendulum. Planet b makes one orbit in the time it takes c to. While this is a very generalised idea of a mean motion resonance, it does not capture the entire idea. Let us denote the mean motion of a planet as ϖ, measured in degrees per unit time. b will make an orbit in twice the amount of time it takes c, so we can say that ϖ_b = 2ϖ_c. One might thus define the resonance as ϕ = ϖ_b – 2ϖ_c = 0 However, secular interactions will occur and the above turns out to be an oversimplification. We must account for the precession of the orbits of the planets. ϕ = ϖ_b – 2ϖ_c + Δω = 0
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