Orbital resonance in Neptune’s moons

Phys.com published an article a couple days ago NASA finds Neptune moons locked in ‘dance of avoidance’. The article is based on the scholarly paper Orbits and resonances of the regular moons of Neptune.

The two moons closest to Neptune, named Naiad and Thalassa, orbit at nearly the same distance, 48,224 km for Naiad and 50,074 km for Thalassa. However, the moons don’t come as close to each other as you would expect from looking just at the radii of their orbits.

Although the radii of the orbits differ by 1850 km, the two moons do not get closer than about 2800 km apart. The reason has to do with the inclination of the two orbital planes with each other, and a resonance between their orbital periods. When the moons approach each other, one dips below the other, increasing their separation.

Assume the orbits of both moons are circular. (They very nearly are, with major and minor axes differing by less than a kilometer.)  Also, choose a coordinate system so that Thalassa orbits in the xy plane. The position of Thalassa at time t is

rT  (cos 2πt/TT, sin 2πt/TT, 0)

where rT is the radius of Thalassa’s orbit and TT is its period.

The orbit of Naiad is inclined to that of Thalassa by an angle u. The position of Naiad at time t is

rN (cos u cos 2πt/TN, sin 2πt/TN, -sin u cos 2πt/TN).

I tried implementing the model above using data from Wikipedia articles on the two moons, but in my calculations the moons get closer than reported. I suspect the parameters have to be set carefully in order to demonstrate the kind of resonance we see in observation, and we don’t know these parameters accurately enough to make a direct geometric approach like this work.

from numpy import *

# Orbital radii in km 
r_T = 50074.44
r_N = 48224.41

# Period in days
T_T = 0.31148444
T_N = 0.29439580

def thalassa(t):

    # frequency
    f = 2*pi/T_T

    return r_T * array([
        cos(f*t),
        sin(f*t),
        0
    ])

def naiad(t):
    
    # inclination in radians relative to Thalassa
    i = 0.082205
    
    # frequency
    f = 2*pi/T_N

    return r_N * array([
        cos(i)*cos(f*t),
        sin(f*t),
        -sin(i)*cos(f*t)
    ])

def dist(t):
    return sum((naiad(t) - thalassa(t))**2)**0.5

d = [dist(t) for t in linspace(0, 10*73*T_N, 1000)]
print(d[0], min(d))

In this simulation, the moons are 4442 km apart when t = 0, but this is not the minimum distance. The code above shows an instance where the distance is 2021 km. I tried minor tweaks, such as adding a phase shift to one of the planets or varying the angle of inclination, but I didn’t stumble on anything that cane closer to reproducing the observational results. Maybe resonance depends on factors I’ve left out. Naiad and Thalassa are practically massless compared to Neptune, but perhaps the simulation needs to account for their mass.

The periods of the two moons are nearly in a ratio of 69 to 73. We could idealize the problem by making the periods rational numbers in exactly a ratio of 69 to 73. Then the system would be exactly periodic, and we could find the minimum distance. Studying a model system close to the real one might be useful. If we do tweak the periods, we’d need to tweak the radii as well so that Kepler’s third law applies, i.e. we’d need to keep the squares of the periods proportional to the cubes of the radii.

Related post: Average distance between planets

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