San José State University

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Thayer Watkins
Silicon Valley
USA

 An Investigation of Whether a Titius-Bode Type Relation Exists for the Satellites of Neptune

A previous study found that there do exist exponential relationships between the orbit periods and the orbit numbers for the planets of four stars, including the Sun, and for the satellite systems of Jupiter, Saturn and Uranus. For the thirteen moons of Neptune such a relationship does not appear to exist. The Neptune case is worth examining because it shows how remarkable are the relationships which exist for the other cases.

The Titius-Bode "Law" is in terms of orbit radii and order numbers, but the fundamental relationship is in terms of orbit periods. The relationship between orbit period T and orbit number n is of the form

Tn = a*bnor, equivalently log(Tn) = log(a) + log(b)*n

It is to be emphasized that n is the order number of the orbit rather than an order number of the planet or satellite. Usually there is no distinction between the two concepts, but there can be. If two planets occupy the same orbit they would have the same orbit order number and the order number would be increased by only one for the next orbit, not two.

A Titius-Bode type relationship exists because of the phenomenon of resonance. Once one planet is formed planetary material is nudged out of resonance orbits. But that material does not go very far before it starts sweeping up the planetary material in nearby orbits and thus forms a planet. Thus satellite bodies are formed near but not at resonance orbits. The resonance orbits may be ones that have orbit periods 1/2, 2/5, 3/5 or some similar simple ratio of the period of one planet for inner orbits or 2, 2.5, 1.67 etc. for outer orbits.

For the Solar System the resonance ratios are mainly 1/2 and 2/5 but for the other stars a resonance ratio of 3/5 is common.

Now consider the orbit period of Neptune.

The Moons of Neptune
Moon
Number
NamePeriod in
Earth days
2Thalassa 0.311
3Despina 0.335
4Galatea 0.429
5Larissa 0.555
6Proteus 1.122
7Triton 5.877
8Nereid 360.136
9Halimede 1879.08
10Sao 2912.72
11Laomedeia 3171.33
12Psamathe 9074.3
13Neso 9740.73

The most obvious thing is that there are two satellite systems. Moons 1 through 7 are close to Neptune and have short orbit periods. Moons 8 through 13 are far away and have long orbit periods. A Titius-Bode relationship would exist only for satellites which formed out of some planetary ring. Satellites which have been randomly captured would not necessarily bear any relationship to one another. Here are the graphs of the logaritms of the periods versus the moon numbers for the two sets of satellites.

There is not much linearity to this relation but a quadratice regression equation provides a remarkably good fit. However there is no plausible basis for such a quadratic relationship. Nevertheless here is the quadratic regression equation.

log10(Tn) = −0.24349 −0.27487n + 0.05773n²

The coefficient of determination (R²) for the equation is 0.9598. The t-ratios for the coefficients of n and n² are −2.7 and 4.6, respectively.

The outer moons are renumbered 1 through 6 for convenience.

This display could be contrued to be linear but it would not be a very good fit. However note how close the logarithms of orbit times are for moons 3 and 4 and for moons 5 and 6. If moons 3 and 4 are taken to be in the same orbit then their orbit numbers would both be 3 and moons 5 and 6 would both be in orbit 4. The data for the outer moons would then be:

 OrbitNumber Name Period inEarth days 1 Nereid 360.136 2 Halimede 1879.08 3 Sao 2912.72 3 Laomedeia 3171.33 4 Psamathe 9074.3 4 Neso 9740.73

The graph and regression of this data are as follows:

log10(Tn) = 2.20628 + 0.44145n

The coefficient of determination (R²) for the equation is 0.96506. The slope of 0.44145 is notably higher than what were found for the satellites of Jupiter, Saturn and Uranus and for the star planetary systems.

With the moderate success in mind for the outer moons in mind that came from modifying the orbit order numbers it is worthwhile to take another look at the data for the inner moons. The logarithms of orbit period for the first three moons are so close that it is not unreasonable to consider them all to be in the same orbit. This makes the orbit of the fourth moon the second orbit and likewise the orbit of the fifth moon is the third orbit and so on. The data for the inner moons of Neptune are then

 MoonNumber Name PeriodEarth days Ratio of Periodto Period for Next Orbit 1 Naiad 0.294 0.68531 1 Thalassa 0.311 0.72494 1 Despina 0.335 0.78089 2 Galatea 0.429 0.77297 3 Larissa 0.555 0.49465 4 Proteus 1.122 0.19091 8 Triton 5.877

The value of 8 in the last line of the table stems from the following line of analysis. The ratio of the period of Proteus to that of Triton is 0.19091. The fourth root of this number is 0.66101. This is roughly the value for a 2/3 resonance. The fourth root corresponds to there being three orbits between that of Proteus and that of Triton. With these modifications of the orbit order numbers the plot of the data is as shown below.

The regression equation is

log10(Tn) = −0.80263 + 0.23891n

The coefficient of determination is 0.9447 and the t-ratio for the regression coefficient is 9.2. The value of 0.23891 for the slope of the regression line is comparable to the value found for the satellites of Uranus.

Conclusions

There is evidence for a Titius-Bode type relationship between orbit period and orbit order number for the two satellite systems of Neptune but allowance but be made for more than one satellite occupying a single orbit.