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 Titius-Bode-Type Rules for Planetary Satellite Systems

Background

There are a number of examples of resonance phenomena in the solar system that suggest that the regularity and magnitude of the parameters of Titius-Bode type of rules for satellite systems should be in terms of the periods of the orbits rather than the average radii of the orbits. For example, among the asteroid there are none having a period which is one third, one half or two fifths of the period of Jupiter. On the other hand there are 19 which have periods which are close to two thirds that of Jupiter. And there are the Trojan and Apollo asteroids which have the same period as that of Jupiter but move in orbits such that they along with the sun and Jupiter form equilateral triangles. Another resonance phenomenon is the Cassini gap in the rings of Saturn. The Cassini gap is a 1700 mile wide gap that corresponds to a region in which particle would have a period of revolution near to one half the period of Saturn's innermost moon, one third the period of the second moon or one fourth the period of the third moon.

Jupiter

The distances and the length of the years of the various satellites of Jupiter are as follows:

Satellite
Discovery
Order
(000s km)
Length of Year
(Earth days)
AmaltheaV1810.498
IoI4221.770
EuropaII6713.554
GanymedeIII10717.166
CallistoIV188316.754
VI11470266.00
VII11740276.67
X11850253
XII21200631
XI22560692
VIII23500735
IX23700758

There appears to be three groups of satellites.

• The inner-most five have a relationship that resembles that of the planets. The inclinations of the orbits of the inner five with respect to the equatorial plane of Jupiter are essentially zero.
• Satellites VI, VII and X form another group. The radii of their orbit are about the same. The inclinations of the second group are all close to 28°.
• The four outermost form a third group. The inclinations of the third group range from 17° to 33° but their motion is clockwise instead of counterclockwise with respect to the north pole of the planet. The radii of their orbits are about the same.

This information indicates that the second and third groups are the results of the capture of asteroids sometime after the formation of the innermost satellites.

For the inner group if one hypothesizes a missing satellite between Amalthea and Io then the plot of the period of revolution versus the order number of the satellites is as shown below:

This suggests an exponential relation of the form

T = abn or, equivalently for more convenient estimation of the parameters a and b log10(T) = log10(a) + n*log10(b)

The plot of log10(T) versus order number:

The regression line is

Saturn

For Saturn the data are:

Satellite
Discovery
Order
(000s km)
Length of Year
(Earth days)
JanusX1570.749
MimasI1840.9424
TethysIII2931.882
DioneIV3742.737
RheaV5234.517
TitanVII121315.945
HyperionX147021.276
IapetusVIII353679.331
PhoebeIX12864550.417

The first observation is that compared with Jupiter's satellite system Saturn really has them packed in. The inner eight seem part of a system and they all have inclinations of their orbits near zero. Definitely the outer moon, Phoebe, is of a different nature. Its distance is of another order of magnitude and it has retrograde motion. Its angle of inclination is 30° with retrograde motion or 150° if its motion is not consider retrograde. The next to the outer moon, Iapetus, has an angle of inclination of 14.7° and its distance make is uncertain whether it is of the same family as the inner group.

First a plot of the basic data:

The data points for the inner six moons do approximate a straight line. The data for the outer two moons were made to fit by hypothesizing two missing moons in the system. This is of dubious justification but not totally devoid of validity in that the last moon in the set more or less fits the line without any further manipulation.

The regression equation for the first six moons is:

log10(T) = -0.3175 + 0.15511n R² = 0.9887

The slope of the regression line for Saturn's inner moons is about half of that for Jupiter's inner moons. For the inner moons of Saturn there appears to be two systems interleaved. If Janus, Enceladus, Tethys and Rhea (System I) are taken as one system and Mimas, Tethys and Titan (System II) taken as another then the data looks as follows. It is hypothesized that there is a missing fifth moon in System I and two missing moons (third and fourth) in System II.

The regression equations for the two systems are:

System I: log10(T) = −0.4363 + 0.2883n, R² = 0.9954 System II: log10(T) = −0.3365 + 0.3076 R² = 1.0000

Again the magnitude of the slope is in the range of 0.25 to 0.30.

Neptune

Nepture has only two moons and the largest of them has retrograde motion. There is not a system of satellite in the nature of what Jupiter, Saturn and Uranus have.

The Planets

The graph of log10(Period) where Period is the Period of Revolution relative to that of Earth versus the order number:

The regression equation is

log10(T) = −0.9920 + 0.3493n R² = 0.9942

The magnitude of the slope is on the same order as the 0.3 found for Jupiter's satellite system.

The estimates of the regression slope coefficients for comparison are:

SystemSlope Estimate
Solar0.3493
Jupiter0.3026
Saturn I0.2883
Saturn II0.3076
Uranus0.2510

The magnitudes are correlated with the mass of the primary bodies in the systems. The plot of the slope coefficients versus the logarithm of the mass of the primary is shown below. (The masses are in units of Earth masses.)

Conclusion

On the basis of the preceding analysis there seems to be a systematic relationship between the periods of planets/satellites revolving around a primary body. The relationship may arise from resonance phenomena that prevents material from achieving a stable orbit if its period is one half, one third or two fifths of the period of an outer body in the system. The resonance could also work in the other direction preventing an outer body forming in orbits with periods equal to two or three times the period of the inner body. Of course if material is prevented from achieving stable orbits in some range that material gets concentrated in orbits of other ranges. Suppose resonance precludes orbits with periods of two fifths or one half of the period of the outer body and so the material gets concentrated an orbit with a period of 0.45 of that of the outer body. The reciprocal of 0.45 is 2.22.

Consider a period/order relationship of the form

Tn = abn and hence log10(Tn) = log10(a) + n*log10(b)

where n is the order number of the planet or satellite. If b=2.22 (the reciprocal of 0.45) then log10(b)=0.3468, essentially the value found for the planets of the solar system. The value of b is marginally different for the Jovian planets' satellite systems and may be weakly a function of the mass of the primary body of the system.

Because of Kepler's Law, R3=αT2, the relationships concerning periods can be converted to relationships involving the radii of orbits; i.e.,

Rn = (αTn)2/3 = (αa)2/3b(2/3)nand hence log10(Rn) = (2/3)log10(αa) + (2/3)log10(b)n.

If b=1/0.45 then b2/3=1.703.

(To be continued.)