The Titius-Bode rule for the orbit radius R of a planet of the Solar System as a function of its order numbers n is:

R₁ = 0.4
R_n = 0.4 + 0.3*2^n-2

where n is the order number of the planet starting with n=1 for Mercury. The relationship posits a planet number 5 where the asteroid belt is. The orbit radii are expressed relative to that of the Earth.

For a history of the Titius-Bode rule and a comparison of the values it gives with the actual values see Titius-Bode. The fit is quite remarkable.

There is however no theoretical justification for the particular form given above. There is a justification for a relationship between the orbit period T and the order number of the form

T_n = a*bⁿ

and Kepler's Law, R=cT^2/3 provides a translation into a relationship between orbit radius and order number.

The theoretical basis for the relationship between orbit period and order number is that in a planetary ring resonance drive planetary material away from resonance orbits. But material nudged out of a resonance orbit moves at a different speed than the surrounding material which results in that surrounding material being swept up into a planet. Thus planets form near the resonance orbits. The resonance orbits are ones in which the ratio of orbit periods are ratio of simple whole numbers. For example, the period of an inner orbit may be 1/2 or 2/5 of the period of the outer orbit.

The asteroids of the Asteroid Belt provide evidence of this resonance phenomena. There is an absence of asteroids in the orbits which involve resonance with the motion of the planet Jupiter. Although there is an absence in the resonance orbits there is increased frequencies near the resonance orbits.

The ratios of the orbit periods of the planets and planetoids of the Solar System are as follows.

The preponderance of the ratios involve the 1/2 and 2/5 resonances. The average ratio is 0.473. Note that the ratios should not be equal to the resonance ratios but be near them.

It is more convenient to use the logarithmic form of the relationship between orbit period and order n for statistical analysis; i.e.,

log₁₀(T_n) = A + Bn

where A=log₁₀(a) and B=log₁₀(b).

The plot of the data for the Solar System is as shown below.

The regression of the logarithm of orbit period on the order number gives the following.

log₁₀(T_n) = −0.9930 + 0.3493n
with a coefficient of determination of
R² = 0.9942

The same analysis applies to the satellite systems of the planets. The statistical results are:

Jupiter:
log₁₀(T_n) = −0.6349 + 0.3026n
R² = 0.99668
Saturn:
System I: log₁₀(T_n) = −0.4363 + 0.2883n,
R² = 0.9954
System II: log₁₀(T_n) = −0.3365 + 0.3076n
R² = 1.0000
Uranus
log₁₀(T_n) = −0.1077 + 0.2510n
R² = 0.9949

It was found reasonable to separate the satellites of Saturn into two systems for the analysis.

Although the regression coefficients for order number are significantly different from the value of 0.3493 found for the planets the values are of the same order of magnitude and reasonably close.

The Planetary Systems
of Other Stars

The development of methods for finding planets orbiting other stars is proving to be quite successful. In the December 6, 2008 issue of Science News it was reported that images of three planets were found for the star HR8799 which lies about 130 light-years from our solar system. The planets are massive, one having 10 times the mass of Jupiter.

These planets lie 25, 40 and 70 astronomical units (A.U.) from their star. This means that, by Kepler's Law, their orbit periods are proportional to 25^3/2, 40^3/2 and 70^3/2; i.e., 125, 253 and 585.7. The constant of proportionality depends on the mass of the star relative to that of the Sun. This constant however does not affect their ratios. The ratio of the period of the middle planet to the outer planet is 0.432. The ratio of the period of the inner planet to that of the middle planet is 0.494. Thus the planets are located near the 0.4 and 0.5 resonance bands. This is essentially the same pattern as prevails for the planets of our solar system.

The regression equation for the above data for the planets of star HR8799 is:

log₁₀(T_n) = 1.75180 + 0.33538n
R² = 0.9975

Note that if the order numbers are off by a constant amount the coefficient for the regression equation is not affected. The effect is entirely in terms of the constant term of the equation.

The similarity of the regression coefficient to that found for the Solar System is quite remarkable.

Here is a plot that shows the similarity.

A study published in the August 13, 2010 issue of Astronomy and Astrophysics by Christophe Lovis et al entitled The HARPS Search for southern extra-solar planets gives estimates of the orbit periods for a system of seven planets orbiting the star HD10180.

It is not clear from the article what the units of the periods are but it does not matter. It is only their relative values that are important.

The method used in finding the planets of HD10180 would miss planets of smaller mass; planets of the size of Earth, for example. Consider the ratio of the periods for planets #4 and #5. If there is an overlooked planet between the two the ratio would be the square a resonance ratio. The square root of 0.32122 is 0.56676, which could represent resonance near 3/5. The cube root of the ratio of 0.21961 for planets #5 and #6 is 0.60333, again notably close to a resonance of 3/5. The fourth root of the ratio of 0.13004 for planets #6 and #7 is 0.60051. These values suggest that there is one overlooked planet between #4 and #5, two between #5 and #6 and three between #6 and #7. This would result in the following revision of the data.

The graph of the logarithms₁₀ versus the order numbers of the planets is quite remarkable.

The linearity comes from the fact that the integral roots of the ratios of planet periods are nearly all approximately 3/5.

The regression equation for the logarithm of periods as a linear function of the order number of the planets is

log₁₀(T_n) = 2.76669 + 0.22421n
R² = 0.9989

In 2010 a team at the ESO La Silla Observatory led by Michel Mayor announced the discovery of three super Earth-sized planets orbiting the star HD40307. Using the High Accuracy Radial velocity Planet Searcher (HARPS) instrument they were able to estimate the orbit periods of these planets.

The ratios are notably close to the 1/2 and 2/5 resonances.

The plot is extraordinarily linear.

The regression equation confirms that linearity.

log₁₀(T_n) = 0.2999 + 0.33814n
R² = 0.9997

In 2010 a team led by Steven Vogt announced the discovery and estimation of the orbit periods for a system of six planets of the star Gliese (GJ)581.

The ratios are notably close to common resonance ratios except for the ratio for the 5th and 6th planet. However the fourth root of that ratio is 0.62688. This suggests that there are three planets falling below the threshold of perception of the methods. This would change the order number of planet 6 to 10.

The graph of the data with this revision is satisfying. The data point for the last planet fits well with the line for the data points of planets 3, 4 and 5.

The regression line differs from those for three previous stars.

log₁₀(T_n) = 0.31875 + 0.23896n
R² = 0.9914

The magnitude of the coefficient is significantly smaller than those for the Sun, HR8799 and HD40307 and closer to those for the satellites of Jupiter, Saturn and Uranus. It is quite close to the value for the satellites of Uranus. It is notable that Gliese 581 is a dwarf star.

Conclusions

The satellite systems of four stars and three solar planets all have Titius-Bode type relationships between orbit periods and order numbers of the form T_n=a*bⁿ and the magnitudes of the regression coefficients are all in the range of 0.22 to 0.35. This confirms that the spacing of the satellites is dictated by resonance phenomena which put the satellites near the resonance orbits.