San José State University

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 The Universality of the Titius-Bode Type Relationship for Planets and Satellites and their Order Number

In 1768 Johann Bode published a remarkable empirical relationship for distances of the planets from the Sun. Bode did not originate the relationship; he only publicized it. Johann Titius in 1766 had earlier stated it in a footnote of his translation of a 1764 book by Charles Bonnet. Apparently the earliest statement of the relationship was by David Gregory in 1715. The relationship of the orbit radii of the planets given as ratios to that of Earth's orbit radius is:

#### R1 = 0.4 Rn = 0.4 + 0.3*2n-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 empirical fit is quite striking.

p
 Mercury Venus Earth Mars AsteroidBelt Jupiter Saturn Uranus Neptune/Pluto OrderNumber 1 2 3 4 5 6 7 8 9 Titius-Bode Law 0.4 0.7 1.0 1.6 2.8 5.2 10.0 19.6 39 ActualValue 0.387 0.723 1.0 1.524 2.7* 5.203 9.539 19.18 30.06/39.52

It is often said that there is no theoretical basis for this relationship. While the form Rn=c+a*bn does not have a theoretical justification there is justification of a relationship of the form Rn=a*bn, where n is the order number of the body and a and b are parameters.

When a physical system is periodically disturbed it becomes perturbed. The physical system has a natural resonance frequency. If the periodicity of the disturbance matches the period of its natural frequency the perturbations grow without bound or until the perturbances change the natural frequency of the system and break the resonance. In the case of planetary or satellite systems resonance nudges masses away from resonant orbits, but they do not need to go too far to break the resonance so planets and satellites form near the resonance orbits. Resonance typically occurs where there is a 1 to 2 or 2 to 5 ratio of the periods two bodies. Resonance can occur for different ratios, such as 1 to 3, but in the Solar System the ratios of 0.4 and 0.5 are overwhelmingly the most common. It seems to be a matter of chance whether a planet or satellite forms near the 0.4 or the 0.5 resonance point and whether it is above or below the resonance ratio. The reciprocals of 0.4 and 0.5 are 2.5 and 2.0, the ratio of the period of next body in the system to a particular one. For more on the details of this analysis see Bode.

When a relation of form T=abn is fitted in logarithmic form to the periods and order numbers of the planets the result is:

#### Tn = 0.094854*(2.280933)n

By Kepler's Law the orbit radius and period are related by the equation R=T2/3. This means that

#### Rn = 0.207987*(1.732773)n

The value of b in the equation for Tn is notably close to the average of the resonance ratios of 2.5 and 2.0; i.e., 2.25.

Similar relationships were found for the satellite systems of Jupiter, Saturn and Uranus. There also one star, designated HR8799, for which orbit distances have been found for three of its planets. The statisical relationships are given in the table below.

The Parameters of the Empirical Relationship T=abn Between Period and Order Number in Satellite Systems
Systemab
Planets
of the Sun
0.0948542.280933
Satellites
of Jupiter
0.23182.0072
Satellites I
of Saturn
0.36621.9422
Satellites II
of Saturn
0.46082.0305
Satellites
of Uranus
0.78041.7824
Planets of HR87991.00132.1646

The variations in the values of the parameter a are of no consequence because they incorporate the scale of the systems. The value of b is the significant parameter. For the planets there is a mix of the 0.4 and 0.5 resonance ratios. For Jupiter, Saturn and Uranus the dominant resonance ratio is 0.5. This is also true for the planets of star HR8799 although the value of b for that system does not reflect this.

The above relationships converted by Kepler's Law into relationships between orbit radii and order numbers are:

The Parameters of the Empirical Relationship R=abn Between Orbit Radius and Order Number in Satellite Systems
Systemab
Planets
of the Sun
0.20801.7328
Satellites
of Jupiter
0.37731.5912
Satellites I
of Saturn
0.51181.5567
Satellites II
of Saturn
.59641.6035
Satellites
of Uranus
0.84761.4701
Planets of HR87991.00091.6734