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A Derivation of Titius-Bode Type Relations for the Planets
of the Solar System, the Satellite Systems of the Solar Planets
and one Exoplanet System

This is a derivation of the relationships of orbit radii and periods of revolution to the order number of the orbits for the planets and their satellite systems. Ever since 1768 when Johann Bode published the remarkable empirical relationship for distances of the planets from the Sun astronomers have puzzled over whether there is any physical justification for it. 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 orbit starting with n=1 for Mercury. The relationship posits a planetary orbit number 5 where the asteroid belt is.

The empirical fit is quite striking.

 Mercury  Venus  Earth  Mars  Asteroid
Belt 
 Jupiter  Saturn  Uranus   Neptune/Pluto 
Order
Number
12345 6789
Titius-
Bode Law
0.40.71.01.62.85.210.019.639
Actual
Value
0.3870.7231.01.5242.7*5.2039.53919.1830.06/39.52
*Ceres

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. This will be given below. For the solar system and some of the satellites of Jupiter, Saturn and Uranus each planet or satellite is roughly twice as far from the system center as the preceding satellite. (For more on the relationships which exists for the satellite systems of Jupiter, Saturn and Uranus see Bode2.) There has to be a physical explanation for this pattern. And there is. The explanation is in terms of resonance.

The explanation is not in terms of radii per se; it is in terms of orbit periods. There is of course a relationship between orbit period and orbit radius, called Kepler's Law, which says the cube of orbit radius is proportional to the square of orbit period. The physical process accounting for the radii at which planets formed involves their orbit periods.

The original system was a planetary ring rotating about the Sun not as a unit but instead with Kepler's Law satisfied. Once one planet was formed any planetary material having an orbit period equal to one half the orbit period of the planet would be nudged out of its orbit. This is the phenomenon of resonance. If any two bits of matter are in orbits such that one makes two revolution about the Sun for every one the other makes then their gravitational fields will ultimately nudge each other out of those resonant orbits. They do not have to move very far to break the resonance.

It is a remarkable property of resonance that the ultimate effect does not depend upon the magnitude of the perturbance but instead only on matching of frequencies.

The phenomenon of resonance occurs both ways; toward an inner band and toward an outer band. For purposes of explanation it is convenient to focus on the inner resonance band created by a planet.

Once the material's orbit period was significantly different from one half of the planet's orbit period the resonance would be broken. Resonance would also occur if the planetary material orbited the Sun three times for every two times the planet orbited it. Likewise for the material having an orbit period 2/5 of the planet's orbit period.

The significant feature of resonance is that the disturbing influence does not have to be strong to eventually have a great effect; it only has to be of the right frequency.

It has long be recognized that resonance has had an important role in the formation of the structure of the solar system. For example, among the asteroids there are none which have periods which are 1/3, 1/2 and 2/5 of that of Jupiter. The absence of asteroids at or near the resonance points are what are known as the Kirkwood Gaps, named after the American astronomer Daniel Kirkwood who discovered the phenomenon in 1886. Here is the plot of the orbit size of about 157 thousand asteroids.

Note that the frequency rises to peaks near the resonance points.

In the rings of Saturn there is a gap of 1700 miles that corresponds to 1/2 the period of the moon Mimas, 1/3 the period of Enceladus and 1/4 the period of Tethys.

However it seemed paradoxical that planets are found close to the forbidden resonance bands. This is no paradox. The planetary material did not have to move very far from the forbiddedn resonance zone to break the resonance. However when the planetary material moved away from its original orbit it would be moving at a velocity different from that of the surrounding material. This led to collisions and agglomeration of material. As illustrated in the diagram the planetary material would be concentrated in the space near the resonance band. Thus planets would form near the resonance bands.

Planetary material close to the forming planet would be swept up by collision and by the gravitational field of the planet. The planets acquired not only mass but angular momentum in this process. See Planetary Sweep.

Here are the ratios of the orbit period of the next closer to the Sun to each planet's orbit period. The ratio of the period for the planet next farther away from the Sun to the period for the planet is called the Reciprocal Ratio in the table.

Planet or
Planetoid
RatioReciprocal
Ratio
Mercury---2.557
Venus0.3911.623
Earth0.6161.900
Mars0.5262.421
Ceres0.4132.609
Jupiter0.3832.458
Saturn0.4072.847
Uranus0.3511.964
Neptune0.5091.503
Pluto0.665---

The average of the nine ratios given is 0.4736. If Pluto is left out the average is 0.4496, notably close the average of 0.4 and 0.5.

Here is the tabulation of the frequencies of the ratios for 0.02 intervals from 0.3 to 0.62. Pluto is left out of the tabulation.

The ratios are generally in the vicinity of 0.4 and 0.5 and the reciprocal ratios in the vicinity of 2.0 or 2.5. The cases where the ratio is not near 0.4 or 0.5 are situations where something besides the ratio is unusual. Pluto is most likely a moon-like object that escaped from its primary. Uranus is the odd case in which its angle of inclination is about 90°. This probably indicates that something catastrophic happened to Uranus. The case of Venus and Earth being so close together is a puzzle, but Venus has retrograde rotation, indicating also that something catastrophic occurred. However even for the odd cases the ratios are close to a resonance. Venus may be a case in which the planet formed near the 3/5 resonance band. Likewise Uranus could have formed near the 1/3 resonance band. And, of course, Pluto could have formed near the 3/2 resonance band for Neptune but there is always the possibility that Pluto simply should be left out of the analysis.

Planet or
Asteroid
RatioResonance
Ratio
DeviationProportional
Deviation
Mercury0.3910.4−0.009−0.0225
Venus0.6160.6+0.016+0.027
Earth0.5260.5+0.026+0.052
Mars0.4130.4+0.013+0.0325
Ceres0.3830.4−0.017−0.0425
Jupiter0.4070.4+0.007+0.0175
Saturn0.3510.333+0.018+0.054
Uranus0.5090.5+0.009+0.018

The asteroid Ceres is used to represent the asteroid belt in the table. However the ratio of the period for Mars to that of Jupiter is 0.158 and the square root of this ratio is 0.398.

Here is the distribution of the proportional deviations.

The average of the proportional deviations from the resonance ratios is 0.017 and the average of the absolute values of the proportional deviations is 0.03325.

Thus planets formed from material having an orbit period of approximately either 0.5 or 0.4 of that of the next outer planet. Whether it was near 0.5, 0.4 or another resonance ratio was a matter of chance. In any case the end result for a sequence of planets is the same as if the planets formed near the midpoint of the 0.4 and 0.5 resonance bands. This means that, on average, the orbit period of a planet would be roughly 0.45 of that of the next outer planet and that the outer planet would have an orbit period of approximately 1/0.45=2.22 times that of the inner planet; i.e., the next closer planet to the Sun. Proto-planets might possibly have formed near both the 0.5 and 0.4 resonance bands and later coalesced into a a single planet or a planet and its satellite.

The prior observations are in terms of orbit periods. Since by Kepler's Law the corresponding orbit radii are (0.5)2/3=0.63 and (0.4)2/3=0.54. The planetary material is spread out over space. Thus, rather than averaging 0.4 and 0.5 for the period times it is more appropriate to average the 0.54 and 0.63 of distances. This gives 0.585, which corresponds to an average orbit period of (0.585)3/2=0.44744. The reciprocal of this value is 2.23494.

Thus according to the above analysis the ratios of the orbit periods of planets should average 2.23494. In other words the relationship of orbit period to order number should be

Tn = a*bn
with b=2.23494

In logarithmic form the above relationship is

log10(Tn) = log10(a) + n*log10(b)

In the above graph it is seen that Pluto represents a deviation from the pattern for the planets but it is not an extreme deviation.

The regression of the logarithms of orbit periods relative to that of Earth on order number for the eight planets and Pluto gives

log10(Tn) = −0.9920 + 0.3493*n
with a coefficient of determination of
R²=0.9942

This means the relationship is

Tn = 0.10186*(2.235116)n

The empirical estimate of b=2.235116 is stunningly close to the value of b=2.23494 based upon planetary material being excluded from the 0.4 and 0.5 resonance bands.

The above results concerning resonance band ratios suggests that it may be of interest to look at the ratio of the period of the second next planet to that of the planet, as in the case of Mars and Jupiter.

Planet or
Planetoid
RatioReciprocal
Ratio
Mercury---4.1511
Venus---3.0844
Earth0.24094.6
Mars0.32426.3158
Ceres0.21746.4130
Jupiter0.15837.0000
Saturn0.15595.5932
Uranus0.1429---
Neptune0.1788---
Pluto0.3387---

In the table Ceres was used as a representative of the asteroid belt. Four of the eight ratios are in the vicinity of 0.16=(0.4)². One is close to 0.25=(0.5)² and one close of 0.2=(0.4)(0.5). Only the ratios for Mars and Pluto are not a combination of 0.4 and 0.5. Their ratios of course corresponds to a 1/3 resonance.

The Exclusion of Pluto from the Analysis

The attempt to find a relationship that includes Pluto distorts the results unnecessarily. Making the relationship fit Pluto results in Uranus and Neptune not fitting very well. Leaving Pluto out of the regression results in the equation

Tn = 0.094854*(2.280933)n

This is the relationship which will be used for further analysis. Here is the comparison of the actual orbit periods and the values computed from the equation.

PlanetOrbit
Period
(years)
Computed
Period
(years)
Mercury0.24090.2164
Venus0.6160.4935
Earth1.00001.1256
Mars1.92.5675
Jupiter12.013.3576
Saturn29.530.4689
Uranus8469.4952
Neptune165158.514

For order number 5 the equation gives a period of 5.856 years, whereas the period for Ceres is 4.6 years. For order number 10 the equation gives a period of 361.56 years, whereas the period for Pluto is 248 years.


The Relationship of the Planets'
Orbit Radii to Their Order Number:
The Titius-Bode Type Relationship

When orbit radii and orbit periods are expressed relative the values for Earth Kepler's Law takes the form

R = T2/3

Thus the corresponding relationship of planet orbit radius and order number is

Rn = 0.207987*(1.732773)n

Using the value of b=2.23494 would give a value of 1.7094 instead of 1.732773. This is notably close. Here is the comparison of the actual orbit radii and the values computed from the equation, both expressed in Astronomical Units (A.U.), distance relative to Earth's orbit radius.

PlanetOrbit
Distance
(A.U.)
Computed
Distance
(A.U.)
Mercury0.3870.3604
Venus0.7230.6245
Earth1.00001.0821
Mars1.5241.8750
Jupiter5.2035.6297
Saturn9.5399.7550
Uranus19.1816.9032
Neptune30.0629.2895

For order number 10 the equation gives a distance of 50.75 whereas Pluto's distance is 39.5. For order number 5 the equation gives a distance of 3.249, whereas the distance of Ceres is 2.7.

Other Satellite Systems

Because of the nature of the derivation this type of relation should apply for any satellite system such as the moons of Jupiter, Saturn and Uranus. The corresponding regression equations for these systems are

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

Although the regression coefficients for order number are significantly different from the value of 0.35 found for the planets the values of the same order of magnitude and reasonably close.

The Planetary Systems of Other Stars

As asserted before, the derivation would apply to any satellite system and that would include the planetary systems of other stars. 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 253/2, 403/2 and 703/2; 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.

Conclusion

There is a physical justification for a Titius-Bode relationship between the order number of a planet and its orbit radius, but that relationship is derived from the relationship of order number and orbit period. The planets formed near the resonance bands as a result of planetary material being nudged away from the resonance band. When planetary material was nudged away from a resonance band its velocity differed from that of the material in its new location. The subsequent collisions aggregated the material into a proto-planet that continued to sweep up material in its band and nearby bands due to the burgeoning gravitational field of the proto-planet. Thus the planets formed near the resonance bands, either above or below just due to chance. Possibly two or more proto-planets may have formed both above and below the resonance band and then later coalesced into one planet at their center of gravity which would have been close to the resonance band.

The resonance bands near which planets were formed were usually the 0.4 and 0.5 resonance bands and the particular one was a matter of chance. On average the planets were formed such that the ratio of the orbit period of the next innermost planet to the orbit period of the planet is about 0.45.

When the orbit periods and orbit radii are expressed relative to those of Earth the relationships are:

Tn = 0.094854*(2.280933)n
and
Rn = 0.207987*(1.732773)n

The evidence from the satellite systems of Jupiter, Saturn and Uranus and also the star HR 8799 is that not only do the different systems have Titius-Bode type relationships between the orbit periods and orbit radii and the order numbers of the satellites but that it is approximately the same relationship.


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