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An Investigation of a Titius-Bode Type
Relation for the Planets of the Star HD10180

Ever since 1768 when Johann Bode published the remarkable empirical relationship for the 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 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.

 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 and Bode4.) 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 or 3/5 of the planet's orbit period.

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 forming planet. The planets acquired not only mass but angular momentum in this process. See Planetary Sweep.

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.

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.

Planet # Period Ratio to
Next Planet
1 1029.340.708233853
2 1453.39 0.481242219
3 3020.08 0.695919073
4 4339.70 0.321222269
5 13509.96 0.219611905
6 61517.43 0.130041012
7 473061.76 

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 0.321222269 is 0.566756, which could represent resonance near 3/5. The cube root of the ratio of 0.219611905 for planets #5 and #6 is 0.603326, again notably close to a resonance of 3/5. The fourth root of the ratio of 0.130041 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.

Order
Number
of Planet
PeriodLogarithm10 of
Orbit Period
1 1029.343.01255885
2 1453.39 3.162382168
3 3020.08 3.480018447
4 4339.70 3.637459708
6 13509.96 4.130654063
9 61517.43 4.788998184
13 473061.76 5.674917843

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

The linearity of the relationship is also quite remarkable. Since the revised order numbers were chosen values there is naturally a suspicion that the linearity comes from the mere choice of those order numbers. That is not the case. If the line were drawn first and the order numbers read from that line they would not generally be integral values. The linearity comes from the fact that that integral roots of the ratios of planet periods are 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(Tn) = 2.76669 + 0.22421n

The coefficient of determination for this equation is 0.99891 and the t-ratio for the slope is 67.8.

The equation for planet period in terms of its order number is then

Tn = 584.3737×(1.67577)n

The slope of the relationship is smaller than the ones found for other satellite systems but of the same order of magnitude.

SystemRegression
Coefficient
Star HD10180
planets
0.2242
Uranus
satellites
0.2510
Saturn
satellites I
0.2883
Saturn
satellites II
0.3076
Jupiter
satellites
0.3026
Star HR8799
planets
0.3354
Sun
planets
0.3493

Conclusion

The periods of the planets of the star HD10180 have a Titius-Bode type relationship. The parameters differ from those of the Sun because the resonance relationship for the HD10180 planets correspond to a 3/5 resonance whereas those of the Sun correspond to a mixture of 1/2 and 2/5 resonances.


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