Table Of ContentA Modern Algorithm to derive the Titius-Bode Rule
by Patrick Bruskiewich
Abstract
The Titius – Bode Rule is an algorithm that models the distances in astronomical
units of the planets in the Solar System from the Sun. A simple three step
algorithm is presented is this short paper to derive a Titius-Bode type rule.
Background
The Titius – Bode Rule fits the data for planetary distance in Astronomical Units
(A.U.) was first published by Johann Daniel Titius (1729 – 1796) and was later
rediscovered independently by Bode. Titius published the rule in the German
edition of Bonnet’s 1781 book Contemplation de la Nature.
The Titius – Bode Rule fits the data for Venus, Earth, Mars, the asteroid belt,
Jupiter, Saturn and Uranus, but not for Neptune nor for Pluto. If you ponder
Neptune and Pluto you might wonder whether Neptune has been perturbed out of
its original orbit (something did happen to that planet in the distant past) and
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whether Pluto may have originally been one of the moons of Neptune and thrown
out by whatever perturbed Neptune out of its orbit. It is also interesting to see that
the planet Uranus spins on its side, indicating something perturbed it in the past.
The Titius – Bode Rule reads the following: The distance D of a planet from the
n
sun in A.U. is given by
where a good fit is given by A = 0.4 B = 0.3 and C = 2 with n which represents the
orbital order of the planets starting with Venus onwards n takes the integer values
0, 1, 2, 3, 4, 5 … . The Titius – Bode Rule generates the following distances to the
planets
Planet Actual Mean n Bode-Titius Law Variation
Distance Distance (A.U.)
(A.U.)
Mercury 0.387 n.a. 0.4 n.a.
Venus 0.723 0 0.4 + 0.3 = 0.7 – 0.032
Earth 1.0 1 0.4+0.3*2 = 1.0 0
Mars 1.523 2 0.4+0.3*4 = 1.6 0.051
Ceres 2.55 3 0.4+0.3*8 = 2.8 0.098
(planetoid)
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Jupiter 5.202 4 0.4+0.3*16 = 5.2 – 0.00038
Saturn 9.538 5 0.4 + 0.3*32 = 0.048
10.0
Uranus 19.181 6 0.4 + 0.3* 64 = 0.022
19.6
Neptune 30.058 7 0.4 + 0.3* 128 =
38.8
Pluto (KBO) 39.44 n.a.
Total Variation: 0.19
Table 1: Measured and Predicted Titius-Bode Rules Distances to Planets
The Titius – Bode Rule compares well to the actual distances: The Titius – Bode
Rule appears to fit Jupiter the best of all planets in the model.
A plausible explanation for the Titius – Bode Rule was proposed in the 1943 in a
physics paper by the German physicist Carl von Weizsäcker when he used
phenomenology to describe the phenomena of gas stability in a rotating nebulous
cloud. Weizsäcker’s paper is a difficult read for a non-physicist. I suspect most
readers would be content with the abridged presentation in any of the popular
science books by George Gamow, such as his One, Two, Three … Infinity or The
Creation of The Universe.
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Our solar system appears to have begun as a flat rotating nebulous cloud of dust
and gas particles. Over the process of about 100 million years the particles in this
primordial nebulous cloud began to coalesce or accrete into small lumps of matter,
which collided forming bigger and bigger planetoids. The truly large big lumps
formed into the planets we have today. As the larger lumps cleared its
surroundings of the smaller accretions, rings began to form, similar to the rings of
Saturn, and in the words of the Astrophysicist and science writer George Gamow
“the radii of successive boundary lines between the neighbouring rings form a
similar geometric progression, each of which being twice as large as the
previous one.” (One, Two, Three … Infinity, p. 309).
Boundary lines and neighboring rings bring Saturn to mind.
Three Step Derivation of the Titius-Bode Algorithm
Let us begin by assuming that the function may be a fit for the data is given by:
By inspection we find that
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Let us use the measured orbital values for Ceres, Mars and Earth (n = 2) to
estimate the value for C. From this measured data we find
Why we would use this data set seems apparent since the planetoid Ceres orbits in
the gap between Mars and Jupiter (the gap between the terrestrial and gas planets).
To find B we see from inspection that
Again let us use the data for Ceres and Mars, from which we get
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where C = 1.98 we find that B = 0.267
By inspection to find A to be
which again for Ceres becomes
For our algorithm we arrive at the algorithm (we round to two significant digits):
Planet Actual Mean n Modern Bode- Variation
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Distance (A.U.) Algorithm (A.U.)
Mercury 0.387 n.a. 0.48 n.a.
Venus 0.723 0 0.48 + 0.27 = 0.75 0.037
Earth 1.0 1 0.48+0.27*2 = 1.02 0.02
Mars 1.523 2 0.48+0.27*4 = 1.56 0.025
Ceres 2.55 3 0.48+0.27*8 = 2.64 0.035
(planetoid)
Jupiter 5.202 4 0.48+0.27*16 = 4.8 – 0.077
Saturn 9.538 5 0.48 + 0.27*32 = – 0.044
9.12
Uranus 19.181 6 0.48 + 0.27* 64 = – 0.072
17.8
Neptune 30.058 7 0.48 + 0.27* 128 =
35.0
Pluto (KBO) 39.44 n.a.
Total Variation: – 0.076
Table 2: Measured and Modern Algorithm Distances to Planets
In our algorithm we find a smaller variation ( – 0.076) compared to the a much
larger variation for the Titius-Bode Rule (0.19). It would be interesting to try to
minimize the total variation for this algorithm for the planets in the solar systems.
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If we now decide to include Neptune into our modern algorithm we find a total
variation of 8.7 %, which is still considerably less than the 19% in the standard
Titius-Bode Rule.
In a separate set of papers the three step algorithm will be used to study several
exo-planetary systems around Sun like G stars.
Conclusion
In this short paper we developed a simple three step technique to derive a Titius-
Bode rule fit for the measured data for the planets Venus, Earth, Mars, Ceres (a
planetoid), Jupiter, Saturn, Uranus and Neptune of
This algorithm is better fit than the standard Titius-Bode Rule for solar system data.
References:
P. Bruskiewich, George Gamow: The Birth of Planets with Guided Notes, Kindle
Direct/ Amazon, 2013
G. Gamow, One, Two, Three … Infinity, Bantam Books, NY, 1965, p. 298 -315
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