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Kepler's Law and Titius*Bode (*BodeTitius) Rule 

For Jeffery
This is to compare and contrast a real physical law, Kepler's Law, with a statistical relationship, the Titius*Bode "Law."
The distances and the length of the years of the various planets are as follows:
Relative to that of Earth 
Relative to Earth's Year 


Mercury  0.387  0.2409 
Venus  0.723  0.616 
Earth  1.0  1.0 
Mars  1.524  1.9 
Jupiter  5.203  12.0 
Saturn  9.539  29.5 
Uranus  19.18  84 
Neptune  30.06  165 
Pluto  39.52  248 
If we call T the length of a planet's year and multiply T times itself (T*T) the result is called "T squared" and it is represented as T^{2}. Call the radius of a planet's orbit R and multiply it times itself three times, R*R*R. The result is called "R cubed" and it is represented as R^{3}. The computations are given below:
Planet  R^{3}  T^{2}  T^{2}/R^{3} 

Mercury  0.05796  0.05803  1.00121 
Venus  0.37793  0.37946  1.00405 
Earth  1.0  1.0  1.00000 
Mars  3.5396  3.61  1.01989 
Jupiter  140.85  144.0  1.02236 
Saturn  867.98  870.25  1.00262 
Uranus  7055.8  7056  1.00003 
Neptune  27162.3  27154.7  0.99972 
Pluto  61723.5  61504  0.99644 
As you notice the ratios of T^{2} to R^{3} are all close to 1.0. If all of the numbers were precisely correct the ratios would all have been exactly 1.0 This means that for each planet the squared of the length of its year is equal to the cube of the radius of its orbit. This is called Kepler's Law. From Kepler's Law if you know how far a planet is from the sun you can tell how long it takes for that planet to go around the sun.
Kepler's Law is exact and can be derived from Newton's laws of the motion of bodies. There is another "law" concerning the planets but it is only approximate and cannot be derived from other laws. It is often called *Bode's Law because it was popularized by *Bode, but it was actually discovered by Titius. It is a rule or formula for finding the orbit radiuses of the planets. The *BodeTitius Rule goes like this:
Take the series of numbers,
0  1  2  4  8  16  32  64  128  256 
and multiply each by 0.3 to get:
0  0.3  0.6  1.2  2.4  4.8  9.6  19.2  38.4  76.8 
To these numbers add 0.4 to get:
0.4  0.7  1.0  1.6  2.8  5.2  10.0  19.6  39  77.2 
Now compare these numbers with the radiuses of the orbits of the planets (relative to the Earth's orbit radius):
0.4  0.7  1.0  1.6  2.8  5.2  10.0  19.6  39 
0.387  0.723  1.0  1.524  5.203  9.539  19.18  30.06/39.52  
Mercury  Venus  Earth  Mars  Asteroid Belt 
Jupiter  Saturn  Uranus  Neptune/Pluto 
The *BodeTitius Law gives a pretty fair approximation of the radiuses of the orbits of the planets. It appears to fail between Mars and Jupiter but that is where there are many asteroids and where they would have combined to form a planet if Jupiter was not so close by. The Law fails to give the right figure for Neptune but Pluto fits the value given by the law quite well. Astronomers think that Pluto is not a true planet but an escaped moon of one of the planets. As remarkable as the *BodeTitius Law is for predicting the orbit radiuses of the planets there is no explanation of the law in terms of other laws of physics.
In April of 2004 a planetlike object of approximately 1000 km in diameter was identified. Its distance is now approximately 86 A.U., not far from the figure of 77 A.U. predicted by the *BodeTitius "Law." Its orbit is quite eccentric so its distance from the sun may range from 75 A.U. to 100 A.U. over the course of its 10,000 Earth year revolution about the sun. Tentative it is being called Sedna, after an Arctic goddess of the sea. It seems like a good choice, in that all planets except Venus and Earth were named after male Roman gods.
If an explanation for the *BodeTitius "Law" were to exist it might be in terms of the size of a radius zone for a planet within which no other planet could form because of the disruptive effects of the gravitational attraction of that planet. There is such a zone for a planet within which a satellite cannot form because of the stresses produced by the gravitational attraction of the planet. This zone is defined by the Roche Limit.
Incidentally, formulas analogous to the *BodeTitius Law applies to the radiuses of the orbits of the satellite systems of the planets Jupiter, Saturn, Uranus and Neptune. For more on this analysis see Titius*Bode Rules for Planetary Satellite Systems, Resonance Phenomena in the Solar System" and Derivation of the Titius*Bode "Law".
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