RECIPROCAL SYSTEM DATABASE Status Report: An Aperiodic Blog

March 22, 2016

Reciprocal System Version of Titius-Bode Relation

Filed under: Science — transpower @ 4:04 pm

In the Reciprocal System, planetary systems result from the supernova explosion of the parent star.  The outward moving (in space), relatively slow speed, matter eventually reconstitutes as a Red Giant and then contracts to the Main Sequence.  The inward moving (outward in time, inward in space), intermediate speed matter constitutes the eventual planets.  In Reciprocal System terms, the low speed (and light) matter moving outward is termed “Substance A” and the intermediate speed (and heavy) matter is termed “Substance B.” There are three speed ranges in the Reciprocal System:  0-1c; 1c-2c; 2c-3c.  The intermediate speed range is 1c-2c.  2c is the dividing point between one-dimensional motion in space (less than c) and one-dimensional motion in time (more than 3c from our perspective).  Between 1c and 2c, the velocity equation flips, and the motion is in time for the individual particles, but the object as a whole remains in 3-D space (and may still have spatial motion, so that the total motion is two-dimension).  The speeds are discretized into units above 1c of 0, 1, 1.5, 2, 3, and 4.  This brings the motion to the neutral point within the range of 1c to 2c.  Beyond that the motion is (4), (3), (2), (1.5), and (1).  There are thus a total of 11 speeds between 1c and 2c and thus 11 possible planets maximum, in which the matter aggregates because the units are at the same relative speed.  For the first half, the inner planets, the semi-major axis is proportional to n^2, whereas for the second half, the outer planets, the semi-major axis is proportional to 1/(n^2).  Let a0 = the semi-major axis of the orbit of the closest planet to the star, in units of AU, and let k = an empirical coefficient–this value depends on the amount of Substance B and the violence of the supernova and so has to be determined by iteration or least squares.  For the inner planets,

an = a0 + k x (n^2)         AU                                                                              (1)

For the outer planets,

an = a0 + k x 288 / (n^2)         AU                                                                   (2)

where n = the speed number given above and the constant 288 comes from matching eqs. (1) and (2) at the neutral point between 1c and 2c.  See Ref. [1] for a more detailed derivation.

Seven stellar systems have been studied, with the following results:

Solar System:  a0 = 0.3871 AU, k = 0.276 (Ref. [1] gives a0 = 0.4 and k = 0.267, but the values given here give a slightly better fit)

Gliese 581:  a0 = 0.0282 AU, k = 0.0124

HD 10180:  a0 = 0.02 AU, k = 0.1100

HD 40307:  a0 = 0.0468 AU, k = 0.033

Kepler 11:  a0 = 0.0910, k = 0.0013

Kepler 186:  a0 = 0.0378, k = 0.02

Tau Cet:  a0 = 0.1050, k = 0.1

Please note that these values are for the initial positions of the planets after formation and do not take into account the migration of the planets over time, due to clearing of the orbits of debris and the gravitational pull of the other planets; also, perhaps the k values could be further tweaked.  Still the agreement is quite stunning.  The equations (1) and (2) can even be used to predict the existence of other planets for these systems.  For example, for Kepler 11, planet positions 3 and 4 are missing:  there should be a planet (or debris) at 0.2178 AU and one at 0.3578 AU.

Ref. [1]  D. Larson, The Universe of Motion (Portland, OR:  North Pacific Publishers, 1984), pp. 96-100.

Study the Reciprocal System and prove it for yourself!  The Reciprocal System:  Microcosmos Database is now available from and  This is the world’s first theoretical computational database for the properties of matterThe Reciprocal System:  Macrocosmos Database (for astronomical properties) is currently a work-in-progress; the above calculations are included!

last updated: 03/24/2016


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