H.H. Voigt in his book breakdown of astronomy states that Copernicus showed that the earth and the planets circle the Sun, Kepler showed how the planets circle the Sun and Newton showed why the planets circle the Sun. I am just a wannabe Kepler in search of Keplers 4th law of planetary motion.As to why my equations should mean anything I have`nt a clue! I will at some point need to find a far smarter poor mans Newton to work out why! Untill then I am quite content to play around with my pocket calculater searching for equations that I, at least find beutifull. To begin with it was like panning for gold, just trying to find rough equations that I would soon abandon in favour of better ones applying the simple logic of accuracy on the one hand against elegance on the other. In constructing a model B, it seems this is more a process of distillation, such that I won`t claim that my answers are right, more that I have come up with some sort of raw material that needs further destillation to find any kind of truth.
N.A.S.A. has calculated planetary distances for two different time periods.I will be using the 3000 BC to 3000 AD values unless otherwise stated.
Here below are the Mars and Venus values that I will be using in my model B.
The picture below is a simple illustration of one astronomical unit, from which all the N.A.S.A. numbers come from and the pentagram, the most obvious depiction of Phi, the golden section.
It had always annoyed me that my Mars and Venus Phi relationship was based on the music interval 1.875 and not 1.5 and 1.25, because it is these two intervals along with 2 the octave that all other music intervals derive.Splitting my equation into three levels though, clearly shows that I am indeed using 1.5 and 1.25.The badly drawn picture below illustates what I mean and I have squared both factors to show there are 3 levels.
Although all of the above equations could be seen as pure coincidence, I find it remarkable that in correcting the music interval power equation to equal Phi squarerd then the resulting changes improve the 2nd equation!
So without any changes the discrepancy was 1.000088909% and after the changes 1.00001185% this is amost a decimal place better.It might be fifty fifty as to it being higher or lower but being more accurate, on a discrepancy so low is something else. The 76th root of the discrepancy would correct the equation.
Below is a picture showing a connection between Phi and musiic intervals, 3.2 devided by 2 is 1.6
Here is another coincidence, which I can't find a use for but it is worth mentioning
Well 1.2 the minor 3rd, is by no means an obscure interval and I could use these values in model B or another model. The problem here is that I would have to use the whole thing as a power which would be even more obscure and complicated
This is very much a work in progress and three days ago I discoverd another coincidence which has given me food for thought. I had never used Phi itself as a power before but the square root of Phi, to the power Phi power 4, is an exelent value for Jupiter 5.202490525 with a discrepancy of 1.000001987%
Well after much thought I couldn`t find any connection with my Mars and Venus values but I have found something much better!
Here is perhaps a better example of the harmonius numbers in this relationship, this is compatable with model B and it is clear to see how easy it is to factor down.
This is the best equation I have found since I started on this project, whereby Jupiter requires only a 1.000001032% correction to fit with the original Mars and Venus values.With my manipulated Mars and Venus values for model B then Jupiter is 5.202479791 and requires only a 1.000000077% correction! a 13th millionth part, too good to be true! Well this has changed everything and so I have decided to delete all the stuff I had originally planned for model B. Here are equations that define Neptune, they can be factored down but I thought the 1.875 interval looks good.
Using both the original values for Mars and Venus then Nepune has a value of 30.06970832 with a decrepancy of 1.000006013%.Using my manipulated values Neptune would be 30.06968236 which is 1.000005149% away from the original value.Having defined Neptune, the following equation will define Uranus
The Uranus value comes out as 19.18806794 which is 1.00000461% away from the N.A.S.A. value.
Turning to Saturn I am somewhat torn between several alternatives, but for model B I will go for the most accurate option.
Here it is plain to see that any Phi power is achievable, whereby Saturn has a value of 9.541497135 which is 1.000000178% again too good to be true at a 5.6th million part!
Problematic is the relationship between Saturn and the Mars Venus equations, such that Saturn power 13, times Mars power 22, devided by Venus power 22, equals Phi power 95. This has not nearly the same elegance or symetry as the Jupiter Mars and Venus equations.
Finally to Mercury which I found difficult in constructing Model A and although 2 or 3 ideas I had for model B, were very accurate I just could not find an elegant relationship with Mars and Venus. I have settled on an old idea I had many years ago which didn`t suit my purpose at the time.
Mercury comes out with a value of .387097867 which is 1.000001453% away from the Nasa value.
So here are the values for my model B where by I have gone for accuracy over elegance.
It was only about ten days ago that I found some relationships between Mercury Venus and Mars that may be something of a turning point! I am going to refer to these equations as my Rocky planet equations fort want of a better title.
I had searched for a Phi value in my normal fashion and using my 1.875 power and root for Mars and Venus I had given Mercury the power of 1.875.Then I arrived at Venus power 14 devided by Mars power 16 devided by Mercury power 9 equals Phi squared, which I thought was quite good. In trying for just Phi I saw the 7 and the 8 but then Mercury would be power 4.5 unless I used the square root of the Mercury power 1.875 and then I had the 7 8 and 9 sequence! Well within the hour I had the 11 10 and 9 sequence wih Phi power 5 and the 19 18 and 17 sequence for Phi 4. I was disappointed with the 17 18 and 15 sequence for the number one! but the 18 for Mercury can be turned into a 16 by 1.125 the whole tone! Mercury has then the power 1.0546875 a little bit obscure but 135 devided by 128 is still mainstream. After further consideration it is far more elegant to turn the Venus power 18 into a 16 whereby the square root of Venus will do the job! So what am I to understand with these chance number sequences? in some ways this is the best pattern I ever found! Could one say that using the the two inner planets I have discoverd the first of the outer planets? and perhaps using Mercury and Venus I could find the next planet Jupiter in a similar manner? Or should I understand that I have found Mercury with the use of Mars and Venus the two planets between which Kepler placed the icosahedron and the dodacahedron? Well this is what I am working on just now! It seems almost unfair that in my model B it is the female planet Venus that carries most of the discrepancy but even if I start with the real value of Venus, then Mars comes out at 1.523728738 or 1.000010703% away from the Nasa value and Mercury with .387097415 which is 1.000002622%. So even choosing the least favourable position to start with, Venus allows Mars a discrepancy of a 93000th part and Mercury with a 381000th part error.
As an after thought, there is a special quality to a Phi sequence such as those above with Mercury but not in the world of devision and multiplication but in the exotic world of addition and subtration. Phi power 1 + Phi power 2 equals Phi power 3 and Phi power 2 plus Phi power 3 equals Phi power 4 this is of course true for all Phi powers. Here is a picture to illustrate what I mean.
The following equation fits with model B and I would like to think that Johannes Kepler would be pleased, that the power 1.5 is drawn from his 3rd law of Planentary motion! I am going to refer to this equation as my Gas giant or Kepler equation.
The power 1.875 could replace 1.5 because 64 times 1.25 is indeed 80. Or the 1.6th root would be Phi power 40 or the 1.333333333 root would be Phi power 32.Things work so well with powers of 2 3 and 5! Using just 1.875 for Neptune would reduce the power of 15 to 12
It is of course inevitable that in constructing equations that I find elegant, I will also end up with equations that I do not find elegant and it is only fair to point out some of them! Firstly though, an equation that does not fit with model B. The reason I mention this equation is becauce it comes tantalizingly close to my Venus value in Model B and is at the same time an equation that, given a pocket calculater, a ten year old child could find.
the Venus value is then .7233045, which is 1.000022839% away from the N.A.S.A value and 1.000004349% away from my Model B value. The problem is I can`t find a good relationship with Mars and Venus and maintain my gas giant Kepler equation thus I will need a Model C to square some circles. Another question is, of course, why am I playing around with music intervals when I have found a Phi value without them? The music intervals are only there to spot patterns that otherwise could or would remain undiscoverd. One may also ask why am I looking for Phi powers? Because of their natural beauty! Keper described the golden section as a precious jewel and I wonder if he meant a diamond, which of course can be used as a cutting tool. It could be the case that, in my speculations, Phi is not so mch the answer but more the cutting tool to find the answer! My model B may be perhaps a house and the Phi powers more the scaffolding. Or perhaps another analagy may be the search for a geometric shape that is only possible with the structure of graph paper. Here is the relationship between Jupiter Mars and Mercury expressed in three equations.
All three equations are compatable with Model B.The first with music intervals and Phi powers, the second with Phi powers but without music intervals and the third without either. The problem is where is the elegance?
The above equation is the best Jupiter Mercury relationship I have found up to now, compatible with Model B, maybe there is something more elegant!
A photo of myself at age about 30 from about 1984