So using two different computer algorithms, we can see by how much that Z-axis alters the effects of the other planets on Mercury's perihelion. This study uses numerous examples with differing starting dates. Various starts are 1773 AD , 1900 AD, and 1940 AD. In addition the various averages are measured with different sample sizes. These scenarios also run at a variety of durations of time quanta within the algorithms. Here we also cross-reference measurements to the Perihelion Precession with measurements to Aphelion Precession. This is done to identify statistical anomalies in any individual data sample, to avoid rounding errors, as well as to potentially detect unknown variations within any single data sample.

The results of Godoi: 5.45 as/Ey and that of utexas.edu being 5.5 as/Ey - are both in close agreement with my 2d algorithm OGS13 which yields 5.4682 as/Ey.

That is an advance to aphelion of about 1.14 degrees after 750 years. This can easily be seen visibly below. The orbits and angular advance to the major axis depicted graphically have been generated precisely and directly from the evolutionary algorithms.

After 1969 orbits the optimal difference of 1968 orbital comparisons is yielded for the precession of the major axis of Mercury.

In 3D Mercury's orbit advances 13% less than in the 2D algorithm.

The 2D algorithm advance is 5.468 as/Ey, whereas in 3D it is now only 4.839 as/Ey.

Thus, the Z-axis alters the precession by -0.63 as/Ey or -63 arc-seconds per century.

But the Relativists have claimed that General Relativity alters the precession by +43 arc-seconds per century using a 2D process. (See Introduction and the section: Perihelion Precession)

It should be quite easy to see that the historical theories and method have NOT calculated for the tilt to the orbit on the Z-axis. Mercury's orbit is 7 degrees away from the ecliptic-plane of the Earth, and Venus is also tilted. This is an intrinsically significant issue.

Because measurements to aphelion are larger and thus easier to see on the screen, I used aphelion in the visible diagrams above. The number of orbits in a sample average is vitally important too, and I'll discuss that at length a bit later. Let us now look at the wider variety of results from various Scenarios in the OGS15 algorithm for Mercury's Perihelion Precession:

It is vital to see, that none of this is my observation. Thus far this is just a discussion on what is the precise computation as to what Newtonian theory predicts the advance to the major axis should be if Newton's theory is the entire correct ontology.

So not only are the historical estimates of what Newton's theory predicts, significantly inaccurate, but when the 'observations' fall short of those crude 2D approximations, they always manage to do so conveniently to suit the predictions of a certain theory.

But because I am using a logical positivist methodology, I want to check all that data against every possible contention. So what do Horizon Ephemeris say about it?

This first data extract (above) shows us that their average time between perihelions for Mercury from 1774 to 2011 was 87.969349533 days.

You should be able to see in the data samples above that on 16 March 1774 at 12:09 Mercury's distance to the sun was lowest. RG is the distance measured in astronomical units. The number 3.075~ is followed by E-01, which is the same as 10^-1, so that is 0.3075 times the distance from Earth to the Sun.

I extract 3 data readings so you can clearly see which reading is the perihelion, given to their closest measurement of 1 minute intervals. I calculate more accurately than 1 minute, extracting the seconds and fractions of seconds, as a proportion. How close is the middle data sample to the two samples either side of it proportionally?

The algorithm I built so that we can easily sort their data is freely available in the section: Sorting Horizons. This I built so that we do not have to go through thousands of records manually to find the least or most distance to the sun in any batch of data.

The next set of data is for the same time-frame, 1774 to 2011, but instead of giving Mercury's duration between perihelions, we are going to measure the duration of Mercury's orbit. To do this we compare the time differences as Mercury's orbit crosses the Y-axis.

It is vital to see that this Y-axis is Mercury's position comparative to the center of the Sun. So the Sun's movement is inherent within these results, and is thus already accounted for.

So we highlight the Y variable when it changes from negative to positive. Then we calculate the average length of the orbit of Mercury for those 984 orbits, or 237 years. (In the  section Jupiter+Saturn I explain why 984 is an optimal number of orbits.)

After this, the last step is then to compare the orbital duration, with the duration between perihelions that we extracted a little earlier.

So the data above tells us that the average duration of Mercury's orbit from 1774 to 2011, was 87.969252248 days.

The seconds and fractions of seconds are also proportioned here. How close are either of the two data readings to zero on the Y-axis?

Those who are paying close attention should realize that it is no coincidence that both those dates that Mercury crosses the Y-axis are January 6, and that both perihelions of Mercury we extracted earlier also take place on the same date of each year: March 16. If you do not appreciate that simply by thinking about it, then you perhaps need to start reading from the Introduction again. Or you may need to read the section: Jupiter+Saturn

So now we simply compare the duration of the orbit, with the average time between perihelions. Then we convert the difference between those two durations into the conventional unit of measurement, which is arc-seconds per Earth-year (as/Ey).

So the calculation above tells us that Horizon Ephemeris gave an average Perihelion Precession of Mercury's orbit from 1774 to 2011, of 5.95 as/Ey. Was this truly observed, or was it calculated? Most certainly it is their best compromise between both of those.

The various 2D Newtonian theoretical processes gave 5.5 as/Ey, which looks moderately good for the 'prediction' of Relativity accounting for a +0.43 as/Ey difference between the Newtonian theory and the observation.

But my 3D model shows a Newtonian calculation of only 4.8375 as/Ey over this period in Scenario [21]. Scenario [31] results in 4.8642 as/Ey. Over twice that period the average is 4.839. So the discrepancy between the Newtonian calculation and the observation is more than double now, at 1.11 as/Ey, and not 0.43 as/Ey as the Relativists predict.

So even if we hypothetically accept Relativity and blindly accept their opaque 'prediction', then we have to conclude that there must be some other factor causing more than the Relativistic effect. But it goes far deeper than this for several reasons. Our choice of which orbits to measure plays a massive role. The other theorists all seem to have ignored this.

We need to carefully note which years are examined, because individual orbits are widely different from the average. Individual alterations to the aphelion being lower than -20 as/Ey for some orbits, but over +50 as/Ey in other orbits. (See graphs a bit further on). But for perihelion the deviations are doubled from -40 as/Ey to +87 or more. So this is another methodological reason for using a wide variety of samples. The aphelion deviates less, and thus gives a better average then the perihelion. It seems obvious that the aphelion should be easier to simply observe anyways.

The errors of other studies are thus not only due to them avoiding even considering the Z-axis. But also because taking a 'nice' round 100 Earth-year sample to Mercury's Perihelion Precession will give different results depending on when we start the 100 year sample.

In the section on Jupiter+Saturn it is shown in detail that when synchronizing the orbits of the planets, 237 Earth-years will give an optimal average, but even a 59-year average is better than a 100-year average due to the 59-year amount being more in-synch with the orbits of Jupiter and Saturn. (5 Jupiter orbits and 2 Saturn orbits). At no point do the other processes acknowledge the importance of selecting the orbits and how that effects the average Perihelion Precession; but also the duration of the orbit. Just a quick glance at the results shows orbital durations for Mercury differing by over a minute of time; which is itself a difference in excess of 10 arc-seconds of angle for the orbit itself - over 40 arc-seconds per Earth-year.

So I wish to be quite clear as to the discrepancies in my own model, and how they differ with others, because I do not want to make the error of finding only what I am expecting to find.

The worst are those who simply ignore data which does not fit theoretical intuitions or populist institutions. I want to be totally honest about my own uncertainties. This is the surest method to dredging up the fact of any matter: Openly and publicly attempt to prove yourself wrong at every opportunity.

For instance using the times from Ephemeris is not the only way to extract Mercury's Perihelion Precession from their data. When I use the X and Y positions, and calculate theta in purely angular terms I get the result of 5.67 as/Ey over those same 237 years, whereas in temporal terms we got the higher result of 5.95 as/Ey. Excluding the Z-axis in angular measurement is a troublesome tiny detail here which is certainly going to raise eyebrows seeing as though this echoes my own critique of the evolution of gravity in 3D terms.

Of course the context of the Z-axis as an angular measurement is entirely different to the role of the Z-axis in evolutionary gravity. This is why I found it better use the dates to extract Mercury's Perihelion Precession from Horizon Ephemeris, as that is simply easier to demonstrate than an angular calculation of the theta angle in 3 dimensions of space.

So to be systematic, I then use the method of the temporal differences between the dates in the OGS15 algorithm Scenario [21]. Now the Perihelion Precession measures an average of 5.07 as/Ey between the years of 1773 and 2011, rather than 4.84 in purely angular terms of X and Y.

So its vital to see that the temporal calculation of Mercury's Perihelion Precession gives a value about 5% more, than the purely angular measurement of changes to theta. This 5% difference occurs in both Horizon Ephemeris and in OGS15:

5.07 / 4.84 = 1.0475 ... (OGS15 = 4.75%)

5.95 / 5.67 = 1.0494 ... (Horizon Ephemeris = 4.94%)

It is not only a matter of including the Z-axis in the evolution of the orbit as it gets effected by gravity. But when we measure the angular difference in Perihelion Precession, we also have to take account of the fact that the orbit itself has a tiny tilt on the Z-axis too. I provide both details to be clear.

So its not just that this analysis is the only process that includes the Z-axis in gravity terms, but also I account for the Z-axis in measurement terms of the angle between perihelions. It is obvious that Horizon Ephemeris have themselves only measured the angle using X and Y variables. This is because their value of 5.67 as/Ey is in-keeping with all such historical accounts, whereas the real angle in their data that includes the Z-axis is the even larger amount of 5.95 as/Ey! So it is clear that they have simply added in the Z-axis onto the orbit after determining its parameters!

As it is, I have honestly baulked at measuring the angle with the inclusion of the Z-axis in purely geometrical terms. I am quite satisfied using the temporal values which must include the Z-axis inherently. That their measurement of Perihelion Precession excludes the Z-axis means that we must be careful to compare proverbial apples and oranges.

I see zero evidence that anyone else measures the angle in purely 3D terms. I offer all permutations so that the reader can see the detail as transparently as possible. Methodologically this is the only route to take where there is a difference between the standard process and what honestly seems to be a better standard to set.

One can get side-tracked in such details indefinitely. Oh how subtle it all gets.

Another self-critique is that the latest version of my models have been made more accurate by very small amounts, so there will be tiny discrepancies between the models themselves and the data displayed on these web-pages in places. This is always an ongoing rigorous part of my own method, whereby I am constantly doubting and re-checking and refining the models to ever more ridiculous degrees of accuracy. Altering the orbit of one planet then causes small changes to the others, so its a never-ending process.

But at no point have those tiny checks altered the essence, logic, or theory of what is said here. If you want the most precise data I have on offer, then it is suggested that you run the most current versions of my models and extract that data yourself.

But it is easiest to compare these results in a table:

This graph shows individual fluctuations in Mercury's aphelion over a 60-year cycle. Scenario [11] of the 3d-n-body algorithm OGS15 (orbit-gravity-sim-15.exe) uses 1.5 virtual seconds per calculation. Over 5 days my old XP PC executed more than a million million logical steps to generate that graph.

The smaller samples results:
Scenario [11] of 59.2 years (246 orbits) = 4.419 as/Ey from 1940 to 1999.
Scenario [62] of 59.2 years (246 orbits) = 4.919 as/Ey from 1900 to 1959.
Scenario [62] of 58.7 years (244 orbits) = 4.676 as/Ey from 1900 to early 1959.

It is clear that the sample size is responsible for the discrepancy between these last two readings, with 2 less orbits in total.

Also supplied in the algorithm download are Scenarios [41]-[44] which are too slow and accurate to run effectively on my little computer. Those with better equipment may get better results. Each one has 10 more quanta to an orbit than the previous, with Scenario [44] using steps of just 0.00015 seconds. But I'm guessing that most likely won't be necessary. They simply serve to demonstrate that the size of the time-quanta is not the cause of the discrepancies.

Most importantly:
It should be very easy to see how individual fluctuations in the major axis for a single orbit completely out-weigh those for the claims from Relativity for their sample size of one century.

Also note the cycles of the other planets: Earth and Venus (color-coded). Most noticeably peaks and troughs appear every half-orbit of Jupiter being slightly less than every 6 Earth-years.

The graph above is Scenario [21] of the 2d-n-body algorithm OGS13 (orbit-gravity-sim-13.exe) using 15 seconds per calculation. By combining it with the graph before it, the two were super-imposed upon one another like this:

The color-coding should be clear to those who are not color-blind. The effect of the 7 degree tilt to Mercury's orbit is marked. But you might notice that at no point does the one graph recede while the other advances. This is partly due to them having the precise same duration of 87.969 days for Mercury's orbit. Both also have peak and trough cycles in synch with half of Jupiter's orbit. The 2d algorithm has greater peaks, a higher average, but also it recedes by larger extremities, but not at every orbit.

But on these graphs it is shown why taking a 100-year sample is just plain woefully bad methodology when considering the discrepancy of 43 arc-seconds per century (0.43 as/Ey) that the Relativists predicted. But it is possible to get a good average with a 100 year sample too.

This graph shows individual fluctuations in Mercury's aphelion over a 100-year cycle. Some orbits advance while others recede. Scenario [11] of the 3d-n-body algorithm OGS15.2 (orbit-gravity-sim-15.exe) uses 15 seconds per calculation. That took a full day to generate using the older more powerful XP computer. It should be very easy to see from the graphs how individual fluctuations to the major axis are in the region of the claims made by Relativity.

It is clear that deviations to individual orbits for Mercury's Perihelion Precession are often double that attributed to Relativity. And also double that of aphelion.

This is certain proof that the Relativists have utilized tragic methodology. Their entire thesis falls neatly within the error-margins of their own methodology here exposed. And they of course did not even know to use a 237 year sample, so they did not know that error-margin even existed. I have not even seen anyone else show the huge variance to individual fluctuations to the major axis, other than that inherent within the Horizon Ephemeris data. I am unaware that anyone else even knows that data is there after considerable internet research now into its 11th consecutive year as a full-time study.

Also depicted in the graph are the numerical counts of the orbits of the other planets: Earth and Venus (color-coded). Most notably peaks and troughs appear every half-orbit of Jupiter being in cycles of slightly less than every 6 Earth-years. This was a bit unexpected. Why should the cycles be for only half-orbits of Jupiter? So because I am uncertain of this, I examine further.

Lets just look at the first group of large advances to aphelion, and explain this in detail:

In the graphic above, an orbit of Mercury with a large advance to the aphelion evolves. The brighter parts of the orbit-lines equate to about 3 weeks. The gray line from the Sun to Mercury is the distance of aphelion.

At this time Mercury is pulled forward by Venus and Earth as they are ahead of Mercury. Jupiter is opposite the Sun from Mercury, and its gravity prevents Mercury from reaching its outermost extremity of aphelion. So for this orbit, Mercury has an aphelion distance a bit less than normal because Jupiter pulls it sunward, which also speeds up its velocity. These effects combine to generate an advance to aphelion several times higher than the average.

Let us now take a group of large advances to the aphelion. In the graph two previously - that of the aphelion starting 1940, note orbits 6-14, which is the first cluster of peaks. From 1941-May-30 to 1943-May-04, we have 8 orbits of Mercury that have a much larger than average advance to aphelion. OGS15 in 3D gives an average advance of 14.6 as/Ey during this time frame, whereas the Horizon Ephemeris gives an average advance of 15.9 as/Ey.

So at least Horizon Ephemeris have calculated that there are large variations to the major axis. But despite the discrepancy between the models of 9%, it is still easy to see that because these rapidly advancing orbits are in groups, that by selecting various starting orbits we can account the missing 0.43 as/Ey blindly attributed to Relativity by 20 times over. Those selected 8 orbits are each about a full 10 as/Ey greater than the average.

The Relativists may want to claim that they have taken into account that individual orbits fluctuate by a similar amount to that attributed to Relativity. But if they had honestly done this properly, then they would not have used a 100 year sample, instead they would have used the 237 or 59 year sample to minimize the effect this has on the overall average.

However the estimates for the duration of Mercury's orbit differ by amounts in excess of 1/1000th of a day. This is a fraction representing 60 as/Ey. We need at least 2 more decimal places to be accurate to within 0.1 as/Ey. So Mercury's orbit varies depending where one looks, and this is contingent on which data is selectively ignored.

For instance, the page https://nssdc.gsfc.nasa.gov/planetary/factsheet/mercuryfact.html is Nasa's "Mercury fact-sheet" which gives the duration of Mercury's orbit of 87.969 days. Whilst https://ssd.jpl.nasa.gov/?planet_phys_par from the Horizon Ephemeris gives the amount of 0.2408467 years, which converts to 87.97079 days. This error occurs because they use a crude 365.25 days per year and neglect to use the proper sidereal value of 365.25636 days for a year of Earth. What woeful methodology that offers a value of 0.2408467 years, to the 7th decimal place, that is based on numbers rounded only to 2 decimal places?

This difference is 155 seconds per orbit or 26.4 as/Ey. This discrepancy for a single orbit is comparable to the amount attributed to Relativity for an entire century: 43 arc-seconds. Once more the error-margins entirely dwarf the thesis of the Relativists. Never mind the supposed accuracy of recent data compared to data from 100 years ago.

There is another aspect of the 100-year observation that is also missing. Mercury's Perihelion Precession has to be measured between two events of either perihelion or aphelion. And 100 years of the Earth does not exactly fit the orbit of Mercury.

A total of 415.21 orbits is not a time between perihelions. Instead the amount would be either 99.949 years between 415 full orbits or 100.190 years between 416 orbits. Here is another gross error: 5% of a year. That is a movement for the Earth of 18 days which is about 20% of an entire orbit for Mercury. Can we just assume that has been accounted for?

Avoiding rounding errors is vital, so why would they not be accurate for the sample size? After all they did not take into account the optimal average cycles of the other planets, or the Z-axis tilt. Perhaps the 100-year rounding was just sloppy writing and nothing more? But the amount that the 416th orbit alters the average will still be significant because only the 237 year sample cancels out the anomalous effects of the other planets, as we shall see in the section Jupiter-Saturn.

How easy it is for the Relativists to find miniscule reasons for tiny perturbations, and yet to neglect massive rounding errors that dwarf their entire thesis. So it should be quite clear, that the 'observations' of the Relativists are incorrect due to radical holes in their calculation process.

So for those who claim to understand Relativity and Mercury's Perihelion Precession, if you have not properly computed a 3d-n-body Newtonian algorithm, then compared it to the 2D variety, then all you 'understand' is how to copy-paste.

But in addition to all of this, if you have not yet seen the radically large effect caused by delaying gravity by insisting that gravity travels at the velocity of light, then you need to look at the previous chapter. Of course, Albert never had a computer like this. He can be forgiven.

Those who continue to copy-paste archaic and arcane pencil-and-paper maths onto the internet or into their academic thesis are less worthwhile than those who take the dates for Creation in Genesis at face value. A Bible-basher at least knows the morals that have survived thousands of years of history. An astrophysicist (or any other type of scientist) without a 3d-n-body-gravity algorithm precisely mapping Newton's formula, is worth less by far. Especially since you likely have the resources, but neglect to use them. This should have been done 20 or 30 years ago. (I was too busy destroying apartheid).

The analytical a priori is a conceptual method that comes to me from Immanuel Kant. The idea of a purely analytical approach seems to have been echoed in ancient Greece, and before that in Sumeria and India. It is based on the notion that by logic alone we can deduce facts without having to make any observations. Of course it is implied that others make observations, and some observations are commonly available. Nevertheless, by analyzing the inherent logic in any system we can figure which observations are correct, simply as a matter of math/logic. We often never need to make any new observations.

Einstein's Relativity is a theory based on the bedrock of Newtonian mechanics. And yet because the conceptualization of Newtonian mechanics made by the Relativists is intrinsically flawed for numerous reasons here outlined, as well as in other chapters, any theoretical conclusions about Relativity thereafter must also collapse, a posteriori.

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