Sidereal Year or Tropical Year

The time from one winter solstice to the next winter solstice is not the time it takes for the Earth to orbit the Sun. Our calendar is based on what is called the tropical year, which reflects the exact time between seasons of 365.2421990 days. This is clearly used by farmers who need to know when is the best time of the season to planet crops.

But when we measure the movement of the Earth in comparisson to the stars, we can see that it takes 365.25636042 days for the Earth to orbit the Sun. This is the sidereal year, and is used by astronomers to understand the precise nature of gravity and other astrophysical phenomena.

So if we mistakenly use the tropical year for measurement of an orbit, we will get a shortfall of about 20 minutes from the real time it takes for the Earth to orbit the Sun. Western tradition suggests this 20 minute difference was first noticed by Hipparchus from ancient Greece in 129 B.C. But the Vedic text from India, the Vedanga Jyotisha, dates this observation to at least 700 B.C.

The diagram below demonstrates this by showing the position of the Earth's axis at two times, half of that cycle of 25772 years apart, but at the same day of the year.

So it is clear that the Earth not only rotates on its axis, as we all know, but that the Earth's axis itself must be rotating every 25772 years. So this is why the seasons arrive 20 minutes sooner for every orbit of the Earth around the Sun. Because 31 million seconds (1 year) divided by 26000 years is about 20 minutes.

The section after this, deals with the mysterious Cause of the Precession of the Equinox, or Axial Precession. In an earlier section we saw how important this Precession of the Equinox
is in measuring the Perihelion Precession of Mercury.  This particular section here details the exact geometry of how Precession of the Equinox can cause an error in observation of Mercury's Perihelion Precession. I have placed these details towards the end of this article because it is seemingly not important in giving the precise account. This is really just a description of an error, but it is an error which is easy to make, so properly comprehending this detail is vital from the perspective of pedagogy.

The Precession of the Equinox will adjust our observation of Mercury's perihelion in spatial terms. But also, a temporal measurement of the length of the year will result in another adjustment during such an observation. So we need to comprehend the results of both these factors.

The spatial adjustment is approximated as follows:
26000 years, the axis of the Earth will rotate 16000 km - an amount equivalent to the arctic circle (previous diagram). Its easy to see this is 0.6 km per year. Or 150m per orbit of Mercury; because Mercury's orbit is about a quarter of the duration of the Earth's.

In the diagram below we can see that our observtion of Mercury's orbit will be proportionally less because Mercury is a fraction of the distance to the Sun than we are. So the spatial adjustment is equal to the proportion of the distance to the Sun.

If the Earth is displaced by distance A from time 1 to time 2, then our observation of Mercury will be displaced by distance B. Mercury's orbit varies from one third to half the distance to the Sun; we will estimate using the larger amount. So because Mercury is at most half the distance to the sun, this will cause an error in our observation of its orbit of just 75m. (Half of the 150m from earlier).

The orbit of Mercury is about 375 million km.
So each arc-second is 1 / 1296000 of that amount.
This means that each arc-second for Mercury's orbit is about: 290km.

So the error caused by spatial dsplacement is trivial:
0.075 / 290 = 0.00026 arc-seconds per orbit of Mercury, which is:
0.1 arc-second per century.
So the spatial adjustment for precession of the equinox per century is negligible.

Next we look at the temporal adjustment.
This is far more relevent, thus more precise calculations will be used. As a point of method it is prudent to do these calculations first with crude averages, then once the process is clear in our minds, we then use the more accurate numbers and get a more precise result.

We can see that 365.25636042 days - 365.2421990 days =
1223.55 seconds between sidereal and tropical year on Earth.

365.2421990 / 87.969252 = 4.15193. The ratio of Earth-orbits to Mercury-orbits.

A period of 25772 years will yield 1296000/25772 = 50.287 arc-seconds per year.
The Precession of the Equinox is then given as 5029 arc-seconds per century for Earth.

Now that 5029 arc-second result seems to fit quite neatly into the Relativist account of how the Precession of the Equinix effects our observation of Mercury's orbit, hmm? We discussed that claim earlier in the section: Precession of the Equinox.

But Mercury orbits 4.15 times faster than the Earth. So it seems that an angular adjustment could be 4.15 times greater for Mercury than it is for the Earth; for any equal amount of time.

So the temporal adjustment might be considered to be:
x 5029 = 20870 arc-seconds per century.
Is that the correct amount for the Earth's axial precession reflected onto an observation of the orbit of Mercury? No its not. That is deliberately misleading to make a point.

Observe this diagram:


In the diagram above, regardless of the difference between Time A and Time B, the angle that the purple line circumscribes will be about 4.15 times greater than the angle that the yellow line circumscribes.

But! We have to be very careful here. Because if the correct point of Mercury's perihelion is at B, but then we use the tropical year instead of the sidereal year, we will mistakenly think that Mercury's perihelion is ahead of us, because we here on Earth are only at the tropical year: Time A.

So the angle between the red line and the green line gives Mercury's perihelion the appearance of being advanced by an equal amount of angle that is the Earth's Axial Preccession: 5029 arc-seconds per century.

It would be very easy to mistakenly think that because we have a 20 minute shortfall in the length of the year, then that means that Mercury will also be behind its expected position. If you cannot visualize all this clearly, you need to contemplate it again. I doubt anyone will get it after just one reading.

Remember, this shortfall occurs because of using the common tropical year of 365.242 days in our dating system, rather than the proper sidereal year which is longer: 365.256 days. So the Precession of the Equinox causes a shortage in time which results in an apparent increase in visual space!

These minutae we neglect at our peril when constructing the algorithm. But methodologically I can only see this, and depict the geometry clearly because I must be able to do this in order to actually build the n-body-gravity algorithm. Bland numbers on a page are fairly meaningless by comparison.

So we must realize that Axial Precession causes us to miscalculate the time it takes for the Earth to move around Sun. So it is not the Axial Precession itself that results in the adjustment to Mercury's orbit. It is  the length of a year that is the issue. That nobody else seems to describe it properly like this shows how the geometry is hardly ever analysed. Its all just by-rote exam regurgitation. I wonder how many were 'failed' over the last 100 years for doing proper geometry?

Every website I have seen just copy-pastes that amount of 5600 arc-seconds per century and just assumes the math is correct to the geometry. I had to go back to original principles and work out everything for myself. This is why I have the spatial calculation first, because the way it is described typically makes no mention that it is an error in the length of the year that causes us to observe Mercury far ahead of where it really is at perihelion. But it is clear that the 5029 arc-second adjustment is an error that has no real place in the physics of the gravity algorithm. But it plays a large role in the observation and measurement process.

Methodologically, unless you understand your opponents errors, you understand nothing. This is why psychologically speaking, the mind that simply dismisses those who 'do not understand', with a grandiose sense of their own superiority will always fall into error. Because the only way you can be certain as to which of you has made the error, is when you properly understand their error.

If you do not explain their error to them, then it is because you cannot; because you do not understand it. And if you do not understand their errors, you are certain to make those or similar errors yourself. That is the difference between dogma and the dialectic. That is why science without philosophy is a disaster waiting to explode. If you cannot have total empathy for the inevitable errors in the learning process, you have no real right to consider yourself a teacher or a researcher, and you know nothing except parroting.

But what was the intuition that led me to this painfully long-winded logical positivist conclusion? That was the absurd notion that the other planets can somehow cause the Earth's axis to wobble every 26000 years. In the next section, I will deconstruct that issue in every possible detail: What is the cause of the Precession of the Equinox?

Sections of this article by web-page

gravity algorithm