Gravitational Explanation of the Glacial Periods and Calculation of the Precession of Mercury's Orbit

Copyright © Dr. Mariano González Ambou, ALicante, Spain

According to the gravitorelative theory, the gravitational acceleration that a reference body experiences due to a mass which moves at a velocity in relation to it is:

(1)

and are the components, perpendicular and parallel to the velocity, of the gravitostatic acceleration .

This formula, which may be considered as the generalization of Newton's Law, is deduced supposing that the gravitational interaction is produced by means of a flux of particles which spread at velocity c. That is to say, a mass creates a flux of particles and the reference body experiences an acceleration depending on the density of the flux (by unit of volume) that it perceives. Due to the Lorentz contraction of the length, the reference body perceives a greater density if the velocity of the mass is perpendicular to the line of centres and less if is parallel. In the general case where the velocity forms an angle with the line of centres, formula (1) is deduced, in which and .

Following the same model of interaction (by means of flux of particles) for the electric case, a similar formula to (1) is deduced which may be considered as the generalization of Coulomb's Law. The magnetic forces are deduced from this formula as shown in.

It must be pointed out that the method used in this theory to describe the interaction is not the usual method of "laboratory reference frame" but rather that of "relation".

In the book (*) "Diálogos sobre Física" the possible mechanism of gravitational interaction is described.

Gravitorelative Effects

The formula (1) is made up of three terms. The first term represents the Newtonian or gravitostatic acceleration. The other two terms are the components (perpendicular and parallel to the velocity) of the so called gravitorelative acceleration which arises when the mass moves in relation to the reference body. Moreover, it turns out that the gravitorelative acceleration component parallel to the velocity is of opposite direction to the component parallel to the velocity of the Newtonian acceleration ( fig 1).

( fig 1)

The radial component of the gravitorelative acceleration contributes to the determination of the average radius of the orbit of a planet.

When the eccentricity varies the radial gravitorelative acceleration also varies. This is, on one hand, because the radial projections of and have opposite directions (the first increases the gravitostatic acceleration while the second decreases it) and, on the other hand, because when the eccentricity decreases, increases and decreases, while when the eccentriciy increases, exactly the opposite occurs. In a circular orbit = 0 and the gravitorelative acceleration coincides with the direction of the gravitostatic acceleration.

The gravitorelative component parallel to the velocity produces a small lengthening of the major axis (near 4 km in the earth's orbit) because as the planet moves away the deceleration of the velocity is less than in the Newtonian case and as it moves nearer the acceleration is less in the same measure so both acceleration and deceleration are balanced but they nevertheless cause this small effect.

The perpendicular gravitorelative component , which in circular orbits coincides in direction with the Newtonian acceleration, in elliptic orbits produces a precession of the orbit.

The Precession of Mercury's Orbit

In elliptic orbits the gravitorelative acceleration perpendicular to the velocity does not change its magnitude, instead it simply changes its direction and with it the trajectory of the planet.

To understand what the trajectory of the planet is, we are going to place ourselves in the Newtonian frame which corresponds to that of a planet that moves in relation to the Sun in a Newtonian elliptic orbit. Now if the planet not only shares the movement of the Newtonian frame but is also affected by the perpendicular component of the gravitorelative acceleration then we will see it moving away continuously from our Newtonian reference frame with the said acceleration (fig 2a). If we place ourselves in the Sun frame the movement of both would be as seen in (fig 2b). (as the factor comes into the gravitorelative acceleration it is very small and must always be taken as a first approximation).

(fig 2a)	(fig 2b)

We will calculate the separation S produced at the end of a period T between the Newtonian frame and the planet on which the gravitorelative acceleration acts.

The separation produced in the interval of time is:

where is the average acceleration which is approximately equal to

, where a is the length of the major semiaxis of the orbit. The value of is calculated in the

data section.

In a complete period T, the separation S from the Newtonian orbit is:

this separation represents a displacement of the perihelion which corresponds to a deviation angle equal to:

where is the radius of the perihelion. The velocity of spin of the perihelion is therefore so in one century the perihelion covers the following angle:

K is the product of the number of seconds in a century by the radian conversion factor in angular seconds, and e is the eccentricity of the orbit. Therefore, in one century the perihelion moves at an angle equal to:

This result admits only a slight error of a few angular seconds due to the use of average values to obtain it. Therefore, the precession of elliptic orbit can also be gathered from the gravitorelative theory. The value obtained in 1973 for the "general precession" of Mercury is approximately 41.4". Whilst the value obtained from the G R theory is 43".

Data:

100 000-year Glacial Periods

Recently it has been possble to establish quite a reliable chronology of the glaciations using various methods, the most important of which is based on the isotopic analysis of the oxygen in marine sediments. It is known that in the last million years some 10 glaciations have taken place, the last of which occurred some 18 000 years ago. The average temperature then was about 5 ºC lower than the present average and ice covered a third of the Earth's surface.

(fig 3)

Variation of the temperature in the last 500 000 years which corresponds to the volume of ice in the glaciars in accordance with the relation of the two oxygen isotopes in marine sediments. (The heavy oxygen isotope O-l8 is more resistent to evaporation so it appears less in glacial ice water than in ocean water. Therefore when continental ice layers are formed, the ocean water and the shells of marine organisms which fall in the ocean sediments appear enriched in oxygen-18).

On the other hand, it is known that due to the small gravitational attraction of the Moon and the other planets, the orbital parametres of the Earth change with time. The inclination of the Earth's axis varies between 22.1 and 24.5 degrees with a period of some 40 000 years. The eccentricity of the orbit varies between 0.005 and 0.006 approximately, with a period of some 100 000 years. The terrestrial axis takes some 26 000 years to describe a complete precession circunference.

The spectral analysis of figure 3 shows a certain number of fundamental frequencies. The frequency of greatest intensity corresponds to a cycle of some 100 000 years, then there are three notable cycles, one of 40 000 and two others of 24 000 and 19 000 years each. The climatic variation of the last three frequencies may be explained from the variations in the inclination of the Earth's axis which produce a variation in average sunshine in the high latitudes of the northern hemisphere (where there is more continental area) in the different seasons (this explanation was given by the Yugoslavian astronomer Milutin Milancovitch and verified by Gerard R. North and his colleagues at NASA with experiments carried out with models of energetic balance). What continues to be an enigma is the main cycle of 100 000 years in the glacial periods. It is thought that this cycle must be linked to the variation of the eccentricity of the terrestrial orbit, but it is not known how .

According to the gravitorelative theory, when the eccentricity of the orbit is increased, the component of radial gravitorelative acceleration is increased in the opposite direction to the gravitostatic acceleration, which makes the radius of the orbit increase. While when the eccentricity is decreased the radial gravitorelative acceleration increases in the same direction as the gravitostatic acceleration, which makes the radius decrease.

For an eccentricity e given the average value of the radial gravitorelative acceleration is equal to:

The two terms of the addition correspond to the radial components of the perpendicular and parallel gravitorelative accelerations respectively, while is the average value of the angle , which varies between the maximum value and the minimum value arccos e, that is to say:

, and is the average gravitorelative acceleration for an eccentricity e given, in which r and v are the average values of the orbital radius and velocity ( is practically equal to 1 as the eccentricity in all cases is small).

From the data that we possess we can see that the curve of variation in time of the ice layers is slanting towards the right which means that ice takes more time in forming that in melting. This can be due to the elastic properties of the bed-rocks over which the ice acumulates, since the weigth of the ice layers causes the collapse (sinks) of the bed-rocks through thousands years, which makes easier the melting of the ice. Afterwards the bed-roks raises again. Thus, the solar irradiation that the Earth recives does not determine the volume of ice in the glaciars but it is, however, the main cause, and the explanation of the great intensity of the 100 000-year glacial periods.

Let us calculate the distance r = r_max - r_min travelled by the Earth in its fall, (r_max and r_min are the average radii that correspond to the maximun and minimum orbits).

In order to simplify this calculation, we are going to consider that the gravito-relative force change in a linear way in time (this is a good aproximation since it changes very slowly), and that there are two phases in the Earth fall: one of acceleration and another of deceleration.

This distance is appoximately equal to:

where t_c is the fall time and is the increase in the radial gravitorelative acceleration (or deceleration) which is equal to:

is the average gravitorelative acceleration in the whole cycle which in turn is equal to:

where v and r are the average velocity and the average radius of the orbit in the whole cycle, that we can take with a good approximation equal to the present average values.

On the other hand, the average sunshine is inversely proportional to the square of the distance, while the temperature is proportional to the average sunshine, that is to say:

where E(r_min ) is the average sunshine in the orbit of minimum average radius, while E(r_max ) is average sunshine in the orbit of maximun average radius, T(r_min ) y T(r_max ) are the respective average temperatures.

Developing the square in brackets we will obtain that the total increase in the temperature
T = T(r_min ) - T(r_max ) is approximately equal to:

Then the total variation of temperature is about 21 ºC.
The variation of temperature in the last 18 000 years can be obtained changing the minimal eccentricity by the actual one, taking the fall time t_c

equal to 18 000 years, and only considering the acceleration phase, that is to say: . We obtain a variation of temperature of about 5.5 ºC. So, in the last 18 000 years the average temperature of the Earth is increased about 5º C which agrees with the estimates made by geologists, which demonstrates Milutin Milancovitch hypothesis that the glacial periods are linked to orbital variations.

Data: