If bodies moved around the Sun only under the influence of its force of attraction, then the motions would be described by Kepler’s laws. Such a movement is called *unperturbed* . In reality, all the bodies of the solar system are attracted to each other. Deviations in the movements of bodies from Kepler’s laws are called *perturbations* , and the real movement of bodies is called *perturbed movement* .

Perturbations are very complex in nature and it is very difficult to take them into account. The values of the perturbations are small, since the total mass of all the planets is less than the mass of the Sun by about 700 times. Perturbations can be considered as the difference between the positions of the star during perturbed and unperturbed motions, and the perturbed motion of a body can be represented as motion according to Kepler’s laws with variable elements of the orbit.

Changes in the elements of the body’s orbit due to its attraction by other bodies, in addition to the central one, are called *perturbations* , or *element inequalities* . Perturbations of elements are divided into *secular* and *periodic* .

The secular perturbations of the bodies of the solar system depend on the relative position of their orbits, which changes very little over very long periods of time.

The longitude of the ascending node b and the longitude of perihelion p are subject to secular perturbations.

Periodic perturbations depend on the relative position of bodies in orbits, which, when bodies move along closed orbits, is repeated at certain intervals. Almost all elements of the orbits are subject to these perturbations.

**Ebb and flow**

A tide is any of the cyclic deformations of one astronomical body caused by the gravitational forces of another.

The dimensions of the Earth are not infinitely small compared to the distance to the Moon and the Sun. The forces of lunar and solar attraction to different points of the Earth are not the same. Therefore, a perturbing force appears, acting on various parts of the Earth’s surface in different ways. In solid masses, the action of force causes tension, while large bodies of water are carried away by gravity and flow from place to place. The tidal effect on the atmosphere is expressed in the appearance of atmospheric currents.

Water tides have been known since ancient times. The geographer Strabo (b. – 66) says that the Phoenicians were well aware of the ebb and flow. In the Mediterranean, the effect is small, but the Phoenicians passed through the Pillars of Hercules and observed it in the ocean. They pointed out that the tides depend on the phases of the moon and are especially intense during full moons and new moons. The Italian Jesuit Cabeo (1585 – 1650) suggested that the moon produces some kind of alcoholic substance on the seabed, which causes the tide. Stevin explained the tide by the attraction of the Moon, but the hump on the far side of the Earth explained that there was another attracting point existing there. Galileo explained the tides by centrifugal force, rejecting gravity. Some researchers have suggested that the Moon produces changes in air pressure that affect sea levels.

The most correct explanation for the phenomenon of ebbs and flows was given by Isaac Newton, using the theory of gravity. He wrote that the Moon pulls water away from the Earth on one side and pulls the Earth away from the water on the other.

If the earth’s surface is covered on all sides by the ocean, then each drop of water has an acceleration proportional to the square of the distance between the particle and the center of the moon.

The resultant acceleration imparted to solid particles passes through the center of the Earth T and is equal to:

,

where m is the mass of the moon, r is the distance of the center of the moon from the center of the earth.

For ocean water, the acceleration at point A is greater than w _{T} , and at point B is less than w _{T} , since:

and ,

where R is the radius of the Earth.

Relative acceleration (relative to the center of the Earth) at point A is:

.

We neglect the small term R ^{2} , and instead of (r – R) we leave r. This acceleration difference is directed away from the center of the Earth.

At points A and B, the action of the Moon weakens the force of gravity on the Earth’s surface. At points F and D, the action of the Moon increases the force of the earth’s gravity.

The action of accelerations at intermediate points leads to the fact that the water in the ocean tends on one half of the Earth to point A, where the Moon is at its zenith, on the other half to point B, where the Moon is at nadir. Under the influence of the Moon, the water shell of the Earth takes the form of an ellipsoid, elongated towards the Moon. Near points A and B there will be high tide, and at points F and D there will be low tide.

During the time interval between two successive lunar culminations, equal to 24 ^{h} 52 ^{m} , tidal protrusions will go around the Earth, and in each place there will be two high tides and two low tides. Under the influence of solar attraction, the water shell of the Earth experiences tidal forces less than lunar forces by 2.2 times. Solar tides are not observed separately, they only change the magnitude of the lunar tides. During new moons and full moons, the forces add up and the tides are larger than usual, the solar ebb falls on the quadrature of the lunar tide, the forces are subtracted and the tides are smaller.

In reality, the Earth is not covered with water everywhere, the bottom of the seas and oceans has a complex relief, the tidal wave experiences great friction. The moment of the tide, therefore, does not coincide with the moment of the culmination of the moon, and is late by up to six hours. This period of time is called the *applied hour* .

The height of the tide in the Black Sea is a few centimeters, in the Bay of Fundy on the Atlantic coast of Canada – 18 meters. The friction of the tidal wave on the solid parts of the Earth causes a systematic slowdown in its rotation. High and low tides affect changes in atmospheric pressure.

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