The smallbodies.ru site was created as a scientific and informational one, so most of the information is accessible to understanding only for sufficiently trained users. In order to attract more interest to the site, we decided, in a popular form, to present the main issues related to the content of the site itself.

It should be noted that the main goal of our research was to study the distribution of small bodies of the solar system approaching the Earth. The main objects of study were short-period comets and asteroids of the Apollo, Amur, and Aton groups.

Short-period comets include comets whose period of revolution around the Sun does not exceed 200 years.

The asteroids of the Apollo, Amur, Aton groups are currently receiving increased attention in connection with the solution of the problem of “asteroid danger”. A comprehensive analysis shows that the problem of «asteroid danger» associated with predicting the collision of large celestial objects with the Earth and preventing catastrophic consequences is complex and far from a final solution.

At present, the characteristic lifetimes before the collision with the Earth of all “dangerous” space objects have been calculated. It is shown that collisions of celestial bodies, such as the Tunguska meteorite, with the Earth occur more often than once every hundred years. Collisions that can cause a global catastrophe such as a «nuclear winter» on Earth occur on average once every several hundred thousand years. Disasters leading to a change in geological epochs occur on average once every several tens of millions of years.

Questions about more accurate estimates of the asteroid hazard are related to our knowledge of the motion and evolutionary processes of small bodies of the solar system, which pose a danger of collision with the Earth. The study of the movement of potentially “dangerous” objects, the cataloging of orbital elements and close proximity to the terrestrial planets is an important step in solving the “asteroid hazard” problem. Regular comprehensive studies of potentially “dangerous” objects will make it possible to predict the collision of an asteroid with the Earth in advance and take appropriate measures to prevent a catastrophe.

The study of the evolution of the orbits of the asteroids of the Apollo, Amur, Aton groups is one of the main tasks in solving the problem of «asteroid danger». The theory of motion of the asteroids of the Apollo, Amur, Aton groups is much more complicated than the theory of planetary motion, since the elliptical orbits of the asteroids are more extended than the orbits of the planets, the planes of the orbits are significantly inclined to the plane of the ecliptic. In addition, the asteroid orbits have close proximity to the orbits of large planets, therefore, the analytical theories of asteroid motion are not accurate and numerical methods are widely used to study the evolution of their orbits. It should be noted that the separation of asteroids into the groups of Apollo, Amur, Aton is very conditional, since asteroids in the process of their evolution can move from one group to another.

For successful work on the solution of this problem, comprehensive studies of both physical and dynamic properties of these objects are required.

The motion of celestial bodies in the solar system is described by various differential equations. The accuracy of predicting the motion of the object under study substantially depends on the choice of specific differential equations.

The study of the motion of large planets, the moon and the sun began in ancient times. Babylonians over the 5th century BC. they knew with great accuracy the duration of the year and month and the periods of revolution of large planets. With the creation of geometry by scientists of ancient Greece, there was a further development of knowledge in the field of astronomy. Greek scientists made a great contribution to the study of the motion of 5 large planets, the moon and the sun. Much more complicated was the case with the planets. According to scientists, ancient Greece, the center of the world was the Earth, and all the planets the Moon and the Sun made circular motions on epicycles at a constant speed. Improving the accuracy of planetary motion led to the introduction of an increasing number of geocentric spheres. Aristotle brought the total number of spheres to 56. But even with this complication, it was impossible to reconcile theoretical calculations with observations.

The following ancient Greek scientists made a great contribution to the development of the geocentric system of the world: Aristotle, Hipparchus, Ptolemy. The most significant contribution to the development of the geocentric system of the world belongs to Ptolemy. In his treatise “Almagest”, the laws of visible planetary movements were first revealed. Ptolemy decomposed the visible motion of each planet into two movements by the deferent and the epicycle. The theory of planetary motion created by Ptolemy existed for 13 centuries, and no one could replace it with a more advanced theory.

The further development of astronomy is associated with the appearance of the Copernicus treatise in 1543, “On the circulation of celestial bodies, VI books.” The great significance of the Copernicus treatise was that occurred a turning point from the geocentric system of the world to the heliocentric of theory. At the same time, astronomy from a purely geometric science has become a physical science.

The transfer of the “center of the world” to the
center of the Sun made it possible to further create new celestial mechanics,
the main subject of which was the study of the motion of celestial bodies in
the solar system. The following outstanding scientists made a great
contribution to its creation: Kepler, Newton, Euler, Lagrange, D’Alembert,
Laplace and others.

Newton’s celestial mechanics is based on
the law of gravity and the laws of motion. The law of universal gravitation
states that the force of mutual attraction of two bodies is directly
proportional to their masses and inversely proportional to the square of the
distance between them.

Now we formulate the three laws of motion

I. There are such reference frames, called inertial, regarding
which the material points, when no forces act on them (or mutually balanced
forces act), are at rest or in uniform rectilinear motion.

II. In an inertial reference frame, the
acceleration that a material point receives with constant mass is directly
proportional to the resultant of all the forces applied to it and inversely proportional
to its mass.

III. Material points interact with each other by forces of the same nature, directed along a straight line connecting these points, equal in magnitude and opposite in direction.

Is
it possible to verify this law from experience?
To do this, it would be necessary to measure three quantities included
in his expression: acceleration, force and mass.

Distracting from the difficulties associated with measuring time, let us assume
that it is possible to measure acceleration.
But how to measure force or mass? We
don’t even know what it is.

What is mass? This, Newton answers, is the product of volume by density. Better to say, Thomson and Tet argue that density is the private of dividing mass by volume. What is power? This, Lagrange answers, is the cause that produces or seeks to produce the movement of the body. This, Kirghoff will say, is the product of mass by acceleration. But then why not say that mass is the private from dividing force by acceleration.

These difficulties are insurmountable. Having defined strength as the cause of the movement, we became on the soil of metaphysics, and if such a definition had to be satisfied, it would be completely fruitless. For a definition to be suitable for anything, it must teach us how to measure power; besides this condition is enough; there is no need for the definition to teach us what force is in itself, or whether it is the cause or effect of the movement ”.

Based on the law of universal gravitation, equations of motion of celestial bodies were obtained, the solution of which made it possible did to predicted their location in space. The most convenient objects for checking theoretical studies are large planets and the Moon, since a significant amount of observations has been accumulated on these objects.

At the beginning of 1850, the French scientist Le Verrier built a theory of the motion of the Sun and seven large planets (Mercury-Neptune) relative to the Earth. The result of his work was the proof of the impossibility to reconcile with the observations of the passage of Mercury across the solar disk on the basis of Newtonian dynamics by any system of orbital elements and the masses of the planets known to him.

Along
with Mercury, the Moon is a more complex object for studying motion by solving
the *n* -body
problem.
Along with the established
empirical difficulties, difficulties associated with the phenomenological
nature of Newtonian theory began to be more and more identified and discussed. Various difficulties caused scientists doubts about the
accuracy of Newton’s law, and numerous attempts were made to amend the exact
formula of the law of gravity. However,
these attempts to improve the law of gravity have been unsuccessful.

The next step in the development of the relativistic
theory of gravitation is the general theory of relativity created by Einstein.
The general theory of relativity is based on the following hypotheses.

I. The space of the general theory of
relativity should be a pseudo-Riemannian four-dimensional space.

II The second hypothesis of the general
theory of relativity is as follows: The energy-momentum tensor is superimposed
on the event space as an additional construction, and it is assumed that the
energy-momentum tensor follows from the pseudo-Riemannian geometry of this space
itself.

General relativity, like Newtonian mechanics, is not
free from flaws. The most important of these is the question of the nature of
gravity. In the framework of the general theory of relativity, just as in the
theory of gravity, Newton is considered purely phenomenologically. Another
drawback of the general theory of relativity is the significant complication of
differential equations in the problem *n*
of bodies. The solution to this problem has to be sought in the form of series
in powers of small parameters, while only the initial terms of the expansion
are taken into account, and the question of the convergence of the series is
not studied at all. The main drawback of both Newtonian dynamics and Einstein’s
theory of gravitation is, according to the authors, the endowment of mass with
a property that generates a gravitational field. In the first case, it is
assumed that the mass has the property of attraction of other material bodies,
in the second case, the mass is endowed with the property to bend the
surrounding space.

As noted earlier by A. Poincare, mass and force are undetectable concepts, like undetectable concepts in mathematics — a point, a line. In the mechanics of Newton and Einstein, these concepts have certain physical properties, which, at least, is not justified. All of the above disadvantages of both Newtonian dynamics and the general theory of relativity are reflected in the degree of accuracy and reliability of the studies based on them. An indispensable advantage of the general theory of relativity is that with the help of a specially selected harmonic coordinate system, it was possible to coordinate the theory of motion of Mercury with observations. However, the motion of the moon in this coordinate system cannot be coordinated with observations by solving one system of differential equations of motion.

A natural question arises: is it possible to obtain equations of motion, the solution of which would allow to coordinate the theory of motion of Mercury and the Moon with observations? The solution to this problem was found on the basis of constructing the simplest model of the interaction of a moving material body with the surrounding space.

The derivation of differential equations of motion (3) is based on the following idea. Suppose that at every fixed point in time a material body occupies a certain volume in space. When the material body moves, the freed space is filled with the environment, thereby compressing the surrounding space by the amount of volume freed by the moving object (see Fig. 1).

Fig. 1

Thus, the following basic hypotheses are the cornerstone of the proposed method: a) space is populated by moving material bodies, with certain sizes and densities; b) the space surrounding material bodies is not an empty vacuum, it has a fairly dense energy and the property of compression relative to moving material bodies.

A natural question arises: how well are these assumptions justified? In classical physics, the concept of empty space is used, in which there are no particles and field. Such an empty space can be considered a vacuum of classical physics. Vacuum in quantum theory is defined as an energy state in which there are no real particles. Moreover, the presence of a field is a prerequisite. According to modern concepts, the universe is dominated by a physical vacuum, in energy density it surpasses all ordinary forms of matter combined. It is believed that the substance comes from a physical vacuum, and its properties stem from the properties of a physical vacuum.

Vacuum has not only a certain energy density, but also pressure. According to the calculations of the Nobel laureate R. Feynman and J. Wheeler, the energy potential of a vacuum is so huge that «in a vacuum enclosed in the volume of an ordinary light bulb, there is so much energy that it would be enough to boil all the oceans on Earth.»

The solution of the new differential equations of
motion of large planets and the Moon was compared with the coordinate data of
these objects located in the DE405 database

Currently, a number of high-precision
numerical theories of the motion of large planets have been developed. The most
famous of them is the numerical theory created by NASA researchers Newhal,
Standers, Williams. Standish created a data bank for the coordinates of large
planets, the Moon and the Sun — DE405 in the time interval from 2305424.5 J.D.
(1599 Dan 5) to 2525008.5 J.D. (2201 Feb 20). The coordinates and velocities of
the inner planets obtained using the DE405 data bank are consistent with radar
observations, and all planets are consistent with optical observations.

A comparison of the coordinates and velocities of large planets and the Moon found by solving new equations with DE405 data differ from each other within the limits of observation errors. It is impossible to achieve a similar accuracy of the agreement of solutions with DE405 by solving the n-body problem or by solving relativistic equations without involving additional equations that take into account the shape of the planets.

The equations of motion, based on the principle of interaction of moving material bodies with the surrounding space, were used to study the motion of asteroids and comets, presented on the site smallbodies.ru. Due to the fact that most asteroids have close proximity to large planets, difficulties arise in the study of their movements by integrating classical and relativistic equations. The information presented on the smallbodies.ru website has a fairly high degree of reliability compared to information obtained on the basis of the gravitational or relativistic model.