A new version explaining the inclination of our satellite's orbit!

Two researchers from the Observatory Cote d'Azur in France Kave Pahlavan and Alessandro Morbidelli put forward new theory, according to which the young Moon left its original orbital plane under the weight of large objects flying by.

Scientists have long believed that the Moon came into being after an object the size of Mars crashed into the young Earth and vomited space great amount garbage, which united and became a satellite of our planet. As a result of this somewhat chaotic process, contrary to the laws of physics, the tilt of the Moon is no more than one degree. For the first time, scientists gave an explanation for this phenomenon.

Complete solar eclipse occurs on Earth about once a year and a half. But imagine if this happened every month. For this to be the case, the Moon must orbit the Earth in the same plane as the Earth travels around the Sun - thus, new Moon will always pass directly between us and the Sun. Instead, the lunar orbit around the Earth is in a slightly different plane, which is tilted 5 degrees with respect to the plane solar system. But earlier the tilt was even greater - about 4.5 billion years ago, when the Moon first formed and did not spend much time under the influence of the Earth's tides, the tilt was 10 degrees.

Kave Pahlevan and Alessandro Morbidelli have compiled a computer model to estimate the effect of objects passing by the Moon during the first 100 million years. They found that no single object would be large enough to yank the Moon out of its expected orbital plane on its own. But the heaviness of many objects in the aggregate could contribute to this. This theory not only explains the strange tilt of the moon, but also explains the abundance of certain metals in earth's crust especially gold and platinum.

So Robin Canup of the Research Institute in Boulder (USA) explained in his essay that both of these precious metals have "strong chemical similarities to iron." If these elements were present during the Earth's early days, the iron that sank into the planet's core would have sucked the gold and platinum along with it. But enough precious metals are on the surface, which means, according to his theory, they arrived here after the core formed.

“Indeed, these metals were probably delivered to our planet by large space objects that were the remains of other planets that form the internal component,” wrote Kanup, who studies the origin of the solar system. “If there were many small objects, some of them must have collided with the moon and left gold and platinum there. The relative scarcity of these precious metals on the Moon strongly suggests that a few large objects, rather than many small ones, have landed on Earth.”

Overall, the data on these metals provide strong evidence in support of Pahlavan and Morbidelli's theory that space bodies passing by the young Moon changed the plane of our satellite's orbit.

All asteroids discovered so far have direct motion: they move around the Sun in the same direction as the large planets (i

The boundaries of the ring are somewhat arbitrary: the spatial density of asteroids (the number of asteroids per unit volume) decreases with distance from the central part. If, as the asteroid moves along its orbit, the mentioned plane zr is rotated (around an axis perpendicular to the ecliptic plane and passing through the Sun) following the asteroid (so that it remains in this plane all the time), then the asteroid in one revolution will describe a certain loop in this plane .

Most of these loops lie within the shaded area, like Ceres and Vesta, moving in slightly eccentric and slightly inclined orbits. In few asteroids, due to the significant eccentricity and inclination of the orbit, the loop, like that of Pallas (i=35o), goes beyond this region or even lies entirely outside it, like that of the Atenians. Therefore, asteroids are also found far outside the ring.

The volume of space occupied by the ring-torus, where 98% of all asteroids move, is huge - about 1.6 1026 km3. For comparison, we point out that the volume of the Earth is only 1012 km e. Asteroids move in orbits with a linear (heliocentric) speed of about 20 km / s, spending from 3 to 9 years for one revolution around the Sun.

Their average daily motion is within 400-1200. The eccentricity of these orbits is small - from 0 to 0.2 and rarely exceeds 0.4. But even with a very small eccentricity, only 0.1, the heliocentric distance of the asteroid during its orbit changes by several tenths of an astronomical unit, and at e = 0.4 by 1.5 - 3 AU. That is, depending on the size of the orbit, the inclination of the orbits to the plane of the ecliptic is usually from 5 ° to 10 °.

But with an inclination of 10°, the asteroid can deviate from the plane of the ecliptic by about 0.5 AU. That is, at an inclination of 30 °, move away from it by 1.5 AU According to the average daily movement, asteroids are usually divided into five groups. Groups I, II, and III, numerous in composition, include asteroids moving, respectively, in the outer (most distant from the Sun), central, and inner zones of the ring.

The central zone is dominated by asteroids of the spherical subsystem, while in the inner zone 3/4 of the asteroids are members of the flat system. As we move from the inner zone to the outer one, there are more and more circular orbits: in group III, the eccentricity e

Only bodies in less eccentric orbits, unattainable for this giant of the solar system, have survived. All asteroids of the ring are, so to speak, in a safe zone. But even they are constantly experiencing perturbations from the planets. The strongest influence on them is, of course, Jupiter. Therefore, their orbits are constantly changing. To be completely strict, it must be said that the path of the asteroid in space is not ellipses, but open quasi-elliptical coils that fit next to each other. Only occasionally - when approaching a planet - do the coils noticeably deviate from one another. The planets, of course, disturb the movement not only of asteroids, but also of each other. However, the perturbations experienced by the planets themselves are small and do not change the structure of the solar system.

They cannot cause the planets to collide with each other. With asteroids, the situation is different. Due to the large eccentricities and inclinations of the orbits of asteroids under the influence of planetary perturbations, they change quite strongly even if there are no approaches to the planets. Asteroids deviate from their path in one direction or the other. The farther, the greater these deviations become: after all, the planets continuously “pull” the asteroid, each towards itself, but Jupiter is stronger than all.

Observations of asteroids cover still too short time intervals to reveal significant changes in the orbits of most asteroids, with the exception of some rare cases. Therefore, our ideas about the evolution of their orbits are based on theoretical considerations. Briefly, they boil down to the following The orbit of each asteroid oscillates around its average position, spending several tens or hundreds of years on each oscillation. Its semi-axis, eccentricity and inclination change synchronously with a small amplitude. Perihelion and aphelion either approach the Sun or move away from it. These fluctuations are included as an integral part in fluctuations of a larger period - thousands or tens of thousands of years.

They have a slightly different character. The semi-major axis does not experience additional changes. On the other hand, the amplitudes of oscillations of the eccentricity and inclination can be much larger. With such time scales, one can no longer consider the instantaneous positions of the planets in their orbits: as in an accelerated film, an asteroid and a planet appear to be smeared in their orbits, as it were.

It becomes reasonable to consider them as gravitating rings. The inclination of the asteroid ring to the plane of the ecliptic, where the planetary rings are located - the source of perturbing forces - leads to the fact that the asteroid ring behaves like a top or a gyroscope. Only the picture is more complicated, because the asteroid's orbit is not rigid and its shape changes over time. The asteroid orbit rotates in such a way that the normal to its plane, restored at the focus where the Sun is located, describes a cone. In this case, the line of nodes rotates in the ecliptic plane with a more or less constant speed clockwise. During one revolution, the inclination, eccentricity, perihelion and aphelion distances experience two oscillations.

When the line of knots coincides with the line aspid (and this happens twice in one revolution), the slope is maximum and the eccentricity is minimum. The shape of the orbit becomes closer to circular, the minor semiaxis of the orbit increases, the perihelion is maximally moved away from the Sun, and the aphelion is close to it (since q+q’=2a=const). Then the line of nodes shifts, the inclination decreases, the perihelion moves towards the Sun, the aphelion moves away from it, the eccentricity increases, and the orbit's minor semiaxis shortens. Extreme values ​​are reached when the line of nodes is perpendicular to the line of the slate. Now the perihelion is closest to the Sun, the aphelion is the farthest from it, and both of these points deviate the most from the ecliptic.

Studies of the evolution of orbits over long periods of time show that the described changes are included in changes of an even longer period, occurring with even greater amplitudes of elemental oscillations, and the aspid line is also included in the movement. So, each orbit continuously pulsates, and besides, it also rotates. For small e and i, their oscillations occur with small amplitudes. Almost circular orbits, which, moreover, lie near the plane of the ecliptic, change hardly noticeably.

For them, it all comes down to a slight deformation and a slight deviation of one or the other part of the orbit from the plane of the ecliptic. But the greater the eccentricity and inclination of the orbit, the stronger the perturbations are manifested over long time intervals. Thus, planetary perturbations lead to continuous mixing of asteroid orbits, and therefore, to mixing of objects moving along them. This makes it possible for asteroids to collide with each other. Over the past 4.5 billion years, since the existence of asteroids, they have experienced many collisions with each other. The inclinations and eccentricities of the orbits lead to the non-parallelism of their mutual motions, and the speed with which the asteroids pass one another (the chaotic velocity component) averages about 5 km/s. Collisions with such speeds lead to the destruction of bodies.

There is another serious argument against the Oort cloud. These are the TILTs of cometary orbits to the plane of the ecliptic (it almost coincides with the plane of the orbit of Jupiter and other major planets). These slopes are mostly small, there are few large slopes, and should be approximately equal. Let's look into this issue.

The orbital velocity in the Oort cloud (100 thousand AU) is approximately 100 m/sec. The departure speed there from the solar system, respectively, is 140 m/sec. In order for a comet to penetrate deep into the solar system and fly to the orbit of Jupiter, its speed (more precisely, the projection of the speed perpendicular to the direction to the Sun) must be less than 1 m/sec. If the speed is equal to 1 m/sec, then near the orbit of Jupiter this speed will increase (the law of conservation of angular momentum) by 20 thousand times and become equal to 20 km/sec. A should be equal to 18 km/sec.

Let's once again recall the traditional path of the comet. 4.5 billion years ago it formed. Then it performs a gravitational maneuver near Jupiter and flies into the Oort cloud. Its speed in the cloud decreases to about 1 m/s. Then the passing star (or several stars) increase the speed of the comet to about 100 m/sec. Then another passing star (or several stars) again reduces this speed to about 1 m/sec. And the comet begins to move towards Jupiter.

A simple question: WHERE will the comet's speed be directed when it decreases to 1 m/sec? Will the vector of this velocity again lie in the plane of the ecliptic?
Of course not.
After a random increase to 100 m/s and a reverse also random decrease to 1 m/s, the direction of this small speed will be RANDOM. It will have some RANDOM angle relative to the plane of the ecliptic. Therefore, after the gravitational maneuver with Jupiter, the comet's orbit will have a certain RANDOM TILT relative to the plane of the ecliptic.

So, we compare two versions of the origin of comets.
1. Comets come from the Oort cloud. In this case, the inclinations of their orbits are random. The angles of inclination are distributed more or less evenly from 0 to 180 degrees.
2. Comets are ejected from the Jupiter system. In this case, the comets will have a predominantly DIRECT motion with small angles, due to the rather high orbital speed Jupiter. Large angles of inclination and even reverse are possible, but UNLIKELY.

Again, we look at Wikipedia for a table of short-period comets:
https://en.wikipedia.org/wiki/List_of_periodic_comets
There are more than a hundred comets in this table. I pressed the "inclination" button and the comets lined up from highest inclination to lowest. Here is what it now represents top part tables (see photo above). Only THREE comets (underlined in red) have a reverse movement (the angle of inclination is more than 90 degrees). Only THREE comets also have a large tilt angle (from 45 to 90 degrees) (underlined in yellow). SEVEN comets already have an average tilt angle (from 30 to 40 degrees) (underlined in green).
Here is part of the table just below:

Here the angles of inclination are from 30 to 20 degrees. There are already TWENTY-NINE such comets.
And here is a fragment of the table even lower:

We see that there are 18 comets in the range of just one degree (8 to 9 degrees).

So, the distribution of inclinations of cometary orbits convincingly proves that these comets COULD NOT come from the Oort cloud. Consequently, they were ejected from the Jupiter system.

ORBIT INCLINE

orbit orientation characteristic celestial body in space; dihedral angle between the plane of this orbit and the main coordinate plane(plane of the ecliptic, for artificial satellite Earth - the plane of the Earth's equator).

Big encyclopedic dictionary. 2012

See also interpretations, synonyms, word meanings and what is ORBIT TILT in Russian in dictionaries, encyclopedias and reference books:

• ORBIT INCLINE
  orbit inclination, inclination of the orbit, value (orbital element) characterizing the orientation of the orbit of a celestial body in space; the angle between the plane of the orbit...
• ORBIT INCLINE in Modern explanatory dictionary, TSB:
  characteristic of the orientation of the orbit of a celestial body in space; dihedral angle between the plane of this orbit and the main coordinate plane (the plane of the ecliptic, for ...
• INCLINE in encyclopedic dictionary:
  , -a, m. 1. see tilt, -sya. 2. Position, the average between vertical and horizontal; sloping surface. Small n. N. orbit ...
• INCLINE in the Big Russian Encyclopedic Dictionary:
  TILT OF THE ORBIT, a characteristic of the orientation of the orbit of a celestial body in space; dihedral angle between the plane of this orbit and the main. coordinate plane (plane ...
• INCLINE in the Full accentuated paradigm according to Zaliznyak:
  tilted "n, tilted" us, tilted "on, tilted" new, tilted "well, tilted" us, tilted "n, tilted" us, tilted "nom, tilted" us, tilted "not, ...
• INCLINE in the Dictionary for solving and compiling scanwords:
  "Pose" of Pisa...
• INCLINE in the Thesaurus of Russian business vocabulary:
  Syn: slope, ...
• INCLINE in the Russian Thesaurus:
  Syn: slope, ...
• INCLINE in the Dictionary of synonyms of Abramov:
  (steep, sloping, gentle), roll, steep, slope, slope, slope, slope, descent, inclination, sloping, sloping, gently sloping; slope, steepness, rapids; climb. "Under the very...
• INCLINE in the dictionary of Synonyms of the Russian language:
  trim, nod, roll, steepness, steep, inclination, slope, flatness, syneclise, declination, ...
• INCLINE in the New explanatory and derivational dictionary of the Russian language Efremova:
  m. 1) Action by value. verb: tilt, tilt. 2) a) The position of the body at an angle between the horizontal and vertical planes. b) ...
• INCLINE in the Dictionary of the Russian Language Lopatin:
  tilted, ...
• INCLINE full spelling dictionary Russian language:
  slope, ...
• INCLINE in the Spelling Dictionary:
  tilted, ...
• INCLINE in the Dictionary of the Russian Language Ozhegov:
  position, the average between vertical and horizontal; sloping surface N. orbits (special). Roll down the slope. incline<= наклонить, …
• INCLINE in the Explanatory Dictionary of the Russian Language Ushakov:
  slope, m. 1. Position between vertical and horizontal; an acute angle formed by a plane with the horizon. The platform forms a slope. 2. surface, ...
• INCLINE in the Explanatory Dictionary of Efremova:
  slope m. 1) Action by value verb: tilt, tilt. 2) a) The position of the body at an angle between the horizontal and vertical planes. …
• INCLINE in the New Dictionary of the Russian Language Efremova:
  m. 1. action according to Ch. tilt, tilt 2. An acute angle formed by some plane with the horizon. ott. Body movement in gymnastics. …
• INCLINE in the Big Modern Explanatory Dictionary of the Russian Language:
  m. 1. the process of action according to Ch. tilt 1., tilt 1. 2. The result of such an action; body movement in gymnastics. 3. Sharp ...
• ORBIT ELEMENTS in the Great Soviet Encyclopedia, TSB:
  orbits in astronomy, a system of quantities (parameters) that determine the orientation of the orbit of a celestial body in space, its size and shape, as well as the position ...
• ORBITS OF HEAVENLY BODIES in the Great Soviet Encyclopedia, TSB:
  celestial bodies, the trajectories along which celestial bodies move in outer space. Forms O. n. tons and speeds with which ...
• ORBITS OF ARTIFICIAL SPACE OBJECTS in the Great Soviet Encyclopedia, TSB:
  artificial space objects, trajectories of spacecraft (SC). They differ from the orbits of the celestial bodies of natures. origin mainly by the presence of active ...
• PARTICLE ACCELERATORS in the Great Soviet Encyclopedia, TSB:
  charged particles - devices for obtaining charged particles (electrons, protons, atomic nuclei, ions) of high energies. Acceleration is done by electric...
• SOLAR SYSTEM in the Great Soviet Encyclopedia, TSB:
  system, a system of celestial bodies (the Sun, planets, satellites of planets, comets, meteoroids, cosmic dust) moving in the area of ​​the prevailing gravitational influence of the Sun. …
• WOOD FAULTS in the Great Soviet Encyclopedia, TSB:
  wood, features and disadvantages of individual sections of wood, worsening its properties and limiting the possibilities of its use. P. d. occur in ...
• THE MOON IS THE EARTH'S SATELLITE) in the Great Soviet Encyclopedia, TSB:
  the only natural satellite of the Earth and the closest celestial body to us; astronomical sign. The movement of the moon. L. moves around the Earth from ...
• MOON in the Great Soviet Encyclopedia, TSB:
  the name of the Soviet program for lunar exploration and a series of automatic interplanetary stations (AMS) launched in the USSR to the Moon since 1959. The first ...
• ICEBREAKER in the Great Soviet Encyclopedia, TSB:
  a ship designed to navigate through ice in order to maintain navigation in freezing basins. The main purpose of L. is the destruction of the ice cover ...
• COMETS in the Great Soviet Encyclopedia, TSB:
  (from the Greek kometes - a star with a tail, a comet; literally long-haired), bodies of the solar system that look like nebulous objects, usually with a light clot ...
• ARTIFICIAL SATELLITES OF THE MOON in the Great Soviet Encyclopedia, TSB:
  Lunar satellites (LUS), spacecraft launched into orbits around the Moon; the motion of the ISL is determined mainly by the attraction of the moon. First ISL...
• ARTIFICIAL EARTH SATELLITES in the Great Soviet Encyclopedia, TSB:
  Earth satellites (AES), spacecraft launched into orbits around the Earth and designed to solve scientific and applied problems. Launch...
• PLANET EARTH) in the Great Soviet Encyclopedia, TSB:
  (from the common Slavic earth - floor, bottom), the third planet in the solar system in order from the Sun, the astronomical sign Å or, +. I...
• DOUBLE STARS in the Great Soviet Encyclopedia, TSB:
  stars, two stars close to each other in space and constituting a physical system, the components of which are connected by mutual gravitational forces. Components refer...
• ASTRODYNAMICS in the Great Soviet Encyclopedia, TSB:
  (from astro- and dynamics), the most common name for the section of celestial mechanics devoted to the study of the motion of artificial celestial bodies - ...
• PHYSICAL ASTRONOMY
  since the time of Kepler, this has been the name of the totality of information and theories about the structure and actual movement in space of celestial bodies, as opposed to ...
• GRAVITY in the Encyclopedic Dictionary of Brockhaus and Euphron:
  Newton's law of universal thermodynamics can be formulated as follows: every atom interacts with every other atom, while the force of interaction ...
• GREENHOUSES AND GREENHOUSES
• TAVRICHESKY PROVINCE in the Encyclopedic Dictionary of Brockhaus and Euphron:
  I is the southernmost of the provinces of European Russia, lies between 47 ° 42 "and 44 ° 25" N. sh. and 49°8" and 54°32" E. d. …
• SOLAR SYSTEM in the Encyclopedic Dictionary of Brockhaus and Euphron:
  The true concept of the S. system, as a set of planets and other celestial bodies moving around the Sun according to known laws, was formed ...
• UTERUS in the Encyclopedic Dictionary of Brockhaus and Euphron.
• THE MOON IS THE EARTH'S SATELLITE in the Encyclopedic Dictionary of Brockhaus and Euphron:
  the closest celestial body to us. The average distance of L. from the Earth is equal to 60.27 equatorial radii of the Earth. Mean equatorial horizontal parallax (see) ...
• SAWING PRODUCTION in the Encyclopedic Dictionary of Brockhaus and Euphron.
• SHOUTING REDISTRIBUTION in the Encyclopedic Dictionary of Brockhaus and Euphron.
• COMETS in the Encyclopedic Dictionary of Brockhaus and Euphron:
  (from ??????? - hairy star). - Celestial bodies, which usually appear as a not sharply limited nebula, called the head of a comet, inside which they distinguish ...
• SPANISH LANGUAGE in the Encyclopedic Dictionary of Brockhaus and Euphron:
  belongs to the Romanesque and comes from Latin, mixed with many other elements. The language of the original inhabitants of Spain (see Iberia) died ...
• SPANISH in the Encyclopedic Dictionary of Brockhaus and Euphron:
  Spanish - belongs to the Romance and comes from Latin, mixed with many other elements. The language of the original inhabitants of Spain died in ...
• ASTEROIDS in the Encyclopedic Dictionary of Brockhaus and Euphron:
  I (planetoids, minor planets) - the essence of the bodies circling around the sun, like large planets, and located in the gap between Mars and ...

), Ptolemy needs to compare the measured position of the moon with the position that an observer in the center of the earth. The latter, of course, must be calculated according to the theory of the moon. The measured position should not be either longitude or right ascension, because they are too change rapidly and are difficult to determine accurately. A slowly changing declination or latitude should be taken as the measured coordinate. Even earlier, Ptolemy obtained all the quantities, except for the inclination of the lunar orbit, necessary to calculate the geocentric position. The inclination of the Moon's orbit is the angle between the plane of the Moon's orbit and the plane of the ecliptic (the plane of the Sun's orbit). In principle, Ptolemy had to make two observations of the position of the moon, the analysis of which includes the inclination of the orbit and parallax. For the sake of convenience, Ptolemy separates the variables and for this he takes the latitude of Alexandria. In this case, he does not increase the accuracy of his results, but only gets rid of the need to solve a system of two equations.

To determine the inclination of the orbit, Ptolemy measures the zenith distance of the Moon [chap. V .12 "Syntax"]. Ptolemy makes the measurement with the instrument just described. At the moment of observation, two conditions must be met simultaneously: the Moon must be at the point of the summer solstice and the latitude of the Moon must be the northernmost. This is equivalent to both the Moon's longitude and its latitude argument being 90°. This, in turn, suggests that the ascending node of the Moon's orbit must be at the vernal equinox.

There is also a third condition. It consists in the fact that the Moon must be in the meridian. But this condition is met once every day. The moon must be clearly visible, i.e., it must be far from the Sun. This means, probably, that the observation should be made between sunset and sunrise. But then the Moon should be between the first and last quarter.

If all these requirements are met, then the declination of the Moon is equal to the inclination of the ecliptic plus the inclination of the orbit. The inclination of the ecliptic is approximately 24°, the inclination of the orbit is approximately 5° according to the approximate readings of the instrument, therefore, the declination is approximately 29°. Thus, the Moon is 29° north of the equator. The latitude of Alexandria is approximately 31°, so the Moon is only 2° from the zenith. In this case, the Moon's parallax is negligible.

Always (αει ), when Ptolemy made observations under these conditions, he obtained a value of zenith distance close to 2 1/8 degrees. Ptolemy, as he claims, from measurements received the latitude of Alexandria equal to 30 ° 58 "(see section V .6). The inclination of the Moon's orbit can be found by subtracting from this value the found zenith distance and the inclination of the ecliptic. For the obliquity of the ecliptic, Ptolemy knew a "verified" value found by Eratosthenes (section III .3). This value is 23°51"20". In his calculations for the obliquity of the ecliptic, Ptolemy uses a value of 23°51" and takes the zenith distance to be 2°7" (he thinks this is equal to 2 1/8 degrees). The inclination of the Moon's orbit is exactly 5°.

The correct values ​​are as follows: the latitude of Alexandria is 31 ° 13 "(section V .6), the inclination of the Moon's orbit is about 5°9", the inclination of the ecliptic at the time of Ptolemy was 23°41". So the zenith distance, which Ptolemy measured all the time, should have been obtained equal to 2 ° 23 "and not 2 ° 7". Therefore, in each such dimension there was an error of about 16", and each time with the same sign. For the method described by Ptolemy, the probable value of the standard deviation is 5".

But Ptolemy not only gets the same value every time. As written at the end of the chapter V .7 "Syntax", both he and Hipparchus by their measurements showed that the inclination of the orbit is 5 °. Ptolemy, apparently, insists on the coincidence of his results with those of Hipparchus to within a minute of an arc. At least that's how you can understand it. But suppose that Ptolemy means coincidence only after rounding to the nearest multiple of 5". Then each of his measurements falls into a predetermined area one standard deviation wide and centered 3.2 standard deviations from the correct value.

Ptolemy does not say how many times "always" is. I think at least three, and most likely more. For the sake of caution, let's assume that Ptolemy made only three measurements, and each value obtained fell into this area. But the probability that such a result was due to errors in the measurement process is less than 1 chance in 10,000,000. In other words, Ptolemy never made these measurements).

Table VIII.1

The inclination of the Moon's orbit on various dates

In vain Ptolemy alludes to multiple measurements. He did not take into account the restrictions imposed on the dates of possible observations by the conditions set. As we have already said, the ascending node of the Moon's orbit is slowly moving along the ecliptic to the west. It completes one rotation in 18 2/3 years. The ascending node after July 24, 126 coincides with the vernal equinox only on March 4, 145 [Part II ]. Both dates fall outside of what is usually considered the period of Ptolemy's astronomical activity. All the observations that Ptolemy, according to his own statements, made were made after July 24, 126 and before March 4, 145.

You also need to make sure that the longitude of the moon is 90 °. The longitude of the Moon was 90° and the node was approximately in the right place only on July 7, 126, August 3, 126, February 20, 145, and March 19, 145 [Part II ]. These days the difference between the declination of the Moon and its maximum value is much less than 1". a month earlier, this error was about 4" (an unacceptable value).

If we assume that Ptolemy could use those observations for which the error due to deviation from ideal conditions was close to 1 "(but not 4"), then we get four possible dates of observations in the summer of 126 and four dates in the winter - in the spring 145. The series of observations could include both the observations of 126 and 145.

I have already noted that various perturbations cause a change in the inclination of the moon's orbit, so Ptolemy could not get the same result every time. To table VIII .1 entered the values ​​that Ptolemy should have received for the corresponding days of observation (four in 126 and four in 145). For any possible set of observations, the values ​​differ by at least 0.25°, or 15". The method that Ptolemy describes allows you to notice such a difference). So Ptolemy's claim that he always got the same value is stronger evidence of forgery than even the probability we got above. The very possible dates for the observations are relevant to the question of Ptolemy's guilt or innocence of deceit. If Ptolemy is not guilty, then he should have instructed the hypothetical assistant to take measurements at the appropriate time, and the assistant should have deceived Ptolemy by falsifying the data. But in the next section, I will show that Ptolemy is unlikely to have wanted a measurement of the inclination of the Moon's orbit made in any of these years. If this is so, then he did not give any instructions to take measurements at all. And when Ptolemy said that the measurements always gave the same result, he knew perfectly well that the measurements had never been taken. In other words, his statement is a deliberate deception. Dates are important to us for another reason as well. Despite all the above, suppose that the measurements were still taken in 145 AD. We know that the measurement of the autumnal equinox of 132 AD is fabricated (see table V .3). And in this case, the observations were faked onfor at least 13 years. If we assume that the measurements were taken in 126, then we can say that the observations were faked for 14 years, since we know that the observations of the spring equinox and summer solstice of 140 are also fakes. In any case, the hypothetical assistant deceived Ptolemy for at least 13 years.

Analyzing the conditions for the joint work of the assistant and Ptolemy (if such an assistant existed), I came to the conclusion [Part II ] that the assistant over this period of 13 years (or even more) should have made at least 100 observations, all with a fake. Too implausible to lie for so long and on such a scale.