To the blessed memory of my teacher - the first dean of the Physics and Mathematics Faculty of Novocherkassk polytechnic institute, Head of the Department "Theoretical Mechanics" Kabelkov Alexander Nikolaevich

Introduction

August, summer is coming to an end. The people violently rushed to the seas, and it is not surprising - the season itself. And on Habré, meanwhile,. If we talk about the topic of this issue of "Modeling ...", then in it we will combine business with pleasure - we will continue the promised cycle and just a little bit with this pseudoscience for the inquisitive minds of modern youth.

And the question is not an idle one - since school years, we are used to thinking that our closest satellite in outer space, the Moon, moves around the Earth with a period of 29.5 days, especially without going into the accompanying details. In fact, our neighbor is a kind and to some extent a unique astronomical object, the movement of which around the Earth is not as simple as some of my colleagues from the near abroad might want.

So, leaving controversy aside, let's try with different sides, to the best of my competence, to consider this certainly beautiful, interesting and very indicative problem.

1. The law of universal gravitation and what conclusions can we draw from it

Discovered back in the second half of the 17th century, by Sir Isaac Newton, the law of universal gravitation says that the Moon is attracted to the Earth (and the Earth to the Moon!) With a force directed along the straight line connecting the centers of the celestial bodies under consideration, and equal in magnitude

where m 1, m 2 - masses, respectively, of the Moon and the Earth; G = 6.67e-11 m 3 / (kg * s 2) - gravitational constant; r 1,2 is the distance between the centers of the Moon and the Earth. If we take into account only this force, then, having solved the problem of the motion of the Moon as a satellite of the Earth and having learned to calculate the position of the Moon in the sky against the background of stars, we will soon be convinced, by direct measurements of the equatorial coordinates of the Moon, that in our conservatory not everything is as smooth as I would like to. And the point here is not in the law of universal gravitation (and at the early stages of the development of celestial mechanics such thoughts were expressed quite often), but in the unaccounted for disturbance of the motion of the Moon from other bodies. Which ones? We look at the sky and our gaze immediately rests on a huge plasma ball weighing as much as 1.99e30 kilograms right under our nose - the Sun. Is the moon attracted to the sun? How, with a force equal in magnitude

where m 3 is the mass of the Sun; r 1,3 is the distance from the Moon to the Sun. Let's compare this force with the previous one.

Let us take the position of the bodies in which the attraction of the Moon to the Sun will be minimal: all three bodies are on one straight line and the Earth is located between the Moon and the Sun. In this case, our formula will take the form:

where, m is the average distance from the Earth to the Moon; , m is the average distance from the Earth to the Sun. Let's substitute real parameters in this formula

This is the number! It turns out that the Moon is attracted to the Sun by a force more than twice the force of its attraction to the Earth.

Such a disturbance can no longer be ignored and it will definitely affect the final trajectory of the moon. Let's go further, taking into account the assumption that the Earth's orbit is circular with a radius of a, we will find the locus of points around the Earth, where the force of attraction of any object to the Earth is equal to the force of its attraction to the Sun. It will be a sphere with a radius

displaced along a straight line connecting the Earth and the Sun in the direction opposite to the direction to the Sun by a distance

where is the ratio of the mass of the Earth to the mass of the Sun. Substituting the numerical values ​​of the parameters, we obtain the actual dimensions of this area: R = 259300 kilometers, and l = 450 kilometers. This area is called sphere of gravity of the Earth relative to the Sun.

The known orbit of the Moon lies outside this area. That is, at any point on its trajectory, the Moon experiences a significantly greater attraction from the Sun than from the Earth.

2. Satellite or planet? Gravitational scope

This information often gives rise to controversy that the Moon is not a satellite of the Earth, but an independent planet. Solar system whose orbit is disturbed by the attraction of the nearby Earth.

Let us estimate the disturbance introduced by the Sun into the trajectory of the Moon relative to the Earth, as well as the disturbance introduced by the Earth into the trajectory of the Moon relative to the Sun, using the criterion proposed by P. Laplace. Consider three bodies: Sun (S), Earth (E), and Moon (M).
Let us assume that the orbits of the Earth relative to the Sun and the Moon relative to the Earth are circular.

Consider the motion of the moon in a geocentric inertial frame of reference. The absolute acceleration of the Moon in the heliocentric frame of reference is determined by the forces of gravity acting on it and is equal to:

On the other hand, according to the Coriolis theorem, the absolute acceleration of the Moon

where is the portable acceleration equal to the acceleration of the Earth relative to the Sun; - the acceleration of the Moon relative to the Earth. There will be no Coriolis acceleration here - the coordinate system we have chosen is moving translationally. From here we get the acceleration of the Moon relative to the Earth

Part of this acceleration, equal to, is due to the attraction of the Moon to the Earth and characterizes its unperturbed geocentric motion. Remaining part

the acceleration of the moon caused by perturbation from the sun.

If we consider the motion of the Moon in a heliocentric inertial frame of reference, then everything is much simpler, acceleration characterizes the unperturbed heliocentric motion of the Moon, and acceleration is the disturbance of this motion from the Earth.

With the parameters of the orbits of the Earth and the Moon existing in the current epoch, at each point of the trajectory of the Moon the inequality is true

what can be checked and direct computation, but I will refer so as not to unnecessarily clutter up the article.

What does inequality (1) mean? Yes, the fact that in relative terms, the effect of the perturbation of the Moon by the Sun (and very significantly) is less than the effect of the attraction of the Moon to the Earth. Conversely, the disturbance by the Earth of the geoliocentric trajectory of the Moon has a decisive influence on the nature of its motion. Influence earth gravity in this case, it is more important, which means that the Moon "belongs" to the Earth by right and is its satellite.

Another thing is interesting - by turning inequality (1) into an equation, one can find the locus of points where the effects of the perturbation of the Moon (and any other body) by the Earth and the Sun are the same. Unfortunately, this is not as simple as in the case of the gravitational sphere. Calculations show that this surface is described by a crazy order equation, but is close to an ellipsoid of revolution. All we can do without unnecessary troubles is to estimate the overall dimensions of this surface relative to the center of the Earth. By solving numerically the equation

relative to the distance from the center of the Earth to the desired surface at a sufficient number of points, we obtain a section of the sought surface by the ecliptic plane

For clarity, the geocentric orbit of the Moon and the sphere of gravity of the Earth relative to the Sun, which we found above, are shown here. The figure shows that the sphere of influence, or the sphere of gravitational action of the Earth relative to the Sun is a surface of rotation about the X axis, flattened along a straight line connecting the Earth and the Sun (along the axis of eclipses). The Moon's orbit is deep within this imaginary surface.

For practical calculations, this surface is conveniently approximated by a sphere centered at the center of the Earth and with a radius equal to

where m is the mass of the smaller celestial body; M is the mass of the larger body, in the gravitational field of which the smaller body moves; a is the distance between the centers of the bodies. In our case

This unfinished million kilometers is the theoretical limit beyond which the power of the old Earth does not extend - her influence on the trajectories of astronomical objects is so small that it can be neglected. This means that it will not work to launch the Moon in a circular orbit at a distance of 38.4 million kilometers from the Earth (as some linguists do), it is physically impossible.

This sphere, for comparison, is shown in the figure with a blue dashed line. In estimated calculations, it is assumed that a body inside a given sphere will be gravitated exclusively from the Earth. If the body is outside this sphere, we assume that the body is moving in the gravitational field of the Sun. In practical cosmonautics, a method of conjugation of conical sections is known, which makes it possible to approximately calculate the trajectory of a spacecraft using the solution of the two-body problem. In this case, all the space that the apparatus overcomes is divided into similar spheres of influence.

For example, it is now clear, in order to be able to theoretically perform maneuvers to enter a circumlunar orbit, spacecraft must fall inside the sphere of action of the Moon relative to the Earth. Its radius can be easily calculated by the formula (3) and it is equal to 66 thousand kilometers.

3. The three-body problem in the classical setting

So, consider the model problem in general setting known in celestial mechanics as the three-body problem. Consider three bodies of arbitrary mass, located arbitrarily in space and moving exclusively under the action of forces of mutual gravitational attraction

We count bodies material points... The position of the bodies will be measured in an arbitrary basis, with which the inertial reference system is associated Oxyz... The position of each of the bodies is set by the radius vector, and, respectively. Each body is acted upon by the force of gravitational attraction from the side of two other bodies, and in accordance with the third axiom of the dynamics of a point (Newton's third law)

Let us write the differential equations of motion of each point in vector form

Or, taking into account (4)

In accordance with the law of universal gravitation, the forces of interaction are directed along the vectors

Along each of these vectors, let us release the corresponding unit vector

then each of the gravitational forces is calculated by the formula

Taking all this into account, the system of equations of motion takes the form

Let us introduce the notation adopted in celestial mechanics

is the gravitational parameter of the attracting center. Then the equations of motion will take the final vector form

4. Normalization of equations to dimensionless variables

Quite a popular technique for mathematical modeling is casting differential equations and other relations describing the process to dimensionless phase coordinates and dimensionless time. Other parameters are also normalized. This allows us to consider, albeit with the use of numerical modeling, but in a sufficient general view a whole class of typical tasks. I leave the question of how justified it is in each problem to be solved, but I agree that in this case this approach is quite fair.

So, let's introduce some abstract celestial body with a gravitational parameter, such that the period of revolution of the satellite in an elliptical orbit with a semi-major axis around it is equal. All these quantities, by virtue of the laws of mechanics, are related by the relation

Let's introduce the replacement of parameters. For the position of the points of our system

where is dimensionless radius vector of the i-th points;
for gravitational parameters of bodies

where is the dimensionless gravitational parameter i-th points;
for time

where is the dimensionless time.

Now let's recalculate the accelerations of the points of the system in terms of these dimensionless parameters. Let us apply direct two-fold differentiation in time. For speeds

For acceleration

When the obtained relations are substituted into the equations of motion, everything elegantly collapses into beautiful equations:

This system of equations is still considered not integrable in analytical functions. Why is it considered and not? Because the successes of the theory of functions of a complex variable led to the fact that a general solution to the three-body problem did appear in 1912 - Karl Sundman found an algorithm for finding the coefficients for infinite series with respect to a complex parameter, which are theoretically common decision tasks of three bodies. But ... to use the Sundman series in practical calculations with the accuracy required for them requires obtaining such a number of members of these series that this task greatly exceeds the capabilities of computers even today.

Therefore, numerical integration is the only way to analyze the solution to equation (5)

5. Calculation of the initial conditions: we extract the initial data

Before starting numerical integration, one should take care of calculating the initial conditions for the problem being solved. In the problem under consideration, the search for the initial conditions turns into an independent subproblem, since system (5) gives us nine scalar equations of the second order, which, when passing to the Cauchy normal form, increases the order of the system by 2 times. That is, we need to calculate as many as 18 parameters - the initial positions and components of the initial velocity of all points of the system. Where can we get data on the position of the celestial bodies of interest to us? We live in a world where a person walked on the Moon - naturally, humanity should have information on how this very Moon moves and where it is.

That is, you say, you, dude, offer us to take thick astronomical reference books from the shelves, blow off the dust ... You didn't guess! I propose to go for this data to those who actually walked on the moon, to NASA, namely to the Jet Propulsion Laboratory, Pasadena, California. Here is the JPL Horizonts web interface.

Here, after spending a little time studying the interface, we will get all the data we need. Let's choose a date, for example, yes we don't care, but let it be July 27, 2018 UT 20:21. Just at this moment, a complete phase was observed lunar eclipse... The program will give us a huge footcloth

Full conclusion for the ephemeris of the Moon on 07/27/2018 20:21 (the origin is at the center of the Earth)

************************************************* ****************************** Revised: Jul 31, 2013 Moon / (Earth) 301 GEOPHYSICAL DATA (updated 2018-Aug-13 ): Vol. Mean Radius, km = 1737.53 + -0.03 Mass, x10 ^ 22 kg = 7.349 Radius (gravity), km = 1738.0 Surface emissivity = 0.92 Radius (IAU), km = 1737.4 GM, km ^ 3 / s ^ 2 = 4902.800066 Density, g / cm ^ 3 = 3.3437 GM 1-sigma, km ^ 3 / s ^ 2 = + -0.0001 V (1.0) = +0.21 Surface accel., m / s ^ 2 = 1.62 Earth / Moon mass ratio = 81.3005690769 Farside crust. thick. = ~ 80 - 90 km Mean crustal density = 2.97 + -. 07 g / cm ^ 3 Nearside crust. thick. = 58 + -8 km Heat flow, Apollo 15 = 3.1 + -. 6 mW / m ^ 2 k2 = 0.024059 Heat flow, Apollo 17 = 2.2 + -. 5 mW / m ^ 2 Rot. Rate, rad / s = 0.0000026617 Geometric Albedo = 0.12 Mean angular diameter = 31 "05.2" Orbit period = 27.321582 d Obliquity to orbit = 6.67 deg Eccentricity = 0.05490 Semi-major axis, a = 384400 km Inclination = 5.145 deg Mean motion, rad / s = 2.6616995x10 ^ -6 Nodal period = 6798.38 d Apsidal period = 3231.50 d Mom. of inertia C / MR ^ 2 = 0.393142 beta (CA / B), x10 ^ -4 = 6.310213 gamma (BA / C), x10 ^ -4 = 2.277317 Perihelion Aphelion Mean Solar Constant (W / m ^ 2) 1414 + - 7 1323 + -7 1368 + -7 Maximum Planetary IR (W / m ^ 2) 1314 1226 1268 Minimum Planetary IR (W / m ^ 2) 5.2 5.2 5.2 *************** ************************************************* ************** *********************************** ******************************************* Ephemeris / WWW_USER Wed Aug 15 20 : 45:05 2018 Pasadena, USA / Horizons *************************************** ************************************** Target body name: Moon (301) (source: DE431mx) Center body name: Earth (399) (source: DE431mx) Center-site name: BODY CENTER ************************** ************************************************* * Start time: AD 2018-Jul-27 20: 21: 00.0003 TDB Stop time: A.D. 2018-Jul-28 20: 21: 00.0003 TDB Step-size: 0 steps ******************************** ********************************************* Center geodetic: 0.00000000 , 0.00000000,0.0000000 (E-lon (deg), Lat (deg), Alt (km)) Center cylindric: 0.00000000,0.00000000,0.0000000 (E-lon (deg), Dxy (km), Dz (km)) Center radii : 6378.1 x 6378.1 x 6356.8 km (Equator, meridian, pole) Output units: AU-D Output type: GEOMETRIC cartesian states Output format: 3 (position, velocity, LT, range, range-rate) Reference frame: ICRF / J2000. 0 Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch ************************************** **************************************** JDTDB XYZ VX VY VZ LT RG RR ** ************************************************* **************************** $$ SOE 2458327. 347916670 = A.D. 2018-Jul-27 20: 21: 00.0003 TDB X = 1.537109094089627E-03 Y = -2.237488447258137E-03 Z = 5.112037386426180E-06 VX = 4.593816208618667E-04 VY = 3.187527302531735E-04 VZ11 = -576.183E-03 = 1.567825598846416E-05 RG = 2.714605874095336E-03 RR = -2.707898607099066E-06 $$ EOE **************************** ************************************************* Coordinate system description: Ecliptic and Mean Equinox of Reference Epoch Reference epoch: J2000.0 XY-plane: plane of the Earth "s orbit at the reference epoch Note: obliquity of 84381.448 arcseconds wrt ICRF equator (IAU76) X-axis: out along ascending node of instantaneous plane of the Earth "s orbit and the Earth" s mean equator at the reference epoch Z-axis: perpendicular to the xy-plane in the directional (+ or -) sense of Earth "s north pole at the reference epoch. Symbol meaning: JDTDB Julian Day Number, Barycentric Dynamical Time X X-component of position vector (au) Y Y-component of position vector (au) Z-component of position vector (au) VX X-component of velocity vector (au / day) VY Y-component of velocity vector (au / day) VZ Z-component of velocity vector (au / day) LT One-way down-leg Newtonian light-time (day) RG Range; distance from coordinate center (au) RR Range-rate; radial velocity wrt coord. center (au / day) Geometric states / elements have no aberrations applied. Computations by ... Solar System Dynamics Group, Horizons On-Line Ephemeris System 4800 Oak Grove Drive, Jet Propulsion Laboratory Pasadena, CA 91109 USA Information: http://ssd.jpl.nasa.gov/ Connect: telnet: // ssd .jpl.nasa.gov: 6775 (via browser) http://ssd.jpl.nasa.gov/?horizons telnet ssd.jpl.nasa.gov 6775 (via command-line) Author: [email protected] *******************************************************************************

Brrr, what is it? Don't panic, for someone who taught astronomy well at school, there is nothing to be afraid of mechanics and mathematics. So, the most important is the final sought coordinates and components of the Moon's velocity.

$$ SOE 2458327.347916670 = A.D. 2018-Jul-27 20: 21: 00.0003 TDB X = 1.537109094089627E-03 Y = -2.237488447258137E-03 Z = 5.112037386426180E-06 VX = 4.593816208618667E-04 VY = 3.187527302531735E-04 VZ11 = -576.183E-03 = 1.567825598846416E-05 RG = 2.714605874095336E-03 RR = -2.707898607099066E-06 $$ EOE
Yes, yes, yes, they are Cartesian! If we carefully read the whole footcloth, then we learn that the origin of this coordinate system coincides with the center of the Earth. The XY plane lies in the plane of the Earth's orbit (the plane of the ecliptic) at epoch J2000. The X axis is directed along the line of intersection of the Earth's equatorial plane and the ecliptic at the vernal equinox. The Z-axis looks in the direction north pole Earth is perpendicular to the plane of the ecliptic. Well, the Y-axis complements all this happiness to the right triplet of vectors. By default, the units of measurement of coordinates: astronomical units (the clever girls from NASA also give the value of the autronomical unit in kilometers). Units of measurement of speed: astronomical units per day, day is taken equal to 86400 seconds. Full stuffing!

We can get similar information for the Earth.

Full output of the Earth's ephemeris on 07/27/2018 20:21 (origin at the center of mass of the solar system)

************************************************* ****************************** Revised: Jul 31, 2013 Earth 399 GEOPHYSICAL PROPERTIES (revised Aug 13, 2018): Vol. Mean Radius (km) = 6371.01 + -0.02 Mass x10 ^ 24 (kg) = 5.97219 + -0.0006 Equ. radius, km = 6378.137 Mass layers: Polar axis, km = 6356.752 Atmos = 5.1 x 10 ^ 18 kg Flattening = 1 / 298.257223563 oceans = 1.4 x 10 ^ 21 kg Density, g / cm ^ 3 = 5.51 crust = 2.6 x 10 ^ 22 kg J2 (IERS 2010) = 0.00108262545 mantle = 4.043 x 10 ^ 24 kg g_p, m / s ^ 2 (polar) = 9.8321863685 outer core = 1.835 x 10 ^ 24 kg g_e, m / s ^ 2 (equatorial) = 9.7803267715 inner core = 9.675 x 10 ^ 22 kg g_o, m / s ^ 2 = 9.82022 Fluid core rad = 3480 km GM, km ^ 3 / s ^ 2 = 398600.435436 Inner core rad = 1215 km GM 1-sigma, km ^ 3 / s ^ 2 = 0.0014 Escape velocity = 11.186 km / s Rot. Rate (rad / s) = 0.00007292115 Surface Area: Mean sidereal day, hr = 23.9344695944 land = 1.48 x 10 ^ 8 km Mean solar day 2000.0, s = 86400.002 sea = 3.62 x 10 ^ 8 km Mean solar day 1820.0, s = 86400.0 Moment of inertia = 0.3308 Love no., K2 = 0.299 Mean Temperature, K = 270 Atm. pressure = 1.0 bar Vis. mag. V (1,0) = -3.86 Volume, km ^ 3 = 1.08321 x 10 ^ 12 Geometric Albedo = 0.367 Magnetic moment = 0.61 gauss Rp ^ 3 Solar Constant (W / m ^ 2) = 1367.6 (mean), 1414 (perihelion ), 1322 (aphelion) ORBIT CHARACTERISTICS: Obliquity to orbit, deg = 23.4392911 Sidereal orb period = 1.0000174 y Orbital speed, km / s = 29.79 Sidereal orb period = 365.25636 d Mean daily motion, deg / d = 0.9856474 Hill "s sphere radius = 234.9 ********************************************** *********************************************** ************************************************* ********** Ephemeris / WWW_USER Wed Aug 15 21:16:21 2018 Pasadena, USA / Horizons *********************** ************************************************* ****** Target body name: Earth (399) (source: DE431mx) Center body name: Solar System Barycenter (0) (source: DE431mx) Center-site name: BODY CENTER ********* ************************************************* ********************* Start time: AD 2018-Jul-27 20: 21: 00.0003 TDB Stop time: A .D. 2018-Jul-28 20: 21: 00.0003 TDB Step-size: 0 steps ******************************** ********************************************* Center geodetic: 0.00000000 , 0.00000000,0.0000000 (E-lon (deg), Lat (deg), Alt (km)) Center cylindric: 0.00000000,0.00000000,0.0000000 (E-lon (deg), Dxy (km), Dz (km)) Center radii : (undefined) Output units: AU-D Output type: GEOMETRIC cartesian states Output format: 3 (position, velocity, LT, range, range-rate) Reference frame: ICRF / J2000.0 Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch ************************************************ ****************************** JDTDB XYZ VX VY VZ LT RG RR ************ ************************************************* ***************** $$ SOE 2458327.347916670 = AD 2018-Jul-27 20: 21: 00.0003 TDB X = 5.755663665315949E-01 Y = -8.298818915224488E-01 Z = -5.366994499016168E-05 VX = 1.388633512282171E-02 VY = 9.678934168415631E-0330 VZ = 3.429188 = 5.832932117417083E-03 RG = 1.009940888883960E + 00 RR = -3.947237246302148E-05 $$ EOE **************************** ************************************************* Coordinate system description: Ecliptic and Mean Equinox of Reference Epoch Reference epoch: J2000. 0 XY-plane: plane of the Earth "s orbit at the reference epoch Note: obliquity of 84381.448 arcseconds wrt ICRF equator (IAU76) X-axis: out along ascending node of instantaneous plane of the Earth" s orbit and the Earth "s mean equator at the reference epoch Z-axis: perpendicular to the xy-plane in the directional (+ or -) sense of Earth "s north pole at the reference epoch. Symbol meaning: JDTDB Julian Day Number, Barycentric Dynamical Time X X-component of position vector (au) Y Y-component of position vector (au) Z Z-component of position vector (au) VX X-component of velocity vector (au / day) VY Y-component of velocity vector (au / day) VZ Z-component of velocity vector (au / day) LT One-way down-leg Newtonian light-time (day) RG Range; distance from coordinate center (au) RR Range-rate; radial velocity wrt coord. center (au / day) Geometric states / elements have no aberrations applied. Computations by ... Solar System Dynamics Group, Horizons On-Line Ephemeris System 4800 Oak Grove Drive, Jet Propulsion Laboratory Pasadena, CA 91109 USA Information: http://ssd.jpl.nasa.gov/ Connect: telnet: // ssd .jpl.nasa.gov: 6775 (via browser) http://ssd.jpl.nasa.gov/?horizons telnet ssd.jpl.nasa.gov 6775 (via command-line) Author: [email protected] *******************************************************************************

Here, the barycenter (center of mass) of the solar system is chosen as the origin of coordinates. Data of interest to us

$$ SOE 2458327.347916670 = A.D. 2018-Jul-27 20: 21: 00.0003 TDB X = 5.755663665315949E-01 Y = -8.298818915224488E-01 Z = -5.366994499016168E-05 VX = 1.388633512282171E-02 VY = 9.678934168415631E-0330 VZ = 3.429188 = 5.832932117417083E-03 RG = 1.009940888883960E + 00 RR = -3.947237246302148E-05 $$ EOE
For the Moon, we need coordinates and speed relative to the barycenter of the Solar System, we can calculate them, or we can ask NASA to give us such data

Full output of the ephemeris of the Moon on 07/27/2018 20:21 (origin at the center of mass of the solar system)

************************************************* ****************************** Revised: Jul 31, 2013 Moon / (Earth) 301 GEOPHYSICAL DATA (updated 2018-Aug-13 ): Vol. Mean Radius, km = 1737.53 + -0.03 Mass, x10 ^ 22 kg = 7.349 Radius (gravity), km = 1738.0 Surface emissivity = 0.92 Radius (IAU), km = 1737.4 GM, km ^ 3 / s ^ 2 = 4902.800066 Density, g / cm ^ 3 = 3.3437 GM 1-sigma, km ^ 3 / s ^ 2 = + -0.0001 V (1.0) = +0.21 Surface accel., m / s ^ 2 = 1.62 Earth / Moon mass ratio = 81.3005690769 Farside crust. thick. = ~ 80 - 90 km Mean crustal density = 2.97 + -. 07 g / cm ^ 3 Nearside crust. thick. = 58 + -8 km Heat flow, Apollo 15 = 3.1 + -. 6 mW / m ^ 2 k2 = 0.024059 Heat flow, Apollo 17 = 2.2 + -. 5 mW / m ^ 2 Rot. Rate, rad / s = 0.0000026617 Geometric Albedo = 0.12 Mean angular diameter = 31 "05.2" Orbit period = 27.321582 d Obliquity to orbit = 6.67 deg Eccentricity = 0.05490 Semi-major axis, a = 384400 km Inclination = 5.145 deg Mean motion, rad / s = 2.6616995x10 ^ -6 Nodal period = 6798.38 d Apsidal period = 3231.50 d Mom. of inertia C / MR ^ 2 = 0.393142 beta (CA / B), x10 ^ -4 = 6.310213 gamma (BA / C), x10 ^ -4 = 2.277317 Perihelion Aphelion Mean Solar Constant (W / m ^ 2) 1414 + - 7 1323 + -7 1368 + -7 Maximum Planetary IR (W / m ^ 2) 1314 1226 1268 Minimum Planetary IR (W / m ^ 2) 5.2 5.2 5.2 *************** ************************************************* ************** *********************************** ******************************************* Ephemeris / WWW_USER Wed Aug 15 21 : 19:24 2018 Pasadena, USA / Horizons *************************************** *************************************** Target body name: Moon (301) (source: DE431mx) Center body name: Solar System Barycenter (0) (source: DE431mx) Center-site name: BODY CENTER ************************* ************************************************* *** Start time: AD 2018-Jul-27 20: 21: 00.0003 TDB Stop time: A.D. 2018-Jul-28 20: 21: 00.0003 TDB Step-size: 0 steps ******************************** ********************************************* Center geodetic: 0.00000000 , 0.00000000,0.0000000 (E-lon (deg), Lat (deg), Alt (km)) Center cylindric: 0.00000000,0.00000000,0.0000000 (E-lon (deg), Dxy (km), Dz (km)) Center radii : (undefined) Output units: AU-D Output type: GEOMETRIC cartesian states Output format: 3 (position, velocity, LT, range, range-rate) Reference frame: ICRF / J2000.0 Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch ************************************************ ****************************** JDTDB XYZ VX VY VZ LT RG RR ************ ************************************************* ***************** $$ SOE 2458327. 347916670 = A.D. 2018-Jul-27 20: 21: 00.0003 TDB X = 5.771034756256845E-01 Y = -8.321193799697072E-01 Z = -4.855790760378579E-05 VX = 1.434571674368357E-02 VY = 9.997686898668805E-0340EZ = -5.1370EZ LT = 5.848610189172283E-03 RG = 1.012655462859054E + 00 RR = -3.979984423450087E-05 $$ EOE **************************** ************************************************* * Coordinate system description: Ecliptic and Mean Equinox of Reference Epoch Reference epoch: J2000.0 XY-plane: plane of the Earth "s orbit at the reference epoch Note: obliquity of 84381.448 arcseconds wrt ICRF equator (IAU76) X-axis: out along ascending node of instantaneous plane of the Earth "s orbit and the Earth" s mean equator at the reference epoch Z-axis: perpendicular to the xy-plane in the directional (+ or -) sense of Earth "s north pole at the reference epoch. Symbol meaning: JDTDB Julian Day Number, Barycentric Dynamical Time X X-component of position vector (au) Y Y-component of position vector (au) Z-component of position vector (au) VX X-component of velocity vector (au / day) VY Y-component of velocity vector (au / day) VZ Z-component of velocity vector (au / day) LT One-way down-leg Newtonian light-time (day) RG Range; distance from coordinate center (au) RR Range-rate; radial velocity wrt coord. center (au / day) Geometric states / elements have no aberrations applied. Computations by ... Solar System Dynamics Group, Horizons On-Line Ephemeris System 4800 Oak Grove Drive, Jet Propulsion Laboratory Pasadena, CA 91109 USA Information: http://ssd.jpl.nasa.gov/ Connect: telnet: // ssd .jpl.nasa.gov: 6775 (via browser) http://ssd.jpl.nasa.gov/?horizons telnet ssd.jpl.nasa.gov 6775 (via command-line) Author: [email protected].gov ********************************************** *******************************

$$ SOE 2458327.347916670 = A.D. 2018-Jul-27 20: 21: 00.0003 TDB X = 5.771034756256845E-01 Y = -8.321193799697072E-01 Z = -4.855790760378579E-05 VX = 1.434571674368357E-02 VY = 9.997686898668805E-0340EZ = -5.1370EZ LT = 5.848610189172283E-03 RG = 1.012655462859054E + 00 RR = -3.979984423450087E-05 $$ EOE
Wonderful! Now you need to lightly process the received data with a file.

6.38 parrots and one parrot wing

First, let's define the scale, because our equations of motion (5) are written in dimensionless form. The data provided by NASA themselves tell us that one astronomical unit should be taken as the coordinate scale. Accordingly, as a reference body, to which we will normalize the masses of other bodies, we will take the Sun, and as a time scale - the period of the Earth's revolution around the Sun.

All this is of course very good, but we did not set the initial conditions for the Sun. "Why?" a linguist would ask me. And I would answer that the Sun is by no means stationary, but also revolves in its orbit around the center of mass of the solar system. This can be seen by looking at NASA data for the Sun

$$ SOE 2458327.347916670 = A.D. 2018-Jul-27 20: 21: 00.0003 TDB X = 6.520050993518213E + 04 Y = 1.049687363172734E + 06 Z = -1.304404963058507E + 04 VX = -1.265326939350981E-02 VY = 5.853475278436883E-03 VZ = 3.1667E-4533 = 3.508397935601254E + 00 RG = 1.051791240756026E + 06 RR = 5.053500842402456E-03 $$ EOE
Looking at the RG parameter, we will see that the Sun revolves around the barycenter of the Solar System, and as of July 27, 2018, the center of the star is at a distance of a million kilometers from it. The radius of the Sun, for reference, is 696 thousand kilometers. That is, the barycenter of the solar system lies half a million kilometers from the surface of the star. Why? Yes, because all other bodies interacting with the Sun also impart acceleration to it, mainly, of course, heavy Jupiter. Accordingly, the Sun also has its own orbit.

Of course, we can choose these data as initial conditions, but no - we are solving the model problem of three bodies, and Jupiter and other characters are not included in it. So, to the detriment of realism, knowing the position and speed of the Earth and the Moon, we will recalculate the initial conditions for the Sun so that the center of mass of the Sun - Earth - Moon system is at the origin. For the center of mass of our mechanical system the equation is valid

We place the center of mass at the origin of coordinates, that is, let us set, then

where

Let us turn to dimensionless coordinates and parameters by choosing

Differentiating (6) with respect to time and passing to dimensionless time, we also obtain the relation for the velocities

where

Now let's write a program that will form the initial conditions in the "parrots" we have chosen. What are we going to write on? Python, of course! After all, as you know, this is the most best language for mathematical modeling.

However, if we get away from sarcasm, then we will really try python for this purpose, and why not? I'll make sure to link to all the code on my Github profile.

Calculation of the initial conditions for the Moon - Earth - Sun system

# # Initial data of the problem # # Gravitational constant G = 6.67e-11 # Masses of bodies (Moon, Earth, Sun) m = # Calculate gravitational parameters of bodies mu = print ("Gravitational parameters of bodies") for i, mass in enumerate (m ): mu.append (G * mass) print ("mu [" + str (i) + "] =" + str (mu [i])) # Normalize gravitational parameters to the Sun kappa = print ("Normalized gravitational parameters" ) for i, gp in enumerate (mu): kappa.append (gp / mu) print ("xi [" + str (i) + "] =" + str (kappa [i])) print ("\ n" ) # Astronomical unit a = 1.495978707e11 import math # Dimensionless time scale, c T = 2 * math.pi * a * math.sqrt (a / mu) print ("Time scale T =" + str (T) + "\ n ") # NASA coordinates for the Moon xL = 5.771034756256845E-01 yL = -8.321193799697072E-01 zL = -4.855790760378579E-05 import numpy as np xi_10 = np.array () print (" Initial position of the Moon, au : "+ str (xi_10)) # NASA coordinates for Earth xE = 5.755663665315949E-01 yE = -8.298818915224488E-01 zE = -5.366994499016168E-05 xi_20 = np.array () print ("Initial position of the Earth, au:" + str (xi_20)) # Calculate starting position Of the Sun, assuming that the origin is at the center of mass of the entire system xi_30 = - kappa * xi_10 - kappa * xi_20 print ("Initial position of the Sun, au:" + str (xi_30)) # Enter constants for calculating dimensionless velocities Td = 86400.0 u = math.sqrt (mu / a) / 2 / math.pi print ("\ n") # starting speed Moon vxL = 1.434571674368357E-02 vyL = 9.997686898668805E-03 vzL = -5.149408819470315E-05 vL0 = np.array () uL0 = np.array () for i, v in enumerate (vL0): vL0 [i] = vL0 * a / Td uL0 [i] = vL0 [i] / u print ("The initial velocity of the Moon, m / s:" + str (vL0)) print ("- // - dimensionless:" + str (uL0)) # The initial velocity of the Earth vxE = 1.388633512282171E-02 vyE = 9.678934168415631E-03 vzE = 3.429889230737491E-07 vE0 = np.array () uE0 = np.array () for i, v in enumerate (vE0): vE0 [i] = v * a / Td uE0 [i] = vE0 [i] / u print ("The initial velocity of the Earth, m / s:" + str (vE0)) print ("- // - dimensionless:" + str (uE0)) # Initial speed of the Sun vS0 = - kappa * vL0 - kappa * vE0 uS0 = - kappa * uL0 - kappa * uE0 print ("Initial speed of the Sun, m / s:" + str (vS0)) print ("- // - dimensionless : "+ str (uS0))

Exhaust program

Gravitational parameters of bodies mu = 4901783000000.0 mu = 386326400000000.0 mu = 1.326663e + 20 Normalized gravitational parameters xi = 3.6948215183509304e-08 xi = 2.912016088486677e-06 xi = 1.0 Time scale T = 31563683.35432710e. -01 -8.32119380e-01 -4.85579076e-05] Initial position of the Earth, AU: [5.75566367e-01 -8.29881892e-01 -5.36699450e-05] Initial position of the Sun, AU: [-1.69738146 e-06 2.44737475e-06 1.58081871e-10] Initial velocity of the Moon, m / s: - // - dimensionless: [5.24078311 3.65235907 -0.01881184] Initial velocity of the Earth, m / s: - // - dimensionless: Initial velocity of the Sun, m / s: [-7.09330769e-02 -4.94410725e-02 1.56493465e-06] - // - dimensionless: [-1.49661835e-05 -1.04315813e-05 3.30185861e-10]

7. Integration of equations of motion and analysis of results

Actually, the integration itself is reduced to a more or less standard SciPy procedure for preparing a system of equations: transforming the ODE system to the Cauchy form and calling the corresponding solver functions. To transform the system to the Cauchy form, recall that

Then, introducing the state vector of the system

reduce (7) and (5) to one vector equation

To integrate (8) with the available initial conditions, we will write a little, quite a bit of code

Integration of the equations of motion in the three-body problem

# # Computing generalized acceleration vectors # def calcAccels (xi): k = 4 * math.pi ** 2 xi12 = xi - xi xi13 = xi - xi xi23 = xi - xi s12 = math.sqrt (np.dot (xi12, xi12)) s13 = math.sqrt (np.dot (xi13, xi13)) s23 = math.sqrt (np.dot (xi23, xi23)) a1 = (k * kappa / s12 ** 3) * xi12 + (k * kappa / s13 ** 3) * xi13 a2 = - (k * kappa / s12 ** 3) * xi12 + (k * kappa / s23 ** 3) * xi23 a3 = - (k * kappa / s13 ** 3 ) * xi13 - (k * kappa / s23 ** 3) * xi23 return # # System of equations in Cauchy normal form # def f (t, y): n = 9 dydt = np.zeros ((2 * n)) for i in range (0, n): dydt [i] = y xi1 = np.array (y) xi2 = np.array (y) xi3 = np.array (y) accels = calcAccels () i = n for accel in accels: for a in accel: dydt [i] = ai = i + 1 return dydt # Initial conditions of the Cauchy problem y0 = # # We integrate the equations of motion # # Initial time t_begin = 0 # End time t_end = 30.7 * Td / T; # The number of trajectory points we are interested in N_plots = 1000 # Time step between points step = (t_end - t_begin) / N_plots import scipy.integrate as spi solver = spi.ode (f) solver.set_integrator ("vode", nsteps = 50000, method = "bdf", max_step = 1e-6, rtol = 1e-12) solver.set_initial_value (y0, t_begin) ts = ys = i = 0 while solver.successful () and solver.t<= t_end: solver.integrate(solver.t + step) ts.append(solver.t) ys.append(solver.y) print(ts[i], ys[i]) i = i + 1

Let's see what we got. The result is the spatial trajectory of the Moon for the first 29 days from our chosen starting point

as well as its projection into the plane of the ecliptic.

“Hey, uncle, what are you selling us ?! It's a circle! "

Firstly, it is not a circle - the projection of the trajectory is noticeably displaced from the origin to the right and down. Secondly, don't you notice anything? No, well, really?

I promise to prepare a justification (based on the analysis of computation errors and NASA data) that the resulting displacement of the trajectory is not a consequence of integration errors. For now, I suggest the reader to take my word for it - this shift is a consequence of the solar disturbance of the lunar trajectory. Let's spin one more turn

How! Moreover, pay attention to the fact that, based on the initial data of the problem, the Sun is located exactly in the direction where the trajectory of the Moon is shifted at each revolution. Yes, this impudent Sun is stealing our beloved satellite from us! Oh, this is the Sun!

It can be concluded that solar gravity affects the Moon's orbit quite significantly - the old woman does not walk the sky twice in the same way. The picture for six months of movement allows (at least qualitatively) to be convinced of this (the picture is clickable)

Interesting? Still would. Astronomy in general is an amusing science.

P.S

At the university where I studied and worked for almost seven years - the Novocherkassk Polytechnic Institute - a zonal Olympiad for students in theoretical mechanics of the universities of the North Caucasus was held annually. We have hosted the All-Russian Olympiad three times. At the opening, our main "Olympian", Professor AI Kondratenko, always said: "Academician Krylov called mechanics the poetry of the exact sciences."

I love mechanics. All the good things that I have achieved in my life and career have happened thanks to this science and my wonderful teachers. I respect mechanics.

Therefore, I will never allow anyone to mock this science and brazenly exploit it for their own purposes, whether he is at least three times a doctor of sciences and four times a linguist, and has developed at least a million curricula. I sincerely believe that writing articles on a popular public resource should provide for their thorough proofreading, normal design (LaTeX formulas are not a whim of the resource developers!) And the absence of errors that lead to results that violate the laws of nature. The latter is generally a must have.

I often tell my students: "The computer frees your hands, but that does not mean that you also need to turn off your brain."

I urge you, my dear readers, to appreciate and respect the mechanics. I will gladly answer any questions, and, as promised, I post the source code of an example of solving the three-body problem in Python in my Github profile.

Thank you for the attention!

student

Name

If the velocity vector of the body is given by the formula shown in the figure, where A and B are some constants, i and j are the unit vectors of the coordinate axes, then the trajectory of the body ...

Straight line.

The ball was thrown into the wall at a speed, the horizontal and vertical components of which are 6 m / s and 8 m / s, respectively. Distance from the wall to the point of throwing L = 4 m. At what point of the trajectory will the ball be located when it hits the wall?

student

Name

student

Name

On the rise.

At what motion of a material point is the normal acceleration negative?

Such movement is impossible.

student

Name

A material point rotates in a circle around a fixed axis. For which dependence of the angular velocity on time w (t) when calculating the angle of rotation is the formula Ф = wt applicable.

The wheel of the machine has a radius R and rotates with an angular velocity w. What time is t

will the car need to travel the distance L without slipping? Please enter the correct formula number. Answer: 2

Frame name

How will the magnitude and direction of the cross product of two non-collinear vectors change when each factor is doubled and their directions are reversed?

	Student response		The module will quadruple, the direction
			Will not change.
	Response time		14.10.2011 15:30:20
	System Assessment
	Frame name

The projection of the acceleration of the material point changes according to the graph shown. The initial velocity is zero. At what moments in time does the speed of a material point change direction?

Student response

Name

student

Name

How can the acceleration vector of a body moving along the depicted trajectory be directed when passing the point P?

At any angle towards the concavity.

The angle of rotation of the flywheel changes according to the law Ф (t) = А · t · t · t, where A = 0.5 rad / s3, t is the time in seconds. To what angular velocity (in rad / s) will the flywheel accelerate in the first second from the moment it starts moving? Answer: 1.5

Name frame205

Name

student

A rigid body rotates with an angular velocity w around a fixed axis. Specify the correct formula for calculating the linear velocity of a point on a body located at a distance r from the axis of rotation. Answer: 2

The Moon revolves around the Earth in a circular orbit so that one side of it is constantly facing the Earth. What is the trajectory of the Earth's center relative to the astronaut on the Moon?

Line segment.

Circle.

The answer depends on the location of the astronaut on the moon.

04.10.2011 14:06:11

Name frame287

Using the above graph of the speed of a moving person, determine how many meters he walked between two stops. Answer: 30

Name frame288

The body is thrown at an angle to the horizon. Air resistance can be neglected at which point of the trajectory the speed changes in magnitude with the maximum speed. List all correct answers.

Student E's answer A

Name frame289

student

Name

The handwheel rotates as shown in the figure. The angular acceleration vector B is directed perpendicular to the plane of the figure to us and is constant in magnitude. How is the angular velocity vector w directed and what is the nature of the flywheel rotation?

The vector w is directed away from us, the flywheel is decelerated.

A material point moves in a circle, and its angular velocity w depends on the time t as shown in the figure. How do its normal An and

student

Name

tangential acceleration At?

An increases, At does not change.

The acceleration of the body has a constant value A = 0.2 m / s2 and is directed along the X-axis. The initial velocity is equal in magnitude to V0 = 1 m / s and is directed along the Y-axis. Find the tangent of the angle between the body's velocity vector and the Y-axis at time t = 10 s. Answer: 2

Name frame257

student

Name

Determine the projection of the displacement Sх for the entire time of movement from the given graph of the projection of speed.

The point moves evenly along the path shown in the figure. At what point (what points) is the tangential acceleration equal to 0?

		All along the trajectory.
	student
	Name

The body rotates around a fixed axis passing through point O perpendicular to the plane of the drawing. The angle of rotation depends on time: Ф (t) = Ф0 sin (Аt), where А = 1rad / s, Ф0 is a positive constant. How does the angular velocity of point A behave at time t = 1 s?

Student's answer Decreases.

Frame260 name

A disc of radius R spins up with constant angular acceleration ε. Specify the formula for calculating the tangential acceleration of point A on the rim of the disk at an angular velocity w. Answer: 5

Frame225 name

The wheel rolls along the road without slipping at an increasing speed. Select the correct formula to calculate the angular acceleration of the wheel if the speed of the wheel center increases in proportion to time. Answer: 4

	Frame name
		If the coordinates of the body change with time t along
		equations x = A t, y = B t t, where A and B are constants, then
		body trajectory ...
	Student response	Parabola.
Name

Original taken from ss69100 in Lunar Anomalies or Fake Physics?

And even in seemingly long-established theories, there are glaring contradictions and obvious errors that are simply hushed up. Let me give you a simple example.

Official physics, which is taught in educational institutions, is very proud of the fact that it knows the relationships between different physical quantities in the form of formulas that are supposedly reliably supported experimentally. As they say, and we stand ...

In particular, all reference books and textbooks state that between two bodies with masses ( m) and ( M), there is an attractive force ( F), which is directly proportional to the product of these masses and inversely proportional to the square of the distance ( R) between them. This ratio is usually presented in the form of the formula "The law of universal gravitation":

where is the gravitational constant equal to approximately 6.6725 × 10 −11 m³ / (kg · s²).

Let's use this formula to calculate what is the force of attraction between the Earth and the Moon, as well as between the Moon and the Sun. To do this, we need to substitute the corresponding values ​​from the dictionaries into this formula:

Moon mass - 7.3477 × 10 22 kg

Mass of the Sun - 1.9891 × 10 30 kg

Earth mass - 5.9737 × 10 24 kg

Distance between the Earth and the Moon = 380,000,000 m

Distance between the Moon and the Sun = 149,000,000,000 m

The force of gravity between the Earth and the Moon = 6.6725 × 10 -11 x 7.3477 × 10 22 x 5.9737 × 10 24 / 380,000,000 2 = 2,028 × 10 20 H

The force of attraction between the Moon and the Sun = 6.6725 × 10 -11 x 7.3477 10 22 x 1.9891 10 30/149000000000 2 = 4.39 × 10 20 H

It turns out that the force of attraction of the moon to the sun is more than twice (!) more than the force of attraction of the moon to the earth! Why, then, does the moon fly around the earth and not around the sun? Where is the agreement between theory and experimental data?

If you can't believe your eyes, please grab a calculator, open the reference books and see for yourself.

According to the formula of "universal gravitation" for this system of three bodies, as soon as the Moon is between the Earth and the Sun, it should leave a circular orbit around the Earth, turning into an independent planet with orbital parameters close to the Earth. However, the Moon stubbornly "does not notice" the Sun, as if it does not exist at all.

First of all, let's ask ourselves what could be wrong with this formula? There are few options here.

From the point of view of mathematics, this formula may be correct, but then the values ​​of its parameters are incorrect.

For example, modern science can be severely mistaken in determining distances in space based on false ideas about the nature and speed of propagation of light; or it is wrong to estimate the masses of celestial bodies, using all the same purely speculative conclusions Kepler or Laplace, expressed as ratios of the sizes of orbits, velocities and masses of celestial bodies; or not at all to understand the nature of the mass of a macroscopic body, as all physics textbooks are very frank about, postulating this property of material objects, regardless of its location and without delving into the reasons for its occurrence.

Also, official science may be mistaken in the reason for the existence and principles of action of the gravitational force, which is most likely. For example, if the masses do not have an attractive effect (which, by the way, there are thousands of visual evidences, only they are hushed up), then this "formula of universal gravitation" simply reflects some idea expressed by Isaac Newton, which turned out to be false.

You can make a mistake in a thousand different ways, but the truth is one. And its official physics deliberately hides, otherwise how to explain the defense of such an absurd formula?

The first and an obvious consequence of the fact that the "formula of universal gravitation" does not work is the fact that the Earth has no dynamic response to the Moon... Simply put, two such large and close celestial bodies, one of which is only four times smaller in diameter than the other, should (according to the views of modern physics) revolve around a common center of mass - the so-called. barycenter... However, the Earth rotates strictly on its axis, and even the ebb and flow in the seas and oceans have absolutely nothing to do with the position of the Moon in the sky.

The Moon is associated with a number of completely outrageous facts of inconsistencies with the established views of classical physics, which in the literature and the Internet bashfully are called "Lunar anomalies".

The most obvious anomaly is the exact coincidence of the period of the Moon's revolution around the Earth and around its axis, which is why it always faces the Earth with one side. There are many reasons for these periods to become more and more out of sync on each orbit of the Moon around the Earth.

For example, no one would argue that the Earth and the Moon are two ideal balls with a uniform distribution of mass inside. From the point of view of official physics, it is quite obvious that the motion of the Moon should be significantly influenced not only by the relative position of the Earth, the Moon and the Sun, but even by the flights of Mars and Venus during periods of the closest approach of their orbits to the Earth's. The experience of space flights in near-earth orbit shows that stabilization similar to the lunar can be achieved only if steer constantly orientation micromotors. But how and how does the Moon steer? And most importantly - for what?

This "anomaly" looks even more discouraging against the background of the little-known fact that the mainstream science has not yet come up with an acceptable explanation. trajectories, along which the moon moves around the earth. Orbit of the moon by no means circular or even elliptical. Strange curve that the Moon describes above our heads is consistent with only the long list of statistical parameters set out in the corresponding tables.

These data are collected on the basis of long-term observations, but by no means on the basis of any calculations. It is thanks to these data that one or another event can be predicted with great accuracy, for example, solar or lunar eclipses, the maximum approach or distance of the Moon relative to the Earth, etc.

So, exactly on this strange trajectory The moon manages to be turned to the Earth by only one side all the time!

Of course, this is not all.

Turns out, Earth orbits the sun by no means not at a uniform speed, as the official physics would like, but makes small slowdowns and jerks forward in the direction of its movement, which are synchronized with the corresponding position of the moon. However, the Earth does not make any movements to the sides perpendicular to the direction of its orbit, despite the fact that the Moon can be located on either side of the Earth in the plane of its orbit.

Official physics not only does not undertake to describe or explain these processes - it is about them just keeps silent! Such a half-month cycle of the earth's tugs correlates perfectly with the statistical peaks of earthquakes, but where and when did you hear about it?

Did you know that in the system of cosmic bodies the Earth-Moon there are no libration points predicted by Lagrange on the basis of the law of "universal gravitation"?

The fact is that the region of gravity of the moon does not exceed the distance 10 000 km from its surface. There are many obvious confirmations of this fact. Suffice it to recall the geostationary satellites, which are not affected by the position of the moon in any way, or the scientific and satirical story with the Smart-1 probe from ESA, with the help of which they were going to take pictures of the Apollo landing sites in 2003-2005.

Probe "Smart-1" was created as an experimental spacecraft with low ion thrust engines, but with a huge operating time. Mission ESA Provided for a gradual acceleration of the spacecraft, launched into a circular orbit around the Earth, so that, moving along a spiral trajectory with a climb, to reach the internal libration point of the Earth-Moon system. According to the predictions of official physics, starting from this moment, the probe should have changed its trajectory, going into a high circumlunar orbit, and begin a long braking maneuver, gradually narrowing the spiral around the Moon.

But everything would be fine if the official physics and the calculations made with its help corresponded to reality. In reality After reaching the libration point, "Smart-1" continued its flight in an unwinding spiral, and on the next orbits did not even think to react to the approaching moon.

From that moment on, around the flight of "Smart-1" began an amazing conspiracy of silence and outright misinformation, until the trajectory of its flight finally made it possible to simply smash it against the surface of the Moon, which semi-official scientific popularizing Internet resources hastened to report under the appropriate informational sauce as a great achievement of modern science, which suddenly decided to "change" the mission of the apparatus and from all over the place to shake tens of millions of foreign exchange money spent on the project on moon dust.

Naturally, on the last orbit of its flight, the Smart-1 probe finally entered the lunar gravitational region, but it could not have slowed down to enter a low lunar orbit with the help of its low-power engine. Calculations of European ballisticians entered a striking contradiction with reality.

And such cases in deep space exploration are by no means isolated, but repeat themselves with enviable consistency, starting from the first attempts to hit the Moon or sending probes to the satellites of Mars, ending with the last attempts to enter orbits around asteroids or comets, whose gravitational force is completely absent even on their surfaces.

But then the reader should have absolutely logical question: how did the rocket and space industry of the USSR in the 60s and 70s of the twentieth century managed to explore the moon with the help of automatic devices, being held captive by false scientific views? How did Soviet ballistics specialists calculate the correct flight path to the Moon and back, if one of the most basic formulas of modern physics turns out to be fiction? Finally, how are the orbits of lunar automatic satellites that take close photographs and scan of the Moon calculated in the 21st century?

Very simple! As in all other cases, when practice shows a discrepancy with physical theories, His Majesty steps in. An experience, which suggests the correct solution to a particular problem. After a series of completely natural failures, empirically ballistics found some correction factors for certain stages of flights to the Moon and other space bodies, which are introduced into on-board computers of modern automatic probes and space navigation systems.

And everything works! But most importantly, there is an opportunity to trumpet the whole world about the next victory of world science, and further teach gullible children and students the formula of "universal gravitation", which has no more to do with reality than the cocked hat of Baron Munchausen to his epic exploits.

And if suddenly a certain inventor comes up with another idea of ​​a new way of travel in space, there is nothing easier than to declare him a charlatan on the simple grounds that his calculations contradict the same notorious formula of "universal gravitation" ... countries work tirelessly.

This is a prison, comrades. A large planetary prison with a slight touch of science to neutralize especially zealous individuals who dare to be smart. It is enough to marry the rest so that, following the apt remark of Karel Chapek, their autobiography ends ...

By the way, all the parameters of the trajectories and orbits of "manned flights" from NASA to the Moon in 1969-1972 were calculated and published precisely on the basis of the assumptions about the existence of libration points and about the fulfillment of the law of universal gravitation for the Earth-Moon system. Doesn't this alone explain why all manned lunar exploration programs after the 70s of the twentieth century were rolled up? Which is easier: quietly leave the topic or confess to the falsification of all physics?

Finally, the moon has a number of amazing phenomena called "Optical anomalies"... These anomalies no longer fit into any gates of official physics that they prefer to be completely silent about them, replacing interest in them with the allegedly constantly recorded UFO activity on the lunar surface.

With the help of the yellow press fictions, fake photo and video materials about flying saucers allegedly constantly moving over the Moon and huge structures of aliens on its surface, the behind-the-scenes owners are trying to cover up with information noise really fantastic reality of the moon, which should definitely be mentioned in this work.

The most obvious and visual optical anomaly of the Moon is visible to all earthlings with the naked eye, so it remains only to be surprised that almost no one pays attention to it. See what the moon looks like in a clear night sky at full moon moments? She looks like flat round body (like a coin) but not like a ball!

A spherical body with rather significant irregularities on its surface, in the case of its illumination by a light source located behind the observer, should shine to the greatest extent closer to its center, and as it approaches the edge of the sphere, the luminosity should smoothly decrease.

Probably the most famous law of optics cries about this, which sounds like this: "The angle of incidence of a ray is equal to the angle of its reflection." But this rule does not apply to the Moon at all. For reasons incomprehensible to official physics, the rays of light falling into the edge of the lunar ball are reflected ... back to the Sun, which is why we see the Moon at the full moon as a kind of coin, but not as a ball.

More confusion in the minds introduces an equally obvious observable thing - the constant value of the luminosity level of the illuminated parts of the Moon for an observer from the Earth. Simply put, if we assume that the Moon has some property of directed scattering of light, then we have to admit that the reflection of light changes its angle depending on the position of the Sun-Earth-Moon system. No one can dispute the fact that even a narrow crescent of a young Moon gives a luminosity exactly the same as the central part of a half Moon corresponding to it in area. And this means that the Moon somehow controls the angle of reflection of the sun's rays so that they are always reflected from its surface to the Earth!

But when the moon is full the luminosity of the moon increases abruptly... This means that the surface of the Moon surprisingly splits the reflected light into two main directions - towards the Sun and the Earth. This leads to another startling conclusion that The moon is virtually invisible to an observer from space, which is not on the straight lines Earth-Moon or Solne-Moon. Who and why needed to hide the Moon in space in the optical range? ...

To understand what the joke is, in Soviet laboratories they spent a lot of time on optical experiments with lunar soil delivered to Earth by the automatic vehicles Luna-16, Luna-20 and Luna-24. However, the parameters of the reflection of light, including solar, from the lunar soil fit well into all the known canons of optics. The lunar soil on Earth did not want to show the wonders that we see on the Moon at all. It turns out that materials on the moon and on earth behave differently?

Quite possible. After all, an unoxidizable film several iron atoms thick on the surface of any objects, as far as I know, in terrestrial laboratories has not yet been obtained ...

The fire was poured by photographs from the Moon, transmitted by Soviet and American machine guns, which were able to land on its surface. Imagine the surprise of the then scientists when all the photographs on the moon were obtained strictly black and white- without a single hint of such a rainbow spectrum familiar to us.

If only the lunar landscape was photographed, evenly covered with dust from meteorite explosions, this would somehow be understandable. But black and white turned out even calibration color plate on the lander body! Any color on the lunar surface turns into the corresponding shade of gray, which is impartially recorded by all photographs of the lunar surface, transmitted by automatic devices of different generations and missions to this day.

Now imagine in what deep ... puddle the Americans are sitting with their white-blue-red Stars and stripes, supposedly photographed on the lunar surface by valiant astronauts, "pioneers".

(By the way, their color pictures and videotapes indicate that Americans generally go there nothing never sent! - Ed.).

Tell me, in their place, would you try hard to resume exploration of the Moon and get to its surface at least with the help of some kind of "pendo rover", knowing that images or videos will only turn out in black and white? Is it possible to quickly paint them, like old films ... But, damn it, in what colors to paint pieces of rocks, local stones or steep mountain slopes!?.

Incidentally, very similar problems awaited NASA on Mars. All researchers have probably already gotten sore by a muddy story with a color mismatch, more precisely, with an obvious shift of the entire Martian visible spectrum on its surface to the red side. When NASA workers are suspected of deliberately distorting images from Mars (supposedly hiding blue skies, green carpets of lawns, blue lakes, crawling locals ...), I urge you to remember the Moon ...

Think, maybe they just act on different planets different physical laws? Then a lot immediately falls into place!

But let's get back to the moon for now. Let's finish with the list of optical anomalies, and then get down to the next sections of Lunar Wonders.

A ray of light passing near the surface of the Moon receives significant scatter in direction, which is why modern astronomy cannot even calculate the time it takes to cover the stars with the body of the Moon.

The official science does not express any ideas why this happens, except for the crazy-delusional electrostatic-style reasons for the movement of lunar dust at high altitudes above its surface or the activity of certain lunar volcanoes, which deliberately throw out dust refracting light exactly in the place where the observation is carried out. this star. And so, in fact, no one has yet observed lunar volcanoes.

As you know, terrestrial science is able to collect information on the chemical composition of distant celestial bodies by studying molecular spectra radiation-absorption. So, for the celestial body closest to the Earth - the Moon - this is a way to determine the chemical composition of the surface does not pass! The lunar spectrum is practically devoid of bands that can provide information about the composition of the moon.

The only reliable information on the chemical composition of the lunar regolith was obtained, as is known, from the study of samples taken by the Soviet "Lunas". But even now, when it is possible to scan the lunar surface from a low circumlunar orbit using automatic devices, reports of the presence of a particular chemical substance on its surface are extremely controversial. Even on Mars - and even then there is much more information.

And one more amazing optical feature of the lunar surface. This property is a consequence of the unique backscattering of light, with which I began my story about the optical anomalies of the Moon. So practically all the light falling on the moon reflected towards the sun and earth.

Let's remember that at night, under appropriate conditions, we can perfectly see the part of the Moon that is not illuminated by the Sun, which, in principle, should be completely black, if not for ... the secondary illumination of the Earth! The earth, when illuminated by the sun, reflects some of the sunlight towards the moon. And all this light that illuminates the shadowy part of the moon returns back to earth!

Hence, it is completely logical to assume that on the surface of the Moon, even on the side illuminated by the Sun, twilight reigns all the time... This guess is superbly confirmed by photographs of the lunar surface taken by Soviet lunar rovers. Look at them carefully on occasion; for everything that can be obtained. They were made in direct sunlight without the influence of distortion of the atmosphere, but they look as if the contrast of the black-and-white picture was tightened in the earthly twilight.

In such conditions, the shadows from objects on the surface of the Moon should be absolutely black, illuminated only by the nearest stars and planets, the level of illumination from which is many orders of magnitude lower than that of the sun. This means that it is not possible to see an object in the shadow of the moon using any known optical means.

To summarize the optical phenomena of the Moon, let us give the floor to an independent researcher A.A. Grishaev, to the author of a book about the "digital" physical world, who, developing his ideas, in another article points out:

“Taking into account the existence of these phenomena provides new, deadly arguments in support of those who believe counterfeits film and photographic materials that allegedly testify to the stay of American astronauts on the lunar surface. After all, we give the keys for the simplest and ruthless independent examination.

If we are shown against the background of sunlit (!) Lunar landscapes of astronauts, on whose spacesuits there are no black shadows from the anti-sun side, or a well-lit figure of an astronaut in the shadow of the "lunar module", or color (!) Frames with a vivid reproduction of the colors of the American flag, then that's all irrefutable evidence screaming falsification.

In fact, we are not aware of a single film or photographic document depicting astronauts on the moon under real moonlight and with a real lunar color “palette”.

And then he continues:

“The physical conditions on the Moon are too abnormal, and it cannot be ruled out that the circumlunar space is destructive for terrestrial organisms. Today we know the only model that explains the short-range action of lunar gravitation, and at the same time the origin of the accompanying anomalous optical phenomena - this is our model of "shaky space".

And if this model is correct, then the vibrations of "unsteady space" below a certain height above the surface of the Moon are quite capable of breaking weak bonds in protein molecules - with the destruction of their tertiary and, possibly, secondary structures.

As far as we know, turtles returned from the lunar space alive on board the Soviet probe "Zond-5", which circled the Moon with a minimum distance of about 2000 km from its surface. It is possible that with the passage of the apparatus closer to the Moon, animals would die as a result of denaturation of proteins in their organisms. If it is very difficult to protect oneself from cosmic radiation, but it is still possible, then there is no physical protection from the vibrations of the "shaky space" ... "

The above excerpt is only a small part of the work, the original of which I strongly recommend to read on the author's website

I also like that the lunar expedition was re-filmed in good quality. And it’s true, it was disgusting to watch. It's the 21st century after all. So welcome, in HD quality "Sleigh rides on Shrovetide".

Let us recall the main characteristics of the Moon's orbit relative to the Earth.

The moon moves around the Earth in an orbit close to circular (the average value of the eccentricity is 0.05). The duration of one revolution of the moon is approximately - 27.3 days. Its distance from the Earth is on average 384,000 km. Due to the existing, albeit insignificant, ellipticity of the orbit, its greatest distance from the Earth (at apogee) reaches 405500 km and the smallest (at perigee) 363000 km. The moon's orbiting speed is approximately 1.02 km / sec. Flying at such a speed, the Moon describes an arc of about 13 ° along the celestial sphere every day. The plane of the Moon's orbit relative to the plane of the Earth's equator constantly changes in the range from 18 ° to 28 °. In 1970, the inclination of the orbital plane was about 28 °. This means that during each month the Moon will be above the equator at an altitude of 28 ° and below it, having also dropped to an angle of 28 °.

The moon can be reached in various ways. To date, the following types of flights to the Moon have been implemented:

Flight near the Moon with the subsequent exit of the spacecraft outside the sphere of the Earth's influence and its transformation into a satellite of the Sun - an artificial planet ("Luna-1", "Pioneer-4");

Flight with a "hard" hit to the moon ("Luna-2", "Ranger-7");

Flight with a soft landing on the Moon without entering the intermediate orbit of its satellite (Luna-9, Surveyor-1);

Flight with the entry into the orbit of the lunar satellite without landing and without returning to the Earth (unmanned - "Luna-10", "Lu-nar-Orbitar-1");

Flight with an entry into the orbit of a satellite of the Moon without landing on the Moon, but with a return to Earth ("Apollo-8");

Orbiting the Moon and returning to Earth ("Probe-5");

Flight with an entry into the orbit of the satellite of the Moon, landing on the Moon and return to the Earth ("Apollo-11", "Luna-16").

From this one can clearly see the general logical purposefulness of the exploration of the Moon and the gradual complication of the flight scheme. Each of these types of flight was of independent interest and made it possible to solve a certain range of scientific and technical problems.

Now let's see what are the general principles that underlie various options for a flight to the Moon. The main criterion that predetermines the method of calculating and choosing flight trajectories to the Moon is the accuracy of the calculation with a minimum expenditure of energy (i.e., fuel) for the implementation of all maneuvers and the possibility of providing the flight with means of a ground or autonomous complex. In accordance with this, a distinction is made between approximate and exact methods of calculating orbits.

Approximate methods are based on the use of the elliptical theory of spacecraft motion. As you know, the Moon is in the sphere of action of the Earth. Therefore, the flight trajectory to the Moon, which lies entirely within the sphere of the Earth's action, can be approximately calculated according to the elliptical theory, assuming that the spacecraft initially performs flight only under the influence of the Earth's gravity. The attraction of the Moon, the Sun and the off-centerness of the Earth's field are neglected in this case. The resulting trajectory extends in the direction of the Moon until the spacecraft enters the Moon's sphere of action, i.e., it is at a distance of 66 thousand km from its center. From this moment on, the trajectory is calculated only taking into account the attraction of the Moon, and the attraction of the Earth and the Sun is neglected. If further the spacecraft, moving away from the Moon, again finds itself at a distance of 66 thousand km from it, then again the influence of the Moon is excluded and subsequently it is considered that the flight occurs only in the field of action of the Earth.

This is how ballistics adapted the elliptical theory to solve the three-body problem. This method is often called the separation of the spacecraft motion according to the spheres of action of celestial bodies. Of course, it is approximate and can only be used for a qualitative analysis of flight trajectories. But in view of its algorithmic simplicity, it finds the widest application in mass research of flights to the Moon. When it comes to real launches, either methods of numerical calculation of trajectories are used, or the theory of elliptical motion, somehow corrected in an artificial way.