In the section on the question What is the rotation speed of the Moon around the Earth? given by the author chevron the best answer is Orbital speed1.022 km/s
Movement of the Moon
To a first approximation, we can assume that the Moon moves in an elliptical orbit with an eccentricity of 0.0549 and a semimajor axis of 384,399 km. The actual motion of the Moon is quite complex; when calculating it, many factors must be taken into account, for example, the oblateness of the Earth and strong influence The Sun, which attracts the Moon 2.2 times stronger than the Earth. More precisely, the movement of the Moon around the Earth can be represented as a combination of several movements:
rotation around the Earth in an elliptical orbit with a period of 27.32 days;
precession (rotation of the plane) lunar orbit with a period of 18.6 years (see also saros);
rotation of the major axis of the lunar orbit (apse line) with a period of 8.8 years;
periodic change in the inclination of the lunar orbit relative to the ecliptic from 4°59′ to 5°19′;
periodic change in the size of the lunar orbit: perigee from 356.41 Mm to 369.96 Mm, apogee from 404.18 Mm to 406.74 Mm;
the gradual removal of the Moon from the Earth (about 4 cm per year) so that its orbit is a slowly unwinding spiral. This is confirmed by measurements carried out over 25 years.

Reply from Suck through[newbie]
Here are the wise guys, Wikipedia Christmas trees. They copied from all sorts of Wikipedias of various insanity and even did not bother to remove references to internal resources like “-” or “(see also saros)”. The elliptical orbit has not yet gone anywhere, but an eccentricity of 0.0549 or a semimajor axis of 384,399 kilometers is already too much.
Well, they would write that the Moon moves around our planet in a rather elongated elliptical orbit and makes rather complex evolutionary movements and librations, that is, slow ones oscillatory movements clearly visible when observed from Earth. Average orbital speed earth's satellite is 1.023 km/s or 3682.8 kilometers per hour. That's it.

Reply from Wake up[newbie]
1.022

Reply from Yoni Tunoff[newbie]
The Moon moves in orbit around the Earth at a speed of 1.02 km per second. If the Moon rotates around its axis at the same speed, then dividing the length of the Moon’s equator by the speed of 1.02 km per second, we find out the time of 1 rotation of the Moon around its axis in seconds. The length of the Moon's equator is 10920.166 km.

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

Full output for the ephemeris of the Moon at 07/27/2018 20:21 (origin 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 (C-A/B), x10^-4 = 6.310213 gamma (B-A/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: A.D. 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 cylindrical: 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 X Y Z VX VY VZ LT RG RR ** **************************************** *************************** $$SOE 2458327. 347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 1.537109094089627E-03 Y = -2.237488447258137E-03 Z = 5.112037386426180E-06 VX = 4.593816208618667E-0 4 VY= 3.187527302531735E-04 VZ=-5.183707711777675E-05 LT = 1.567825598846416E-05 RG= 2.714605874095336E-03 RR=-2.707898607099066E-06 $$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 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] *******************************************************************************

$$SOE 2458327.347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 1.537109094089627E-03 Y = -2.237488447258137E-03 Z = 5.112037386426180E-06 VX = 4.593816208618667E-0 4 VY= 3.187527302531735E-04 VZ=-5.183707711777675E-05 LT = 1.567825598846416E-05 RG= 2.714605874095336E-03 RR=-2.707898607099066E-06 $$EOE
Yes, yes, yes, they are Cartesian! If we carefully read the entire footcloth, we will learn that the origin of this coordinate system coincides with the center of the Earth. The XY plane lies in the plane earth's orbit(ecliptic plane) for the J2000 epoch. The X axis is directed along the line of intersection of the Earth's equatorial plane and the ecliptic at the point of the vernal equinox. The Z axis points in the direction north pole The Earth is perpendicular to the ecliptic plane. Well, the Y axis complements all this happiness to the right three vectors. The default coordinate units are astronomical units (the smart guys at NASA also give the value of the autronomical unit in kilometers). Speed ​​units: astronomical units per day, a day is taken to be 86400 seconds. Complete stuffing!

We can obtain similar information for the Earth

Full output of the Earth's ephemeris as of 07/27/2018 20:21 (origin at the center of mass 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: A.D. 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 cylindrical: 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 X Y Z VX VY VZ LT RG RR ** **************************************** *************************** $$SOE 2458327.347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 5.755663665315949E-01 Y =-8.298818915224488E-01 Z =-5.366994499016168E-05 VX= 1.388633512282171E-0 2 VY= 9.678934168415631E-03 VZ= 3.429889230737491E-07 LT = 5.832932117417083E-03 RG= 1.009940888883960E+00 RR=-3.947237246302148E-05 $$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] *******************************************************************************

$$SOE 2458327.347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 5.755663665315949E-01 Y =-8.298818915224488E-01 Z =-5.366994499016168E-05 VX= 1.388633512282171E-0 2 VY= 9.678934168415631E-03 VZ= 3.429889230737491E-07 LT = 5.832932117417083E-03 RG= 1.009940888883960E+00 RR=-3.947237246302148E-05 $$EOE
For the Moon, we will need coordinates and velocity 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 as of 07/27/2018 20:21 (origin of coordinates 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 (C-A/B), x10^-4 = 6.310213 gamma (B-A/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: A.D. 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 cylindrical: 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 X Y Z VX VY VZ LT RG RR ************ **************************************** ***************** $$SOE 2458327. 347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 5.771034756256845E-01 Y =-8.321193799697072E-01 Z =-4.855790760378579E-05 VX= 1.434571674368357E-0 2 VY= 9.997686898668805E-03 VZ=-5.149408819470315E-05 LT= 5.848610189172283E-03 RG= 1.012655462859054E+00 RR=-3.979984423450087E-05 $$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 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] *******************************************************************************

6. 38 parrots and one parrot wing

All this is of course very good, but we did not set the initial conditions for the Sun. "For what?" - some linguist would ask me. And I would answer that the Sun is not at all motionless, but also rotates in its orbit around the center of mass of the Solar System. You can see this by looking at NASA data for the Sun.

$$SOE 2458327.347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 6.520050993518213E+04 Y = 1.049687363172734E+06 Z =-1.304404963058507E+04 VX=-1.265326939350981E-0 2 VY= 5.853475278436883E-03 VZ= 3.136673455633667E-04 LT = 3.508397935601254E+00 RG= 1.051791240756026E+06 RR= 5.053500842402456E-03 $$EOE
Looking at the RG parameter, we see that the Sun rotates around the barycenter of the Solar System, and as of July 27, 2018, the center of the star is located 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 a model three-body problem, 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 of coordinates. For our center of mass mechanical system the equation is valid

Let's place the center of mass at the origin of coordinates, that is, set , then

where

Let's move on to dimensionless coordinates and parameters by choosing

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

Where

Now let’s write a program that will generate the initial conditions in the “parrots” we have chosen. What will we write on? In Python, of course! After all, as you know, this is the most best language for mathematical modeling.

However, if we move away from sarcasm, we will actually try python for this purpose, and why not? I'll be sure to link to all the code in my Github profile.

Calculation of 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 the 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 the 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, a.u. : " + str(xi_10)) # NASA coordinates for the 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 of coordinates 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") # Initial speed Moons 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] = v * a / Td uL0[i] = vL0[i] / u print("Initial velocity of the Moon, m/s: " + str(vL0)) print(" -//- dimensionless: " + str(uL0)) # The initial speed of the Earth VXE = 1.3863351282171E-02 VYE = 9.678934168415631E-03 VZE = 3.42989230737491E-07 VE0 = NP.Array () UE0 = NP.Array () for i, v in in Enumerate (VE0): VE 0 [i] = v * a / Td uE0[i] = vE0[i] / u print("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))

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 = 31563683.35432583 Initial position of the Moon, AU: [ 5.77103476e -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 speed of the Moon, m/s: -//- dimensionless: [ 5.24078311 3.65235907 -0.01881184] Initial speed of the Earth, m/s: -//- dimensionless: Initial speed 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

Integration of equations of motion in the three-body problem

# # Calculation of 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 normal form Cauchy # 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] = a i = i + 1 return dydt # Initial conditions Cauchy problem y0 = # # Integrating 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 is 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

Firstly, it is not a circle - there is a noticeable shift in the projection of the trajectory from the origin to the right and down. Secondly, don’t you notice anything? No, really?

We can conclude that solar gravity affects the orbit of the Moon quite significantly - the old woman does not walk the same way across the sky twice. A picture of six months of movement allows (at least qualitatively) to be convinced of this (picture is clickable)

Interesting? Of course. Astronomy in general is an interesting science.

P.S

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, even if he is a doctor of science three times and a linguist four times, and has developed at least a million educational programs. I sincerely believe that writing articles on a popular public resource should include their careful proofreading, normal formatting (LaTeX formulas are not a whim of the resource’s developers!) and the absence of errors leading 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 doesn’t mean you have to turn off your brain.”

I urge you, my dear readers, to appreciate and respect mechanics. I will be happy to answer any questions, and the source text of an example of solving the three-body problem in Python, as promised, Add tags

LIBRATION OF THE MOON: The Moon completes a revolution around the Earth in 27.32166 days. In exactly the same time, it makes a revolution around its own axis. This is not a coincidence, but is associated with the influence of the Earth on its satellite. Since the period of revolution of the Moon around its axis and around the Earth is the same, the Moon should always face the Earth with one side. However, there are some inaccuracies in the rotation of the Moon and its movement around the Earth.

The rotation of the Moon around its axis occurs very uniformly, but the speed of its revolution around our planet varies depending on the distance to the Earth. The minimum distance from the Moon to the Earth is 354 thousand km, the maximum is 406 thousand km. The point of the lunar orbit closest to the Earth is called perigee from “peri” (peri) - around, about, (near and “re” (ge) - earth), the point of maximum distance is apogee [from the Greek “apo” (aro) - above, above and “re”. At closer distances from the Earth, the speed of the Moon’s orbit increases, so its rotation around its axis “lags” somewhat. As a result, a small part of the far side of the Moon, its eastern edge, becomes visible to us. In the second half of its near-Earth orbit, the Moon slows down, as a result of which it “hurries” a little to rotate around its axis, and we can see a small part of its other hemisphere from the western edge to a person who watches the Moon through a telescope from night to night. it seems that it slowly oscillates around its axis, first for two weeks in the eastern direction, and then for the same amount of time in the western direction (However, such observations are practically difficult because part of the surface of the Moon is usually obscured by the Earth. - Ed.) Lever scales. also oscillate around the equilibrium position for some time. In Latin, scales are “libra”, therefore the apparent vibrations of the Moon, due to the unevenness of its motion in its orbit around the Earth while rotating uniformly around its axis, is called libration of the Moon. Librations of the Moon occur not only in the east-west direction, but also in the north-south direction, since the axis of rotation of the Moon is inclined to the plane of its orbit. Then the observer sees a small section of the far side of the Moon in the areas of its north and south poles. Thanks to both types of libration, almost 59% of the Moon's surface can be seen from the Earth (not simultaneously).

GALAXY

The Sun is one of many hundreds of billions of stars gathered in a giant lens-shaped cluster. The diameter of this cluster is approximately three times its thickness. Our Solar System is located in its outer thin edge. Stars look like individual bright points scattered in the surrounding darkness of deep space. But if we look along the diameter of the lens of the assembled cluster, we will see an innumerable number of other star clusters that form a ribbon shimmering with soft light, stretching across the entire sky.

The ancient Greeks believed that this “path” in the sky was formed by drops of spilled milk, and called it a galaxy. "Galakticos" is in Greek milky from "galaktos" which means milk. The ancient Romans called it "via lactea", which literally means the Milky Way. As soon as regular telescope research began, nebulous clusters were discovered among distant stars. English astronomers father and son Herschel, as well as French astronomer Charles Messier, were among the first to discover these objects. They were called nebulae from the Latin “nebula” (nebula) fog. This Latin word was borrowed from the Greek language. In Greek, “nephele” also meant cloud, fog, and the goddess of clouds was called Nephele. Many of the discovered nebulae turned out to be dust clouds that covered some parts of our Galaxy, blocking light from them.

When observed, they looked like black objects. But many "clouds" are located far beyond the boundaries of the Galaxy and are clusters of stars as large as our own cosmic "home". They seem small only because of the gigantic distances that separate us. The closest galaxy to us is the famous Andromeda nebula. Such distant star clusters are also called extragalactic nebulae “extra” (extra) in Latin means the prefix “outside”, “above”. To distinguish them from the relatively small dust formations inside our Galaxy. There are hundreds of billions of these extragalactic nebulae - galaxies, as we now speak of galaxies in the plural. Moreover: since galaxies themselves form clusters in outer space, they speak of galaxies of galaxies.

INFLUENZA

The ancients believed that the stars influenced the destinies of people, so there was even a whole science that was dedicated to determining how they do this. We are talking, of course, about astrology, the name of which comes from the Greek words “aster” (aster) - star and “logos” (logos) - word. In other words, an astrologer is a “star talker.” Usually “-logy” is an indispensable component in the names of many sciences, but astrologers have discredited their “science” so much that they had to find another term for the true science of the stars: astronomy. The Greek word “nemein” means routine, pattern. Therefore, astronomy is a science that “orders” the stars, studying the laws of their movement, emergence and extinction. Astrologers believed that the stars emit a mysterious force that, flowing down to Earth, controls the destinies of people. In Latin, to pour in, flow down, penetrate - “influere”, this word was used when they wanted to say that star power “flows” into a person. In those days, the true causes of illness were not known, and it was quite natural to hear from a doctor that the illness that visited a person was a consequence of the influence of the stars. Therefore, one of the most common diseases, which we know today as influenza, was called influenza (literally, influence). This name was born in Italy (Italian influenca).

The Italians noticed the connection between malaria and swamps, but overlooked the mosquito. To them he was just a small annoying insect; They saw the real reason in the miasma of bad air over the swamps (it was undoubtedly “heavy” due to high humidity and gases released by decaying plants). The Italian word for something bad is “mala,” so they called the bad, heavy air (aria) “malaria,” which eventually became the generally accepted scientific name for the well-known disease. Today, in Russian, no one, of course, will call the flu influenza, although in English it is called that way, however, in colloquial speech it is most often shortened to the short “flu”.

Perihelion

The ancient Greeks believed that celestial bodies move in orbits that are perfect circles, because a circle is an ideal closed curve, and the celestial bodies themselves are perfect. The Latin word “orbita” means track, road, but it is derived from “orbis” - circle.

However, in 1609, the German astronomer Johannes Kepler proved that each planet moves around the Sun in an ellipse, at one of the foci of which the Sun is located. And if the Sun is not in the center of the circle, then the planets at some points in their orbit approach it more than at others. The point of the orbit of a celestial body orbiting around it closest to the Sun is called perihelion.

In Greek, “peri-” is part of a compound word meaning near, around, and “helios” means the Sun, so perihelion can be translated as “near the Sun.” In a similar way, the Greeks began to call the point of greatest distance of a celestial body from the Sun “aphelios” (archeliqs). The prefix “apo” (aro) means away, from, so this word can be translated as “far from the Sun.” In the Russian program, the word “aphelios” turned into aphelion: the Latin letters p and h next to each other are read as “f”. The Earth's elliptical orbit is close to a perfect circle (the Greeks were right here), so the Earth has a difference between perihelion and aphelion of only 3%. Terms for celestial bodies describing orbits around other celestial bodies were formed in a similar way. Thus, the Moon revolves around the Earth in an elliptical orbit, with the Earth located at one of its foci. The point of the Moon's closest approach to the Earth was called perigee "re", (ge) in Greek Earth, and the point of greatest distance from the Earth was called apogee. Astronomers are familiar with double stars. In this case, two stars rotate in elliptical orbits around a common center of mass under the influence of gravitational forces, and the greater the mass of the companion star, the smaller the ellipse. The point of closest approach of the orbiting star to the main star is called periastron, and the point of greatest distance is called apoaster from the Greek. “astron” – star.

Planet - definition

Even in ancient times, people could not help but notice that the stars occupy a constant position in the sky. They moved only in a group and made only small movements around a certain point in the northern sky. It was very far from the sunrise and sunset points where the Sun and Moon appeared and disappeared.

Every night there was an inconspicuous shift in the entire picture of the starry sky. Each star rose 4 minutes earlier and set 4 minutes earlier compared to the previous night, so in the west the stars gradually disappeared from the horizon, and new ones appeared in the east. A year later the circle closed and the picture was restored. However, there were five star-like objects in the sky that shone as brightly, or even brighter, than the stars, but did not follow the general pattern. One of these objects could be located between two stars today, and tomorrow it could shift, the next night the displacement would be even greater, etc. Three such objects (we call them Mars, Jupiter and Saturn) also made a full circle in the heavens, but in a rather complicated way. And the other two (Mercury and Venus) did not move too far from the Sun. In other words, these objects “wandered” between the stars.

The Greeks called their vagrants “planetes”, so they called these celestial vagrants planets. In the Middle Ages, the Sun and Moon were considered planets. But by the 17th century. Astronomers have already realized the fact that the Sun is the center of the solar system, so celestial bodies that revolve around the Sun began to be called planets. The Sun lost its status as a planet, and the Earth, on the contrary, acquired it. The Moon also ceased to be a planet, because it revolves around the Earth and only goes around the Sun together with the Earth.

The Earth and Moon are in continuous rotation around their own axis and around the Sun. The moon also revolves around our planet. In this regard, we can observe numerous phenomena in the sky associated with celestial bodies.

Nearest cosmic body

The Moon is a natural satellite of the Earth. We see it as a luminous ball in the sky, although it itself does not emit light, but only reflects it. The source of light is the Sun, the radiance of which illuminates the lunar surface.

Each time you can see a different Moon in the sky, its different phases. This is a direct result of the Moon's rotation around the Earth, which in turn revolves around the Sun.

Lunar exploration

The Moon was observed by many scientists and astronomers for many centuries, but the real, so to speak “live” study of the Earth’s satellite began in 1959. Then the Soviet interplanetary automatic station Luna-2 reached this celestial body. Then this device did not have the ability to move along the surface of the Moon, but could only record some data using instruments. The result was a direct measurement of the solar wind - the flow of ionized particles emanating from the Sun. Then a spherical pennant depicting the coat of arms of the Soviet Union was delivered to the Moon.

The Luna 3 spacecraft, launched a little later, took the first photograph from space of the far side of the Moon, which is not visible from Earth. A few years later, in 1966, another automatic station called Luna-9 landed on the earth’s satellite. She was able to make a soft landing and transmit television panoramas to Earth. For the first time, earthlings saw a television show directly from the Moon. Before the launch of this station, there were several unsuccessful attempts at a soft “lunar landing.” With the help of research carried out using this apparatus, the meteor-slag theory about the external structure of the Earth's satellite was confirmed.

The trip from Earth to the Moon was carried out by Americans. Armstrong and Aldrin were lucky enough to be the first people to walk on the moon. This event happened in 1969. Soviet scientists wanted to explore the celestial body only with the help of automation; they used lunar rovers.

Characteristics of the Moon

The average distance between the Moon and Earth is 384 thousand kilometers. When the satellite is closest to our planet, this point is called Perigee, the distance is 363 thousand kilometers. And when there is a maximum distance between the Earth and the Moon (this state is called apogee), it is 405 thousand kilometers.

The Earth's orbit has an inclination relative to the orbit of its natural satellite - 5 degrees.

The Moon moves in its orbit around our planet at an average speed of 1.022 kilometers per second. And in an hour it flies approximately 3681 kilometers.

The radius of the Moon, in contrast to the Earth (6356), is approximately 1737 kilometers. This is an average value as it may vary at different points on the surface. For example, at the lunar equator the radius is slightly larger than average - 1738 kilometers. And in the area of ​​the pole it is slightly less - 1735. The Moon is also more of an ellipsoid than a ball, as if it had been “flattened” a little. Our Earth has the same feature. The shape of our home planet is called “geoid”. It is a direct consequence of rotation around an axis.

The mass of the Moon in kilograms is approximately 7.3 * 1022, the Earth weighs 81 times more.

Moon phases

Moon phases are the different positions of the Earth's satellite relative to the Sun. The first phase is the new moon. Then comes the first quarter. After it comes the full moon. And then the last quarter. The line separating the illuminated part of the satellite from the dark one is called the terminator.

The new moon is the phase when the Earth's satellite is not visible in the sky. The Moon is not visible because it is closer to the Sun than our planet, and accordingly, its side facing us is not illuminated.

The first quarter - half of the heavenly body is visible, the star illuminates only its right side. Between the new moon and the full moon, the moon “grows”. It is at this time that we see a shining crescent in the sky and call it the “growing month.”

Full Moon – The Moon is visible as a circle of light that illuminates everything with its silver light. The light of the heavenly body at this time can be very bright.

The last quarter - the Earth's satellite is only partially visible. During this phase, the Moon is called “old” or “waning” because only its left half is illuminated.

You can easily distinguish the waxing month from the waning moon. When the moon wanes, it resembles the letter "C". And when it grows, if you put a stick on the month, you get the letter “R”.

Rotation

Since the Moon and Earth are quite close to each other, they form a single system. Our planet is much larger than its satellite, so it influences it with its gravitational force. The Moon faces us with one side all the time, so before space flights in the 20th century, no one saw the other side. This happens because the Moon and Earth rotate on their axis in the same direction. And the revolution of the satellite around its axis lasts the same time as the revolution around the planet. In addition, together they make a revolution around the Sun, which lasts 365 days.

But at the same time, it is impossible to say in which direction the Earth and Moon rotate. It would seem that this is a simple question, either clockwise or counterclockwise, but the answer can only depend on the starting point. The plane on which the Moon's orbit is located is slightly inclined relative to that of the Earth, the angle of inclination is approximately 5 degrees. The points where the orbits of our planet and its satellite intersect are called the nodes of the lunar orbit.

Sidereal month and Synodic month

A sidereal or sidereal month is the period of time during which the Moon revolves around the Earth, returning to the same place from where it began to move, relative to the stars. This month lasts 27.3 days on the planet.

A synodic month is the period during which the Moon makes a full revolution, only relative to the Sun (the time during which the lunar phases change). Lasts 29.5 Earth days.

The synodic month is two days longer than the sidereal month due to the rotation of the Moon and Earth around the Sun. Since the satellite rotates around the planet, and that, in turn, rotates around the star, it turns out that in order for the satellite to go through all its phases, additional time is needed beyond a full revolution.