student
Name
If the velocity vector of a 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.
A ball is thrown at a wall with a speed whose horizontal and vertical components are 6 m/s and 8 m/s, respectively. The distance from the wall to the throwing point is L = 4 m. At what point of the trajectory will the ball be 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 a movement is impossible.
student
Name
A material point rotates in a circle around a fixed axis. For what dependence of angular velocity on time w(t) is the formula Ф = wt applicable when calculating the angle of rotation?
The car wheel has a radius R and rotates with an angular velocity w. What time is it
will the car need to travel a distance L without slipping? Please indicate the number of the correct formula. Answer:2
Frame name
How will the magnitude and direction of the vector product of two non-collinear vectors change when each of the factors is doubled and their directions are reversed?
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The module will increase four times, direction |
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Will not change. |
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14.10.2011 15:30:20 |
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System evaluation |
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The projection of the acceleration of a material point changes in accordance with the depicted graph. The initial speed is zero. At what instants of time does the velocity of a material point change direction?
Student answer |
Name
student
Name
How can the acceleration vector of a body moving along the depicted trajectory be directed when passing 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 А = 0.5 rad/s3, t is 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 angular velocity w around a fixed axis. Give 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 with one side constantly facing the Earth. What is the trajectory of the center of the Earth relative to an astronaut on the Moon?
Straight segment.
Circle.
The answer depends on the astronaut's location on the Moon.
04.10.2011 14:06:11
Name frame287
Using the given graph of the speed of a moving person, determine how many meters he walked between two stops. Answer: 30
Name frame288
A body is thrown at an angle to the horizontal. Air resistance can be neglected. At which point in the trajectory does the velocity change in magnitude with maximum speed? Please indicate all correct answers.
Student E's answer A
Name frame289
student
Name
The flywheel rotates as shown in the figure. The angular acceleration vector B is directed perpendicular to the plane of the drawing towards us and is constant in magnitude. What is the direction of the angular velocity vector w and what is the nature of the rotation of the flywheel?
Vector w is directed away from us, the flywheel is slowing down.
A material point moves in a circle, and its angular velocity w depends on time t as shown in the figure. How does its normal An and
student
Name
tangential At acceleration?
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 value to V0 = 1 m/s and is directed along the Y axis. Find the tangent of the angle between the body velocity vector and the Y axis at time t = 10 s. Answer: 2
Name frame257
student
Name
Using the above velocity projection graph, determine the projection of displacement Sx for the entire time of movement.
The point moves uniformly along the trajectory shown in the figure. At what point(s) is the tangential acceleration equal to 0?
Along the entire trajectory. |
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student |
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Name |
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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 answer Decreases.
Name frame260
A disk of radius R spins with constant angular acceleration ε. Give the formula for calculating the tangential acceleration of point A on the rim of the disk at angular velocity w. Answer:5
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If the coordinates of the body change with time t by |
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equations x = A·t, y = B·t·t, where A and B are constants, then |
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body trajectory... |
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Parabola. |
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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 taught in educational institutions, is very proud that she knows the relationships between various physical quantities in the form of formulas, which are supposedly reliably supported experimentally. As they say, that’s where we stand...
In particular, in all reference books and textbooks it is stated that between two bodies having masses ( m) And ( M), an attractive force arises ( F), which is directly proportional to the product of these masses and inversely proportional to the square of the distance ( R) between them. This relationship is usually presented as the formula "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 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 reference books 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 Earth and Moon = 380,000,000 m
Distance between the Moon and the Sun = 149,000,000,000 m
The force of attraction between the Earth and the Moon = 6.6725 × 10 -11 x 7.3477 × 10 22 x 5.9737 × 10 24 / 380000000 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 gravitational force of the Moon on 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 don't believe your eyes, please take a calculator, open the reference books and see for yourself.
According to the formula of “universal gravity” for a given system of three bodies, as soon as the Moon is between the Earth and the Sun, it should leave its circular orbit around the Earth, turning into an independent planet with orbital parameters close to the Earth’s. 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 a mathematical point of view, this formula may be correct, but then the values of its parameters are incorrect.
For example, modern science can make serious mistakes in determining distances in space based on false ideas about the nature and speed of light; or it is incorrect to estimate the masses of celestial bodies using the same purely speculative conclusions Kepler or Laplace, expressed in the form of ratios of orbital sizes, velocities and masses of celestial bodies; or not at all understand the nature of the mass of a macroscopic body, which all physics textbooks talk about very frankly, postulating this property of material objects, regardless of its location and without delving into the reasons for its occurrence.
Also, official science may be wrong about the reason for the existence and principles of action of the force of gravity, which is most likely. For example, if masses do not have an attractive effect (for which, by the way, there are thousands of visual evidence, only they are hushed up), then this “formula of universal gravitation” simply reflects a certain idea expressed by Isaac Newton, which in fact turned out to be false.
You can make thousands of mistakes different ways, but there is only one truth. And official physics deliberately hides it, otherwise how can one explain the upholding of such an absurd formula?
First and the obvious consequence of the fact that the "gravitational formula" does not work is the fact that the Earth has no dynamic reaction 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) rotate around a common center of mass - the so-called. barycenter. However, the Earth rotates strictly around its axis, and even the ebbs and flows in the seas and oceans have absolutely nothing to do with the position of the Moon in the sky.
Associated with the Moon whole line absolutely blatant facts of inconsistencies with the established views of classical physics, which are in the literature and the Internet bashfully are called "lunar anomalies".
The most obvious anomaly is the exact coincidence of the period of revolution of the Moon 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 increasingly out of sync with each orbit of the Moon around the Earth.
For example, no one would argue that the Earth and the Moon are two ideal spheres with a uniform distribution of mass inside. From the point of view of official physics, it is quite obvious that the movement 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 passages of Mars and Venus during periods of maximum convergence of their orbits with the Earth’s. The experience of space flights in near-Earth orbit shows that it is possible to achieve lunar-type stabilization only if constantly taxi orientation micromotors. But what and how does the Moon steer? And most importantly - for what?
This “anomaly” looks even more discouraging against the backdrop of the little-known fact that official science has not yet developed an acceptable explanation trajectories, along which the Moon moves around the Earth. Moon orbit not at all circular or even elliptical. Strange curve, which the Moon describes above our heads, is consistent only with a long list of statistical parameters set out in the corresponding tables.
These data were collected on the basis of long-term observations, but not on the basis of any calculations. It is thanks to these data that it is possible to predict certain events 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 with only one side all the time!
Of course, this is not all.
Turns out, Earth does not move in orbit around the Sun not at a uniform speed, as 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 on any 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 he's just keeping silent! Such a semi-monthly cycle of jerks globe correlates well with statistical earthquake peaks, but where and when did you hear about it?
Did you know that in the Earth-Moon system of cosmic bodies there are no libration points, predicted by Lagrange on the basis of the law of “universal gravitation”?
The fact is that the Moon’s gravitational region does not exceed the distance 10 000 km from its surface. There is a lot of obvious evidence of this fact. Suffice it to recall 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 casually photograph the Apollo lunar landing sites back in 2003-2005.
Probe "Smart-1" was created as an experimental spacecraft with low ion thrust engines, but with a long operating time. Mission ESA the gradual acceleration of the apparatus, launched into a circular orbit around the Earth, was envisaged in order to, moving along a spiral trajectory with an increase in altitude, reach the inner libration point of the Earth-Moon system. According to the predictions of official physics, starting from this moment, the probe was supposed to change its trajectory, moving to a high lunar orbit, and begin a long braking maneuver, gradually narrowing the spiral around the Moon.
But everything would be fine if official physics and the calculations made with its help corresponded to reality. In fact, after reaching the libration point, “Smart-1” continued its flight in an unwinding spiral, and on the next orbits it did not even think about reacting to the approaching Moon.
From that moment on, an amazing event began around the flight of Smart-1. conspiracy of silence and outright misinformation, until the trajectory of its flight finally allowed it to simply crash on the surface of the Moon, which official popular science Internet resources hastened to report under the appropriate information sauce as a great achievement modern science, which suddenly decided to “change” the mission of the device and, with all its might, throw tens of millions of foreign currency money spent on the project into lunar 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 using its low-power engine. The calculations of European ballisticians entered into a striking contradiction with real reality.
And such cases in deep space exploration are by no means isolated, but are repeated with enviable regularity, starting from the first attempts to hit the Moon or sending probes to the satellites of Mars, ending last attempts enter orbit around asteroids or comets, the force of attraction of which is completely absent even on their surface.
But then the reader should have a completely legitimate question: How did the rocket and space industry of the USSR in the 60s and 70s of the twentieth century manage to explore the Moon with the help of automatic vehicles, being in captivity of false scientific views? How did Soviet ballisticians calculate the correct flight path to the Moon and back, if one of the most basic formulas of modern physics turns out to be a fiction? Finally, how in the 21st century are the orbits of automatic lunar satellites that take close photographs and scans of the Moon calculated?
Very simple! As in all other cases, when practice shows a discrepancy with physical theories, His Majesty comes into play 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 cosmic bodies, which are entered into on-board computers of modern automatic probes and space navigation systems.
And everything works! But most importantly, there is an opportunity to trumpet to the whole world about another victory of world science, and then to teach gullible children and students the formula of “universal gravity,” which has no more to do with reality than Baron Munchausen’s cocked hat has to do with his epic exploits.
And if suddenly some inventor comes up with yet another idea for a new method of transportation in space, there is nothing easier than declaring him a charlatan on the simple grounds that his calculations contradict the same notorious formula of “universal gravity”... Commission for Combating Pseudoscience at the Academies of Sciences different countries work tirelessly.
This is a prison, comrades. A large planetary prison with a slight touch of science to neutralize particularly zealous individuals who dare to be smart. For the rest, it’s enough to get married so that, following Karel Capek’s apt remark, 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 assumptions about the existence of libration points and the fulfillment of the law of universal gravitation for the Earth-Moon system. Doesn’t this alone explain why all programs for manned exploration of the Moon after the 70s of the twentieth century were rolled up? What is easier: to quietly move away from the topic or to admit to falsifying all of physics?
Finally, the Moon has a number of amazing phenomena called "optical anomalies". These anomalies are so out of step with official physics that it is preferable to remain silent about them completely, replacing interest in them with the supposedly constantly recorded activity of UFOs on the surface of the Moon.
With the help of fabrications from the yellow press, fake photos and videos about flying saucers supposedly constantly moving over the Moon and huge alien structures on its surface, the behind-the-scenes masters are trying to cover it up with information noise. truly 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 one can only 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 a round body (such as a coin), but not like a ball!
A spherical body with quite significant irregularities on its surface, if illuminated by a light source located behind the observer, should glow to the greatest extent closer to its center, and as it approaches the edge of the ball, the luminosity should gradually decrease.
This is probably the most famous law of optics, 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. For reasons unknown to official physics, rays of light hitting the edge of the lunar ball are reflected... back to the Sun, which is why we see the Moon on a full moon as a kind of coin, but not as a ball.
Even more confusion in our minds introduces an equally obvious observable thing - a constant value of the luminosity level of the illuminated areas of the Moon for an observer from Earth. Simply put, if we assume that the Moon has a certain property of directional 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 the narrow crescent of the young Moon gives a luminosity exactly the same as the corresponding central section of the half Moon. 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 towards the Earth!
But when the full moon comes, Luminosity of the Moon increases abruptly. This means that the surface of the Moon miraculously splits the reflected light into two main directions - towards the Sun and the Earth. This leads to another startling conclusion: The Moon is virtually invisible to an observer from space, which is not located on straight lines Earth-Moon or Sun-Moon. Who and why needed to hide the Moon in space in the optical range?...
To understand what the joke was, Soviet laboratories spent a lot of time on optical experiments with lunar soil delivered to Earth by the Luna-16, Luna-20 and Luna-24 automatic devices. However, the parameters of the reflection of light, including solar light, from the lunar soil fit well into all known canons of optics. The lunar soil on Earth did not at all want to show the wonders that we see on the Moon. It turns out that Materials on the Moon and on Earth behave differently?
Quite possible. After all, as far as I know, a non-oxidizable film thickness of several iron atoms on the surface of any objects, as far as I know, has not yet been obtained in terrestrial laboratories...
Photos from the Moon, transmitted by Soviet and American machine guns that managed to land on its surface, added fuel to the fire. Imagine the surprise of the scientists of that time when all the photographs on the Moon were obtained strictly black and white- without a single hint of the rainbow spectrum so familiar to us.
If only the lunar landscape was photographed, evenly strewn with dust from meteorite explosions, this could somehow be understood. But it even turned out black and white calibration color plate on the body of the lander! Any color on the surface of the Moon turns into a corresponding gradation of gray, which is impartially recorded by all photographs of the surface of the Moon transmitted by automatic devices of different generations and missions to this day.
Now imagine what a deep... puddle the Americans are sitting in with their white-blue-red Stars and stripes, allegedly photographed on the surface of the Moon by the valiant “pioneer” astronauts.
(By the way, their color pictures And video recordings indicate that Americans generally go there Nothing never sent! - Ed.).
Tell me, if you were in their place, would you try very hard to resume exploration of the Moon and get to its surface at least with the help of some kind of “pendo-descent”, knowing that the images or videos will only turn out in black and white? Unless you quickly paint them, like old films... But, damn it, what colors should you paint pieces of rocks, local stones or steep mountain slopes with!?
By the way, very similar problems awaited NASA on Mars. All researchers have probably already set their teeth on edge about the murky story with the color discrepancy, or more precisely, with a clear shift of the entire Martian visible spectrum on its surface to the red side. When NASA employees are suspected of deliberately distorting images from Mars (allegedly hiding the blue sky, 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 of things immediately fall into place!
But let's return to the Moon for now. Let's finish with the list of optical anomalies, and then move on to the next sections of Lunar Wonders.
A ray of light passing near the surface of the Moon receives significant variations in direction, which is why modern astronomy cannot even calculate the time required for the stars to cover the Moon’s body.
Official science does not express any ideas why this happens, except for the wildly delusional electrostatic reasons for the movement of lunar dust at high altitudes above its surface or the activity of certain lunar volcanoes, which deliberately emit dust that refracts light exactly in the place where observations are being made. given star. And so, in fact, no one has observed lunar volcanoes yet.
As is known, earthly science is able to collect information about chemical composition distant celestial bodies through the study of 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 doesn't work! The lunar spectrum is practically devoid of bands that can provide information about the composition of the Moon.
The only reliable information about the chemical composition of lunar regolith was obtained, as is known, from the study of samples taken by the Soviet Luna probes. But even now, when it is possible to scan the surface of the Moon from low lunar orbit using automatic devices, reports of the presence of a particular chemical substance on its surface are extremely contradictory. Even on Mars there is much more information.
And about 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 not illuminated by the Sun, which, in principle, should be completely black, if not for... the secondary illumination of the Earth! The Earth, being illuminated by the Sun, reflects part of the sunlight towards the Moon. And all this light that illuminates the shadow of the Moon, returns back to Earth!
From here 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 perfectly confirmed by photographs of the lunar surface taken by Soviet lunar rovers. Look at them carefully if you have the chance; for everything that can be obtained. They were made in direct sunlight without the influence of atmospheric distortions, but they look as if the contrast of the black and white picture was increased in the earthly twilight.
Under such conditions, shadows from objects on the surface of the Moon should be completely black, illuminated only by nearby 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 located on the Moon in the shadow using any known optical means.
To summarize the optical phenomena of the Moon, we give the floor to an independent researcher A.A. Grishaev, the author of a book about the “digital” physical world, who, developing his ideas, points out in another article:
“Taking into account the fact of the presence of these phenomena provides new, damning arguments in support of those who believe fakes film and photographic materials that allegedly indicate the presence of American astronauts on the surface of the Moon. After all, we provide the keys for conducting the simplest and merciless independent examination.
If we are shown, against the background of lunar landscapes flooded with sunlight (!), astronauts whose spacesuits do not have black shadows on the anti-solar side, or a well-lit figure of an astronaut in the shadow of the “lunar module,” or color (!) footage with a colorful rendering of the colors of the American flag, then that's all irrefutable evidence screaming of falsification.
In fact, we are not aware of any film or photographic documentation depicting astronauts on the Moon under real lunar lighting 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 cislunar space is destructive for terrestrial organisms. Today we know the only model that explains the short-term effect of lunar gravity, and at the same time the origin of accompanying anomalous optical phenomena - this is our “unsteady space” model.
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 alive from cislunar space on board the Soviet Zond-5 spacecraft, which flew around the Moon with a minimum distance from its surface of about 2000 km. It is possible that with the passage of the apparatus closer to the Moon, the animals would have died as a result of the denaturation of proteins in their bodies. If it is very difficult to protect yourself from cosmic radiation, but still possible, then there is no physical protection from vibrations of “unsteady space.”
The above excerpt is only a small part of the work, the original of which I strongly recommend that you read on the author’s website
I also like that the lunar expedition was reshot 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 Maslenitsa.”
Let us recall the main characteristics of the orbit of the Moon relative to the Earth.
The Moon moves around the Earth in a nearly circular orbit (the average 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 405,500 km and the smallest (at perigee) 363,000 km. The Moon's orbital speed is approximately 1.02 km/sec. Flying at such a speed, the Moon describes an arc of about 13° across the celestial sphere every day. The plane of the Moon's orbit relative to the plane of the Earth's equator continuously 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, also descending at 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 beyond the Earth's sphere of influence and its transformation into a satellite of the Sun - an artificial planet ("Luna-1", "Pioneer-4");
Flight with a “hard” hit on 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 entry into orbit of a lunar satellite without landing and without returning to Earth (unmanned - "Luna-10", "Lu-nar-Orbitar-1");
Flight with entry into orbit of a lunar satellite without landing on the Moon, but with a return to Earth (Apollo 8);
Flying around the Moon with a return to Earth (Zond-5);
Flight with entry into orbit of a lunar satellite, landing on the Moon and return to Earth (Apollo 11, Luna 16).
This clearly shows the general logical purposefulness of the exploration of the Moon and the consistent complication of the flight pattern. 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 flight to the moon. The main criterion that predetermines the method of calculating and selecting flight trajectories to the Moon is the accuracy of the calculation with minimal energy consumption (i.e. fuel) to carry out all maneuvers and the ability to support the flight using ground-based or autonomous systems. In accordance with this, there are approximate and exact methods for calculating orbits.
Approximate methods are based on the use of the elliptical theory of spacecraft motion. As you know, the Moon is within the sphere of influence of the Earth. Therefore, the flight trajectory to the Moon, which lies entirely within the sphere of influence of the Earth, can be approximately calculated using the elliptical theory, assuming that the spacecraft initially flies only under the influence of the Earth’s gravity. The attraction of the Moon, the Sun and the non-centrality 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 sphere of influence of the Moon, i.e., it is at a distance of 66 thousand km from its center. From this moment on, the trajectory of motion is calculated only taking into account the gravity of the Moon, and the gravity 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 influence of the Earth.
So ballisticians adapted the elliptical theory to solve the three-body problem. This method is often called dividing the motion of a spacecraft into spheres of action of celestial bodies. Of course, it is approximate and can only be suitable for a qualitative analysis of flight trajectories. But due to its algorithmic simplicity, it finds the widest application in mass studies 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 is somehow artificially corrected.
To the blessed memory of my teacher - the first dean of the Faculty of Physics and Mathematics of the Novocherkassk Polytechnic Institute, head of the Department of Theoretical Mechanics, Alexander Nikolaevich Kabelkov
Introduction
August, summer is coming to an end. People frantically rushed to the seas, and it’s not surprising - it’s the season. 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 fight with this very pseudoscience for the inquisitive minds of modern youth.But the question is really not an idle one - since school years we have become accustomed to believing 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 peculiar and to some extent unique astronomical object, the movement of which around the Earth is not as simple as some of my colleagues from neighboring countries might like.
So, leaving polemics aside, let’s try from different angles, to the best of our competence, to consider this undoubtedly beautiful, interesting and very revealing task.
1. The law of universal gravitation and what conclusions we can draw from it
Discovered 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 in question, and equal in magnitudewhere m 1, m 2 are the masses of the Moon and Earth, respectively; G = 6.67e-11 m 3 /(kg * s 2) - gravitational constant; r 1.2 - 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 movement of the Moon as a satellite of the Earth and learned to calculate the position of the Moon in the sky against the background of stars, we will soon be convinced, through 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 in the early stages of the development of celestial mechanics such thoughts were expressed quite often), but in the unaccounted disturbance of the movement of the Moon from other bodies. Which ones? We look at the sky and our gaze immediately rests on a hefty plasma ball weighing as much as 1.99e30 kilograms right under our noses - the Sun. Is the Moon attracted to the Sun? Just like that, with a force equal in magnitude
where m 3 is the mass of the Sun; r 1.3 - 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 the same 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 - the average distance from the Earth to the Sun. Let's substitute real parameters into 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 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 radius a, we will find the geometric location 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. This will be a sphere with a radius
displaced along the straight line connecting the Earth and the Sun in the direction opposite to the direction of 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 = 259,300 kilometers, and l = 450 kilometers. This area is called sphere of gravity of the Earth relative to the Sun.
The orbit of the Moon known to us lies outside this region. That is, at any point in its trajectory, the Moon experiences significantly greater attraction from the Sun than from the Earth.
2. Satellite or planet? Gravitational scope
This information often gives rise to disputes that the Moon is not a satellite of the Earth, but an independent planet in the solar system, the orbit of which is disturbed by the gravity of the nearby Earth.Let us evaluate 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: the Sun (S), the Earth (E) and the Moon (M).
Let us accept the assumption that the orbits of the Earth relative to the Sun and the Moon relative to the Earth are circular.
Let us consider the motion of the Moon in a geocentric inertial reference frame. The absolute acceleration of the Moon in the heliocentric reference frame is determined by the gravitational forces 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; - acceleration of the Moon relative to the Earth. There will be no Coriolis acceleration here - the coordinate system we have chosen moves forward. From here we get the acceleration of the Moon relative to the Earth
An equal part of this acceleration is due to the attraction of the Moon to the Earth and characterizes its undisturbed geocentric motion. Remaining part
acceleration of the Moon caused by disturbance from the Sun.
If we consider the movement of the Moon in a heliocentric inertial reference frame, then everything is much simpler: acceleration characterizes the undisturbed heliocentric movement of the Moon, and acceleration characterizes the disturbance of this movement from the Earth.
Given the existing parameters of the Earth and Moon orbits in the current era, at each point of the Moon’s trajectory the following inequality is true:
which can be verified by direct calculation, but I will refer to it so as not to unnecessarily clutter the article.
What does inequality (1) mean? Yes, that in relative terms the effect of the Moon’s disturbance by the Sun (and very significantly) is less than the effect of the Moon’s attraction to the Earth. And vice versa, the Earth’s disturbance of the geoliocentric trajectory of the Moon has a decisive influence on the nature of its movement. The influence of Earth's gravity in this case is more significant, which means the Moon “belongs” to the Earth by right and is its satellite.
Another interesting thing is that by turning inequality (1) into an equation, you can find the locus of points where the effects of 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 sphere of gravity. Calculations show that this surface is described by an equation of crazy order, but is close to an ellipsoid of revolution. All we can do without unnecessary problems is to estimate the overall dimensions of this surface relative to the center of the Earth. 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 desired 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. It is clear from the figure that the sphere of influence, or sphere of gravitational action of the Earth relative to the Sun, is a surface of rotation relative to the X axis, flattened along the straight line connecting the Earth and the Sun (along the eclipse axis). The Moon's orbit lies deep within this imaginary surface.
For practical calculations, it is convenient to approximate this surface by a sphere with a center at the center of the Earth and a radius equal to
where m is the mass of the smaller celestial body; M is the mass of the larger body in whose gravitational field 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 - its influence on the trajectories of astronomical objects is so small that it can be neglected. This means that it will not be possible 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 blue in the figure. dotted line. In estimation calculations, it is generally accepted that a body located inside a given sphere will experience gravity exclusively from the Earth. If the body is located outside this sphere, we assume that the body moves in the gravitational field of the Sun. In practical astronautics, the method of conjugating conic sections is known, which allows one to approximately calculate the trajectory of a spacecraft using the solution of the two-body problem. At the same time, the entire space that the device overcomes is divided into similar spheres of influence.
For example, it is now clear that in order to be theoretically able to perform maneuvers to enter lunar orbit, the spacecraft must fall within the sphere of influence of the Moon relative to the Earth. Its radius is easy to calculate using formula (3) and is equal to 66 thousand kilometers.
3. Three-body problem in the classical formulation
So, let us consider the model problem in a general formulation, known in celestial mechanics as the three-body problem. Let us consider three bodies of arbitrary mass, located arbitrarily in space and moving exclusively under the influence of the forces of mutual gravitational attraction
We consider bodies to be material points. The position of the bodies will be measured in an arbitrary basis to which the inertial reference system is associated Oxyz. The position of each body is specified by the radius vector , and , respectively. Each body is subject to the force of gravitational attraction from two other bodies, and in accordance with the third axiom of the dynamics of a point (Newton’s 3rd law)
Let's write down the differential equations of motion of each point in vector form
Or, taking into account (4)
In accordance with the law of universal gravitation, interaction forces are directed along the vectors
Along each of these vectors we issue 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
- gravitational parameter of the attracting center. Then the equations of motion will take the final vector form
4. Normalization of equations to dimensionless variables
A fairly popular technique in mathematical modeling is to reduce 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 using numerical modeling, but in a fairly general form, a whole class of typical problems. I leave open the question of how justified this is in each problem being 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 semimajor axis around it is equal to . All these quantities, by virtue of the laws of mechanics, are related by the relation
Let us introduce a change of parameters. For the position of the points of our system
where is the dimensionless radius vector of the i-th point;
for gravitational parameters of bodies
where is the dimensionless gravitational parameter of the i-th point;
for time
where is dimensionless time.
Now let's recalculate the accelerations of the points of the system through these dimensionless parameters. Let us apply direct double differentiation with respect to time. For speeds
For accelerations
When substituting the resulting relations 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 Sundmann found an algorithm for finding coefficients for infinite series with respect to a complex parameter, which theoretically are a general solution to the three-body problem. But... to use Sundmann series in practical calculations with the required accuracy requires obtaining such a number of terms 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 initial conditions: obtaining initial data
Before starting numerical integration, you should take care of calculating the initial conditions for the problem being solved. In the problem under consideration, the search for initial conditions turns into an independent subtask, since system (5) gives us nine scalar second-order equations, which, when moving to the normal Cauchy form, increases the order of the system by another factor of 2. 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 do we get data on the position of the celestial bodies we are interested in? We live in a world where man walked on the Moon - naturally, humanity should have information about how this very Moon moves and where it is located.That is, you say, you, dude, are suggesting that we take thick astronomical reference books from the shelves and blow off the dust from them... You didn’t guess! I suggest going for this data to those who actually walked on the Moon, to NASA, namely the Jet Propulsion Laboratory, Pasadena, California. Here - JPL Horizonts web interface.
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 [email protected]
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Brrr, what is this? Don’t panic, for someone who studied astronomy, mechanics and mathematics well at school, there is nothing to be afraid of. So, the most important thing is the final desired coordinates and components of the Moon’s velocity.
$$SOE 2458327.347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 1.537109094089627E-03 Y = -2.237488447258137E-03 Z = 5.112037386426180E-06 VX = 4.593816208618667E-0 4 VY= 3.187527302531735E-04 VZ=-5.183707711777675E-05 LT = 1.567825598846416E-05 RG= 2.714605874095336E-03 RR=-2.707898607099066E-06 $$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 of the Earth's orbit (the ecliptic plane) at 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 of the Earth's north pole perpendicular to the ecliptic plane. Well, the Y axis complements all this happiness to the right three vectors. By default, the coordinate units are astronomical units (the smart guys from 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 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: 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 [email protected]
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Here the barycenter (center of mass) of the Solar System is chosen as the origin of coordinates. Data we are interested in
$$SOE 2458327.347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 5.755663665315949E-01 Y =-8.298818915224488E-01 Z =-5.366994499016168E-05 VX= 1.388633512282171E-0 2 VY= 9.678934168415631E-03 VZ= 3.429889230737491E-07 LT = 5.832932117417083E-03 RG= 1.009940888883960E+00 RR=-3.947237246302148E-05 $$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. [email protected]
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$$SOE 2458327.347916670 = A.D. 2018-Jul-27 20:21:00.0003 TDB X = 5.771034756256845E-01 Y =-8.321193799697072E-01 Z =-4.855790760378579E-05 VX= 1.434571674368357E-0 2 VY= 9.997686898668805E-03 VZ=-5.149408819470315E-05 LT= 5.848610189172283E-03 RG= 1.012655462859054E+00 RR=-3.979984423450087E-05 $$EOE
Wonderful! Now you need to lightly process the obtained data with a file.
6. 38 parrots and one parrot wing
First, let's decide on the scale, because our equations of motion (5) are written in dimensionless form. The data provided by NASA itself tells us that the coordinate scale should be taken as one astronomical unit. Accordingly, we will take the Sun as the reference body to which we will normalize the masses of other bodies, and the period of revolution of the Earth around the Sun as the time scale.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 the center of mass of our mechanical system, the following 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, au : " + 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 the initial 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 of the Moon vxL = 1.434571674368357E-02 vyL = 9.997686898668805 E-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 speed of the Moon, m/s: " + str(vL0)) print(" -//- dimensionless: " + str(uL0)) # Initial speed 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("Initial speed 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 = 31563683.35432583 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
Actually, the integration itself comes down 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, we recall thatThen, introducing the system state vector
we reduce (7) and (5) to one vector equation
To integrate (8) with the existing initial conditions, we will write a little, very little code
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 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] = a i = i + 1 return dydt # Initial conditions of the 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 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 was the spatial trajectory of the Moon for the first 29 days from our chosen starting point
as well as its projection into the ecliptic plane.
“Hey, uncle, what are you selling us?! It’s a circle!”
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?
I promise to prepare a justification for the fact (based on an analysis of calculation errors and NASA data) that the resulting trajectory shift is not a consequence of integration errors. For now, I invite the reader to take my word for it - this displacement is a consequence of the solar disturbance of the lunar trajectory. Let's spin one more turn
Wow! 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 Moon’s trajectory shifts at each revolution. Yes, this impudent Sun is stealing our beloved satellite from us! Oh, this is the Sun!
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? Still would. Astronomy in general is an interesting science.
P.S
At the university where I studied and worked for almost seven years - Novocherkassk Polytechnic Institute - an annual zonal Olympiad for students in theoretical mechanics of universities in the North Caucasus was held. Three times we hosted the All-Russian Olympiad. At the opening, our main “Olympian”, Professor A.I. 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, 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’ll be happy to answer any questions, and, as promised, I’ll post the source text of an example of solving the three-body problem in Python on my Github profile.
Thank you for your attention!