From Earth to the Moon. History and math. Part 1



    If we go back to the history of studying flight paths from the Earth to the Moon, then you need to go back one and a half centuries ago, in 1865, when Jules Verne’s new novel “From the Earth to the Moon in a direct way in 97 hours 20 minutes” was published. Of course, this was far from the first book that described such a flight, but it was the first novel in which the author decided to approach the description of travel from a scientific point of view.

    Moreover, the level of preparation of this text is especially well understood now, since for the first time space travel was not just shown, but all the nuances, sometimes very small, were analyzed in detail. Now they like to criticize the novel for those mistakes that the author made. But with such a volume of work, it would be surprising if there were no errors at all! And now it is not his mistakes that amaze him more, but what he was right about. For example, in theory, oxygen purification systems on both the Columbiada shell and the Soyuz spacecraft are very similar. Only Jules Verne had sodium hydroxide, and now lithium hydroxide is used.



    Even more interesting are the moments in which he was right, despite the incorrect assumptions. For example, Perelman criticized him for providing erroneous data on the time of flight from Earth to the Moon and even put them in the name of the novel. The mathematician conducted more accurate calculations by the specified method, having received a figure of the order of six days, two days more than in the novel. And his calculations, within the framework of this method, were indeed correct. It is really possible to fly to the Moon only in four days (actually, many devices reached the Moon in about this time), but in six days it is already impossible. The calculation method was chosen by the science fiction writer incorrectly, and the figure turned out to be correct!

    Even, for example, shortly after the release of the novel “Around the Moon”, many doubted the possibility of such a trajectory of a close flyby of the Moon under the influence of only gravitational forces. Only at the beginning of the twentieth century, when evaluating orbits in binary star systems, was the possibility of the existence of such orbits really shown. Here is what the English journal “Knowledge” wrote at the beginning of the 20th century in the article “Astronomy by Jules Verne”:

    “The curve described by the shell of J. Verne gives rise to very interesting questions. It begins and ends on Earth and represents a closed orbit of the projectile. Such an orbit should be described by a particle ejected from the Earth and subject to the action of only gravitational forces. But under the simultaneous action of the Earth and the Moon, the shape of the orbit, obviously, should change. This was indicated for the first time, it seems, by Burro, who investigated various particular cases related here. George Darwin also investigated similar cases. But only recently was a general theory of this question given by Professor F. R. Multon, who proved the existence of such orbits and found the necessary initial conditions for their existence. "Jules Verne anticipated all these studies for a long time, and of course, he must be credited with convincing development of this issue."

    In other words, the tasks that Jules Verne tried to solve at that time were not just at the forefront of science - they often posed questions that science then could not answer.

    However, one must not forget that he did not do all this alone. When writing all of his books, the author tried to turn to specialists in any field. According to his diaries, the calculations for the flight from Earth to the Moon were carried out by his cousin Henri Garce (1815-1871), a mathematics teacher at the Lyceum of Napoleon (now the Lyceum of Henry IV). By the way, here you can see his book in French, dedicated to the Cosmography of Leçons nouvelles de Cosmographie (1854). And his calculations were quite accurate. For example, here, according to the data in the book, I calculated the exact start date for the crew of Columbiada.

    image

    Henri Garcet (1815–1871)

    However, we probably would not have known about his calculations if Jules Verne had not placed them at the very beginning of the book “Around the Moon”, using them in the conversation of the crew of the Columbiada shell. They write that readers of the newspaper "Debates" dated November 7, 1869 were literally amazed to see complex mathematical calculations - the publication of the novel began in this newspaper.

    This is how the mathematical model of the flight was described through the lips of Michel Ardan and Barbicane:

    “Do you know, Barbicane, what have I been thinking about all night?”
    - About what? The chairman asked.
    “I kept thinking about our Cambridge friends.” Of course, you noticed that I didn’t understand a damn thing in mathematics. So: I just can’t understand how our scientists at the observatory could calculate the speed that a projectile must have in order to reach the moon.
    “You want to say,” interrupted Barbicane, “to that neutral point where the forces of gravity and lunar gravity are the same, because from this point, which is almost nine-tenths of the total distance between the two planets, the shell will fly to the moon by itself, due to its own gravity.
    “Well, yes, that is exactly what I had in mind,” said Michel. “But how did they calculate this speed?”
    - Nothing is easier.
    “Would you be able to do this calculation yourself?”
    ...
    “Yes, dear friend.” Taking into account all the known conditions of the problem: the
    distance from the center of the earth to the center of the moon, the radius of the earth, the mass of the earth, the mass of the moon, I can accurately determine the initial velocity of our projectile, and at the same time using the simplest formula.
    - What is the formula?
    - But you see. But only I will not cross out the curve described by our shell between the Moon and the Earth, given their relative motion around the Sun. Suppose both planets are motionless. That will be quite enough.
    - Why?
    - Because this is how the tasks called “three-body tasks” are solved, the integral method for solving such problems has not yet been sufficiently developed.


    After a while, the formula was also introduced, which was presented to readers:

    image

    where r is the radius of the Earth, is the distance between the centers of the Earth and the Moon, x is the distance of the core from the Earth. V 0 is the initial speed of the core. V is the speed at a distance of x.

    And the flight looked like this

    image

    Here I would like to note that this formula with the indicated initial data is indeed true. And it really accurately displays the gravitational effect of the Earth and the Moon on the device at any given time. It is quite easy to derive from the law of conservation of energy.

    Derivation of the Verne-Garce formula
    Amount given kinetic and potential energy is as

    mv 2 /2 - γM W m / R W - γM R m / R L = const

    where m and M W , M A - mass of the projectile, Earth and Moon,

    γ - Universal gravitational constant , R L is the distance to the center of the moon, R 3 is the distance to the center of the Earth

    We pass to the notation of the novel. And on the right side we take the data at the time of launch.

    mv 2 /2 - γM W m / x- γM A m / (Dx) = mv 0 2 /2 - γM Hm / r - γM A m / (dr)

    is reduced by m, tolerated

    v 2 /2 -v 0 2 /2 = γM W / x + γM L / (Dx) - γM H / r - γM L / (dr)

    1 / 2 * (v 2 -v 0 2 ) = γM З (1 / х-1 / r + M Л / M З * 1 / (dx) -M Л / M З * 1 / (dr))

    1/2 * (v 2 -v 0 2 ) = γM З / r (r / х-1 + M Л / M З (r / (dx) -r / (dr))

    Well, since from mg = mγM З / r2 follows γM З = g * r 2

    we obtain the initial formula
    1/2 * (v 2 -v 0 2 ) = gr (r / х-1 + M Л / M З (r / (dx) -r / (dr) )

    So what's the deal? The problem is that it does not take into account the dynamics of the system. The moon rotates around the Earth at a speed of about 1 km / s. This is precisely the problem of this problem. As a result, when you try to fly according to the above scheme, when the station flies to a neutral point between the Earth and the Moon, it will not be captured at all by the Moon’s field, since the latter will fly away from the projectile at a speed of a kilometer per second, and the projectile will begin to fall to Earth. But then there was hope that the capture of the device by the Moon would nevertheless happen, and a similar scheme, to a first approximation, accurately shows the dynamics of the flight. With the light hand of Jules Verne, a similar method of calculating flight soon became very popular. It began to be used not only in art books, but also in serious scientific works.

    It should be noted that then astronomers just understood the importance of taking dynamics into account. Yes, and differential equations that take into account both gravity and dynamics, then could easily be composed. Just also could not find a simple solution. Actually, as is now well understood, the three-body problem does not have a common analytical solution. The problem could be solved only numerically, which meant a very, very large number of routine calculations, which could spend months or even years of life. At the time of publication of the novel, a similar problem was solved numerically only twice. Moreover, the first attempt was more than a hundred years before the publication of the novel, in 1759.

    It is known that Edmund Halley, compiling a catalog of comets, noticed that several comets that appeared in the sky at different times have very similar parameters. Having verified them, he decided that it could be the same comet. Then she will be called by his name. Comet Halley appeared in the sky in 1531, 1607 and 1682. That is, the circulation period was 75-76 years, and the scientist predicted the next appearance for 1758. And everything would be wonderful, if not for one “but”: at the estimated time, the comet in the sky never showed up.

    Then the French mathematician Cleo decided to calculate its trajectory as accurately as possible. Including, taking into account perturbations of its trajectory due to the attractiveness of Jupiter and Saturn. It was a very tiring and long work, although he did not do this work alone: ​​he was assisted by the astronomer J. Lalande and the mathematician Madame Lepot. It turned out that the difference with previous estimates of the period of revolution of the comet is 618 days, and it will arrive at its perihelion on April 13, 1759. With a possible error for a month. The assessment turned out to be very accurate: the comet passed its perihelion on March 12. It was both a victory for mathematicians and a very good confirmation of Newton's laws.

    About this calculation, Laland later wrote: "We calculated six months from morning to night, sometimes without even looking up from food, and the consequence of this was that I upset my health for the rest of the days of my life. Madame Lepot’s help was such that without her we would never dare to undertake this enormous work, which consisted of calculating the distance of the comet from two planets - Jupiter and Saturn - for each degree of the celestial sphere for 150 years. "



    You can see the trajectory of Halley’s comet relative to the orbits of planets The

    second time a similar method was used in 1829, again when calculating the trajectory of Halley’s comet. Estimates gave the date of the comet’s return to the Sun on November 15, 1835. The comet returned on November 16, only a day late.

    As a result, a similar numerical method has proved its accuracy. But from the above story, you can clearly see how much and tedious work was required. Six months of work of three mathematicians took to calculate only one trajectory with well-known initial parameters! But to calculate the trajectories of a flight from Earth to the Moon, it was necessary to evaluate dozens, if not hundreds of trajectories, with different initial parameters. Otherwise, it is impossible to get, say, the exact speed, launch date, trajectory characteristics, possible errors, flight parameters of the station near the moon, etc. This could take years of life. That is, this solution was considered very difficult. Especially for a task that does not then have much practical meaning.

    At the same time, there was also confidence then that an analytical solution to the three-body problem could be found. At least allowing, in a first approximation, to evaluate the flight parameters.

    And the method proposed by Jules Verne seemed to give an answer to this question. After all, mathematically, within the framework of initial assumptions, he was impeccable. Yes, he did not take into account the dynamics, but then many expected that this assumption was normal, and the real trajectory of the projectile from Earth to the Moon would look something like this.

    image

    Calculation from the book “Spaceships. (Interplanetary communications in the fantasies of novelists) ”1928

    And the mathematical beauty, coupled with the great popularity of the novel, played a cruel joke: using this method, practically all the pioneers of space exploration - Goddard , Sternfeld , Obert , Max Valle and many others - evaluated flight . Sometimes, extremely rarely, they also estimated not a direct flight, but by the ellipse of the Earth’s satellite. But this method also had its drawbacks. For example, for some reason many were sure that to reach the moon it was enough to fall into its sphere of action; further, they say, the attraction of the moon should do the rest. Although, as was said above, the possibility of flying around in the three-body problem has already been proved.

    A good illustration would be Werner von Braun’s article in the weekly journal of October 18, 1952, in which von Braun described how, in his opinion, an expedition to the moon would look like.





    The article itself can be downloaded here.

    From the illustration it is clearly visible that it is precisely Jules Verne’s scheme that is presented there: first, reaching a neutral region between the Earth and the Moon, and subsequently falling onto the Moon under the influence of its attraction. The cannon has long been replaced by a rocket, but the basic principle has remained unchanged. Although almost a hundred years have passed since the publication of the novel.

    But soon the approach to flights from Earth to the Moon began to change. Including, thanks to the development of electronic computers.

    To be continued .

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