From Earth to the Moon. History and mathematics. Part 2

Continued. The first part is here .

The solution of the problem

Attitude towards flights to the Moon began to change at the end of 1953, when the head of the department of applied mathematics at the Mathematical Institute of the USSR Academy of Sciences Mstislav Keldysh summoned a postgraduate student Vsevolod Egorov and instructed him to calculate the trajectory of the flight to the Moon. And as close as possible to reality. When Egorov asked him about the dates, Keldysh answered: " Hurry. Results are needed today."
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The department was originally created to calculate the thermonuclear warhead. Then Keldysh will receive the “Hero of Socialist Labor” for this work. But just in 1953 it was allocated in a separate organization. This allowed Keldysh to more widely vary the tasks that he could set for his staff and graduate students. Fortunately, besides the calculation of thermonuclear ammunition and nuclear reactors, the institute also resolved issues related to ballistic missile flight support, helping missilemen to solve seemingly insignificant, but for really important tasks like the assessment of guaranteed fuel reserves.

I would also like to note that the team then was both very strong and very young. Employees in the memoirs are often called "Keldysh boys." And it is not surprising that it was almost completely formed in 1952-1955 from students of close courses in the degree of Moscow State University. As a result, almost everyone in this team was between 20 and 30 years old. If you select only the team that dealt with rocket-space problems, then over 30 years there were only two people: Actually, Mstislav Keldysh and Dmitry Ohotsimsky .

The team was young. Cosmos stirred blood, and tasks that had not been solved over the centuries seemed easy and understandable. Among other things, just in 1953, Egorov also organized a permanent seminar devoted to space at the institute.

As a result, according to the articles, in 1953-1955, Egorov successfully solved the “flat” task of flying to the moon, and in 1956-1957 - spatial.

Of course, he would hardly have managed in such a short time if Keldysh had not handed over to him a so-called specialized digital machine - SPM. Actually, this fact itself is quite remarkable, since the beginning of the 50s is the dawn of digital technology. In particular, our first computer MESM was officially launched into regular operation only on December 25, 1951. But any experts understood that such a task could not be solved without a computer.

It would be very appropriate to put a photo of this computer. But, alas, I never found it. It is good that its characteristics were indicated in an article devoted to flying around the moon. The speed of ~ 100 operations per second, with 64 memory RAM, permanent memory on magnetic drums. Modern search on sites also allows you to say that SKB-245 developed it, and Malinovsky took part in its development, in early 1952.

Although, perhaps, in these characteristics lies the answer, why it was transferred to the calculation of the flight paths to the Moon and why it is now so difficult to find.

Just 100 operations per second is a weak result for those times. For example, the BESM-1 had a speed of 8,000–10,000 operations per second, with a RAM of 2,047 cells, and a Strela-1 had 2,000 operations per second, a RAM of 2,048 words.

Serious computers began to solve serious problems, but these middling were transferred to solve various additional problems, and then completely forgotten.

But in any case, computers were then at the forefront of technology, required to solve many problems, and the transfer of computers to such a task speaks volumes.

Silver winners

Chronologically, the work of Egorov was, indeed, the first. And, most importantly, it was not just a theoretical study. It was thanks to this work that they paved their way to our natural satellite "Luna-1", "Luna-2" and "Luna-3" in 1959.

But, as often happens, close ideas come to different people at about the same time. So it was here: in the 50s, the problem of flying to the moon was solved by several more people.
In the USSR, it was Professor Gleb Chebotaryov . Then he worked (and in 1964 became its director) at the Institute of Theoretical Astronomy of the USSR Academy of Sciences. It was a specialized institute created to study celestial mechanics. I, alas, do not have the exact text of his work; judging by the references in other materials, he then considered several special cases. But his work is still interesting, since he most likely did not have a computer. However, it is also possible that he had specialized electromechanical differential analyzers at his disposal. Then they were used for such purposes.

Since the work of Egorov was then “closed”, and Chebotarev, on the contrary, worked in a completely open civil institute, the work of the latter in 1955-1957 became much more famous. She was written about in popular science literature, and later claimed that it was she who influenced the trajectory of "Luna-3" and much more.

For example, when in the USA after the launch of the satellite, the Soviet media rushed to analyze, they decided that the USSR was already conducting three different projects related to the flight to the moon. And the first project is led by Professor Chebotaryov, and the second is a graduate student Egorov. Here is an article about this:
andreyplumer.livejournal.com/227077.html

So the situation was in the USSR, but in the USA there were works on flights to the Moon.

USA

The first swallow of exact mathematical research was in 1956. Earlier this year, Robert Burchem of the RAND Corporation offered to use the Tor-Eable rocket then being developed for the lunar mission. On May 28, 1956, the secret report "General Report on the carrier of lunar probes" was released. It considered the possibility of launching to the moon using the Atlas launch vehicle. Interesting, but this report on the site «RANDcorporation» is not yet. But there are two following, from June 1956. Actually, it was this organization that was engaged in lunar trajectories in 1956-1958, until after the creation of NASA this was not assigned to JPL. And they worked out in detail the idea of ​​lunar satellites, which later became the first Pioneer probes.

You also need to mention Erica Krafft (1917-1984). He was one of the specialists who worked during the Second World War in Peenemünde. Like many other German specialists, he later came to the United States. Krafft is best known as the author of the Republic of Belarus "Centaurus". But he conducted and very serious theoretical studies. His multi-volume "Space Flight" in the 60s and released with us. It is believed that he did not participate in the Apollo program only because at one time he quarreled with von Braun. So he, too, in the period 1955-1957, clearly gained access to the computer and analyzed many features of the "lunar" trajectories. He is also one of those few people who was buried in space.

So what became clear with the help of a computer?

To begin with, a rather unexpected fact became obvious: there can be no capture by the Moon for an object launched from the Earth, in the sphere of its operation. At least on the first turn. The flight speeds inside the sphere of the moon were greater than the local parabolic. In other words, the vehicle launched to the moon can either get to the moon, or fly past it at a hyperbolic (relative to the moon) speed, after which it can either return to the earth or become a satellite of the sun.

The second fact concerned the analysis of possible flight paths. Look at the diagram



This is a conditional view from above on the plane of motion of the moon. The arrows indicate the rotation of the Earth around its axis (O) and the orbital motion of the Moon. Theoretically, all specified flight paths are possible. Including direct flight AB, with sufficient speed. That's just it is the most sub-optimal of all. The most energetically favorable trajectory on the scheme is VG. Just because it makes the most of the rotation of the Earth. After all, the angular velocity of rotation of the Earth is not so small. At the equator, it is 460 m / s. In the plane of the moon is somewhat smaller. But still, 300-400 m / s is not at all an extra addition to the initial speed, as when flying to the moon, even tens of meters per second sometimes change the picture of the flight. For example, again, you can remember the "scheme" of Jules Verne. According to Garce's calculations, the lowest possible speed of flight to the Moon, with a rather serious assumption, is 11051 m / s. In this case, the second cosmic velocity (that is, the speed at which the projectile will fly infinitely far) under those conditions would be 11 188 m / s. The difference is only 137 m / s.

If we translate the above into the language of mathematics, then a more optimal trajectory is one that has a larger angle between the starting point, the center of the Earth and the direction to the Moon. That is the angle of the VOA in the diagram.

The above is the so-called plane mission to the moon. That is, the task that considers flights in the plane of the moon. Since it requires some simpler calculations, it was solved first. And immediately after the decision, it became obvious that the chances of a real flight inside the orbital plane of the Moon are quite small. Just because for this it is necessary that the cosmodrome be on this very plane. At the same time, the moon's plane changes its inclination to the Earth’s equator from 18 degrees 18 minutes to 28 degrees 36 minutes with a period of 18.6 years.

But any cosmodrome located on the territory of the USSR will be guaranteed outside the orbital plane of the moon. So, have to fly outside of its plane. Again, from the point of view of mathematics, for this it is necessary that the plane of motion of the apparatus simply intersects at the desired point the plane of motion of the moon.

Below, for example, the diagram of the flight station "Luna-2"



As often happens, such a scheme had its own problems. In particular, it is more demanding of energy. But, worst of all, with direct flights, the most optimal phase angle is simply not achievable.



Here is a diagram. To simplify it, the polar orbit of the ship on the way to the Moon was chosen, and the section of the figure passes through the axis of rotation of the Earth and the plane of the Moon's orbit. So suppose that the spaceport is located at the latitude of AB. Theoretically, you can fly along the BS curve, but thanks to the rotation of the Earth, it is always possible to adjust the launch moment under the AC curve. But, alas, as you can see, even in this case, the phase angle of the AOW is far from optimal. Moreover, since the Moon rotates around the Earth with a period of about 28 days, at some points its location allows you to fly only along the DB curve. And the energy curves of the AU and the database are very different.

For example, according to a ballistic report prepared for the Luna-3 flight, when launched on October 4-6, 1959, the weight loss in the load relative to the ideal case was only 6-26 kg. But when you start October 17-19, the loss was already 418-444 kg. In the case of Luna-3 (launched on October 4, 1959), the total weight of the entire payload was 435 kg. So on certain days, the third step of the "seven" might not even bring itself to the Moon. In other words, the optimal start date in this method is once a month.

An even more unpleasant nuance was that, as mentioned above, the angle between the plane of rotation of the moon and the earth's equator is constantly changing, with a period of 18 years. And with such a flight, the best start dates will be only once in 18 years.

Enough is the fact that just 1959 was the “worst” year for launching from Baikonur and favorable for launching from Cape Canaveral. But since we then had quite powerful rockets, this fact remained almost unnoticed.

Also, when calculating such orbits, it turned out that it is necessary to take into account the influence of the Sun, and not just the Moon. Already at the first calculations, the importance of the inclination of the orbit became obvious. And also why the orbits of all the planets are approximately in the same plane (the plane of the ecliptic). Just the orbit data is stable. For example, Lidov carried out such a calculation. Imagine that the Moon is in orbit with the same size of the major semiaxis, eccentricity, orbital period, etc., only at an inclination of 90 degrees to the plane of motion of the Earth. And then what will happen to her? It turned out that she would fall to the Earth very, very soon. Just 55 months. This result was very, very surprised both by astronomers and mathematicians. But already in 1959-1960, the "Luna-3" confirmed the correctness of the calculations, falling to Earth under the influence of this effect.

It was along such trajectories that flew to the moon in 1958-1960. But rather quickly, a new method was proposed, which at the same time made it possible to maximize the payload, and at any latitude of the cosmodrome, and greatly reduce the waiting time for the launch window. If you had to wait for 18 years with a direct “cannon” flight, and with a direct launch from the cosmodrome, the window opened once a month, then with the new method it was possible to launch rockets at least every day. Even twice a day.

And from the point of view of mathematics, it is very simple. You just need not try to immediately get off the ground toward the moon when you start from Earth. You can first go into Earth orbit, wait for the phase angle to be optimal, then go to the Moon.



Here is a diagram. Point A is the moment of launch. AB - an exit to a low orbit of the satellite of the Earth. BV - free flight in orbit. And at point B, the transition to the flight path to the Moon. It can be seen that the BOC angle is ideal, which means that the method provides the maximum payload. Actually, now almost all the vehicles to the Moon fly just like that.

This method we have proposed Eneev . And it was developed in detail at the end of 1959.

Despite his beauty in terms of mathematics, he demanded quite complex technical solutions. It was necessary to develop a rocket block, which could launch in zero gravity, a vacuum and after tens of minutes of free flight in Earth orbit. And all this time he had to maintain a strictly defined orientation.

In order to accurately convey the importance of the method to the rocket men, a small plan was even developed. Here is how the story of Platonov remembers:

The report on the new flight scheme was supposed to be conducted by Okhotsimsky in relation to starts to Venus and Mars.

“Dmitry Evgenievich brilliantly, and in his own way, solved the problem of the gradual introduction into the consciousness of S.P. The queen and the chief designers of understanding the inevitability of the required design development. A poster was prepared for the meeting at Keldysh with a large number (about 8) of calculated “bad” variants of flights to Venus and Mars with unusable payload weights of these options and at the end - with two acceptable options marked with an asterisk. It must be said that this poster was made in a purely academic style - black ink on the floor of a straightened and so-folded sheet of paper paper that these “good” options, marked with an asterisk, were not sitting in the “main” soft armchairs.

Dmitry Evgenievich would not have been himself if he had immediately switched to the last two options. Instead, he took turns, one by one, to describe in detail, from top to bottom, all the calculated boundary-value problems and explain all the ballistic shortcomings of the continuous overclocking method. In principle, everything became clear to those present after analyzing the second option, and somewhere in the middle of the story about the third, which is not very suitable flight option, Sergey Pavlovich asked: “Dmitry Evgenievich! And what do you have down there, with an asterisk? “I received a very polite answer:“ Sergei Pavlovich, I will definitely say this later, ”and the narrative of the circumstances of the next version continued. After another one or two options, the situation repeated itself, and it repeated for the third time. This time, Sergei Pavlovich did not ask, but was very annoyed (he just blew up) demanded: “Dmitry Evgenievich! Yes, tell me, finally, what do you have there with an asterisk “?”. The atmosphere is very heated, and in the ensuing silence came the quiet and calm voice of Mstislav Vsevolodovich: “Dmitry Evgenievich! Well, please follow Sergei Pavlovich's request! ”A detailed story about the“ asterisk ”followed. And then after some silence a detailed discussion took place, and a purely epochal decision was made — to make a star! "

Again, one point needs to be clarified. So far, from time to time, there are ideas about assembling lunar missions in Earth orbit (for example, on the ISS). In many ways, this is the legacy of the ideas of the 50s (Werner von Braun and others), who assessed such a flight, still not knowing the features of the lunar trajectories. Or, alternatively, they are talking about launching the lunar stations into Earth's orbit by passing cargo, followed by launching to the Moon. As you can see, the launch on the first orbit to the Moon is possible only with very accurate phasing of the orbital plane of the Earth satellite with the flight path to the Moon. There is practically no chance that an orbit destined for another device will allow it. So, you need to wait in the orbit of the right time. Given the trajectory of the moon, a similar window opens only twice a month. And taking into account the requirements for the illumination of the moon - even once a month. Moreover, such a window may not be optimal either, since it may well happen that at the moment of coincidence of the planes the station will not be at the required point B, but in B or on the other side of the Earth. And this will greatly change the phase angle and increase energy.

As a result, the required starting time can be expected within a few months. And it is necessary that the device was designed for such modes of operation. With that, to the moon, in fact, fly only a few days.

In other words, launching to the Moon from an arbitrary orbit of the Earth’s satellite is not at all the best solution. Of course, if a tug with a nuclear or propulsion, which can compensate for many errors during breeding, is waiting for the Earth orbit, this option is acceptable :) But in all other cases it is better to start from Earth.

Something like an afterword

I am sure that the essay above rather accurately describes the approach that was at that time to fly to the moon. For this, we had to analyze a lot of documents in different languages, and the overall picture was just that. Until the 50s, most of the authors evaluated the flight according to the scheme of Jules Verne. And after the 50s, everyone started to refer to the calculations of the above authors. But anyway. After all, the formulation of the problem was known until the 50s. Numerical methods for solving differential equations were known, there were a variety of instruments to speed up calculations. From arithmometers to specialized differential calculators. So, theoretically, there could be a person who decided to put the years of his life on such calculations. And I learned the truth about such flights long before the advent of a computer. But if he was in reality?

It may very well be. History, as usual, is much more complicated than it first appears.
Reading the article by Friedrich Zander "The Theory of Interplanetary Travel" of 1922-1925 from this collection , I noticed a rather interesting footnote under the words "The device left to itself describes a complex curve" devoted to the trajectory of the flyby of the Moon.

“These curves are partially explored by Stromgren in Copenhagen by mechanical quadrature. His research has been going on for 12 years. ”

Since there are no words "Approx. Editor’s ”is clearly Zander’s note from those years. Speech about the Swedish-Danish astronomer Svante Strømgren (1870-1947). He was a professor of astronomy at the University of Copenhagen and director of the Copenhagen Observatory.

Alas, I did not find any such works of his. Only modest references in other works. Perhaps, if he did a similar job, he did not publish. Maybe a war prevented her from publishing, and then death. Well, or published in a completely unknown journal. In any case, it should be recognized that if the work was done, it had no effect on the ideas of those years about the flights to the moon.

Actually this material is written on the basis of my book dedicated to the moon. And I am very grateful
lozga and Zelenyikot for support. If you like, I will try to publish on this resource posts about landing on the moon and on various private issues.