ON THE LOCATION OF SPACECRAFT IN A GIVEN NUMBER OF ORBITS Текст научной статьи по специальности «Математика»

UDC 519.21

Doi: 10.31772/2587-6066-2020-21-2-170-175

For citation: Egorythev G. P., Shiryaeva Т. A., Shlepkin A. K., Filippov K. A., Savostyanova I. L. On the location of spacecraft in a given number of orbit. Siberian Journal of Science and Technology. 2020, Vol. 21, No. 2, P. 170-175. Doi: 10.31772/2587-6066-2020-21-2-170-175

Для цитирования: О распределении космических аппаратов по заданному числу орбит / Г. П. Егорычев, Т. А. Ширяева, А. К. Шлепкин и др. // Сибирский журнал науки и технологий. 2020. Т. 21, № 2. С. 170-175. Doi: 10.31772/2587-6066-2020-21-2-170-175

ON THE LOCATION OF SPACECRAFT IN A GIVEN NUMBER OF ORBITS

G. P. Egorythev1, Т. A. Shiryaeva2, A. K. Shlepkin2*, K. A. Filippov2, I. L. Savostyanova3

1 Siberian Federal University 70, Svobodny Av., Krasnoyarsk, 660074, Russian Federation 2 Krasnoyarsk State Agrarian University 90, Mira Av., Krasnoyarsk, 660049, Russian Federation 3 Reshetnev Siberian State University of Science and Technology 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037, Russian Federation *E-mail: ak_kgau@mail.ru

Space vehicles are an expensive product. For example, just putting such a device into orbit costs at least one hundred million dollars plus the cost of the satellite itself and scientific equipment it carries. However, the current state of human civilization does not allow us to do without the presence of satellites in orbit. There were 2,062 active satellites in the international database as of March 2019. Compared to 2018, the number of new devices increased by 15 %. Experts warn that in the coming years, the world is expecting a «satellite boom» with a projected increase in the number of devices of about 15-30 % annually. All these satellites are rather different. Currently, several orbits are used for placing satellites on them, depending on the tasks they solve. A geostationary orbit is used for live television broadcasting. Low satellite orbits are used for communication between satellite phones. There are some orbits for navigation systems (GPS, Navstar, GLONASS). Naturally, under these conditions, there is a problem of placing spacecraft over a given number of orbits, with some restrictions on the location of the spacecraft in certain orbits, depending on the purpose of the spacecraft. The solution to this problem is considered on the condition that the number of spacecraft coincides with the number of possible orbits in which they can be placed with some additional restrictions on the possibility of their placement in orbit. Several solutions to this problem are obtained that allow us to calculate the number of possible combinations for such placement of spacecraft over a given number of orbits.

Keywords: satellite, orbit, substitution, permanent.

О РАСПРЕДЕЛЕНИИ КОСМИЧЕСКИХ АППАРАТОВ ПО ЗАДАННОМУ ЧИСЛУ ОРБИТ Г. П. Егорычев1, Т. А. Ширяева2, А. К. Шлепкин2*, К. А. Филлипов2, И. Л. Савостьянова3

1 Сибирский федеральный университет Российская Федерация, 660074, г. Красноярск, просп. Свободный, 70

2 Красноярский государственный аграрный университет Российская Федерация, 660049, г. Красноярск, просп. Мира, 90

3Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

*E-mail: ak_kgau@mail.ru

Космические аппараты - дорогостоящий продукт. Например, только вывод такого аппарата на орбиту обходится минимум в сто миллионов долларов плюс стоимость самого спутника и научной аппаратуры, которую он несет. Однако современное состояние человеческой цивилизации уже не позволяет обходиться без наличия космических аппаратов на орбите. В международной базе данных на март 2019 г. числилось 2062 действующих спутника. По сравнению с 2018 г. рост числа новых аппаратов составил 15 %. Эксперты предупреждают, что в ближайшие годы мир ожидает «спутниковый бум» с прогнозируемым приростом количества аппаратов порядка 15-30 % ежегодно. Все эти космические аппараты сильно отличаются друг от друга. В настоящее время используется несколько орбит для размещения на них спутников в зависимости от решаемых ими задач. Геостационарная орбита используется для прямого телевещания. Низкие спутниковые орбиты используются для связи между спутниковыми телефонами. Свои орбиты существуют для спут-

ников систем навигации (GPS, Navstar, ГЛОНАСС), военных спутников, спутников для различных научных исследований. Естественно, в этих условиях возникает задача распределения космических аппаратов по заданному числу орбит при некоторых ограничениях на нахождении космического аппарата на некоторых орбитах в зависимости от назначения космического аппарата. Рассматривается решение данной задачи при условии, что число космических аппаратов совпадает с числом возможных орбит, на которых эти космические аппараты могут находиться при некоторых дополнительных ограничениях на возможность расположения спутника на орбите. Получено несколько решений этой задачи, позволяющих вычислить число возможных комбинаций для таких распределений космических аппаратов по заданному числу орбит.

Ключевые слова: спутник, орбита, подстановка, перманент.

1. Introduction. In the international database as for March 2019, there were 2062 active satellites. Compared to 2018, the growth of the number of new devices is 15 %. Experts warn that in the coming years the world is expecting a "satellite boom" with a projected increase of the number of devices of about 15-30 % annually. At present, some companies such as SpaceX and OneWeb are implementing an extensive program to launch satellites into low Earth orbits to put into action the plans for creating a global Internet broadcasting network.

In this case, a number of problems naturally arise [1-6], in particular, the problem of choosing the most favorable orbits for satellites of one class or another, as well as the problem of allocating a given number of satellites over a given set of orbits with existing prohibitions on placing a satellite in some orbit.

The article provides a solution to the second problem under the following conditions: the number of satellites n coincides with the number of possible orbits in which they are placed; each satellite is forbidden to be in exactly one orbit; two satellites cannot be in the same orbit. There is a formula that allows us to calculate the number of possible combinations for such satellite distributions in orbits.

The mathematical model of the problem. By permutation we mean the dimension matrix 2 x n

1 2-

i h '

A permutation is called regular if k ф ik , i. e.

i1 Ф1, i2 Ф 2,

, i„ Ф n.

(1)

(2)

Theorem 1. If n = 1, then D1 = 0 . If n = 2, then D2 = 1. If n > 2, then

Dn = n!- ClnD[n_l) - C2n D(n_2) - - - Cnn~2)Dn-(n-2) -1

Proof. The first two conclusions of the theorem are a direct consequence of the definition of a regular permutation. Let n > 2. Let us consider the irregular permutation

b-ni ••• nk-

a1 ••• n1 ••• nk

n

an

Thus, in accordance with the restrictions indicated above, exactly one regular permutation of degree n will correspond to each option for the placement of satellites in orbits. The problem of finding the number of regular permutations of degree n belongs to an extensive class of problems of enumerating permutations with restrictions on positions (with forbidden positions), and it is referred to as the problem of derangements (le problem des recontres); [7]).

It is equivalent to the classical problem of enumerating Latin rectangles having the size 2 x n ; for it, as the problem of combinatorics, many solutions of various types are known ([8], chapters 7, 8). The article presents several such solutions made using various methods, each of which has an independent value.

2. Recurrence formula for calculating the number of regular permutations. The number of regular permutations of degree n will be denoted by Dn.

which holds fixed precisely k symbols of n1,n2,---,nk, and it moves all the remaining symbols (in quantity k -1). At fixed n1,n2, —, nk the number of such permutations is Dn-k . Since the number of different sets without reiteration of the length k from the range {1, - • •, n} equals Cnk , then the total number of irregular permutations of this kind equals Ck . To complete the proof of the theorem, it is enough to subtract all irregular permutations of the form considered above for all k from the total number of permutations of the degree n (and this is exactly n!).

The theorem is proved.

Let us verify the conclusions of the theorem for some initial values n = 1, 2, 3, 4, 5, 6 . The values D1 = 0 and D2 = 1 follow directly from the definition

D3 = 3!-C\D2 -1 = 6-3 -1 = 2, D4 = 4!- C1D3 - C42D2 -1 = 24 - 4 • 2 - 6-1-1 = 9, D5 = 5!- C5 D4 - C52 D3 - C53 D2 -1 = = 120 - 5 • 9 -10 • 2 -10 -1 = 44,

D6=6!- c D5 - c6 D4 - c| D3 - c4 D2 -1= = 720 - 6 • 44 -15 • 9 - 20 • 2 -15-1 -1 = 265.

The results have been confirmed by listing all the regular permutations for the specified values n = 1, 2, 3, 4, 5, 6 .

3. The formulae for calculating the number of regular permutations based on the concept of a permanent matrix. Here we consider the original task as an enumeration problem when calculating the number of the permutations with fixed positions and we use the means of permanents (cyclic) (0,1) of incidence matrices [9; 10]. By the permanent of a square n x n matrix A = ( ai j) over a commutative ring we mean the expression

per (A)=X a1a(1) ana(n^ (3)

ae Sn

where summation is carried out for all Sn permutations

ct = (ct(1),--,o(n)) n-of the range {1,2,•••,n}, as well

as on all the diagonals of the matrix A.

Let In be identity n x n matrix, Jn - n x n -the matrix, in which each element equals 1/n, and An = (nJn - In) - the following matrix of order n:

Л =

( 0 1 1 0

1 1 1 1

1 1 I 1 1

0 1 1 0

(4)

D' = n!(1 " 1!+2!"-+(("1)'7,

n = 2,3,

(6)

Proof. By the "permanent" Laplace formula, decomposing per (An) into the elements of the first line, we have

Dn = Per (An ) = £per ( A,

k=2

' = 3,4,

(7)

Here per (An ((k)) (k = 2,3, — , n) - permanental minors from (0,1) matrices An (^) of the order (n -1),

which differ only in the arrangement of the same (0,l)-lines.

Herewith each matrix An ((k) contains precisely

(n - 2) zeros, and no two of them are contained in one line or in one column. It follows that at any k = 2,3, — , n

the following equation is valid

Per (An (lk)) = Per (An-i) - per (4,-2 ), and according to (7) and (5) we obtain

D' =(' -1)( per (A'_! )- per (An-2 )) =

(8)

Theorem 2. The following formula for the number Dn is valid

Dn = Per (An ) = per (nJn - In ), n = 2,3, — . (5)

Proof. The incidence matrix An = (■) of the order n

in (4) is constructed in the way that in accordance with the conditions (2) it has aiJ = 0, if i = j , and aiJ = l, if

i ^ j . We can easily see that for each "identity" diagonal of the matrix An in (4), each element of which equals 1, the permutation of the type (2) corresponds one-to-one. On the other hand, it is obvious that the multiplication of n units equal to 1 corresponds to each such diagonal in the permanent sum (3) for per(An). At the same time, the corresponding multiplications of the elements for each diagonal of the matrix An containing at least one zero element equal zero.

The theorem is proved.

Theorem 3. The following formula for the number Dn is valid

= (n - l)(Dn-i + Dn - 2 ), n = 3,4, — , n.

Therefore, by induction, taking into account the initial conditions Dl = 0, D2 = l, the formula (6) follows.

From here it follows that

The theorem is proved.

4. Derivation of the formula for calculating the number of regular permutations based on the inclusion-exclusion formula. This solution is related to the classical inclusion-exclusion formula (please see, for example, [10], § 3.2) by breaking the set Sn into disjoint

sets P0,P, — ,Pn, where Pi - set of permutations from

Sn with precisely i fixed positions, i = l, 2, —, n or, the

same with (n - i) constraint positions. Since

Sn = n! = || Til, |P| = [ n) Dn-i,

' I n!

i I i

!(n - i ))

i = 1,2, —, n,

We have

Let

1 = SD'

i !(n - i )!

(9)

Dz =X

i=0

Dl i!

- generating function of a power (exponential) type for a

D

numerical sequence { y^ ! j, and consequently,

Dl = i !rez \ D(z)z-i-1 \,i = 1,2

(10)

Here by formal deduction operator resz {A (z)} for the power Laurent series

A (z) = , r e Z,

k=r

we mean the coefficient at z 1 of the range A ( z ), i. e., by convention,

resz {A (z)} = (11)

Further, according to the coefficient method [11], we have in (9)

1= res'w {eWw-г-1},= reSz {^^(z)z>"1 ^,

i ! ^ >'(n -1)!

and we sequentially obtain

1 = £(resw {eww-i-1}xreSz {D(zH =

=0 V

i=0

where in the last equation the added elements of the sum equal 0 by the definition of the operator resz. Further, by regrouping the terms of the sum and putting the operator £ under the symbol of the operator resz, we obtain

i =o

1 =

~ ад

resz{D (z)z( {eWw-i-1}])} =

(summation in square brackets by index i: substitution rule, substituting w = z )

= resz{D(z)z-n-1 [ewWz} = resz \ez D(z) That is, we have

D (z )

-n-1

= 1 resz

1 - z

1 -z-n-1 У, n = 0,1,-

It immediately follows that

ezD(z) = —, D(z)=-V ' 1 - z V ' 1

and according to (10)

Dn = nlresz JD(z)

z n-1 } = n \res, <j ez —— z-n-1

1 - z

= n\resz

1 + XH) ^ X 1 + X(-1)hz

V i=0 1 \ У V i=0 у

k=0

k=0

i. e. the recurrence formula (8) taking into account the initial conditions of the form

D = 0, D2 = 1,

(13)

System solution (12)-(13). If we divide both sides of the equality (12) by ( n -1) !, multiply by zn 1, sum up at n > 3 , then taking into account the notation

Dn = Dn/,, n > 1

n /n!' we obtain the following equation

m D m D m D

y_D^zn-1 = z y Dn-1 zn-1 + z y Dn-2 zn-2 (14)

y (n -1)! ¿=3 (n - 2)! X (n - 2)! ( )

Further, from (14), taking into account the initial conditions (13) and the new notation

D n = V„ n > 2, D (z ):=y zn Dy = X zn,

n=1 n! n=2

we obtain

ад n M d

zn-1 - z ¿-^n-V zn-2 =

n=3

(n -1)\

n=3

(n - 2 )\

Dn

(according to the operator resz)

= n !(1 - - + — - - + |-1)n — |, n = 1,2,3,-.

1! 2! ^ n!)

5. Derivation of a formula for calculating the number of regular permutations based on combinatorial methods. This solution is connected to the use of traditional means of combinatorial analysis and generating functions of a power type related to the use of operations on power series (please see, for example [8-12]). Let Qn - the set of all n-permutations of type (2), and

Qn (k) - the set of these permutations from Qn that have

1 ^ k (k = 2,3, — , n). Obviously, the class of sets

Qn (k) - the partition of the set Sn . Let

Qn (k ) = Q1 (k)u Q2 (k), where the set Qln (k) contains all the permutations from Qn (k) whose k does not image into 1, and the set Q2 (k) contains all the permutations from Qn (k) whose k images into 1.

It stands to reason that for each fixed k we have

Q (k)| = Dn-!,|q2 (k)| = Dn-2, and consequently,

Qn (k) = Dn-1 + Dn-2.

Thus, we have

Dn |Qn| = ¿Qn (k) = X(Dn-1 + Dn-2),n = 3,4, (12)

= - z2 D2 +(1 - z )V-i^zn-1 =

2 V n 3 (n-1)\

= - z 2 +(1 - z )—{¿Г Dnzn } =

v —z П=з n\ 1 = - z 2 +(1 - z )—z {d (z )-D2 z 2 j= (15)

= - z 2 +(1 - z )—\d D (z )}-(! - z )^2 z = = - z 2 +(1 - z )—j D (z )-D2 z 2 ^ =

=- z+(1 - z 1D (z . It follows from here that

m j-\ m ~

zXAV zn-2 = zX Dn-2 zn-2 =

n=3 (n - 2)! n=3

m ~

n-2

= zXDn-2 zn-2 = z{D(z)-zD1 z У = zD(z).

(16)

at n > 3 is valid.

Thus, the solution to the system (12) - (13) by virtue of (14) - (16) reduces to solving the following first order differential equation:

(1 - z)d-D(z) j - zD(z)- z = 0,D (0) = D'(0) = 0. (17)

The equation (17) has the following unambiguous solution

D (z ) = -1 + e-z—, V ' 1 - z

which is verified by direct substitution in (17). The end of the proof is carried out exactly as in Section 4.

6. Notices. From the Maclaurin decomposition for the number e-1 the known next important estimate of the number Dn follows immediately:

lim Dn =1. (19)

n! e

(18)

n=3

In 1979, M. Aigner found the following unexpected probabilistic interpretation of that asymptotic estimate of the number Dn : «Let us suppose that n pages of a manuscript were mixed up by a gust of wind and then put arbitrarily.

The formula (19) states that for bigger n, the probability that there is not a single page in its place is higher

than Y3.» [13].

Another, more difficult classical combinatorial problem of the same type is the following: The problem of married couples: in how many ways can couples be arranged around a round table so that women and men alternate, and none of wives is sitting next to her husband?

In 1934, by listing enumerated n -permutations of the corresponding type, Jacques Toucheard found the following formula for the number Un - ways to place married couples of the above type [14; 15]:

Un = per (nJn - In - Pn ) =

2n | n-'

'=0

2n-1

(n -1)!'

(20)

where the matrix Pn - permutation matrix of the degree n with l on the places (l,2),(2,3), — ,...

Let us note that the formula (20), as well as the formula (6), can also be used to calculate options for the placement of satellites on a given number of orbits with a different set of conditions.

Conclusion. The problem of the possible distribution of satellites over a given number of orbits, depending on the purpose of a satellite, is solved. It is assumed that the number of satellites coincides with the number of orbits on which they can be placed. Several solutions to this problem have been obtained, which make it possible to calculate the number of combinations for distributing satellites over a given number of orbits.

References

1. Dr. Kelso T. S. Basics of the Geostationari Orbit. Available at: http://www.celestrak.com/columns. (accessed 26.03.2020).

2. Treaty on principles governing the activities of States in the exploration and use of outer space, including the moon and other celestial bodies. Available at: http://www.un.org/ru/documents/decl_conv/conventions/ outer_space_goveming.html (accessed 26.03.2020)

3. Fateev V. F., Minkov S. [New direction of development of remote sensing of the Earth]. Izv. vuzov. Instrument making. 2004, Vol. 47, No. 3, P. 18-22 (In Russ.).

4. Lebedeva A. A., Nesterenko O. P. Kosmicheskie sistemy nablyudeniya. Sintez i modelirovanie [Space observation systems. Synthesis and modeling]. Moscow, Mashinostroenie Publ., l99l, 224 p.

5. Pob'ezdkov Yu. A. Kosmicheskaya s"emka Zemli 2006-2007 gg. [Space survey of the Earth 2006-2007]. Moscow, Radio engineering Publ., 2008, 275 p.

6. Nevdyaevl L. M., Smirnova A. A. Personal'naya sputn'kovaya svyaz [Personal satellite communication]. Moscow, Eco-Trends Publ., 1998, 216 p.

7. Fitken A. C. Determinants and Matrices. Edinburgh, 1939, 201 p.

8. Riordan J. An introductions to combinatorial analysis. John Wiley & Sons, Inc., New York, 1982, 288 p.

9. Minc H. Permanents. Encycloped'a of Mathematics and Its Appl'cat'ons. 1978, Vol. 6, P. 65-70 p.

10. Egorychev G. P. D'skretnaya matemat'ka. Perma-nenty [Discrete Math. Permanents]. Krasnoyarsk, Siberian Federal University Publ., 2007, 272 p.

11. Egorychev G. P. Integralnoe predstavlen'e ' vy-ch'slen'e komb'natornyh summ [Integral representation and computation of combinatorial Math.]. Novosibirsk, Nauka Publ., 1977.

12. Kuzmin O. V. Introduction to combinatorial methods of discrete mathematics. Irkutsk, ISU Publishing House, 2012, 113 p.

13. Aigner M. Combinatorial theory, Springer-Verlag, New York, 1979, 90 р.

14. Touchard J. Sur un proble'me de permutations. Ed. C. R. Acad. Sci. Paris, 1934.

15. Kaplansky I. Solution of the proble'me des me'nages. Bull. Amer. Math. Soc. 1943, Vol. 49, P. 784-785 p.

Библиографические ссылки

1. Dr. Kelso T. S. Basics of the Geostationari Orbit [Электронный ресурс]. URL: http://www.celestrak.com/ columns (дата обращения: 26.03.2020).

2. Договор о принципах деятельности государств по исследованию и использованию космического пространства, включая Луну и другие небесные тела (1967) [Электронный ресурс] / Офиц. cайт ООН. URL: http://www.un.org/ru/documents/decl_conv/ con-ventions/outer_space_goveming.html (дата обращения: 26.03.2020).

3. Фатеев В. Ф., Миньков С. Новое направление развития МКА дистанционного зондирования Земли. // Изв. вузов. Приборостроение. 2004. Т. 47, № 3. С. 18-22.

4. Лебедев А. А., Нестеренко О. П. Космические системы наблюдения. Синтез и моделирование. М. : Машиностроение, 1991. 224 с.

5. Подъездков Ю. А. Космическая съемка Земли 2006-2007 гг. М. : Радиотехника, 2008. 275 с.

6. Невдяев Л. М., Смирнов А. А. Персональная спутниковая связь. М. : Эко-Трендз, 1998. 216 с.

7. Fitken A. C. Determinants and Matrices. Edinburgh, 1939. 201 с.

8. Riordan J. An introductions to combinatorial analysis // John Wiley & Sons, Inc., New York, 1982. 288 p.

9. Minc H. Permanents // Encyclopedia of Mathematics and Its Applications. 1978. Vol. 6. P. 65-70.

10. Егорычев Г. П. Дискретная математика. Перманенты. Красноярск : Изд-во СФУ, 2007. 272 с.

11. Egorychev G. P. Интегральное представление и вычисление комбинаторных сумм. Новосибирск : Наука, 1977.

12. Kuzmin O. V. Introduction to combinatorial methods of discrete mathematics. Irkutsk : ISU Publishing House, 2012. 113 p.

13. Aigner M. Combinatorial theory. SpringerVerlag, New York, 1979. 90 р.

14. Touchard J. Sur un proble'me de permutations / ed. C. R. Acad. Sci. Paris, 1934.

15. Kaplansky I. Solution of the proble'me des me'nages // Bull. Amer. Math. Soc. 1943. Vol. 49. Р. 784-785.

Egorythev G. P., Shiryaeva Т. A., Shlepkin A. K., Filippov K. A., Savostyanova I. L., 2020

Egorychev Georgiy Petrovich - D. Sc., Professor, Siberian Federal University. E-mail: egorychev@sfu-kras.ru.

Shiryaeva Tamara Alekseevna - Cand. of physical and mathematical Sc., Associate Professor, associate Professor of the Department Information technologies and mathematical support of information systems; Krasnoyarsk State Agrarian University. E-mail: info@kgau.ru.

Shlepkin Anatoly Konstantinovich - D. Sc., Professor, Professor of the Department Higher mathematics and computer modeling; Krasnoyarsk State Agrarian University. E-mail: ak_kgau@mail.ru.

Filippov Konstantin Anatolevich - D. Sc., Associate Professor, Professor of the Department Information technologies and mathematical support of information systems; Krasnoyarsk State Agrarian University. E-mail: info@kgau.ru.

Savostyanova Irina Leonidovna - Cand. Sc., Associate Professor of the Department of IES; Siberian State University of Science and Technology. E-mail: savostyanova@sibsau.ru.

Егорычев Георгий Петрович - доктор физико-математических наук, профессор; Сибирский федеральный университет. E-mail: egorychev@sfu-kras.ru.

Ширяева Тамара Алексеевна - кандидат физико-математических наук, доцент; Красноярский государственный аграрный университет. E-mail: tas_sfu@mail.ru.

Шлепкин Анатолий Константинович - доктор физико-математических наук, профессор; Красноярский государственный аграрный университет. E-mail: ak_kgau@mail.ru.

Филиппов Константин Анатольевич - доктор физико-математических наук, доцент; Красноярский государственный аграрный университет. E-mail: filippov_kostya@mail.ru.

Савостьянова Ирина Леонидовна - кандидат педагогических наук, доцент кафедры ИЭС; Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева. E-mail: savost-yanova@sibsau.ru.