RU2558959C2 - Method for monitoring collocation at geostationary orbit - Google Patents

Abstract

FIELD: physics; control.

SUBSTANCE: invention relates to controlling movement of a group (cluster) of spacecraft, primarily geostationary satellites. According to the method, the nodal lines and apsidal lines of orbits of monitoring spacecraft and adjacent spacecraft are kept orthogonal. The sum of eccentricities of the orbits must approximately 0.0004, and the inclination of the orbit of the monitoring spacecraft relative to the orbit of the adjacent spacecraft must not be less than (14-15) arcseconds. To this end, regular adjustments are made to keep the ends (phase) of inclination and eccentricity vectors in the required aiming regions. Longitudes (orbital periods) are also adjusted so that the origin of the coordinate axes (deviations along the orbit and on the radius vector) coincides within given boundaries with the centre of the distancing ellipse from the adjacent spacecraft. The centres of the aiming regions are redefined when adjusting the strategy of controlling movement of the centre of mass of the adjacent spacecraft. When the reception level, at the monitoring spacecraft, of radiation of antennae mounted on the adjacent spacecraft falls, a switch is made to a mode for receiving information from ground-based antennae for the adjacent spacecraft. In case of consistent reception, at the monitoring spacecraft, of signals of said antennae of the adjacent spacecraft, direct around-the-clock monitoring of the adjacent spacecraft with two monitoring spacecraft is carried out within 12 hours. Said monitoring spacecraft are located at diametrically opposite sides of said distancing ellipse.

EFFECT: maintaining spacecraft at an operating position without interfering with other spacecraft and monitoring of adjacent spacecraft.

3 cl, 6 dwg

Description

The present invention relates to the field of space technology, and to achieve a technical result, it uses the settings of the orbits of spacecraft (SC) in the phase planes of the inclination and eccentricity vectors.

As a rule, the collocation of spacecraft is carried out according to agreed schemes. All analogue circuits are reduced to equidistance of the aiming points of vectors e _n [e _n , (Ω + ω) _n ] (n = 1, 2, ...) and i _n [i _n , Ω _n ] (n = 1, 2, ... ) in the corresponding phase planes of the spacecraft and maintaining the ends of the vectors e _n and i _n inside the corresponding regions of the selected radii, the centers of which are the corresponding aiming points. An ideal option for two spacecraft is the separation of the longitudes of the ascending nodes (Ω _n ) and right ascensions of the perigee (Ω + ω) _n of the aiming points by 180 °, and the arguments of the latitude of the perigee of the spacecraft should be close to zero. For three spacecraft, the figure 180 is replaced by 120. This principle of collocation is well known, it follows from the prior art. However, behind the apparent simplicity of the schemes lies a complex and costly procedure for managing collocation vectors.

A known method of controlling a cluster of satellites in geostationary orbit (RU 2284950 C2, IPC B64G 1/10, B64G 1/24, B64G 1/44), which is taken as a prototype. According to this method, which includes measuring the orbital parameters of each spacecraft, determining from them the current values of the orbital elements of each spacecraft, comparing them with the required ones and making corrections to the orbital period, inclination and eccentricity of the orbit, maneuvers on each spacecraft are performed using small thrust engines, translating the vectors inclination spacecraft i _n (n - number of conditioned SC) in spaced relation to each other annular regions of allowable changes so that the angle between the line connecting the current position of the end of each vector with q ntrom its corresponding annular region, and the direction of the sun is equal to plus 180 ° magnitude of the RA Sun simultaneously conduct the correction of the eccentricity e _n vectors in order to convert these vectors in spaced relation to each other annular regions of allowable changes so that the line connecting the current the position of each vector with the center of the corresponding annular region, (hereinafter options):

1 - lagged behind the direction to the Sun by half the angular distance when the eccentricity vector moves around the circle of the natural drift within the annular region, then throughout the flight, the relative distance between the spacecraft is changed within the required limits due to compensation of the quasi-century increment of the inclination vector of each spacecraft in combination with correction of the eccentricity vector, in which at the moment the eccentricity vector passes the middle of the interval between the entry point of the natural drift circle to the annular region of the permissible change in the eccentricity vector and the exit point from this region, the line connecting the center of the circle of the natural drift and the center of the corresponding annular region of the permissible change in the eccentricity coincided with the direction to the Sun, thereby leading to a relative relative inclination and eccentricity vectors between the spacecraft;

2 - coincided with the direction to the Sun, then throughout the flight, the relative distance between the spacecraft is changed within the required limits due to compensation of the quasi-century increment of the inclination vector of each spacecraft without correction of the eccentricity vector, thereby leading to a relative relative inclination and eccentricity vector between the spacecraft.

Here, the "circle of natural drift" is the circle of the radius of the stable eccentricity (Appendix 1, figure 1).

The essence of this method is to synchronize the motion of the ends of the inclination vectors and the eccentricity of the SC orbit in the corresponding phase planes [i _x ; i _y ] and [e _x ; e _y ], where i _x = i · cosΩ; i _y = i sinΩ; e _x = e cos (Ω + ω); e _y = e · sin (Ω + ω); Ω is the longitude of the ascending node of the spacecraft orbit; ω is the latitude argument of perigee; (Ω + ω) = α _π is the right ascension of the direction to the perigee of the SC orbit, moreover, the synchronization of the motion of the SC in both planes, the synchronization is forced, since the relative orientation of the relative vectors of eccentricity and inclination ΔE and ΔI of 2-3 SC is not preserved with the annual cycle of evolution the eccentricity vector around the circle of natural drift and existing, subject to compensation only secular perturbations of the inclination vector of the semi-annual cyclic evolution of the end of the inclination vector around the circle of natural drift (When dix 2). Both cycles are dominant and owe their existence exclusively to the Sun, because the initial and current orientations of the inclination and eccentricity vectors relative to the Sun are necessary conditions for achieving a technical result. “As a rule, they do not use the elimination of only the secular perturbation in the joint control of satellites, since with inconsistent positions of the satellites with respect to the semi-annual perturbing members of the inclination vectors, their difference can be 0.05 °. However, a special choice of the initial positions of the satellite inclination vectors and with a small periodicity of the correction of inclination (i.e., satellites with relatively low thrust) can provide, when correcting only the secular part, the in-phase evolution of the position of the satellite inclination vectors along circles of radius 0.025 ° in such a way that the vector of their difference will keep close to a constant direction ”(prototype).

The disadvantages of the prototype are:

1 - the lack of the mathematical formula "a special choice of the initial positions of the inclination vectors"; no formula - no clear idea of what is being done;

2 - “a special choice of the initial positions of the inclination vectors” implies a special choice of the centers of the annular areas (aiming points) or proceeds from it, which also requires mathematical justification, but if we simply (this is not said in the prototype), the direct ascent of the Sun ( season) and the initial inclination vector (inclination modulus and the ascending node of the spacecraft’s orbit) determine the current position of the hodograph of the circular motion of the end of the inclination vector (“solar circle”) and the current vector of the change in the inclination increment, in Possible centers of annular areas located on locus circle of radius with center at the end of the initial inclination vector (Appendix 2);

3 - if the in-phase following of the ends of the inclination vectors of the SC orbits in the inertial space along the circles of natural drift, or along the boundaries of the annular regions of their permissible change takes place: from n points (the initial conditions of the SC) in the phase plane begins exactly in-phase (with or without secular perturbation correction ) motion along n paths, then the in-phase following of the eccentricity vectors of the SC orbits along their natural drift circles, or along the boundaries of the annular regions of their permissible changes by choosing the initial conditions ( rum as combined eccentricity vector of the spacecraft in magnitude and direction) can not be organized, in principle, even if the radius of the circle of forced movement is sustainable eccentricity. This obviously follows from the consideration of the formula (12 ') of Appendix 1. For example, with diametrically located perigee, the movement is not only not in phase, but also directed towards each other; from n points (initial spacecraft conditions) in the phase plane, non-phase (with or without retention correction) movement along n paths begins, since the angles θ at a single point in time differ from each other by the amount of directional mismatch on the perigee). Everyone is convinced of the impossibility of the in-phase motion of the ends of the eccentricity vectors of the SC orbits based on their own calculations. Only by the frequency of holding retention corrections, entailing significant energy costs, can the desired result be achieved;

4 - this is the main thing - observing, ideally, the constancy of the difference of the vectors (the distance between the ends of the inclination vectors, the eccentricity of all (n) SCs and the constancy of the distances between the ends of the inclination vectors and the eccentricity of each SC) is not a necessary and sufficient collocation factor providing guaranteed separation of the SC in true space and in phase planes. To ensure high-quality collocation, it is necessary to maintain the constancy of the spacing of the inclination and eccentricity vectors in Ω, and α _π , since for close and even intersecting regions of permissible changes in the ends of the vectors i _n and e _n, the discrepancies in Ω and α _π corresponding to their synchronous motion can reach about 90 ° . This is because, although the motion of the ends of the inclination vectors during natural drift with compensation of secular perturbations is uniform, the centers of the "solar circles" in the general case are not the origin of the phase plane [i _x ; i _y ]. If we take into account the differences in the average velocities of the inclination and eccentricity vectors, then the options are possible when

$${\Omega }_{\mathrm{one}}\approx {\mathrm{<figure-callout\; id="2"\; label="\Omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\Omega </figure-callout>}}_2\text{and:}{\omega }_{\text{one}}\approx \pi /2{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\text{2}}\approx \frac{3}{2}\pi ;\text{or}{\omega }_{\text{one}}\approx \frac{3}{2}\pi ,{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\text{2}}\approx \pi /2\left(\text{one}\right)$$

or

$${\Omega }_{\mathrm{one}}\approx {\Omega }_2+\pi \text{and}{\omega }_{\text{one}}\approx {\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2\approx \pm \pi /2\left(\text{one'}\right)$$

or (for i ₁ ≈i ₂ ≈0)

$${\Omega }_{\mathrm{one}}\approx {\Omega }_2+\pi \text{and:}{\omega }_{\text{one}}\approx {\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2\approx \pi ;\text{or}{\omega }_{\text{one}}\approx {\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2\approx 0\left(\text{one''}\right)$$

those. when the focal parameters p of the SC orbits, the dependence of which on the eccentricity is negligible [p = a (1-e ² ), where a is the semi-major axis of the orbit], practically coincide, as a result of which critical approach in the true space of two SC is inevitable the values of the difference between the inclination modules and the eccentricity modules.

Collocation in the prototype, as well as in analogues, is considered as a way to control the movement of the centers of mass, guaranteeing against spacecraft collisions. This problem is relevant, but only in principle, and is satisfactorily solved for two spacecraft (even at zero inclinations) under the conditions:

$${\Omega }_{\mathrm{one}}\approx {\Omega }_2\text{and:}{\omega }_{\text{one}}\approx 0{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\text{2}}\approx \pi ;\text{or}{\omega }_{\text{one}}\approx \pi ,{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\text{2}}\approx 0\left(\text{2}\right)$$

those. then, when the ascending nodes of the orbits are equal, for each of the orbits the line of nodes coincides with the line of apse, the directions to the ascending node and the perigee of one of the orbits coincide, the other are mutually opposite. The guaranteed minimum inter-satellite distance, with a real spacecraft eccentricity of ~ 0.00015, is 12.6 km.

Another task of collocation is not to interfere with nearby satellites to work for their intended purpose. If we focus on conditions (2), in the regions of the orbit nodes, with almost identical periods of revolution (deviation from stellar days rarely when more than 5 s), mutually alternating interferences between the spacecraft and the Earth arise.

And such a problem is best solved for two spacecraft under the conditions:

$${\Omega }_{\mathrm{one}}\approx {\Omega }_2\pm \pi /2\mathrm{and}:{\omega }_{\mathrm{one}}\approx {\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2\approx 0;\mathrm{and}l\mathrm{and}{\omega }_{\mathrm{one}}\approx {\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2\approx \pi ,\left(\text{3}\right)$$

those. then, for each of the orbits, the line of nodes is perpendicular to the line of the apse, and the line of nodes is mutually perpendicular. The control centers of all spacecraft located in a single area of latitude and longitude retention follow a unified collocation strategy by exchanging ballistic information.

However, for guaranteed collocation, a permanent process of exchange of ballistic information between spacecraft control centers is required. Such a process may fail, and failures will certainly occur. In addition, the fundamental impossibility of interaction between spacecraft control centers cannot be ruled out. It is easier to be in a state of autonomous collocation: when other spacecraft and their control centers are not involved in the collocation process. When setting such a task, it should be borne in mind that the line of nodes and the line of apses of the orbit of an adjacent spacecraft (SCA) can intersect at an arbitrary angle. Further, under the text, “autonomous” spacecraft means a spacecraft that “took” all responsibility for collocation in a given area of retention in latitude and longitude.

The idea of a low-cost autonomous collocation that does not impose any any significant obligations on the SKA control center (which means the presence or absence of actions to implement an agreed collocation strategy on the part of such a control center), which allows, by adjusting the inclination and eccentricity vectors, to bypass the rays from of all antennas, including global ones, on the SKA, without thereby creating shielding effects, it seems relevant and most effective. The idea of autonomous collocation (self-collocation) has no analogues.

It is possible to obtain and solve ballistic information about the SKA and the task of separating the inclination and eccentricity vectors in the self-collocation mode, for example, using the data - the results of measuring the orbit parameters from the NORAD international satellite tracking system, revealing the tactics and strategy of holding the SKA. This system works without errors of rotational prediction, the main component of which is the implementation of spacecraft retention using the engines of the correction system. Errors in e, i, ω are more than satisfactory.

The task of autonomous collocation, as shown by the geometry of the arrangement of the constituent elements of the inclination and eccentricity vectors in inertial space (figure 2) and the calculation of inter-satellite distances (figure 3), is optimally solved under the conditions:

$${\Omega }_{\mathrm{one}}\approx {\Omega }_2\pm \pi /2\mathrm{and}{\omega }_{\mathrm{one}}\approx {\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2,\left(\text{four}\right)$$

where index “1” corresponds to “autonomous” spacecraft, index “2” corresponds to SKA, i.e. then, when the lines of the nodes of the orbits of the "autonomous" and SKA intersect at a right angle, the lines of the apses of the orbits of the "autonomous" and SKA intersect at a right angle, the mismatch angle (ξ, Fig. 2) between the right ascension of the perigee and the ascending node of the orbit of the SKA is equal to the mismatch angle between the right ascension of perigee and the ascending node of the orbit of the “autonomous” spacecraft.

In figure 2, the numbers indicate:

1 - the orbit of the "autonomous" spacecraft;

2 - SKA orbit;

3 - ascending node of the "autonomous" spacecraft;

4 - perigee SKA;

5 - ascending node SKA;

6 - Earth;

7 - perigee of the “autonomous” spacecraft.

Summary results of calculations of inter-satellite distances, with the initial conditions of the spacecraft motion taken as a basis:

- sidereal circulation periods - 86164.1 s;

- eccentricity of the orbits - 0,00020;

- the inclination of the orbits is 1.5 arcmin,

are shown in figure 3. In figure 3, the numbers 1-4 indicate the lines of the minimum inter-satellite distances under conditions (4) and:

1- (i ₁ = 1'30``; e ₁ = 0,00020; i ₂ = 1'30 ''; e ₂ = 0,00020; ξ∈ [0-2π]);

2- (i ₁ = 1'30``; e ₁ = 0,00030; i ₂ = 1'30 ''; e ₂ = 0,00010; ξ∈ [0-2π]);

3- (i ₁ = 1'30``; e ₁ = 0,00030; i ₂ = 0'10 ''; e ₂ = 0,00010; ξ∈ [0-2π]);

4- (i ₁ = 0'10``; e ₁ = 0.00015; i ₂ = 1'30 ''; e ₂ = 0.00015; ξ∈ [0-2π]);

index “1” corresponds to “autonomous” spacecraft.

From figure 3 it follows that compliance with conditions (4) and tolerance for a minimum inter-satellite distance of 8 km is technically feasible.

Autonomous collocation on the principles of (4) also allows a mismatch according to any of the conditions (4) with respect to the nominal value of 90 ° to 25 °.

The idea of monitoring one spacecraft by another spacecraft, requiring the "autonomous" spacecraft to be at a safe technological and physical distance from the SKA and engage in direct monitoring of the spacecraft at the daily interval for the maximum possible time, is also in demand. Direct monitoring of the SKA at the daily interval for the maximum possible time leads to additional over-agreed costs for managing the center of mass of the "autonomous" spacecraft. Hereinafter referred to as an “autonomous” spacecraft, monitoring or (and) managing another spacecraft, is a monitoring spacecraft (MCA).

Ballistic information about the SKA and the problem of collocating with it, in addition to the above version of determining the orbit parameters using the international satellite tracking system, it is possible to obtain and solve by giving the ICA navigation and motion control system a range of transceiver radio equipment for measuring range and an optical star sensor of angular position.

In principle, the MCA can be located on the side with deliberate longitude spacing relative to the SCA, so that without interruption, the inter-satellite distance greater than the minimum allowed can be guaranteed. But there is a problem. The existing regulation on the standing of geostationary spacecraft presupposes retention in a region of no more than 0.1 ° in longitude relative to the nominal working position, then for sure separation of two spacecraft in longitude, both spacecraft should be in areas of less than ± 0.1 °, and the distance between them should be about 0.1 ° (74 km), but at such a distance from the SCA, the ICA can fall into the area where there will be a third-party (third) spacecraft with which it is also necessary to be in a state of collocation. This is a technically insoluble task. The collocation problem also becomes unsolvable by removing (spacing) the MCA from the SKA in longitude in the total area of SKA ± 0.05 ° relative to the nominal working position - the area is too narrow for relative movement maneuvers, especially if the SKA implements a maneuver plan inconsistent with the ICA control center . Given the population of the geostationary orbit, it is possible initially - perhaps later, from the long-range collocation scheme only by removing the MCA from the SCA in length. Since it is necessary to rely on the region ± 0.05 ° in latitude and longitude relative to the nominal orbital position that is uniform for SKA and MCA, it is necessary to be able to collocate the field in longitude and latitude on the same SKA, while performing the function of active tracking of SKA. This means that the parameters of autonomous collocation should be both the deviation of the MCA from the SCA in longitude, and the eccentricity, and the inclination of the orbit of the MCA. A collocation of eccentricity, inclination, and longitude “on your own” has not yet been considered.

It is not possible for one MCA to be around the clock in the SKA broadcast area. Although, in the presence of the necessary fuel supply (an order of magnitude greater than that usually produced in a stationary orbit), this is possible. With the diversity in longitude, you can count on round-the-clock reception of signals from the SKA “perpendicularly”. But from the above, this option is not feasible. However, you can be effectively within 12 hours / day "under SKA".

The stationary orbit is filled with signals from the Earth for a large number of devices that are not even in a stationary orbit, and the location of the MCA near the SCA does not guarantee the receipt of information intended specifically for the SCA. The second - information for SKA, most likely, will have more than one degree of protection, the keys to which will change according to an unknown (without the efforts of ground services) law and even at a speed that fundamentally excludes adaptation to the flow of information. Most of the problem can be removed only by removing some of the information from the SKA. So the way to find the MCA “under the SKA” is relevant, and there is no qualitative alternative to it (remember that it is unlikely to stand aside and receive signals “perpendicular”). In fact, this is a very likely and serious problem that must be borne in mind when deploying the system. Information from the SKA will help identify target information from the Earth, and both amounts of information will be useful to complement each other and present a single daily file that does not have information loss.

The invention consists of two parts:

1) - the ballistic part, offering the combination of SKA and MCA in longitude and the organization of collocation autonomous from the SKA to ensure that the ICA is “under the SKA”;

2) - the radio engineering part, which provides (the global antenna is 8.65 ° X8.65 °) the maximum period of time when it is possible to take information from the SKA, being under it, and justifying the quality and advisability of round-the-clock information collection via the Earth-SKA lines "And" SKA-Earth ".

The aim of the invention is the implementation of the above ideas.

This goal is achieved by the fact that in the monitoring collocation method in a geostationary orbit, which includes translating the inclination and eccentricity vectors to the boundaries of the aiming regions spaced apart from each other (areas of permissible variation of the inclination and eccentricity vectors), measuring the orbit parameters of each spacecraft, determining the current orbital values from them elements of each spacecraft and carrying out with the help of small thrust engines corrections of the period of revolution, inclination and eccentricity of the orbit, new operations consisting in the fact that, in order to organize collocation autonomous from the SKA, the latitude (inclination) and longitude according to the data of independent trajectory measurements, determine the strategy of controlling the motion of the center of mass of the SKA according to the data of independent trajectory measurements, in the process of holding, the position of the center of the aiming region is determined according to the inclination of the SKA, by making corrections of the inclination, the inclination vector of the orbit of the MCA in the phase plane is set so that the line of nodes of the orbit of the MCA becomes perpendicular to the line of nodes of the orbit of the SCA and the inclination e (i _min ) of the MCA orbit relative to the SCA orbit was at least (14-15) arcsec, eccentricity corrections are carried out to remove the direction to the perigee from the direction to the ascending node of the ICA orbit by the value of the mismatch angle between the directions to the perigee and the ascending node of the SCA orbit and maintaining such a position of the perigee within the specified limits of the aiming region according to the eccentricity, carry out regular complex corrections of the inclination of the orbit of the MCA to maintain a right angle between the lines of the nodes of the SC orbits within the specified limits of the scope Bani of inclination, to eliminate the age-old care component of inclination is exceeded and i _min longitude correction is performed (period) to the origin [ΔL; ΔR - respectively, the deviation along the orbit and the deviation along the radius vector] coincided within the specified limits with the center of the distance ellipse from the SCA, the centers of the aiming areas are redefined on the MCA according to the inclination and eccentricity of the MCA orbit when adjusting the strategy for controlling the motion of the center of mass of the adjacent SC and when the mismatch angles increase between the apses lines and the lines of the orbits of the ICA and SKA, in cases of a dangerous approach of the spacecraft, deviations are corrected, which are simultaneous corrections of longitude and eccentricity the orbits, after the autonomous collocation is organized by the passage of the MCA under the SCA, when the radius vector of the MCA is less than the radius vector of the SCA, the daily time interval is specified when the level of reception on the MCA of the radiation of antennas installed on the SCA is satisfactory, with a decrease in the level of reception on the MCA, they switch to reception mode information for SCA from ground-based antennas, in the case of reliable continuous reception of radiation from antennas installed on the SCA at the MCA, direct monitoring of the SCA is carried out by two MCAs installed on the diameter within 12 hours cial distancing opposite sides of the ellipse.

As the calculations show (which everyone can carry out), the relative motion of two spacecraft has quite definite laws, namely:

1 - projection of the relative motion of one spacecraft on the orbit plane of another spacecraft - an ellipse;

2 - the ratio of the minor axis to the major axis of the distance ellipse is 1: 2 (figure 5);

3 - a shift by ΔL along the orbit (in longitude in the Greenwich Coordinate System, a positive direction to the east) of one of the spacecraft leads to a shift with the same sign of the center of the old distance ellipse by AL along the coordinate axis "Deviation along the orbit".

4 - the distance ellipse is always oriented by the semi-major axis along the coordinate axis “Deviation along the orbit”;

5 - when the eccentricities of the spacecraft orbits change, the major and minor semi-axes of the distance ellipse are determined by the relations:

$$a"=a\text{'}\text{'}\frac{e_{\mathrm{one}}^{\text{'}\text{'}}+e_2^{\text{'}\text{'}}}{e_{\mathrm{one}}^{"}+{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">e</figure-callout>}}_2^{"}};\left(\text{5}\right)$$

$$b"=b\text{'}\text{'}\frac{e_{\mathrm{one}}^{\text{'}\text{'}}+e_2^{\text{'}\text{'}}}{e_{\mathrm{one}}^{"}+{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">e</figure-callout>}}_2^{"}};\left(\text{6}\right)$$

where the indices "'" and "' '" refer to the time, respectively, before and after orbital maneuvers;

a ', a' '- respectively, major semi-axes before and after orbital maneuvers, km;

b ', b' '- respectively, the minor semiaxes before and after the orbital maneuvers, km;

e ₁ , e ₂ are the eccentricities of the orbits of the first and second spacecraft, respectively.

Figure 5 shows the distance ellipse calculated according to the inter-satellite distance program for the next set of initial conditions (lines of nodes are orthogonal, lines of apses are orthogonal).

NU 1:

- the longitude of the ascending node is 0;

- argument latitude perigee - 0;

- inclination - (0-1.5) ang. min .;

;- eccentricity $$e_{\mathrm{one}}^{\text{'}\text{'}}=\mathrm{0,0002}$$

;

NU 2:

- longitude of the ascending node - 90 °;

- argument latitude perigee - 0 °;

- inclination - (0-1.5) ang. min .;

.- eccentricity $$e_2^{\text{'}\text{'}}=\mathrm{0,0004}$$

.

If in the above initial conditions change the eccentricities to

и $$e_2^{"}=\mathrm{0,0003}$$

, $$e_{\mathrm{one}}^{"}=0.0001$$

and $${\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">e</figure-callout>}}_2^{"}=\mathrm{0,0003}$$

,

we get:

- maximum relative deviations along the orbit of 22.4 km in t.1 (figure 5) and minus 29.9 km in t.3;

- maximum relative deviations along the radius vector of 13 km in vol. 2 and 12.6 km in vol. 4.

If in the above initial conditions the eccentricities are as follows:

и $$e_2^{"}=\mathrm{0,0002}$$

, $$e_{\mathrm{one}}^{"}=\mathrm{0,0002}$$

and $${\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">e</figure-callout>}}_2^{"}=\mathrm{0,0002}$$

,

we get:

- maximum relative deviations along the orbit of 20.6 km in t.1 (figure 5) and minus 28.3 km in t.3;

- maximum relative deviations along the radius vector of 12.5 km in vol. 2 and 12 km in vol. 4.

If the MCA to move in longitude by 0.4 ang. min (4.9 km for the geostationary orbit) to the west with e ₁ = e ₂ = 0,0002, we get:

- maximum relative deviations along the orbit of 20 km in t.1 (figure 5) and minus 17.5 km in t.3;

- the maximum relative deviations along the radius vector of 9.2 km in t.2 and 9 km in t.4, i.e. when conditions (4) are fulfilled, one of the major semiaxes of the former distance ellipse changes by the amount of double departure along the orbit, and the small axis of the ellipse is two times smaller than the major axis.

The main condition. The lines of nodes and lines of the apses of the orbits of the MCA and SKA should be orthogonal and the sum of the eccentricities of the orbits should be about 0.0004.

For example, Fig. 6 shows the distance ellipse for the following set of initial conditions (the lines of nodes are orthogonal, the perigee of the orbits of the MCA and SCA are diametrically opposite).

NU 1:

- the longitude of the ascending node is 0;

- argument latitude perigee - 0;

- inclination - (0-1.5) ang. min .;

- eccentricity e ₁ = 0,0002;

NU 2:

- longitude of the ascending node - 90 °;

- argument latitude perigee - 90 °;

- inclination - (0-1.5) ang. min .;

- eccentricity e ₂ = 0,0004.

If the MCA to move in longitude by 0.8 angles. min (9.8 km for the geostationary orbit) to the west with e ₁ = 0.0002, e ₂ = 0.0004, we obtain:

- maximum relative deviations along the orbit of 24.2 km in t.1 (Fig.6) and minus 80.8 km in t.3;

- maximum relative deviations along the radius vector of 26.2 km in t.2 and 25.8 km in t.4,

those. when the conditions for Fig.6 are fulfilled, the major axis of the old distance ellipse is shifted by the amount of departure along the orbit, and the minor axis of the ellipse (two times less than the major axis) remains the same.

With the combined radius vectors SCA and MCA (when the MCA, SCA and Earth are on the same line) and the real eccentricities of the orbits ~ 0.0004 and ~ 0.0002, the inter-satellite distance will be about 19-20 km.

With an exclusion radius r (19–20) km and an angle of 8.65 °, we will have a minimum inclination of the MCA orbit relative to the SCA orbit r · sin (8.65) / 42164.1 = (14-15) angular sec. This is supported promptly and “for free,” since exactly so much is required daily to compensate for centuries-old inclination drifts.

To assess the feasibility of the proposed method, we determine the relative radiation levels that the MCA can receive, turning around the investigated SCA, as well as the approximate values of the angle "Earth-SCA-MCA" (ZSM), within which the MCA can perform the task assigned to it. Since modern communication and data relay satellites use a large nomenclature of airborne antennas with a beam width (LH) of mainly 1 ° to 18 °, we begin our consideration with the narrowest rays.

In accordance with the idea of the proposed method, the MCA should be outside the Earth’s radio-visibility cone with the SCA, the vertex of which is the point where the SCA is in the geostationary orbit, and the generators of the cone are almost tangent to the Earth’s surface, drawn from the top of this cone. All antenna beams used for communication with the Earth have such a spatial orientation that their DNs do not go beyond the specified cone at the half power level (namely, at this level), the angle at the apex of which is 17.3 °. In this case, when conducting an analysis, for example, for a beam with a beam width of 1 °, one should be guided by the fact that the MCA will have to receive radiation from such a beam at the level of high order side lobes. It is advisable to use the reference radiation pattern of satellite antennas provided in the Radio Regulations, Volume 2, Edition of 2008, p.578, mainly intended for assessing electromagnetic compatibility with other communication satellites (Fig. 4). The diagram in figure 4 in the form of a curve 1 gives an idea of the relative gain of the satellite antenna at levels from the main lobe to the first side, and line 3 corresponds to the level of the distant side lobes of a higher order. (Curve 2 is not used for the following analysis, since it characterizes the radiation level of the antenna at orthogonal polarization). The angle φ _o in this diagram corresponds to the beam width in terms of half power, and the angle φ corresponds to the angle of deviation from the axial radiation of the beam (in our case, this is the ZSM angle). From figure 4 it follows that for values of the relative angle (φ / φ _o ) from 10 to 60, the radiation level of the antenna can be taken at a level 43 dB lower (or minus 43 dB) relative to the radiation level along the axis of the beam. In relation to the beam considered as an example with φ _o = 1 °, it can be said that radiation at the level of minus 43 dB will be observed in the range of angles φ from 10 ° to 60 ° relative to the axis of the beam. In the future it will be shown that there are potential opportunities for increasing the size of this range. Obviously, for wider SKA rays, radiation at this level will be observed in a proportionally wider sector of angles φ.

However, it is necessary to take into account that for values of φ from 90 ° to 180 ° it is extremely difficult to estimate the radiation level for antennas installed on the SKA case due to the shadowing of this radiation by the satellite body and large structural elements. At the same time, antennas located, for example, on remote rods in this sector of angles φ are capable of creating radiation in the surrounding space, the level of which, to a first approximation, can also be of the order of minus 43 dB.

Now it is necessary to evaluate the possibility of receiving SKA radiation received above a level, i.e. 43 dB below the maximum level created along the axis of the beam. To do this, we will proceed from the fact that the SCA creates at the border of its service area (at the line of intersection of the radio visibility cone with the Earth) the power flux density (PPM) of such a value that is required for reliable reception by earth stations of signals from the SKA. In this case, PPM = EIIM _SKA / 4πd ² , where EIIM _SKA is the equivalent isotropically radiated power of the SKA, equal to the product of the transmitter power by the antenna gain, ad is the length of the radio line. Since the specified MRP is created for communication conditions at a distance of 35.8 thousand km, then if this distance is reduced to the estimated 20 km (distance between the MCA and the SKA), the MRP for the MCA would have to increase in (35800/20) ² = 3.2 * 10 ⁶ times or 65 dB (i.e. 101g 3.2 * 10 ⁶ ) if the MCA were within the coherent beam of the SKA antenna. However, as was determined above, the gain of the SCA antenna in the direction of the MCA will have a value of 43 dB lower than for stations on the Earth's surface. However, as a result, the value of the MRP for the MCA is in this case 65-43 = 22 dB more.

The obtained energy gain in the MRP level can be used to increase the probability of receiving signals from SKA and expand the sector of reception of these signals. This is due to the fact that the shape of the radiation pattern of real antennas is characterized by both the main lobe of the beam and the side lobes, the maximum level of which gradually decreases with increasing angle φ. Line 3 in figure 4 to some extent represents just the maximum level of the far side lobes. At the same time, between the maxima of the adjacent side lobes, there are relative minimums of MDs with levels approximately 10 dB lower than the level of the previous maximum, therefore, the energy gain obtained above can be used both to compensate for signal level losses in the minimums of DNs, and to compensate for a gradual decrease in maximum levels distant side lobes.

To a certain extent, the energy gain can, if necessary, be used to vary the distance between the MCA and SCA.

Thus, the assessment of the feasibility of the proposed method indicates the possibility of receiving radiation from the SKA on the MCA in a relatively wide sector of ZSM angles even when using rather narrow antenna beams with a width of about 1 ° on the SKA.

To ensure that the MCA performs the tasks in accordance with the proposed method, it will be necessary to install two blocks of receiving antennas on it, the first of which is facing the SKA and receiving signals emitted by the SKA, and the other is facing the Earth and intercepting signals intended for the SKA. Accordingly, a transmitting antenna should also be installed on board the MCA to transmit monitoring results to Earth.

The type and characteristics of the mentioned groups of receiving antennas are selected on the basis of data on the radio technical characteristics of the SKA (frequency ranges, levels of transmitted signals) and data on the relative position of the MCA and the SKA (viewing sectors), which will cover the entire spectrum of monitored emissions in the entire sector of the SKA monitoring angles. What kind of antennas they are, how they provide reception of signals in the given monitoring sectors - whether using a wide radiation pattern or by scanning in these sectors with a narrow beam - it is not so important, it is important that the first unit has a mechanical drive that pumps it in the range of ± 45 ° for better reception of signals from SKA.

In general, two MCAs solve the problem of round-the-clock information retrieval from SKA at the maximum permissible angle of the ZSM of 90 °.

Whenever possible, it is advisable to use the optical wavelength range to reset monitoring data to Earth. Thus, there is no need to obtain radio frequency assignments for the MCA and, in addition, optical radio lines, thanks to very narrow transmitting beams, provide almost absolute secrecy of information transmission and its protection from interception. The optical wavelength range can also be used to monitor and control MCAs. The necessary transceiver equipment for this is currently being used on a number of foreign and domestic spacecraft. The Earth’s atmosphere is not an obstacle if the receiving station is located in the highlands (for example, there is now a station for receiving information from Russian ISS blocks transmitted via an optical channel). There is a clear distinction: SKA is monitored by the MCA in the radio range, data is dumped to the Earth and the MCA is controlled in the optical range. Both functions (monitoring and reset; control) receive almost perfect electromagnetic compatibility (isolation).

If it is not practical to use the optical range, a radio range with a standard electromagnetic compatibility circuit is used.

ANNEX 1

Hodograph of the eccentricity vector

на эксцентриситет e и аргумент широты перигея ω описывается соотношениями [1], К. Эрике «Космический полет», т.II, часть 1, стр.388 (см. фиг.1, цифрами на которой обозначены: 1 - Солнце; 2 - Земля; 3 - КА; 4 - орбита КА на текущую эпоху; 5 - перигей орбиты КА; 6 - орбита КА в эпоху, когда центр Земли, перигей и Солнце находятся на одной линии; 7 - радиус большого круга е_уст), учитывающими тангенциальное и нормальное возмущения в плоскости орбиты:The effect of a small speed pulse (speed increments per second) $${\dot{\vartheta }}_0$$

on eccentricity e and the latitude argument of perigee ω is described by the relations [1], K. Erique “Space Flight”, vol. II, part 1, p. 388 (see figure 1, the numbers on which denote: 1 - the Sun; 2 - Earth; 3 - spacecraft; 4 - spacecraft orbit for the current epoch; 5 - spacecraft orbit perigee; 6 - spacecraft orbit in an era when the center of the Earth, perigee and the Sun are on the same line; 7 - large circle radius e _mouth ), taking into account the tangential and normal disturbances in the orbit plane:

$$\frac{de}{dt}=2\cdot \left(e+\cos \eta \right)\cdot \frac{{\dot{\vartheta }}_0}{\vartheta }\cdot \sin \alpha -\frac{{\dot{\vartheta }}_0}{\vartheta }\cdot \sin \eta \cdot \cos \alpha ;\left(\mathrm{one}\right)$$

$$\frac{d\omega }{dt}=\frac{2\sin \eta }{e}\cdot \frac{{\dot{\vartheta }}_0}{\vartheta }\cdot \sin \alpha +\frac{2e+\cos \eta }{e}\cdot \cos \alpha ,\left(2\right)$$

where ϑ - the average speed of the spacecraft, 3074 m / s;

η = α-θ - in - true anomaly;

θ is the angle between the direction to the Sun and to the perigee of the spacecraft orbit.

Then, substituting the expression η in (3) and using the formulas of the difference of two angles, we obtain:

$$\begin{array}{l}\frac{de}{dt}=\frac{{\dot{\vartheta }}_0}{\vartheta }\cdot (2e\cdot \sin \alpha +2\cos \alpha \cdot \cos \theta \cdot \sin \alpha +\\ +2\sin \alpha \cdot \sin \theta \cdot \sin \alpha -\sin \alpha \cdot \cos \theta \cdot \cos \alpha +\cos \alpha \cdot \sin \theta \cdot \cos \alpha )=\\ =\frac{{\dot{\vartheta }}_0}{\vartheta }\cdot \left(2e\cdot \sin \alpha +\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2\alpha \cdot \cos \theta -0.5\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2\alpha \cdot \cos \theta +\underset{\sin \theta \cdot \left(\ddot{s}i{n}^2\alpha +\mathrm{one}\right)}{\underbrace{2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\alpha \cdot \sin \theta +{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\alpha \cdot \sin \theta }}\right)=\\ =\frac{{\dot{\vartheta }}_0}{\vartheta }\cdot \left(2e\cdot \sin \alpha +0.5\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2\alpha \cdot \cos \theta +{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\alpha \cdot \sin \theta +\sin \theta \right).\left(3\right)\end{array}$$

The first term is at least two orders of magnitude smaller than the rest and is not a constant member, then

$$\frac{de}{dt}=\frac{{\dot{\vartheta }}_0}{\vartheta }\cdot \left(\frac{\mathrm{one}}{2}\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2\alpha \cdot \cos \theta +{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\alpha \cdot \sin \theta +\sin \theta \right).\left(\mathrm{four}\right)$$

When θ = 0

$$\frac{de}{dt}=\frac{\mathrm{one}}{2}\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2\eta \cdot \frac{{\dot{\vartheta }}_0}{\vartheta }.\left(5\right)$$

Arguing in a similar way, we will have perigee for the speed of movement (latitude argument):

$$\frac{d\omega }{dt}=\frac{\mathrm{one}}{e}\cdot \frac{{\dot{\vartheta }}_0}{\vartheta }\cdot \left(-\frac{\mathrm{<figure-callout\; id="2"\; label="one"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">one</figure-callout>}}{2}\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2\alpha \cdot \sin \theta +{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\alpha \cdot \cos \theta +\cos \theta \right)\text{.}\left(\text{6}\right)$$

When θ = 0

$$\frac{d\omega }{dt}=\frac{{\dot{\vartheta }}_0}{e\cdot \vartheta }\cdot \left(\mathrm{one}+{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\eta \right).\left(7\right)$$

Further,

$$dt=\frac{d\alpha }{n-{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="00000074.png,00000077.png"\; state="\{\{state\}\}">n</figure-callout>}}_0},\left(\text{8}\right)$$

- среднее движение KA, с^-1;Where $$n=\frac{2\pi }{T}=\mathrm{0,000073}$$

- average movement KA, s ^-1 ;

- среднее движение Солнца, с^-1 _, $$n=\frac{0.01745}{86400}$$

- the average movement of the Sun, s ^-1 _,

then

$$de=\frac{{\dot{\vartheta }}_0}{\vartheta \cdot \left(n-n_0\right)}\cdot \left(\frac{\mathrm{one}}{2}\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2\alpha \cdot \cos \theta +{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\alpha \cdot \sin \theta +\sin \theta \right)da;\left(9\right)$$

$$d\omega =\frac{{\dot{\vartheta }}_0}{e\cdot \vartheta \cdot \left(n-n_0\right)}\cdot \left(-\frac{\mathrm{<figure-callout\; id="2"\; label="one"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">one</figure-callout>}}{2}\cdot \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2\alpha \cdot \sin \theta +{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\alpha \cdot \cos \theta +\cos \theta \right)da.\left(10\right)$$

We integrate on a daily basis (on a revolution):

$$\Delta e_{cym}={\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\left(\mathrm{eleven}\right)\right)da=3\pi \cdot \frac{{\dot{\vartheta }}_0}{\vartheta \cdot \left(n-n_0\right)}\cdot \sin \theta ,<t\mathrm{about}hn\mathrm{about}>\left(\mathrm{eleven}\right)}$$

at θ = 0 $$\Delta e_{\mathrm{from}\mathrm{at}t}=0;\left(\text{eleven'}\right)$$

$$\Delta {\omega }_{cym}={\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\left(12\right)\right)d\alpha =3\pi }\cdot \frac{{\dot{\vartheta }}_0}{e\cdot \vartheta \cdot \left(n-n_0\right)}\cdot \cos \theta ,<t\mathrm{about}hn\mathrm{about}>\left(12\right)$$

at θ = 0 $$\Delta {\omega }_{\mathrm{from}\mathrm{at}\mathrm{<figure-callout\; id="3"\; label="t"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">t</figure-callout>}}3\pi \cdot \frac{{\dot{\vartheta }}_0}{e\cdot \vartheta \cdot \left(n-n_0\right)}.\left(\text{12'}\right)$$

. Световое давление описывается формулой We give an estimate $${\dot{\vartheta }}_0$$

. Light pressure is described by the formula

$$P=\frac{S\left(\mathrm{one}+A\right)}{c},\left(\text{13}\right)$$

where S is the power of the light wave incident on 1 m ^{2 the} surface of the body, W / m ² ,

A is the reflection coefficient (A = 0 for a completely black body);

s is the speed of light, km / s.

S = 1.4 · 10 ³ W / m ² .

The value of A depends on the reflectivity of the details of the spacecraft construction and in the context of this technical solution should include (conditionally) the gravitational influence of the Sun as "±" relative to the position when the Laplace vector is directed at the Sun. For real spacecraft at the height of the stationary orbit, the values of A are in the range [0.28-0.44]. Based on A = 0.44, we will have P equal to 6.72 · 10 ^-6 n / m ² . Since the light pressure force is F = S '· P, where S' is the mid-sectional area, then

$${\dot{\vartheta }}_0=a=\frac{S\text{'}\text{'}}{m}\cdot P.\left(\text{fourteen}\right)$$

для современных отечественных геостационарных КА более или менее постоянно и равно порядка (2,3-2,6)·10^-2. Тогда, к примеру, при k=0,0259 $${\dot{\vartheta }}_0=\mathrm{0,174}\cdot {10}^{-6}м/{с}^2$$

. Как показывает численное интегрирование, период цикличности для эксцентриситета составляет несколько больше года - порядка 390 суток. Это происходит из-за того, что возмущения движения перигея при устойчивом эксцентриситете от гравитационного поля Солнца имеют не годовой, а полугодовой период с амплитудой колебания, как показывает раздельное интегрирование, порядка 0,00005. Подстановка $${\dot{\vartheta }}_0$$

уравнение (12') дает при $$\frac{2\pi }{390}$$

значение e=e_уст порядка 0,00045, т.е. для того, чтобы скорость движения перигея равнялась скорости движения Солнца, необходимо иметь устойчивый эксцентриситет. Оригинальный вывод формул (11-12') приведен для раскрытия сущности понятия устойчивого эксцентриситета. Энергозатраты на поддержание е_уст практически отсутствуют: каждый год необходимо проводить коррекцию эксцентриситета на величину его векового ухода Δe=0,000052 за счет нецентральности гравитационного поля Земли. Это очень малая величина, соизмеримая с суточными уходами эксцентриситета за счет совокупного влияния всех пассивных возмущающих факторов, однако, если вековой уход не компенсировать, текущий эксцентриситет к концу срока активного существования будет далек от устойчивого.Attitude $$k=\frac{S\text{'}\text{'}}{m}$$

for modern domestic geostationary spacecraft more or less constant and equal to the order of (2.3-2.6) · 10 ^-2 . Then, for example, with k = 0.0259 $${\dot{\vartheta }}_0=0.174\cdot {10}^{-6}m/{\mathrm{from}}^2$$

. As numerical integration shows, the cycle period for eccentricity is slightly more than a year - about 390 days. This is due to the fact that perturbation motion perturbations with a stable eccentricity from the gravitational field of the Sun do not have a yearly, but a six-month period with an oscillation amplitude, as shown by separate integration, of the order of 0.00005. Substitution $${\dot{\vartheta }}_0$$

equation (12 ') gives for $$\frac{2\pi }{390}$$

the value e = e _{mouth is of the} order of 0,00045, i.e. in order for the speed of perigee to be equal to the speed of the sun, it is necessary to have a stable eccentricity. The original derivation of formulas (11-12 ') is given to reveal the essence of the concept of stable eccentricity. Energy costs for maintaining the _{mouth are} practically absent: every year it is necessary to correct the eccentricity by the value of its secular departure Δe = 0.000052 due to the off-center gravity of the Earth's gravitational field. This is a very small value, commensurate with the daily eccentricity drifts due to the combined influence of all passive disturbing factors, however, if the secular care is not compensated, the current eccentricity will be far from stable by the end of the active life.

When the rotational analysis is possible, for example, based on the fact that

$$\frac{de}{d\omega }=e\cdot tg\theta ,\left(\text{fifteen}\right)$$

get the formula:

$$\ln \left(\frac{e_2}{e_{\mathrm{one}}}\right)=\ln \left(\frac{\cos {\theta }_{\mathrm{one}}}{\cos {\theta }_2}\right)+A\left(\sqrt{\mathrm{one}-B\cdot {\theta }_{\mathrm{one}}^2}-\sqrt{\mathrm{one}-\mathrm{<figure-callout\; id="2"\; label="B\; \cdot \; \theta "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">B\; </figure-callout>}\cdot {\theta }_{}^{}{}_2^2}\right),\left(\text{16}\right)$$

Where $$A=\frac{a}{9{\mathrm{<figure-callout\; id="3"\; label="b"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">b</figure-callout>}}^3}\cdot \left(2+9b^2\right);\left(\text{17}\right)$$

$$B={\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">b</figure-callout>}}^2;\left(\text{eighteen}\right)$$

$$a=\{\begin{array}{l}\begin{array}{l}\frac{e}{e+2ctg\left(\frac{\mathrm{<figure-callout\; id="2"\; label="\pi "\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\pi </figure-callout>}}{2}\cdot \frac{e_0}{e_{ycm}}\right)},\\ \\ \frac{2e_{ycm}}{e_{ycm}+{\mathrm{<figure-callout\; id="0"\; label="e"\; filenames="00000074.png,00000077.png"\; state="\{\{state\}\}">e</figure-callout>}}_0},\end{array}\hfill \\ \hfill \end{array}\begin{array}{l}dl\mathrm{I\; am}\mathrm{TO}\mathrm{BUT}\text{whose orbit eccentricity}\hfill \\ \text{held in optimal aiming area}\hfill \\ (\mathrm{about}bl\mathrm{but}\mathrm{from}t\mathrm{and},\text{where the eccentricity vector is deviated from}\hfill \\ n\mathrm{but}PR\mathrm{but}\mathrm{at}len\mathrm{and}\mathrm{I\; am}n\mathrm{but}\mathrm{FROM}\mathrm{about}lnceneb\mathrm{about}lee\text{than 90}°);\hfill \\ dl\mathrm{I\; am}\text{KA}\text{whose orbit eccentricity}\hfill \\ \text{held in suboptimal area}\hfill \\ \text{aiming;}\hfill \end{array}\begin{array}{ll}\hfill & \hfill \\ \hfill & \hfill \\ \hfill & \hfill \\ \hfill & \left(19\right)\hfill \\ \hfill & \hfill \\ \hfill & \hfill \\ \hfill & \hfill \end{array}$$

$$b=\frac{8}{9}\cdot \frac{e_{ycm}}{e_{ycm}-{\mathrm{<figure-callout\; id="0"\; label="e"\; filenames="00000074.png,00000077.png"\; state="\{\{state\}\}">e</figure-callout>}}_0};\left(\text{twenty}\right)$$

e ₀ - the value of the eccentricity in the middle of the retention cycle;

e is the base of the natural logarithm.

On the retention interval of up to 20-30 days, you can use the simplified formula:

$$\frac{e_{\mathrm{one}}}{{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">e</figure-callout>}}_2}={\left[\frac{\cos {\theta }_2}{\cos {\theta }_{\mathrm{one}}}\right]}^{\left(\mathrm{one}+k\right)}\mathrm{and}l\mathrm{and}\text{ln}=\left(\frac{{\text{e}}_{\text{one}}}{{\text{e}}_{\text{2}}}\right)=\left(\mathrm{one}+k\right)\cdot \ln \left(\frac{\cos {\theta }_2}{\cos {\theta }_{\mathrm{one}}}\right),\left(\text{21}\right)$$

Where $$k=\frac{e_0}{e_{ycm}-{\mathrm{<figure-callout\; id="0"\; label="e"\; filenames="00000074.png,00000077.png"\; state="\{\{state\}\}">e</figure-callout>}}_0}.\left(\text{22}\right)$$

APPENDIX 2

Hodograph of the inclination vector

Projections of the inclination vector:

$$i_x=\sin i\cdot \cos \Omega ;\left(\text{one}\right)$$

$$i_y=\sin i\cdot \sin \Omega ;\left(\text{2}\right)$$

for small i

$${\dot{i}}_x=-i\cdot \sin \Omega \cdot \frac{d\Omega }{dt}+\cos \Omega \cdot \frac{di}{dt};\left(3\right)$$

$${\dot{i}}_y=-i\cdot \cos \Omega \cdot \frac{d\Omega }{dt}+\sin \Omega \cdot \frac{di}{dt};\left(\mathrm{four}\right)$$

на скорости изменения Ω и i, при малом i, описываются формулами:The effect of lateral acceleration $$\dot{V}$$

at the rate of change, Ω and i, for small i, are described by the formulas:

$$\dot{\Omega }=\frac{\dot{V}}{V}\cdot \frac{\sin \Theta }{i};\left(5\right)$$

$$\dot{i}=\frac{\dot{V}}{V}\cdot \cos \Theta ,\left(6\right)$$

where Θ is the argument of the latitude of the Sun;

V = 3.074 km / s - the average speed of the spacecraft in geostationary orbit.

$${\dot{i}}_x=-i\cdot \frac{\dot{V}}{V}\cdot \sin \Omega \cdot \frac{\sin \Theta }{i}+\frac{\dot{V}}{V}\cdot \cos \Omega \cdot \cos \Theta =\frac{\dot{V}}{V}\cdot \cos \left(\Omega +\Theta \right);\left(7\right)$$

$${\dot{i}}_y=-i\cdot \frac{\dot{V}}{V}\cdot \cos \Omega \cdot \frac{\sin \Theta }{i}+\frac{\dot{V}}{V}\cdot \sin \Omega \cdot \cos \Theta =\frac{\dot{V}}{V}\cdot \sin \left(\Omega +\Theta \right);\left(8\right)$$

Ω + Θ = α _c - right ascension of the Sun;

$${\dot{i}}_x=\frac{\dot{V}}{V}\cdot \cos {\alpha }_c;\left(9\right)$$

$${\dot{i}}_y=\frac{\dot{V}}{V}\cdot \sin {\alpha }_c;\left(10\right)$$

Based on the general vector perturbation equation

$$\Delta {\overline{q}}_{KA}={\overline{q}}_{c-KA}-{\overline{q}}_c,\left(\mathrm{eleven}\right)$$

average acceleration per revolution:

$$\dot{V}={\mu }_c\cdot \left[\left(\frac{\mathrm{one}}{{\rho }_{\min }^2}-\frac{\mathrm{one}}{{\rho }_c^2}\right)-\left(\frac{\mathrm{one}}{{\rho }_{\max }^2}-\frac{\mathrm{one}}{{\rho }_c^2}\right)\right]\cdot \underset{\stackrel{"}{-\sin {\delta }_c}}{\underbrace{\sin \epsilon \cdot \sin {\alpha }_c}}\cdot k;\left(12\right)$$

$${\rho }_{\max }={\rho }_c+r_{\mathrm{from}t}\cdot \cos \left({\mathrm{50,5}}^0\right);\left(13\right)$$

$${\rho }_{\min }={\rho }_c-r_{\mathrm{from}t}\cdot \cos \left({\mathrm{50,5}}^0\right);\left(\mathrm{fourteen}\right)$$

ρ _c = 149600000 km;

cos (50.5 °) - weight average cosine;

δ _с - declination of the Sun.

The effectiveness of any correction is estimated by the formula:

$$k=\frac{\sin \left(\frac{\Delta U}{2}\right)}{\frac{\mathrm{<figure-callout\; id="2"\; label="\Delta \; U"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">\Delta \; </figure-callout>}U}{2}},\left(\mathrm{fifteen}\right)$$

where U is the active section of the movement;

in the diurnal interval k≈0.6366≈cos (50.5 °), at U∈ [0-180 °].

$$i_x=i_{x0}+\frac{\mathrm{one}}{V}\cdot {\displaystyle \int \dot{V}}\cdot \cos {\alpha }_c\cdot dt;\left(16\right)$$

$$i_y=i_{y0}+\frac{\mathrm{one}}{V}\cdot {\displaystyle \int \dot{V}}\cdot \sin {\alpha }_c\cdot dt.\left(17\right)$$

Imagine the day (time) through the average movement of the Sun:

$$dt=\frac{d{\alpha }_c}{0.017453}\left(\mathrm{eighteen}\right)$$

and denote the constant part of equations (16), (17), taking into account (18), by A:

$$A={\mu }_c\cdot \left(\frac{\mathrm{one}}{{\rho }_{\min }^2}-\frac{\mathrm{one}}{{\rho }_{\max }^2}\right)\cdot k\cdot \sin \epsilon \cdot \frac{\mathrm{one}}{V\cdot 0.017453},\left(19\right)$$

Then:

$$i_x=i_{x0}+\frac{A}{2}{\displaystyle \int \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}2{\alpha }_c}\cdot d\alpha ;\left(\mathrm{twenty}\right)$$

$$i_y=i_{y0}+A{\displaystyle \int {\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="00000072.png,00000073.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\alpha }_c}\cdot d\alpha ;\left(21\right)$$

A = 0.00175;

$$R=A/\mathrm{four}=90\mathrm{at}gl.\mathrm{from};\left(24\right)$$

$$L=\frac{A}{2}{\alpha }_c=3\mathrm{at}gl.\mathrm{from}-\mathrm{at}e\mathrm{to}\mathrm{about}\mathrm{at}\mathrm{about}\mathrm{th}hlen;\left(25\right)$$

every day Δα _c ≈ 0.017453.

The system of equations (22) and (23) without the secular term (25) represents the equation of the circle (“solar circle”) in parametric form, where R is the radius of the circle, i _x0 , i _y0 are the components of the end of the vector of the aiming point by inclination, calculated by formulas (1), (2). Knowing the direct ascent of the Sun and choosing the coordinates of the aiming point [i _x0 ; i _y0 ], determine the position of the end of the inclination vector at the beginning of the next retention cycle in accordance with the departure of the vector i SKA.

Claims (3)

1. A method for monitoring collocation in a geostationary orbit, including translations of the inclination and eccentricity vectors to the boundaries of aiming regions spaced apart from each other (areas of permissible variation of the inclination and eccentricity vectors), measuring the orbit parameters of each spacecraft (SC), and determining the current values of orbital elements from them each spacecraft and carrying out with the help of small thrust engines the corrections of the period of revolution, inclination and eccentricity of the orbit, characterized in that for the organ of collocation autonomous from an adjacent spacecraft (SCA), before the monitoring spacecraft (MCA) is brought into a predetermined holding area in latitude (inclination) and longitude, the strategy for controlling the motion of the center of mass of the SCA is determined according to independent trajectory measurements, and in the process of holding, the position of the center of the aiming region is specified SKA of inclination, inclination correction vector carrying MCA inclination orbits exhibit phase plane so that the line of nodes perpendicular to the MCA was the orbit line of nodes and SKA orbit inclination (i _min) OR ICA Ity relative orbital SKA is not less than (14-15) angular seconds is carried eccentricity correction vector:
- so that the sum of the eccentricities of the orbits of the MCA and SCA is about 4 · 10 ^-4 ,
- to remove the direction to the perigee from the direction to the ascending node of the ICA’s orbit by the value of the mismatch angle between the directions to the perigee and the ascending node of the SCA’s orbit,
- to maintain this position of the perigee within the specified limits of the aiming region for eccentricity,
carry out regular complex correction of the inclination of the orbit of the ICA:
- to maintain a right angle between the lines of the nodes of the orbits of the spacecraft in the specified limits of the aiming region by inclination,
- to eliminate the age-old component of care for inclination,
- to exceed i _{min the} specified value,
longitude (rotation period) is corrected so that the origin (ΔL, ΔR is respectively the deviation along the orbit and the deviation along the radius vector) coincides within the specified limits with the center of the distance ellipse from the SKA, the centers of the aiming areas are redefined on the ICA by inclination and eccentricity MCA orbits, when adjusting the strategy for controlling the motion of the center of mass of the SCA and with an increase in the mismatch angles between the lines of the apses and the lines of the nodes of the orbits of the MCA and the SCA, in cases of a dangerous approach of the SC, the corrections are deviated which are simultaneous corrections of the longitude and eccentricity of the orbit, after organizing an autonomous collocation by the passage of the MCA under the SCA, when the radius vector of the MCA is less than the radius vector of the SCA, the time interval when the reception level of the radiation of antennas installed on the SCA on the SCA is satisfactory is daily updated, when the reception level at the MCA decreases, they switch to the information reception mode for SCA from ground-based antennas, in the case of reliable continuous reception of radiation from antennas installed on the SCA to the MCA within 12 hours, round-the-clock monitoring of SKA is carried out by two MCAs installed on the diametrically opposite sides of the distance ellipse. 2. The method according to claim 1, characterized in that the block of receiving antennas on the MCA facing the SKA has a drive swinging it in the range of ± 45 ° for better reception of signals from the SKA. 3. The method according to claim 1, characterized in that the transmission of monitoring data from the MCA to the Earth, as well as monitoring and control of the MCA, if possible, is carried out using the optical wavelength range.

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