Spacecraft motion in a low circular orbit in establishing intersatellite link

Introduction. Intersatellite links can increase the efficiency in achieving the principle objective of satellite systems – providing communications for subscribers distributed globally. Besides, operating IL helps to solve the problems of parallel control of all spacecraft in the constellation without numerous ground tracking stations [1].

There are several satellite systems which provide monitoring and data transmission, such as “Globalstar”, “Iridium”. Development of such intersatellite systems is also essential for solving various problems of data transmission, monitoring and locating moving objects. Technical solutions of these systems provide the basis of monitoring facilities development [2; 3].

The aim of IL is to provide radio exchange of subscribers not in the line of sight of the same spacecraft -that requires information exchange both between SC in the same plane and between ones in different planes. The use of IL significantly increases the efficiency of radio links [4]:

• –    the service area is increased by interconnection of all radio visibility zones (RVZ) of SC into an integrated RVZ of the orbital constellation;

• –    the load on the satellite links is reduced due to instant information exchange;

• –    the reliability and stability of satellite links is increased.

Establishing IL in OC of SC. For the experiment we assume that there is an OC of 24 SC in a near-circular orbit of 1500 km altitude, and from the point of view of the OC ballistic formation and SC motion, we review the ways of plotting the IL for the given satellite system, both in one plane and between adjacent planes. A similar IL configuration is already introduced in “Iridium” satellite system [5]. “Iridium” is a low-orbit system operating in circular orbits. The orbit altitude is about 700 km; the standard OC includes 66 interlinked SC [6].

When an IL are made in a circular orbit of 1500 km altitude, the optimal mutual visibility of SC in the orbital plane is achieved when the number of SC in the plane is more than or equals 6 [7].

It must be noted that for IL correct operation in one plane, orbit correction must be carried out to keep the orbital position of all SC stable, otherwise the distance between the SC can alter too much and cause operational failure.

We can determine the station-keeping zone and positioning of SC in the orbital plane for IL operation in reference to 5, 6, 7 SC. It is necessary to view the orbit configuration of SC 1 and SC 2 in more detail (fig. 1).

The symbols in fig.1 are: a – Earth radius; b – halfdistance between SC 1 and SC 2; с – major semi-axis of SC orbit; ∠β – half of the center angle between SC 1 and SC 2.

We take the basic relation of a right-angled triangle [8] to calculate 2∙∠β – the limit angle of keeping SC in the line of sight a = 6378.16 km – Earth radius [9];

с = 1500 + 6378.16 = 7878,16 km – major semi-axis of SC orbit;

sin α = a / c ;

∠α = 54.056°;

∠β = 35.944°;

2∙ ∠β = 71.888°.               (1)

According to the formula (1), for a number of SC composition variants in the orbital plane, we obtain the

values of the angle between the nearest SC and of the station-keeping zone (tab. 1).

Fig. 1. IL within an orbital plane

Рис. 1. МЛС внутри орбитальной плоскости

The results in tab. 1 show that for IL at 1500 km altitude with five SC in the plane, to provide mutual visibility it is necessary to keep the SC in orbit with an accuracy of 0.5° by latitude argument, which is quite demanding technically.

Therefore, an architecture of 6 SC in the plane with ± 5° station-keeping zone relative to the SC orbital position can be applicable. In this case, the distance between two adjacent SC with an IL will depend on the accuracy of keeping the SC in the orbital positions with respect to the latitude argument. Half-distance value between adjacent SC with IL (parameter b in fig. 1) when there are 6 SC in the orbital plane will make b = с∙sinβ. (2)

In case 6 SC are evenly positioned in the plane, (the center angle between these SC is 60 °) and the stationkeeping accuracy is ± 5º, the expression (2) gives the 7878 km distance in the nominal position. The distance calculations for the center angle of 50° and 70° are listed in tab. 2.

We form the chosen for the experiment standard OC architecture of 24 SC for modeling and studying the ballistic parameters of IL as follows: four orbital planes of 6 SC each (tab. 3).

Absolute longitude divergence of the ascending nodes of adjacent orbital planes is 46°. The SC phase distribution is taken for the moment of SC 11 passing the ascending node of the orbit.

Fig. 2 represents 24 SC in four orbital planes and the track of every SC in flight period of a few minutes.

Table 1

Station-keeping zone for a SC with 1.500 km orbit altitude operating IL

SC number in the orbital plane	SC positioning in the plane with respect to the argument of latitude, °	Station-keeping zone, °
5	71.9	±0.5
6	60.0	±5.0
7	51.4	±9.3

Table 2

	Distance between SC, 2∙ b , km	Center angle between SC, 2∙ ∠β , °
Minimal	6658	50
Nominal	7878	60
Maximal	9037	70

Table 3

Distribution for OC of 24 SC

Plane number	SC number	Ascending node longitude, °	Argument of latitude, °	Plane number	SC number	Ascending node longitude, °	Argument of latitude, °
1	11	0	0	3	31	92	50
	12	0	60		32	92	110
	13	0	120		33	92	170
	14	0	180		34	92	230
	15	0	240		35	92	290
	16	0	300		36	92	350
2	21	46	25	4	41	138	75
	22	46	85		42	138	135
	23	46	145		43	138	195
	24	46	205		44	138	255
	25	46	265		45	138	315
	26	46	325		46	138	15

Fig. 2. Routes of 24 SC in the OC

Рис. 2. Трассы ОГ из 24-х КА

Values of distance and center angle between SC in the plane

ма                     md                    so                    е                     «                    ■                    uo                    «о                    мо

IL is a unique element of “Iridium” communication system, every SC of which is interconnected with 4 adjacent ones, 2 of them in front and in aft position in the same orbital plane, and 2 on the left and on the right in adjacent orbital planes [10].

For the experimental OC architecture of 24 SC we can outline the following IL configurations:

• –    Configuration 1 – IL between the SC in one orbital plane;

• –    Configuration 2 – IL between SC of adjacent planes № 1 and 2, № 2 and 3, № 3 and 4 (parallel motion);

• -    Configuration 3 - IL between SC in planes № 1 and

• 3, № 2 and 4 (orthogonal motion in the moment of mutual visibility);

• -    Configuration 4 - IL between SC in planes № 1 and 4 (cross-movement in the moment of mutual visibility).

Operating IL dictates the need of continuous communication during the SC flight. The analysis of this aspect is presented in [11] for “CubeSat” system.

Below, there is an analysis of some specified SC from every orbital plane in the experimental OC of 24 SC, as well as of ballistic parameters variation for each of the IL configurations, including the periods of SC mutual visibility.

Here is an analysis of the following ballistic parameters:

• –    the SC positioning range and the rate of its changing;

• –    declination and the rate of its changing;

• –    elevation and the rate of its changing;

• –    period of mutual visibility in an orbit pass.

These parameters allow to determine SC antenna control characteristics in IL communication, as well as the mutual visibility periods of SC within IL zone.

Configuration 1 – IL between the SC in one orbital plane . Fig. 3 presents an IL variant in one plane, when all RVZ of six SC (at an elevation angle of 10 °) are networked into a common RVZ of the orbital plane. Common RVZ will significantly increase the coverage zone.

Further we review the requirements for IL-providing equipment on the example of two most closely positioned sC of the first orbital plane SC № 11 and SC № 12. Fig. 4 presents the SC, their subsatellite points and flight route.

Tab. 4 presents the initial data - SC №11 and SC № 12 reference. MDT - Moscow decree time in the ascending node, the SC coordinates and velocities are in the Greenwich rotating coordinate system (GRCS).

Let us review the information transfer for SC № 11 to SC № 12 in the nominal position during one orbit pass (115 min).

In fig. 5–7 are shown ballistic parameter variations (where subitem a) parameter, b) parameter change rate):

• –    Range – line-of-sight distance between SC [12]. Range change rate;

• –    Declination – antenna direction angle from one SC to another measured from the direction to the Earth’s center. Declination change rate;

• –    Elevation – antenna direction angle from one SC to another measured from velocity vector in the clockwise direction. Elevation change rate.

We can put the calculations of SC antenna pointing ballistic parameters together. The range of parameter changing for IL in an orbit pass between SC № 11 and SC № 12 is listed in tab. 5.

We must point out that in the orbit pass SC № 11 and SC № 12 are constantly in the zone of mutual visibility -that is confirmed by the parameter of 100 %.

Fig. 3. Common RVZ in one orbital plane with IL within that plane

Рис. 3. Общая ЗРВ одной орбитальной плоскости при МЛС внутри плоскости

Fig. 4. The motion of two close-positioned SC № 11 and 12 in the first orbital plane

Рис. 4. Движение двух ближайших КА № 11 и 12 первой орбитальной плоскости

Table 4

SC № 11 and SC № 12 reference

Parameter	SC № 11	SC № 12
Date	31.05.2019	31.05.2019
MDT	11:59:57	12:19:14
Coordinate Х, km	7232.954011	6936.407866
Coordinate Y, km	–3120.071909	–3714.968713
Coordinate Z, км	0,0	0.0
Velocity Vx, km/s	0.140334	0.171858
Velocity Vy, km/s	0.325291	0.310911
Velocity Vz, km/s	7.052965	7.060811

а                                                          b

Fig. 5. Ballistic parameter range:

а – range changing between SC № 11 and 12; в – range change range

Рис. 5. Баллистический параметр «дальность»:

а – изменение дальности между КА № 11 и 12;

б – скорость изменения дальности

a                                                          b

Fig. 6. Ballistic parameter declination:

а – declination changing between SC № 11 and 12; в – declination change rate

Fig. 7. Ballistic parameter elevation:

а – elevation change between SC № 11 and 12; в – elevation change rate

Рис. 6. Баллистический параметр «склонение»:

а – изменение склонения между КА № 11 и 12; б – скорость изменения склонения

b

Рис. 7. Баллистический параметр «восхождение»:

а – изменение восхождения между КА № 11 и 12; б – скорость изменения восхождения

Fig. 5–7 and tab. 5 data analysis demonstrates in the first place the stability of the basic parameters. Elevation is close to 0° in signal transmission from SC № 12 to SC № 11, and close to 180° in transmission from SC № 11 to SC № 12. The greatest variations are demonstrated by the SC antenna declination parameter.

Configuration 2 – IL between SC of adjacent planes № 1 and 2, № 2 and 3, № 3 and 4. Here is an analysis of IL configuration for SC moving in adjacent planes № 1 and 2, on example of SC № 12, 22, 23.

Tab. 6 presents the reference of SC № 22 and SC № 23, SC № 12 – in tab. 4.

Changing of IL ballistic parameters between SC within the orbital plane

Table 5

Parameter	Range of parameter changing
Range, km	from 6658 to 9037
Range change rate, km/s	from –0.03 to 0,03
Declination, °	from 54.37 to 65.36
Declination change rate, °/s	from –0.001 to 0.001
Elevation, °	from 179.99 to 180.01
Elevation change rate, °/с	from –0.00006 to 0.00006
SC mutual visibility period in an orbit pass, min	115.9 (100 %)

SC № 22 and № 23 reference

Table 6

Parameter	SC № 22	SC № 23
Date	31.05.2019	31.05.2019
MDT	12:09:37	12:28:54
Coordinate Х, km	7386,336533	7591.458385
Coordinate Y, km	2724.447324	2091.536252
Coordinate Z, km	0.0	0.0
Velocity Vx, km/s	–0.127584	–0.086993
Velocity Vy, km/s	0.341148	0.352760
Velocity Vz, km/s	7.055772	7.054474

Changing of IL ballistic parameters between SC in adjacent planes № 1 и 2

Table 7

Parameter	Range of parameter changing
Range, km	from 3800 to 7200
Range change rate, km/s	from –3 to 3
Declination, °	from 63 to 77
Declination change rate, °/с	from –0.05 to 0.05
Elevation, °	from –60 to 60; from 120 to 240
Elevation change rate, °/с	from –0.4 to 0.15; from –0.15 to 0.15
SC mutual visibility period in an orbit pass, min	115.9 (100 %)

Fig. 8. SC motion in adjacent planes № 1 and 2

Рис. 8. Движение КА в соседних плоскостях № 1 и 2

SC № 31, 32 and 33 reference

Fig. 9. SC orthogonal motion in planes № 1 and 3

Table 8

Parameter	SC № 31	SC № 32	SC № 33
Date	31.05.2019	31.05.2019	31.05.2019
MDT	11:40:43	12:00:00	12:19:17
Coordinate Х, km	2233.447140	2864,993939	3469.973938
Coordinate Y, km	7544.938345	7337.622768	7062.144241
Coordinate Z, km	0.0	0.0	0,0
Velocity Vx, km/s	–0.354350	–0.342044	–0.328272
Velocity Vy, km/s	0.099865	0.133551	0.166529
Velocity Vz, km/s	7.059052	7.051298	7.059063

Fig. 8 shows the OC fragment, consisting of SC № 12 from the first orbital plane and SC № 22, SC № 23 from the second orbital plane, for the moment of their positioning in the equator and pole areas. During the full orbit pass SC № 12 motion is mostly parallel to SC № 22, SC № 23 is also parallel; SC routes get crossed only in the pole areas.

We can put together the calculations of SC antenna pointing parameters for planes № 1 and 2. The range of IL parameter changing is listed in tab. 7.

We must point out that the sampled SC № 12, SC № 22 and SC № 23 continuously remain in the mutual visibility zone.

The IL in tab. 7 compared to the IL of one orbital plane (tab. 5) shows the increasing dynamics of declination and elevation angles changing. Here the declination change rate increased by 50 times (0.001 and 0.05), and the elevation change rate increased from 0.00006 °/s to 0.15°/s.

We have the best conditions for information transmission in the equator area. There the information can be transmitted in chain order to all 4 planes of the system.

It should be noted that for the sampled OC of 24 SC at 1500 km altitude, the common IL including configurations 1 and 2 is self-supporting (provides general linking of all SC in the OC).

Configuration 3 – IL between SC in planes № 1 and 3, № 2 and 4. Let us review the IL configuration, in which SC move almost transversely, at the example of orbital planes № 1 and 3.

Tab. 8, in addition to Tab. 4, presents SC № 31, 32, 33 reference.

Here the mutual visibility of SC in different planes occurs at high latitudes.

Each SC of one orbital plane can see from two to three SC of another orbital plane. We can point out that the time of simultaneous visibility of three SC is not long. We analyze the example of IL for SC № 11 of the first orbital plane, and SC № 31, 32 and 33 of the third orbital plane (fig. 9).

IL parameter changes between SC № 11 and SC № 31, 32, 33 are listed in tab. 9.

In motion of SC № 11 relative to SC № 31, 32 and 33, mutual visibility period or IL operational time of the SC is rather long, mainly close to revolution half-period.

Comparison of this IL parameters (tab. 9) with IL in one orbital plane (tab. 5) and IL in adjacent orbital planes (tab. 7) demonstrates still more growing dynamics when the declination and elevation angles get changed. Now the declination change rate comes up to 0.17 °/s, and elevation change rate – to 0.55 °/s.

Configuration 4 - IL between SC in planes № 1 and 4. Here is an analysis of IL configuration in planes № 1 and 4. Its main feature is the SC oncoming motion.

During the revolution half-period a SC of one orbital plane runs across all other SC of another orbital plane.

The dynamics of the SC route intersection depends on the Earth's surface latitude, over which the SC route intersection takes place. In this case, there may be three variants: equator zone, middle latitudes and high latitudes.

Let us review the IL on the example of SC № 11 of the first orbital plane and all six SC of the fourth orbital plane.

Tab. 10 shows the reference of SC № 41-46, SC № 11 - in tab. 4.

SC of one orbital plane can see from one to three SC of another orbital plane at the same time. Fig. 10 shows SC positioning in equator and pole areas.

Рис. 9. Ортогональное движение КА в плоскостях № 1 и 3

Table 9

Parameter	Range of parameter changing
	SC № 11 and 31	SC № 11 and 32	SC № 11 and 33
Range, km	from 6000 to 9200	from 500 to 9200	from 5000 to 9200
Range change rate, km/s	from –6 to 6	from -9 до +9	from –6.3 to +6.3
Declination, °	from 54 to 68	from 54 to 88	from 54 to 71
Declination change rate, °/с	from –0.14 to 0.14	from –0.17 to 0.17	from –0.15 to 0.15
Elevation, °	from 7 to –93; from 170 to 267	from 60 to –120; from 140 to 310	from 82 to 190; from –20 to 100
Elevation change rate, °/с	from –0.5 to 0.05	from –0.18 to 0.06	from –0.05 to 0.55
SC mutual visibility period in an orbit pass, min	50.1 (43.2 %)	71.1 (61.3 %)	56.0 (48.3 %)

Changing of IL ballistic parameters between SC in planes № 1 and 3

Table 10