Characteristics and Applications of Common-Track Constellation in One Orbit Plane

LIN Xiaoxiong,QI Yu,LI Xingang,HOU Fenglong,PEI ShengweiInstitute of Telecommunication and Navigation Satellites,Beijing 100094

Abstract:Common-track constellation is a special kind of constellation,all ground tracks of satellites are in one track.With recursive orbit or quasi-recursive orbit,the ground track of the constellation is a closed loop with a repeating track.In particular,if the constellation is in one orbit plane,the constellation can meet the requirement for a multi-satellite launch mission and fast satellite-satellite link establishment.The characteristics and coverage properties of a common-track constellation in one orbit plane are studied in this paper.The simulation results show that a common-track constellation in one orbit plane can provide favorable coverage to a target station in the polar areas.

Key words:polar communication,one orbit plane,common-track constellation,constellation coverage

1 INTRODUCTION

A satellite constellation is a group of satellites working together as a system,which can provide much more converage than a single satellite.Regional coverage means a continuous coverage to every point of a large scale area at any time with not less than one satellite[1].A common-track constellation is a special kind of constellation,which benefits from Earth rotation and a design based on a longitude difference of ascending node between adjacent orbit planes and phase difference between adjacent satellites in one plane,where all ground tracks of satellites are in one track.In particular,with a recursive orbit or quasi-recursive orbit,the ground track of the constellation is a closed loop with the track repeating.If the constellation is in one orbit plane,the constellation can meet the requirement of a multi-satellite launch mission and a fast satellite-satellite link establishment.

A class of circular orbit geometries that has become popular is the Walker Delta constellation.This has an associated notation to describe it which was proposed by John Walker[2].The notation isN/P/F,whereNis the total number of satellites,Pis the number of equally spaced planes,Fis the relative spacing between satellites in adjacent planes.

WU Tingyong[3]proposed a coded notation which can describe a common-track constellation entirely and the restricted equivalent relationship of the Walker Delta constellation proposed based on a thorough investigation on the parametric characteristics of the common-track constellation.The optimized method of a common-track constellation for regional coverage and some optimal constellation schemes for China were studied at the same time.The Walker Delta constellation in this paper was a uniform distribution ofNsatellites inPorbit planes.In this paper,one satellite per orbit plane was a prerequisite for the equivalence between the Walker Delta constellation and the common-track constellation.Suppose Δλis longitude difference of ascending node between adjacent orbit plane,Δωis phase difference between adjacent satellites in one plane,the relationship between Δλand Δωwas shown in Equation (1).

where Δλis longitude difference of ascending node between adjacent orbit planes,ωeis angular velocity of Earth rotation,Δωis phase difference between adjacent satellites in one plane,ωsis angular velocity of satellites.

The Walker Sigma constellation is a special kind of Walker Delta constellation,all ground tracks are in one track which is recursive inDdays with closedLloops.ZHANG Yasheng[4]studied the relationship between ground track separation values and the Walker Sigma constellation configuration and built up a spatial geometrical model of the Walker Sigma constellation,and proposed a Walker Sigma constellation design and analysis method,the results show a huge prospect for Walker Sigma constellation design for high-track communications satellite constellations and warning satellite constellations.

Reference [4] verified that Walker Sigma constellation with a ground track separation value of 1 satisfied Equation (2).

If a closed loop looks like a sine wave,theLloops andDrecursive days satisfy:

whereSis the number of satellites per orbit plane,Dis the recursive period,Fis the ground track separation value,Lis loops in the recursive period,bis a positive integer.H[D,N] is the greatest common divisor ofDandN,which satisfies:

wherebandaare co-prime numbers.

The characteristics and coverage properties of a common-track constellation in one orbit plane are studied in this paper,the simulation results showed that a common-track constellation in one orbit plane can provide favorable coverage for a target station in the polar areas.

2 CHARACTERISTICS OF A COMMON-TRACK CONSTELLATION IN ONE ORBIT PLANE

2.1 Parameters Satisfying a Common-Track Constellation in One Orbit Plane

ALloop recursive orbit inDdays means thatLandDare co-prime numbers,i.e.,

If all satellites are in one orbit plane,the number of equally spaced planesPis 1 and ground track separation valueFis 0.

According to Equation (6),we obtainS=D.With Equation(5),we obtain

Equation (7) indicates that,for a common-track constellation in one orbit plane with a recursive loop,ifSsatellites are distributed in the orbit plane,the recursive period must beSdays.

Meanwhile,concerning the constraint of the Earth radius,the recursive loops per day cannot be more than 17,i.e.,L≤17D.Thus we obtain

All feasible solutions are shown in Table 1 according to Equation (8).If uniform distributed satellites are needed,letS*be the actual satellites distributed in the orbit plane,we can deduceS*as a common factor with recursive periodS,i.e.,for a common-track constellation in one orbit plane with a recursive loop in 15 days,the number of uniform distributed satellites is 3,5 or 15.

Table 1 Parameters of a common-track constellation in one orbit plane with a recursive loop

2.2 Analysis of Constellation

The results of Equation (8) are analyzed under three situations (L＜D,L=D+1,L＞D+1) as follows.

1) Case 1:L＜D.In this situation,the semi-major axis of the orbit is larger than GSO.Figure 1 shows the configuration of constellation for (S=3,D=3,L=2).

2) Case 2:L=D+1.In this situation,the results are similar to reference [1].The closed loop looks like a sine wave.Figure 2 shows the configuration of constellation for (D=3,L=4).

3) Case 3:L＞D+1.In this situation,the ground trace of satellites overlaps.Figure 3 shows the configuration of the constellation for (D=3,L=7) and Figure 4 shows the configuration of constellation for (S=3,D=3,L=10).

Figure 1 Configuration of constellation for (S=3,D=3,L=2)

Figure 2 Configuration of constellation for (D=3,L=4)

Figure 3 Configuration of constellation for (D=3,L=7)

Figure 4 Configuration of constellation for (S=3,D=3,L=10)

According to the analyses above,characteristics of a common-track constellation in one orbit plane are shown below.

1) With the benefits of all of the satellites in one orbit plane,the constellation can meet the requirement for a multi-satellite launch mission,which reduces the propellant consumption for orbit plane transfer;

2) The ground trace of the constellation can be used in the coverage of the belt-shaped region as shown in Figures 2 -4.Selected key targets can be revisited and covered with proper design of the ascending node of the orbit.

3) With the benefits of all of the satellites in one orbit plane,it is not that difficult for fast satellite-satellite link establishment.In particular,if the eccentricity is approximately equal to 0,a stable satellite-satellite link can be established which reduces the complexity.

4) Trace procedure of the station can be designed as a predefined period curve with the properties of repeating the trace of the satellite,which is easy to manage at the satellite operation control center.

3 APPLICATIONS OF A COMMON-TRACK CONSTELLATION IN ONE ORBIT PLANE

In this section,an example of the application of a common-track constellation in one orbit plane is proposed.In order to communicate with targets between the North Pole region and domestic areas in real time,the coverage and a link between North Pole region and domestic areas should be built.

Input parameters are shown below.

1) The target is a station in South Pole region;

2) Elevation angle of the station is 10°;

3) Satellite-satellite link should be established.

Design objectives are shown below.

1) Continious real-time communications between the targets;

2) Minimum number of satellites;

3) Minimum range of link.

3.1 Methods

Three steps are proposed to realize communications between the North Pole region and domestic areas in real time.First,mean inclination and mean eccentricity should be designed according to the mission requirements;Second,the constraint forL/Dshould be designed according to satellite-satellite link requirements;Third,an ergonic algorithm should be proposed from least satellites to more satellites.For every (S,D,L) set,the semi-major axis should be designed first.Finally,optimization for access rate should be proposed with the variable mean argument of latitude.

Step 1:

In order to form a frozen apsidal line of orbit,and in consideration of high altitude properties of the South Pole region,the mean inclination of the orbit was chosen as 63.43°.For a stable satellite-satellite link and reduced range of pointing assembly,the mean eccentricity of the orbit was chosen as 0.

Step 2:

If two satellites are distributed in the plane,there is no satallite-satellite link.So the number of satellites in the plane must be at least 3.In consideration of the nearest distance between the link and Earth surface being 500 km,theD/Lrelationship and the minimum of semi-major axis of the orbit are constrained.The results are shown in Table 2.

Table 2 Relationship between the number of satellites and maximum of (D/L)

Step 3:

D/L,the mean inclination and mean eccentricity are given above.In order to form a strict recursive orbit,the mean semi-major axis of the orbit can be calculated.Reference [6]gives a method to calculate semi-major axis concerningJ2perturbation.

Reference [7] converts the precise revisit problem to the coincidence problem in the inertial coordinate system of a motion vector in the inertial coordinate system and a constant vector in the moving coordinate system,so different target functions are constructed,while orbit elements are obtained by searching the zeros of the corresponding function.References[8-9] investigates high regressive accuracy by combining the iterative adjustments and applying a high precision orbit dynamic.In this paper,prediction is calculated according to references[6-7],then strict regressive orbit elements are established with the high precision orbit dynamic (EGM 2008 model) according to references [8-9],giving the results shown in Table 3.

Table 3 Solution of (S,D,L)

Arrange the Equation (9),and letwe get:

SubstituteJ2,i,ReandRG,we get the results shown as Figure 5.

Figure 5 (D/L) at different inclinations and different (Re/a)

If (S,D,L) is given,the variable under optimized is the mean argument of latitudeλ.As is known to all,the lower altitude of the orbit,the higher access loss rate,so if there is no continuous solution at a altitude,then it is not necessary to increaseL.

3.2 Simulation Results

The results are shown in Table 3,(S,D,L)=(4,4,9) and(S,D,L)=(5,5,14) are the preferred solutions with fewer satellites and lower orbit altitude.For the solution of (S,D,L)=(4,4,9),4 satellites are distributed in one orbit plane,each satellite runs 9 loops in the recursive period of 4 days.For the solution of (S,D,L)=(5,5,14),5 satellites are distributed in one orbit plane,each satellite runs 14 loops in the recursive period of 5 days.With more satellites,the reliability of satellite-satellite link is decreased dramatically.The constellation of (S,D,L)=(4,4,9)is shown in Figure 6.Figure 7 shows the elevation angle from a target in the South Pole region to a satellite in the constellation,Figure 8 shows the range from the target to a satellite in the constellation.

Note:access loss rate is the ratio of access lost time to recursive period in a recursive period

Figure 9 shows the elevation angle and azimuth angle of the station with the highest elevation angle.The trace procedure of the station can be designed based on the curve in Figure 9.

Figure 6 Configuration of constellation for (D=4,L=9)

Figure 7 Elevation angle from target in South Pole region to a satellite in the constellation

Figure 8 Range from target in South Pole region to a satellite in the constellation

Figure 9 Elevation angle and azimuth angle of the station with the highest elevation angle

In conclusion,a common-track constellation in one orbiplane can be used to establish continuous real-time communications between the targets.The constellation supports a stable satellite-satellite link and a convenient way to trace the satellite with a predefined curve.

4 CONCLUSION

The parameter characteristics and coverage propities of common-track constellation in one orbit plane are studied in this paper.An example of the constellation are proposed.To sum up,for the common-track constellation in one orbit plane,the multi-satellite launch mission can reduce the propellant consumption for orbita plane transfer.Moreover,it is not that difficult for the fast satellite-satellite link establishment.In addition,the ground trace of the constellation can be used in the coverage of belt-shaped region.Finally,trace procedure of the satellite operation control center can be designed easily as a predefined period curve.