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Introduction
The long term and highly accurate orbit estimation, especially rise-and-set time computation, plays a key part in the pre-request information for mission analysis and on-board resources management in more general communication, Earth observation and scientific spacecraft, there has been a big trend to use low cost, fast access and multi-functional small satellites to provide and exchange information for a wide range of military and civil applications such as communication in a very remote area for changing conditions, low cost store-and-forward communication, disaster warning for global shipping service and some Earth Observation missions. Therefore, this requires accurate estimation of when the satellites will start to be visible (rise) to a given location on the Earth or to other satellite and the time when the satellite disappears from the horizon (set) over a time-scale of months in some cases.
The rise/set problem may be defined as the process of determining the times at which a satellite rises and sets with respect to a ground location. The easiest solution uses a numerical method to determine visibility periods for the site and satellite by evaluating UK position vectors of each. It advances vectors by a small time increment, Δt, and checks visibility at each step. A drawback to this method is computation time, especially when modeling many perturbations and processing several satellites. Escobal [1], [2] proposed a faster method to solve the rise/set problem by developing a closed-form solution for unrestricted visibility periods about an oblate Earth. He assumes infinite range, azimuth, and elevation visibility for the site. Escobal transforms the geometry for the satellite and tracking station into a single transcendental equation for time as a function of eccentric anomaly. He then uses numerical methods to find the rise and set anomalies, if they exist. Lawton [3] has developed another method to solve for satellite-satellite and satellite-ground station visibility periods for vehicles in circular or near circular orbits by approximating the visibility function, by a Fourier series.
More recently, Alfano, Negron, and Moore [4] derived an analytical method to obtain rise/set times of a satellite for a ground station and includes restrictions for range, azimuth, and elevation. The algorithm uses pairs of fourth-order polynomials to construct functions that represent the restricted parameters (range, azimuth, and elevation) versus time for an oblate Earth. It can produce these functions from either uniform or arbitrarily spaced data points. The viewing times are obtained by extracting the real roots of localized quantic.
Palmar [5], introduced a new method to predict the passes of satellite to a specific target on the ground which is useful for solving the satellite visibility problem. he firstly described a coarse search phase of this method including two-body motion, secular perturbation and atmospheric drag, then he described the second phase (refinement), which uses a further developed controlling equation F (a) = 0 based on the epicycle equations
In this work, a fast method for satellite-satellite visibility periods for the rise-and-set time prediction for two satellites in terms of classical orbital elements of the two satellites versus time were established. The secular variations of the orbital elements due to Earth Oblateness were taken into account in order to consider the changes in the nodal period of satellite and the changes in the long term prediction of maximum elevation angle. In the following description we will introduce the formulae for satellite rise-and-set times of the two satellites. The derived visibility function provides high accuracy over a long period, and provides direct computation of rise-set times.
Visibility Analysis
The location of a satellite is determined by the Kepler’s laws, in addition, orbit perturbations due to Earth Oblateness are considered. A set of six orbital parameters is used to fully describe the position of a satellite in a point in space at any given time: semi-major axis a, eccentricity e, inclinations of the orbit plane i, right ascension of the nodeΩ, the argument of perigee ω, and true anomaly f. The above parameters are shown in the Figure1.
The links between two satellites are determined by the visibility analysis presented as follows:
Referring to Fig.1, the position vectors of satellites 1, and 2 with respect to the ECI coordinate system are ${{\vec{r}}_{1}}$and ${{\vec{r}}_{2}}.$The position vector from satellite 1 to satellite 2 will be denoted by $\vec{\rho }={{\vec{r}}_{2}}-{{\vec{r}}_{1}}.$
Let h = OP = Re +Δh, be the perpendicular from the dynamical center of the earth to the range vector , cutting the earth surface at Q and the range vector $\vec{\rho },$where Re is the mean radius of the Earth, and Δh is the thickness of the atmosphere above the surface of the Earth to P. From the geometry of Fig. 1, we have the following relations:
Where ψ is the angle between ${{\vec{r}}_{1}}\,\,\,\text{and}\,\,\,{{\vec{r}}_{2}}$the two satellites. The condition for direct visibility between the two satellites can be determined from Eq. 3 by putting
After the determination of the locations of satellites at any time in the space, they can only achieve visibility when they are both above the same plane which is tangent to the earth surface. The extreme situation is that both of them are in the tangent plane.
When Δh > 0, the two satellites can achieve visibility. Otherwise, there is no visibility.
If the position relation between two satellites satisfies the visibility conditions, two satellites can communicate with each other over interstellar links.
Construction of The Visibility Function
The position vector of each satellite in the geocentric coordinate system,$\vec{r}=\left( x,y,z \right)$, can be calculated by the following formula [6]:
Where σ1 = cos (i1/2) and γ1 = sin (i1/2), with similar expressions for the other satellite. In order to obtain the visibility function as an explicit function of time, we transform the true anomaly f, to the mean anomaly M, using the following transformation formulas Brouwer [6] up to O(e4)
Where the coefficients As, s = 0,1, …,164 are functions of (aj , ej , i j), j = 1,2 are given in Appendix A.
Adding Perturbing Forces
We shall consider the effect of perturbation on the orbital elements due to the Earth Oblateness. So, the orbital elements of the two satellites can be written in the form:
Where(Δσj)obl denote the first order perturbation in the orbital elements, and j = 1, 2 denotes satellites 1 and 2.
The expansion of the perturbed visibility function about some epoch time t0 can be obtained by Taylor expansions about the osculating elements (σ0j,e0j,i0j,Ω0j,ω0j,M0j) up to the first order as:
The symbols σs, represent any of the orbital elements. The summation ranges from s = 1 to 3 represents the elements (Ω1,ω1,M1) and from s = 4 to 6 represents (Ω2,ω2,M2) respectively. The quantities Δσs, s = 1, 2,..,6 represent the secular variations in the corresponding orbital elements due to the perturbation.
The effect of Earth Oblateness
A satellite under the influence of an inverse square gravitational law has truly constant orbital elements. In reality, however, there is a gradual change in the orbital elements due to the Earth’s Oblateness. The principal effect of this is to introduce a short period oscillation of the orbital elements, which we can ignore in most cases. The argument of perigee ω, the longitude of the ascending node Ω , and the Mean anomaly M, however, experience a secular drift which significantly changes the long term prediction of maximum elevation angle. Using the method of variation of parameters to take proper account of all these secular variations due to earth oblateness up to J2. The perturbation method is explained in many standard textbooks on [7].
The gravitational potential, U, of a satellite including the contribution of J2, is given by [7]
Since we consider only the secular variation, so we average the perturbative function, with respect to the mean anomaly M . The derivation and solution are given in many text books for example [8].
In what follows the visibility function were tested for some examples to obtain the mutual visibility between two Earth Satellites whatever the types of their orbits may be. Classical orbital elements for some satellites from the Center for Space Standards & Innovation were used as test data for this study, and are listed in Tables 1 and 2.
Norad Two - Line Element Sets For The Satellites AQUA, ARIRANG-2, HST and ODIN
Satellite Orbital Elements
1-AQUA
2-ARIRANG-2
3-HST
4-Odin
Equivalent altitude (Km)
699.588
682.6205
543.2687
540.5256
a (Km)
7077.725
7060.757
6921.405
6918.662
n (rev/min)
0.010408
0.010445
0.010779
0.010769
e
0.000286
0.001669
0.000256
0.001057
i (degree)
98.2031
98.0676
28.4705
97.591
Ω (degree)
121.6097
76.9906
17.611
200.4958
ω (degree)
54.081
258.4665
301.12
186.4076
M (degree)
125.1605
101.4671
170.9719
173.7019
ρ (kg/km3)
3.63E-05
4.6E-05
0.000354
0.000369
ρo (kg/km3)
0.000145
0.000145
0.000697
0.000697
h0(Km) (kg/km3)
600
600
500
500
H (Km)
71.835
71.835
63.822
63.822
Epoch Year & Julian Date
18180. 59770749
18180. 82019665
18182.935593
18182.93790454
time of data (min)
2018 06 29
2018 06 29
2018 07 01
2018 07 01
13:31:30
19:41:03.004
21:57:32.134
22:30:32.994
The visibility intervals are shown in the following Figures 2,4,6 without any perturbing force and with Oblateness force in Figures 3, 5, 7, calculated in Tables 1, 2 and 3, respectively.
Norad Two - Line Element Sets For The Satellites CFESAT and MTI
Satellite Orbital Elements
5-CFESAT
6-MTI
Equivalent altitude (Km)
468.8831
412.5092
a (Km)
6847.02
6790.646
n (rev/min)
0.010953
0.011074
e
0.000582
0.000812
i (degree)
35.4247
97.5789
Ω (degree)
203.043
17.7612
ω (degree)
183.8662
345.6071
M (degree)
176.2019
143.5229
ρ (kg/km3)
0.001162
0.003008
ρo (kg/km3)
0.001585
0.003725
h0(Km) (kg/km3)
450
400
H (Km)
60.828
58.515
Epoch Year & Julian Date
18182.5017322
18182.7746284
time of data (min)
2018 07 01
2018 07 01
12:02:28.526
18:02:08.608
Visibility Intervals Between AQUA and ARIRANG 24 Houres
Without Earth Oblateness Force
With Earth Oblateness Force
Rise
Set
visibility time
Rise
Set
visibility time
m
s
m
s
1
17.1074
44.8663
27
45.534
-
2.35384
-
-
2
66.4086
94.0482
27
38.376
14.4534
51.6719
37
13.1
3
115.669
143.318
27
38.94
64.843
100.955
36
6.72
4
164.973
192.499
27
31.56
114.976
150.222
35
14.76
5
214.236
241.768
27
31.92
164.951
199.507
34
33.36
6
263.542
290.948
27
24.36
214.755
248.754
33
59.94
7
312.807
340.216
27
24.54
264.522
298.042
33
31.2
8
362.115
389.396
27
16.86
314.144
347.279
33
8.1
9
411.382
438.663
27
16.86
364.522
396.57
32
2.88
10
460.692
487.842
27
8.94
413.301
445.8
32
29.94
11
509.962
537.108
27
8.76
462.871
495.093
32
13.32
12
559.276
586.286
27
6
512.306
544.32
32
0.84
13
608.547
635.552
27
0.3
561.825
593.615
31
47.4
14
657.863
684.729
26
51.96
611.207
642.839
31
37.92
15
707.135
733.994
26
51.54
660.689
692.135
31
26.76
16
756.453
783.171
26
43.08
710.033
741.357
31
19.44
17
805.728
832.435
26
42.42
759.488
790.655
31
10.02
18
855.048
881.611
26
33.78
808.804
839.876
31
4.32
19
904.325
930.874
26
32.94
858.238
889.174
30
56.16
20
953.647
980.049
26
24.12
907.532
938.394
30
51.72
21
1002.93
1029.31
26
22.8
965.95
987.694
30
44
22
1052.25
1078.49
26
14.4
1006.23
1036.91
30
40.8
23
1101.35
1127.75
26
13.2
1055.63
1086.21
30
34.8
24
1150.86
1176.92
26
3.6
1104.9
1135.43
30
31.8
25
1200.14
1226.18
26
2.4
1154.29
1184.73
30
26.4
26
1249.47
1275.36
25
53.4
1203.54
1233.95
30
24.6
27
1298.75
1324.62
25
52.2
1252.93
1283.25
30
19.2
28
1348.08
1373.79
25
42.6
1302.17
1332.47
30
18
29
1397.37
1423.05
25
40.8
1351.55
1381.77
30
13.2
30
-
-
-
-
1400.79
1430.99
30
12
Conclusions
An analytical method for the rise -set time prediction for two satellites were derived through a visibility function in terms of classical orbital elements of the two satellites versus time. The secular variations of the orbital elements due to Earth Oblateness were taken into account in order to consider the changes in the nodal period of satellite and the changes in the long term prediction of maximum elevation angle.
In the Table 3: The Visibility Intervals Between AQUA and ARIRANG 2, it is noticed from the first column of (The Function of Visibility without any perturbation) that the time of visibility periods oscillates in a periodic fashion till the ninth period then it decreases gradually. Also, from the second column we conclude that the effect of Earth Oblateness is the increasing the number of periods significantly. But the time intervals of the visibility decreases gradually.
In the Table 4: The Visibility Intervals Between HST and ODIN, It is noticed from the first column of (The Function of Visibility without any perturbation) that the time of visibility periods oscillates in a periodic manner, and then it decreases gradually. It is also noticed that the second column explains the Earth Oblateness which affects the number of period’s increases significantly and clearly. It is also observed that the time of periods of the visibility ascending increases till the ninth period then it increases and decreases in an oscillating manner and over time, stability occurs and the time period stabilizes.
Visibility Intervals Between HST and ODIN 24 Houres
Without Earth Oblateness Force
With Earth Oblateness Force
visibility time
visibility time
Rise
Set
m
s
Rise
Set
m
s
1
39.3255
44.4432
5
7.062
-
5.86362
-
-
2
87.2035
92.0758
4
52.338
14.1397
22.0401
7
54.024
3
134.82
139.913
5
5.52
28.8463
43.6288
14
46.59
4
182.699
187.545
4
50.76
57.927
91.3591
33
25.926
5
230.316
235.383
5
4.02
105.468
139.084
33
36.96
6
278.195
283.014
4
49.14
153.145
186.832
33
41.22
7
325.812
330.852
5
2.4
200.854
234.559
33
42.3
8
373.691
378.843
5
9.12
248.578
282.308
33
43.8
9
421.307
426.322
5
0.9
296.07
330.036
33
57.96
10
469.188
473.952
4
45.84
344.041
377.786
33
44.7
11
516.803
521.792
4
59.34
391.776
425.514
33
44.28
12
564.684
569.421
4
44.22
439.514
473.263
33
44.9
13
612.298
617.261
4
57.72
487.252
520.92
33
40.08
14
660.18
664.98
4
48
534.992
568.741
33
44.94
15
707.794
712.731
4
56.22
582.731
616.47
33
44.43
16
755.676
760.359
4
40.98
630.473
664.219
33
44.76
17
803.289
808.201
4
54.72
678.212
711.948
33
44.16
18
851.172
855.828
4
39.18
725.954
759.697
33
44.58
19
898.785
903.67
4
53.1
773.694
807.426
33
43.92
20
964.668
951.297
4
37.74
821.437
855.175
33
44.28
21
994.28
999.14
4
51.6
869.178
902.904
33
43.65
22
1042.16
1046.77
4
36.6
916.92
950.653
33
43.98
23
1089.78
1094.61
4
49.8
964.661
998.382
33
43.62
24
1137.66
1142.23
4
34.2
1012.4
1046.13
33
43.8
25
1185.27
1190.08
4
48.6
1060.15
1093.86
33
42.6
26
1233.16
1237.7
4
32.4
1107.89
1141.61
33
43.2
27
1280.77
1285.55
4
46.8
1155.63
1189.34
33
42.6
28
1328.65
1333.17
4
31.2
1203.37
1237.09
33
43.2
29
1376.26
1381.02
4
45.6
1251.11
1248.82
33
42.6
30
1424.15
1428.64
4
29.4
1298.86
1332.56
33
42
31
-
-
-
-
1346.6
1380.29
33
41.4
32
-
-
-
-
1394.34
1428.04
33
42
In the Table 5: The Visibility Intervals Between CFESAT and MTI, It is noticed from the first column of (The Function of Visibility without any perturbation) that the time of visibility periods has gradually obvious increases. Also the periods’ time of the visibility function increases significantly and clearly. The second column explains the Earth Oblateness which affects the number of periods increase significantly and clearly. It is also observed that the time of periods of the visibility increases and decreases in an oscillating manner.
Visibility Intervals Between CFESAT and MTI 24 Houres