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Satellite constellations in sliding ground track orbits
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Satellite constellations in sliding ground track orbits Christian Circi ∗ , Emiliano Ortore, Federico Bunkheila Department of Astronautical, Electrical and Energetic Engineering, Sapienza University of Rome, Via Salaria 851, 00138 Rome, Italy
a r t i c l e
i n f o
Article history: Received 5 March 2013 Received in revised form 31 March 2014 Accepted 18 April 2014 Available online xxxx Keywords: Orbital transfer Satellite constellation Ground track pattern
a b s t r a c t This paper deals with repeating ground track orbits, taking into consideration the case of a single satellite and of constellations composed of one or more orbital planes. First, the orbital relationships able to guarantee a uniform ground track pattern on the Earth’s surface are provided. Then, the sliding ground track concept, based on the possibility to shift from a ground track pattern to another by small corrective manoeuvres, is introduced. This technique, which can be applied to both single satellite and satellite constellations, allows the fulfillment of several objectives in the course of the same mission. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction The characteristics of a satellite observation system are strictly related both to the instruments installed on board of satellites and to the choice of the orbits. In particular, the repeating ground track orbits, which allow the formation of regular ground track patterns and therefore cyclic observations of the Earth’s surface and of the atmosphere, represent an essential element in the Earth observation field [4,5]. Very often this requirement is associated with the one related to the synchronism between the apparent motion of the Sun around the Earth and orbital plane displacement (SunSynchronous Orbits – SSOs), so as to gain approximately the same illumination conditions during the observations of a given region. In addition, recent studies have shown the possibility to use (for the observation of the Earth and Mars) the so-called Multi-SunSynchronous Orbits, in which also the solar illumination conditions cyclically vary according to the choice of the orbit elements [11,3]. When the use of a single satellite prevents the observation of a given area with simultaneous satisfactory observation frequency and ground spatial resolution (requirements inversely correlated one to the other), it is necessary to resort to a constellation. A great number of papers have been written on satellite constellations. Some examples of these are represented by the kinematically regular satellite systems [6,7], the Walker constellation [17,18], the circular polar constellations for continuous single or multiple coverage [1], the homogeneous constellations for regional coverage [14], the Rosette constellation [2], up to the more recent studies
*
Corresponding author. E-mail addresses: [email protected] (C. Circi), [email protected] (E. Ortore), [email protected] (F. Bunkheila). http://dx.doi.org/10.1016/j.ast.2014.04.010 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.
performed with the aim of gaining a continuous coverage of the Earth by using kinematically symmetrical satellite systems in circular orbits [8], a multiple continuous coverage by systems with linear structure [13,12], a continuous coverage by using elliptic orbits with a quasi-stationary condition in an orbit arc around the apogee [9,10] and by constellations of Molniya type orbits [15,16]. This paper proposes a technique to satisfy several objectives in the course of the same mission. In fact, by means of low-cost manoeuvres, it is possible to modify more times the orbit elements and, accordingly, the observational characteristics of a satellite so as to optimize its performance for specific and changing goals. Although these manoeuvres are characterized by small velocity variations, they allow the shift from a ground track pattern to another that can be significantly different (thus the definition of “sliding” pattern), opening the way to a dynamical model of satellite and of satellite constellation. The article is organized as follows: Section 2 illustrates the orbital relationships that have to be satisfied, by a single satellite and by single and multi-plane constellations, to obtain regular cycles of observation of the Earth with a uniform ground track pattern (equally spaced tracks on the equator). In the case of constellations, the satellites have to be located over orbits with the same shape and inclination and opportunely arranged over orbital planes properly phased in longitude and containing the same number of satellites. In Section 3 the concept of sliding ground track orbits is described and the feasibility of carrying out corrective manoeuvres to properly modify the ground track, shifting from one orbital solution to another, is investigated. The analysis is applied to both the single satellite, taking into consideration SSOs as a case study, and to the satellite constellations.
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Fig. 1. Curves of periodicity.
2. Repeating ground tracks A ground track is defined as periodic (repeating) if it repeats itself at regular time intervals allowing a cyclic observation of the Earth. In the case of circular orbit, considering the zonal harmonic J 2 of the gravitational field and indicating with r and i the radius and the inclination of the orbit respectively, the mathematical condition identifying a repeating ground track can be written as [3]:
r 3 .5 + b 1 r 2 + b 2 = 0 ,
(1)
where the coefficients have the following expressions:
√ b1 = − b2 =
μE , qω E
3 J 2 R 2E
√
2qω E
μ E
1 − 4 cos2 i + q cos i ,
and where μ E is the gravitational constant of the Earth, q is the repetition factor, ω E is the angular velocity of the Earth around its axis and R E is the mean equatorial radius of the Earth. The repetition factor q is given by the ratio R /m, where m is the integer number of nodal days after which the track is repeated and R is the integer number of nodal periods accomplished by satellite in m nodal days (R and m are prime to one another). Once the values of i and q have been assigned, the physically acceptable solution of Eq. (1) provides the value of the radius r of the corresponding orbit. The same equation, for each value of q, links the values of the orbit altitude h and the inclination defining a curve referred to as “curve of periodicity”. Fig. 1 shows the solutions of Eq. (1), with m going from 1 to 5, in the case of Low Earth Orbits (LEOs). All points of a curve, being R and m prime one to the other, refer to orbits having the same revisit time over a given area. The parameter q, representing the number of orbital periods per day (both nodal), can be written as sum of an integer number N i plus a fractional part N f = k/m, where the parameter k is an integer number which is prime with m and ≤ m − 1. In particular, if k = 0 ( N f = 0) then q = N i and m = 1. For example, for the orbit P 1 in
Fig. 1, in which k = 0 and q = 14, it is m = 1 and R = 14 · 1 = 14, while for the orbit P 2 , in which the satellite accomplishes also another portion of orbit, q = 14 + 1/5 (k = 1), it is m = 5 and R = N i m + k = 14 · 5 + 1 = 71. The equatorial longitudinal separation between two successive ground tracks is S t = 2π /q while, after a repetition period (m nodal days), the minimum equatorial ground track distance reduces to the value S m = S t /m. In fact, each interval S t is subdivided in m intervals S m by the m − 1 tracks related to orbits successive to the N i -th (in particular, if m = 1, the (N i + 1)-th track will coincide with the first one). The parameter k determines the way in which such a subdivision is carried out. In fact, the minimum ground track distance S 1 (ascending node passes) related to consecutive nodal days can be expressed by the following relationships:
S 1 = − Smk
for k <
m 2
(2a)
,
S 1 = S m (m − k) for k >
m 2
.
(2b)
Fig. 2 shows, as an example, all the possible cases with m = 5 (the distance S t is subdivided in 5 equally spaced intervals). In the case k = 1 the interval S 1 is equal to the minimum distance S m and the tracks related to the successive nodal days (numbered with 2, 3, 4) are eastward shifted of S m . In the case k = 2, being S 1 = 2S m , the tracks related to the successive nodal days are eastward shifted of 2S m . The cases k = 3 and k = 4 are similar respectively to the cases k = 2 and k = 1 but the track progression inside the interval S t is westward. In fact, in Eqs. (2a), (2b) it is S 1 > 0 if the progression is westward, S 1 < 0 if the progression is eastward. If also the descending node passes are taken into consideration, the minimum equatorial distance between tracks can be reduced to the value S t /2m and it occurs if m and R are not both odd numbers, otherwise ascending and descending nodes coincide. To obtain Eqs. (2a), (2b) the longitudes λi and λi +1 corresponding to tracks N i -th (last track of the first nodal day) and ( N i + 1)-th (first of the second nodal day) are taken into account. Indicating with λ0 the longitude of the first track:
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Fig. 2. Track sequence for m = 5.
Fig. 3. Daily ground track displacement.
λi = λ0 + N i S t = λ0 − S m k,
(3a)
λi +1 = λ0 + ( N i + 1) S t = λ0 + S m (m − k),
(3b)
the daily spacing S 1 can be identified by the minimum value out of the following two differences: |λi − λ0 |, |λi +1 − λ0 | and it is |λi − λ0 | < |λi +1 − λ0 | for k < m/2, |λi − λ0 | > |λi +1 − λ0 | for k > m/2 (such differences are equal in the case m = 2). Therefore, the daily spacing can be assumed as an eastbound (sign −) displacement if k < m/2 and as a westbound (sign +) displacement if k > m/2 (Fig. 3). The condition k = m/2 occurs only for m = 2 (k = 1) and in such a case it is possible to refer to both Eqs. (2). A useful relationship that summarizes the track progression in the interval [λ0 , λ0 + S t ] can be expressed as:
λd = λ0 + S t · 1 + int(dN f ) − dN f ,
d = 1, . . . , m − 1,
to guarantee the required coverage, resolution and/or repetition of observation, it is necessary to resort to a constellation. In particular, with a single satellite, it is not possible to reduce the repetition of observation to values less than one nodal day. 2.1. Satellite constellations In the case of N satellites orbiting on the same orbit, indicating with λ1,0 the longitude of the ascending node pass of satellite 1 at the day 0, the ground track progression in the interval [λ1,0 − St , λ1,0 + S2t ] can be described by the following relationship: 2
λi ,d = λ1,0 + S t · mod(x),
i = 2 , . . . , N , d = 0, . . . , m ,
(5)
where λi ,d is the longitude of the ascending node pass of satellite
(4)
where λd is the longitude after d nodal day (with d integer number) and the operator int(dN f ) provides the integer part of dN f . With Eq. (4), all the displacements are westward counted (1 + int(dN f ) − dN f > 0), according to the fact that the first track of the day d is westward shifted with respect to the first track of the day 0. In order to achieve a complete longitudinal coverage of the Earth in m nodal days, the swath L of the instrument of observation must be ≥ S m · sin i (Fig. 4). Since the swath L is inversely correlated to the ground spatial resolution (the number of pixels is fixed as a characteristic of the instrument), observation frequency (value of m) and spatial resolution are also inversely correlated. Thus, if a higher spatial resolution is required, it is necessary to opportunely increase the value of m so as to reduce the track spacing S m and vice versa. However, when the couple of parameters S m and m are not able
M
i at the day d, x = dN f + 2π i , M i is the mean anomaly phasing between satellites i and 1 and the operator mod(x) is defined as:
mod(x) = int frac(x) +
1 2
− frac(x),
(6)
with frac(x) = fractional part of x and int( y ) = integer part of y. If M i = 0, Eq. (5) becomes:
λd = λ0 + S t · mod(dN f ),
d = 0, . . . , m
(7)
and describes the track progression for the single satellite case. It represents an alternative to Eq. (4). If the goal is to increase the repetition of observation, in order to make satellite i passes on the same track of satellite 1, it must be λi ,d = λ1,0 . It occurs when the value of x in Eq. (5) is an integer number. Since the considered constellations refer to the case in which the N satellites are equally shifted in mean anomaly one from another:
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If the goal is the increase of repetition of observation, the condition of coincidence of ground track patterns originating by different planes can be expressed as:
lcm(m, N P )
M R 2π
+
Ω 2π
= I,
q
(10)
where I is an integer number, M R (constant) represents the mean anomaly phasing between the i-th satellite of the p-th plane and the i-th satellite of the (p − 1)-th plane, and Ω (constant) is the shift in RAAN between the orbital planes (p = 1, . . . , P f , with P f = number of planes devoted to the increase of observation frequency; i = 1, . . . , N P , with N P = number of satellites for each plane). In this way, the minimum interval between passes of satellites over the same ground track will be reduced to the following value: Fig. 4. Minimum required swath.
M i =
2π N
(i − 1),
i = 2, . . . , N ,
t = (8)
the aforesaid condition is satisfied if the ratio m/ N is an integer number or, in other words, when N = m or N is a sub-multiple of m. In particular, if N = m the observations occur at a regular interval of one nodal day (N > m is meaningless: the position of satellite (N + 1)-th would coincide with the position of satellite 1, the position of the (N + 2)-th with the one of satellite 2, and so on). If the goal is the reduction of the minimum track spacing, an N satellite constellation allows a reduction of this distance to the following value:
S m, N =
St lcm( N , m)
(9)
,
where lcm( N , m) is the least common multiple between N and m (N can be smaller, equal or greater than m). The maximum reduction occurs when N and m are prime to one another and all the satellites describe different tracks. Table 1 summarizes all the possible situations for the single plane case in terms of time interval between two observations over the same area (t), expressed in nodal days, and of minimum track spacing. From the general expressions it is possible to retrieve the particular cases in which all the satellites perform the same tracks (third row of Table 1) and all the satellites perform different tracks (fourth row of the table). In the other cases (e.g. N = 6, m = 8), the result is both an increase of the repetition of observation and a reduction of the minimum track distance. In the case of a single plane constellation with N satellites satisfying Eq. (8), the time t elapsing between two passes over the same ground track is constant and a uniform ground track distribution is guaranteed (equally spaced tracks). 2.1.1. Multi-plane constellations In order to reduce the observation frequency to intervals lower than one nodal day, it is necessary to take into account satellites orbiting on different orbital planes. The use of more orbital planes allows also a further reduction of the minimum track spacing distance.
lcm(m, N P ) P f NP
(11)
.
The value of t is not necessarily an integer number of nodal days (as in the single plane case) and then intervals of revisit lower than one nodal day can be obtained. In particular, considering Ω = 2π / P f in Eq. (10), the observation of given area occurs at regular intervals (t = constant). Finally, if m = N P all satellites of every plane describe the same ground tracks and such a constant value reaches its absolute minimum: t = 1/ P f = Ω/2π . If the goal is the reduction of the minimum track spacing distance, it is necessary to impose the following condition to ensure equally spaced ground tracks:
lcm(m, N P )
M R 2π
+
Ω 2π
q
=I+
1 Ps
,
(12)
where P s (> 1) is the number of planes devoted to thickening the ground track pattern and the minimum ground track spacing will be reduced to the value:
S m, N , P =
St P s · lcm(m, N P )
(13)
.
In order to obtain the lowest value of S m, N , P , it is necessary to take N P and m prime one to the other so that all satellites perform different tracks. 3. Sliding ground tracks Each ground track pattern, obtained by a satellite or a constellation, can be considered (instead of a steady characteristic of a space mission: one mission, one track pattern) as a step that can be gained during a mission (one mission, more patterns). This concept, arising from the possibility of “sliding” from a repeating ground track pattern to another, leads to a dynamical model of satellite and constellation. In fact, by applying small velocity variations V , it is possible to transfer a satellite from a periodic solution of Fig. 1 to another one, which is characterized by a different value of m, i or both. These transfers from an orbit to another allows the modification of the observational parameters thus satisfying different mission requirements in terms of revisit time and coverage. Moreover, the considering of constellations composed of a limited number of satellites that suitably change their orbital
Table 1 Revisit frequency and minimum equatorial track spacing for single plane constellations.
t
S m, N
All satellites perform the same tracks (N = m, N sub-multiple of m)
lcm(m, N ) N m N
All satellites perform different tracks (N and m prime one to the other)
m
St lcm(m, N ) St m St mN
General expressions
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Fig. 5. Selected repeating ground track solutions.
configuration in the course of mission allows the fulfillment of different objectives with a small cost. Fig. 5 shows repeating ground track solutions, considering m from 1 to 5, together with the paths related to the transfers from one point to another. The considered orbits, having an inclination from 20 to 40◦ , find their best employment in the observation of Earth mid-low latitudes. The characteristics of such points are reported in Table 2. The considered orbits are, in general, different in terms of revisit time and ground track spacing and Table 3 shows the velocity variations needed to transfer a satellite from one point to another, taking into account Hohmann transfers and, where necessary, orbital plane changes carried out at the apogee of the Hohmann orbits. The vertical displacements in Fig. 5, corresponding to transfers along orbits with the same inclination, are characterized by the lowest values of V (in plane manoeuvres). For example, the vertical transfers from X 1 to the other selected solutions X i and Y i , with i = 1, 2, 3, 4, require a V from 38 to 293 m/s. On the contrary, the transfer from X 1 to the Z i solutions needs a V from 913 to 979 m/s (out of plane manoeuvres). In general, the results show how with moderate variations V it is possible to gain a significant change of the mission requirements in terms of revisit time, coverage or both. 3.1. Application to the single satellite In this section, the concept of sliding ground track orbits is applied to the single satellite, taking into consideration, as a case study, SSOs, which are the classical orbits used for Earth observation. In particular, the intersections of the curves of periodicity (Fig. 1) with the Sun-Synchronism curve, in which the link radiusinclination is such that the angular velocity of the nodal line due ˙ is equal to the angular velocity of the Sun in its apto J 2 (Ω) ˙ S ), define the Periodic parent motion with respect to the Earth (Ω Sun-Synchronous Orbits (PSSOs). These orbits, as well known, allow the repeated observation of an area at the same local time and are very often exploited in the remote sensing field. Fig. 6 shows three PSSO solutions, deriving from the intersection of the Sun-Synchronism curve with the curves of periodicity corresponding to q = 13, 14 and 15. Considering all the curves of periodicity, the related intersections with the Sun-Synchronism curve lead to a high number of solutions, summarized in Fig. 7 in terms of m, h, q and i.
Fig. 6. Examples of PSSO solutions.
Given that each point of Fig. 7 is characterized by a value of orbit altitude, inclination, m, R and k, it can be exploited to observe the Earth satisfying specific mission requirements. For example, point P in Fig. 7 represents a PSSO which allows the satellite a repeated observation of a given region every m = 8 nodal days at an altitude of about 1600 km. Since q = N i + k/m, the integer part N i = 12. The value of k, which is prime with m and goes from 1 to m − 1, can be retrieved by Fig. 7 going from right to left. In this case k = 1 and therefore R = qm = 97 (q = 12 + 1/8). Since the velocity variations V required to transfer a satellite from a point of Fig. 7 to another one can be low, while the corresponding observational parameters can significantly vary, the services offered by a satellite can be modified during the mission to satisfy different objectives. Fig. 8 shows some PSSO solutions located in the orbit inclination range between 98.20◦ and 98.28◦ and Table 4 reports the related characteristic parameters. Thanks to the closeness of the orbit inclination and altitude values among the selected PSSO solutions in Table 3, the velocity variations V needed to transfer a satellite from one point to another of Table 3 are minimized (quasi-in plane manoeuvres). Table 5 reports the required velocity variations to transfer a satellite from one solution to another of Table 4, taking into account Hohmann type transfers with orbital plane changes carried out at the apogee of the Hohmann orbits. As an example, Fig. 9 shows the change of ground track pattern related to the couple of orbits A, I. The big change in the ground track pattern is obtained by a velocity variation of only 4.09 m/s. This optimal ground track sliding from case A to case I (and vice versa) allows a change of the revisit time from 2 to 29 and a variation of the minimum ground track spacing from 1381.9 to 95.2 km.
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Table 2 Characteristic parameters for the selected solutions. Orbit
i (◦ )
h (km)
m
k
R
S m (km)
O X1 X2 X3 X4 Y1 Y2 Y3 Y4 Z1 Z2 Z3 Z4
30 30 30 30 30 30 30 30 30 20 20 40 40
817.94 1115.02 1038.08 962.96 889.61 747.90 679.43 612.45 546.93 1191.33 814.79 1197.72 822.65
1 5 5 5 5 5 5 5 5 1 1 1 1
0 1 2 3 4 1 2 3 4 0 0 0 0
14 65 65 65 65 70 70 70 70 13 14 13 14
2862.5 616.5 616.5 616.5 616.4 572.5 572.5 572.5 572.5 3082.7 2862.5 3082.7 2862.5
Table 3 Required velocity variations to transfer a satellite from one point to another.
V (m/s)
O
X1
X2
X3
X4
Y1
Y2
Y3
Y4
Z1
Z2
Z3
Z4
O X1 X2 X3 X4 Y1 Y2 Y3 Y4 Z1 Z2 Z3 Z4
0 149 111 74 37 36 73 109 144 994 918 995 918
149 0 38 75 112 186 222 258 293 913 979 914 976
111 38 0 37 75 148 184 220 256 933 962 934 960
74 75 37 0 37 110 147 182 218 953 947 954 945
37 112 75 37 0 73 109 145 181 973 932 974 930
36 186 148 110 73 0 36 72 108 1015 935 1016 937
73 222 184 147 109 36 0 36 72 1036 954 1037 956
109 258 220 182 145 72 36 0 36 1057 973 1059 975
144 293 256 218 181 108 72 36 0 1079 993 1081 994
994 913 933 953 973 1015 1036 1057 1079 0 188 1783 1878
918 979 962 947 932 935 954 973 993 188 0 1881 1829
995 914 934 954 974 1016 1037 1059 1081 1783 1881 0 187
918 976 960 945 930 937 956 975 994 1878 1829 187 0
Fig. 7. Intersections between periodicity and Sun-Synchronism curves.
3.2. Application to the satellite constellations The concept of sliding ground track can be extended to the satellite constellations leading to a further improvement of the mission performances. Similar to the case of a single satellite, the mission can be planned considering a series of orbital transfers which allows the achievement of various configurations able to satisfy different requirements. Even taking into account a constellation of only two satellites, many possibilities to plan the mission arise. In order to clarify the
advantages related to the proposed technique, a simple example of mission (composed of three steps) is here presented: starting from a initial configuration in which the satellites are performing the same ground tracks, the constellation achieves a second configuration in which the satellites are performing different ground tracks and then a third configuration in which the satellites are again performing the same tracks, but those are different from the ones related to the initial configuration. For example, in the first configuration the satellites can both be located over orbit A. Since this orbit is characterized by a revisit time of two nodal
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Table 4 Characteristic parameters for the selected PSSOs. Orbit
i (◦ )
h (km)
m
k
R
S m (km)
A B C D E F G H I
98.274 98.200 98.223 98.235 98.207 98.242 98.247 98.205 98.251
720.00 701.89 707.45 710.40 703.69 712.22 713.46 703.13 714.36
2 9 13 17 20 21 25 29 29
1 5 7 9 11 11 13 16 15
29 131 189 247 291 305 363 422 421
1381.9 305.9 212.0 162.2 137.7 131.4 110.4 95.0 95.2
Table 5 Needed V to transfer a satellite from a PSSO solution to another.
V (m/s)
A
B
A B C D E F G H I
0 13.14 9.08 6.95 11.86 5.66 4.76 12.25 4.09
13.14 0 4.06 6.20 1.28 7.49 8.38 0.90 9.06
C 9.08 4.06 0 2.14 2.78 3.43 4.32 3.16 5.00
D
E
6.95 6.20 2.14 0 4.92 1.29 2.18 5.30 2.86
F
11.86 1.28 2.78 4.92 0 6.21 7.10 0.39 7.78
G 5.66 7.49 3.43 1.29 6.21 0 0.89 6.59 1.57
4.76 8.38 4.32 2.18 7.10 0.89 0 7.48 0.68
H
I
12.25 0.90 3.16 5.30 0.39 6.59 7.48 0 8.16
4.09 9.06 5.00 2.86 7.78 1.57 0.68 8.16 0
Fig. 8. Selected PSSOs.
day (Table 4) and the mean anomaly phasing between the satellites is equal to 180◦ (according to Eq. (8)), the revisit quency and the minimum equatorial track spacing of this lcm(m, N ) configuration are given by (Table 1): t = = 1 nodal N
two frefirst day,
St S m, N = lcm(Smt , N ) = m , which is equal to the one of a single satellite, being the satellites performing the same ground tracks (N = m). The second configuration can be gained shifting one of the two satellites in orbit I by means of a velocity variation of 4.09 m/s (Table 5). This quasi-coplanar transfer allows the achievement of a condition in which a satellite passes every two nodal days over the same areas (orbit A) and the other one offers a very thick ground track pattern (orbit I). Fig. 10 shows the ground tracks of orbit A and of orbits A + I (tracks of A plus tracks of I). The third configuration can be obtained shifting also the other satellite in orbit I by means of a second V = 4.09 m/s (both satellites in orbit I with a mean anomaly phasing equal to 180◦ ). Being in this case N and m prime one to the other, this configuration is characterized by a revisit frequency equal to the one related to orbit I (29 nodal days) but by a minimum equatorial
Fig. 9. Sliding ground track pattern for the cases A, I.
track spacing reduced to the following value: S m, N = St 58
St lcm(m, N )
=
= 47.6 km.
The sliding ground track technique can be applied to a generic constellation and different ground track patterns can be gained with a limited number of satellites. In fact, according to Eqs. (10), (11), (12), (13) satellites located over different orbital planes can
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number of satellites, guarantees the fulfillment of several objectives in the course of the same mission. Conflict of interest statement We confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. References
Fig. 10. Application of the sliding concept to a two-satellite constellation.
be re-allocated many times by small V thus satisfying different mission requirements. 4. Conclusions After having presented useful relationships, for single and multi-plane satellite constellations, to gain ground tracks uniformly distributed on the Earth’s surface, a low- V technique to meet different mission requirements has been proposed introducing the concept of sliding ground track pattern. This technique, which is based on the shift from one ground track pattern to another, allows indeed the optimization of both the frequency of observation over a given area and the coverage of the Earth. In the inclination-altitude plane, optimal manoeuvres to perform the orbital transfers and thus opportunely modify the ground track pattern have been presented. The analysis has highlighted how small modifications of the orbit elements, which for repeating ground track Sun-Synchronous Orbits can be obtained by V from 0.39 to 13.14 m/s, have allowed great variations in the track pattern. This result, together with an appropriate arrangement of a limited
[1] W.S. Adams, L. Rider, Circular polar constellations providing continuous single or multiple coverage above a specified latitude, J. Astronaut. Sci. 35 (2) (1987) 155–492. [2] A.H. Ballard, Rosette constellations of Earth satellites, IEEE Trans. Aerosp. Electron. Syst. AES–16 (5) (1980) 656–673. [3] C. Circi, E. Ortore, F. Bunkheila, C. Ulivieri, Elliptical multi-sun-synchronous orbits for Mars exploration, Celest. Mech. Dyn. Astron. 113 (4) (2012) 215–227. [4] R.G. Hopkins, Long-term revisit coverage using multi-satellite constellations, AIAA paper No. 88-4276-CP, in: AAS/AIAA Astrodynamics Specialist Conference, Minneapolis, USA, 1988. [5] J.C. King, Quantization and symmetry in periodic coverage patterns with applications to Earth observation, J. Astronaut. Sci. 24 (4) (1976) 347–363. [6] G.V. Mozhaev, Problem of continuous coverage of the Earth and kinematically regular satellite systems I, II, Kosm. Issled. 10 (6) (1972) 833–840. [7] G.V. Mozhaev, Problem of continuous coverage of the Earth and kinematically regular satellite systems, Kosm. Issled. 11 (1) (1973) 59–69. [8] G.V. Mozhaev, Capabilities of kinematically regular satellite systems with symmetric groups of the second type in the problem of continuous single coverage of the Earth, Kosm. Issled. 43 (3) (2005) 215–223 [Cosmic Res. 205–212]. [9] E. Ortore, C. Ulivieri, A small satellite constellation for continuous coverage of mid-low Earth latitudes, J. Astronaut. Sci. 56 (2) (2008) 185–198. [10] E. Ortore, C. Ulivieri, F. Bunkheila, Satellite constellations in inclined multistationary orbits, in: Proceedings of the Institution of Mechanical Engineers, Part G, J. Aerosp. Eng. 225 (9) (2011) 1050–1060. [11] E. Ortore, C. Circi, F. Bunkheila, C. Ulivieri, Earth and Mars observation using periodic orbits, Adv. Space Res. 49 (1) (2012) 185–195. [12] V.K. Saulskiy, Multisatellite systems with linear structure and their application for continuous coverage of the Earth, Kosm. Issled. 43 (1) (2005) 36–53 [Cosmic Res., 34–51]. [13] V.K. Saulskiy, Multiple continuous coverage of the Earth based on multisatellite systems with linear structure, Kosm. Issled. 47 (2) (2009) 164–175 [Cosmic Res., 144–155]. [14] C. Ulivieri, G. Laneve, S.M. Hejazi Moghaddam, A. Foni, UPH constellations for continuous regional coverage, 1997, pp. 347–356, AAS Paper 97-622, AAS paper (97). [15] Yu.P. Ulybyshev, Design of satellite constellations with continuous coverage on elliptic orbits of Molniya type, Kosm. Issled. 47 (4) (2009) 343–354 [Cosmic Res., 310–321]. [16] Yu.P. Ulybyshev, Elliptic orbits and satellite constellations for coverage of illuminated areas, Kosm. Issled. 50 (2) (2012) 205–208 [Cosmic Res., 199–202]. [17] J.G. Walker, Some circular orbit patterns providing continuous whole Earth coverage, J. Br. Interplanet. Soc. 24 (11) (1971) 369–384. [18] J.G. Walker, Satellite constellations, J. Br. Interplanet. Soc. 37 (1984) 559–571.
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Satellite constellations in sliding ground track orbits Christian Circi ∗ , Emiliano Ortore, Federico Bunkheila Department of Astronautical, Electrical and Energetic Engineering, Sapienza University of Rome, Via Salaria 851, 00138 Rome, Italy
a r t i c l e
i n f o
Article history: Received 5 March 2013 Received in revised form 31 March 2014 Accepted 18 April 2014 Available online xxxx Keywords: Orbital transfer Satellite constellation Ground track pattern
a b s t r a c t This paper deals with repeating ground track orbits, taking into consideration the case of a single satellite and of constellations composed of one or more orbital planes. First, the orbital relationships able to guarantee a uniform ground track pattern on the Earth’s surface are provided. Then, the sliding ground track concept, based on the possibility to shift from a ground track pattern to another by small corrective manoeuvres, is introduced. This technique, which can be applied to both single satellite and satellite constellations, allows the fulfillment of several objectives in the course of the same mission. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction The characteristics of a satellite observation system are strictly related both to the instruments installed on board of satellites and to the choice of the orbits. In particular, the repeating ground track orbits, which allow the formation of regular ground track patterns and therefore cyclic observations of the Earth’s surface and of the atmosphere, represent an essential element in the Earth observation field [4,5]. Very often this requirement is associated with the one related to the synchronism between the apparent motion of the Sun around the Earth and orbital plane displacement (SunSynchronous Orbits – SSOs), so as to gain approximately the same illumination conditions during the observations of a given region. In addition, recent studies have shown the possibility to use (for the observation of the Earth and Mars) the so-called Multi-SunSynchronous Orbits, in which also the solar illumination conditions cyclically vary according to the choice of the orbit elements [11,3]. When the use of a single satellite prevents the observation of a given area with simultaneous satisfactory observation frequency and ground spatial resolution (requirements inversely correlated one to the other), it is necessary to resort to a constellation. A great number of papers have been written on satellite constellations. Some examples of these are represented by the kinematically regular satellite systems [6,7], the Walker constellation [17,18], the circular polar constellations for continuous single or multiple coverage [1], the homogeneous constellations for regional coverage [14], the Rosette constellation [2], up to the more recent studies
*
Corresponding author. E-mail addresses: [email protected] (C. Circi), [email protected] (E. Ortore), [email protected] (F. Bunkheila). http://dx.doi.org/10.1016/j.ast.2014.04.010 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.
performed with the aim of gaining a continuous coverage of the Earth by using kinematically symmetrical satellite systems in circular orbits [8], a multiple continuous coverage by systems with linear structure [13,12], a continuous coverage by using elliptic orbits with a quasi-stationary condition in an orbit arc around the apogee [9,10] and by constellations of Molniya type orbits [15,16]. This paper proposes a technique to satisfy several objectives in the course of the same mission. In fact, by means of low-cost manoeuvres, it is possible to modify more times the orbit elements and, accordingly, the observational characteristics of a satellite so as to optimize its performance for specific and changing goals. Although these manoeuvres are characterized by small velocity variations, they allow the shift from a ground track pattern to another that can be significantly different (thus the definition of “sliding” pattern), opening the way to a dynamical model of satellite and of satellite constellation. The article is organized as follows: Section 2 illustrates the orbital relationships that have to be satisfied, by a single satellite and by single and multi-plane constellations, to obtain regular cycles of observation of the Earth with a uniform ground track pattern (equally spaced tracks on the equator). In the case of constellations, the satellites have to be located over orbits with the same shape and inclination and opportunely arranged over orbital planes properly phased in longitude and containing the same number of satellites. In Section 3 the concept of sliding ground track orbits is described and the feasibility of carrying out corrective manoeuvres to properly modify the ground track, shifting from one orbital solution to another, is investigated. The analysis is applied to both the single satellite, taking into consideration SSOs as a case study, and to the satellite constellations.
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Fig. 1. Curves of periodicity.
2. Repeating ground tracks A ground track is defined as periodic (repeating) if it repeats itself at regular time intervals allowing a cyclic observation of the Earth. In the case of circular orbit, considering the zonal harmonic J 2 of the gravitational field and indicating with r and i the radius and the inclination of the orbit respectively, the mathematical condition identifying a repeating ground track can be written as [3]:
r 3 .5 + b 1 r 2 + b 2 = 0 ,
(1)
where the coefficients have the following expressions:
√ b1 = − b2 =
μE , qω E
3 J 2 R 2E
√
2qω E
μ E
1 − 4 cos2 i + q cos i ,
and where μ E is the gravitational constant of the Earth, q is the repetition factor, ω E is the angular velocity of the Earth around its axis and R E is the mean equatorial radius of the Earth. The repetition factor q is given by the ratio R /m, where m is the integer number of nodal days after which the track is repeated and R is the integer number of nodal periods accomplished by satellite in m nodal days (R and m are prime to one another). Once the values of i and q have been assigned, the physically acceptable solution of Eq. (1) provides the value of the radius r of the corresponding orbit. The same equation, for each value of q, links the values of the orbit altitude h and the inclination defining a curve referred to as “curve of periodicity”. Fig. 1 shows the solutions of Eq. (1), with m going from 1 to 5, in the case of Low Earth Orbits (LEOs). All points of a curve, being R and m prime one to the other, refer to orbits having the same revisit time over a given area. The parameter q, representing the number of orbital periods per day (both nodal), can be written as sum of an integer number N i plus a fractional part N f = k/m, where the parameter k is an integer number which is prime with m and ≤ m − 1. In particular, if k = 0 ( N f = 0) then q = N i and m = 1. For example, for the orbit P 1 in
Fig. 1, in which k = 0 and q = 14, it is m = 1 and R = 14 · 1 = 14, while for the orbit P 2 , in which the satellite accomplishes also another portion of orbit, q = 14 + 1/5 (k = 1), it is m = 5 and R = N i m + k = 14 · 5 + 1 = 71. The equatorial longitudinal separation between two successive ground tracks is S t = 2π /q while, after a repetition period (m nodal days), the minimum equatorial ground track distance reduces to the value S m = S t /m. In fact, each interval S t is subdivided in m intervals S m by the m − 1 tracks related to orbits successive to the N i -th (in particular, if m = 1, the (N i + 1)-th track will coincide with the first one). The parameter k determines the way in which such a subdivision is carried out. In fact, the minimum ground track distance S 1 (ascending node passes) related to consecutive nodal days can be expressed by the following relationships:
S 1 = − Smk
for k <
m 2
(2a)
,
S 1 = S m (m − k) for k >
m 2
.
(2b)
Fig. 2 shows, as an example, all the possible cases with m = 5 (the distance S t is subdivided in 5 equally spaced intervals). In the case k = 1 the interval S 1 is equal to the minimum distance S m and the tracks related to the successive nodal days (numbered with 2, 3, 4) are eastward shifted of S m . In the case k = 2, being S 1 = 2S m , the tracks related to the successive nodal days are eastward shifted of 2S m . The cases k = 3 and k = 4 are similar respectively to the cases k = 2 and k = 1 but the track progression inside the interval S t is westward. In fact, in Eqs. (2a), (2b) it is S 1 > 0 if the progression is westward, S 1 < 0 if the progression is eastward. If also the descending node passes are taken into consideration, the minimum equatorial distance between tracks can be reduced to the value S t /2m and it occurs if m and R are not both odd numbers, otherwise ascending and descending nodes coincide. To obtain Eqs. (2a), (2b) the longitudes λi and λi +1 corresponding to tracks N i -th (last track of the first nodal day) and ( N i + 1)-th (first of the second nodal day) are taken into account. Indicating with λ0 the longitude of the first track:
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Fig. 2. Track sequence for m = 5.
Fig. 3. Daily ground track displacement.
λi = λ0 + N i S t = λ0 − S m k,
(3a)
λi +1 = λ0 + ( N i + 1) S t = λ0 + S m (m − k),
(3b)
the daily spacing S 1 can be identified by the minimum value out of the following two differences: |λi − λ0 |, |λi +1 − λ0 | and it is |λi − λ0 | < |λi +1 − λ0 | for k < m/2, |λi − λ0 | > |λi +1 − λ0 | for k > m/2 (such differences are equal in the case m = 2). Therefore, the daily spacing can be assumed as an eastbound (sign −) displacement if k < m/2 and as a westbound (sign +) displacement if k > m/2 (Fig. 3). The condition k = m/2 occurs only for m = 2 (k = 1) and in such a case it is possible to refer to both Eqs. (2). A useful relationship that summarizes the track progression in the interval [λ0 , λ0 + S t ] can be expressed as:
λd = λ0 + S t · 1 + int(dN f ) − dN f ,
d = 1, . . . , m − 1,
to guarantee the required coverage, resolution and/or repetition of observation, it is necessary to resort to a constellation. In particular, with a single satellite, it is not possible to reduce the repetition of observation to values less than one nodal day. 2.1. Satellite constellations In the case of N satellites orbiting on the same orbit, indicating with λ1,0 the longitude of the ascending node pass of satellite 1 at the day 0, the ground track progression in the interval [λ1,0 − St , λ1,0 + S2t ] can be described by the following relationship: 2
λi ,d = λ1,0 + S t · mod(x),
i = 2 , . . . , N , d = 0, . . . , m ,
(5)
where λi ,d is the longitude of the ascending node pass of satellite
(4)
where λd is the longitude after d nodal day (with d integer number) and the operator int(dN f ) provides the integer part of dN f . With Eq. (4), all the displacements are westward counted (1 + int(dN f ) − dN f > 0), according to the fact that the first track of the day d is westward shifted with respect to the first track of the day 0. In order to achieve a complete longitudinal coverage of the Earth in m nodal days, the swath L of the instrument of observation must be ≥ S m · sin i (Fig. 4). Since the swath L is inversely correlated to the ground spatial resolution (the number of pixels is fixed as a characteristic of the instrument), observation frequency (value of m) and spatial resolution are also inversely correlated. Thus, if a higher spatial resolution is required, it is necessary to opportunely increase the value of m so as to reduce the track spacing S m and vice versa. However, when the couple of parameters S m and m are not able
M
i at the day d, x = dN f + 2π i , M i is the mean anomaly phasing between satellites i and 1 and the operator mod(x) is defined as:
mod(x) = int frac(x) +
1 2
− frac(x),
(6)
with frac(x) = fractional part of x and int( y ) = integer part of y. If M i = 0, Eq. (5) becomes:
λd = λ0 + S t · mod(dN f ),
d = 0, . . . , m
(7)
and describes the track progression for the single satellite case. It represents an alternative to Eq. (4). If the goal is to increase the repetition of observation, in order to make satellite i passes on the same track of satellite 1, it must be λi ,d = λ1,0 . It occurs when the value of x in Eq. (5) is an integer number. Since the considered constellations refer to the case in which the N satellites are equally shifted in mean anomaly one from another:
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If the goal is the increase of repetition of observation, the condition of coincidence of ground track patterns originating by different planes can be expressed as:
lcm(m, N P )
M R 2π
+
Ω 2π
= I,
q
(10)
where I is an integer number, M R (constant) represents the mean anomaly phasing between the i-th satellite of the p-th plane and the i-th satellite of the (p − 1)-th plane, and Ω (constant) is the shift in RAAN between the orbital planes (p = 1, . . . , P f , with P f = number of planes devoted to the increase of observation frequency; i = 1, . . . , N P , with N P = number of satellites for each plane). In this way, the minimum interval between passes of satellites over the same ground track will be reduced to the following value: Fig. 4. Minimum required swath.
M i =
2π N
(i − 1),
i = 2, . . . , N ,
t = (8)
the aforesaid condition is satisfied if the ratio m/ N is an integer number or, in other words, when N = m or N is a sub-multiple of m. In particular, if N = m the observations occur at a regular interval of one nodal day (N > m is meaningless: the position of satellite (N + 1)-th would coincide with the position of satellite 1, the position of the (N + 2)-th with the one of satellite 2, and so on). If the goal is the reduction of the minimum track spacing, an N satellite constellation allows a reduction of this distance to the following value:
S m, N =
St lcm( N , m)
(9)
,
where lcm( N , m) is the least common multiple between N and m (N can be smaller, equal or greater than m). The maximum reduction occurs when N and m are prime to one another and all the satellites describe different tracks. Table 1 summarizes all the possible situations for the single plane case in terms of time interval between two observations over the same area (t), expressed in nodal days, and of minimum track spacing. From the general expressions it is possible to retrieve the particular cases in which all the satellites perform the same tracks (third row of Table 1) and all the satellites perform different tracks (fourth row of the table). In the other cases (e.g. N = 6, m = 8), the result is both an increase of the repetition of observation and a reduction of the minimum track distance. In the case of a single plane constellation with N satellites satisfying Eq. (8), the time t elapsing between two passes over the same ground track is constant and a uniform ground track distribution is guaranteed (equally spaced tracks). 2.1.1. Multi-plane constellations In order to reduce the observation frequency to intervals lower than one nodal day, it is necessary to take into account satellites orbiting on different orbital planes. The use of more orbital planes allows also a further reduction of the minimum track spacing distance.
lcm(m, N P ) P f NP
(11)
.
The value of t is not necessarily an integer number of nodal days (as in the single plane case) and then intervals of revisit lower than one nodal day can be obtained. In particular, considering Ω = 2π / P f in Eq. (10), the observation of given area occurs at regular intervals (t = constant). Finally, if m = N P all satellites of every plane describe the same ground tracks and such a constant value reaches its absolute minimum: t = 1/ P f = Ω/2π . If the goal is the reduction of the minimum track spacing distance, it is necessary to impose the following condition to ensure equally spaced ground tracks:
lcm(m, N P )
M R 2π
+
Ω 2π
q
=I+
1 Ps
,
(12)
where P s (> 1) is the number of planes devoted to thickening the ground track pattern and the minimum ground track spacing will be reduced to the value:
S m, N , P =
St P s · lcm(m, N P )
(13)
.
In order to obtain the lowest value of S m, N , P , it is necessary to take N P and m prime one to the other so that all satellites perform different tracks. 3. Sliding ground tracks Each ground track pattern, obtained by a satellite or a constellation, can be considered (instead of a steady characteristic of a space mission: one mission, one track pattern) as a step that can be gained during a mission (one mission, more patterns). This concept, arising from the possibility of “sliding” from a repeating ground track pattern to another, leads to a dynamical model of satellite and constellation. In fact, by applying small velocity variations V , it is possible to transfer a satellite from a periodic solution of Fig. 1 to another one, which is characterized by a different value of m, i or both. These transfers from an orbit to another allows the modification of the observational parameters thus satisfying different mission requirements in terms of revisit time and coverage. Moreover, the considering of constellations composed of a limited number of satellites that suitably change their orbital
Table 1 Revisit frequency and minimum equatorial track spacing for single plane constellations.
t
S m, N
All satellites perform the same tracks (N = m, N sub-multiple of m)
lcm(m, N ) N m N
All satellites perform different tracks (N and m prime one to the other)
m
St lcm(m, N ) St m St mN
General expressions
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Fig. 5. Selected repeating ground track solutions.
configuration in the course of mission allows the fulfillment of different objectives with a small cost. Fig. 5 shows repeating ground track solutions, considering m from 1 to 5, together with the paths related to the transfers from one point to another. The considered orbits, having an inclination from 20 to 40◦ , find their best employment in the observation of Earth mid-low latitudes. The characteristics of such points are reported in Table 2. The considered orbits are, in general, different in terms of revisit time and ground track spacing and Table 3 shows the velocity variations needed to transfer a satellite from one point to another, taking into account Hohmann transfers and, where necessary, orbital plane changes carried out at the apogee of the Hohmann orbits. The vertical displacements in Fig. 5, corresponding to transfers along orbits with the same inclination, are characterized by the lowest values of V (in plane manoeuvres). For example, the vertical transfers from X 1 to the other selected solutions X i and Y i , with i = 1, 2, 3, 4, require a V from 38 to 293 m/s. On the contrary, the transfer from X 1 to the Z i solutions needs a V from 913 to 979 m/s (out of plane manoeuvres). In general, the results show how with moderate variations V it is possible to gain a significant change of the mission requirements in terms of revisit time, coverage or both. 3.1. Application to the single satellite In this section, the concept of sliding ground track orbits is applied to the single satellite, taking into consideration, as a case study, SSOs, which are the classical orbits used for Earth observation. In particular, the intersections of the curves of periodicity (Fig. 1) with the Sun-Synchronism curve, in which the link radiusinclination is such that the angular velocity of the nodal line due ˙ is equal to the angular velocity of the Sun in its apto J 2 (Ω) ˙ S ), define the Periodic parent motion with respect to the Earth (Ω Sun-Synchronous Orbits (PSSOs). These orbits, as well known, allow the repeated observation of an area at the same local time and are very often exploited in the remote sensing field. Fig. 6 shows three PSSO solutions, deriving from the intersection of the Sun-Synchronism curve with the curves of periodicity corresponding to q = 13, 14 and 15. Considering all the curves of periodicity, the related intersections with the Sun-Synchronism curve lead to a high number of solutions, summarized in Fig. 7 in terms of m, h, q and i.
Fig. 6. Examples of PSSO solutions.
Given that each point of Fig. 7 is characterized by a value of orbit altitude, inclination, m, R and k, it can be exploited to observe the Earth satisfying specific mission requirements. For example, point P in Fig. 7 represents a PSSO which allows the satellite a repeated observation of a given region every m = 8 nodal days at an altitude of about 1600 km. Since q = N i + k/m, the integer part N i = 12. The value of k, which is prime with m and goes from 1 to m − 1, can be retrieved by Fig. 7 going from right to left. In this case k = 1 and therefore R = qm = 97 (q = 12 + 1/8). Since the velocity variations V required to transfer a satellite from a point of Fig. 7 to another one can be low, while the corresponding observational parameters can significantly vary, the services offered by a satellite can be modified during the mission to satisfy different objectives. Fig. 8 shows some PSSO solutions located in the orbit inclination range between 98.20◦ and 98.28◦ and Table 4 reports the related characteristic parameters. Thanks to the closeness of the orbit inclination and altitude values among the selected PSSO solutions in Table 3, the velocity variations V needed to transfer a satellite from one point to another of Table 3 are minimized (quasi-in plane manoeuvres). Table 5 reports the required velocity variations to transfer a satellite from one solution to another of Table 4, taking into account Hohmann type transfers with orbital plane changes carried out at the apogee of the Hohmann orbits. As an example, Fig. 9 shows the change of ground track pattern related to the couple of orbits A, I. The big change in the ground track pattern is obtained by a velocity variation of only 4.09 m/s. This optimal ground track sliding from case A to case I (and vice versa) allows a change of the revisit time from 2 to 29 and a variation of the minimum ground track spacing from 1381.9 to 95.2 km.
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Table 2 Characteristic parameters for the selected solutions. Orbit
i (◦ )
h (km)
m
k
R
S m (km)
O X1 X2 X3 X4 Y1 Y2 Y3 Y4 Z1 Z2 Z3 Z4
30 30 30 30 30 30 30 30 30 20 20 40 40
817.94 1115.02 1038.08 962.96 889.61 747.90 679.43 612.45 546.93 1191.33 814.79 1197.72 822.65
1 5 5 5 5 5 5 5 5 1 1 1 1
0 1 2 3 4 1 2 3 4 0 0 0 0
14 65 65 65 65 70 70 70 70 13 14 13 14
2862.5 616.5 616.5 616.5 616.4 572.5 572.5 572.5 572.5 3082.7 2862.5 3082.7 2862.5
Table 3 Required velocity variations to transfer a satellite from one point to another.
V (m/s)
O
X1
X2
X3
X4
Y1
Y2
Y3
Y4
Z1
Z2
Z3
Z4
O X1 X2 X3 X4 Y1 Y2 Y3 Y4 Z1 Z2 Z3 Z4
0 149 111 74 37 36 73 109 144 994 918 995 918
149 0 38 75 112 186 222 258 293 913 979 914 976
111 38 0 37 75 148 184 220 256 933 962 934 960
74 75 37 0 37 110 147 182 218 953 947 954 945
37 112 75 37 0 73 109 145 181 973 932 974 930
36 186 148 110 73 0 36 72 108 1015 935 1016 937
73 222 184 147 109 36 0 36 72 1036 954 1037 956
109 258 220 182 145 72 36 0 36 1057 973 1059 975
144 293 256 218 181 108 72 36 0 1079 993 1081 994
994 913 933 953 973 1015 1036 1057 1079 0 188 1783 1878
918 979 962 947 932 935 954 973 993 188 0 1881 1829
995 914 934 954 974 1016 1037 1059 1081 1783 1881 0 187
918 976 960 945 930 937 956 975 994 1878 1829 187 0
Fig. 7. Intersections between periodicity and Sun-Synchronism curves.
3.2. Application to the satellite constellations The concept of sliding ground track can be extended to the satellite constellations leading to a further improvement of the mission performances. Similar to the case of a single satellite, the mission can be planned considering a series of orbital transfers which allows the achievement of various configurations able to satisfy different requirements. Even taking into account a constellation of only two satellites, many possibilities to plan the mission arise. In order to clarify the
advantages related to the proposed technique, a simple example of mission (composed of three steps) is here presented: starting from a initial configuration in which the satellites are performing the same ground tracks, the constellation achieves a second configuration in which the satellites are performing different ground tracks and then a third configuration in which the satellites are again performing the same tracks, but those are different from the ones related to the initial configuration. For example, in the first configuration the satellites can both be located over orbit A. Since this orbit is characterized by a revisit time of two nodal
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Table 4 Characteristic parameters for the selected PSSOs. Orbit
i (◦ )
h (km)
m
k
R
S m (km)
A B C D E F G H I
98.274 98.200 98.223 98.235 98.207 98.242 98.247 98.205 98.251
720.00 701.89 707.45 710.40 703.69 712.22 713.46 703.13 714.36
2 9 13 17 20 21 25 29 29
1 5 7 9 11 11 13 16 15
29 131 189 247 291 305 363 422 421
1381.9 305.9 212.0 162.2 137.7 131.4 110.4 95.0 95.2
Table 5 Needed V to transfer a satellite from a PSSO solution to another.
V (m/s)
A
B
A B C D E F G H I
0 13.14 9.08 6.95 11.86 5.66 4.76 12.25 4.09
13.14 0 4.06 6.20 1.28 7.49 8.38 0.90 9.06
C 9.08 4.06 0 2.14 2.78 3.43 4.32 3.16 5.00
D
E
6.95 6.20 2.14 0 4.92 1.29 2.18 5.30 2.86
F
11.86 1.28 2.78 4.92 0 6.21 7.10 0.39 7.78
G 5.66 7.49 3.43 1.29 6.21 0 0.89 6.59 1.57
4.76 8.38 4.32 2.18 7.10 0.89 0 7.48 0.68
H
I
12.25 0.90 3.16 5.30 0.39 6.59 7.48 0 8.16
4.09 9.06 5.00 2.86 7.78 1.57 0.68 8.16 0
Fig. 8. Selected PSSOs.
day (Table 4) and the mean anomaly phasing between the satellites is equal to 180◦ (according to Eq. (8)), the revisit quency and the minimum equatorial track spacing of this lcm(m, N ) configuration are given by (Table 1): t = = 1 nodal N
two frefirst day,
St S m, N = lcm(Smt , N ) = m , which is equal to the one of a single satellite, being the satellites performing the same ground tracks (N = m). The second configuration can be gained shifting one of the two satellites in orbit I by means of a velocity variation of 4.09 m/s (Table 5). This quasi-coplanar transfer allows the achievement of a condition in which a satellite passes every two nodal days over the same areas (orbit A) and the other one offers a very thick ground track pattern (orbit I). Fig. 10 shows the ground tracks of orbit A and of orbits A + I (tracks of A plus tracks of I). The third configuration can be obtained shifting also the other satellite in orbit I by means of a second V = 4.09 m/s (both satellites in orbit I with a mean anomaly phasing equal to 180◦ ). Being in this case N and m prime one to the other, this configuration is characterized by a revisit frequency equal to the one related to orbit I (29 nodal days) but by a minimum equatorial
Fig. 9. Sliding ground track pattern for the cases A, I.
track spacing reduced to the following value: S m, N = St 58
St lcm(m, N )
=
= 47.6 km.
The sliding ground track technique can be applied to a generic constellation and different ground track patterns can be gained with a limited number of satellites. In fact, according to Eqs. (10), (11), (12), (13) satellites located over different orbital planes can
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number of satellites, guarantees the fulfillment of several objectives in the course of the same mission. Conflict of interest statement We confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. References
Fig. 10. Application of the sliding concept to a two-satellite constellation.
be re-allocated many times by small V thus satisfying different mission requirements. 4. Conclusions After having presented useful relationships, for single and multi-plane satellite constellations, to gain ground tracks uniformly distributed on the Earth’s surface, a low- V technique to meet different mission requirements has been proposed introducing the concept of sliding ground track pattern. This technique, which is based on the shift from one ground track pattern to another, allows indeed the optimization of both the frequency of observation over a given area and the coverage of the Earth. In the inclination-altitude plane, optimal manoeuvres to perform the orbital transfers and thus opportunely modify the ground track pattern have been presented. The analysis has highlighted how small modifications of the orbit elements, which for repeating ground track Sun-Synchronous Orbits can be obtained by V from 0.39 to 13.14 m/s, have allowed great variations in the track pattern. This result, together with an appropriate arrangement of a limited
[1] W.S. Adams, L. Rider, Circular polar constellations providing continuous single or multiple coverage above a specified latitude, J. Astronaut. Sci. 35 (2) (1987) 155–492. [2] A.H. Ballard, Rosette constellations of Earth satellites, IEEE Trans. Aerosp. Electron. Syst. AES–16 (5) (1980) 656–673. [3] C. Circi, E. Ortore, F. Bunkheila, C. Ulivieri, Elliptical multi-sun-synchronous orbits for Mars exploration, Celest. Mech. Dyn. Astron. 113 (4) (2012) 215–227. [4] R.G. Hopkins, Long-term revisit coverage using multi-satellite constellations, AIAA paper No. 88-4276-CP, in: AAS/AIAA Astrodynamics Specialist Conference, Minneapolis, USA, 1988. [5] J.C. King, Quantization and symmetry in periodic coverage patterns with applications to Earth observation, J. Astronaut. Sci. 24 (4) (1976) 347–363. [6] G.V. Mozhaev, Problem of continuous coverage of the Earth and kinematically regular satellite systems I, II, Kosm. Issled. 10 (6) (1972) 833–840. [7] G.V. Mozhaev, Problem of continuous coverage of the Earth and kinematically regular satellite systems, Kosm. Issled. 11 (1) (1973) 59–69. [8] G.V. Mozhaev, Capabilities of kinematically regular satellite systems with symmetric groups of the second type in the problem of continuous single coverage of the Earth, Kosm. Issled. 43 (3) (2005) 215–223 [Cosmic Res. 205–212]. [9] E. Ortore, C. Ulivieri, A small satellite constellation for continuous coverage of mid-low Earth latitudes, J. Astronaut. Sci. 56 (2) (2008) 185–198. [10] E. Ortore, C. Ulivieri, F. Bunkheila, Satellite constellations in inclined multistationary orbits, in: Proceedings of the Institution of Mechanical Engineers, Part G, J. Aerosp. Eng. 225 (9) (2011) 1050–1060. [11] E. Ortore, C. Circi, F. Bunkheila, C. Ulivieri, Earth and Mars observation using periodic orbits, Adv. Space Res. 49 (1) (2012) 185–195. [12] V.K. Saulskiy, Multisatellite systems with linear structure and their application for continuous coverage of the Earth, Kosm. Issled. 43 (1) (2005) 36–53 [Cosmic Res., 34–51]. [13] V.K. Saulskiy, Multiple continuous coverage of the Earth based on multisatellite systems with linear structure, Kosm. Issled. 47 (2) (2009) 164–175 [Cosmic Res., 144–155]. [14] C. Ulivieri, G. Laneve, S.M. Hejazi Moghaddam, A. Foni, UPH constellations for continuous regional coverage, 1997, pp. 347–356, AAS Paper 97-622, AAS paper (97). [15] Yu.P. Ulybyshev, Design of satellite constellations with continuous coverage on elliptic orbits of Molniya type, Kosm. Issled. 47 (4) (2009) 343–354 [Cosmic Res., 310–321]. [16] Yu.P. Ulybyshev, Elliptic orbits and satellite constellations for coverage of illuminated areas, Kosm. Issled. 50 (2) (2012) 205–208 [Cosmic Res., 199–202]. [17] J.G. Walker, Some circular orbit patterns providing continuous whole Earth coverage, J. Br. Interplanet. Soc. 24 (11) (1971) 369–384. [18] J.G. Walker, Satellite constellations, J. Br. Interplanet. Soc. 37 (1984) 559–571.