A ground track-based approach to design satellite constellations

Accepted Manuscript A ground track-based approach to design satellite constellations

Emiliano Ortore, Marco Cinelli, Christian Circi

PII: DOI: Reference:

S1270-9638(17)31240-3 http://dx.doi.org/10.1016/j.ast.2017.07.006 AESCTE 4100

To appear in:

Aerospace Science and Technology

Received date: Revised date: Accepted date:

5 October 2015 27 June 2017 5 July 2017

Please cite this article in press as: E. Ortore et al., A ground track-based approach to design satellite constellations, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.07.006

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A ground track-based approach to design satellite constellations

Emiliano Ortore1, Marco Cinelli1, Christian Circi1

1

Department of Astronautical, Electrical and Energy Engineering, Sapienza University of Rome,

Via Salaria, 851, 00138, Rome, Italy.

[email protected] [email protected] [email protected]

Abstract. Following an approach based on ground track analysis, original and compact relationships which permit the construction of ground track patterns and the determination of satellite arrangements able to generate appropriate track distance and revisit frequency over a given area are presented. These equations are valid in the general case of elliptical orbit and can easily be implemented in computer codes devoted to the design of single and multi-plane satellite constellations.

Keywords: satellite constellations; ground track; periodic orbit.

1. Introduction Satellite constellations represent a very wide field of research which involves telecommunication, navigation, Earth observation and science missions. On these topics, a big quantity of studies and projects have been developed and numerous papers have been written. As far as the orbit design is concerned, from the beginning several researches have focused on the determination of the most suitable configurations to guarantee appropriate performances of coverage by a minimum number of satellites [1-3], as in the case of the Walker constellations [4], which are based on the consideration of regular distributions of satellites and orbital planes. About coverage, many other studies have been conducted, as those which have considered the possibility of obtaining continuous multiple coverage (e.g. [5-9]), or those which have approached the problem through an analysis based on the ground tracks performed by satellites in repeating ground track orbits (e.g. [10-17]). Moreover, starting from such studies, a great number of applications, involving telecommunication, navigation, Earth observation and science missions have been carried out [18]. In particular, concerning repeating ground track orbits (also called periodic orbits), [10, 11] describe the general concept of repeating ground track orbit and the related distribution of ascending and descending nodes; [12] applies such concepts to a real mission design; [13] presents useful relationships to retrieve the longitudes of ascending and descending node crossings and analyses the ground track spacing as a function of the latitude; [14] reconsiders the relationships presented in [13] to provide the inter-orbit and inter-plane angular shifts between, respectively, satellites and orbital planes; [15] and [16] use equations able to determine, respectively, circular and elliptical periodic orbits under the influence of planetary oblateness; [17] takes into account the case of uniformly phased satellites in mean anomaly to apply useful relationships concerning the arrangement of satellites over periodic orbits belonging to one or more orbital planes. In this paper, a further investigation of such a topic has been carried out, extending the matter discussed in [17] to the general case in which the satellites can be arranged without constraints of 2

uniformity. Thus, following a ground track-based approach, new relationships for designing single and multi-plane satellite constellations in periodic orbits have been determined. The proposed equations, which are valid in the general case of elliptical orbit and for both prograde and retrograde orbits, allow, in an original and compact way, the construction of ground track patterns and the determination of satellite arrangements able to generate appropriate track distance and revisit frequency over a given area. Thanks to their simplicity, these new relationships can easily be implemented in computer codes devoted to the design of satellite constellations. The paper is organised as follows: Section 2 describes the proposed relationships for the case of a single satellite, Section 3 extends the concepts to the case of single plane constellations and Section 4 presents the general case of multi-plane constellations.

2. The case of a single satellite Repeating ground track orbits allow a satellite to observe the same region of the Earth from the same position at regular time intervals, taking into account the variations of the orbit elements due to the effects of orbital perturbations [19-23]. In fact, if Dn is the nodal day of the Earth, given by

 temporal  ) with ω angular velocity of the Earth around its polar axis and Ω Dn = 2π / (ω E − Ω E variation of Right Ascension of the Ascending Node – RAAN), the ground track pattern completes itself in an integer number (m) of nodal days and, after this time ( m Dn ), the satellite returns to the same location with respect to a given point on the Earth. Indicating with Tn the nodal period of satellite ( Tn = 2π / ξ , where ξ is the argument of latitude, given by the sum of argument of pericentre and true anomaly of satellite) and with R the number of nodal revolutions (of period Tn) accomplished by the satellite in m nodal days, the ground track periodicity condition can be expressed as m Dn = RTn , where m and R are prime numbers one to the other.

3

The description of the ground track distribution in the time m Dn is expressed here by means of an original and compact way, which allows the simple retrieval of the longitudes related to the satellite equatorial crossings (separated by a time interval equal to Tn). In fact, indicating with λ0 the initial geographical longitude of the satellite ascending node (measured from the Greenwich meridian at time t0), the longitudes related to the ascending node crossings ( λt ), in the interval [180 deg West, 180 deg East], are given by:

λt = λ0 − S t [ceil ( x) − q int(t )]

(1)

where: •

t is the time normalised with respect to the nodal day (time/Dn);

•

S t = 2π / q is the longitudinal distance between temporarily subsequent ascending node crossings;

•

q = R / m = Dn / Tn = N i + N f ;

•

Ni is the integer number of nodal revolutions accomplished by the satellite in 1 nodal day;

•

N f = k / m is the fractional part of q;

•

k is an integer number that is prime with m and ” m-1;

•

x =qt;

•

ceil(x) is a function which provides the smallest integer number greater than (or equal to) x;

•

int(t) is a function which provides the integer part of t.

More in detail, Eq. (1) is structured in such a way that, for t = t0, t0+1/q, t0+2/q, … (instants in which the satellite crosses an ascending node) it provides the ascending node longitude at time t, while introducing a generic time t it gives the longitude of the subsequent (temporarily) ascending 4

node crossing. The ground track displacement is taken as positive if it occurs eastwards. Thus, given that λt is westwards shifted with respect to λ0 , in Eq. (1) it appears the sign – out of the square brackets. As an example, the lower part of Fig. 1 shows the ascending node crossings for a repeating ground track orbit with q = 14+2/3 (m = 3), a = 7045.687 km, e = 0 and i = 99 deg. Including the perturbative effect due to the Earth’s oblateness, the minimum ground track longitudinal distance (after m = 3 nodal days) is S m = St / m = 24.55/3 deg.

Fig. 1 Ascending node crossings for a circular orbit with a = 7045.687 km and i = 99 deg (ߣ0 = 0).

Given that the ground tracks repeat themselves at a regular time interval of m nodal days, after this time the ground track pattern is completely defined and Eq. (1) describes the related ground track succession. In particular, considering only integer values of t (t = d is multiple of the nodal day), a simpler equation can be gained from Eq. (1):

λd = λ0 − S t [ceil ( x ) − x ]

(2)

d = 0,..., m

where: •

λd is the longitude at time d; 5

•

x=qd.

While Eq. (1) provides the longitudes related to all crossings, Eq. (2) allows, in a simple way, the retrieval of only the longitudes falling in the interval [λ0 , λ0 − S t ] , which refer to crossings of different nodal days: from the first crossing related to the first nodal day to the first crossing related to the (m-1)-th nodal day (the first crossing related to the m-th nodal day will occur again in λ0 ). In fact, once defined the Ni (integer part of q) ground tracks of the first nodal day, separated by an interval St (lower part of Fig. 1), m-1 tracks will fall inside each interval St in the m-1 successive days, thus leading to the achievement of the minimum spacing Sm (upper part of Fig.1). The distribution of these tracks in the generic interval St will follow an order depending on the value of k (fractional part of q) [10, 11, 13, 17]. For a given value of m, when k increases then q also increases (lower orbit altitude) and therefore St = 2π / q decreases.

3. Satellites over the same orbit Extending the discussion to the case of N satellites lying on the same orbit (with different values of mean anomaly), the ascending node longitude of the i-th satellite ( λi ,t ) can be found by using the following compact equation:

λi ,t = λ0 − St [ceil ( x) − q int(t ) − y] i = 1,..., N where: •

λ0 is the initial longitude of satellite 1 (reference satellite);

•

x =qt+ y;

6

(3)

ΔM i ; 2π

•

y=

•

ΔM i is the mean anomaly phasing between the i-th satellite and satellite 1.

Similar to Eq. (1), for a generic value of t Eq. (3) provides the longitude related to the next ascending node crossing of the i-th satellite. Then, considering for t only the integer multiples (d) of the nodal day, Eq. (3) becomes:

λi ,d = λ0 − St [ceil ( x) − x] i = 1,..., N d = 0,..., m

(4)

where: •

λi,d is the longitude of the i-th satellite at time d;

•

x=qd + y.

This last relationship allows the retrieval of only the longitudes which fall in the interval

[λ0 , λ0 − S t ] . Note that, assuming y = 0 in Eqs. (3) and (4), the single satellite case is re-obtained. With respect to the single satellite, the consideration of more satellites permits both a reduction of the ground track distance, making the satellite perform different ground track patterns, and an increase in the number of observations of the same area, making the satellite perform the same ground track pattern.

3.1 Reduction of ground track distance With N satellites lying on the same orbit and performing uniformly distributed (equally spaced in longitude one to the other) ground track patterns, the equatorial track spacing is reduced to the value 7

S m / N (Sm = single satellite case). To achieve this condition a simple relationship has been found here, which provides the mean anomaly phasing of satellites:

I L · § ΔM i = 2π ¨1 − − ¸ © mN m ¹ I = 0,..., N − 1

(5)

L = 1,..., m

In Eq. (5), I and L are two integer numbers. In particular, while each satellite must be characterised by a value of I different from the one of all the other satellites (N satellites, N different values of I), the value of L can be arbitrarily chosen (even the same value for all the satellites). This is because, while the parameter I determines, in the generic interval St, the locations of the ground tracks, the parameter L is associated only with the order in which these tracks are performed. Thus, for assigned values of m and N, Eq. (5) provides, in a compact way, the description of all the configurations associated with the same (uniform) ground track pattern. Once fixed the longitude of satellite 1 at time t =0 ( λ1, 0 = λ 0 ), for this satellite it is I = 0 and L=m (ΔMi = 0) and the mean anomaly phasing of the other satellites is counted from this reference longitude. Then, considering all the combinations of the possible values of I and L, the number of satellite configurations (NC) producing uniformly spaced ground tracks (and therefore satisfying Eq. (5)) can be found as N C = m N −1 . To obtain Eq. (5), it is possible to consider the first ascending node crossing related to a generic satellite, phased by ΔM i with respect to the reference satellite. This crossing, which will be westwards shifted with respect to the initial longitude ( λ 0 ) of the satellite reference, will occur at the following longitude:

8

§ ΔM i · Δλi = St ¨1 − ¸ 2π ¹ ©

(6)

But, in order to gain uniformly distributed ground track patterns, this crossing will have to be located in one of the following positions:

Sm + L Sm N I = 0,..., N − 1

Δλi = I

(7)

L = 1,..., m

Thus, equalling Eqs. (6) and (7), Eq. (5) is retrieved. As an example, taking into account the orbit related to Fig. 1 and considering a constellation of N = 4 satellites, there are NC = 34-1= 27 configurations able to offer a uniform ground track distance and equal to S m / N = 2.0455 deg . Such configurations correspond to all the combinations which come out freely choosing one value for each column of Table 1, where, according to Eq. (5), the values of ΔMi assignable to every satellite are reported (there are m= 3 possible values for each satellite). In particular, the choice of the values 90, 180, 270 deg leads to a configuration of four uniformly phased satellites over the same orbit.

Table 1 Mean anomaly phasing with respect to a reference satellite. Satellite 2

Satellite 3

Satellite 4

ΔM1= 210 deg (I=1, L=1)

ΔM2= 180 deg (I=2, L=1)

ΔM3= 150 deg (I=3, L=1)

ΔM1= 90 deg (I=1, L=2)

ΔM2= 60 deg (I=2, L=2)

ΔM3= 30 deg (I=3, L=2)

ΔM1= 330 deg (I=1, L=3)

ΔM2= 300 deg (I=2, L=3)

ΔM3= 270 deg (I=3, L=3)

9

All these configurations will produce, at the end of m nodal days, the same overall ground track pattern. Fig. 2 shows the corresponding ascending node crossings in the interval [λ0 , λ0 − S t ] .

Fig. 2 Ascending node crossings for a four-satellite constellation.

However, the ground tracks of these configurations will be performed in a different chronological order. As a matter of fact, Figs. 3 and 4 show the ground track patterns described by the two foursatellite constellations reported in Table 2, after one nodal day. Although the two configurations produce, at the end of m nodal days, the same ground track pattern (whose minimum longitudinal ground track spacing is S m / N = 2.0455 deg ), the consideration of different values for the parameter L leads to different patterns in the first nodal day.

10

Fig. 3 Ground track pattern of Configuration 1 after one nodal day.

Fig. 4 Ground track pattern of Configuration 1 after one nodal day. 11

Table 2 Four-satellite single plane constellations.

Δ M1

Δ M2

ΔM 3

Δ M4

Configuration 1 0° (I=0, L=m) 90° (I=1, L=2) 180° (I=2, L=1) 270° (I=3, L=3) Configuration 2 0° (I=0, L=m) 90° (I=1, L=2)

60° (I=2, L=2)

30° (I=3, L=2)

3.2 Reduction of revisit time On the other hand, if the goal of a satellite constellation is to increase the revisit frequency of a given area, it is necessary to impose that the satellites should describe coincident ground track patterns ( λi,t = λ0 in Eq. (3)). To express this condition, the following simple equation has been found:

L· § ΔM i = 2π ¨1 − ¸ © m¹ L = 1,..., m

(8)

Thus, if Eq. (8) is satisfied the i-th satellite describes the same ground track as the reference satellite (identified by L = m, which leads to ΔMi = 0). According to this equation, there are m values of mean anomaly which allow the fulfillment of this goal (in this case each satellite must be characterised by a value of L different from that of all the other satellites). By using all these values, and therefore N = m satellites satisfying Eq. (8), all the satellites will perform the same ground tracks at a regular time interval of one nodal day. In this case the satellites will be uniformly phased in mean anomaly by 2π/N = 2π/m.

12

More in general, if a constant time interval between two observations of the same area is desired, the general expression for this time interval is given by lcm ( N , m) / N , where lcm ( N , m) represents the least common multiple between N and m. To obtain this result the satellites have to be uniformly phased in mean anomaly one to the other (by 2π/N) and the minimum time interval is obtained (as mentioned above) when N and m are equal.

4. Satellites over orbits phased in Right Ascension of the Ascending Node The relationships found for the single plane case can be generalised taking into consideration several orbital planes. In this case, the satellites will have to be arranged over orbits presenting the same values of semi-major axis, eccentricity, inclination and argument of perigee, while different values of RAAN and mean anomaly can be selected. Thus, considering a constellation of N satellites distributed on P planes, the ascending node longitude of the i-th satellite of the p-th plane ( λ p ,i ,t ) can be retrieved by:

λ p ,i ,t = λ0 − St [ceil ( x) − q int(t ) − y ] p = 1,..., P

(9)

i = 1,..., N p where: ΔM i , p

•

x = qt+

•

N p is the number of satellites of the p-the plane;

•

Δ M i , p is the mean anomaly phasing between the i-th satellite of the p-th plane and satellite 1

2π

;

of plane 1;

•

y=

ΔM i , p 2π

+

ΔΩ p St

; 13

•

ΔΩ p is the longitudinal separation between the p-th plane and plane 1 (Ω = RAAN).

Similar to Eqs. (1) and (3), for a generic value of t Eq. (9) provides the longitude related to the next ascending node crossing of the i-th satellite of the of the p-th plane. Besides, considering for t only the integer multiples (d) of the nodal day, Eq. (9) becomes:

ª

λ p ,i ,d = λ0 − St «ceil ( x) − x − ¬

ΔΩ p º » St ¼

p = 1,..., P

(10)

i = 1,..., N p d = 0,..., m where:

•

λ p ,i , d is the longitude of the i-th satellite at time d;

•

x=qd +

ΔM i , p 2π

.

4.1 Reduction of ground track distance As in the single plane case, N satellites lying on P planes and performing uniformly distributed ground track patterns lead to a ground track spacing equal to S m / N . Also in this case, a simple equation to achieve this condition has been found:

I L § · ΔM i , p − qΔΩ p = 2π ¨1 − − −J¸ © mN m ¹ I = 0,..., N − 1 L = 1,..., m J = 0,..., R − 1

14

(11)

Such a relationship allows the obtaining of the pair of values Δ M i , p and ΔΩ p (mean anomaly phasing between satellites and RAAN phasing between orbital planes) that have to be assigned to gain the aforesaid condition (once fixed the value of ΔΩ p Eq. (11) allows the determination of the value

of Δ M i , p , and vice versa). In particular, while each satellite must be characterised by a value of I different from that of all the other satellites, the values of L and J (associated with the order in which the tracks are described) can be arbitrarily chosen. Once fixed, at time t =0, the longitude of the satellite corresponding to p = 1 and i = 1 ( λ1,1, 0 = λ 0 ), for this satellite it is I = 0, L=m, J = 0. Thus, once assigned the ΔΩ p for every plane, the number of possible values for ΔM i , p is still equal to m and the number of configurations producing uniformly spaced ground tracks (and therefore satisfying Eq. (11)) is still given by N C = m N −1 . Similar to Eq. (5), to obtain Eq. (11) it is possible to consider the first ascending node crossing related to a generic satellite, phased by Δ M i , p and ΔΩ p with respect to the reference satellite:

§ ΔM i , p Δλi , p = S t ¨¨1 − 2π ©

· ¸¸ + ΔΩ p ¹

(12)

Given that, to gain uniformly distributed ground track patterns, this crossing will have to be located in one of the following positions:

Sm + L S m + J St N I = 0,..., N − 1

Δλi , p = I

L = 1,..., m J = 0,..., R − 1

15

(13)

Eq. (11) is then retrieved by equalling Eqs. (12) and (13) ( q = 2π / S t ). Considering, for instance, the same orbit as in the previous examples, with a constellation of N = 12 satellites equally distributed over P = 3 planes (Np = 4 for each plane), identified by ΔΩ1 = 0 , ΔΩ 2 = 120 deg , ΔΩ 3 = 240 deg (values of ΔΩ p ), and assuming λ0 = 0 for the satellite with p = 1

and i = 1 at the initial time, the number of configurations able to make the ground track distance equal to S m / N is 311= 177147. Examples of possible configurations are reported in Table 3.

Table 3 Multi-plane configurations reducing the ground track spacing. Satellite

p

i

ΔΩ

ΔM

ΔΩ

ΔM

1

1

1

0

0

0

0

2

1

2

0

110

0

230

3

1

3

0

220

0

220

4

1

4

0

90

0

90

5

2

1

120

120

120

120

6

2

2

120

350

120

230

7

2

3

120

100

120

100

8

2

4

120

90

120

210

9

3

1

240

120

240

240

10

3

2

240

110

240

110

11

3

3

240

100

240

340

12

3

4

240

330

240

330

All these configurations provide, at the end of the cycle (m nodal days), the same overall ground track pattern. An analogous pattern is obtained also choosing a higher number of planes (even P=N=12) and not necessarily considering the same number of satellites for each plane.

16

4.2 Reduction of revisit time Alternatively, to gain a reduction of the time elapsing between two observations of the same area, the ground track patterns of satellites must coincide ( λ p,i ,t = λ0 in Eq. (9)). This occurs if, at time t0,

λp,i ,0 = HSm + λ0 , where H is an integer number from 1 to R (H=R+1 is equivalent to H= 1 and so on). Such a condition translates into the fact that the term in parenthesis in Eq. (9) must be equal to an integer number (H) and this leads to the following expression:

ΔM i , p + qΔΩ p = 2π

H m

(14)

H = 1,..., R

Once fixed the value of Δ M i , p , R different values of ΔΩ p are obtained in correspondence to the R possible values of H, whereas once fixed the value of ΔΩ p , only m values of Δ M i , p , phased by

2π / m , can be gained for H ranging from 1 to R. With more orbital planes, unlike the single plane case (Section 3), it is possible to gain a constant revisit time, between two observations of the same area, less than one nodal day. If the same number of satellites is considered for each plane (Np=NS), the general expression for this time interval is given (in nodal days, similar to the single plane case), by lcm( N S , m) /( PN S ) , whose minimum value is m/N. To obtain this value, by assigning an index j to each satellite to indicate the chronological order in which the satellites cross the equator at the ascending node λ0 (j= 1 means that satellite 1 is the first satellite to cross λ0 ), the following condition has to be fulfilled:

ª § R ·º ΔM j , j −1 = 2π «1 + int ¨ ¸» − qΔΩ j , j −1 © N ¹¼ ¬ j = 2,..., N 17

(15)

where Δ M j , j −1 is the mean anomaly phasing between satellite j and satellite j-1 and ΔΩ j , j −1 is the RAAN phasing between the plane containing satellite j and the plane containing satellite j-1. By assuming 0 ≤ Δ M j , j −1 ≤ 2π , the values of ΔΩ j , j −1 considerable in Eq. (15) will have to be confined to one interval S t given by:

§R· St ⋅ int ¨ ¸ ≤ ΔΩ j , j −1 ≤ St ©N¹ j = 2,..., N

ª § R ·º «1 + int ¨ N ¸» © ¹¼ ¬

(16)

Thus, once fixed an admissible value for ΔΩ j , j −1 according to Eq. (16), it will be possible to find the corresponding value of Δ M j , j −1 by Eq. (15). In particular, considering NS = m satellites over each orbital plane, the revisit time reaches the minimum value m / N = 1 / P (nodal days). To obtain this result, it is necessary to consider satellite constellations in which all the following conditions have to be satisfied:

•

the satellites are uniformly phased in mean anomaly (by 2π/NS) over each plane;

•

the planes are uniformly shifted in RAAN (by 2π/P).

•

the i-th satellite of the p-th plane and the i-th satellite of the (p-1)-th plane present a constant mean anomaly phasing, given by:

ª q º ΔM R = ΔM i , p − ΔM i , p −1 = 2π «1 − frac §¨ ·¸ » © P ¹¼ ¬ q q where frac §¨ ·¸ is the fractional part of . P ©P¹

18

(17)

As an example, Table 4 shows two constellations, where one satisfies the conditions mentioned above Eq. (17) (homogeneous constellation) while the other does not fulfil such conditions (nonhomogeneous constellation). These constellations are associated with the same (constant) revisit time, equal to m/N nodal days (which, for the homogeneous case is also equal to 1/P). The example is based on the already considered orbit, having q=14+2/3, while the RAAN and the mean anomaly of the reference satellite (j = 1) have been set at 0 for both constellations. Given that N=12, the revisit time is Δt=m/N=3/12=0.25 nodal days (in the homogeneous case, it is P=12/3=4). In particular, in the non-homogeneous constellation, the RAAN phasing between two consecutive satellites (j, j-1) has to be set according to Eq. (16), thus obtaining 73.6364 ≤ ΔΩ j , j −1 ≤ 98.1818 . Then, once chosen a RAAN phasing in this interval, the corresponding mean anomaly phasing between two consecutive satellites (j, j-1) has to be computed by Eq. (15).

Table 4 Example of constellations with the same revisit time.

Non-homogeneous constellation

Homogeneous constellation

j-th satellite

Ω

M

ğj,j-1

ΔMj,j-1

Ω

M

ğj,j-1

ΔMj,j-1

1

0.00

0.00

-

-

0.00

0.00

-

-

2

97.46

10.64

97.46

10.64

90.00

120.00

90.00

120.00

3

194.58

26.12

97.13

15.48

180.00

240.00

90.00

120.00

4

280.13

211.44

85.55

185.32

270.00

0.00

90.00

120.00

5

13.41

283.39

93.28

71.95

360.00

120.00

90.00

120.00

6

90.52

232.35

77.12

308.96

90.00

240.00

90.00

120.00

7

174.51

80.55

83.99

208.20

180.00

0.00

90.00

120.00

8

270.62

110.84

96.12

30.29

270.00

120.00

90.00

120.00

9

3.70

185.72

93.08

74.88

360.00

240.00

90.00

120.00

10

100.89

200.32

97.19

14.60

90.00

0.00

90.00

120.00

11

190.61

324.33

89.73

124.01

180.00

120.00

90.00

120.00

12

265.12

311.57

74.51

347.24

270.00

240.00

90.00

120.00

19

As evidence shows, in the case of homogeneous constellation, in which the RAAN phasing is equal to 2π / P , the values of Δ M j , j −1 are all constant and equal to the one provided by Eq. (17) for

ΔM R . This consideration is valid, in general, for any homogeneous constellation. Moreover, by replacing the values of ΔΩ j , j−1 and Δ M j , j −1 of Table 4 in place of the values of, respectively, ΔΩ p and Δ M i , p of Eq. (14), Eq. (14) itself provides, for all satellites (excluding the reference one), the same value H = 12 for both constellations.

4.3 Summary of the results and ground track reconstruction Tables 5 and 6 summarise the obtained results for, respectively, the ground track distance and the revisit time (both constant), showing the conditions that have to be fulfilled.

Table 5 Ground track distance and related conditions.

Single satellite Single plane constellations Multi-plane constellations

Minimum value

Conditions

Sm

----

Sm N Sm N

I L · § ΔM i = 2π ¨1 − − ¸ © mN m ¹ I L § · ΔM i , p − qΔΩ p = 2π ¨1 − − −J¸ © mN m ¹

20

Table 6 Revisit time and related conditions. Minimum value

Conditions

Single satellite

m

----

Single plane constellations

1

N =m ΔM i = (i − 1)

Multi-plane constellations

2π , m

i = 1,..., m

ª § R ·º ΔM j , j −1 = 2π «1 + int ¨ ¸ » − qΔΩ j , j −1 © N ¹¼ ¬

m N

§R· St ⋅ int ¨ ¸ ≤ ΔΩ j , j −1 ≤ St ©N¹ j = 2,..., N

ª § R ·º «1 + int ¨ N ¸ » © ¹¼ ¬

To determine the sensor swath ( S S ) needed to gain complete longitudinal coverage, it is necessary to consider, besides the ground track distance (in general S m / N ), the value of the so-called apparent inclination i′ , defined as the angle that the ground track forms with the equator. In fact, the sensor swath must be greater than (or at least equal to) the equatorial ground track distance (which decreases as the latitude increases), measured in cross-track direction: S S ≥ ( S m / N ) sin i ′ for prograde orbits, S S ≥ ( S m / N ) sin(π − i ′) for retrograde orbits. In the case of circular orbit, the apparent inclination can be computed by the following relationship ( i ′ ≥ i ) [24]: tgi ′ = sin i /(cos i − 1 / q ) . As is well-known, to reconstruct the ground track as a function of time, it is necessary to determine the temporal variation of both longitude (λ) and latitude (φ) of the sub-satellite point (radial

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projection of the instantaneous satellite position on the Earth’s surface). To this end, the following formulae can be used:

φ = sin −1 (sin i sin ξ )

(18)

λ = λ0 + tan −1 ( cos i tan ξ ) − 2π t

(19)

λ = λ0 − π + tan −1 ( cos i tan ξ ) − 2π t

(20)

where, as said in Section 1, t is the time normalised with respect to the nodal day, so that  ) D t . In particular, Eq. (19) has to be considered in the ascending phase of the orbit, 2π t = (ω E − Ω n

for which

4K + 1 4K + 3 with K natural number, while Eq. (20) has to be taken into account in
the descending one. In the particular case of periodic circular orbit, for which the argument of latitude ξ can be expressed as:

ξ = ξtDn =

2π tDn = 2π qt Tn

(21)

Eqs. (18)-(20) become:

φ (t ) = sin −1 [sin i sin(2π qt )]

(22)

λ (t ) = λ0 + tan −1 [ cos i tan(2π qt ) ] − 2π t

(23)

λ (t ) = λ0 − π + tan −1 [ cos i tan(2π qt ) ] − 2π t

(24)

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Thus, to retrieve the latitude of a satellite phased by Δ M i , p with respect to a reference satellite, whose latitude is identified by Eq. (22), the following relationship can be considered:

φ (t ) = sin −1 ª¬sin i sin(2π qt + ΔMi , p )º¼

(25)

On the other hand, the RAAN phasing ( ΔΩ p ) does not influence the latitude, for which the multiplane case coincides with the single plane one. Similarly, to determine the ground track longitude of a satellite phased by Δ M i , p and ΔΩ p with respect to the satellite reference, whose ground track longitude is identified by Eqs. (23)-(24), the following formulae can be used:

λ (t ) = λ0 + ΔΩ p + tan −1 ª¬ cos i tan(2π qt + ΔM i , p ) º¼ − 2π t

(26)

λ (t ) = λ0 − π + ΔΩ p + tan −1 ¬ª cos i tan(2π qt + ΔM i , p ) ¼º − 2π t

(27)

where Eq. (26) has to be considered in the ascending phase of the orbit, for which

(4 K + 1)

π 2

< t + ΔM i , p < (4 K + 3)

π 2

, while Eq. (27) in the descending one. Thus, in conclusion, Eqs.

(25)-(27) allow the reconstruction of the ground track of a constellation of satellites arranged over periodic circular orbits lying on different planes.

5. Conclusions With reference to repeating ground track orbits, original and compact relationships involving ascending node longitude, mean anomaly and Right Ascension of the Ascending Node are proposed to describe the ground track patterns and to arrange the satellites of a constellation so as to gain appropriate ground track distance and revisit frequency over a given area. 23

The problem has been faced first considering a single satellite, then extending the matter to the case in which several satellites are moving over the same orbit and finally taking into account constellations composed of satellites lying over different orbital planes. While in the case of satellites distributed over the same orbit a constant revisit time over a given area, whose minimum value is equal to one nodal day, can be obtained only considering satellites uniformly phased in mean anomaly, in the case of several orbital planes a constant revisit time, even less than one nodal day, can be gained taking into account both uniformly and non-uniformly phased satellites. On the other hand, a constant ground track distance can be obtained by both uniformly and non-uniformly phased satellites and by both single and multi-plane constellations.

References [1] R.D. Luders, Satellite networks for continuous zonal coverage, American Rocket Society Journal, 31(2) (1961) 179-184. [2] R.D. Luders, L.J. Ginsberg, Continuous zonal coverage: a generalized analysis, NASA STI/Recon Technical Report, N 75 (1974): 30218. [3] D.C. Beste, Design of satellite constellations for optimal continuous coverage, IEEE Transactions on Aerospace and Electronic Systems, AES-14(3) (1978) 466-473. [4] J.G. Walker, Some circular orbit patterns providing continuous whole Earth coverage, Journal of the British Interplanetary Society, 24 (1971) 369-384. [5] J.E. Draim, Three-and four-satellite continuous-coverage constellations, Journal of Guidance, Control, and Dynamics, 8(6) (1985) 725-730. [6] J.E. Draim, A common-period four-satellite continuous global coverage constellation, Journal of Guidance, Control, and Dynamics, 10(5) (1987) 492-499.

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[7] W.S. Adams, L. Rider, Circular polar constellations providing continuous single or multiple coverage above a specified latitude, The Journal of the Astronautical Sciences, 35(2) (1987) 155-192. [8] T.J. Lang, Symmetric circular orbit satellite constellations for continuous global coverage, Astrodynamics 1987, Proceedings of the AAS/AIAA Astrodynamics Conference, Kalispell, MT, Aug. 1013, 1987. Part 2 (A89-12626 02-12). San Diego, CA, Univelt, Inc., 1988, 1111-1132. [9] Y. Ulybyshev, Near-polar satellite constellations for continuous global coverage, Journal of spacecraft and rockets, 36(1) (1999) 92-99. [10] R.D. Luders, The satellite trace repetition parameter Q, Aerospace Corp. TOR- 1001(2307)-3, 17 August, 1966. [11] J.C. King, Quantization and symmetry in periodic coverage patterns with applications to Earth observation, The Journal of the Astronautical Sciences, 24(4) (1976) 347-363. [12] D.L. Farless, The application of periodic orbits to TOPEX mission design, Astrodynamics 1985, Advances in the Astronautical Sciences, Proceedings of the AAS/AIAA Astrodynamics Conference, Vail, Colorado, August 12–15, 58(Part I) (1986) 13-36. [13] R.G. Hopkins, Long-term revisit coverage using multi-satellite constellations, AIAA paper N. 88-4276CP, AIAA/AAS Astrodynamics Conference, Minneapolis, USA, 1988. [14] J.M. Hanson, M.J. Evans, R.E. Turner, Designing good partial coverage satellite constellations, Proceedings of the AIAA/AAS Astrodynamics Conference, Portland, OR, Aug. 1990, 214-231. [15] E. Ortore, C. Circi, F. Bunkheila, C. Ulivieri, Earth and Mars observation using periodic orbits, Advances in Space Research, 49(1) (2012) 185-195. [16] C. Circi, E. Ortore, F. Bunkheila, C. Ulivieri, Elliptical multi-sun-synchronous orbits for Mars exploration, Celestial Mechanics and Dynamical Astronomy, 114(3) (2012) 215-227. [17] C. Circi, E. Ortore, F. Bunkheila, Satellite constellations in sliding ground track orbits Aerospace Science and Technology, 39 (2014) 395-402.

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[18] J.R. Wertz, W.J Larson, Space mission analysis and design (SMAD), 3rd Ed., Eds. Wertz, J. R. and Larson, W.J., Space Technology Library, Published Jointly by Microcosm Press (E1 Segundo, California), K1uwer Academic PubUsbers (Dordrecbt/Boston/London), 1999. [19] M. Lara, Searching for repeating ground track orbits: A systematic approach, The Journal of the Astronautical Sciences, 47(3-4) (1999) 177–188. [20] M. Lara, Repeat ground track orbits of the Earth tesseral problem as bifurcations of the equatorial family of periodic orbits, Celestial Mechanics and Dynamical Astronomy, 86(2) (2003) 143–162. [21] M. Lara, R. Russell, Fast design of repeat ground track orbits in high-fidelity geopotentials, The Journal of the Astronautical Sciences, 56(3) (2008) 311–324. [22] X. Fu, M. Wu, Y. Tang, Design and maintenance of low-Earth repeat-ground-track successive-coverage orbits, Journal of Guidance, Control, and Dynamics, 35(2) (2012) 686-691. [23] M. Aorpimai, P.L. Palmer, Repeat-ground track orbit acquisition and maintenance for Earth-observation satellites, Journal of guidance, control, and dynamics, 30(3) (2007) 654-659. [24] M. Capderou, Satellites - Orbits and Missions, Springer-Verlag (2005) 182-186.

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