A Novel Technique to Compute the Revisit Time of Satellites and Its Application in Remote Sensing Satellite Optimization Design

This paper proposes a novel technique to compute the revisit time of satellites within repeat ground tracks. Different from the repeat cycle which only depends on the orbit, the revisit time is relevant to the payload of the satellite as well, such as the tilt angle and swath width. The technique is discussed using the Bezout equation and takes the gravitational second zonal harmonic into consideration. The concept of subcycles is defined in a general way and the general concept of “small” offset is replaced by a multiple of the minimum interval on equator when analyzing the revisit time of remote sensing satellites. This technique requires simple calculations with high efficiency. At last, this technique is used to design remote sensing satellites with desired revisit time and minimum tilt angle. When the side-lap, the range of altitude, and desired revisit time are determined, a lot of orbit solutions which meet the mission requirements will be obtained fast. Among all solutions, designers can quickly find out the optimal orbits. Through various case studies, the calculation technique is successfully demonstrated.


Introduction
Satellite missions devoted to the observation of the Earth as well as navigation satellites commonly use repeat ground track orbits [1].With the development of civilian satellites technology, constellation composed of a number of satellites plays an important role in remote sensing [2].The Earth observation missions often require the constant solar illumination, the same ground resolution, and small repeat cycles, which often results in the design of Repeat Sun-Synchronous Orbit (RSSO) satellites as the most suitable one.This kind of satellites allows the observation of a given region of the Earth at the same local time after a time interval [3,4].The approach to design repeat ground track orbit for the Earth observations (EO) is quite mature [5,6].As well, the RSSO satellites are also used for Mars observations [7][8][9][10].
Repeat ground track (RGT) orbits allows a satellite to reobserve the same area after a repeat cycle.Some articles have shown the various uses of RGT orbits.Fu et al. [11] presented a strategy for design and maintenance of low RGT successive-coverage orbits and their analysis is based on the drift over the entire ground track.Li et al. [12] introduced a special repeat coverage orbit which is a special class of RGT orbit, such orbits can visit a target site at both the ascending and descending stages in one revisit cycle.Circi et al. [13] showed the concepts of sliding ground track pattern, which allows one RGT orbit to transfer to another RGT orbit using a low-ΔV technique.This technology guarantees the fulfillment of several objectives in the course of the same mission.Recent studies have shown the possibility of using the Periodic Multi-Sun-Synchronous orbits (PMSSOs) for Earth and Mars observation; these orbits allow the observation of the same area under different solar illumination conditions and have a repetition period of the solar illumination conditions which is multiple of the repeat cycle [9,10].Wang et al. [14] divided the region by latitude stripes.The relationship between the cumulative coverage and the altitude can be quickly got, which is helpful for orbit designer to select optimal orbits for Earth observation.
Revisit time (RT) of a single satellite is the time elapsed between two successive observations of the same ground point on the surface of the Earth [15].Different from the repeat cycle which is only relevant to the satellite orbits, the revisit time is relative to both of the orbit and the payload of satellite, such as tilt angle and swath width [16].Most of the previous papers which aim to design orbits for remote sensing concentrate on the repeat cycle and near-repeat cycle [4,16].However, they have not considered the revisit time when designing satellites.In general, the orbit design should be treated as a multidisciplinary process, which needs to consider the satellite system such as payload properties.Saboori et al. [17] proposed a multiobjective optimization tool to design repeat sun-synchronous orbits for remote sensing satellites; they considered the revisit time as a function of the tilt angle and side-lap of the satellite.However, the calculation in their work does not consider the subcycles of the orbit.Pie and Schutz [18] introduced the subcycles of repeat ground track orbits and the charts of subcycles are used to decompose the repeat cycle into three main subcycles.
Nadoushan and Assadian [19] presented a novel technique to design RGT orbit with desired revisit time and optimal tilt angle; their calculation is based on the analyses of subcycles.However, their approach could be used only when the repeat cycle is prime relative to the revisit time.When the repeat cycle is not prime relative to the revisit time, the design approach could not be used.As a result, a lot of feasible orbits which meet the mission requirements are ignored.As a design tool, it is preferable to conduct an ergodic performance on the altitude.
This paper proposes a novel technique to compute the revisit time and minimum revisit time of remote sensing satellites.The relationship between the revisit time and the tilt angle of a satellite can be computed fast by this technique.Compared to the approach proposed by Nadoushan and Assadian [19], when the swath width, side-lap, the range of altitude, and desired revisit time are given, a lot of orbit solutions which meet the mission requirements will be obtained fast.It will be helpful for the orbit designer to select the best orbits in all solutions.
Section 2 illustrates the orbital relationships that have to be satisfied to obtain regular cycles of observation of the Earth with a uniform ground track pattern.And the subcycles of repeat ground track orbit are considered and developed using Bezout's lemma.In the third section, the procedure of the proposed technique is raised to calculate the RT of satellite.At last, the technique is used to compute and select the best orbits according to the mission requirements.Finally, some cases are investigated for evaluation of the technique.

Repeating Sun-Synchronous
Orbits and Subcycles 2.1.Repeating Sun-Synchronous Orbits.The repeating orbits are also known as the repeat ground track orbits, which are defined as orbits with periodic repeating ground tracks.Their ground track will repeat after a whole number of revolutions  in  nodal days.These orbits have good appearances for the Earth's coverage, which is good for remote sensing.
For optical observations, it is important to ensure that illumination conditions remain the same or vary as little as possible when observing the same ground area; this is known as sun-synchronous orbit.We consider an intersection of the equator and a satellite's descending (ascending) ground track.For a satellite, the interval between two successive equatorial crossings of the ground track on the equator is where   is the Earth's rotation rate with respect to the vernal equinox.ω is the variation rates of argument of perigee and Ṁ is the variation rates of mean anomaly.  is the nodal period of the motion of the satellite, which is expressed as The condition for repeating ground track orbits can be written as or where  and  are positive integers and they are prime to one another.  is nodal day; it is expressed as If the orbit is sun-synchronous orbit as well, the nodal precession rate Ω equals the rotational angular speed of the Earth   .A nodal day of the repeating sun-synchronous orbit is a solar day which is equal to 86400 s.This is a useful relationship unique to RSSOs when evaluating how fast the ground track advances in longitude as a function of time [4].
In engineering practice, the repeating factor  represents the number of orbits completed per day. determines the location and sequence of all ground traces, which is defined as In (6),  can be written as an integer number  plus a fractional part /, where  is an integer number which is prime to  and 0 ≤  < .
This paper only considers the descending node passes or ascending node passes.The terms related to this article are defined as follows.
Definition 2. Minimum interval   :   is the minimum interval on equator between two ground tracks after a repeat cycle.  =   /.
Table 1 shows the parameters of some repeating sunsynchronous satellite orbits.
Table 1: Parameters of some repeating sun-synchronous orbits.Ground track analyses play an important role in designing the repeating sun-synchronous orbits.Every fundamental interval   is divided into  minimum fundamental intervals by ( − 1) time-successive ground tracks.The parameter  determines the way where these subdivisions are carried out.The minimum ground track distance  0 on equator in consecutive nodal days can be expressed as follows: Depending on the parameters , the ground traces can be considered direct or skipping.If  is 1 or  − 1, then each successive trace falls next to the one which is before it.Take into account orbits named E, F in Table 1 as an example.The fundamental interval   is divided into 5 equally spaced intervals.For the case of orbit F, the parameter  equals 1.The daily interval  0 is equal to   and the ground tracks related to the successive nodal days are eastward shifted of   .For the case of orbit E, the parameter  is equal to 2. The daily interval  0 = 2  and the ground tracks related to the successive nodal days are eastward shifted of 2  .Figure 1 shows the ground track sequences of the satellite when the parameter  = 7.In fact, if  0 > 0, the progress of ground trace is eastward and if  0 < 0, the progress is westward.

Subcycles in a Repeat
Cycle.According to the paper [20,21], a subcycle is an integer value of days after which the ground track of the satellite nearly repeats itself within a small offset.A subcycle of a satellite can be viewed as a near-repeat cycle which is equal to an integer number of nodal days.
From Figure 1, we can get a conclusion: for any repeat ground track orbits whose values of  and  are decided, there exists a time of interval  1 nodal days after which the ground track of the satellite passes at a minimum interval   of the original node to the east.The ground track will pass at a minimum interval   of the original node to the west in ( −  1 ) days.
In Lim and Schutz's definition, there are no specifics given for the offset, other than being "small" [21].The offset is always considered to be equal to a minimum interval   in some literatures [16,18].
However, for different purpose, it may be required to obtain the subcycle of a RGT orbit with a specific offset.The analyses for subcycles have a significant role in designing RGT orbits; in this paper, the small offset is redefined and replaced by a multiple of minimum intervals   .For a specific value , there exists a large range of different subcycle patterns due to the value .The approach to calculate subcycles of a satellite is presented in the paper [18].Basically, the definition of subcycles is based on Bezout lemma, which can be expressed as [22]  ⋅  −  ⋅  = , where , , and  ̸ = 0 are integer numbers.An integer solution  exists if and only if the parameter  is a multiple of the great common divisor of  and .Since the value  and value  are coprime to one another, so if  is an integer number, the Bezout lemma ensures the existence of the solution.In (8), a subcycle   is the number of days after which the ground track will repeat itself with an offset equal to  ⋅   .The subcycles can be, respectively, labeled (  ,  ⋅   ).For example, the first subcycle  1 is the number of days required for the ground track to pass an offset   of the original descending node.Since the offset can be east or west, so the first subcycle  1 has two values  1min ,  1max and  1max >  1min .And two values are relative to each other by Orbit data of HJ-1A is used in this section, HJ-1A is a Chinese satellite of satellite constellation which aims to the environment monitoring and forecasting, and the design goal is to cover the Earth within four days [6].Choose the orbit of HJ-1A as an example of which the parameter  = 14 + 23/31.In Table 2, we present the subcycles of HJ-1A when setting different values of .As illustrated in Table 2, when the required offset is twice as long as   , the corresponding subcycles are 23 days or 8 days.The 23 days of subcycle is relative to east offset which is labeled (23, 2  ), while the 8  days is relative to a west offset which is labeled (8, −2  ).The temporal order of subsatellite trackes is illustrated in Figure 2.

Revisit Time and Optimization in the Satellite Design
In this section, the revisit time of a satellite is discussed.Revisit time (RT) of a single satellite is the time elapsed between two successive observations of the same ground point using off-nadir pointing of the payload or the attitude of the satellite [15].The revisit time is relevant to both of the orbits and the payload of the satellite, such as tilt angle and swath width [16].In practice, a satellite's optical instrument can view an area (using off-nadir observation) before and after the orbit passes over it.Off-nadir observation of a ground area is possible in two ways.The first one is the swath width of a sensor which is wide enough to cover the adjacent ground tracks, which is illustrated in Figure 3, where  is the half of field of view (FOV) angle.Another one is using the attitude maneuvers (tilt angles) ± of the satellite with a small swath width, which is illustrated in Figure 4, and where  is the tilt angle of the satellite.In either case, the satellite in adjacent orbits can observe the ground target, thus making the revisit time less than the repeat cycle.

Maximum Revisit Time.
It should be noted that, in the case of a satellite with a swath width less than the minimum interval   and with no tilting capacity, the revisit time (or equivalently maximum revisit time) is equal to the repeat cycle.For these orbits, the revisit time of ground point at equator covered by the field of regard (FOR) of the satellite payload is equal to the repeat cycle.In addition, complete coverage at equator cannot be provided by the satellite and the revisit time is only related to the points which can be covered by the sensor.FOR is the area covered by a detector or sensor when pointing to all mechanically possible positions which is different from the FOV.The FOV is the area covered by the sensor or detector when pointing to one position [17].
The maximum cone angle  where the surface of the Earth can be observed by a satellite is calculated as follows [19]: where  is the altitude of orbit and   is the mean radius of the Earth.

Calculation of Revisit Time.
The calculation of the revisit time is divided into two kinds of cases.The first case is where the sensor has a wide swath width with no tilting capacity; another case is that the sensor with tilting capacity has a small swath width.In either case, the maximum revisit time is smaller than the repeat cycle of the orbit.In both two cases, we can get the swath width  corresponding to FOV, which can be calculated as follows [4]: When calculating the swath width corresponding to the FOR, the FOV in (11) should be replaced by the FOR.In order  to get a more accurate result, the apparent inclination   is introduced in this paper.Apparent inclination is the angle between the equator and the ground track of the satellite in Earth-centered Earth-Fixed Coordinate (ECEF) system.And for circular orbits,   is defined as [22] tan In some cases, in order to get a better spatial resolution, a margin of safety must be taken into account.Assuming a 10% overlap is required, so the effective swath width  of the satellite should be replaced by 0.9.
The swath of the satellite on the equator   is depicted in Figure 5 and it can be calculated as follows [22]: In order to achieve a complete longitude coverage on the equator in a repeat cycle ,   must be greater than the minimum interval   .When the swath width of the satellite is wide enough to cover several minimum intervals, the half of number of minimum separations  which can be covered is calculated as follows: where function int() is to get the integer part of .According to (13), the number of minimum separation intervals which can be covered by the satellite payload is obtained.In this case, the satellite can observe the ground target with an offset which is smaller than  ⋅   , thus making the satellite in adjacent tracks able to observe the ground target as well.The feasible number  of minimum intervals   which can be covered by the satellite forms a set , and  = {−, − + 1, . . ., −1, 0, 1, . . .,  − 1, }.
Choose an element  in the set , and bring it into (8); then the subcycle   can be obtained.Through this step, a set of subcycles  corresponding to the set  can be derived as Next, the set of subcycles  should be sorted from small to large; the sorted set is named as   .Assume that   = { 0 ,  1 ,  2 , . . .,  2−1 ,  2 }.The revisit time (equivalently maximum revisit time RT max ) and the minimum revisit time RT min can be computed as Choose the orbit of HJ-1A as an example of which  = 14 + 23/31.The swath on the equator of the satellite is 6 times as long as minimum interval   and  = 3 according to (13).So the set  = {−3, −2, −1, 0, 1, 2, 3}.The set of corresponding subcycles  = {12, 8, 4, 0, 27, 23, 19} which is depicted in Table 2.According to (15a) and (15b), the revisit time of HJ-1A is 7 nodal days and the minimum revisit time is 4 nodal days.
Actually, different revisit time corresponds to the different tilt capacity and side-lap; Table 3 shows the different revisit time of P6 satellite [19] with different tilt capacity and required side-lap.
There are some analyses for Table 3 as follows: (1) The revisit time and minimum revisit time of a satellite vary with different tilt angle and required side-lap.
(2) In some cases, an increase of tilt angle could not result in reduction of the RT.For example, when required side-lap is 5% and the tilt angle ranges from 8.39 ∘ to 23.63 ∘ , the revisit time remains 5 days.
The procedure to calculate the revisit time of a satellite with specific parameters is shown in Figure 6.  the performance of a satellite can be represented in terms of resolution, revisit time, repeat cycle, tilt angle, and overlapping width [16].Those characteristics should be taken into account in the orbit design of remote sensing satellites.

Design for
According to the approach shown in the last section, when the orbit and payload parameters are determined, the relationships between the tilt angle and the revisit time can be quickly obtained.This approach needs simple calculations and has high efficiency.In this section, the proposed approach is used to design and select the optimal orbits for remote sensing satellites.
When designing satellites for the Earth observation, the basic inputs are the orbital altitude range, the required sidelap, desired revisit time, and the maximum repeat cycle.The output of this approach is the minimum tilt angle in the premise of the desired revisit time and available side-lap constraint.This approach makes an ergodic search on the range of altitude and generates the orbital parameters which meet all of the operation constraints.
The calculation procedure of designing a satellite with required revisit time and optimal tilt angle are given as follows.
Step 1. Input the range of altitude, required side-lap, and desired revisit time RT des .
Step 2. Input the maximum repeat cycle  max .
Step 3. Generate the set  composed of all the feasible orbits which is in the altitude range (the approach can be seen in [3]).
Step 4. Calculate the repeating factor  and inclination  of each orbit.Since the orbit is sun-synchronous, so the inclination of orbit is determined.In this type of orbits,  is only a function of the orbital altitude (the approach can be seen in [4]).
Step 5. Generate the maximum cone angle  corresponding to each orbital altitude using (9).
Step 6. Choose an orbit of which the revisit time and tilt angle are not calculated in the solution set .
Step 7. Search all the tilt angles which are smaller than maximum cone angle , and get the corresponding revisit rime (RT).
Step 8.If the RT is equal to RT des , then record minimum tilt angle and the orbit in the set   .
Step 9. Turn to Step 6 when an orbit of which the revisit time is not computed in the set  exists.
Step 10.Output all the orbits in the solution set   .
Step 11.Compare the solutions and choose the optimal orbits in set   .

Case Studies
4.1.Calculation of Revisit Time.The proposed technique is applied to compute the revisit time of some remote sensing satellites.HJ-1A [2] and P6 satellite [19] are chosen to verify the method.The orbital parameters and mission characteristics are presented in Table 4.
It could be found that the former satellite has a wide swath width while the latter satellite has a tilting capacity with a narrow swath.Utilizing the proposed technique, the revisit time of two satellites are calculated and presented in Table 5.
From Table 5, the maximum time gap between two successive observations of the HJ-1A satellite is 5 days.Actually, the revisit time of the HJ-1A is designed as 4 days [2], which can be seen in Table 4.At the same time, from Table 5, the revisit time of the P6 satellite is 5 days, which equals the values given in Table 4.These two cases demonstrate the accuracy of the proposed technique.

Optimization Design for Remote Sensing Satellites.
In this section, some optimizations will be implemented on the P6 satellite.Since the altitude of the P6 satellite is 816.964km, so we search all the feasible orbits in the altitude range from 810 km to 820 km.In Step 1, the swath width of satellite is 141 km, the required side-lap is set to 0%, and the required revisit time is equal to 5 nodal days.Table 6 illustrates the number of the orbits in set  and   with different maximum repeat cycle  max .
In Step 2, the maximum repeat cycle is set to 100 days.Compute all the orbits in set  and all the orbit solutions which meet the mission requirements will be obtained.Figure 7 illustrates the feasible orbit solutions within RT des = 5 when  max = 100; Figure 8 illustrates the feasible orbit solutions within RT des = 5 when  max = 200.In Figures 7 and 8, some orbit solutions termed, respectively, as Sol.1, Sol.2 in all feasible orbit solutions are shown in Table 7.
The most remarkable solutions are Sol.1,Sol.2, and the case of P6 satellite.Of course, Sol.1 could be the best choice since its tilt angle is the smallest among three solutions.
Figures 9 and 10 illustrate all the orbits which can meet the conditions where revisit time is 4 days in the altitude range of 810 km-820 km.In Figures 7 and 8, some orbit solutions termed, respectively, as Sol.3, Sol.4, and Sol.5 in all feasible orbit solutions are shown in Table 8.
The most remarkable solutions are Sol.3,Sol.4, and Sol.5 and the case of p6 satellite.Of course, Sol.3 could be the best choice since its tilt angle is the smallest among three solutions.Comparing the feasible orbit solutions in Figures 7 and  10, there are some analyses as follows: (1) With the increase of maximum repeat cycle, the number of feasible orbit solutions will increase as well.
At the same time, the best orbit of which the desired revisit time is 5 days is not the same as the best orbit of which the desired revisit time is 4 days.(2) Sol.3, Sol.4, and Sol.5 have different repeat cycles and ground tracks, but all of them can observe the same ground target within 4 days.According to the paper [13], when the orbital altitude is approximately the same, one orbit can transfer to another orbit using a low ΔV technique.This technology guarantees the fulfillment of several objectives in the course of the same mission, which is useful for different task.

Conclusion
A novel technique which takes into account the orbits and the payload is proposed to calculate the revisit time of a remote sensing satellite.The technique is discussed using the Bezout equation and takes the gravitational second zonal harmonic into consideration.The relationships between altitude and minimum tilt angle can be quickly obtained when the revisit time is determined.So, the proposed technique can make an ergodic search on the altitude and a lot of orbit solutions which meet the mission requirements will be got fast as well.Therefore, the approach can be used in the optimization design for remote sensing satellites.It will be helpful for orbit designer to calculate and select the best orbits in all solutions.

Figure 3 :Figure 4 :
Figure 3: Observe the ground target by a sensor with a wide swath width.

Figure 5 :
Figure 5: Apparent inclination   and the swath on the equator   .

Figure 6 :
Figure 6: The procedure to compute the revisit time of a satellite.

Table 3 :
Different revisit time of P6 satellite with different tilt capacity and side-lap.
Remote Sensing Satellites.Orbital parameters design is a very important task during the mission analysis and design phase of a satellite.For remote sensing satellites, International Journal of Aerospace Engineering

Table 4 :
The orbital parameters and mission characteristics of two satellites.

Table 5 :
The revisit time of two satellites.

Table 6 :
The number of the orbits in sets  and   with different  max .

Table 7 :
Parameters and characteristics of some orbit solutions within RT des = 5.

Table 8 :
Parameters and characteristics of some orbit solutions within RT des = 4.