International Journal of Aerospace Engineering

Volume 2017, Article ID 6469439, 9 pages

https://doi.org/10.1155/2017/6469439

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

^{1}School of Computer, China University of Geosciences, Wuhan 430074, China^{2}Hubei Key Laboratory of Intelligent Geo-Information Processing, China University of Geosciences, Wuhan 430074, China^{3}Institute of Computer Sciences, Heidelberg University, 69120 Heidelberg, Germany

Correspondence should be addressed to Maocai Wang; nc.ude.guc@gnawcm

Received 15 October 2016; Revised 30 November 2016; Accepted 15 January 2017; Published 31 January 2017

Academic Editor: Christian Circi

Copyright © 2017 Xin Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. 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 [7–10].

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- 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.

#### 2. 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 iswhere is the Earth’s rotation rate with respect to the vernal equinox. is the variation rates of argument of perigee and 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 orwhere 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 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 .

This paper only considers the descending node passes or ascending node passes. The terms related to this article are defined as follows.

*Definition 1. *Fundamental interval : is the interval on equator between two successive ground tracks. .

*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 sun-synchronous satellite orbits.