Orbital Perigee Deviation under Inclination Window for Sun Synchronized Low Earth Orbits

Data processing related to the Earth’s changes, gathered from different platforms and sensors implemented worldwide and monitoring the environment and structure represents Earth observation (EO). Environmental monitoring includes changes in Earth’s vegetation, atmospheric gas content, ocean state, melting level in the ice fields, etc. This process is mainly performed by satellites. The Earth observation satellites use Low Earth Orbits (LEO) for their missions. These missions are accomplished mainly based on photo imagery. Thus, the relative Sun’s position related to the observed area, it is very important for the photo imagery, in order the observed area from the satellite to be treated under the same lighting (illumination) conditions. This could be achieved by keeping a constant Sun position related to the orbital plane due to the Earth’s motion around the Sun. This is called Sun synchronization for low Earth orbits, the feature which is applied for satellites dedicated for the Earth observation. Nodal regression is the phenomenon which is utilized for low circular orbits providing to them the Sun synchronization. Nodal regression refers to the shift of the orbit’s line of nodes over time as Earth revolves around the Sun, caused due to the Earth’s oblateness. Nodal regression depends on orbital altitude and orbital inclination angle. For the in advance defined range of altitudes stems the inclination window for the satellite low Earth orbits to be Sun synchronized. For analytical and simulation purposes, the altitudes from 600km to 1200km are considered. Further for the determined inclination window of the Sun synchronization it is simulated the orbital perigee deviation for the considered altitudes and the eventual impact on the satellite’s mission.


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Abstract-Data processing related to the Earth's changes, gathered from different platforms and sensors implemented worldwide and monitoring the environment and structure represents Earth observation (EO).Environmental monitoring includes changes in Earth's vegetation, atmospheric gas content, ocean state, melting level in the ice fields, etc.This process is mainly performed by satellites.The Earth observation satellites use Low Earth Orbits (LEO) for their missions.These missions are accomplished mainly based on photo imagery.Thus, the relative Sun's position related to the observed area, it is very important for the photo imagery, in order the observed area from the satellite to be treated under the same lighting (illumination) conditions.This could be achieved by keeping a constant Sun position related to the orbital plane due to the Earth's motion around the Sun.This is called Sun synchronization for low Earth orbits, the feature which is applied for satellites dedicated for the Earth observation.Nodal regression is the phenomenon which is utilized for low circular orbits providing to them the Sun synchronization.Nodal regression refers to the shift of the orbit's line of nodes over time as Earth revolves around the Sun, caused due to the Earth's oblateness.Nodal regression depends on orbital altitude and orbital inclination angle.For the in advance defined range of altitudes stems the inclination window for the satellite low Earth orbits to be Sun synchronized.For analytical and simulation purposes, the altitudes from 600km to 1200km are considered.Further for the determined inclination window of the Sun synchronization it is simulated the orbital perigee deviation for the considered altitudes and the eventual impact on the satellite's mission.

I. INTRODUCTION
LEO satellites move at around 7.5km/s velocity relative to a fixed point on the Earth surface (ground station).Two main characteristics of LEOs are: the shortest distance from the Earth compared with other orbits and consequently less time delay.Satellites in these orbits have an orbital period in range of (90-110) minutes.The communication time between the satellite and the ground station takes (5-15) minutes, (6)(7)(8) times during the day [1], [2].
These satellites provide opportunities on investigations for which alternative techniques are difficult to be applied, thus, it may be expected that such scientific missions will be further developed in the near future.Among others, such investigations are specifically applied through remote sensing process [3].
Remote sensing as a process encompasses the Published on October 25, 2019.S. Cakaj is with Faculty of Information Technology, Polytechnic University of Tirana, Albania, Adr.Nena Tereza, Nr.3 (email: Shkelzen.cakaj@fulbrigjtmail.org).examination, measurement and analysis related to an object or area, without being in touch with object or presence in the examined area.Environmental monitoring by detecting changes in the Earth's vegetation, atmospheric trace gas content, sea and ocean color and state, ice melting, disaster risk reduction are nowadays examples of the remote sensing processes [4].For example, the 2002 oil spill off the northwest coast of Spain was watched carefully by the European ENVISAT, which, though not a weather satellite, flies an instrument (Advanced Synthetic Aperture Radar-ASAR) which can see changes in the sea surface [5], [6].Considering small coverage of LEO satellites, in order to achieve continuous coverage of larger areas (for example: oceans survey), these satellites can be organized as a train constellation as presented in Fig. 1. [7], [8].
The Sun synchronization is a specific feature of LEO satellites utilized for remote sensing and Earth observation.These are known also as "frozen" orbits, because the Sun synchronous orbit keeps constant geometry to the Sun representing an advantage for remote sensing process and the appropriate instruments.The mathematical background of the Sun synchronization process for low Earth orbits is based on terrestrial potential expression through zonal harmonics and it is presented by [9] where the methodology of inclination window determination for the Sun synchronization is also given.The following elaboration moves a step ahead by calculating the perigee deviation within an already determined inclination window for the Sun synchronized orbits.The satellites orbiting around the Earth would be faced with two types of problems, the first, affected by space environment, such as the technical problems of satellite's structure and payload, and the second are the deviations of Orbital Perigee Deviation under Inclination Window for Sun Synchronized Low Earth Orbits Shkelzen Cakaj, and Bexhet Kamo the in advanced defined satellite orbital parameters.An example, for the diagnose of such problems (anomalies), of the first nature, for LEO satellites are analyzes by using electron data from the Medium Energy Proton and Electron Detector onboard the National Oceanic and Atmospheric Administration (NOAA-15) satellite [10].Not only technical issues impact the satellite's operation, but also different environment orbit perturbations which are mainly caused by atmospheric drag and non-homogeneity of the Earth's mass, which belong under the second nature [11].Among these perturbations is the perigee deviation, what is analyzed in this paper specifically for the case of the Sun Synchronized orbits.The satellite's position in space is defined by space orbital parameters, what is too shortly described at second section.Further for the determined inclination window of the Sun synchronization under [9] it is calculated the orbital perigee deviation under different altitudes within the inclination window range.

II. SPACE ORBITAL PARAMETERS AND SUN SYNCHRONIZATION
The path of the satellite's motion is an orbit.In order to describe the satellite's movement within its orbit in space, a few parameters are required to be defined.These are known as space orbital parameters schematically presented in Fig. 2 and defined under items a), b), c) and d) [1]- [3].

A. The position of the orbital plane in space
This is specified by means of two parameters -the inclination i and the right ascension of the ascending node Ω. Inclination i represents the angle of the orbital plane with respect to the Earth's equator.The right ascension of the ascending node Ω defines the location of the ascending and descending orbital crossing nodes (these two nodes make a line of nodes) with respect to a fixed direction in space.The fixed direction is Vernal equinox.Vernal equinox is direction of line joining the Earth's center and the Sun on the first spring's day [1]- [3].

B. Location of the orbit in orbital plane
Normally an infinite number of orbits can be laid within an orbital plane.So, the orientation of the orbit in its plane is defined by the argument of perigee ω.This is the angle, taken positively from 0º to 360º in the direction of the satellite's motion, between the direction of the ascending node and the direction of perigee [1]- [3].

C. Position of the satellite in the orbit
The position of the satellite in orbit is determined by the angle θ called the true anomaly, which is the angle measured positively in the direction of satellite's movement from 0º to 360º, between the direction of perigee and the position of the satellite [1]- [3].

D. The shape of orbit
The shape of orbit is presented by the semi-major axis a which defines the size of orbit and the eccentricity e which defines the shape of the orbit.
Nodal regression depends on orbital radius (r) and orbital inclination angle (i), (See Fig. 2) [12].Through these two parameters could be achieved Sun synchronization, deeply elaborated under [9].Sun-synchronization is typical for LEO orbits, since for MEO and GEO orbits (too much higher radius) the nodal regression tends to be infinitesimal, thus MEOs and GEOs could not be Sun synchronized.The Sun synchronous LEO orbits ensure that: 1. the satellite passes over a given location on Earth every time at the same local solar time, thereby guaranteeing almost the same illumination conditions, varying only with seasons, and 2. the satellite ensures the coverage of the whole surface of the Earth, being quasi-polar in its nature.Satellites in sunsynchronous orbits are particularly suited to applications like passive remote sensing, meteorological, military reconnaissance and atmospheric studies.Radarsat is an example of a satellite in a low sun-synchronous orbit.Radarsat is in orbit of 798 km above the Earth, at an angle of inclination of 98.6 degrees, circling the globe from pole to pole [13].
Since the nodal regression effect is typical for LEO orbits, under manuscript [9] it is elaborated the range of the inclination for different LEO orbit altitudes for Sun synchronization to be achieved.Considering Van Allen belt effect (above 1400km) [14], for simulation purposes are considered attitudes from 600km up to 1200km.LEOs have too low eccentricity which one can be considered 0  e .Thus, for attitudes from 600km up to 1200km and considering Earth's radius as 6400 km yields out the orbits' radius range from 7000km up to 7600km.The Sun synchronization is achieved only when daily nodal regression (ΔΩ) is equal of 0.9856[º/day], represented with the straight line in the Fig. 3, identified with 0.986 above it [9].For orbital altitude of 600km (the lowest altitude considered under simulation conditions) consequently stems 7000   r a km under no eccentricity ( 0  e ) will get inclination for sun synchronization as:

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. 97 ( From Fig. 3, for nodal regression of 0.986º/day for LEO attitudes of 600km to 1200km (radius from 7000km up to 7600km), yields out the inclination range from 97.9º up to 100.5º, orbit Sun synchronization to be achieved [9].This range is nominated as Inclination Window for Sun synchronization.Further goal of this paper is to analyze and conclude what happens with orbital perigee deviation within the inclination window of the Sun synchronized orbits.

III. ORBITAL PERIGEE DEVIATION
The orientation of the orbit within orbital plane is defined by the argument of perigee ω, [Fig.2].The argument of perigee determines the position of the major axis.This is the angle, taken positively from 0º to 360º in the direction of the satellite's motion, between the direction of the line of nodes and the direction of the orbital perigee.Like the line of nodes shifting due to equatorial bulge of Earth, argument of perigee undergoes natural perturbation.This is defined as orbital perigee deviation.Further goal of this paper is calculate this orbital perigee deviation for Sunsynchronized orbits, or better saying how much the major axis deviates for one satellite pass for Sun-synchronized orbit.This perturbation may be visualized as the movement of the elliptical orbit in a fixed orbital plane, as presented in Fig. 4 (Figure presents the case of only perigee deviation under unchangeable line of nodes position).This deviation happens in a fixed orbital plane (Two orbits in Fig. 4 lie on the same plane).Obviously, the apogee and perigee points change in position, manifested as a major axis drift, deviated by ∆.The question is how much?
This deviation is a function of the satellite altitude and orbital plane inclination angle.This drift is analyzed under all serious satellite books, and for the simulation purposes of this paper is applied [15] expressing the perigee deviation per orbit in [º/orbit] as: For the inclination lower than 63.4° then ∆ is positive, so the perigee deviation occurs in the same direction as the satellite motion, and for the inclination greater than 63.4° then ∆ is negative, so the perigee deviation occurs in the opposite direction as the satellite's motion.Also, closer the satellite to the center of the Earth, the larger is the deviation.

IV. ORBITAL PERIGEE DEVIATION FOR SUN SYNCHRONIZED ORBIT
For simulation purposes are considered altitudes from 600km up to 1200km, and considering Earth's radius as 6400  E R km, then the orbits' radius ranges from 7000km up to 7600km.By previous section, based on [9] the range of orbital inclination lies from 97.9° up to 100.5° in order the Sun-synchronization to be attained.The inclination within this range is always greater than 63.4°, thus the result of (3) is always negative, what means that the orbital major axis will drift in the opposite direction to the satellite's motion.
LEOs are most of the time circular orbits, so the eccentricity is 0. Also,   =  =a, where a is semi major axis, and D= 2RE where 6400  E R km is Earth's radius.Applying these statements at (3), yields out the perigee deviation for low Earth Sun-synchronized orbits.
For the Sun-synchronization the orbital inclination lies on range: 97.9°≤  ≤ 100.5° Finally, the orbital perigee deviation for any Sunsynchronized orbit will lie in the range of: ∆ (=97.9)≤ ∆ ≤ ∆ (=100.5) The calculation of orbital deviation and appropriate results are presented in Table 1.The negative sign indicates that the orbital perigee deviation shifts in opposite side to the satellite motion.From the table it is obvious that the largest perigee deviation appears at altitude of 7000 km at inclination of 97.9° and the lowest perigee deviation appears at altitude of 7600km at inclination of 100.5°.These values converted in minutes, represent the deviation of 13.1' and 10.3' per orbit, respectively for the largest and lowest case.
Finally, the orbital perigee deviation expressed in ['/orbit], for LEO Sun-synchronized orbits is mathematically expressed as: This calculation leads toward the thrust vector to be applied in order to keep argument of perigee under in advance defined value, respectively unchangeable over time.The vector intensity depends on absolute value of perigee deviation.Further work is planned the thrust vector analysis to be applied in order to avoid perigee deviation.
V. CONCLUSION Sun-synchronized orbits are very useful for Earth's observation (scientific satellites) applications.Nodal regression is the feature which is especially utilized for LEO circular orbits providing to them the Sun synchronization.Sun synchronized orbits are always retrograde, since the inclination is greater than 90º.
Due to equatorial bulge of the Earth and natural perturbations, also the argument of the perigee deviates.This is known as orbital perigee deviation, expressed in ['/orbit].For the considered LEO altitudes under the Sun synchronized inclination window the orbital perigee deviation ranges 10.3['/orbit] to 13.1['/orbit]', always in the opposite direction to the satellite motion.

Fig. 3 .
Fig. 3. Inclination window for Sun synchronization [9].For orbital altitude of 1200km (the highest altitude considered under simulation conditions) consequently stems 7600   r a km under no eccentricity ( 0  e ) will get