## Mathematical Methods Applied to the Celestial Mechanics of Artificial Satellites 2013

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Antonio Fernando Bertachini de Almeida Prado, "Searching for Orbits with Minimum Fuel Consumption for Station-Keeping Maneuvers: An Application to Lunisolar Perturbations", *Mathematical Problems in Engineering*, vol. 2013, Article ID 415015, 11 pages, 2013. https://doi.org/10.1155/2013/415015

# Searching for Orbits with Minimum Fuel Consumption for Station-Keeping Maneuvers: An Application to Lunisolar Perturbations

**Academic Editor:**Maria Zanardi

#### Abstract

The present paper has the goal of developing a new criterion to search for orbits that minimize the fuel consumption for station-keeping maneuvers. This approach is based on the integral over the time of the perturbing forces. This integral measures the total variation of velocity caused by the perturbations in the spacecraft, which corresponds to the equivalent variation of velocity that an engine should deliver to the spacecraft to compensate the perturbations and to keep its orbit Keplerian all the time. This integral is a characteristic of the orbit and the set of perturbations considered and does not depend on the type of engine used. In this sense, this integral can be seen as a criterion to select the orbit of the spacecraft. When this value becomes larger, more consumption of fuel is required for the station keeping, and, in this sense, less interesting is the orbit. This concept can be applied to any perturbation. In the present research, as an example, the perturbation caused by a third body is considered. Then, numerical simulations considering the effects of the Sun and the Moon in a satellite around the Earth are shown to exemplify the method.

#### 1. Introduction

The problem of orbital maneuvers is one of the most important topics in orbital mechanics. It has been under study for a long time. It has several aspects to be considered, like the fuel consumption, the maneuvering time, and so forth. One of the first and more important results is the one obtained by Hohmann in 1925 [1], which solved the problem of transfers between two coplanar circular orbits with minimum variation of velocity applied to the spacecraft. This transfer would be generalized later to the elliptic case by Marchal in 1965 [2].

After those initial researches, the problem of two-impulse transfers received more attention in the literature. Specific situations were considered, like the case where the magnitudes of the impulses are fixed [3, 4].

Another important step was to consider that the maneuver can be performed by using three impulses. This concept was introduced by Hoelker and Silber [5] and Shternfeld [6], both in 1959. They showed, independently and simultaneously, that a bielliptical transfer between two circular orbits can be more economical than the Hohmann transfer, depending on the initial and final orbits involved. Other studies available in the literature considering impulsive maneuvers are available in [7–19].

The situation changes when the control available to make the maneuvers consist of a low thrust. In this case, the approach used to solve the problem is based in optimal control theory. Some papers that use this technique can be seen in [20–28].

In more recent years, two other techniques were used in the problem of orbital maneuvers, based on the concepts of swing-by and gravitational capture. Both approaches are based on the use of the gravitational force of a third body to replace engines, thus reducing the fuel consumption. Some references that study the swing-by problem are [29–42]. Considering the gravitational capture, some examples are [43–48].

Regarding station-keeping maneuvers that is more related to the topic of the present research, the literature also has several publications, like [49–58].

The present paper has the objective of studying a new criterion to measure the consumption required by a specific orbit with respect to the fuel required for station keeping. The idea behind it is that it is possible to consider the existence of an ideal propulsion system that can deliver a force that has the same magnitude of the perturbations that are acting in the satellite but in the opposite direction. So, this measurement is based on the integral of the perturbation suffered by the spacecraft over the time and can be applied to orbits around any primary and subjected to any type of perturbations. In the majority of the cases it has to be evaluated numerically, since no closed form for the integral can be found. To calculate this index, the perturbations are written in the equations of motion and integrated over the time to see their cumulative effects as a function of the initial conditions. The integral of this force over the time represents the equivalent variation in velocity that the propulsive system needs to deliver, since it represents the integral of the acceleration received by the spacecraft. This technique can be used in any dynamical system, including planets and planetary satellites. In the present paper, this idea is applied to study orbits of satellites around the Earth for a satellite that is perturbed by the Sun and the Moon. The goal is to find the potential cost to perform station keeping in those orbits, it is able to show the best orbits to place a satellite, with respect to the fuel required by the station-keeping maneuvers. It does not mean that the satellite has to be constrained to a Keplerian orbit all the time. Regarding station-keeping maneuvers, in some situations, if the mission requirements allow, it is possible to let the spacecraft deviates from its orbit and then return to it after some time. This technique uses the flexibility of the satisfaction of the constraints to reduce the fuel consumption. The idea here is that, if the orbit has a lower value for this integral, it is a good indication that it is a better orbit regarding station-keeping consumption, independent of which technique will be used to control the orbit in a real situation.

As far as the third-body perturbation is concerned, there are also several papers available in the literature studying this point. In particular, the effects of the gravity attractions of the Sun and the Moon in orbits of artificial satellites of the Earth have been studied by several researches. Kozai [59, 60] writes down the Lagrange planetary equations considering the disturbing function due to the Moon or the Sun, including secular and long periodic terms. It gives explicit expressions only for the secular terms. Blitzer [61] obtains estimates for the perturbations by using methods of classical mechanics. Only secular terms are included. It shows that the main effect is the precession of the orbital plane. Musen [62] shows two sets of equations to determine the long periodic perturbations. The first one uses a theory developed by Gauss and gives a numerical treatment for the very long periodic effects. The second one develops the disturbing function as a function of the Legendre polynomials and obtains the long periodic terms and its influence on the stability of the orbits. Kaula [63] also studies the lunisolar perturbation, obtaining general terms for the disturbing function. They use equatorial elements for the Moon.

After that, Giacaglia [64] obtains the disturbing function for the perturbation of the Moon using equatorial elements for the satellite and ecliptic elements for the Moon. Hough [65] averages the Hamiltonian and study periodic perigee motion near the critical inclinations of 63.4° and 116.6°. Delhaise and Morbidelli [66] study lunisolar effects on a geosynchronous satellite staying near the critical inclination. He shows that there are no resonance phenomena.

In more recent times, Broucke [67], Prado [68], and Moraes et al. [69] obtain general forms for the disturbing function due to the third body, using a double average technique to eliminate the periodic terms due to the perturbed body and the perturbing body, with the expansion truncated after the second- and fourth-order terms, respectively.

This problem was also studied considering a single average that eliminates the periodic terms only due to the perturbed body. Some researches on that line are in Solórzano and Prado [70, 71] and Domingos et al. [72]. Those ideas are also generalized to be used for orbits around Europe, in Carvalho et al. [73], and around the Moon, in Carvalho et al. [74].

#### 2. Mathematical Models

This section shows the equations given by the mathematical model used to describe the problem. It is assumed the existence of a main body with mass , fixed in the center of the reference system . This body is the Earth, in the simulations shown here. There is also a body that is perturbing the motion of the satellite (the Sun or the Moon in the simulations shown here), with mass , that is assumed to be in a circular orbit with semimajor axis and mean motion () around the central body. The spacecraft is assumed to be in a three-dimensional elliptic orbit that has orbital elements: (semimajor axis), (eccentricity), (inclination), (argument of perigee), and (longitude of the ascending node), and the mean motion is (). The magnitude of the perturbation force by unit of mass acting in the spacecraft can be obtained by the disturbing potential [75]: where the coordinates of the Sun are (, , ), the coordinates of the Moon are (, , ), the coordinates of the spacecraft are (, , ), is the gravitational constant, and are the masses of the Moon and the Sun, respectively, and , , , and are the distances between the Moon and the Earth, between the Sun and the Earth, between the spacecraft and the Moon, and between the spacecraft and the Sun, respectively. The Earth is assumed to be in the center of the reference system. Those distances are given by

In this way, the integral of the magnitude of the force over the time for one period of the spacecraft , that will be called PI (that stands for “perturbation integral”), is given by where stands for the gradient of the potential, which represents the force due to the disturbing body. Using the fact that the mean anomaly of the spacecraft is given by , we have

Note that the perturbing body is assumed to be in circular orbit, and the orbit of the spacecraft is assumed to be Keplerian all the time, because there is an engine compensating the perturbations at every instant of time. In the same way performed in Prado [68], the variable can be replaced by the eccentric anomaly () to perform the calculations. In order to evaluate this integral and similar ones that appear later in the present paper, the potential is written in terms of the keplerian elements of the spacecraft, the Sun, and the Moon. The equations are not shown explicitly here because they are too long. To proceed in this way, the following equations are required:

Using those relations, the integral PI becomes

In this form the integral PI can be evaluated by any method, and it shows how difficult, in terms of fuel consumption, is to keep the orbit Keplerian. This new criterion has the following characteristics.(1)It is a dynamical criterion. So, the index calculated depends on the specific orbit of the spacecraft and on the force model adopted.(2)Since the orbits are Keplerian all the time, it means that it is possible to calculate this index for each perturbation individually. In this way, the effect of each force is evaluated regarding its integral effect for one period of the nominal orbit desired for the spacecraft, and it is possible to compare those numbers to decide which forces need to be taken into account for the motion of the spacecraft, according to the accuracy required by the study.(3)For a given pair of orbits (perturbed and perturbing bodies), this index also depends on the initial position of the bodies. So, to have a complete view of those numbers, considering that the spacecraft will stay in orbit for several periods of the primaries, it is interesting to make an average over the initial true anomaly () of the perturbing body.(4)This index measures the amount of variation of velocity that the perturbation causes in the spacecraft, so it can be related to the fuel consumption required to keep the orbit of the spacecraft Keplerian. Although there are engineering reasons to be considered in maneuvers like that (propulsion nonideality of many types, as well as strategy of maneuvers, that explore the possibility of allowing instantaneous deviation from the nominal orbit to occur, etc.), this number identifies which orbits have potential to require less consumption of fuel for the maneuvers. In this way, it points out the more economical orbits to place a spacecraft.

#### 3. Results

The idea is to show the evolution of the perturbing force and its integral over the time for a satellite around the Earth perturbed by the Moon and the Sun. To make this study, the Sun and the Moon are assumed to be in circular orbits around the Earth with semimajor axis of 384399 km for the Moon and 149597870 km for the Sun. The inclination of the orbit of the Sun is 23.5 degrees and the inclination of the orbit of the Moon (with respect to Earth’s equator) varies from 18 to 28 degrees.

The mass of the Moon is assumed to be , so , and the mass of the Sun is assumed to be , so .

An important point, as said before, of this criterion is that it depends on the initial configuration of the system. In the present case, it means the positions of the perturbing bodies when the motion starts. To take into account this fact, in all the simulations made in the present paper, the study of the effects of changing each orbital element of the spacecraft is made as a function of the initial mean anomaly of the perturbing bodies. Then, an average technique is applied, which means that the integral is evaluated with respect to the initial true anomaly of the perturbing body from zero to . This operation is made to take into account the variation of this variable and gives results that have an average meaning, which is very important because the spacecraft will stay in orbit for a large number of periods of the perturbing body. It means that the following integrals are evaluated: where is the initial anomaly of the Sun, to obtain the mean effect of the Sun in the perturbation, where is the initial anomaly of the Moon, to obtain the mean effect of the Moon in the perturbation and to obtain the mean effect of the Sun and the Moon in the spacecraft.

So, a double integral is evaluated over the eccentric anomaly of the spacecraft (from 0 to ) to obtain the total effect per revolution and over the initial position of the perturbing body (from 0 to ) to take into account the total mean effect of this perturbing body for one orbit of the spacecraft. A triple integral is used when considering two perturbing bodies.

First of all, to illustrate the importance of the initial position of the perturbing body in the evaluation of the integral PI, a study is made to show this effect with respect to the inclination of the spacecraft considering only the perturbation of the Moon. The Moon is assumed to be in an orbit inclined by 18 degrees. Similar studies were made for other inclinations of the orbit of the Moon in the range from 18 to 28 degrees, but the results are equivalent. Figure 1 shows the results. It plots the value of the PI for one orbit of the spacecraft as a function of the inclination of the orbit of the spacecraft. The other orbital parameters are assumed to be: , , and . Figure 1 shows the values of PI as a function of the inclination of the spacecraft for the following initial positions of the Moon (true anomaly at time zero): zero (dotted line), (dot-dashed line), (dashed line), and (continuous line). It is clear that the main effect is a shift of the curves. The differences in the magnitude are small, in the order of 5% or less. So, if the spacecraft stays in orbit for several periods of the perturbing body, the average effects is almost independent of this initial condition. Note also that results for the initial true anomaly of the Moon equal to zero and are very similar, as well as for and . This fact happens due to the symmetry of the system regarding these angles.

Figures 2 to 15 show the effects of changing each orbital element of the spacecraft. It is important to note that, since this integral is taken over one period of the spacecraft, it is necessary to define one value for the semimajor axis (so, one period of revolution) to be a reference for the calculations. This orbit is chosen to be circular with semimajor axis of 42164 km, which corresponds to a geostationary orbit, which is an orbit well known by suffering important lunisolar perturbations.

##### 3.1. Effects of Changing the Semimajor Axis of the Orbit of the Spacecraft

Figure 2 shows the evolution of the magnitude of the perturbing force due to the Moon as a function of the semimajor axis of the orbit of the spacecraft covering the range from 70,000 m to 45,000,000 m, that is, a little above the geostationary orbit. This simulation was made assuming an inclination of 18 degrees for the orbit of the Moon. Other values in the range from 18 to 28 degrees were used, with no significant difference in the results. The initial values for the other orbital elements are , , and . The reference orbit has a semimajor axis of 42164 km, so the value of the integral is calculated for a period of 86163 s, which is the period of an orbit with this value for the semimajor axis. It shows that the effects increase linearly with the altitude. It means that the effects of the perturbing body can be modeled very well by the relation: where is the semimajor axis of the orbit of the spacecraft in meters and . This simple equation can be used to estimate the fuel consumption for station keeping as a function of the semimajor axis of the orbit of the spacecraft.

After that the range of values of semimajor axis were extended to 1,000,000,000 m, to reach orbits that are beyond the orbit of the Moon. The curve is shown in Figure 3. It has an interesting behavior, combining the initial linear dependence with the semimajor axis shown in Figure 2, with a fast growing near the orbit of the Moon. This result is expected, since the spacecraft is approaching the orbit of the Moon, and so their mutual distance is reduced, causing a strong increase in the perturbation. After passing the orbit of the Moon the perturbation decreases again, since the spacecraft is getting far from the Moon.

Figure 4 shows the perturbing effect of the Sun for the same range of semimajor axis used in Figure 3. It is also clear a linear dependence, so the effects can be modeled by where .

Figures 5 and 6 show the effects of both perturbing bodies (Sun and Moon) acting together in the spacecraft. Figure 5 shows a linear relation for values of the semimajor axis from 7,000,000 m until 45,000,000 m. So, it can be modeled by where .

A comparison of those values shows that the effects of the perturbation due to the Moon are about 2.52 stronger then the effects of the perturbation due to the Sun. It is also noted that the effect of the Moon is stronger than the total effects of the Sun and the Moon. It indicates that there are some compensation of the effects, and the Sun acts to reduce the fuel consumption for station keeping, helping the control system. It is a consequence of the fact that, sometimes, there are components of the force of the Sun that is acting in the opposite direction with respect to the force of the Moon. So, a dynamical system formed by the Sun and the Moon requires less effort from the control then a dynamical system formed only by the Moon.

Figure 6 shows the same behavior noticed in Figure 3 of increasing the value of the perturbation integral when reaching the orbit of the Moon and then decreasing after that. The difference is that now there is a new increase in the effect of the Moon, caused by the fact that the spacecraft is getting closer to the orbit of the Sun. This combined effect causes an interesting point of minimum effect behind the Moon. In other words, if it is necessary to place a satellite behind the Moon, there is an optimum distance located near the semimajor axis of 620,000,000 m from the Earth with respect to the fuel consumption for station keeping to compensate the lunisolar perturbations.

##### 3.2. Effects of Changing the Eccentricity of the Orbit of the Spacecraft

Figure 7 shows the evolution of the magnitude of the integral of the perturbing force due to the Moon as a function of the eccentricity of the orbit of the spacecraft. The initial values for the other orbital elements are (the reference orbit), , and . This figure was made using an inclination of 18 degrees for the orbit of the Moon, but simulations were made for several values of this variable in the range between 18 and 28 degrees, and the results are very similar, so they are omitted here. The whole interval of eccentricities were plotted to show the mathematical behavior of the equations developed, but it is necessary to remember that the maximum value for the eccentricity that does not imply a crash with the Earth or a passage by regions of high-density atmosphere is around 0.83 that is equivalent to a perigee altitude of 7168 km. So, this is the limit of practical applications for this variable.

It is clear that the effect increases with the eccentricity, so circular orbits require less fuel consumption for station keeping due to the third-body perturbation. The difference is not negligible, since it can reach the order of 50%. It is possible to explain this result based on the geometry of the system. In an eccentric orbit, the spacecraft reaches higher altitudes (so, closer to the orbit of the Moon) at the apogee of the orbit. It is true that it also remains some time in lower altitudes (so, far away from the orbit of the Moon) at the perigee, but the increase of the integral during the passage of the spacecraft by the apogee is larger than the correspondent decrease due to its passage by the perigee, and the net result, after making the average over the initial position of the Moon, is an increase in the integral. Of course this effect is increased when the eccentricity increases.

Figure 8 shows the same results considering the perturbation of the Sun, and the results are similar, just reduced by a scale factor around 2.13. The same difference of up to 50% in the value of the integral from circular to elliptic orbits is found for the perturbation of the Sun. Figure 9 shows the combined effects of the Sun and the Moon, with the same order of the differences between circular and elliptic orbits. The relationship between the effects is about the same as the one obtained in the study of the effects of the semimajor axis and the perturbation due to that the effect of the Moon is more than twice stronger than the effect of the Sun. The combined effects (Sun and Moon) are smaller than the sum of both effects individually, for the same physical reasons already explained.

##### 3.3. Effects of Changing the Inclination of the Orbit of the Spacecraft

Figure 10 shows the evolution of the magnitude of the perturbing force due to the Moon as a function of the initial inclination of the orbit of the spacecraft. The initial values for the other orbital elements are (the reference orbit), , and . Three simulations are made to cover different values of the inclination of the orbit of the Moon: 18, 23, and 28 degrees. This time the differences of the results are not negligible, and three curves are plotted in Figure 10. It is noticed that, for equatorial orbits (inclination zero and ), the effects are stronger when the inclination of the orbit of the Moon is 18 degrees, because this is the orbit that is closer to the orbit of the spacecraft. Of course the situation with minimum effects is for inclination of 28 degrees for the orbit of the Moon. The difference between the two extreme cases is about 0.01 m/s, which correspond to 2%.

Regarding the general behavior, some important points to note are described later. The values of the PI have considerable changes, in the order of 15%, so the inclination plays an important role in the station-keeping maneuvers. The higher values for the mean PI appear for the cases where the orbit is coplanar with the Moon, either prograde or retrograde. They have about the same values, which mean that, regarding costs for station keeping maneuvers, prograde and retrograde orbits are similar. The orbits with minimum values are the ones that lie in a plane that is perpendicular to the plane of the orbit of the Moon. This is expected because the coplanar orbits are the ones that pass closer to the Moon compared with the inclined orbits. Of course, the perpendicular orbit is the one that makes the spacecraft stay at a longer distance from the Moon.

Figure 11 shows the same effects but now considers only the perturbation due to the Sun. It is the same pattern, just reduced by a factor scale of about 2.2 and shifted to take into account the inclination of the orbit of the Sun. Figures 12 and 13 show the effects of the Sun and the Moon combined, for the extreme cases of the inclination of the orbit of the Moon. It has the same pattern of Figures 10 and 11, but once again the total effects are smaller than the sum of the individual effects, for the reasons explained before. The same stronger effect in equatorial orbits when the orbit of the Moon is inclined by 18 degrees is also noted here, as expected, for the same reasons already explained.

##### 3.4. Effects of Changing the Longitude of the Perigee of the Orbit of the Spacecraft

Figure 14 shows the evolution of the magnitude of the perturbing force due to the Moon and the Sun as a function of the argument of the perigee of the orbit of the spacecraft (rad). The initial values for the other orbital elements are (the reference orbit), , and . The results show that this parameter has a periodic, but very little effect on the perturbation integral. Compared with the effects due to the other terms, it can be neglected in a first analysis. Simulations were also made considering the effects of the Moon and the Sun individually, but they are omitted here because the effects are too small to be shown.

##### 3.5. Effects of Changing the Longitude of the Ascending Node of the Orbit of the Spacecraft

Figure 15 shows the evolution of the magnitude of the perturbing force due to the Moon and the Sun as a function of the argument of the ascending node of the orbit of the spacecraft. The initial values for the other orbital elements are (the reference orbit), , and . Similarly to what happened for the longitude of the perigee, these effects are also too small and can be neglected in a first analysis. Simulations considering the individual effects of the Moon and the Sun are omitted because they are negligible. Note that this figure is very similar to the previous one. This similarity is due to the fact that changing the longitude of the ascending node and changing the longitude of the perigee have the same effect on the orbit, because the orbits are planar and those angles are equivalent.

#### 4. Conclusions

This paper showed a definition of a new criterion to choose orbits for a space mission, focused in the fuel consumption for station-keeping maneuvers, which considers the effects of the perturbations suffered by the spacecraft by means of evaluating the integral over the time of the perturbations.

This criterion is then applied to the perturbation of a third body included in the dynamics, and numerical results are shown for the lunisolar perturbations.

The results showed the dependence of this index on the initial relative geometry of the bodies, so a study was made considering an average over the initial positions of the perturbing bodies, which are specified by the true anomalies of the Sun and the Moon at the initial time.

The effects of the Moon are larger, by a factor in the range between 2 and 3, when compared to the effects of the Sun, depending on which orbital element of the orbit of the spacecraft is under study. It is also noticed that the effect of the combined effects of the Sun and the Moon is smaller than the sum of the effects individually.

It is also shown that there is a linear relation linking the semimajor axis of the orbit of the spacecraft and the effects of the third-body perturbation. Another characteristic found here is that, if it is necessary to place a satellite behind the orbit of the Moon, there is point with minimum value for the third-body perturbation, which is located near the position of 620,000,000 m from the Earth. The effects tend to a very large values when the spacecraft reaches orbits near the orbit of the Moon.

Regarding the eccentricity of the orbit of the spacecraft, it was shown that circular orbits require less fuel consumption for station-keeping maneuvers when compared to elliptic orbits and that this difference is very large, in the of the order of 50%.

The inclination of the orbit of the spacecraft plays an important role in the costs for station keeping, with a difference of the order of 15% between the maximum and the minimum. The higher values for the effects appear for the cases where the orbit of the spacecraft is coplanar with the Moon, either prograde or retrograde, and the minimum occurs for perpendicular orbits.

The effects due to the variations of the argument of the ascending node and the longitude of the perigee of the orbit of the spacecraft are negligible.

The variation of the inclination of the orbit of the Moon in the range from 18 to 28 degrees has no significant difference in the results, except when studying the inclination of the orbit of the spacecraft. In this situation, the difference between the two extreme cases is about 0.01 m/s, which corresponds to 2%.

#### Acknowledgments

The author is grateful to National Council for Scientific and Technological Development (CNPq), Brazil and to Foundation to Support Research in São Paulo State (FAPESP).

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Copyright © 2013 Antonio Fernando Bertachini de Almeida Prado. 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.