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.

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 [

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 [

Another important step was to consider that the maneuver can be performed by using three impulses. This concept was introduced by Hoelker and Silber [

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 [

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 [

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

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 [

After that, Giacaglia [

In more recent times, Broucke [

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 [

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

In this way, the integral of the magnitude of the force over the time for one period of the spacecraft

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 [

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.

It is a dynamical criterion. So, the index calculated depends on the specific orbit of the spacecraft and on the force model adopted.

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.

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 (

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.

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

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

So, a double integral is evaluated over the eccentric anomaly of the spacecraft (from 0 to

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

Perturbation integral (m/s) for one period of the spacecraft as a function of its inclination (rad) considering only the Moon as the perturbing body. The true anomalies of the Moon at time zero are zero (dotted line),

Figures

Perturbation integral (m/s) as a function of the semimajor axis of the orbit (m) considering the perturbation of the Moon.

Figure

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

Perturbation integral (m/s) as a function of the semimajor axis of the orbit (m) considering the perturbation of the Moon for orbits beyond the Moon.

Figure

Perturbation integral (m/s) as a function of the semimajor axis of the orbit (m) considering the perturbation of the Sun for orbits beyond the Moon.

Figures

Perturbation integral (m/s) as a function of the semimajor axis of the orbit (m) considering the perturbations of the Sun and the Moon.

Perturbation integral (m/s) as a function of the semimajor axis of the orbit (m) considering the perturbations of the Sun and the Moon for orbits beyond the Moon.

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

Figure

Perturbation integral (m/s) as a function of the eccentricity of the orbit considering the perturbation of the Moon.

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

Perturbation integral (m/s) as a function of the eccentricity of the orbit considering the perturbation of the Sun.

Perturbation integral (m/s) as a function of the eccentricity of the orbit considering the perturbation of the Sun and the Moon.

Figure

Perturbation integral (m/s) as a function of the inclination (rad) of the orbit considering the perturbation of the Moon for inclinations of the orbit of the Moon of 18 degrees (dotted line), 23 degrees (dashed line), and 28 degrees (continuous line).

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

Perturbation integral (m/s) as a function of the inclination (rad) of the orbit considering the perturbation of the Sun.

Perturbation integral (m/s) as a function of the inclination (rad) of the orbit considering the perturbation of the Sun and the Moon with the orbit of the Moon inclined by 18 degrees.

Perturbation integral (m/s) as a function of the inclination (rad) of the orbit considering the perturbation of the Sun and the Moon with the orbit of the Moon inclining by 28 degrees.

Figure

Perturbation integral (m/s) as a function of the argument of the perigee (rad) of the orbit considering the perturbation of the Sun and the Moon.

Perturbation integral (m/s) as a function of the argument of the ascending node (rad) of the orbit considering the perturbation of the Sun and the Moon.

Figure

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%.

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).