1 Introduction

The possibility of achieving complete spatial and continuous temporal coverage of Mars takes on a role of key importance in view of the future Mars missions, involving also human exploration, which are being planned by the space agencies. In this context, the use of the Martian Areostationary Orbit (MAO), which is analogous of the Geostationary Earth Orbit (GEO), would appear as the most promising option, although the high altitude is associated with a low resolution for the science instruments and with a higher power for the communications. In fact, the MAO, whose orbital period coincides with the temporal interval that Mars takes to accomplish a complete rotation around its polar axis, allows a very wide coverage of the planet and provides, at the same time, a stationary condition with respect to the surface of the planet. In this regard, the design and the performances of the Mars Areostationary Relay Satellite (MARSAT) have been described in Edwards et al. (2007), Hastrup et al. (2003), Edwards and De Paula (2007).

The possibility of having continuous contact with the ground instruments would lead to important advantages in terms of navigation and telecommunications for the landing phase and for the ground assets, like rovers and human bases. For example, about the Mars Exploration Rover Opportunity mission, the use of the non-volatile “flash” memory for holding data overnight has been discontinued in the last year and the storage of daily measurements of the rover has been limited. In this case, the use of the MAO would have allowed a larger data storage. This benefit could also be exploited for the future Interior Exploration using Seismic Investigation, Geodesy and Heat Transport (InSight) and Mars 2020 missions. Moreover, the MAO would allow remote sensing applications devoted to the acquisition of change detection datasets to monitor weather and other natural phenomena, at low spatial resolution.

However, due to the orbital perturbations, the feasibility of maintaining a probe within a given tolerance interval around its nominal position entails periodic corrective manoeuvres, which, besides making the MAO a very expensive solution (Silva and Romero 2013), imply limited life-times for the orbiting probe.

It is therefore important to provide alternative solutions which allow the gaining of a wide coverage of Mars without requiring considerable propellant consumption. For GEO satellites, the same problem can be faced considering satellites orbiting, at geosynchronous altitude, over circular orbits lying on the Laplace plane (Ulivieri et al. 2013; Rosengren et al. 2014). Concerning Mars, in Castellini et al. (2010) several solutions guaranteeing a complete and continuous coverage of Mars have been proposed, considering probes orbiting the planet or positioned in HALO orbits; then, to fulfil telecommunication constraints, a Walker constellation (Walker 1971) with six probes distributed over two orbital planes (three for each plane) has been chosen; in Ortore et al. (2015) long dwell time orbits have been proposed to obtain quasi-stationary coverage of a region of the planet and therefore long contact times with a Mars lander.

In this paper, the problem of gaining complete and continuous coverage of an extended equatorial belt has been taken into account and solved by exploiting three probes moving over circular orbits in the neighbourhood of the MAO. The goal has been achieved after having analysed the coverage obtainable by the MAO and investigated the perturbative effects in the operational environment in which the probes move.

2 Coverage by Martian Areostationary Orbit

As happens with the GEO, three probes placed in MAO and phased by 120° are able to guarantee complete and continuous coverage of the planet, excluding the polar caps. However, unlike the GEO case, for the MAO the most important perturbative effect is due to the elliptical shape of the equator, causing an East–West oscillation (Alvarellos 2010). In fact, the gravitational attractions of the natural satellites Phobos and Deimos do not significantly influence the probe motion and the Sun does not provoke a considerable North–South displacement (Romero et al. 2015). Consequently, while for the GEO the North–South and East–West manoeuvres require, respectively, annual velocity variations of 40–50 m/s, depending on the Moon’s position, and 0–2 m/s, depending on the satellite longitude (Soop 1983), for the MAO the situation is reversed. In fact, in MAO, the North–South manoeuvres need about 2 m/s per year, while the East–West ones require from 0 to about 38 m/s per year (Silva and Romero 2013).

Due to these East–West manoeuvres, the feasibility of gaining a global coverage of Mars using a three-probe constellation in MAO would be associated with a high total propellant consumption. In absence of such manoeuvres, each probe would be subjected to a different oscillatory motion around the nearest stable point, with a consequent loss of complete longitudinal coverage.

Regarding this, an investigation about the velocity variations needed to obtain complete and continuous coverage of Mars, by using a three-probe constellation in MAO, has been carried out here. This analysis has led to the conclusion that, under the J 22 harmonic effect of the gravitational field, the configuration minimising the propellant consumption is the one shown in Fig. 1, where a probe is located in a stable equilibrium point and the other two are phased by 120°. The required total variation of velocity for this configuration would be equal to 60 m/s/year. In absence of corrective manoeuvres, the time after which there would be a loss of complete longitudinal coverage is about one terrestrial month.

Fig. 1
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Optimal configuration in Martian Areostationary Orbit

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3 Resonance Phenomenon Around the Martian Areostationary Orbit

The analysis performed in Sect. 1 highlights how the use of the MAO for gaining a global coverage of Mars is unavoidably associated with an important propellant consumption. After all, as for the GEO, because of the equality between probe orbital period and period of rotation of the planet around its axis, the MAO is affected by a resonance phenomenon associated with the J 22 harmonic. As is well known, this causes a semi-major axis oscillation, which leads to a periodic motion of the probe around the nearest stable point.

For the purpose of investigating the possibility to determine a Martian constellation able to provide complete and continuous coverage, the semi-major axis oscillations associated with this resonance phenomenon have been studied, as a function of both the longitudinal distance from a stable point and of the orbit altitude. The results of this analysis, in the neighbourhood of the MAO, are illustrated in Figs. 2, 3, 4, 5, and 6, where Δλ represents the initial longitudinal distance from the nearest stable point (see also Fig. 1), a is the initial value of semi-major axis and Δa the corresponding total variation (double the amplitude of the oscillations). The investigation has been conducted numerically, using the Goddard Mars Model 2B (GMM-2B) (Lemoine et al. 2001) with (2 × 2) harmonics.

Fig. 2
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J22 effect at Δλ = 5° from a stable point

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Fig. 3
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J22 effect at Δλ = 25° from a stable point

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Fig. 4
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J22 effect at Δλ = 45° from a stable point

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Fig. 5
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J22 effect at Δλ = 65° from a stable point

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Fig. 6
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J22-resonant effect at Δλ = 85° from a stable point

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The figures show how the Martian resonance zone extends over a wide region surrounding the MAO, whose orbit radius is 20,428 km, and how the corresponding semi-major axis variations are very large. In all figures, in correspondence of the MAO altitude, the Δa presents a local minimum. In fact, once fixed the initial longitude, when the initial altitude of the probe moves away from that of the MAO, the amplitude of the oscillations of the probe around the stable longitude increases (symmetrically with respect to the MAO condition). This phenomenon, which disappears at a certain distance from the stable point, is significantly more marked when the values of Δλ are small (it reduces when Δλ increases). The oscillation periods of the semi-major axis, depending on the value of a, increase with Δλ and with the difference between the initial altitude and the MAO altitude. In particular, for the MAO Probes 2 and 3 of Fig. 1, the oscillation period is equal to 170 terrestrial days.

4 No-Station Keeping Configurations

The results obtained in Sect. 2 lead to the conclusion that, to gain global coverage of Mars without station keeping manoeuvres, it is recommended to leave the East–West resonance region which surrounds the MAO. In fact, inside all the resonance region, the probes undergo strong and diverse variations of altitude, according to the specific initial longitude, and are therefore subject to a relative shift entailing a loss of coverage in longitude.

This unavoidably leads to the choice of either a lower or a higher altitude with respect to that of the MAO, where the considered semi-major axis variations are small enough to determine an acceptable relative shift among the probes. Thus, considering a constellation of three probes located over a circular orbit with a radius sufficiently higher (or lower) than that of the MAO and phased by 120°, global coverage of Mars can be ensured, without requiring any corrective manoeuvre. In fact, out of the resonance region, the initial longitudes of the probes are not influent. On the other hand, given that each probe is characterised by a longitudinal shift with respect to the planet’s surface, it has to be tracked from Mars by using a mobile antenna, although the slow shift makes the tracking very simple.

For example, assuming a minimum elevation angle of 5° for the acquisition of each probe, continuous coverage of the belt extending between ±20° of latitude can be gained by a circular orbit with initial semi-major axis at least equal to 22,893 km. In fact, the aforesaid belt can be covered for a period of three terrestrial years, although the probes still present different semi-major axis variations. Figure 7 shows the coverage obtainable at the initial instant (nominal coverage, Fig. 7a) and after three years (worst condition, Fig. 7b), in this case. While in the nominal configuration the coverage circles are perfectly spaced one by the other, after three years the circles present different longitudinal distances and two of them intersect at a latitude of ±20°. Elapsed three years, the coverage circles will begin to present intersections at lower latitudes and then, after a certain time, they will show a longitudinal gap. Therefore, if a longer time of coverage is desired, a suitably higher nominal orbit altitude has to be chosen.

Fig. 7
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(Systems Tool Kit software: www.agi.com)

Nominal (a) and final (b) coverage with an initial semi-major axis of 22,893 km

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For the case of Fig. 7a, the elevation angle trends obtained considering both an equatorial antenna (solid line) and an antenna placed at a latitude of 20° (dashed line) are reported in Fig. 8, where the time is expressed in hours. As evidence shows, the elevation rate is very slow (less than 3° per hour). As for the azimuth, while for the case at 20° of latitude the maximum rate is 6.6° per hour, for the equatorial case the rate is null but an instantaneous azimuth variation of 180° occurs when the elevation is 90°.

Fig. 8
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Evolution of the elevation angle with a = 22,893 km

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Figure 9 illustrates the correlation between the initial values of the semi-major axis and the latitudinal extension of the equatorial belt (symmetrical with respect to the equator) in which complete and continuous coverage can be guaranteed by the proposed three-probe constellation, for a period of three terrestrial years. Higher values of semi-major axis allow the obtaining of a wider coverage in latitude. On the other hand, they are associated with a faster variation of the antenna tracking angles.

Fig. 9
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Initial value of the semi-major axis as a function of the latitudinal coverage

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To complete the analysis, an investigation into the perturbing effects deriving from both the higher harmonics of Mars’ gravitational field and Sun’ gravitational attraction has also been carried, numerically, considering the model Lemoine et al. (2001) with (40 × 40) harmonics for Mars and the ephemerides data DE430 for the Sun. This study has highlighted how, at the considered altitudes, such perturbing effects do not entail significant differences, on the variations of the orbit elements and on the associated obtainable coverage, with respect to an analysis performed taking into account a simplified model with (2 × 2) harmonics and neglecting the perturbing effect of the Sun.

5 Conclusions

The problem of gaining complete and continuous coverage of an extended equatorial belt of Mars has been investigated. The analysis has demonstrated how, once left the range of altitude, around the Martian Areostationary Orbit, in which the perturbative effect due to the asymmetry of Mars’ equator shows a resonance region, the above-mentioned goal can be fulfilled without performing corrective manoeuvres. In fact, due to the weak effect of the Sun’s attraction, the orbit altitude can be increased appropriately and three-probe configurations in circular orbit can be considered to provide the desired coverage in latitude. However, presenting a longitudinal motion with respect to Mars’ surface, such configurations require the use of a mobile antenna to be tracked from the ground.