Analysis of the optimization for an Earth to Mars areostationary mission

Abstract
Areostationary Relay Satellites are the most robust candidates to meet the coming set of Mars science missions. Areostationary satellites combine both the continuous coverage over the same Mars localization and the return of engineering and scientific instrument data in almost real time needed for the tele-robotic exploration. We present a procedure to minimize the total required impulses magnitudes in the major mission phases for an areostationary satellite. This mission design optimization problem involves two main problems. First, the determination of the launch and arrival dates for an optimal minimum energy Earth-Mars transfer trajectory. Second, the optimal minimum thrust manoeuvres to capture the spacecraft from the hyperbolic arrival trajectory to Mars and place it in the areostationary orbit.

1. Introduction
An areostationary satellite would orbit Mars in an areoequatorial circular orbit with a semi-major axis of a_s = 20428 km to remain over a fixed point on the Martian surface with respect to the rotating Mars with a period of P=88775.244 s (Montabone, 2020). The purpose of this work is to analyse the elliptic heliocentric transfer and the hyperbolic areocentric orbits to achieve the final areostationary position with the minimal impulses cost. To this end, first the elliptic trajectory optimization using genetic metaheuristic algorithms is performed. Then, an analysis of accuracy of the matched solutions for the elliptic and the hyperbolic orbits using an iterative procedure at the Mars influence sphere is accomplished. Finally, the Mars arrival manoeuvres needed to capture the probe and to place it in an areostationary orbit are optimized.

2. Earth to Mars elliptic transfer optimization
In this minimum-energy problem, we search a solution minimizing the equation:

where the characteristic energy  at the boundary of the sphere of influence of the Earth, C₃= V²_∞E, and  the Mars  arrival velocity V_∞M are weighted according to W_C3 and W_∞M, respectively. The minimum C in equation (1) tends to give lower values of the total impulsive manoeuvres required, first to give to the space vehicle an orbital velocity greater than  the parabolic Earth escape velocity, and after, at the Mars arriving hyperbolic orbit, to reduce the  hyperbolic excess velocity to capture the probe.

Determining the optimal elliptic orbital transfer parameters by minimizing C by exhaustive search or analytical methods is infeasible as for each Earth departure date and Mars arrival date combination the Lambert's problem needs to be solved:

being  and  the heliocentric Earth and Mars position vectors and  the areocentric vector position of the spacecraft in an ecliptic reference system.

Genetic algorithms are metaheuristic algorithms that explore different solutions amongst a defined population, finding a possible minimum solution. Lambert's problem shall be solved for each member of the population following the procedure in Gooding (1988), having the orbital elements of the spacecraft's orbit once the solution is achieved. Table 1 shows different simulation scenarios to compare genetic algorithms performance minimizing the cost function C. Figure 1 shows an optimal transfer for launch on 20th July 2020 and arrival on 1st February 2021.

3. Accuracy analysis for the conic matching procedure
To match the heliocentric and the areocentric orbits, the following model is used:

The function f corresponds to the Lambert's problem (2), g and h are given in Battin (1999). Then, the following iterative method is implemented:

that converges when the following condition is met:

which is equivalent to:

Simulations are performed introducing different initial inclination conditions. The method converges, after 3 iterations, to the objective values of the periapsis radius and the inclination of the hyperbolic orbit.

4. Mars arrival optimization
Finally, we present the procedure for minimizing the total impulse needed to capture the probe and to place it in an areostationary orbit:

where  is the capture manoeuvre to avoid the probe leaving its sphere of influence on a flyby trajectory;  and  are the Hoffman transfer maneuvers from the capture orbit to target the areostationary orbit; and  is  the inclination correction manoeuvre to reach the final zero desired inclination.
5. Summary and Conclusions
Optimal dates which provide minimum hyperbolic excess velocity at Earth and the Mars arrival velocity are determined to obtain the optimal transfer orbit from Earth to Mars. Then, an iterative procedure is proposed to match the heliocentric and areocentric orbits at Mars sphere of influence which converges to the objective values to position the spacecraft in the areostationary orbit. Optimal Mars arrival manoeuvres are obtained when hyperbolic periapsis is fixed to be the semi-major axis of the areostationary orbit with the minimum inclination constrained by the areocentric declination of the arrival velocity direction V_∞M. 

Acknowledgements
Part of this research has been funded by the Government of Spain (Project ESP2016-79135-R).

References

Montabone, L. et al, Observing Mars from Areostationary Orbit: Benefits and Applications, 2020, MEPAG

Gooding, R.H., On the solution of Lambert's orbital boundary-value problem, 1988, Royal Aerospace Establishment

Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, 1999, AAIA Educational Series