10000 SERVICIOS PERSONALES

A test spacecraft has been considered with an initial acceleration in GEO of 7.6453e-4 m/s2, m0= 1000 kg and Isp= 2000 s. With these fix values various

iterations have been performed with increasing inclination of the target LLO (one of the most critical parameters) and with different initial guesses on the length of the three phases and of the Lagrange multipliers.

The departing orbit is defined in terms of [aE, eccE, inclE, ωE, ΩE], the

departing true longitude νE is not fixed in advance. For the GEO case, it

reduces to: [aE = 35768 km, eccE = 0, inclE = 0 deg], ωE and ΩE are not

defined neither required. The arrival LLO, in a similar fashion, is defined by means of [aM, eccM, inclM, ωM, ΩM] that are inputs depending on the specific

test case, while the arrival true anomaly νM (or the equivalent anomaly) is

not a-priori fixed, but found by the numerical scheme.

Two typical results are presented in Fig. 6.6, where a “short” transfer towards a circular, 25 deg inclination selenocentric orbit is represented, and in Fig. 6.7 where a ”long” solution to a circular, 45 deg inclined target orbit is shown. Both final lunar orbits have aM = 5000 km but different ΩM.

The “short” transfer of Fig. 6.6 takes 41 days and delivers in the target LLO 885 kg of mass. The Earth escape phase takes 27 days and consumes 92 kg of propellant mass, obviously the largest part. After, the coasting phase takes for approximately one week and finally 6 days and 22 kg are required to reach the 25 deg inclined target LLO.

The “long” transfer of Fig. 6.7 requires 59 days and delivers in the target LLO 872 kg of mass. Thus, in this case, the transfer is 13 kg more expansive than the “short” solution of Fig. 6.6 and its transfer time is 20 days longer. The Earth escape phase takes 29 days and consumes 98 kg of propellant mass, again the largest part for the whole trajectory. After, the coasting phase takes approximately 20 days and finally 9 days and 30 kg are required to reach the 45 deg inclined LLO.

Figure 6.6: Three views and mass evolution of a GEO to LLO“short”solution (aM = 5000 km, eccM = 0, ΩM = 90 deg, inclM = 25 deg).

The main difference of the “short” and “long” solution lies in the length of the ballistic arc, related to the whole transfer time, while the propellant mass consumption is not significantly affected by this difference.

Moreover, if the transfer time is the mission driving factor, also solutions without any ballistic phase can be obtained. As the transfer has been divided a-priori into three phases, simply setting zero the duration of the ballistic phase and removing this from the iteration scheme results in a thrust-thrust solution. Here the first powered phase, still in Earth centered frame, is in- tended for the escape and the second one for the lunar capture. This kind of transfers are usually approximately 10% more expensive in terms of pro- pellant mass fraction if compared with the thrust-coast-thrust strategy here presented.

The current analysis lacks of a global optimization and thus the resulting transfers have to be intended only as locally minimum propellant solutions; a complete different guess of the elements of the control vector can result (for converging simulations) in more/less expensive solutions. This is a classical issue of each local optimization scheme, in particular of the hybrid one. For the present analysis the initial guesses have been generated by means of a grid of initial conditions, in particular on the durations of the three phases that can vary from few weeks up to few months.

In Fig. 6.8 and Fig. 6.9 the evolutions of some interesting parameters are shown. The Lagrange multipliers vary quite a lot during the transfer and the

Figure 6.7: Three views and mass evolution of a GEO to LLO “long” solution (aM = 5000 km, eccM = 0, ΩM = 0 deg, inclM = 45 deg).

magnitude of the one associated with the radial direction is usually two or three orders of magnitude larger than the others, both for the Earth escape and the Moon capture phase. The geocentric position and velocity respect the same color convention of the trajectories in Fig. 6.6 and Fig. 6.7. Also the evolution of the in-plane and out-of-plane thrust angles is shown where they exhibit the typical evolution given by the spirals of low thrust arcs.

Solutions up to 80 deg of inclination of the LLO have been obtained, al- though the numerical sensitivity with respect to the initial guess (especially for the adjoints) increases for increasing inclinations. For increasing inclina- tions of the target LLO, the result of a previous transfer with lower inclM

has been used as initial guess; this resemble a sort of continuation procedure. Its main limit is that in a chaotic system, like the CR3BP, the dynamics can change abruptly but still giving a converging solution with a completely different escape/ballistic/capture phase. This issue is present especially in the electric Moon capture phase and a very small step in inclM is required

to drive the solution to converge with the same dynamics of the initial guess. In Fig. 6.10 an application of such continuation procedure for a Earth- Moon low thrust transfer is shown. It presents a lunar swing-by during the capture phase. This swing-by can be the last part of the powered phase or it can take place during the electric de-spiralling. On the left, Fig.6.10(a), the first planar trajectory used as “initial seed” of the procedure is plotted and on the right, Fig. 6.10(b), the continuation carried out on the LLO inclination

Figure 6.8: Time evolutions of geocentric radius, velocity and thrusting angles for the transfer of Fig. 6.6.

up to inclM = 50 deg is shown. The continuation could still be carried on but

a very small change in the target inclination (continuation step) is required as the Moon swing-by tends to disappear.

From Fig. 6.10(b)), it comes out that the role of the out-of-plane motion results in a slightly more expensive transfer, where only the 0.4% of additional propellant, if compared with the planar solution, is required to reach a 50 deg inclined LLO[88]. In Fig. 6.10(b), moreover, it is shown also the transfer time required to reach the LLO for increasing inclinations. This does not change significantly at least until the dynamics used in the transfer (in this case a short ballistic phase and the lunar swing-by) remain the same.

For a planar case (inclM = 0), it is possible to deliver more than the 92%

of the initial mass in a 5000 km circular LLO. For the three dimensional case this value also depends on the orientation (actually ΩM) of the target LLO

besides from the initial guesses, for the adjoints and for the durations. This method is very effective to design a generic preliminary geocentric- to-selenocentric transfer where the initial and/or the final orbits have to stick some specific mission requirements. Starting from a GEO (or GEO-like) orbit up to a circular lunar orbit, thousands of kilometers above its surface, requires a transfer time that can span a quite large range. It can vary from slightly

Figure 6.9: Time evolutions of geocentric radius, velocity and thrusting angles for the transfer of Fig. 6.7.

more than one month to more than three months and depends, in particular, on the duration of the coasting phase. A longer ballistic phase allows also exploring different regions of the geospace while shorter and slightly more expensive transfers are obtainable simply considering a thrusting-thrusting solution without any coasting phase. The Earth escape phase takes at least 25 days while the moon capture can be obtained in at least 5 days.

Finally, the same approach can also be used to design low thrust transfers between two geocentric/selenocentric orbits. However, in this context, the resulting control laws given by this approach are not very accurate, especially when the starting and the ending orbits differ for a high ∆i and a large number of revolutions is required.