In-plane components
6.3 Variable transfer orbits: an overview
Figure 6.12: Primer vector profile, in common to all departure and arrival orbits of Fig.
6.11.
dimensions of the search space: BC in direction of ∆Vs (α and β), eccentricity of the transfer orbit and BC in position (ν0 and νf). Specifically, for this case the values are:
α = 30 deg, β = 60 deg, e = 0.5, ν0 = 5 deg and νf = 180 deg.
The semi-major axes of the transfer orbit is allowed to vary from 0.01 DU to 1.0 DU, whereas the magnitudes of ∆V0,f go from 1% to 80% of the magnitudes of v0,f (see Eq. 4.4). On the top line of Fig. 6.12 the Keplerian elements of the possible departure orbits are plotted, whereas on the bottom line the equivalent parameters for the arrival orbits are shown.
With the chosen intervals of aT and ∆V0,f, the eccentricities of the departure orbits result in a possible variation from roughly 0.4 to 0.8, whereas the ones of the arrival orbits have higher values (0.5 to ≈1). On the other hand, the span of values of the semi-major axes of the two orbits are roughly comparable, but they vary differently with respect to aT and ∆V0,f.
The main important conclusion of this result is that all kind of departure and arrival orbits summarised in Fig. 6.11 meet the same optimisation objective: in fact they all correspond to the same primer vector profile (shown in Fig. 6.12), that in this case is optimal.
6.3 Variable transfer orbits: an overview
In light of the novel results obtained in the context of the primer vector theory, the research presented in this Chapter, related to the application of the model, has been focused on fixed transfer orbits. To be thorough, an overview on variable transfer orbits is presented in this section.
In order to obtain a set of different transfer trajectories, that do not share the
6.3. Variable transfer orbits: an overview
same optimality properties, the discriminating factor is their eccentricity.
In fact, as demonstrated and underlined several times along the thesis, the primer vector problem is independent of the semi-major axis of the transfer trajectory. Whereas, the other two variables related to the transfer orbit, which determine the dimension of the search space, are the true anomalies at the boundaries. However, having different ν0 or νf does not actually vary the orbit itself; their choice influences the optimality of the transfer arc, as discussed in both Chapters 4 and 5.
Therefore, this section aims to present an overview of how the profile of the primer vector, and therefore optimality of the transfer arc, can vary for different values of the eccentricity of the transfer orbit.
This is shown in Fig. 6.13 and, for this example, the fixed parameters are ν0 =0 deg and νf =160 deg; whereas, for the first rows of plots β is fixed (60 deg) and for the second one α is fixed (60 deg). The eccentricity of the transfer orbits include all possible elliptical trajectories, in fact it goes from 0.01 to ≈0.9.
For the fixed β orbits, the profiles refer to sub-optimal or non-optimal (top-right) trajectories, whereas in the cases where α is fixed, the behaviour tends to be optimal almost everywhere.
Overall, the results show very different behaviour even if a sort of orderliness, as the one of Fig. 6.7, can be observed. In fact, it can be seen how in the neighbourhood of a specific eT, the primer vector profile does not have a sudden variation with respect to other transfer orbit’s eccentricities, but it tends to change systematically.
As a final remark it can be confirmed that transfer orbits with different eccentricities have, in general, different primer vector profiles and, therefore, different optimality’s characteristics.
6.3. Variable transfer orbits: an overview
Figure6.13:Primervectorprofilevspropagationstepfordifferenttransferorbitswithvaryingvalueseccentricity. Thelightgrayplanerepresentsp=1,boundarybetweenoptimalandnon-optimalsolutions.Fixedparametersare:ν=0deg,νf=160deg.
6.4. Summary
6.4 Summary
This Chapter complements the analysis of Chapter 5. In fact, it conducts a study on the in-plane components of the primer vector.
The decoupling between the out-of-plane and in-plane components allows to find an approximate analytic solution through a similarity with the Hamiltonian of Hill’s problem. The exploitation of this property brings to the usage of the Palmer coordinates which simplify the mathematical structure of the differential equations.
The solution of the primer vector equation for the in-plane components is found as an analytic approximation, in the form of multiplication of matrices. Given the boundary conditions of the problem, the primer vector can be then fully propagated along the transfer arc.
It has been also demonstrated that the Palmer coordinates, the primer vector and its derivative are all independent of the semi-major axis of the transfer orbit.
For the circular transfer case it has been demonstrated that Hill’s solution is also a solution for the primer vector problem.
Furthermore, fixing the transfer orbit, it has been established that the dimen-sions of the search space can be reduced to 5. An overview of the three dimensional case has been shown and then the planar case has been examined through different examples.
At first the systematic evolution of the primer vector profile for a case with variable boundary conditions has been presented; later the ‘optimality maps’ have been intro-duced. Within this context, two approaches have been used, in order to examine the likelihood of the optimality of a specific set of transfers trajectories.
A parametrisation of the semi-major axis of the transfer orbit and magnitudes of the
∆Vs has been shown.
To conclude the analysis of the planar case with fixed transfer orbit, a set of departure and arrival trajectories which share the same optimality conditions has been presented.
Finally, a brief overview to the scenario with variable transfer orbit has been shown.