Three-Dimensional QSOs

(a)C0= 2.9547

(b)C0= 2.9550

(c)C0= 2.9553

Figure 4.11.FLI Maps for 3D QSOs in the Mars-Phobos system with initial conditionsx0 = 0and tan- gential and vertical velocity. The stability regions decrease with the increase of the modified Jacobi integral.

(a)xvs.y (b)xvs.z

(c)yvs.z

Figure 4.12.Trajectory of a 3D QSO with initial conditionsy0 =−100km,x˙0 =−15m/s, andz˙0 = 14.5 m/s (C0= 2.9553). The orbit is represented in the three planes:x-y,x-z, andy-z.

z0Vs. y˙0

Whereas the previous three-dimensional QSOs were obtained by the means of an initial vertical ve- locity, the injection into 3D QSOs can also be performed with an initial heightz₀. The entry in a 3D QSO by the means of an initial height has implications on its stability that are analyzed in this section.

We are interested in the analysis of the variation of the initial height with the variation of the velocity

˙

y0, parameter that so far appears to be the most sensible to initial variations. The initial distancey0 is set to 100 km,x˙0is derived from the modified Jacobi integral and the remainder of the parameters are set to zero.

The first remark upon observation of the FLI maps on figure 4.13 is that after a small value of the initial height, positive or negative (the map is symmetric in thezcoordinate), the range of values ofy˙0starts to decrease until the end of the stability region near 100 km. This suggests that three-dimensional QSOs with large amplitudes in thezdirection are more sensible to variations of velocity in theydirection.

(a)C0= 2.9547

(b)C0= 2.9550

(c)C0= 2.9553

Figure 4.13.FLI Maps for 3D QSOs in the Mars-Phobos system with initial conditionsx0 = 0,y0 = 100 km,z˙= 0, and withz0andy˙0as varying parameters. The velocity onxis derived fromC0.

z0Vs. z˙0

There are three distinct initial situations that result in a three-dimensional QSO: the presence of an initial vertical velocity, an initial height, or the conjunction of both. We analyze in which of these larger inclinations can be obtained. In figure 4.14, the FLI maps for the variation of the initial height and initial vertical velocity are plotted where one can distinguish the different relations between positive and negative velocities and heights. Notice that forC0= 2.9553, the stability region is smaller and the relation between initial vertical velocity and initial height is symmetric.

We are interested in analyzing which of the described initial situations lead to larger values of the ratio between the amplitudes of the motion in the x-y plane and in thez direction,q = γ/α. In figure 4.15 sample QSOs of the limiting cases (largerq’s found) for each initial situation that lead to 3D QSOs are presented in thex-zplane. The plotted trajectories indicate that similar values are obtained for all three cases although the case with initial vertical speed and no initial height seems to present slightly larger q’s.

Three-dimensional QSOs present a different behavior for large values of the amplitude in the x-y plane, α. Figure 4.16 presents an example of an orbit of large amplitude α that along other similar orbits suggest that these orbits present an almost constant angle β, unlike small amplitude QSOs. As discussed in section 3.3.2, the ratioq is more restricted forβ =±π/2and it is expected to find larger values ofqforβ = 0orβ=π.

Figure 4.16(c) supports the hypothesis that, when analyzing the different initial situations that lead to 3D QSOs, the maximumqis achieved for an angleβclose to 0 corresponding to an orbit with both initial height and initial vertical velocity. Note that this maximum valueqmax≈0.75is farther from the predicted value predicted of 0.96 than the value ofq_maxobtained for small amplitude QSOs.

In the case with no initial vertical velocity for whichβ ≈π/2is obtained, which is expected to be the worst-case scenario, the result isq_max≈0.25, a much smaller value. This supports the hypothesis that larger values of the ratioq=γ/αare obtained for anglesβnear zero orπ. In QSOs with small amplitude αthe QSO plane rotates and, in a QSO orbiting for a long enough period, the angleβ assumes values in its whole domain (0to2π) and this analysis does not apply to these orbits.

From the definition of the osculating elements in section 3.2.4, recall thatβ =ψ−φand notice that in the negative y semi-negative axis we haveφ =−π/2whereas in thexaxisφ =πorφ= 0. When the spacecraft is injected in a QSO in thex-axis, or in any initial position other thanx0= 0,β might not present the same behavior and this analysis must be repeated. For small amplitude orbits,β varies in its whole domain and such analysis is not required.

Despite the advantage of obtaining large values of the ratioqwithβ = 0, this angle also implies that whenz = 0, we gety ≈ ±2α±δ_y. In contrast, for β =π/2 we getx≈ ±α±δ_x whenz = 0, which represents a smaller minimum distance. In space mission design, it might be in the designer’s best interest to have the spacecraft pass as close as possible to the moon, thus, the choice of the angleβ represents a trade-off between maximum QSO inclination and minimum distance to the moon.

(a)C0= 2.9547

(b)C0= 2.9550

(c)C0= 2.9553

Figure 4.14.FLI Maps for 3D QSOs in the Mars-Phobos system with initial conditionsx0 = 0,y0 = 100 km,z˙= 0, and withz0andy˙0as varying parameters. The velocity onxis derived fromC0.

(a)z0= 0km,z˙= 14.5m/s (b)z0= 50km,z˙= 0m/s

(c)z0= 30km,z˙= 10m/s

Figure 4.15.Trajectories of 3D QSOs in thex-zplane with initial conditionsy0 =−100km,y˙0 = 0m/s, andx˙0computed from the invariant relation (C0= 2.9553).

(a)z0= 0km,z˙= 22m/s (b)z0= 57km,z˙= 0m/s

(c)z0= 50km,z˙= 32m/s

Figure 4.16.Trajectories of 3D QSOs in thex-zplane with initial conditionsy0 =−100km,y˙0 = 0m/s, andx˙0computed from the invariant relation (C0= 2.9547).