Evolution of the $\mathcal {L}_1$ halo family in the radial solar sail circular restricted three-body problem

Abstract

We present a detailed investigation of the dramatic changes that occur in the $\mathcal {L}_1$ halo family when radiation pressure is introduced into the Sun–Earth circular restricted three-body problem (CRTBP). This photo-gravitational CRTBP can be used to model the motion of a solar sail orientated perpendicular to the Sun-line. The problem is then parameterized by the sail lightness number, the ratio of solar radiation pressure acceleration to solar gravitational acceleration. Using boundary-value problem numerical continuation methods and the AUTO software package (Doedel et al. in Int J Bifurc Chaos 1:493–520, 1991) the families can be fully mapped out as the parameter $\beta$ is increased. Interestingly, the emergence of a branch point in the retrograde satellite family around the Earth at $\beta \approx 0.0387$ acts to split the halo family into two new families. As radiation pressure is further increased one of these new families subsequently merges with another non-planar family at $\beta \approx 0.289$, resulting in a third new family. The linear stability of the families changes rapidly at low values of $\beta$, with several small regions of neutral stability appearing and disappearing. By using existing methods within AUTO to continue branch points and period-doubling bifurcations, and deriving a new boundary-value problem formulation to continue the folds and Krein collisions, we track bifurcations and changes in the linear stability of the families in the parameter $\beta$ and provide a comprehensive overview of the halo family in the presence of radiation pressure. The results demonstrate that even at small values of $\beta$ there is significant difference to the classical CRTBP, providing opportunity for novel solar sail trajectories. Further, we also find that the branch points between families in the solar sail CRTBP provide a simple means of generating certain families in the classical case.

Appendix

In this appendix we present examples of the families discussed in Sects. 4–6. Examples of the L1, C2 and RS families for several values of $\beta$ are shown in Figs. 12, 13 and 14. Examples of the H1, HR, H1B, H1C and H1R families for the Earth–Sun mass ratio in the RSCRTBP are shown in Figs. 15 and 16. A three-dimensional plot of the HR family for the Earth–Moon CRTBP ($\mu =0.01215$, $\beta =0$) is shown in Fig. 17.

Fig. 12

Examples of the L1 family for different values of $\beta$ in the Earth–Sun RSCRTBP, with branch points marked in blue. The location of the Earth ($m_2$) is shown as a black dot

Fig. 13

Examples of the C2 family for different values of $\beta$ in the Earth–Sun RSCRTBP, with branch points marked in blue. The location of the Earth ($m_2$) and Sun ($m_1$) are shown as black dots. The families continue out to infinity in all cases. The C2 family remains relatively unchanged as the sail lightness number increases

Fig. 14

Examples of the RS family for different values of $\beta$, with branch points marked in blue, in the Earth–Sun RSCRTBP. The locations of the Earth ($m_2$) and Sun ($m_1$) are shown as black dots. Each family ends in a collision with the Sun. A close up of the two extra branch points near the Earth is also shown for the $\beta =0.05$ case

Fig. 15

The north branch of the H1 family in the Earth–Sun CRTBP (i.e. $\beta =0$) from a the L1 branch point and b the C2 branch point, and c the south branch of the HR family, shown in the $x$–$z$ plane. Order-0 instability orbits are shown in green, others in gray. Branching orbits are shown in blue, period-doubling bifurcations in purple and folds in yellow. The positions of the Earth and Sun are also marked. Note that different scales are used in each plot

Fig. 16

Examples of the H1, H1B, H1C and H1R families shown in the $x$–$z$ plane for various values of $\beta$ in the Earth–Sun RSCRTBP. Order-0 instability orbits are shown in green, others in gray. Branching orbits are shown in blue, period-doubling bifurcations in purple, folds in yellow and the approximate locations of Krein collisions in red. The positions of the Earth and Sun are also marked. Note that different scales are used in each plot

Fig. 17

The southern branch of the HR family in the classical CRTBP for the Earth–Moon mass ratio