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�0���gT(17m�vt2�5Hl��ń��Y�E����ޅ�%�,0j�^I! \end{equation}, \begin{eqnarray} \end{eqnarray}, \begin{eqnarray} \sigma _{11}&=&\frac{\beta _{12}\beta _{21}-\beta _{11}\beta _{22}}{\beta _{21}^2+\beta _{22}^2}, endobj However, they ignored the drag force, although Eapen & Sharma (2014) have studied the halo orbits at the Sun–Mars L1 Lagrangian point in the photogravitational restricted three-body problem and have found that as the radiation pressure increases, the transition from the Mars-centric path to the heliocentric path is delayed. \end{eqnarray}, \begin{eqnarray} <>stream β = 0), then the trajectory of the Lissajous orbit completes one period approximately at t = 3.0, whereas if β increases with sw = 0.35, it does not complete its period at the same time. \nu _{35}&=&\frac{3}{2}(c_{4}+d_{4})-\frac{2(1+sw)\beta (1-\mu )\mu l}{c D_{1}^6}, 230 0 obj (2013). \end{equation}, \begin{equation} \nu _{34}&=&\frac{3}{2}(c_{4}+d_{4})-\frac{6(1+sw)\beta (1-\mu )\mu l}{cD_{1}^6}, (-1)^{{(n-1)}/{2}} \cos 3\tau _{1}, & n=1,3 \nonumber \end{array}\right.\\ endobj &&+\,\alpha _{26}\sin (2\tau _{2}+\tau _{1})+\alpha _{27}\sin (2\tau _{2}-\tau _{1})+\alpha _{28}\sin 3\tau _1, \end{equation}, In order to calculate the Lagrangian equilibrium points, we solve equations (, \begin{eqnarray} \end{equation*}, \begin{equation} You must log in or register to reply here. \delta _{2}&=&\frac{\nu _{22}A_{x}^2-(\nu _{25}+\nu _{23}\kappa )\kappa A_{x}^2\nu _{10}}{2}-2\lambda ^2\nu _{23}\kappa ^2 A_{x}^2, (-1)^{({n-1})/{2}}(\alpha _{34}\cos 3\tau _1-\alpha _{36}\sin 3\tau _1), n=1,3 \end{array}\right.\right].\nonumber \\ z^{\prime \prime }+\lambda ^2 z&=&0. When β increases, then the trajectory shrinks. endstream 353 0 obj 349 0 obj &&{{-\,(\nu _{11}+\lambda ^2) [\alpha _{11}+2\omega _{2}\lambda ^2 A_{x}(2\kappa -1)] +2\lambda [\alpha _{22}}}\nonumber \\ <>stream She obtained orbits that increase in size when increasing the mass parameter μ. Clarke (2003, 2005) discussed a discovery mission concept that utilizes occultations from a lunar halo orbit by the Moon to enable detection of terrestrial planets. <>stream 337 0 obj Halo orbit at n = 0 and n = 2 at β = 0.18. <>stream &&\qquad+\,\left. \end{eqnarray}, \begin{eqnarray} \end{eqnarray}, \begin{eqnarray} 132 0 obj \end{eqnarray}, \begin{eqnarray} For more information on halo orbits, we refer to the three-dimensional periodic halo orbits near the collinear Lagrangian points in the restricted three-body problem obtained by Howell (1984). a^*&=&\frac{(1-\beta )(1-\mu )}{r_{1}^{*3}}+\frac{\mu }{r_{2}^{*3}}, \end{equation}, \begin{eqnarray} \end{eqnarray}, \begin{eqnarray} x���� x�S�*B�.C 4T0���31T04�37W��ҏ�42Q04TI���350�4�F�� ;�@��S�`�}P�E+y�EU+�q��9����O����Ӛ���}��HK�瀵��ܿ�sC�cJ٣��JH5_�>F�.� \left\lbrace \begin{array}{@{}l@{\quad }l@{}}(-1)^{{n}/{2}}(\alpha _{34}\sin 3\tau _1+\alpha _{36}\cos 3\tau _1),n=0,2\\ &&+\,\frac{(1+sw)\beta (1-\mu )}{c}\left[ \frac{\mu l(1-\gamma )}{D_{1}^4}+\frac{l}{D_{1}^2}\right],