^{1}

^{2}

Evolution of periodic orbits in Sun-Mars and Sun-Earth systems are analyzed using Poincare surface of section technique and the effects of oblateness of smaller primary on these orbits are considered. It is observed that oblateness of smaller primary has substantial effect on period, orbit’s shape, size and their position in the phase space. Since these orbits can be used for the design of low energy transfer trajectories, so perturbations due to planetary oblateness has to be understood and should be taken care of during trajectory design. In this paper, detailed stability analysis of periodic orbit having three loops is given for A_{2} = 0.0001.

A low-energy transfer trajectory [

However, it is to be noted that the classical approaches to spacecraft design are like Hohmann transfer for Apollo Moon landings and swing bys of outer planets for voyager. But these missions were costly in terms of fuel. A very high fuel requirement for space missions may cause it infeasible. A new class of low energy trajectories has been introduced recently in which low fuel burn is required to control the trajectory from initial position to the targeted final position. A proper understanding of low-energy trajectory can be seen in [

The planar restricted three body problem (PRTBP) describes the motion of a body of negligible mass compared to two massive bodies, called primaries that are moving in circular orbits due to their mutual gravitational attraction. The two primaries could be the Sun and Earth, or the Sun and Mars, or any other pair of heavenly bodies that can produce considerable gravitational field. We take the total mass of the primaries normalized to 1. Mass of bigger primary is taken as 1 − μ and that of smaller primary is μ. The two main bodies rotate in the plane in circular orbits counter clockwise about their common center of mass and with angular velocity which is also normalized to 1. The third body, the spacecraft, has an infinitesimal mass and is free to move in the plane. Generalization to the 3-dimensional problem is of course important, but many of the essential dynamics can be captured well with the planar model. [

[

As per Kolmogorov-Arnold-Moser (KAM) theory (Moser, 1966), a fixed point on the Poincare surface of section represents a periodic orbit in the rotating frame, and the closed curves around the point correspond to the quasi-periodic orbits. [

A large number of periodic orbits were generated by [

Recently, [

Periodic orbits of spacecraft around two primaries are used to construct low energy trajectory. [_{2} and Jacobi constant C for Sun-Mars and Sun-Earth system. It has been found that A_{2} and C has substantial effect on the position in phase space, shape and size of the orbits and hence must be considered during low energy trajectory design.

Let m_{1} and m_{2} be two masses of first primary and second primary bodies. Mass of first primary is greater than mass of second primary. The perturbed mean motion n of second primary body is given by,

where

Here A_{2} is oblateness coefficient of second primary. r_{e} and r_{p} represent equatorial and polar radii of second primary and R is the distance between two primaries. The unit of mass is chosen equal to the sum of the primary masses and the unit of length is equal to their separation. The unit of time is such that the Gaussian constant of gravitation is unity in the unperturbed case. The usual dimensionless synodic coordinate system Oxy is used to express the motion.

Choose a rotating coordinate system with origin at the center of mass, the primaries lie on the x-axis at the points (−μ, 0) and (1 − μ, 0), respectively, where mass factor

where

Here

and

Equations (3) and (4) lead to the first integral

where C is Jacobi constant of integration given by

These equations of motion are integrated in (x, y) variables using a Runge-Kutta Gill fourth order variable or fixed step-size integrator. The initial conditions are selected along the x-axis. By defining a plane, say y = 0, in the resulting three dimensional space the values of x and

We shall consider two systems, the Sun-Mars system and the Sun-Earth system. The mass of Sun, Earth and Mars considered in the study are 1.9881 × 10^{30} kg, 5.972 × 10^{24} kg and 6.4185 × 10^{23} kg, respectively, [_{2} = 2.42405 × 10^{−12} and A_{2} = 5.21389 × 10^{−13} respectively. Though these values of oblateness are negligible, they definitely affect orbits with close proximity. In order to study the effect of oblateness on periodic orbit around both primaries, we take different values of oblateness that can make observable changes in different parameters. For a given A_{2}, selection of C is not arbitrary. By solving Equation (8) w.r.t.

We have studied the effect of oblateness on the location and period of Sun-Mars system for different values of Jacobi constant C using PSS. _{2}, C) is (0.0005, 2.93) by taking value of x from the interval [0.8, 1] with interval of x differencing as 0.001. Also, time span t = 10,000 time units and interval of time differencing is taken as 0.001. So, for each x equations of motions are integrated using Runge-Kutta-Gill method. Each solution is plotted as a point in

In a similar way, we can obtain PSS for Sun-Earth which is shown in _{2}, C) given by (0.0005, 2.93). Our aim is to make a comparative study of the effect of oblateness on different parameters of the orbits of Sun-Earth and Sun-Mars systems using PSS technique. Mass factor μ of Sun-Earth is greater than Sun-Mars.

For Sun-Mars system, the numerical values of location of periodic orbit and left end

A_{2} | Maximum value of C | Value of C greater than maximum value | Excluded region (where | Size of excluded region |
---|---|---|---|---|

0.00001 | 3.00 | 3.001 | 0.984 to 0.999 | 0.016 |

0.00005 | 3.00 | 3.001 | 0.984 to 0.998 | 0.015 |

0.0001 | 3.00 | 3.001 | 0.985 to 0.997 | 0.013 |

0.0005 | 3.001 | 3.002 | 0.981 to 0.995 | 0.015 |

A_{2} | Maximum value of C | Value of C greater than maximum value | Excluded region (where | Size of excluded region |
---|---|---|---|---|

0.00001 | 3.00 | 3.001 | 0.988 to 0.992 | 0.005 |

0.00005 | 3.001 | 3.002 | 0.978 to 0.995 | 0.018 |

0.0001 | 3.001 | 3.002 | 0.979 to 0.993 | 0.015 |

0.0005 | 3.002 | 3.003 | 0.976 to 0.990 | 0.015 |

(L) and right end (R) of corresponding island for C = 2.93, 2.94, 2.95, 2.96 and for oblateness A_{2} = 0.00001, 0.00005, 0.0001 and 0.0005 are displayed in _{2} in the location of periodic orbits and left end (L) and right end (R) of corresponding island for the Sun-Earth system are studied and the numerical estimates of the changes are displayed in

The periodic orbits starting from single-loop to five loops in the Sun-Mars and Sun-Earth systems for A_{2} = 0.0005 and C = 2.93 are shown in Figures 3(a)-(j). It can be observed, from

NO. OF LOOPS | C | A_{2} = 0.00001 | A_{2} = 0.00005 | A_{2} = 0.0001 | A_{2} = 0.0005 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Location (x) | L | R | Location (x) | L | R | Location (x) | L | R | Location (x) | L | R | ||

1 | 2.96 | 0.98295 | 0.9829 | 0.983 | 0.98285 | 0.9827 | 0.983 | 0.98272 | 0.9825 | 0.9829 | 0.9816 | 0.9815 | 0.9817 |

2.95 | 0.9679 | 0.9678 | 0.968 | 0.9678 | 0.9676 | 0.968 | 0.96765 | 0.9669 | 0.9686 | 0.9666 | 0.966 | 0.967 | |

2.94 | 0.95324 | 0.953 | 0.9535 | 0.95314 | 0.953 | 0.9533 | 0.953 | 0.953 | 0.953 | 0.95196 | 0.9519 | 0.952 | |

2.93 | 0.93896 | 0.9389 | 0.939 | 0.93887 | 0.9387 | 0.939 | 0.93875 | 0.9386 | 0.9389 | 0.93772 | 0.9376 | 0.9378 | |

2 | 2.96 | 0.94149 | 0.9406 | 0.9425 | 0.94136 | 0.9411 | 0.9414 | 0.94119 | 0.941 | 0.9414 | 0.93985 | 0.9397 | 0.94 |

2.95 | 0.92251 | 0.9224 | 0.9226 | 0.92239 | 0.9223 | 0.9225 | 0.92223 | 0.922 | 0.9225 | 0.92096 | 0.9209 | 0.921 | |

2.94 | 0.9045 | 0.9042 | 0.905 | 0.90438 | 0.9041 | 0.905 | 0.90424 | 0.904 | 0.9045 | 0.90302 | 0.903 | 0.903 | |

2.93 | 0.88734 | 0.887 | 0.8878 | 0.88722 | 0.887 | 0.8874 | 0.88708 | 0.887 | 0.8872 | 0.88592 | 0.8858 | 0.886 | |

3 | 2.96 | 0.91529 | 0.915 | 0.9156 | 0.91514 | 0.915 | 0.9153 | 0.91496 | 0.9149 | 0.915 | 0.91348 | 0.9133 | 0.9137 |

2.95 | 0.89425 | 0.894 | 0.8945 | 0.89412 | 0.894 | 0.8942 | 0.89395 | 0.8939 | 0.894 | 0.89258 | 0.8922 | 0.893 | |

2.94 | 0.87463 | 0.8743 | 0.8748 | 0.87449 | 0.8743 | 0.8747 | 0.87433 | 0.874 | 0.8747 | 0.87305 | 0.873 | 0.8731 | |

2.93 | 0.85616 | 0.856 | 0.8563 | 0.85604 | 0.856 | 0.8561 | 0.85588 | 0.8558 | 0.856 | 0.85467 | 0.8543 | 0.855 | |

4 | 2.96 | 0.89703 | 0.897 | 0.8971 | 0.89688 | 0.8968 | 0.897 | 0.89668 | 0.8965 | 0.8969 | 0.89515 | 0.895 | 0.8953 |

2.95 | 0.8749 | 0.8748 | 0.875 | 0.87476 | 0.8745 | 0.875 | 0.87458 | 0.874 | 0.875 | 0.87316 | 0.873 | 0.8733 | |

2.94 | 0.85447 | 0.8544 | 0.8545 | 0.85433 | 0.854 | 0.8547 | 0.85417 | 0.8541 | 0.8543 | 0.85284 | 0.8527 | 0.853 | |

2.93 | 0.8354 | 0.835 | 0.836 | 0.83528 | 0.835 | 0.8357 | 0.83512 | 0.835 | 0.8353 | 0.83388 | 0.8338 | 0.834 | |

5 | 2.96 | 0.88363 | 0.8832 | 0.8834 | 0.88347 | 0.883 | 0.8838 | 0.88328 | 0.883 | 0.8836 | 0.8817 | 0.8814 | 0.882 |

2.95 | 0.8609 | 0.8608 | 0.861 | 0.86075 | 0.8605 | 0.861 | 0.86057 | 0.8602 | 0.8609 | 0.85912 | 0.859 | 0.8593 | |

2.94 | 0.84004 | 0.84 | 0.8401 | 0.83991 | 0.8398 | 0.84 | 0.83974 | 0.8394 | 0.8401 | 0.83839 | 0.8383 | 0.8385 | |

2.93 | 0.82068 | 0.8204 | 0.821 | 0.82056 | 0.8205 | 0.8206 | 0.8204 | 0.82 | 0.8208 | 0.81914 | 0.819 | 0.8193 |

NO. OF LOOPS | C | A_{2} = 0.00001 | A_{2} = 0.00005 | A_{2} = 0.0001 | A_{2} = 0.0005 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Location (x) | L | R | Location (x) | L | R | Location (x) | L | R | Location (x) | L | R | ||

1 | 2.96 | 0.983 | 0.983 | 0.983 | 0.9829 | 0.9828 | 0.983 | 0.9828 | 0.9828 | 0.9828 | 0.9817 | 0.9817 | 0.9817 |

2.95 | 0.9679 | 0.9678 | 0.968 | 0.96782 | 0.9676 | 0.968 | 0.9677 | 0.9674 | 0.968 | 0.9666 | 0.9662 | 0.967 | |

2.94 | 0.95325 | 0.953 | 0.9535 | 0.95315 | 0.953 | 0.9533 | 0.95304 | 0.9530 | 0.9531 | 0.952 | 0.952 | 0.952 | |

2.93 | 0.939 | 0.939 | 0.939 | 0.9389 | 0.9388 | 0.939 | 0.93877 | 0.9385 | 0.939 | 0.93775 | 0.9375 | 0.938 | |

2 | 2.96 | 0.94155 | 0.941 | 0.942 | 0.9414 | 0.941 | 0.9418 | 0.94125 | 0.941 | 0.9415 | 0.9399 | 0.9398 | 0.94 |

2.95 | 0.92255 | 0.9221 | 0.923 | 0.92244 | 0.922 | 0.923 | 0.92226 | 0.922 | 0.9225 | 0.921 | 0.921 | 0.921 | |

2.94 | 0.90455 | 0.904 | 0.9051 | 0.90442 | 0.904 | 0.9048 | 0.90426 | 0.904 | 0.9045 | 0.90305 | 0.903 | 0.9031 | |

2.93 | 0.88737 | 0.887 | 0.8877 | 0.88725 | 0.887 | 0.8875 | 0.8871 | 0.887 | 0.8872 | 0.88595 | 0.8859 | 0.886 | |

3 | 2.96 | 0.91535 | 0.915 | 0.9157 | 0.9152 | 0.915 | 0.9154 | 0.915 | 0.915 | 0.915 | 0.91354 | 0.9131 | 0.9141 |

2.95 | 0.8943 | 0.894 | 0.8946 | 0.89416 | 0.894 | 0.8943 | 0.894 | 0.894 | 0.894 | 0.89262 | 0.8922 | 0.893 | |

2.94 | 0.87465 | 0.8743 | 0.875 | 0.87453 | 0.8741 | 0.875 | 0.87438 | 0.874 | 0.8748 | 0.87309 | 0.873 | 0.8732 | |

2.93 | 0.8562 | 0.8554 | 0.857 | 0.85608 | 0.856 | 0.8562 | 0.85592 | 0.8558 | 0.856 | 0.8547 | 0.8544 | 0.855 | |

4 | 2.96 | 0.8971 | 0.897 | 0.8972 | 0.89695 | 0.8969 | 0.897 | 0.89675 | 0.8965 | 0.897 | 0.8952 | 0.895 | 0.8954 |

2.95 | 0.87495 | 0.8749 | 0.875 | 0.87482 | 0.8746 | 0.875 | 0.87464 | 0.8743 | 0.875 | 0.87321 | 0.873 | 0.8734 | |

2.94 | 0.8545 | 0.854 | 0.855 | 0.85438 | 0.854 | 0.8548 | 0.85421 | 0.854 | 0.8544 | 0.85289 | 0.8528 | 0.853 | |

2.93 | 0.83544 | 0.835 | 0.8359 | 0.83532 | 0.835 | 0.8356 | 0.83516 | 0.835 | 0.8353 | 0.83392 | 0.8338 | 0.834 | |

5 | 2.96 | 0.8837 | 0.8834 | 0.884 | 0.88353 | 0.883 | 0.8841 | 0.88334 | 0.883 | 0.8837 | 0.88175 | 0.8815 | 0.882 |

2.95 | 0.86095 | 0.8609 | 0.861 | 0.8608 | 0.8606 | 0.861 | 0.86062 | 0.8602 | 0.861 | 0.85918 | 0.859 | 0.8594 | |

2.94 | 0.8401 | 0.84 | 0.8402 | 0.83995 | 0.8399 | 0.84 | 0.83978 | 0.8396 | 0.84 | 0.83844 | 0.838 | 0.8389 | |

2.93 | 0.82072 | 0.8204 | 0.821 | 0.8206 | 0.82 | 0.821 | 0.82044 | 0.82 | 0.8209 | 0.81918 | 0.819 | 0.8194 |

around the second primary (Mars or Earth) in addition to orbiting both primaries. Further the secondary body is closest to Mars or Earth in the single-loop closed orbit. Such orbits may be useful in the study of different aspects of both primaries. In many models available in literature not many closed orbits possess this kind of nature. Further the position of the orbit in the case of odd number of loops approaches the first primary, namely the Sun. This is true in the case of even number of loops also. It can be observed that the period of the orbit remains unchanged due to change in oblateness or mass factor, but period of the orbit increases with increment in number of loops.

We have studied the variation of position of periodic orbits around Sun-Mars and Sun-Earth system due to the variation in oblateness and Jacobi constants C. In

Similar kind of conclusion can be drawn from

We have studied the effect of A_{2} and C on the position of the orbits having loops varying from 1 to 5 for both Sun-Mars and Sun-Earth systems. The results of these observations for 5 loops closed periodic orbit in both Sun-Mars and Sun-Earth systems are shown in

From

We have analyzed stability of periodic orbits from loop 1 to 5 for A_{2} = 0.00001, 0.00005, 0.0001 and 0.0005. Since stability behavior is similar for all these orbits, in this paper stability analysis for three loops orbit corresponding to A_{2} = 0.0001 is given. _{2} = 0.0001 for three loop orbits. The left and right tips of the island are plotted by red and green curves, respectively. From

body required few amount of fuel than Hohmann transfer. _{2} = 0.0001. It can be observe that there are two separatrices at C = 2.95 and 2.96 where stability of the periodic orbit is zero as the size of the island is zero. For C = 2.94 we get maximum stability which is 0.0008. _{2} = 0.0001 which is 0.0008 whereas

seen that size of this island is zero. _{2} = 0.0001.

In _{1} and D_{2} are the distance of secondary body from Mars and Sun respectively. V is in kms^{−1}, D_{1} and D_{2} are

in km. Similar notations are used in _{1}.

V can be obtained using Equation (9).

The conversion from units of distance (I) and velocity (J) in the normalized dimension less system to the dimensionalized system is given by,

Distance

Velocity

where L is the distance between the centers of both primaries in km. O is the orbital velocity of second primary around first primary [^{−1} and 29.78 km∙sec^{−1} respectively.

It can be observed from _{1} increase while D_{2} decreases. For a given C and given

No. of loop | C | A_{2} = 0.00001 | A_{2} = 0.00005 | A_{2} = 0.0001 | A_{2} = 0.0005 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

x | V (km∙sec^{−1}) | D_{1} (10^{7} km) | D_{2} (10^{8} km) | x | V (km∙sec^{−1}) | D_{1} (10^{7} km) | D_{2} (10^{8} km) | x | V (km∙sec^{−1}) | D_{1} (10^{7} km) | D_{2} (10^{8} km) | x | V (km∙sec^{−1}) | D_{1} (10^{7} km) | D_{2} (10^{8} km) | ||

1 | 2.96 | 0.98295 | 28.52 | 0.3886 | 2.2405 | 0.98285 | 28.53 | 0.3909 | 2.2403 | 0.98272 | 28.53 | 0.3938 | 2.2400 | 0.9816 | 28.54 | 0.4194 | 2.2374 |

2.95 | 0.9679 | 28.84 | 0.7316 | 2.2062 | 0.9678 | 28.85 | 0.7339 | 2.2060 | 0.96765 | 28.85 | 0.7373 | 2.2056 | 0.9666 | 28.86 | 0.7613 | 2.2032 | |

2.94 | 0.95324 | 29.16 | 1.0658 | 2.1728 | 0.95314 | 29.16 | 1.0681 | 2.1725 | 0.953 | 29.16 | 1.0713 | 2.1722 | 0.95196 | 29.18 | 1.0950 | 2.1698 | |

2.93 | 0.93896 | 29.48 | 1.3913 | 2.1402 | 0.93887 | 29.48 | 1.3933 | 2.1400 | 0.93875 | 29.48 | 1.3961 | 2.1397 | 0.93772 | 29.49 | 1.4196 | 2.1374 | |

2 | 2.96 | 0.94149 | 28.08 | 1.3336 | 2.1460 | 0.94136 | 28.08 | 1.3366 | 2.1457 | 0.94119 | 28.08 | 1.3405 | 2.1453 | 0.93985 | 28.11 | 1.3710 | 2.1422 |

2.95 | 0.92251 | 28.52 | 1.7662 | 2.1027 | 0.92239 | 28.53 | 1.7690 | 2.1024 | 0.92223 | 28.53 | 1.7726 | 2.1021 | 0.92096 | 28.55 | 1.8016 | 2.0992 | |

2.94 | 0.9045 | 28.96 | 2.1768 | 2.0617 | 0.90438 | 28.96 | 2.1795 | 2.0614 | 0.90424 | 28.96 | 2.1827 | 2.0611 | 0.90302 | 28.99 | 2.2105 | 2.0583 | |

2.93 | 0.88734 | 29.38 | 2.5679 | 2.0226 | 0.88722 | 29.39 | 2.5706 | 2.0223 | 0.88708 | 29.39 | 2.5738 | 2.0220 | 0.88592 | 29.41 | 2.6003 | 2.0193 | |

3 | 2.96 | 0.91529 | 28.06 | 1.9308 | 2.0863 | 0.91514 | 28.06 | 1.9342 | 2.0859 | 0.91496 | 28.07 | 1.9383 | 2.0855 | 0.91348 | 28.10 | 1.9721 | 2.0821 |

2.95 | 0.89425 | 28.59 | 2.4104 | 2.0383 | 0.89412 | 28.59 | 2.4134 | 2.0380 | 0.89395 | 28.59 | 2.4172 | 2.0376 | 0.89258 | 28.62 | 2.4485 | 2.0345 | |

2.94 | 0.87463 | 29.09 | 2.8576 | 1.9936 | 0.87449 | 29.09 | 2.8608 | 1.9933 | 0.87433 | 29.10 | 2.8645 | 1.9929 | 0.87305 | 29.12 | 2.8936 | 1.9900 | |

2.93 | 0.85616 | 29.58 | 3.2786 | 1.9515 | 0.85604 | 29.58 | 3.2814 | 1.9512 | 0.85588 | 29.58 | 3.2850 | 1.9508 | 0.85467 | 29.61 | 3.3137 | 1.9480 | |

4 | 2.96 | 0.89703 | 28.15 | 2.3470 | 2.0446 | 0.89688 | 28.15 | 2.3505 | 2.0443 | 0.89668 | 28.15 | 2.3550 | 2.0438 | 0.89515 | 21.18 | 2.3899 | 2.0404 |

2.95 | 0.8749 | 28.72 | 2.8515 | 1.9942 | 0.87476 | 28.72 | 2.8547 | 1.9939 | 0.87458 | 28.73 | 2.8588 | 1.9935 | 0.87316 | 28.76 | 2.8911 | 1.9902 | |

2.94 | 0.85447 | 29.27 | 3.3172 | 1.9476 | 0.85433 | 29.27 | 3.3203 | 1.9473 | 0.85417 | 29.27 | 3.3240 | 1.9469 | 0.85284 | 29.30 | 3.3543 | 1.9439 | |

2.93 | 0.8354 | 29.79 | 3.7518 | 1.9042 | 0.83528 | 29.79 | 3.7546 | 1.9039 | 0.83512 | 29.80 | 3.7582 | 1.9035 | 0.83388 | 29.82 | 3.7865 | 1.9007 | |

5 | 2.96 | 0.88363 | 28.25 | 2.6525 | 2.0141 | 0.88347 | 28.25 | 2.6561 | 2.0137 | 0.88328 | 28.26 | 2.6605 | 2.0133 | 0.8817 | 28.29 | 2.6965 | 2.0097 |

2.95 | 0.8609 | 28.86 | 3.1706 | 1.9623 | 0.86075 | 28.86 | 3.1740 | 1.9619 | 0.86057 | 28.86 | 3.1781 | 1.9615 | 0.85912 | 28.89 | 3.2112 | 1.9582 | |

2.94 | 0.84004 | 29.43 | 3.6461 | 1.9147 | 0.83991 | 29.43 | 3.6490 | 1.9144 | 0.83974 | 29.43 | 3.6529 | 1.9141 | 0.83839 | 29.46 | 3.6837 | 1.9110 | |

2.93 | 0.82068 | 29.98 | 4.0874 | 1.8706 | 0.82056 | 29.98 | 4.0901 | 1.8703 | 0.8204 | 29.98 | 4.0937 | 1.8700 | 0.81914 | 30.01 | 4.1225 | 1.8671 |

No. of loop | C | A_{2} = 0.00001 | A_{2} = 0.00005 | A_{2} = 0.0001 | A_{2} = 0.0005 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

x | V (km∙sec^{−1}) | D_{1} (10^{7} km) | D_{2} (10^{8} km) | x | V (km∙sec^{−1}) | D_{1} (10^{7} km) | D_{2} (10^{8} km) | x | V (km∙sec^{−1}) | D_{1} (10^{7} km) | D_{2} (10^{8} km) | x | V (km∙sec^{−1}) | D_{1} (10^{7} km) | D_{2} (10^{8} km) | ||

1 | 2.96 | 0.983 | 35.32 | 0.2542 | 1.4705 | 0.9829 | 35.32 | 0.2557 | 1.4704 | 0.9828 | 35.33 | 0.2572 | 1.4702 | 0.9817 | 35.36 | 0.2737 | 1.4686 |

2.95 | 0.9679 | 35.70 | 0.4801 | 1.4479 | 0.96782 | 35.70 | 0.4813 | 1.4478 | 0.9677 | 35.70 | 0.4831 | 1.4476 | 0.9666 | 35.72 | 0.4996 | 1.4460 | |

2.94 | 0.95325 | 36.09 | 0.6993 | 1.4260 | 0.95315 | 36.09 | 0.7008 | 1.4259 | 0.95304 | 36.09 | 0.7024 | 1.4257 | 0.952 | 36.11 | 0.7180 | 1.4241 | |

2.93 | 0.939 | 36.47 | 0.9125 | 1.4047 | 0.9389 | 36.47 | 0.9140 | 1.4045 | 0.93877 | 36.48 | 0.9159 | 1.4044 | 0.93775 | 36.50 | 0.9312 | 1.4028 | |

2 | 2.96 | 0.94155 | 34.75 | 0.8743 | 1.40856 | 0.9414 | 34.75 | 0.8766 | 1.4083 | 0.94125 | 34.75 | 0.8788 | 1.4081 | 0.9399 | 34.78 | 0.8990 | 1.4060 |

2.95 | 0.92255 | 35.30 | 1.1586 | 1.3801 | 0.92244 | 35.30 | 1.1602 | 1.3799 | 0.92226 | 35.30 | 1.1629 | 1.3797 | 0.921 | 35.33 | 1.1817 | 1.3778 | |

2.94 | 0.90455 | 35.83 | 1.4278 | 1.3532 | 0.90442 | 35.84 | 1.4298 | 1.3530 | 0.90426 | 35.84 | 1.4322 | 1.3527 | 0.90305 | 35.87 | 1.4503 | 1.3509 | |

2.93 | 0.88737 | 36.36 | 1.6848 | 1.3275 | 0.88725 | 36.36 | 1.6866 | 1.3273 | 0.8871 | 36.36 | 1.6889 | 1.3271 | 0.88595 | 36.39 | 1.7061 | 1.3253 | |

3 | 2.96 | 0.91535 | 34.72 | 1.2663 | 1.3693 | 0.9152 | 34.73 | 1.2685 | 1.3691 | 0.915 | 34.73 | 1.2715 | 1.3688 | 0.91354 | 34.77 | 1.2933 | 1.3666 |

2.95 | 0.8943 | 35.37 | 1.5812 | 1.3378 | 0.89416 | 35.37 | 1.5833 | 1.3376 | 0.894 | 35.38 | 1.5857 | 1.3374 | 0.89262 | 35.41 | 1.6063 | 1.3353 | |

2.94 | 0.87465 | 35.99 | 1.8751 | 1.3084 | 0.87453 | 36.00 | 1.8769 | 1.3083 | 0.87438 | 36.00 | 1.8792 | 1.3080 | 0.87309 | 36.03 | 1.8985 | 1.3061 | |

2.93 | 0.8562 | 36.60 | 2.1512 | 1.2808 | 0.85608 | 36.60 | 2.1529 | 1.2807 | 0.85592 | 36.60 | 2.1553 | 1.2804 | 0.8547 | 36.63 | 2.1736 | 1.2786 | |

4 | 2.96 | 0.8971 | 34.83 | 1.5393 | 1.34206 | 0.89695 | 34.83 | 1.5415 | 1.3418 | 0.89675 | 34.83 | 1.5445 | 1.3415 | 0.8952 | 34.87 | 1.5677 | 1.3392 |

2.95 | 0.87495 | 35.54 | 1.8707 | 1.3089 | 0.87482 | 35.54 | 1.8726 | 1.3087 | 0.87464 | 35.54 | 1.8753 | 1.3084 | 0.87321 | 35.58 | 1.8967 | 1.3063 | |

2.94 | 0.8545 | 36.21 | 2.1766 | 1.2783 | 0.85438 | 36.21 | 2.1784 | 1.2781 | 0.85421 | 36.22 | 2.1809 | 1.2779 | 0.85289 | 36.25 | 2.2007 | 1.2759 | |

2.93 | 0.83544 | 36.86 | 2.4617 | 1.2498 | 0.83532 | 36.86 | 2.4635 | 1.2496 | 0.83516 | 36.87 | 2.4659 | 1.2494 | 0.83392 | 36.90 | 2.4845 | 1.2475 | |

5 | 2.96 | 0.8837 | 34.95 | 1.7398 | 1.3220 | 0.88353 | 34.96 | 1.7423 | 1.3217 | 0.88334 | 34.96 | 1.7451 | 1.3214 | 0.88175 | 35.00 | 1.7689 | 1.3191 |

2.95 | 0.86095 | 35.70 | 2.0801 | 1.2879 | 0.8608 | 35.71 | 2.0823 | 1.2877 | 0.86062 | 35.71 | 2.0850 | 1.2874 | 0.85918 | 35.75 | 2.1066 | 1.2853 | |

2.94 | 0.8401 | 36.41 | 2.3920 | 1.2567 | 0.83995 | 36.41 | 2.3943 | 1.2565 | 0.83978 | 36.42 | 2.3968 | 1.2563 | 0.83844 | 36.46 | 2.4168 | 1.2543 | |

2.93 | 0.82072 | 37.09 | 2.6819 | 1.2278 | 0.8206 | 37.09 | 2.6837 | 1.2276 | 0.82044 | 37.10 | 2.6861 | 1.2273 | 0.81918 | 37.13 | 2.7050 | 1.2254 |

number of loop as A_{2} increases initial velocity increases and D_{1} increases so, D_{2} decreases. So, the effect of Jacobi constant C and oblateness A_{2} is opposite in nature. For given value of A_{2} and C as number of loops increases, D_{1} increases and D_{2} decreases where as V decreases up to loops 1 - 3 and then increases from 3 - 5-loop orbits. From _{2} = 0.00001 is closest to Mars and this distance is 3.886 × 10^{7} km. which is obtained using C = 2.96. Similar notations are used in

From _{2} = 0.00001 is closest to Earth and this distance is 2.542 × 10^{7} km. which is obtained using C = 2.96. Here D_{1} is the distance of secondary body from Earth.

The locations of single-loop and two loops periodic orbits obtained for different values of C for Sun-Mars system A_{2} = 0.00001 from PSS is displayed in

Number of loop | C | X |
---|---|---|

1 | 2.96 | 0.98295 |

2.95 | 0.9679 | |

2.94 | 0.95324 | |

2.93 | 0.93896 | |

2 | 2.96 | 0.94149 |

2.95 | 0.92251 | |

2.94 | 0.9045 | |

2.93 | 0.88734 |

Number of loop | C | Predicted | Exact | Error |
---|---|---|---|---|

1 | 2.92 | 0.92372 | 0.92508 | 0.00136 |

2.925 | 0.93105 | 0.93198 | 0.00093 | |

2.935 | 0.94571 | 0.94605 | 0.00034 | |

2.945 | 0.96037 | 0.9605 | 0.00013 | |

2.955 | 0.97503 | 0.9754 | 0.00037 | |

2.965 | 0.98969 | 0.99069 | 0.001 | |

2 | 2.92 | 0.86768 | 0.87092 | 0.00324 |

2.925 | 0.8767 | 0.87904 | 0.00234 | |

2.935 | 0.89474 | 0.89582 | 0.00108 | |

2.945 | 0.91278 | 0.9134 | 0.00062 | |

2.955 | 0.93082 | 0.931875 | 0.001055 | |

2.965 | 0.94886 | 0.9514 | 0.00254 |

^{2} = 0.999, whereas, for two- loop periodic orbit regression line is ^{2} = 0.999.

Similar calculations are made for one-loop and two loops orbits for A_{2} = 0.0005 and the relevant estimates are displayed in

The regression curve for single-loop periodic orbit is given by

^{2} = 1 and for two-loops periodic orbit regression curve is ^{2} = 1.

It can be observed that as C increases the error between the predicted and exact values of the position of periodic orbits increases. The PSS together with regression analysis will help one to locate the position of the periodic orbit with less effort, using the predicted positions from the analysis.

The variation of position x with oblateness A_{2} for Sun-Mars system for fixed value of C = 2.96 using PSS is shown in ^{2} = 1, where as for two loops orbit ^{2} = 1.

Number of loop | C | X |
---|---|---|

1 | 2.96 | 0.9816 |

2.95 | 0.9666 | |

2.94 | 0.95196 | |

2.93 | 0.93772 | |

2 | 2.96 | 0.93985 |

2.95 | 0.92096 | |

2.94 | 0.90302 | |

2.93 | 0.88592 |

Number of loop | C | Predicted | Exact | Error |
---|---|---|---|---|

1 | 2.92 | 0.9244 | 0.92387 | −0.00053 |

2.925 | 0.931288 | 0.93075 | −0.00054 | |

2.935 | 0.945348 | 0.9448 | −0.00055 | |

2.945 | 0.959787 | 0.95922 | −0.00057 | |

2.955 | 0.974607 | 0.97404 | −0.00057 | |

2.965 | 0.989808 | 0.98929 | −0.00052 | |

2 | 2.92 | 0.87044 | 0.86955 | −0.00089 |

2.925 | 0.878422 | 0.87904 | 0.000618 | |

2.935 | 0.895057 | 0.89582 | 0.000763 | |

2.945 | 0.912587 | 0.9134 | 0.000813 | |

2.955 | 0.931012 | 0.931875 | 0.000863 | |

2.965 | 0.950332 | 0.9514 | 0.001068 |

The variation of position x for different values of C for one-loop and two loops orbit for A_{2} = 0.00001 for Sun-Earth system is shown in

Using regression analysis we have displayed predicted and exact values of position together with error estimates are shown in ^{2} = 0.999, and for two loops orbit regression line ^{2} = 0.999.

Similar estimates are displayed for A_{2} = 0.00005 in

For single-loop orbit regression curve is given by

Number of loop | A_{2} | X |
---|---|---|

1 | 0.00001 | 0.98295 |

0.00005 | 0.98285 | |

0.0001 | 0.98272 | |

0.0005 | 0.9816 | |

2 | 0.00001 | 0.94149 |

0.00005 | 0.94136 | |

0.0001 | 0.94119 | |

0.0005 | 0.93985 |

Number of loop | A_{2} | Predicted | Exact | Error |
---|---|---|---|---|

1 | 0.000025 | 0.982937187 | 0.98293 | −7.187E−06 |

0.000075 | 0.982809684 | 0.9828 | −9.683E−06 | |

0.00025 | 0.982343706 | 0.98233 | −1.3706E−05 | |

0.00075 | 0.980843356 | 0.98095 | 0.000106644 | |

2 | 0.000025 | 0.940916751 | 0.94145 | 0.000533249 |

0.000075 | 0.940750105 | 0.94128 | 0.000529895 | |

0.00025 | 0.940165304 | 0.94069 | 0.000524696 | |

0.00075 | 0.938481234 | 0.93902 | 0.000538766 |

Number of loop | C | X |
---|---|---|

1 | 2.96 | 0.983 |

2.95 | 0.9679 | |

2.94 | 0.95325 | |

2.93 | 0.939 | |

2 | 2.96 | 0.94155 |

2.95 | 0.92255 | |

2.94 | 0.90455 | |

2.93 | 0.88737 |

Number of loop | C | Predicted | Exact | Error |
---|---|---|---|---|

1 | 2.92 | 0.92272 | 0.9251 | 0.00238 |

2.925 | 0.93005 | 0.932 | 0.00195 | |

2.935 | 0.94471 | 0.9461 | 0.00139 | |

2.945 | 0.95937 | 0.96055 | 0.00118 | |

2.955 | 0.97403 | 0.97542 | 0.00139 | |

2.965 | 0.98869 | 0.99072 | 0.00203 | |

2 | 2.92 | 0.8686 | 0.87095 | 0.00235 |

2.925 | 0.877625 | 0.87909 | 0.001465 | |

2.935 | 0.895675 | 0.89588 | 0.000205 | |

2.945 | 0.913725 | 0.91342 | −0.0003 | |

2.955 | 0.931775 | 0.9319 | 0.000125 | |

2.965 | 0.949825 | 0.95147 | 0.001645 |

Number of loop | C | X |
---|---|---|

1 | 2.96 | 0.9817 |

2.95 | 0.9666 | |

2.94 | 0.952 | |

2.93 | 0.93775 | |

2 | 2.96 | 0.9414 |

2.95 | 0.92244 | |

2.94 | 0.90442 | |

2.93 | 0.88725 |

Number of loop | C | Predicted | Exact | Error |
---|---|---|---|---|

1 | 2.92 | 0.9226 | 0.92501 | 0.00241 |

2.925 | 0.929453 | 0.93189 | 0.002437 | |

2.935 | 0.943478 | 0.94599 | 0.002512 | |

2.945 | 0.957928 | 0.96044 | 0.002512 | |

2.955 | 0.972803 | 0.97532 | 0.002517 | |

2.965 | 0.988103 | 0.99062 | 0.002517 | |

2 | 2.92 | 0.87964 | 0.87083 | −0.00881 |

2.925 | 0.887672 | 0.87896 | −0.00871 | |

2.935 | 0.904407 | 0.89573 | −0.00868 | |

2.945 | 0.922037 | 0.91333 | −0.00871 | |

2.955 | 0.940562 | 0.9318 | −0.00876 | |

2.965 | 0.959982 | 0.9513 | −0.00868 |

with R^{2} = 1 and for two-loops orbit regression curve is given by

^{2} = 1.

The variation of position x with oblateness A_{2} for Sun-Earth system is shown in

The predicted and exact value of position together with error estimates are shown in

The equation of the best fit curve for single-loop is

^{2} = 1, where as for two-loops orbit regression curve is ^{2} = 1.

In this paper, we have studied the effect of oblateness on the position, shape and size of closed periodic orbit with loops varying from 1 to 5 for Sun-Mars and Sun-Earth systems, respectively. It is concluded that for given number of loops and given C, as oblateness increases, location of periodic orbit moves towards Sun. For given C and given oblateness, as number of loops increases, location of periodic orbits shifts towards Sun. Also, for given value of oblateness and given number of loops, as C decreases, location of periodic orbit moves towards Sun. Also, period of the orbit increases as number of loops increases. It is also observed that single-loop orbit is closest to second primary body. Further, as number of loops decreases, width of the orbit increases. The distance of closest approach of the infinitesimal particle from the smaller primary increases with oblateness and number of loops for a given C. Thus, the present analysis of the two systems―Sun-Mars and Sun-Earth systems―using PSS technique reveals that A_{2} and C has

Number of loop | A_{2} | X |
---|---|---|

1 | 0.00001 | 0.983 |

0.00005 | 0.9829 | |

0.0001 | 0.9828 | |

0.0005 | 0.9817 | |

2 | 0.00001 | 0.94155 |

0.00005 | 0.9414 | |

0.0001 | 0.94125 | |

0.0005 | 0.9399 |

Number of loop | A_{2} | Predicted | Exact | Error |
---|---|---|---|---|

1 | 0.000025 | 0.982946715 | 0.98294 | −6.715E−06 |

0.000075 | 0.982836185 | 0.98283 | −6.185E−06 | |

0.00025 | 0.98240775 | 0.9824 | −7.75E−06 | |

0.00075 | 0.98082725 | 0.981 | 0.00017275 | |

2 | 0.000025 | 0.940916838 | 0.9415 | 0.000583162 |

0.000075 | 0.940750295 | 0.94133 | 0.000579705 | |

0.00025 | 0.940165088 | 0.94074 | 0.000574912 | |

0.00075 | 0.938473293 | 0.93906 | 0.000586707 |

substantial effect on the position, shape and size of the obit. The PSS together with regression analysis will help one to locate the position of the periodic orbit with less effort, using the predicted positions from the analysis.

It can be observed that for given oblateness and given number of loops, as Jacobi constant decreases, initial velocity of infinitesimal particle (spacecraft) and distance of spacecraft from second primary increase and distance of spacecraft from first primary body decreases. For given Jacobi constant and given number of loops, as oblateness increases, initial velocity increases and distance of spacecraft from second primary increases. So, distance of spacecraft from first primary decreases. Thus, the effect of Jacobi constant C and oblateness coefficient A_{2} is opposite in nature. For given value of oblateness coefficient and Jacobi constant, as number of loops increases, distance of spacecraft from second primary increases and distance of spacecraft from first primary decreases where as initial velocity of spacecraft decreases up to loops 1 - 3 and then increases from 3 - 5. It is further observed that for Sun-Mars system, single-loop orbit for A_{2} = 0.00001 and C = 2.96 is closest to Mars and this distance is 3.886 × 10^{7} km. Whereas for Sun-Earth system, single-loop orbit for A_{2} = 0.00001 and C = 2.96 is closest to Earth and this distance is 2.542 × 10^{7} km. Since stability of this class of periodic orbits is very low, it can be used for designing low-energy trajectory design for space mission.

The authors thank Mrs. Pooja Dutt, Applied Mathematics Division, Vikram Sarabhai Space Centre (ISRO), Thiruvananthapuram, India for her constructive comments.

Pathak, N. and Thomas, V.O. (2016) Analysis of Effect of Oblateness of Smaller Primary on the Evolution of Periodic Orbits. International Journal of Astronomy and Astrophysics, 6, 440-463. http://dx.doi.org/10.4236/ijaa.2016.64036