^{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 solar radiation pressure of bigger primary and actual oblateness of smaller primary on these orbits are considered. It is observed that solar radiation pressure of bigger 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 solar radiation pressure has to be understood and should be taken care of during trajectory design. It is also verified that stability of such orbits are negligible so they can be used as transfer orbit. For each pair of solar radiation pressure q and Jacobi constant C we get two separatrices where stability of island becomes zero. In this paper, detailed stability analysis of periodic orbit having two loops is given when q = 0.9845.

A Hohmann transfer (Hohmann, 1925) is effectively used to transfer a spacecraft from one circular orbit to another in a most fuel efficient way. The “low-energy” transfer trajectory [

The “low-energy” transfer orbits are important in the study of practical problems regarding transfer of orbits. Low energy transfer of orbit to moon has been studied by [

Study of orbits of bodies in RTBP has been described in detail by [

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. [

Periodic orbits of spacecraft around the two massive primaries are used to construct low energy trajectory. In this paper, we have studied periodic orbits around both primaries in the frame work of RTBP using PSS method. [

Let m_{1} and m_{2} be two masses of first and second primary bodies, respectively. Mass of first primary is greater than mass of second primary. The solar radiation pressure force F_{p} changes with the distance by the same law as the gravitational attraction force F_{g} and acts opposite to it. It is possible to consider that the result of the action of this force leads to reducing the effective mass of the sun [

where

We follow the notation and terminology of [

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 of the secondary body (space craft).

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

System of Equation (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

We shall consider two systems, the Sun-Mars system and the Sun-Earth system. The masses of Sun, Earth and Mars are taken as 1.9881 × 10^{30} kg, 5.972 × 10^{24} kg, and 6.4185 × 10^{23} kg, respectively [

q | Maximum value of C | Value of C greater than maximum value | Excluded region (where y is complex) | Size of excluded region |
---|---|---|---|---|

1 | 3.0 | 3.001 | x = 0.988 to x = 0.994 | 0.006 |

0.995 | 2.99 | 2.991 | x = 0.985 to x = 0.994 | 0.009 |

0.99 | 2.98 | 2.981 | x = 0.983 to x = 0.995 | 0.012 |

0.9845 | 2.969 | 2.970 | x = 0.98 to x = 0.995 | 0.015 |

q | Maximum value of C | Value of C greater than maximum value | Excluded region (where y is complex) | Size of excluded region |
---|---|---|---|---|

1 | 3.0 | 3.001 | x = 0.984 to x = 1.0 | 0.016 |

0.995 | 2.99 | 2.991 | x = 0.982 to x = 1.0 | 0.018 |

0.99 | 2.98 | 2.981 | x = 0.980 to x = 1.0 | 0.020 |

0.9845 | 2.969 | 2.970 | x = 0.978 to x = 0.995 | 0.022 |

integrated using Runge-Kutta-Gill method. Each solution is plotted as a point in the

In a similar way, we have obtained PSS for Sun-Earth system as shown in

The numerical values of location and period of periodic orbit of spacecraft for C = 2.93, 2.94, 2.95, 2.96 and for q = 1, 0.995, 0.99 and 0.9845 are displayed in

Number of loops | C | q = 1 | q = 0.995 | q = 0.99 | q = 0.9845 | ||||
---|---|---|---|---|---|---|---|---|---|

Location (x) | Period (T) | Location (x) | Period (T) | Location (x) | Period (T) | Location (x) | Period (T) | ||

1 | 2.96 | 0.983 | 13 | 0.9967 | 13 | - | - | - | - |

2.95 | 0.96791 | 13 | 0.98115 | 13 | 0.9949 | 13 | - | - | |

2.94 | 0.95327 | 13 | 0.96607 | 13 | 0.97935 | 13 | 0.99455 | 13 | |

2.93 | 0.939 | 13 | 0.95139 | 13 | 0.96424 | 13 | 0.97891 | 13 | |

2 | 2.96 | 0.94152 | 19 | 0.95985 | 19 | 0.97957 | 19 | - | - |

2.95 | 0.92255 | 19 | 0.93971 | 19 | 0.95807 | 19 | 0.97992 | 19 | |

2.94 | 0.90454 | 19 | 0.92071 | 19 | 0.93792 | 19 | 0.95825 | 19 | |

2.93 | 0.88737 | 19 | 0.90269 | 19 | 0.91891 | 19 | 0.93795 | 19 | |

3 | 2.96 | 0.91532 | 26 | 0.9363 | 26 | 0.95957 | 26 | 0.98875 | 26 |

2.95 | 0.89429 | 26 | 0.91353 | 26 | 0.93455 | 26 | 0.96036 | 26 | |

2.94 | 0.87465 | 26 | 0.89248 | 26 | 0.91175 | 26 | 0.93506 | 26 | |

2.93 | 0.85619 | 26 | 0.87283 | 26 | 0.89069 | 26 | 0.91204 | 26 | |

4 | 2.96 | 0.89707 | 32 | 0.91959 | 32 | 0.9451 | 32 | 0.9783 | 32 |

2.95 | 0.87494 | 32 | 0.89529 | 32 | 0.91785 | 32 | 0.9462 | 32 | |

2.94 | 0.8545 | 32 | 0.87314 | 32 | 0.89353 | 32 | 0.91857 | 32 | |

2.93 | 0.835433 | 32 | 0.852695 | 32 | 0.871375 | 32 | 0.89397 | 32 | |

5 | 2.96 | 0.883675 | 38 | 0.90711 | 38 | 0.93405 | 38 | 0.9701 | 38 |

2.95 | 0.86093 | 38 | 0.8819 | 38 | 0.90538 | 38 | 0.93535 | 38 | |

2.94 | 0.84007 | 38 | 0.85915 | 38 | 0.88016 | 38 | 0.90623 | 38 | |

2.93 | 0.820715 | 38 | 0.83829 | 38 | 0.8574 | 38 | 0.88069 | 38 |

observed from the table that a change in C in the range (2.93, 2.96) affects the location but has no effect on the period and number of loops of the orbit. Solar radiation pressure also affects the location and period of the orbit. Similarly, the effects of C and q in the location and period for the Sun-Earth system are studied and the numerical estimates of the changes are displayed in

A noticeable difference observed in both the systems is that for C = 2.96, q = 0.995, single-loop periodic orbit, for C = 2.96, q = 0.99 two loops orbit, for C = 2.96, q = 0.9845, three loops periodic orbit does not exist for Sun-Earth system where as it exists for Sun-Mars system.

The periodic orbits starting from single-loop to five loops in the Sun-Mars system for q = 0.9845 and 0.995 are shown in Figures 3(a)-(j). It can be observed, from

Number of loops | C | q = 1 | q = 0.995 | q = 0.99 | q = 0.9845 | ||||
---|---|---|---|---|---|---|---|---|---|

Location (x) | Period (T) | Location (x) | Period (T) | Location (x) | Period (T) | Location (x) | Period (T) | ||

1 | 2.96 | 0.98305 | 13 | - | - | - | - | - | - |

2.95 | 0.96795 | 13 | 0.9812 | 13 | 0.9949603 | 13 | - | - | |

2.94 | 0.9533 | 13 | 0.9661 | 13 | 0.9794 | 13 | 0.994601 | 13 | |

2.93 | 0.93904 | 13 | 0.95142 | 13 | 0.96425 | 13 | 0.97895 | 13 | |

2 | 2.96 | 0.94157 | 19 | 0.9599 | 19 | - | - | - | - |

2.95 | 0.92259 | 19 | 0.93975 | 19 | 0.95813 | 19 | 0.98 | 19 | |

2.94 | 0.90456 | 19 | 0.92075 | 19 | 0.93796 | 19 | 0.9583 | 19 | |

2.93 | 0.8874 | 19 | 0.90273 | 19 | 0.91895 | 19 | 0.938 | 19 | |

3 | 2.96 | 0.91538 | 26 | 0.93635 | 26 | 0.95965 | 26 | - | - |

2.95 | 0.89434 | 26 | 0.91359 | 26 | 0.93462 | 26 | 0.96045 | 26 | |

2.94 | 0.87469 | 26 | 0.89252 | 26 | 0.91181 | 26 | 0.93513 | 26 | |

2.93 | 0.85623 | 26 | 0.87287 | 26 | 0.89074 | 26 | 0.9121 | 26 | |

4 | 2.96 | 0.89714 | 32 | 0.91965 | 32 | 0.9452 | 32 | 0.97844 | 32 |

2.95 | 0.87499 | 32 | 0.89535 | 32 | 0.91793 | 32 | 0.9463 | 32 | |

2.94 | 0.85454 | 32 | 0.8732 | 32 | 0.89359 | 32 | 0.91865 | 32 | |

2.93 | 0.83547 | 32 | 0.85274 | 32 | 0.87142 | 32 | 0.89403 | 32 | |

5 | 2.96 | 0.88373 | 38 | 0.90719 | 38 | 0.93415 | 38 | 0.9703 | 38 |

2.95 | 0.86099 | 38 | 0.88195 | 38 | 0.90546 | 38 | 0.93547 | 38 | |

2.94 | 0.84012 | 38 | 0.8592 | 38 | 0.88022 | 38 | 0.9063 | 38 | |

2.93 | 0.82075 | 38 | 0.83833 | 38 | 0.85745 | 38 | 0.88075 | 38 |

closest approach of the space craft receeds from the second primary [

where,

Using the set of Equations (10)-(14), for different number of loops and selected values of Jacobi constant C and solar radiation pressure q, the location of the orbit, the semi major axis a and eccentricity e of the periodic orbits are calculated and numerical values are given for Sun-Mars system in

From

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

Similar kind of conclusion can be drawn from

We have studied the effect of q and

No. of loop | C | q = 1 | q = 0.995 | q = 0.99 | q = 0.9845 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

x | a | e | x | a | e | x | a | e | x | a | e | ||

1 | 2.96 | 0.983 | 1.5880 | 0.3809 | 0.9967 | 1.57075 | 0.36546 | - | - | - | - | - | - |

2.95 | 0.96791 | 1.58760 | 0.39033 | 0.98115 | 1.56765 | 0.37412 | 0.9949 | 1.55014 | 0.35818 | - | - | - | |

2.94 | 0.95327 | 1.58765 | 0.39957 | 0.96607 | 1.56716 | 0.38355 | 0.97935 | 1.54801 | 0.36735 | 0.99455 | 1.52940 | 0.34971 | |

2.93 | 0.939 | 1.58755 | 0.40852 | 0.95139 | 1.56664 | 0.39272 | 0.96424 | 1.54709 | 0.37674 | 0.97891 | 1.52685 | 0.35887 | |

2 | 2.96 | 0.94152 | 1.31045 | 0.28153 | 0.95985 | 1.29725 | 0.26009 | 0.97957 | 1.28508 | 0.23774 | - | - | - |

2.95 | 0.92255 | 1.31045 | 0.296010 | 0.93971 | 1.29672 | 0.27532 | 0.95807 | 1.28401 | 0.25384 | 0.97992 | 1.27131 | 0.22921 | |

2.94 | 0.90454 | 1.31044 | 0.30974 | 0.92071 | 1.29630 | 0.28974 | 0.93792 | 1.28318 | 0.26907 | 0.95825 | 1.26990 | 0.24541 | |

2.93 | 0.88737 | 1.31042 | 0.32284 | 0.90269 | 1.29594 | 0.30345 | 0.91891 | 1.28245 | 0.28347 | 0.93795 | 1.26869 | 0.26069 | |

3 | 2.96 | 0.91532 | 1.21146 | 0.24445 | 0.9363 | 1.19999 | 0.21974 | 0.95957 | 1.18956 | 0.19334 | 0.98875 | 1.17987 | 0.16199 |

2.95 | 0.89429 | 1.21146 | 0.26180 | 0.91353 | 1.19959 | 0.23846 | 0.93455 | 1.18863 | 0.21376 | 0.96036 | 1.17776 | 0.18459 | |

2.94 | 0.87465 | 1.21143 | 0.278004 | 0.89248 | 1.19919 | 0.25576 | 0.91175 | 1.18783 | 0.23242 | 0.93506 | 1.17645 | 0.20519 | |

2.93 | 0.85619 | 1.21144 | 0.29324 | 0.87283 | 1.19880 | 0.27191 | 0.89069 | 1.18710 | 0.24969 | 0.91204 | 1.17526 | 0.22397 | |

4 | 2.96 | 0.89707 | 1.16044 | 0.22696 | 0.91959 | 1.14979 | 0.20021 | 0.9451 | 1.14015 | 0.17108 | 0.9783 | 1.13112 | 0.13511 |

2.95 | 0.87494 | 1.16043 | 0.24602 | 0.89529 | 1.14929 | 0.22107 | 0.91785 | 1.13925 | 0.19434 | 0.9462 | 1.12931 | 0.16214 | |

2.94 | 0.8545 | 1.16042 | 0.26363 | 0.87314 | 1.14898 | 0.24008 | 0.89353 | 1.13847 | 0.21515 | 0.91857 | 1.12798 | 0.18565 | |

2.93 | 0.835433 | 1.16042 | 0.28006 | 0.852695 | 1.14862 | 0.25763 | 0.871375 | 1.13772 | 0.23411 | 0.89397 | 1.12678 | 0.20662 | |

5 | 2.96 | 0.883675 | 1.12928 | 0.21749 | 0.90711 | 1.11910 | 0.18943 | 0.93405 | 1.10990 | 0.15844 | 0.9701 | 1.10120 | 0.11906 |

2.95 | 0.86093 | 1.12926 | 0.237620 | 0.8819 | 1.11869 | 0.21167 | 0.90538 | 1.10902 | 0.18362 | 0.93535 | 1.09956 | 0.14934 | |

2.94 | 0.84007 | 1.12925 | 0.25608 | 0.85915 | 1.11830 | 0.23173 | 0.88016 | 1.10824 | 0.20581 | 0.90623 | 1.09825 | 0.17485 | |

2.93 | 0.820715 | 1.12926 | 0.27323 | 0.83829 | 1.11794 | 0.25015 | 0.8574 | 1.10749 | 0.22582 | 0.88069 | 1.09705 | 0.19722 |

No. of loop | C | q = 1 | q = 0.995 | q = 0.99 | q = 0.9845 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

x | a | e | x | a | e | x | a | e | x | a | e | ||

1 | 2.96 | 0.98305 | 1.59310 | 0.38293 | - | - | - | - | - | - | - | - | - |

2.95 | 0.96795 | 1.59008 | 0.39126 | 0.9812 | 1.57215 | 0.37589 | 0.9949603 | 1.56800 | 0.36546 | - | - | - | |

2.94 | 0.9533 | 1.58919 | 0.40013 | 0.9661 | 1.56937 | 0.38440 | 0.9794 | 1.55205 | 0.36896 | 0.994601 | 1.54580 | 0.35658 | |

2.93 | 0.93904 | 1.58875 | 0.40895 | 0.95142 | 1.56808 | 0.39326 | 0.96425 | 1.54897 | 0.37749 | 0.97895 | 1.53065 | 0.36044 | |

2 | 2.96 | 0.94157 | 1.31144 | 0.28204 | 0.9599 | 1.29882 | 0.26095 | - | - | ||||

2.95 | 0.92259 | 1.31111 | 0.29634 | 0.93975 | 1.29762 | 0.27579 | 0.95813 | 1.28554 | 0.25469 | 0.98 | 1.27517 | 0.23148 | |

2.94 | 0.90456 | 1.31087 | 0.30996 | 0.92075 | 1.29693 | 0.29006 | 0.93796 | 1.28404 | 0.26953 | 0.9583 | 1.27138 | 0.24626 | |

2.93 | 0.8874 | 1.31080 | 0.32301 | 0.90273 | 1.29643 | 0.30368 | 0.91895 | 1.28305 | 0.28378 | 0.938 | 1.26958 | 0.26118 | |

3 | 2.96 | 0.91538 | 1.21204 | 0.24477 | 0.93635 | 1.20078 | 0.22022 | 0.95965 | 1.19115 | 0.19436 | - | ||

2.95 | 0.89434 | 1.21187 | 0.26202 | 0.91359 | 1.20015 | 0.23878 | 0.93462 | 1.18946 | 0.21426 | 0.96045 | 1.17944 | 0.18568 | |

2.94 | 0.87469 | 1.21174 | 0.27816 | 0.89252 | 1.19956 | 0.25597 | 0.91181 | 1.18837 | 0.23273 | 0.93513 | 1.17729 | 0.20570 | |

2.93 | 0.85623 | 1.21170 | 0.29337 | 0.87287 | 1.19910 | 0.27207 | 0.89074 | 1.18748 | 0.24990 | 0.9121 | 1.17580 | 0.22428 | |

4 | 2.96 | 0.89714 | 1.16088 | 0.22720 | 0.91965 | 1.15036 | 0.20057 | 0.9452 | 1.14124 | 0.17179 | 0.97844 | 1.13506 | 0.13800 |

2.95 | 0.87499 | 1.16073 | 0.24618 | 0.89535 | 1.14980 | 0.22130 | 0.91793 | 1.13986 | 0.19471 | 0.9463 | 1.13042 | 0.16289 | |

2.94 | 0.85454 | 1.16065 | 0.26375 | 0.8732 | 1.14930 | 0.24024 | 0.89359 | 1.13886 | 0.21538 | 0.91865 | 1.12859 | 0.18603 | |

2.93 | 0.83547 | 1.16061 | 0.28015 | 0.85274 | 1.14886 | 0.25776 | 0.87142 | 1.13800 | 0.23426 | 0.89403 | 1.12717 | 0.20685 | |

5 | 2.96 | 0.88373 | 1.12961 | 0.21768 | 0.90719 | 1.11959 | 0.18973 | 0.93415 | 1.11072 | 0.15899 | 0.9703 | 1.10404 | 0.12116 |

2.95 | 0.86099 | 1.12953 | 0.23776 | 0.88195 | 1.11899 | 0.21185 | 0.90546 | 1.10950 | 0.18392 | 0.93547 | 1.1004 | 0.14994 | |

2.94 | 0.84012 | 1.12947 | 0.25619 | 0.8592 | 1.11854 | 0.23187 | 0.88022 | 1.10856 | 0.20599 | 0.9063 | 1.09871 | 0.17514 | |

2.93 | 0.82075 | 1.12942 | 0.27331 | 0.83833 | 1.11813 | 0.25024 | 0.85745 | 1.10773 | 0.22595 | 0.88075 | 1.09736 | 0.19740 |

observations for 5 loops closed periodic orbit in both Sun-Mars and Sun-Earth systems are shown in

From

periodic orbit increase. Also, as solar radiation pressure increases from 1 to 0.9845, location of periodic orbit moves towards second primary.

We have examined the effect of solar radiation pressure q for different values of Jacobi constants C on the semi major axis of periodic orbits with single loop and 5-loops of Sun-Mars and Sun-Earth systems. The results of these effects are displayed in Figures 11-14. While there are significant changes in the semi major axis due to solar radiation pressure, the effect of C on the semi-major axis is less in comparison with that of q. Figures 11-14 indicate that the semi major axis decreases as solar radiation pressure increases from 1 to 0.9845.

The variation of semi major axis against the number of loops is examined for different solar radiation pressure and the results of the study are displayed in

From

radiation pressure increases from 1 to 0.9845 semi major axis of periodic orbit for given loop decreases.

The variations of the other geometric parameter, viz, eccentricity e with respect to solar radiation pressure q and Jacobi constant C for single and five loops periodic orbits

are shown in Figures 17-20 for both Sun-Mars and Sun-Earth systems. The eccentricity decreases as q increases from 1 to 0.9845 and decreases as C increases for single loop periodic orbit for both systems Sun-Mars and Sun-Earth as shown in

corresponding to variation in q for given C. green curve shows non smooth behavior corresponding to periodic orbit for C = 2.94, q = 0.9845 which is located at x = 0.994601, loses its periodicity after t = 110.

We have also analyzed the effect of number of loops on the eccentricity of the orbit. These effects are depicted in

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 (11).

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

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} increases and D_{2} decreases. For given C and given number of loop as q increases from 1 to 0.9845 initial velocity V and D_{1} decreases and D_{2} increases. So, the effect of Jacobi constant C and solar radiation pressure

No. of loop | C | q = 1 | q = 0.995 | q = 0.99 | q = 0.9845 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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 | 28.52 | 0.38749 | 2.24065 | 0.9967 | 28.17 | 0.07521 | 2.27187 | - | - | - | - | - | - | - | - |

2.95 | 0.96791 | 28.84 | 0.73145 | 2.20625 | 0.98115 | 28.48 | 0.42965 | 2.23643 | 0.9949 | 28.12 | 0.11624 | 2.26777 | - | - | - | - | |

2.94 | 0.95327 | 29.16 | 1.06515 | 2.17288 | 0.96607 | 28.80 | 0.77339 | 2.20206 | 0.97935 | 28.44 | 0.47068 | 2.23233 | 0.99455 | 28.04 | 0.12421 | 2.26697 | |

2.93 | 0.939 | 29.47 | 1.39042 | 2.14035 | 0.95139 | 29.12 | 1.10800 | 2.16859 | 0.96424 | 28.76 | 0.81510 | 2.19788 | 0.97891 | 28.35 | 0.48071 | 2.23132 | |

2 | 2.96 | 0.94152 | 28.08 | 1.33298 | 2.14610 | 0.95985 | 27.57 | 0.91517 | 2.18788 | 0.97957 | 27.05 | 0.46567 | 2.23283 | - | - | - | - |

2.95 | 0.92255 | 28.52 | 1.76538 | 2.10286 | 0.93971 | 28.04 | 1.37424 | 2.14197 | 0.95807 | 27.53 | 0.95574 | 2.18382 | 0.97992 | 26.95 | 0.45769 | 2.23363 | |

2.94 | 0.90454 | 28.96 | 2.17590 | 2.06180 | 0.92071 | 28.48 | 1.80732 | 2.09866 | 0.93792 | 27.99 | 1.41504 | 2.13789 | 0.95825 | 28.05 | 0.95164 | 2.18423 | |

2.93 | 0.88737 | 29.38 | 2.56728 | 2.02267 | 0.90269 | 28.92 | 2.21807 | 2.05759 | 0.91891 | 28.44 | 1.84835 | 2.09456 | 0.93795 | 27.90 | 1.41436 | 2.13796 | |

3 | 2.96 | 0.91532 | 28.06 | 1.93018 | 2.08638 | 0.9363 | 27.47 | 1.45197 | 2.13420 | 0.95957 | 26.84 | 0.92155 | 2.18724 | 0.98875 | 26.09 | 2.56425 | 2.25375 |

2.95 | 0.89429 | 28.59 | 2.40954 | 2.03844 | 0.91353 | 28.02 | 1.97098 | 2.08230 | 0.93455 | 27.43 | 1.49185 | 2.13021 | 0.96036 | 26.73 | 0.90354 | 2.18904 | |

2.94 | 0.87465 | 29.09 | 2.85722 | 1.99367 | 0.89248 | 28.55 | 2.45080 | 2.03431 | 0.91175 | 27.98 | 2.01156 | 2.07824 | 0.93506 | 27.32 | 1.48023 | 2.13137 | |

2.93 | 0.85619 | 29.58 | 3.27799 | 1.95160 | 0.87283 | 29.05 | 2.89870 | 1.98952 | 0.89069 | 28.51 | 2.49160 | 2.03023 | 0.91204 | 27.88 | 2.00495 | 2.07890 | |

4 | 2.96 | 0.89707 | 28.14 | 2.34617 | 2.04478 | 0.91959 | 27.49 | 1.83285 | 2.09611 | 0.9451 | 26.79 | 1.25138 | 2.15426 | 0.9783 | 25.92 | 4.94622 | 2.22993 |

2.95 | 0.87494 | 28.72 | 2.85061 | 1.99433 | 0.89529 | 28.12 | 2.45080 | 2.03431 | 0.91785 | 27.45 | 1.87251 | 2.09214 | 0.9462 | 26.67 | 1.22630 | 2.15676 | |

2.94 | 0.8545 | 29.27 | 3.31651 | 1.94774 | 0.87314 | 28.68 | 2.89163 | 1.99023 | 0.89353 | 28.06 | 2.42686 | 2.03671 | 0.91857 | 27.34 | 1.85610 | 2.09378 | |

2.93 | 0.835433 | 29.79 | 3.75113 | 1.90428 | 0.852695 | 29.23 | 3.35766 | 1.94363 | 0.871375 | 28.64 | 2.93187 | 1.98621 | 0.89397 | 27.96 | 2.41684 | 2.03771 | |

5 | 2.96 | 0.883675 | 28.25 | 2.65150 | 2.01424 | 0.90711 | 27.56 | 2.11732 | 2.06766 | 0.93405 | 26.80 | 1.50325 | 2.12907 | 0.9701 | 25.85 | 6.81533 | 2.21124 |

2.95 | 0.86093 | 28.85 | 3.16995 | 1.96240 | 0.8819 | 28.11 | 2.38675 | 2.04072 | 0.90538 | 27.52 | 2.15676 | 2.06372 | 0.93535 | 26.68 | 1.47362 | 2.13203 | |

2.94 | 0.84007 | 29.43 | 3.64543 | 1.91485 | 0.85915 | 28.82 | 3.21052 | 1.95834 | 0.88016 | 28.17 | 2.73162 | 2.00623 | 0.90623 | 27.40 | 2.13738 | 2.06566 | |

2.93 | 0.820715 | 29.98 | 4.08672 | 1.87072 | 0.83829 | 29.39 | 3.68601 | 1.91079 | 0.8574 | 28.78 | 3.25041 | 1.95435 | 0.88069 | 28.06 | 2.71954 | 2.00744 |

No. of loop | C | q = 1 | q = 0.995 | q = 0.99 | q = 0.9845 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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 | 0.98305 | 35.32 | 0.25352 | 1.47064 | - | - | - | - | - | - | - | - | - | - | - | - |

2.95 | 0.96795 | 35.70 | 0.47942 | 1.44805 | 0.9812 | 35.26 | 0.28120 | 1.46787 | 0.9949603 | 34.88 | 0.07535 | 1.48846 | - | - | - | - | |

2.94 | 0.9533 | 36.09 | 0.69858 | 1.42614 | 0.9661 | 35.64 | 0.50709 | 1.44529 | 0.9794 | 35.20 | 0.30813 | 1.46518 | 0.994601 | 34.77 | 0.08072 | 1.48792 | |

2.93 | 0.93904 | 36.47 | 0.91191 | 1.40480 | 0.95142 | 36.03 | 0.72671 | 1.42332 | 0.96425 | 35.59 | 0.53477 | 1.44252 | 0.97895 | 35.10 | 0.31486 | 1.46451 | |

2 | 2.96 | 0.94157 | 34.74 | 0.87406 | 1.40859 | 0.9599 | 34.13 | 0.59985 | 1.43601 | - | - | - | - | - | |||

2.95 | 0.92259 | 35.30 | 1.15800 | 1.38019 | 0.93975 | 34.69 | 0.90129 | 1.40587 | 0.95813 | 34.07 | 0.62633 | 1.43336 | 0.98 | 33.38 | 2.99155 | 1.46608 | |

2.94 | 0.90456 | 35.83 | 1.42773 | 1.35322 | 0.92075 | 35.24 | 1.18553 | 1.37744 | 0.93796 | 34.64 | 0.97803 | 1.39819 | 0.9583 | 33.96 | 0.62378 | 1.43362 | |

2.93 | 0.8874 | 36.36 | 1.68445 | 1.32755 | 0.90273 | 35.78 | 1.45511 | 1.35048 | 0.91895 | 35.19 | 1.21246 | 1.37475 | 0.938 | 34.53 | 0.92747 | 1.40325 | |

3 | 2.96 | 0.91538 | 34.72 | 1.26587 | 1.36941 | 0.93635 | 33.99 | 0.95215 | 1.40078 | 0.95965 | 33.22 | 6.03591 | 1.43564 | - | - | - | - |

2.95 | 0.89434 | 35.37 | 1.58062 | 1.33793 | 0.91359 | 34.67 | 1.29264 | 1.36673 | 0.93462 | 33.94 | 0.97803 | 1.39819 | 0.96045 | 33.08 | 5.91623 | 1.43683 | |

2.94 | 0.87469 | 35.99 | 1.87459 | 1.30854 | 0.89252 | 35.32 | 1.60785 | 1.33521 | 0.91181 | 34.62 | 1.31927 | 1.36407 | 0.93513 | 33.81 | 0.97041 | 1.39895 | |

2.93 | 0.85623 | 36.60 | 2.15075 | 1.28092 | 0.87287 | 35.95 | 1.90181 | 1.30581 | 0.89074 | 35.27 | 1.63448 | 1.33255 | 0.9121 | 34.50 | 1.31493 | 1.36450 | |

4 | 2.96 | 0.89714 | 34.82 | 1.53874 | 1.34212 | 0.91965 | 34.02 | 1.20199 | 1.37580 | 0.9452 | 33.15 | 8.19763 | 1.41402 | 0.97844 | 32.11 | 3.22492 | 1.46375 |

2.95 | 0.87499 | 35.53 | 1.87010 | 1.30898 | 0.89535 | 34.78 | 1.56551 | 1.33944 | 0.91793 | 33.97 | 1.22772 | 1.37322 | 0.9463 | 33.01 | 8.03307 | 1.41566 | |

2.94 | 0.85454 | 36.21 | 2.17603 | 1.27839 | 0.8732 | 35.49 | 1.89688 | 1.30631 | 0.89359 | 34.73 | 1.59184 | 1.33681 | 0.91865 | 33.83 | 1.21695 | 1.37430 | |

2.93 | 0.83547 | 36.86 | 2.46132 | 1.24986 | 0.85274 | 36.16 | 2.20296 | 1.27570 | 0.87142 | 35.44 | 1.92351 | 1.30364 | 0.89403 | 34.59 | 1.58526 | 1.33747 | |

5 | 2.96 | 0.88373 | 34.95 | 1.73935 | 1.32206 | 0.90719 | 34.10 | 1.38839 | 1.35716 | 0.93415 | 33.17 | 9.85071 | 1.39749 | 0.9703 | 32.01 | 4.44267 | 1.45157 |

2.95 | 0.86099 | 35.70 | 2.08089 | 1.28791 | 0.88195 | 34.71 | 1.41427 | 1.35457 | 0.90546 | 34.05 | 1.41427 | 1.35457 | 0.93547 | 33.01 | 9.65323 | 1.39946 | |

2.94 | 0.84012 | 36.41 | 2.39175 | 1.25682 | 0.8592 | 35.65 | 2.10632 | 1.28536 | 0.88022 | 34.85 | 1.79186 | 1.31681 | 0.9063 | 33.91 | 1.40170 | 1.35582 | |

2.93 | 0.82075 | 37.09 | 2.68153 | 1.22784 | 0.83833 | 36.36 | 2.41853 | 1.25414 | 0.85745 | 35.60 | 2.13250 | 1.28274 | 0.88075 | 34.72 | 1.78393 | 1.31760 |

q is opposite in nature. For given value of q and C as number of loops increases, D_{1} increases and D_{2} decreases.

From ^{7} km. which is obtained using C = 2.96 for q = 0.995. Similar notations are used in

Here D_{1} is the distance of secondary body from Earth. From ^{5} km. This orbit is obtained by taking C = 2.95.

We have analyzed stability of periodic orbits from loop 1 to 5 for q = 1, 0.995, 0.99 and 0.9845. Since stability behavior is similar for all these orbits, in this paper stability analysis for two loop orbit corresponding to q = 0.9845 is given.

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 0.0006.

of first separatrix at C = 2.93 which is looks like a straight line where as for “f” family orbit it is triangular due to third order resonance [

maximum, where as for C = 2.95 again size of island becomes zero, which is second separatrix as shown in

We have investigated the effect of solar radiation pressure 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. A noticeable difference observed in both the systems is that for C = 2.96, q = 0.995, single-loop periodic orbit, for C = 2.96, q = 0.99, two loops orbit, for C =

2.96, q = 0.9845, three loops periodic orbit does not exist for Sun-Earth system whereas it exists for Sun-Mars system.

The distance of closest approach of the infinitesimal particle from the smaller primary decreases with increase in solar radiation pressure from 1 to 0.9845 and distance between smaller primary and infinitesimal particle increases as number of loops increases for a given C and q. It is found that the eccentricity decreases as number of loops increases. For a given number of loops, the eccentricity is found to decrease as solar radiation pressure increases from 1 to 0.9845. Thus, the present analysis of the two systems-Sun-Mars and Sun-Earth systems-using PSS technique reveals that q and C has substantial effect on the position, shape and size of the obit.

It can be observed that for given solar radiation pressure and given number of loops, as Jacobi constant decreases, initial velocity of infinitesimal particle (space craft) and distance of infinitesimal particle (space craft) from second primary increase and distance of infinitesimal particle (space craft) from first primary body decreases. For given Jacobi constant and given number of loops, as solar radiation pressure increases from 1 to 0.9845, initial velocity decreases and distance of infinitesimal particle (space craft) from second primary decreases. So, distance of infinitesimal particle (space craft) from first primary increases. Thus, the effect of Jacobi constant C and solar radiation pressure q is opposite in nature. For given value of solar radiation pressure q and Jacobi constant, as number of loops increases, distance of infinitesimal particle (space craft) from second primary increases and distance of infinitesimal particle (space craft) from first primary decreases. It is further observed that for Sun-Mars system, single loop orbit for q = 0.995 and C = 2.96 is closest to Mars and this distance is 7.521 × 10^{5} km, whereas for Sun-Earth system, single loop orbit for q = 0.99 and C = 2.95 is closest to Earth and this distance is 7.535 × 10^{5} km. Since these orbits can be used for designing low-energy space mission. Hence detailed study is presented in this paper.

Stability analysis of this family of orbit indicates that these orbits having smaller stability region in comparison to “f” family orbit. So, these orbits can be used as a transfer trajectory as less amount of fuel required for transferring of satellite from one orbit to another orbit. For each pair of (q, C), there are two separatrices exist where stability of periodic orbit becomes zero.

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 Solar Radiation Pressure of Bigger Primary on the Evolution of Periodic Orbits. International Journal of Astronomy and Astrophysics, 6, 464-493. http://dx.doi.org/10.4236/ijaa.2016.64037