^{1}

^{2}

We analyze the periodic orbits of “f” family (simply symmetric retrograde periodic orbits) and the regions of quasi-periodic motion around Saturn in the photo gravitational Sun-Saturn system in the framework of planar circular restricted three-body problem with oblateness. The location, nature and size of these orbits are studied using the numerical technique of Poincare surface of sections (PSS). In this paper we analyze these orbits for different solar radiation pressure (q) and actual oblateness coefficient of Sun Saturn system. It is observed that as Jacobi constant (C) increases, the number of islands in the PSS and consequently the number of periodic and quasi-periodic orbits increase. The periodic orbits around Saturn move towards the Sun with decrease in solar radiation pressure for given value of “C”. It is observed that as the perturbation due to solar radiation pressure decreases, the two separatrices come closer to each other and also come closer to Saturn. It is found that the eccentricity and semi major axis of periodic orbits at both separatrices are increased by perturbation due to solar radiation pressure.

Restricted three body problem (RTBP) describes the motion of massless body which moves under the gravitational effect of two finite masses called primaries. The primaries are supposed to move in circular orbits around their center of mass on account of their mutual attraction. Usually Sun and anyone of its planets are taken as primaries. The secondary body is taken as the satellite of the primary planet or asteroid or comet or artificial satellite.

The RTBP has many applications in the solar system dynamics. [

The solar radiation pressure force Fp changes with the distance by the same law as the gravitational attraction force Fg 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

minology of [

where,

Here AE and AP represent equatorial and polar radii of Saturn and R is the distance between Sun and Saturn. 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. The origin of the system is positioned at the center of mass of the primaries while the more massive and less massive primaries always lie on the Ox axis at P(−μ, 0) and Q(1−μ, 0), respectively. Following [

And

where,

Here

And

Also,

which is Jacobi constant of integration.

The Poincare surface of section (PSS) technique is a method for determining the regular or chaotic nature of the trajectory. The numerical technique of PSS has been utilized by a large number of researchers to generate periodic and quasi-periodic orbits [

For Sun-Saturn system the mass of Sun m_{1} = 1.9881 × 10^{30} kg, mass of Saturn m_{2} = 568.36 × 10^{24} kg [^{ }

Thus,

Also equatorial radius of Saturn is 60,268 km, polar radius of Saturn is 54,364 km. and distance between Sun and Saturn is 1,433,000,000 km. So, according to the equation 3 the oblateness coefficient is calculated as 6.59158 × 10^{−}^{11}.

q = 1 means that there is no perturbation due to solar radiation pressure and q < 1 indicates that we are including perturbation due to solar radiation pressure. For a given q, selection of C is not arbitrary. By solving Equation 8 w.r.t.

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

1 | 3.018 | 3.019 | x = 0.946 to x = 0.964 | 0.018 |

3.02 | x = 0.941 to x = 0.968 | 0.027 | ||

3.021 | x = 0.937 to x = 0.970 | 0.033 | ||

3.022 | x = 0.934 to x = 0.972 | 0.038 | ||

3.023 | x = 0.930 to x = 0.974 | 0.044 | ||

3.024 | x = 0.928 to x = 0.975 | 0.047 | ||

0.9845 | 2.985 | 2.986 | x = 0.948 to x = 0.960 | 0.012 |

2.987 | x = 0.941 to x = 0.965 | 0.024 | ||

2.988 | x = 0.936 to x = 0.968 | 0.032 | ||

2.989 | x = 0.933 to x = 0.971 | 0.038 | ||

2.990 | x = 0.929 to x = 0.972 | 0.043 | ||

2.991 | x = 0.926 to x = 0.975 | 0.049 | ||

0.9645 | 2.943 | 2.944 | x = 0.945 to x = 0.957 | 0.012 |

2.945 | x = 0.938 to x = 0.963 | 0.025 | ||

2.946 | x = 0.933 to x = 0.966 | 0.033 | ||

2.947 | x = 0.929 to x = 0.969 | 0.04 | ||

2.948 | x = 0.926 to x = 0.972 | 0.046 | ||

2.949 | x = 0.923 to x = 0.972 | 0.049 | ||

0.9 | 2.807 | 2.808 | x = 0.929 to x = 0.951 | 0.022 |

2.809 | x = 0.923 to x = 0.956 | 0.033 | ||

2.81 | x = 0.918 to x = 0.960 | 0.042 | ||

2.811 | x = 0.914 to x = 0.964 | 0.05 | ||

2.812 | x = 0.910 to x = 0.965 | 0.055 | ||

2.813 | x = 0.907 to x = 0.967 | 0.06 |

The semi-major axis a and eccentricity e of the Saturn centered orbits are given by [

where,

And

Period T of Saturn’s orbit is given by

using Equation 2. For Saturn, T = 0.283185306868965 and A_{2} = 6.59158 × 10^{−}^{11}. From Kepler’s third law semi major axis of Saturn’s orbit is obtained as a = 1.000867863227069.

From

In

From

q | C | x | a | e | D | T | L | R | A |
---|---|---|---|---|---|---|---|---|---|

1 | 2.8 | 0.56455 | 1.0022 | 0.43694 | 0.8705 | 6.272 | 0.5438 | 0.577 | 0.0332 |

1 | 2.795 | 0.5594 | 1.0021 | 0.442046 | 0.8816 | 6.27 | 0.5396 | 0.5792 | 0.0396 |

1 | 2.79 | 0.55435 | 1.0021 | 0.44706 | 0.8917 | 6.28 | 0.547 | 0.559 | 0.012 |

1 | 2.785 | 0.54935 | 1.0021 | 0.45202 | 0.9017 | 6.28 | 0.5487 | 0.55 | 0.0013 |

1 | 2.78 | 0.5444 | 1.0020 | 0.4569 | 0.9116 | 6.28 | 0.5418 | 0.5468 | 0.005 |

0.9645 | 2.8 | 0.6362 | 0.9220 | 0.31052 | 0.7038 | 6.26 | 0.6324 | 0.639 | 0.0066 |

0.9645 | 2.795 | 0.6296 | 0.9209 | 0.31052 | 0.7038 | 6.28 | 0.629 | 0.6302 | 0.0012 |

0.9645 | 2.79 | 0.62315 | 0.9198 | 0.32305 | 0.7299 | 6.28 | 0.6213 | 0.6253 | 0.004 |

0.9645 | 2.785 | 0.61685 | 0.9188 | 0.32912 | 0.7422 | 6.28 | 0.613 | 0.6236 | 0.0106 |

0.9645 | 2.78 | 0.61065 | 0.9177 | 0.3350 | 0.7554 | 6.28 | 0.606 | 0.614 | 0.008 |

0.9345 | 2.8 | 0.72165 | 0.8834 | 0.18435 | 0.5123 | 6.22 | 0.6972 | 0.74 | 0.0428 |

0.9345 | 2.795 | 0.7124 | 0.8814 | 0.1928 | 0.5316 | 6.23 | 0.6853 | 0.7334 | 0.0481 |

0.9345 | 2.79 | 0.70345 | 0.8793 | 0.20111 | 0.5495 | 6.23 | 0.6752 | 0.725 | 0.0498 |

0.9345 | 2.785 | 0.6948 | 0.8773 | 0.20909 | 0.5662 | 6.24 | 0.6679 | 0.7165 | 0.0486 |

0.9345 | 2.78 | 0.68645 | 0.8754 | 0.2168 | 0.5826 | 6.24 | 0.6658 | 0.707 | 0.0412 |

q | C | x | a | E | D | T | L | R | A |
---|---|---|---|---|---|---|---|---|---|

1 | 2.6 | 0.4 | 1.0011 | 0.60048 | 1.2 | 6.281 | 0.394 | 0.4 | 0.006 |

2.7 | 0.47339 | 1.0015 | 0.52744 | 1.0536 | 6.2781 | 0.4687 | 0.4758 | 0.0072 | |

2.8 | 0.56455 | 1.0022 | 0.43694 | 0.8705 | 6.272 | 0.5438 | 0.577 | 0.0332 | |

2.9 | 0.6888 | 1.0041 | 0.31455 | 0.6222 | 6.249 | 0.6652 | 0.7111 | 0.0458 | |

2.91 | 0.7045 | 1.0044 | 0.29920 | 0.5915 | 6.239 | 0.6836 | 0.7254 | 0.0418 | |

2.93 | 0.7392 | 1.0056 | 0.26566 | 0.5218 | 6.22 | 0.7124 | 0.758 | 0.0456 | |

2.95 | 0.7796 | 1.0075 | 0.227144 | 0.4414 | 6.175 | 0.7568 | 0.798 | 0.0412 | |

2.97 | 0.83 | 1.012 | 0.181147 | 0.341 | 6.05 | 0.8109 | 0.8501 | 0.0392 | |

2.975 | 0.8452 | 1.0142 | 0.16807 | 0.3108 | 5.973 | 0.8290 | 0.8622 | 0.0332 | |

2.98 | 0.8622 | 1.0177 | 0.15436 | 0.2768 | 5.86 | 0.8401 | 0.877 | 0.0369 | |

2.985 | 0.8816 | 1.0234 | 0.14036 | 0.2374 | 5.638 | 0.8610 | 0.894 | 0.033 | |

2.99 | 0.9047 | 1.0349 | 0.1277 | 0.1914 | 5.2 | 0.894 | 0.9107 | 0.0167 | |

2.995 | 0.93334 | 1.0648 | 0.1255 | 0.1337 | 4.1 | 0.9297 | 0.94 | 0.0103 | |

3.006 | 0.97585 | 1.2361 | 0.2116 | 0.0482 | 1.2 | 0.9559 | 0.986 | 0.0301 | |

3.01 | 0.98117 | 1.3009 | 0.2466 | 0.0368 | 0.9 | 0.9614 | 0.9934 | 0.0320 | |

3.014 | 0.98465 | 1.3649 | 0.2793 | 0.0303 | 0.7 | 0.9713 | 0.9948 | 0.0235 | |

3.018 | 0.987101 | 1.4302 | 0.31045 | 0.0249 | 0.492 | 0.9753 | 0.9960 | 0.0207 | |

0.9845 | 2.57 | 0.39587 | 0.9374 | 0.5777 | 1.1981 | 6.28 | 0.3786 | 0.412 | 0.0334 |

2.67 | 0.46929 | 0.9486 | 0.5054 | 1.0507 | 6.277 | 0.4456 | 0.4906 | 0.0450 | |

2.77 | 0.5606 | 0.9590 | 0.4157 | 0.8684 | 6.267 | 0.5399 | 0.5811 | 0.0412 | |

2.87 | 0.6853 | 0.9701 | 0.2941 | 0.6197 | 6.245 | 0.6637 | 0.7073 | 0.0436 | |

2.89 | 0.718 | 0.9729 | 0.2627 | 0.554 | 6.3 | 0.693 | 0.7386 | 0.0456 | |

2.91 | 0.75555 | 0.9764 | 0.2270 | 0.4795 | 6.2 | 0.734 | 0.7757 | 0.0417 | |

2.93 | 0.80075 | 0.9812 | 0.1851 | 0.38925 | 6.2 | 0.7775 | 0.8221 | 0.0446 | |

2.95 | 0.8612 | 0.9924 | 0.1340 | 0.2688 | 5.9 | 0.8363 | 0.8771 | 0.0408 | |

2.97 | 0.9609 | 1.1171 | 0.1416 | 0.0721 | 1.99 | 0.954 | 0.9646 | 0.0106 | |

2.975 | 0.9736 | 1.1905 | 0.1834 | 0.0474 | 1.174 | 0.954 | 0.9838 | 0.0298 | |

2.98 | 0.9804 | 1.2595 | 0.2226 | 0.0346 | 0.78 | 0.9646 | 0.9911 | 0.0265 | |

2.985 | 0.98455 | 1.3269 | 0.25884 | 0.0275 | 0.555 | 0.973 | 0.994 | 0.0210 | |

0.9645 | 2.443 | 0.33585 | 0.8428 | 0.6014 | 1.3042 | 6.28 | 0.3211 | 0.347 | 0.0259 |

2.543 | 0.39855 | 0.8672 | 0.5405 | 1.1785 | 6.276 | 0.3777 | 0.4155 | 0.0378 | |

2.643 | 0.4737 | 0.8893 | 0.4675 | 1.0283 | 6.273 | 0.4514 | 0.49 | 0.0386 | |

2.743 | 0.5681 | 0.9101 | 0.3761 | 0.8399 | 6.268 | 0.5487 | 0.5903 | 0.0416 | |

2.843 | 0.69995 | 0.9319 | 0.2496 | 0.5761 | 6.235 | 0.6796 | 0.72 | 0.0404 | |

2.86 | 0.72995 | 0.9363 | 0.2213 | 0.517 | 6.215 | 0.7058 | 0.7483 | 0.0425 | |

2.88 | 0.771 | 0.9427 | 0.1833 | 0.4350 | 6.2 | 0.7493 | 0.789 | 0.0397 | |

2.9 | 0.8225 | 0.9519 | 0.1376 | 0.3325 | 6.1 | 0.8043 | 0.8409 | 0.0366 | |

2.92 | 0.9017 | 0.9819 | 0.0848 | 0.1763 | 4.94 | 0.896 | 0.9048 | 0.0088 | |

2.943 | 0.98125 | 1.2328 | 0.2051 | 0.0297 | 0.618 | 0.9694 | 0.99 | 0.0206 | |

0.9345 | 2.18 | 0.2265 | 0.6523 | 0.6526 | 1.5027 | 6.284 | 0.215 | 0.239 | 0.0240 |

2.28 | 0.27231 | 0.6965 | 0.6089 | 1.4108 | 6.283 | 0.2583 | 0.2877 | 0.0294 | |

2.38 | 0.32535 | 0.7367 | 0.5583 | 1.3048 | 6.282 | 0.311 | 0.339 | 0.028 | |

2.48 | 0.38755 | 0.7737 | 0.4991 | 1.1805 | 6.282 | 0.3664 | 0.4062 | 0.0398 | |

2.58 | 0.4622 | 0.8080 | 0.4282 | 1.0308 | 6.279 | 0.4432 | 0.4821 | 0.0389 | |

2.68 | 0.55585 | 0.8409 | 0.3394 | 0.8442 | 6.278 | 0.5318 | 0.5773 | 0.0455 | |

2.78 | 0.68645 | 0.8754 | 0.2168 | 0.5826 | 6.276 | 0.6658 | 0.707 | 0.0412 | |

2.8 | 0.7217 | 0.8835 | 0.1843 | 0.5123 | 6.3 | 0.6976 | 0.7423 | 0.0447 | |

2.82 | 0.7631 | 0.8927 | 0.1468 | 0.4299 | 6.2 | 0.7419 | 0.7803 | 0.0384 | |

2.84 | 0.8156 | 0.9054 | 0.1017 | 0.3264 | 6.1 | 0.7982 | 0.8333 | 0.0351 | |

2.86 | 0.899 | 0.9417 | 0.0509 | 0.1649 | 4.8 | 0.898 | 0.9 | 0.002 | |

2.88 | 0.97591 | 1.2109 | 0.1923 | 0.0331 | 0.72 | 0.966 | 0.9895 | 0.0235 |

Each point in the island around its center corresponds to a quasi-periodic orbit. The largest of these islands is the one with the maximum amplitude of oscillation that are still stable. The regular regions of PSS are defined by a periodic orbit surrounded by an area of quasi-periodic orbits. The regular regions can be interpreted as regions of stability in the sense that outside them the motion is certainly unstable (chaotic) and inside them the motion is in general regular.

Figures 7-10 give the size of the stability region for q = 1, 0.9845, 0.9645 and 0.9345 respectively. The left and right tips of the island are plotted by green and red curves, respectively. It is seen that in each figure there are two separatrices where stability of the periodic orbit is zero as the size of the island is zero at separatrix. For each PSS corresponding to (q, C), we get two separatrices. We have analyzed the stability curve for three stages, which are divided by two separatrices. In the first stage, the quasi periodic orbit oscillates around the periodic

orbits in such a way that the farthest point from the Saturn is in the line of conjunction is a pericenter. Stability of periodic orbit decreases up to first separatrix. At the first separatrix stability becomes zero as island disappears. In the second stage, island reappear and consequently the stability increases. Then size of the island starts decreasing up to second separatrix where again the island disappears. In the third stage, the quasi periodic orbit oscillates around the periodic orbit in such a way that the closest point to the Saturn in the line of conjunction is pericenter. After second separatrix, island reappears and consequently the stability increases.

From Figures 11-20, it can be shown that there is a transition in the way the quasi periodic orbit oscillates about the periodic orbit before and after the separatrix due to third order resonance. The disappearance of the region of stability is caused by the intersection of the central periodic orbit and the unstable periodic orbit lying at the three corners of the triangular stability region [

For q = 1, size of stability region increases for values of C in the range [2.6 2.9]. Stability is maximum at C = 2.9 and then decrease till first separatrix obtained at C = 2.9928. Again size of stability region increases and then decrease for small interval of C. Second separatrix obtained at C = 2.999. Again size of stability region increases.

Figures 11-15 shows PSS of islands for q = 1corresponding to C = 2.991, 2.9928, 2.995, 2.999 and 3.0 respectively.

For q = 0.9345 size of stability region increases for values of C in the range [2.18, 2.68]. For q = 0.9345 at C = 2.86 and 2.867.

Reduction in q increases perturbation due to solar radiation pressure which affects location, shape, size and stability of periodic orbits located at center of separatrix.

orbits at both separatrices decreases as q drops to the value 0.9845 and then slightly increases. The effect of solar radiation pressure on the semi-major axis and eccentricity of the periodic orbits are shown in

The present paper analyses the effect of radiation pressure and oblateness on family “f” of periodic orbit (PO) in the Sun-Saturn system. It is observed that radiation pressure has significant influence on evolution of family “f” of PO. As radiation pressure increases the admissible value of C decreases and PO shifts towards Saturn. Also the geometric parameters of the orbits such as diameter, eccentricity and semi major axis decrease. An increment in C increases the semi-major axis of f family of periodic orbits, the eccentricity decreases up to certain

Solar radiation pressure q | Jacobi constant C | Location of periodic orbit x | Diameter of periodic orbit | Semi major axis of periodic orbit | Eccentricity of periodic orbit |
---|---|---|---|---|---|

1 | 2.9928 | 0.92 | 0.161 | 1.0476 | 0.1238 |

1 | 2.999 | 0.95585 | 0.0881 | 1.11897 | 0.1474 |

0.9845 | 2.962 | 0.91725 | 0.1574 | 1.02272 | 0.1055 |

0.9845 | 2.968 | 0.95275 | 0.0872 | 1.0878 | 0.1261 |

0.9645 | 2.9215 | 0.9106 | 0.1594 | 0.98936 | 0.0828 |

0.9645 | 2.928 | 0.9499 | 0.0851 | 1.0542 | 0.1015 |

0.9345 | 2.86 | 0.8991 | 0.1649 | 0.9418 | 0.05103 |

0.9345 | 2.867 | 0.94345 | 0.0845 | 1.0036 | 0.0642 |

value of C and then shows a sudden increase in its value. Diameters of these periodic orbits decrease slowly, but after certain value of C, there is a sudden decrement in value of diameter of periodic orbits. It is also observed that as solar radiation pressure decreases, the location of PO orbits at both separatrices moves towards Saturn. Also, as q moves towards 1, value of Jacobi constant corresponding to both separatrices moves towards 3. It is concluded that the difference between corresponding Jacobi constant decreases as q decreases up to 0.9845 and then increases slightly. It can be seen that the difference between location of two periodic orbits at both separatrices decreases as q drops to the value 0.9845 and then slightly increases. In other words, as the perturbation due to solar radiation pressure decreases, the two separatrices come closer to each other and also come closer to Saturn. It is found that the eccentricity and semi major axis of periodic orbits at both separatrices are increased by perturbation due to solar radiation pressure. The evolution of the family of periodic orbits can be divided into three stages separated by two separatrices. These are caused by the 3:1 resonance. That is, the bifurcation of the basic family f with the family f 3 of triple periodic orbits branching from f at these values of the Jacobi constant. There is a change in the direction of the islands around the periodic orbits before and after these separatrice. In other words, there is a change in the way the quasi-periodic orbits oscillate around the periodic orbits before and after these separatrices. Family f can be used for patching of trajectory of satellite, that is, joining of two or more orbits to obtain a trajectory.

The authors thank Pooja Dutt, Applied Mathematics Division, Vikram Sarabhai Space Centre (ISRO), Thiruvananthapuram, India for her constructive comments for separatix analysis. They also thank the associate editor and referees for their constructive comments, which helped in bringing this paper to the present form.

Niraj Pathak,V. O. Thomas, (2016) Evolution of the “f” Family Orbits in the Photo Gravitational Sun-Saturn System with Oblateness. International Journal of Astronomy and Astrophysics,06,254-271. doi: 10.4236/ijaa.2016.63021