Analysis of Effect of Oblateness of Smaller Primary on the Evolution of Periodic Orbits

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1. Introduction

A low-energy transfer trajectory [1] is a trajectory in space that allows spacecraft to change the existing orbits using very small amount of fuel in comparison with Hohmann transfer orbit. To construct low-energy transfer trajectories, we require periodic orbits of spacecrafts around both primary bodies. Space agencies are trying to place telescopes into deep space like NASA’s Terrestrial Planet Finder and ESA’s Darwin project, to find Earth-like planets around other stars exhibiting life. In order to make space mission to explore different aspects of far away objects, it is required to determine special types of orbits that cannot be found by classical approaches.

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 [2] .

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. [1] studied low energy transfer to the Moon. In this study, they considered four bodies Sun-Earth-Moon- Spacecraft as coupled three body systems, namely Sun-Earth-spacecraft and Earth-Moon- spacecraft. They constructed low energy transfer trajectories from the earth which executed ballistic capture at the Moon.

[3] is a fundamental book on the RTBP. Because of the non-linear nature of the equations of motion involved in the study of the system, analytical methods may not give illuminating insights into the solution. So, Poincare surface of section (PSS) is widely used for finding the orbits of the infinitesimal particle (spacecraft) and analyzing periodic, quasi periodic and chaotic orbits. Solar system dynamics by [4] provide a detailed analysis of periodic orbits using PSS technique.

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. [5] analyzed the PSS for Earth-Moon system without considering any perturbation. [6] also had analysed Sun-Mars system by incorporating perturbations due to solar radiation. They have identified periodic, quasi-periodic solutions and chaotic regions from the PSS. Similar studies have been made by [7] for Saturn Titan system. [8] analyzed the family “f” orbits around smaller primary under RTBP for 14 systems under ideal conditions. [9] had studied Liapunov orbits in the photogravitational RTBP with oblates. In this paper they study periodic motion around collinear equilibrium points for two and three dimension. Using numerical analysis they studied the effect of solar radiation pressure and oblateness on location and stability. [10] studied periodic orbits of the Hill problem by considering solar radiation and oblateness as perturbing forces. [11] had studied equilibrium points and periodic orbits of planar restricted three-body problem when both primaries are tri axial rigid bodies. [12] analyzed Periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body. [13] studied Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body. [14] computed series of horizontally critical symmetric periodic orbits of the six basic families of the restricted three-body problem when the more massive primary is an oblate spheroid. The vertical stability of the horizontally critical orbits is also computed.

A large number of periodic orbits were generated by [15] in the framework of RTBP numerically. He classified the orbits into ten families. His work widely referred in literature. These studies explore the regions of the phase space that contain sensitive dependence on initial conditions. [16] [17] and [18] made extensive study on phase space by exploring large portion of the phase plane with Poincare surfaces of section (PSS) [19] . This technique was also used by [20] [21] [22] and [23] to explore the phase space of a system with the Sun-Jupiter mass ratio.

Recently, [24] analyzed Sun and Saturn centered periodic orbits with solar radiation pressure and oblateness and found that there is a substantial effect of solar radiation pressure on position and geometry of infinitesimal particle orbit. [25] had studied family “f” orbits and their stability for Sun-Saturn system with perturbation. In their study Sun was taken as a source of radiation and Saturn was considered as oblate spheroid. By taking the actual coefficient of oblateness for Saturn and different values of solar radiation pressure, the family “f” orbits have been analyzed in detail.

Periodic orbits of spacecraft around two primaries are used to construct low energy trajectory. [26] established three classes with orbits around both primaries depending on motion of spacecraft is prograde or retrograde in the rotating system as well as fixed system. In this paper we have analyzed periodic orbits around both primaries with retrograde motion in rotating system and analyzed periodic orbits having number of loops from 1 to 5 for different pairs of oblateness coefficient A_{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.

2. Equations of Motion

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,

(1)

where

(2)

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

. Following [27] , the equations of motion of the spacecraft are

(3)

where

(4)

Here

(5)

and

(6)

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

(7)

where C is Jacobi constant of integration given by

(8)

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 can be plotted every time the particle has y = 0, whenever the trajectory intersects the plane in a particular direction, say > 0. We have constructed Poincare surface section (PSS) on the x, plane. The initial values were selected along the Ox-axis by using intervals of length 0.001. By giving different value of C we can plot the trajectories, and then analysis of orbits can be done.

3. Results and Discussion

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, [5] . Thus, for the Sun-Earth and Sun-Mars systems, mass factor μ are 0.000003002 and 0.0000003212 respectively. Equatorial and polar radii of Earth are 6378.1 kms. and 6356.8 kms. and that of Mars are 3396.2 kms. and 3376.2 kms., respectively. The distance between Sun and Earth is taken as 149,600,000 kms. and distance between Sun and Mars is 227,940,000 kms. So, oblateness coefficient calculated from Equation (2) for Sun-Earth and Sun-Mars have values A_{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. and by setting in to this equation y = 0 and > 0 the resulting inequality defines the regions allowed to motion on the PSSs. Thus, we eliminate the possibility of excluded region for secondary body between Sun and Earth (or Mars). This maximum value of C is considered as admissible value of C.

Table 1 and Table 2 show range of admissible values of C for Sun-Mars and Sun- Earth systems. It can be observed that for both systems as oblateness increases, admissible range of C increases. But this increment is larger for Sun-Earth system than Sun- Mars system. So, we can say that as mass factor increases, effect of oblateness increases and due to that admissible range of C increases.

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. Figure 1 is PSS constructed for Sun-Mars system when (A_{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 Figure 1. The arcs of PSS are known as islands whose center gives periodic orbit.

In a similar way, we can obtain PSS for Sun-Earth which is shown in Figure 2. This PSS is also constructed for the pair (A_{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

Table 1. Admissible range of C for Sun-Mars system.

Table 2. Admissible range of C for Sun-Earth system.

Figure 1. PSS for A_{2} = 0.0005 and C = 2.93, for x = [0.8, 1], t = 10,000 for Sun-Mars system.

Figure 2. PSS for A_{2} = 0.0005 and C = 2.93, for x = [0.8, 1], t = 10,000 for Sun-Earth system.

(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 Table 3. It is observed from the table that a change in C in the range (2.93, 2.96) affects the location of the periodic orbits. Oblateness also affects the location of the periodic orbit. Similarly, the effects of C and A_{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 Table 4. Size of the island gives stability of the corresponding orbit.

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 Figure 3 that the width of the orbit decreases continuously as the number of loops increases. Further in all cases the infinitesimal particle (spacecraft) orbits

Table 3. Analysis of periodic orbit for different pairs of A_{2} and C for Sun-Mars system.

Table 4. Analysis of periodic orbit for different pairs of A_{2} and C for Sun-Earth system.

(a) Sun-Mars system C = 2.93, A_{2} = 0.0005, period 13. (b) Sun-Earth system C = 2.93, A_{2} = 0.0005, period 13.(c) Sun-Mars system C = 2.93, A_{2} = 0.0005, period 19. (d) Sun-Earth system C = 2.93, A_{2} = 0.0005, period 19.(e) Sun-Mars system C = 2.93, A_{2} = 0.0005, period 26. (f) Sun-Earth system C = 2.93, A_{2} = 0.0005, period 26.(g) Sun-Mars system C = 2.93, A_{2} = 0.0005, period 32. (h) Sun-Earth system C = 2.93, A_{2} = 0.0005, period 32.(i) Sun-Mars system C = 2.93, A_{2} = 0.0005, period 38. (j) Sun-Earth system C = 2.93, A_{2} = 0.0005, period 38.

Figure 3. Periodic orbits around both primaries for Sun-Mars and Sun-Earth systems.

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 Figure 4 we have shown the variation of position of closed periodic orbit with single-loop for oblateness in the range (0, 0.0005) for Sun-Mars system corresponding to Jacobi constants C = 2.93, 2.94, 2.95 and 2.96. From Figure 5 it is clear that the position of the orbits recedes away from Mars when the oblateness increases and C decreases.

Similar kind of conclusion can be drawn from Figure 5 for the Sun-Earth system.

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 Figure 6 and Figure 7.

From Figure 8 and Figure 9, it can be observed that for given oblateness, location of periodic orbit moves away from second primary as number of loops in periodic orbit increase. Also, as oblateness increases, location of periodic orbit moves away from second primary.

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. Figure 10 shows stability region for A_{2} = 0.0001 for three loop orbits. The left and right tips of the island are plotted by red and green curves, respectively. From Figure 10 it is clear that size of stability region is very small in comparison to “f” family orbit [9] . So, these periodic orbits can be used as a transfer orbits as they are not stable. So, secondary

Figure 4. Variation in location of single-loop periodic orbit around Sun-Mars system due to oblateness.

Figure 5. Variation in location of single-loop periodic orbit around Sun-Earth due to oblateness.

body required few amount of fuel than Hohmann transfer. Figure 11 shows amplitude for three loops orbit when A_{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. Figure 12 shows size of the island for C = 2.94 for three loops orbit when A_{2} = 0.0001 which is 0.0008 whereas Figure 13 shows PSS of first separatrix at C = 2.95 which is looks like a straight line where as for “f” family orbit it is triangular due to third order resonance [9] . It can be

Figure 6. Variation in location of five loops periodic orbit around Sun-Mars system due to oblateness.

Figure 7. Variation in location of five loops periodic orbit around Sun-Earth system due to oblateness.

seen that size of this island is zero. Figure 14 shows three loops orbit corresponding to first separatrix when A_{2} = 0.0001. Figure 15 shows for C = 2.955 again size of island increases and it becomes 0.0006, where as for C = 2.96 again size of island becomes zero, which is second separatrix as shown in Figure 16. Three loops orbit corresponding to second separatrix is given in Figure 17.

In Table 5 initial velocity of secondary body is denoted by V, D_{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

Figure 8. Variation in location of periodic orbit of secondary body around Sun and Mars for C = 2.93 due to number of loops for different oblateness A_{2}.

Figure 9. Variation in location of periodic orbit of secondary body around Sun-Earth system for C = 2.93 due to number of loops for different oblateness A_{2}.

in km. Similar notations are used in Table 6 also. In Table 6 distance of secondary body from Earth is denoted by D_{1}.

(9)

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,

Figure 10. Stability analysis for three loops orbit for Sun-Earth system when A_{2} = 0.0001.

Figure 11. Amplitude for three loops orbit for Sun-Earth system when A_{2} = 0.0001.

Distance (10)

Velocity (11)

where L is the distance between the centers of both primaries in km. O is the orbital velocity of second primary around first primary [2] . For Sun-Mars and Sun-Earth system L = 227,940,000 and 149,600,000 km respectively. Mean orbital velocity of Mars around Sun and Earth around Sun are 24.07 km∙sec^{−1} and 29.78 km∙sec^{−1} respectively.

It can be observed from Table 5 and Table 6 that for given oblateness and given loop orbit as C decreases, V and D_{1} increase while D_{2} decreases. For a given C and given

Figure 12. Enlarge view of PSS for C = 2.94 when A_{2} = 0.0001.

Figure 13. Enlarge view of PSS of first separatrix for two loops orbit for C = 2.95 when A_{2} = 0.0001.

Figure 14. Orbit at first separatrix corresponding to C = 2.95, A_{2} = 0.0001 and x = 0.894.

Figure 15. Enlarge view of PSS for C = 2.955 when A_{2} = 0.0001.

Figure 16. Enlarge view of PSS of second separatrix for three loops orbit for C = 2.96 when A_{2} = 0.0001.

Figure 17. Orbit at second separatrix corresponding to C = 2.96, A_{2} = 0.0001, and x = 0.915.

Table 5. Location, velocity and distance of orbit from both primaries for A_{2} = 0.00001, 0.00005, 0.0001, 0.0005 for Sun-Mars system.

Table 6. Location, velocity and distance of orbit from both primaries for A_{2} = 0.00001, 0.00005, 0.0001, 0.0005 for Sun-Earth system.

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 Table 5, it is observed that single-loop orbit for A_{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 Table 6 also.

From Table 6, it is observed that single-loop orbit for A_{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.

4. Prediction of Orbit through Regression Analysis

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 Table 7. Using regression analysis we have displayed the predicted position of the orbit for different values of C. The predicted and exact values of positions together with error estimates are displayed in Table 8. The best fit curve for single-loop is a straight line with equation

Table 7. Variation in location of periodic orbit when A_{2} = 0.00001 for Sun-Mars system.

Table 8. Prediction and error for periodic orbit when A_{2} = 0.00001 for Sun-Mars system.

and the coefficient of determination R^{2} = 0.999, whereas, for two- loop periodic orbit regression line is and R^{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 Table 9 and Table 10.

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

with R^{2} = 1 and for two-loops periodic orbit regression curve is with R^{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 Table 11. The predicted and exact values of the position of orbits together with error estimates using regression analysis are shown in Table 12. For single-loop the best fit regression curve is with R^{2} = 1, where as for two loops orbit with R^{2} = 1.

Table 9. Variation in location of periodic orbit when A_{2} = 0.0005 for Sun-Mars system.

Table 10. Prediction and error for periodic orbit when A_{2} = 0.0005 for Sun-Mars system.

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 Table 13.

Using regression analysis we have displayed predicted and exact values of position together with error estimates are shown in Table 14. As in the case of Sun-Mars system the error estimates decreases, with increase in C for both single-loop and two-loops orbits. The best fit curve for single-loop orbit is given by with R^{2} = 0.999, and for two loops orbit regression line with R^{2} = 0.999.

Similar estimates are displayed for A_{2} = 0.00005 in Table 15 and Table 16, for Sun- Earth system.

For single-loop orbit regression curve is given by

Table 11. Variation in location of periodic orbit when C = 2.96 for Sun-Mars system.

Table 12. Prediction and error for periodic orbit when C = 2.96 for Sun-Mars system.

Table 13. Variation in location of periodic orbit when A_{2} = 0.00001 for Sun-Earth system.

Table 14. Prediction and error for periodic orbit when A_{2} = 0.00001 for Sun-Earth system.

Table 15. Variation in location of periodic orbit when A_{2} = 0.00005 for Sun-Earth system.

Table 16. Prediction and error for periodic orbit when A_{2} = 0.00005 for Sun-Earth system.

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

with R^{2} = 1.

The variation of position x with oblateness A_{2} for Sun-Earth system is shown in Table 17 for a fix value of C = 2.96.

The predicted and exact value of position together with error estimates are shown in Table 18.

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

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

5. Conclusions

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

Table 17. Variation in location of periodic orbit when C = 2.96 for Sun-Earth system.

Table 18. Prediction and error for periodic orbit when C = 2.96 for Sun-Earth system.

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.

Acknowledgements

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

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