The Circular Restricted Three-Body Problem (CRTBP) with more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries is considered to generate the halo orbits around L1 and L2 for the seven satellites (Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus) of Saturn in the frame work of CRTBP. It is found that the oblateness effect of Saturn on the halo orbits of the satellites closer to Saturn has significant effect compared to the satellites away from it. The halo orbits L1 and L2 are found to move towards Saturn with oblateness.
Three-Body Problem formulated by Newton provided route to the analysis of closed form analytical solution. This solution remains elusive even today, as one has never been found for the three-body problem. Euler developed the restricted problem using a rotating frame in the 1770s and located collinear points. Along with Euler, Lagrange considered this form of the three-body problem and calculated the locations of equilateral points, often known as libration or Lagrange points. Jacobi studied the circular- restricted problem (CRTBP) and found that an integral of motion exists. Plummer [
The subject of periodic solutions of the CRTBP has received enormous attention in the past few decades. Since the late twentieth century until today, enormous amount of research has enriched the study of CRTBP, but the influence of the various perturbing forces has not been studied in many of such interesting problems. The classical model does not account for some of the perturbing forces such as oblateness, solar radiation pressure, Poynting-Robertson drag effects and variation in the mass of the primaries. Some of significant works in RTBP with oblateness effects are done by Sharma and Subba Rao [
In the present study, we consider the restricted three-body problem by considering the more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion (Sharma and Subba Rao [
The equations of motion for the restricted three-body problem are considered with
The origin of the co-ordinate system is the barycentre of the two primaries with the more massive primary lying to the left of the origin and the smaller primary to the right as shown in
AE and AP are dimensional equatorial and polar radii of the more massive primary and R is the distance between the primaries.
The non-dimensional mass ratio
The parameter
The three-dimensional equations of motion are:
where
In the above expression
From the equations of motion (1)-(3), it is apparent that an equilibrium solution exists relative to the rotating frame when the partial derivative of the pseudo potential function are all zero, i.e.
Richardson’s third-order approximation provides a deep qualitative insight. The approximate solution is sufficient for generating accurate motion near L1 and L2. Analytical approximation need to be combined with numerical techniques to generate a halo orbit accurate enough for mission design. In the present study to generate the halo orbits, we use analytical approximation as the first guess for the differential correction process, we have modified the third-order approximation of Thurman and Worfolk [
For obtaining an analytical solution, following Tiwary and Kushvah [
The equations of motion can be written as
where
The upper sign in the above equations depicts the Lagrangian point L1 and the lower sign corresponds to L2.
The usage of Legendre polynomials can result in some computational advantages, when non-linear terms are considered. The distance between these Lagrangian points and the smaller primary is considered to be the normalized unit as in Koon et al. [
The non-linear terms are expanded by using the following formula as given by [
where
The above formula is used for expanding the non-linear terms in the equations of motion. The equations of motion after substituting the values of the non-linear terms and carrying out some algebraic manipulations by defining a new variable cm after expanding up to m = 2 become
with
Neglecting the non-linear higher-order terms in Equations (5)-(7), we get
It is clear that the z-axis solution, obtained by putting X = Y = 0, does not depend upon X and Y and c2 > 0. Hence we can conclude that the motion in Z-direction is simple harmonic. The motion in XY-plane is coupled. A fourth degree polynomial is obtained which gives two real and two imaginary roots as eigenvalues:
The solution of the linearized Equations (7)-(9), as derived in [
where
There is a necessity to introduce frequency and amplitude terms to perform the Lindstedt-Poincaré method. The solution of the linearized equations is again written in terms of amplitudes (Ax and Az) and phases (in-plane phase, ϕ and out-of-plane phase, ψ) and the frequencies (λ and
The amplitudes
where l1 and l2 depend upon the roots of the characteristic equation of the linear equation. The correction term,
Hence, any halo orbit can be characterized by specifying a particular out-of-plane amplitude
The phases ϕ and ψ are related as
When Ax is greater than certain value, the third-order solution bifurcates. This bifurcation is manifested through the phase-angle constraint. The solution branches are obtained according to the value of m. For m = 1, Az is positive and we have the northern halo (z > 0) and for m = 3, Az is negative and we have the southern halo (z < 0).
Lindstedt-Poincaré method involves successive adjustments of the frequencies to avoid secular terms and allows one to obtain approximate periodic solution. The equations of motion with non-linear terms up to third-order approximation as in Richardson and Thurman and Worfolk are
A new independent variable τ = ωt is introduced, where
It is to be noted that the values of
The coefficients
We continue the perturbation analysis by assuming the solutions of the form:
where
Substituting Equations (14)-(16) in Equations (11)-(13), and equating the coefficients of the same order of
The first-order equations are obtained by taking the coefficients of the term
The periodic solution to the above equations is
where k = k2.
By collecting the terms of
where
To remove the secular terms, we need to set the value ω1 = 0. The particular solution of the second-order equations are obtained with the help of Maxima software as
The coefficients are given in the Appendix.
By collecting the terms of
where
From the above equations, it is not possible to remove the secular terms by setting ω2 = 0. Hence, the phase relation is used here to remove the secular terms.
The solution of the third-order equation is obtained as
The coefficients are given in the Appendix. Thus, the third-order analytical solution is developed.
The mapping,
The coefficients are given in the Appendix.
The method of differential correction is a powerful application of Newton’s method that employs the state transition matrix (STM) to solve various boundary value problems. Differential correction method is used to determine the initial conditions of the halo orbits from the initial guess [
The equations of motion and state transition matrix are integrated numerically until the trajectory crosses the xz-plane again. The desired final condition is of the form
It is obtained by modifying the known parameters of the initial state vector. Then the orbit will be periodic with period
Saturn, the sixth planet from the Sun, is home to a vast array of intriguing and unique satellites. It is also the largest oblate body in the solar system. Hence, studying the halo orbits about the moons of Saturn is interesting to observe the oblateness effect on the periodic orbits about the collinear points of the Saturn-Satellite systems. Christian Huygens discovered Titan, the first known moon of Saturn in 1655. Jean-Dominique
Cassini made the next four discoveries: Iapetus (1671), Rhea (1672), Dione (1684), and Tethys (1684). Mimas and Enceladus were both discovered by William Herschel in 1789. We have used these known moons of Saturn for our study. The mass parameter μ and the oblateness coefficient A1 of the systems under study are presented in
Mimas is the closest moon to Saturn which is considerably larger than other moons closer to Saturn. Mimas has an enormous crater on one side, the result of an impact that nearly split the moon apart. We have computed the initial guesses for the differential correction process provided in
S.No | System | μ | A1 |
---|---|---|---|
1 | Saturn-Mimas | 0.0000000659 | 0.0042349996 |
2 | Saturn-Enceladus | 0.0000001480 | 0.0025865767 |
3 | Saturn-Tethys | 0.0000010950 | 0.0016835857 |
4 | Saturn-Dione | 0.0000020390 | 0.0010308526 |
5 | Saturn-Rhea | 0.0000032000 | 0.0005275432 |
6 | Saturn-Titan | 0.0002461294 | 0.0000981153 |
7 | Saturn-Iapetus | 0.0000039400 | 0.0000115606 |
Case | x | Z | Time period | ||
---|---|---|---|---|---|
L1 | A1 = 0 | 0.999148645816 | 0.002697118688 | 0.003631032924 | 0.012962186612 |
with A1 | 0.997756317041 | 0.002698311236 | 0.003221853710 | 0.013961626242 | |
L2 | A1 = 0 | 1.001494387394 | 0.002668571267 | 0.004130733335 | 0.051026202231 |
with A1 | 1.001334728492 | 0.002669710410 | 0.004159604147 | 0.047929194009 |
and the L1 halo orbit shifts towards Saturn by 260 km and L2 halo orbit shifts by 11.6 km. We also find that the non-dimensional time period of the halo orbits increases at L1 and decreases at L2 of the Saturn-Mimas system.
Enceladus is the second closest moon next to Mimas at a distance of 237,948 km from Saturn. It displays evidence of active ice volcanism. Cassini observed warm fractures where evaporating ice evidently escapes and forms a huge cloud of water vapour over the South Pole. Similar to Saturn-Mimas system, we compute the halo orbits about the L1 and L2 of Saturn-Enceladus system with the initial guesses provided in
We observe that the halo orbits about L1 and L2 shift towards Saturn with oblateness by 153 km and 25 km, respectively. Similar to Saturn-Mimas system, the non-dimen- sional time period of the halo orbit increases at L1 and decreases at L2.
Case | x | Z | Time period | ||
---|---|---|---|---|---|
L1 | A1 = 0 | 0.997637174347 | 0.002523536751 | 0.002967906138 | 0.009187401493 |
with A1 | 0.996995119904 | 0.002523943758 | 0.002866920875 | 0.012994111232 | |
L2 | A1 = 0 | 1.001898926264 | 0.002482616839 | 0.003088643942 | 0.162015872405 |
with A1 | 1.001794977728 | 0.002488450813 | 0.003107834204 | 0.133201604509 |
Tethys is the third closest and interesting satellite of Saturn at a distance of 294,619 km. Tethys has a huge rift zone called Ithaca Chasma that runs nearly three-quarters of the way around the moon. Telesto and Calypso are the two moons occupying the two Lagrangian points of Saturn-Tethys system. With the help of the initial guess, we compute the halo orbits about L1 and L2 of Saturn-Tethys system using the parameters of
Dione is the fourth closest moon of Saturn with considerable higher mass than Tethys, and is 377,396 km from Saturn. Similar to Tethys, Helene and Poly deuces are the two other moons occupy the corresponding Lagrangian points of Dione in Saturn-Dione system. Using
Rhea is the moon of Saturn orbiting about 527,108 km from Saturn. It experiences the oblateness effect of Saturn very less compared to the previous moons. Using
Case | X | Z | Time period | ||
---|---|---|---|---|---|
L1 | A1 = 0 | 0.994072883928 | 0.003397738416 | 0.003679026495 | 0.019397166689 |
with A1 | 0.993731440476 | 0.003397956122 | 0.003655841907 | 0.021581205672 | |
L2 | A1 = 0 | 1.005479499139 | 0.003310157713 | 0.003821612540 | 0.440763838604 |
with A1 | 1.005469580625 | 0.003312429828 | 0.003818185981 | 0.427826024799 |
Case | X | Z | Time period | ||
---|---|---|---|---|---|
L1 | A1 = 0 | 0.992572578340 | 0.003979398600 | 0.004276068694 | 0.023996523692 |
with A1 | 0.992368151952 | 0.003979543225 | 0.004263465270 | 0.025297934225 | |
L2 | A1 = 0 | 1.006945979742 | 0.003827419168 | 0.004564847025 | 0.712807021300 |
with A1 | 1.006945850052 | 0.003829629598 | 0.004558905658 | 0.701629441557 |
and
Titan is the solar system’s second-largest moon with 5150 km diameter after Ganymede (5362 km) of Jupiter. Titan hides its surface beneath a thick, nitrogen-rich atmosphere. Cassini’s instruments have revealed that Titan possesses many parallels to Earth-clouds, dunes, mountains, lakes, and rivers. Titan’s atmosphere is approximately 95 percent nitrogen with traces of methane. While Earth’s atmosphere extends about 60 km into space, Titan’s extends nearly by 600 km (10 times that of Earth’s atmosphere) into space. Effect of oblateness of Saturn on Titan in the frame work of restricted three-body problem was previously studied by Beevi and Sharma [
Case | X | Z | Time period | ||
---|---|---|---|---|---|
L1 | A1 = 0 | 0.990483630180 | 0.003038157227 | 0.003133921797 | 0.029033398230 |
with A1 | 0.990388234447 | 0.003038192546 | 0.003132075003 | 0.029690594615 | |
L2 | A1 = 0 | 1.008973973795 | 0.002907084824 | 0.003849027800 | 1.188870485719 |
with A1 | 1.008972498613 | 0.002910487598 | 0.003820135275 | 1.160933403200 |
Case | x | z | Time period | ||
---|---|---|---|---|---|
L1 | A1 = 0 | 0.957987711866 | 0.008197464523 | 0.008308377495 | 0.122381274243 |
with A1 | 0.957985353543 | 0.0081893443451 | 0.008308123980 | 0.123434542342 | |
L2 | A1 = 0 | 1.035106494273 | 0.007326224792 | 0.023183141982 | 3.079932656019 |
with A1 | 1.035116122187 | 0.007327676268 | 0.023150570604 | 3.072760388593 |
Oblateness attracts L1 halo orbit towards Saturn by 2.88 km and L2 halo orbit by 0.1 km.
Iapetus has one side as bright as snow and other side as dark as black velvet, with a huge
ridge running around most of its dark-side equator. Iapetus is 3,560,820 km from Saturn and is smaller than Titan, Rhea and larger than Mimas, Enceladus. Initial conditions for computation of L1 and L2 halo orbits of Saturn-Iapetus system are given in
Halo orbits in the vicinity of L1 and L2 collinear points in Saturn-Satellites systems in the frame work of circular restricted three-body problem with more massive primary Saturn as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries are considered. The halo orbits with oblateness effect of Saturn for seven largest satellites of Saturn are computed through the differential correction method of Mireles [
Case | x | z | Time period | ||
---|---|---|---|---|---|
L1 | A1 = 0 | 0.989102546102 | 0.000561734312 | 0.000564566003 | 0.032123470396 |
with A1 | 0.989101982635 | 0.000561697239 | 0.000564432349 | 0.031342342225 | |
L2 | A1 = 0 | 1.009284587510 | 0.000511851697 | 0.005023822252 | 2.965141856279 |
with A1 | 1.009270089379 | 0.000512126964 | 0.004999698237 | 2.945465601853 |
System | L1 | L2 | ||
---|---|---|---|---|
Δx (km) | ΔTime Period(s) | Δx (km) | ΔTime Period(s) | |
Saturn-Mimas | 260 | 12.3689 | 11.6 | 38.3282 |
Saturn-Enceladus | 153 | 73.2844 | 25 | 554.7144 |
Saturn-Tethys | 101 | 57.0621 | 3 | 338.02465 |
Saturn-Dione | 77.15 | 48.3183 | 0.48 | 414.9974 |
Saturn-Rhea | 50.3 | 40.6694 | 0.3 | 1728.7311 |
Saturn-Titan | 2.88 | 231.7356 | 0.1 | 1578.0121 |
Pushparaj, N. and Sharma, R.K. (2016) Oblateness Effect of Saturn on Halo Orbits of L1 and L2 in Saturn-Satellites Restricted Three-Body Problem. International Journal of Astronomy and Astrophysics, 6, 347-377. http://dx.doi.org/10.4236/ijaa.2016.64029
Coefficients for the second and third-order equations and solution:
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