SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology10.17485/IJST/v13i16.401Periodic orbits in the planar restricted photo-gravitational problem when the smaller primary is an oblate spheroidArohanRiteshritesharohan8@gmail.com1SharmaRam Krishanramkrishansharma@gmail.com1Department of Aerospace Engineering, Karunya Institute of Technology and SciencesCoimbatore, Tamil Nadu, 641114India13162020Abstract
Background/Objectives: This study deals with the stationary solutions of the planar circular restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The objective is to study the location of the Lagrangian points and to find the values of critical mass. Also, to study the periodic orbits around the Lagrangian points. Methods: A new mean motion expression by including the secular perturbation due to oblateness utilized by 1, 2 is used in the present studies. The characteristic roots are obtained by linearizing the equation of the motion around the Lagrangian points. Findings: The critical mass parameter µcrit3, 4, which decreases radiation force, whereas it increases with oblateness when we consider the value of new mean motion. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case µ = µcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness, although there is slight variation in L2 location.
KeywordsRestricted three body problemLagrangian pointsEccentricityOblatenessCritical massRadiation forceMean motion.NoneIntroduction
The three-body problem, in general, is about the study of three massive bodies (m1, m2, m3) in space which affect one another by their gravitational forces. In the restricted three-body problem, one of these three bodies are considered to be very small and that it has a very less mass which does not affect the motion of the other two bodies called primaries with masses m1 and m2, but it is affected by their mutual forces. The circular restricted three-body problem (CR3BP) studies the motion of a body with a negligible mass under the influence of two bigger bodies, called the primaries, which revolve around their center of mass in ‘n’ circular orbit. To make the problem more applicable to real cases, perturbations are also included. Here, time (t) is the independent variable on which the other variables depend. When the restricted three- body problem is solved, it has been observed that there are 5 points on the plane of the primaries where the gravitational forces of the primaries nullify each other and there is no gravitational
Since the Sun — the source of radiation — is almost spherical, and some of the planets like Saturn and Jupiter whose equatorial and polar radii are approximately 60400, 54600 and 71400, 67000 km, respectively, and sufficiently oblate, hence a modification is considered that the smaller primary (planet) is an oblate spheroid. The more massive primary, the Sun, is a source of radiation. It is considered that the equatorial plane of the smaller primary is coincident with the plane of motion and only the planar case has been studied.
In this paper, case considered when the gravitation prevails. As 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. Thus, the Sun’s resultant force acting on the particle is
F=Fg-Fp=Fg1-FpFg=qFg,
Where q=1-Fp/Fg is the mass reduction factor constant for a given particle. It can be expressed in terms of particle radius (a), density (δ), and the solar radiation pressure efficiency factor k (in CGS units) as
q=1-5.6×10-5aδk
Many of the celestial bodies do not have an exact sphere shape. Instead, they have an oblate spheroid shape. An oblate spheroid is obtained by rotating an ellipse about its minor axis i.e., the equatorial radius becomes longer than the polar radius. This oblateness of the planets render a change to the mean motion of orbit of the primaries, because of the variation of gravitation. The effect of oblateness is quantified by the term J2, which varies for each planet. Oblateness A2=AE2-AP25R2; AE, AP being the equatorial and polar radii of the smaller primary, while R is the distance between the primaries.
In5 the authors study the periodic orbits numerically for fixed values of the mass parameter and oblateness coefficient of the smaller primary and by changing the radiation pressure and the energy constant. However, the value of oblateness coefficient was taken from 3. In6in the frame work of the perturbed photo-gravitational restricted three-body problem, the first order exterior resonant orbits and the first, third and fifth order interior resonant periodic orders were analysed. The location, eccentricity and periodic of the first order exterior and interior resonant orbits are investigated in the unperturbed and perturbed cases for a specific value of Jacobi constant. In7 a version of the relativistic restricted three-body problem which includes the effects of oblateness of the secondary and radiation of the primary was considered to determine the positions and analyse the stability of the triangular points. In 8 interior resonance periodic orbits around the Sun in the Sun-Jupiter photogravitational restricted three-body problem by including the oblateness of Jupiter was studied using the method of Poincaré surface of section. In 9 a passive micron size particle in the field of radiating binary stellar system in the framework of circular restricted three body problem influenced from radial radiation pressure and Poynting-Robertson drag (PR drag) on the equilibrium points and their stability in the binary stellar systems RW-Monocerotis and Krüger-60 was studied. In 10 the motion in the vicinity of triangular equilibrium points of the circular restricted three-body problem of a passively gravitating dust particle in the gravitational field for the binaries system (Kruger 60 and Achird) is investigated. The two bodies of the binary are both oblate radiating stars possessing P-R drag.
We have calculated the value of the critical mass parameter, μcrit. It is found to decrease with radiation force of the more massive primary and increase with oblateness of the smaller primary.
Equation of Motion
Following the terminology and notation of 11 and the unit of mass is considered equivalent to the sum of the primary masses, the unit of length is equivalent to their separation and the unit of time is such that the Gaussian constant of gravitation is unity. Similar to as in 3, 6, the equation of motion in the dimensionless barycentric-synodic coordinate system (x, y) arex¨-2ny˙=dΩdx (1)
y¨+2nx˙=dΩdy
where the force function equation is given by,
Ω=n221-μr12+μr22+q1-μr1+μr2+μA22r23
where r1 and r2 are the magnitudes of the position of the spacecraft or satellite from the massive and smaller primary, respectively; q is the mass reduction factor of the massive primary and A2 is the oblateness coefficient of the smaller primary. The magnitudes of the position of the satellite or spacecraft are given as
r12=x-μ2+y2, r22=x+1-μ2+y2 (3)
The Jacobian integral of Equations (1) and (2) is
x2+y2=2Ω-c ; (4)
C is the Jacobian constant.
The curves of zero-velocity are given by
2Ωx,y=c ;
The curves are symmetric with respect to the x-axis, since Ωx,y=Ωx,±y. The singularities of the manifold of the states of motion are located at those points of the curves of zero-velocity, where
dΩdx=0=dΩdy,
i.e.,
n2x-q1-μx-μr13-μx+1-μr23-3μA2x+1-μ2r25=0, (5)
yn2-q1-μr13-μr23-3A2μ2r25=02.1 Mean Motion
In this study, we have derived mean motion expression including the secular perturbation effects. From 12, 13 for planar motion (inclination i = 0).
Solving the equations (5) and (6), where y≠0 we get
r13=q/n2, r2=1 (8)
By the help of equation (7), we can locate the other two points L4&L5. These points forming isosceles triangles with the primaries are called as triangular points. And r1≤1.
Stability of the libration points
Replacing x=a+ξ, y=b+ηin the equations of motion (1) for studying the motion near any of the equilibrium points L(a,b), we get the first variational equations as :
ξ"-2nη'=Ωxxa,bξ+Ωxya,bη, (9)
η"+2nξ'=Ωxy(a,b)ξ+Ωyy(a,b)η
The Characteristic equation of Equations (9) is
λ4+(4n2-Ωxx-Ωyy)λ2+ΩxxΩyy-Ωxy2=04.1 Stability of the collinear points
At the collinear points, we get
Ωxx=n2+2q1-μr13+2μr23+6μA2r25>0,
Ωxy=0,
Ωyy=n2-q1-μr12-μr23-3A2μ2r25.
With some calculation, it can be proved that Ωyy<0at L1,2,3. Consequently,
ΩxxΩyy-Ωxy2<0
We can note that the roots λi(i=1,2,3,4) of the Characteristic Equation (10) are
λ1,2=±-β1+β12+β221212=±λ ,
λ3,4=±-β1-β12+β221212=±is ,
where
β1=2n2-(Ωxx+Ωyy)/2,
β22=-ΩxxΩyy>0.
The General solution of Equations (9) can be written as
ξ=∑αieλiti=14, η=∑γieλiti=14 , (11)
And
(λi2-Ωxx)αi=(2nλi+Ωxy)γi .
It is noted that λ1,2are real and λ3,4 are pure imaginary. Hence, in general case the collinear equilibria are unstable. However, according to 4, we can choose the initial conditions ξ0,η0 such that α1,2=0 and then Equations (11) represent an ellipse whose eccentricity is given by
1-β3-2
respectively, where
β3=(s2+Ωxx)/2ns
Considering a theoretical problem. Where, µ=0.1,0.2,0.3;ε=0,0.01 and A2 (oblateness)=0,0.01.
Properties and nature of L1.
Locationof L1 and eccentricity of conditional periodic orbits around L1.
μ
ε
A2
ξ
x1
Eccentricity (e)
0.1
0
0
0.35969983
-1.25969983
0.92543667
0.1
0.01
0
0.35889138
-1.25889138
0.92572111
0.1
0.01
0.01
0.36100212
-1.26100212
0.94471622
0.2
0
0
0.47104869
-1.27104869
0.91600058
0.2
0.01
0
0.47035553
-1.27035553
0.91619536
0.2
0.01
0.01
0.46785205
-1.26785205
0.93117171
0.3
0
0
0.55673469
-1.25673469
0.90787718
0.3
0.01
0
0.55613915
-1.25613915
0.90802208
0.3
0.01
0.01
0.55080979
-1.25080979
0.92163148
We can see that, eccentricity increases with oblateness and slightly increases with radiation pressure. And orbit is elliptical.
Properties and nature of L2.
Location of L2 and eccentricity of conditional periodic orbits around L2.
μ
ε
A2
ξ
x2
Eccentricity (e)
0.1
0
0
0.29096488
-0.60903511
0.96842307
0.1
0.01
0
0.29223645
-0.60776354
0.96813137
0.1
0.01
0.01
0.30720465
-0.59279534
0.97304460
0.2
0
0
0.36192404
-0.43807595
0.97132709
0.2
0.01
0
0.36319101
-0.43680898
0.97111656
0.2
0.01
0.01
0.37481534
-0.42518465
0.97382094
0.3
0
0
0.41387021
-0.28612978
0.97276616
0.3
0.01
0
0.41511790
-0.28488209
0.97092712
0.3
0.01
0.01
0.42467365
-0.27532634
0.97438134
We can see that, eccentricity increases with oblateness and eccentricity decreases with radiation force. And orbit is elliptical.
Properties and nature of L3.
Location of L3 and eccentricity of conditional periodic orbits around L3.
μ
ε
A2
ξ
x3
Eccentricity (e)
0.1
0
0
0.94160890
1.04160890
0.86835862
0.1
0.01
0
0.93840960
1.03840960
0.86836855
0.1
0.01
0.01
0.91962433
1.01962433
0.87972477
0.2
0
0
0.88283946
1.08283946
0.87750511
0.2
0.01
0
0.87979735
1.07979735
0.87333010
0.2
0.01
0.01
0.86147227
1.06147227
0.88140032
0.3
0
0
0.82320559
1.12320559
0.87932469
0.3
0.01
0
0.82033371
1.12033371
0.87936718
0.3
0.01
0.01
0.80254908
1.10254908
0.88560716
We see that, eccentricity increases with oblateness, and eccentricity increases slightly with radiation force. And orbit is elliptical.
Considering some real examples from our solar system
Properties and Nature of L1.
Location of L1 and eccentricity of conditional periodic orbits around L1 for Sun-planet systems.
System
μ
A2
ε
x1
Eccentricity(e)
Sun-Mars
3.2127*10-7
5211.14188981*10-16
0 0.001 0.01
-1.00475597 -1.00464773 -1.00386702
0.94985821 0.92214540 0.96819065
Sun-Jupiter
9.5333*10-4
211.71783905*10-12
0 0.001 0.01
-1.02506012 -1.02495138 -1.02401151
0.99599793 0.99604663 0.99645211
Sun-Saturn
2.85639*10-4
65.276555555*10-12
0 0.001 0.01
-1.04606463 -1.04595722 -1.04501136
0.94712293 0.94736490 0.94949755
Sun-Uranus
4.36431*10-5
6713.99789137*10-16
0 0.001 0.01
-1.02596188 -1.02585315 -1.02491275
0.94255895 0.94300822 0.94691164
Sun-Neptune
5.14805*10-5
2037.88920652*10-16
0 0.001 0.01
-1.04117099 -1.04094671 -1.03897779
0.89538651 0.89592653 0.90098599
Properties and nature of L2.
Location of L2 and eccentricity of conditional periodic orbits around L2 for Sun-planet systems.
System
μ
A2
ε
x2
Eccentricity(e)
Sun-Mars
3.2127*10-7
5211.14188981*10-16
0 0.001 0.01
-0.99525841 -0.99514427 -0.99384034
0.95049170 0.94809550 0.92227849
Sun-Jupiter
9.5333*10-4
211.71783905*10-12
0 0.001 0.01
-0.93237854 -0.93226238 -0.93119681
0.95468743 0.95452551 0.95303883
Sun-Saturn
2.85639*10-4
65.276555555*10-12
0 0.001 0.01
-0.95475377 -0.95463906 -0.95357825
0.95320893 0.95296564 0.95071447
Sun-Uranus
4.36431*10-5
6713.99789137*10-16
0 0.001 0.01
-0.97574465 -0.97563121 -0.97455947
0.95180148 0.95134427 0.94702807
Sun-Neptune
5.14805*10-5
2037.88920652*10-16
0 0.001 0.01
-0.97437855 -0.97426504 -0.97319526
0.95189331 0.95146075 0.95327614
Properties and nature of L3.
Location of L3 and eccentricity of conditional periodic orbits around L3 for Sun-planet systems.
System
μ
A2
ε
x3
Eccentricity(e)
Sun-Mars
3.2127*10-7
5211.14188981*10-16
0 0.001 0.01
1.00000013 0.99966668 0.99665562
0.86602540 0.86602540 0.86602512
Sun-Jupiter
9.5333*10-4
211.71783905*10-12
0 0.001 0.01
1.00039722 1.00006390 0.99705404
0.86602570 0.86602570 0.86602570
Sun-Saturn
2.85639*10-4
65.276555555*10-12
0 0.001 0.01
1.00011901 0.99978561 0.99677490
0.86602543 0.86602543 0.86602543
Sun-Uranus
4.36431*10-5
6713.99789137*10-16
0 0.001 0.01
1.00001818 0.99968474 0.99667373
0.86602540 0.86602540 0.86602540
Sun-Neptune
5.14805*10-5
2037.88920652*10-16
0 0.001 0.01
1.00002144 0.99968801 0.99667701
0.86602540 0.86602540 0.86602540
Periodic orbits around the Sun in Sun-Jupiter system for L1 is generated by MATLAB for the mass ratio of µ =0.00095333 and solar radiation pressure q = 1 and 0.99, as shown in [Figure 1 & Figure 2].
Periodic orbit around the Sun in Sun-Jupiter system for (L1) where, semi-major axis = 0.0011881893731 and semi-major axis = 1x10-4
Periodic orbit around the Sun in Sun-Jupiter system for (L1)where, semi-major axis = 0.0011881893731 and semi-major axis = 1x10-4
Periodic orbits around the Sun in Sun-Jupiter system for L2 is generated by MATLAB for the mass ratio of µ =0.00095333 and solar radiation pressure q = 1 and 0.99, as shown in [Figure 3 & Figure 4 ].
Periodic orbit around the Sun in Sun-Jupiter system for (L2) where, semi-major axis = 3.3600985537x10-4 and semi-major axis = 1x10-4
Periodic orbit around the Sun in Sun-Jupiter system for (L2) where, semi-major axis = 3.3019851583x10-4 and semi-major axis = 1x10-4
Periodic orbits around the Sun in Sun-Jupiter system for L3 is generated by MATLAB for the mass ratio of µ =0.00095333 and solar radiation pressure q = 1 and 0.99, as shown in [Figure 5 & Figure 6].
Periodic orbit around the Sun in Sun-Jupiter system for (L3) where, semi-major axis = 2.00000208265x10-4 and semi-major axis = 1x10-4
Periodic orbit around the Sun in Sun-Jupiter system for (L3) where, semi-major axis = 2.00000209358x10-4 and semi-major axis = 1x10-4
4.2 Stability of the triangular points
At the triangular points L4 and L5,we have
Ωxx=f,
Ωxy=±y[f(x-μ)+g(x+1-μ)],
Ωyy=y2(f+g)>0,
where
f=3q(1-μ)r15=3(1-μ)n2r12>0, g=3μ(1+52A2)>0
The characteristics Equation (10) becomes
Λ2+(4n2-fr12-gr22)Λ+y2fg=0
with Λ=λ2,
Λ1,2=123μA2-n2±n2-3μA22-36μ1-μn21+52A2×1-1412
We observe that the roots
λ1=Λ112, λ2=-Λ112, λ3=Λ212, λ4=-Λ212 (13)
are functions of μ,q and A2and their nature depends upon the nature of the discriminant (D).
D=(n2-3μA2)2-36μ(1-μ)n21+52A21-r124 ,
And three cases can be discussed as,
When D is positive, we note that Λ1,2 are negative and roots (13), written as
λ1,2=±i-Λ112=±is4 , λ3,4=±i-Λ212=±is5 ,
Shows the triangular points to be linearly stable.
The solution of Equation (9) in this case can be easily seen to consist of short - and long - period terms with angular frequencies s5and s4respectively. As in 5, 8, 9the short-or-long-period terms can be eliminated from the solution with proper selection of initial conditions. In both cases, the motion is along a retrograde ellipse whose eccentricity and the orientation of the major axis are independent of the initial conditions.
When D is negative, the real parts of two of the four roots equation (13) are positive and equal and, hence the equilibria are unstable. However, with suitable selection of initial conditions, periodic motion can be achieved in the linear sense which approaches the equilibrium point asymptotically.
Critical Mass
The discriminant of the quadratic Equation (12) is zero when
First value of oblateness in (µcrit) or Critical mass is been found out for different values of mean motions (n2).
(µcrit)or Critical mass parameter is been found out for different values of meanmotions (n2).
n2
μcrit (only first term of A)
1+32A2
-0.06277956
1+3A2
-0.0020507319
1+4A2
0.038435157
1+5A2
0.07892104618
1+6A2
0.119406935
Graph explaining the mean motion and first value of oblateness in critical mass.
Graph showing the mean motion vs. first value of Oblateness in Critical mass parameter.
Conclusion
In earlier case of mean motion ‘n’ given by (n2)=1+3/2A2 from 3, only secular effect of oblateness on the mean motion was considered 3. However, when the secular effect of the oblateness on the mean motion is considered, argument of perigee and right ascension of the ascending node 1, 2, the resulting mean motion ‘n’ of the primaries is given by n2=1+6A2 is included in the present studies.
In this study, mean motion is increasing since the effect of oblateness on mean motion, argument of perigee and right ascension of ascending node have been considered. All these three parameters are included to get the mean motion of primaries n2=1+6A2. The value obtained of the critical mass μcrit higher than the unperturbed value of μc = 0.0385209 in 3 with oblateness of the smaller primary. It is a very interesting result, because the zone of mass parameter μ providing stable solutions at the triangular points increases with oblateness. It can be observed from [Figure 7 ] and [Table 7 ] that the increase in oblateness increases the mean motion n and increases the value of critical mass parameter μcrit.
ReferencesJohnsonAncySharmaRam KrishanLocations of Lagrangian points and periodic orbits around triangular points in the photo gravitational elliptic restricted three-body problem with oblateness20197225382312-741410.14419/ijaa.v7i2.29377Science Publishing CorporationJencyA. ArantzaSharmaRam KrishanLocation and stability of the triangular Lagrange points in photo-gravitational elliptic restricted three body problem with the more massive primary as an oblate spheroid20197257622312-741410.14419/ijaa.v7i2.29814Science Publishing CorporationSharmaRam KrishanThe linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid198713522712810004-640X, 1572-946X10.1007/bf00641562Springer Science and Business Media LLCRaoP V SubbaSharmaR KA note on the stability of the triangular points of equilibrium in the restricted three-body problem197543381383http://adsabs.harvard.edu/abs/1975A%26A....43..381SMittalAmitAhmadIqbalBhatnagarK. B.Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body2009323165730004-640X, 1572-946X10.1007/s10509-009-0038-2Springer Science and Business Media LLCPatakV O NirajThomasElbaz IAbouelmagdThe perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamic Systems-S2019124 & 584987510.3934/dcdss.2019057BelloNakoneSinghJagadishOn the Stability ofL4,5in the Relativistic R3BP with Oblate Secondary and Radiating Primary201520151121687-7969, 1687-797710.1155/2015/489120Hindawi LimitedPushparajNishanthSharmaRam KrishanInterior resonance periodic orbits around the Sun in the Sun-Jupiter photogravitational restricted three-body problem using the method of Poincaré surface of section2017212534DasM. K.NarangPankajMahajanS.YuasaM.Effect of radiation on the stability of equilibrium points in the binary stellar systems: RW-Monocerotis, Krüger 60200831442612740004-640X, 1572-946X10.1007/s10509-008-9765-zSpringer Science and Business Media LLCSinghJagadishAmudaTajudeen OluwafemiEffects of Poynting-Robertson (P-R) drag, radiation, and oblateness on motion around the L4,5 equilibrium points in the CR3BP20171521772001726-037X, 2169-005710.1080/1726037x.2017.1411043Informa UK LimitedSzebehelyVTheory of OrbitsAcademic PressNew York1967a1967005530319670055303DanbyJ M AFundamentals of Celestial Mechanics2ndWillmann-Bell, Inc1988CurtisH DOrbital Mechanics for Engineering Students3rdElsevier2014ChernikovYu AThe photogravitational restricted three-body problem197047217223