- Email: [email protected]
Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory
Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory
Journal Pre-proof
Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory A. Narayan, A. Chakraborty, A. Dewangan PII: DOI: Reference:
S1384-1076(19)30252-0 https://doi.org/10.1016/j.newast.2019.101323 NEASPA 101323
To appear in:
New Astronomy
Received date: Accepted date:
2 August 2019 22 October 2019
Please cite this article as: A. Narayan, A. Chakraborty, A. Dewangan, Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory, New Astronomy (2019), doi: https://doi.org/10.1016/j.newast.2019.101323
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Highlights • The stability in the case of linear resonance are analyzed based on the Floquet’s theory. • The effect of oblateness on linear stability is explored. • The critical value of µ for the stability boundary for parametric excitation is dependent on the oblateness
1
Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory A. Narayan1 , A. Chakraborty∗1 , and A. Dewangan1 1
Bhilai Institute of Technology, Durg, 491001, C. G., India
Abstract This paper deals with the stability analysis of the triangular equilibrium points for the generalized problem of the photogravitational restricted three body where both the primaries are radiating. The problem is generalized in the sense that the eccentricity of the orbits and the oblateness due to both the primaries and infinitesimal are considered. The stability in the case of linear resonance are analyzed based on the Floquet’s theory for finding the characteristic exponent for a system containing periodic coefficients. It was found that the critical value of µ for the stability boundary for parametric excitation is dependent on the oblateness of the primaries as well as infinitesimal. Key words:Oblateness, Triangular equilibrium points, Transition curves, Floquet’s theory.
1
Introduction
In celestial mechanics the study of three body system is an intensively studied part. It has been given a more approachable form by restricting it in the sense that one of the body is considered to be infinitesimal bearing no effect on the motion of other two bodies.The practical applicability of these problems has been enhanced by considering the orbits of the participating bodies as elliptic with non-zero eccentricity. The study of ∗
Corresponding author, Email address: [email protected]
2
equilibrium points and the stability for elliptic restricted three body problem (ERTBP) was undertaken by many researchers (see [1, 2] and the references therein). The continuous discovery of more and more Trojan asteroids in our Solar system has increased the interest of the researchers in studying the existence and stability of the triangular equilibrium points. If two bodies in restricted three body problem are assumed to be placed at the two vertices of an equilateral triangle respectively, then the third vertex will be the equilibrium point for the infinitesimal body as shown by Lagrange. This equilibrium point is known as triangular equilibrium points. The existence of thousand Trojan follower of Jupiter, Trojan asteroid and moons of other major planets has been reported till date. The existence of such a large number of celestial bodies at the triangular equilibrium point in our solar system strongly indicates the possibility of existence of such bodies in other exoplanetary or even binary systems. In ERTBP, the existence of two free parameters, the mass ratio µ and the orbital eccentricity e of the primaries makes the stability analysis much more difficult. Danby[2] employed the Floquet’s theory [3] of differential equation with periodic coefficient and numerically obtained the boundaries separating the stable and unstable region. Bennett [4, 5] employed the same method to analytically define the structure of the unstable region. Alfriend and Rand[6] used a two variable expansion method for treating singular perturbation problems and to obtain algebraic expressions for the transition curves upto O(e3 ). Transition curves, separating stable and unstable domains, were determined by Tschauner [7] using analytical methods, where as Meire[8] used numerical methods for the same. Rajnai etal.[9] determined the stable and unstable domains in µ − e plane by computing the characteristic exponents. They have also presented the frequencies of motion around the triangular point L4 both in the stable and unstable domains, and hence derived the appropriate functions for the frequencies depending on the mass parameter and the eccentricity. In the paper [10], we have explored the effect of the oblateness of the three participating bodies on the pulsating zero velocity surfaces and the fractal basin of the equilibrium points in photogravitational elliptic restricted three body problem. This paper aims at the analysis of the stability of triangular equilibrium point for ERTBP when all the participating bodies are oblate and two massive bodies are radiating as well. The paper is organized as follows: Section 1 gives the introduction. The equations of motion and the variational equations are described in section 2. In section 3, the characteristic exponents for the solution according to Floquet’s theory are presented. In section 4, the results are presented in the form of the expression giving the relation between µ and e and the graphical simulations are discussed. Finally, conclusions are drawn in section 5.
3
2
Variational Equations of Motion
The equation of motion of infinitesimal in ERTBP where all three bodies are oblate and the two primaries are radiating in the barycentric rotating pulsating coordinate system are given as follows [1]: x¯00 − 2¯ y 0 = φUx¯ ; y¯00 + 2¯ x0 = φUy¯;
(1)
where, U=
and
x2 + y 2 , 2 1 q1 q1 A1 + A3 q2 q2 A2 + A3 + 2 (1 − µ) +µ , + + n r¯1 2¯ r13 r¯2 2¯ r23
φ=
1 . 1 + e cos f
(2)
(3)
In this frame of reference, the distance between the primaries and gravitational constant are unity. Also the sum of the masses of the primaries is taken to be unity and mass 2 ratio is given as µ = m1m+m . The prime 0 denotes differentiation with respect to true 2 anomaly f . Since both the primaries are assumed to be luminous bodies, q1 and q2 are assumed to be mass reduction factors of the two primaries. If we assume the oblateness of primary, secondary and infinitesimal bodies are given by the factors A1 , A2 and A3 respectively, where 0 ≤ Ai < 1, i = 1, 2, 3. Also s denotes the semimajor axis of the elliptic orbit followed by the primaries, then the mean motion n is given by[11] 3 2 1 2 (4) n = 3 1 + (e + A1 + A2 ) . s 2
4
Then the coordinates of the triangular equilibrium points are obtained as: 1 δ A1 A2 ∗ x¯ = − µ + + 1−δ − 1+δ , 2 2 2 2 2 1 2/3 ∗ y¯ = ± δ1 − 1 + δ) + A3 4 A1 2/3 2 + 1 + δ − 2δ1 2 1/2 A2 2/3 2 + 1 + δ + 2δ − δ1 , 2 where, 2/3
(5)
2/3
δ = δ1 − δ2 . δi = aqi , i = 1, 2. The linear stability of the Lagrangian equilibrium points are studied using the first varaiational equations: X0 = P X
(6)
where,
P (f, e) =
X =
0 0 1 0 0 0 (0) (0) φUxx φUxy 0 (0) (0) φUyx φUyy −2 ξ η . ξ0 η0
0 1 , 2 0
(7)
where ξ and η are the infinitesimal displacement in the position coordinates of the Lagrangian equilibrium point and the suffix (0) denotes that the partial derivatives are to be computed at the equilibrium point.
3
Determination of characteristic exponent
Exploiting the Floquet’s theory and using the theoretical method developed by Bennett [4], we shall determine the characteristic exponents. For applying the Floquets theory the solution of the system of Eqs.(6) is written in the form: Xk = Yk eλk f 5
(8)
where Yk is the periodic coefficient with period 2π and λk is the characteristic exponent. Dropping the suffixes for the solution in general form, the derivative of the solution with respect to f is given as: Y 0 = (P − λI)Y
(9)
where, I is the unit matrix. Now taking the expansion of the coefficient Y , the characteristic exponent λ and the matrix P in terms of the eccentricity of orbit e as: Y = Y (0) + eY (1) + e2 Y (2) + · · · λ = λ0 + eλ1 + e2 λ2 + · · ·
P (e, f ) = P (0) + eP (1) + e2 P (2) + · · ·
(10) (11) (12)
where,
P (0)
and
0 0 1 0 0 0 = (0) (0) Uxx Uxy 0 (0) (0) Uyx Uyy −2
0 1 , 2 0
P (m) =(− cos f )m C,
(13)
(14)
such that
0 0 0 0 C = (0) (0) Uxx Uxy (0) (0) Uyx Uyy
1 0 0 0
0 1 , 0 0
(15)
Substituting the values of Y and P from Eqs. (10) and (12) in (9) and equating the coefficient of all powers of e, we get the set of equations: Y 0(0) + (Iλ0 − P (0) )Y (0) =0
Y 0(1) + (Iλ0 − P (0) )Y (1) =(−C cos f − Iλ1 )Y (0)
Y 0(2) + (Iλ0 − P (0) )Y (2) =(−C cos f − Iλ1 )Y (1)
Y 0(n) + (Iλ0 − P (0) )Y (n)
+ (C cos2 f − Iλ2 )Y (0) ··· ··· n X = (C(− cos f )m − Iλm )Y (n−m) m=1
6
(16)
The solution of the zeroth order equation will be a constant vector and the nth order n . Then the particular equation with non-homogeneous terms have frequencies upto 2π solution is given as Y
(n)
=
n X
a(n,k) eikf ; n = 0, 1, 2
(17)
k=−n
where,
a
(n,k)
=
(n,k)
a1 (n,k) a2 (n,k) a3 (n,k) a4
.
(18)
Substituting the values from Eqs. (17) and (18) in Eq. (16), we obtain from the first equation of system Eq. (16): (Iλ0 − P (0) )a(0,0) = 0 For the existence of the term a(0,0) , it is necessary that det(Iλ0 − P (0) ) = 0, that is λ0 0 −1 0 0 λ0 0 −1 =0 (0) (0) −Uxx −U λ −2 xy 0 (0) (0) −U −Uyy 2 λ0 yx
or λ40 − Qλ20 + R = 0 where, 0 0 Q = Uxx + Uyy − 4,
0 0 0 2 R = Uxx Uyy − (Uxy ).
Solving Eq. (19) we obtain the value of λ0 as follows: p Q ± Q2 − 4R 2 λ0 = 2 Now, solving the second equation of the system Eqa. (16), we obtain (Iλ0 − P (0) )a(1,0) = −λ1 a(0,0) 7
(19)
That is det(Iλ0 − P (0) ) + λ1 a(0,0) vanishes. Since λ1 appears as a factor in each term, therefore we get λ1 (det(Iλ0 − P (0) ) + a(0,0) ) = 0
That is
λ1 = 0. Next on solving the third equation of the system Eqs. (16), we get the following set of equations: C (Iλ0 + i − P (0) )a(1,1) = − a(0,0) (20) 2 C (21) (Iλ0 − i − P (0) )a(1,−1) = − a(0,0) 2 C (Iλ0 − P (0) )a(2,0) = − λ1 a(1,0) + − − Iλ2 a(0,0) 2 C − (a(1,1) + a(1,−1) ) (22) 2 Substituting the values of a(1,1) and a(1,−1) from Eq.(20) and Eq.(21) in Eq.(22) and solving we get C C (Iλ0 − P (0) )a(2,0) = Re[I(λ0 + i) − P (0) ]−1 + − Iλ2 a(0,0) (23) 2 2 Eliminating a(2,0) by multiplying the adjoint of (Iλ0 − P (0) ) to both sides of the Eq. (23) and simplifying further, the value of λ2 in terms of λ0 is given as: λ2 = Aλ0 ;
(24)
where, A=− and
(Q2 − 4R − 16)λ20 + A0 F0 + A1 F1 + A2 F2 4(Q2 − 4Q − 4R)λ20 + 32R
A0 =((Q + 4)2 (Q − 4) − 4QR)λ20 − R(Q + 4) − 4R2 1 (Q + 1)(λ20 + 1) + 2R F0 = N A1 = − 8λ0 R(2λ20 − (Q + 4)) λ0 F1 = − 2λ20 + Q + 3 N A2 = − λ20 R(Q2 − 4R − 16) 1 2 F2 = − λ0 − Q − 1 N 2 N =λ0 4Q2 + 8Q + 4 − 16R + (Q + 1)2 − 4R 8
(25)
(26)
Hence the characteristic exponents expanded uoto second order of e is given as λ = λ0 (1 + Ae2 )
4
(27)
Results and discussions
The transition curves separating the stable and unstable region for the triangular equilibrium points in the µ − e plane can be found by equating the expression for the characteristic exponents to the value for periodic solution in the range of 0 < µ << 12 . The characteristic exponent λj are given by νj = exp(λj T ), j = 1, 2, 3, 4;
(28)
where, νj are the eigen values and T is the period of revolution of the bigger primary. Thus we get λj =
Φj + 2πk ln|νj | + i( ), k = 0, ±1, ±2, · · · T T
(29)
where νj = |νj |exp(iΦj ) and the value of λj is within the multiple of 2πi . The real part T of the characteristic exponent determines whether the solution is bounded or not. So, the value of λ with in the specified range of 0 < µ << 21 is the frequency corresponding to the mean motion resonance λ∗ = ± 2i . Using the value of λ∗ in Eq. (27), we get the relation between µ and e, which gives the transition curves in the µ − e plane. The Figs. 1 to 3 show the transition curves in the µ − e plane varying the values of A3 , A1 and A2 respectively when the oblateness of the other two bodies are assumed to be zero. The shaded regions in the plots depicts the stable region. The Fig. 4 shows the the transition curves in the µ − e plane varying the values of A3 , when A1 = 0.001 and A2 = 0. Fig. 5 shows the the transition curves in the µ − e plane varying the values of A3 , when A1 = 0 and A2 = 0.001 where as Fig. 6 shows the the transition curves in the µ − e plane varying the values of A3 , when A1 = 0.001 and A2 = 0.001.
5
Conclusions
Based on the Floquet’s theory, the stability of oblate infinitesimal’s motions about the triangular equilibrium points in the elliptic restricted three body problem with radiating and oblate primaries has been investigated using the analytical technique. The transition curves in the µ − e plane has been plotted in Fig.1-3. It was found that when all the other perturbing forces (radiation pressure and oblateness) are neglected than both the values of µ, i.e. µa = 0.02859 (the value corresponding to the bifurcation point on the µ-axis) and µb = 0.03852 (limiting the stability domain on the µ-axis) 9
Figure 1: The transition curve in µ−e plane, taking A3 = 0, A2 = 0, q1 = 0.9999, q2 = 0.9999
Figure 2: The transition curve in µ−e plane, taking A1 = 0, A3 = 0, q1 = 0.9999, q2 = 0.9999
10
Figure 3: The transition curve in µ−e plane, taking A1 = 0, A2 = 0, q1 = 0.9999, q2 = 0.9999
Figure 4: The transition curve in µ − e plane, taking A1 = 0.001, A2 = 0, q1 = 0.9999, q2 = 0.9999
11
Figure 5: The transition curve in µ − e plane, taking A1 = 0, A2 = 0.001, q1 = 0.9999, q2 = 0.9999
Figure 6: The transition curve in µ − e plane, taking A1 = 0.001, A2 = 0.001, q1 = 0.9999, q2 = 0.9999
12
agrees exactly with the classical results [2, 6]. For the cases when the oblateness of the second primary or the infinitesimal is increased the values of µa and µb decreases slightly. However it remains same in the case when the oblateness of the heavier primary is increased. It was also observed when the oblateness of the second primary is taken to be non-zero, the values of both µa and µb further decreases for varying values of A3 . However the oblateness is again not found to have much effect on the values.
6
Acknowledgment
The financial assistance from Chattisgarh Council of Science and Technology (Endt. no. 2260/CCOST/MRP/2015) is duly acknowledged with gratitude.
References [1] V. Szebehely, Theory of Orbits, The Restricted problem of three bodies (Academic Press, New York )(1967) [2] J. M. A. Danby, Stability of the triangular points in the elliptic restricted problem of three bodies, Astron. J. 69(2), 165-172 (1964), doi:10.1086/109254. [3] G. Floquet, Sur les ´equations diff´erentielles lin´eaires a´ coefficients p´eriodiques, An´ nales de l’Ecole Normale Sup´erieure 12 47-88 (1883), doi : 10.24033/asens.220. [4] A. Bennett, Methods in astrodynamics and celestial mechanics(eds) R. L. Duncombe and V. G. Szebehely (Montery, California), 101 (1965) [5] A. Bennett, Characteristic exponents of the five equilibrium solutions in the elliptically restricted problem, Icarus 4(2), 177(1965) [6] K. T. Alfriend and R. H. Rand, Stability of the triangular points in the elliptic restricted problem of three bodies, AIAA J. 7(6), 1024-1028 (1969) [7] J. Tschauner, Die Bewegung in der N¨ahe der Dreieckspunkte des Elliptischen Eingeschr¨ankten Dreik¨orper-problems, Celest. Mech. 3(2) 189-196 (1971) [8] R. Meire, The stability of the triangular points in the elliptic restricted problem, Celest. Mech. 23(1) 89(1981), doi.org/10.1007/BF01228547 ´ [9] R. Rajnai, I. Nagy and B. Erdi, Frequencies and resonances around L4 in the elliptic restricted three-body problem, MNRAS 443, 1988-1998 (2014), doi:10.1093/mnras/stu1212. 13
[10] A. Narayan, A. Chakraborty and A. Dewangan, Pulsating zero velocity surfaces and fractal basin of oblate infinitesimal in the elliptic restricted three body problem Few-Body Syst. 2018 59:43 (2018), doi.org/10.1007/s00601-018-1368-9. [11] A Narayan, A Chakraborty and A Dewangan, Dynamics of oblate test particle under the influence of oblate and radiating primaries in elliptic restricted three body problem J. of Informatics and Mathl. Sci. 10(1) 73-92 (2018)
14
Journal Pre-proof
Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory A. Narayan, A. Chakraborty, A. Dewangan PII: DOI: Reference:
S1384-1076(19)30252-0 https://doi.org/10.1016/j.newast.2019.101323 NEASPA 101323
To appear in:
New Astronomy
Received date: Accepted date:
2 August 2019 22 October 2019
Please cite this article as: A. Narayan, A. Chakraborty, A. Dewangan, Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory, New Astronomy (2019), doi: https://doi.org/10.1016/j.newast.2019.101323
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Highlights • The stability in the case of linear resonance are analyzed based on the Floquet’s theory. • The effect of oblateness on linear stability is explored. • The critical value of µ for the stability boundary for parametric excitation is dependent on the oblateness
1
Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory A. Narayan1 , A. Chakraborty∗1 , and A. Dewangan1 1
Bhilai Institute of Technology, Durg, 491001, C. G., India
Abstract This paper deals with the stability analysis of the triangular equilibrium points for the generalized problem of the photogravitational restricted three body where both the primaries are radiating. The problem is generalized in the sense that the eccentricity of the orbits and the oblateness due to both the primaries and infinitesimal are considered. The stability in the case of linear resonance are analyzed based on the Floquet’s theory for finding the characteristic exponent for a system containing periodic coefficients. It was found that the critical value of µ for the stability boundary for parametric excitation is dependent on the oblateness of the primaries as well as infinitesimal. Key words:Oblateness, Triangular equilibrium points, Transition curves, Floquet’s theory.
1
Introduction
In celestial mechanics the study of three body system is an intensively studied part. It has been given a more approachable form by restricting it in the sense that one of the body is considered to be infinitesimal bearing no effect on the motion of other two bodies.The practical applicability of these problems has been enhanced by considering the orbits of the participating bodies as elliptic with non-zero eccentricity. The study of ∗
Corresponding author, Email address: [email protected]
2
equilibrium points and the stability for elliptic restricted three body problem (ERTBP) was undertaken by many researchers (see [1, 2] and the references therein). The continuous discovery of more and more Trojan asteroids in our Solar system has increased the interest of the researchers in studying the existence and stability of the triangular equilibrium points. If two bodies in restricted three body problem are assumed to be placed at the two vertices of an equilateral triangle respectively, then the third vertex will be the equilibrium point for the infinitesimal body as shown by Lagrange. This equilibrium point is known as triangular equilibrium points. The existence of thousand Trojan follower of Jupiter, Trojan asteroid and moons of other major planets has been reported till date. The existence of such a large number of celestial bodies at the triangular equilibrium point in our solar system strongly indicates the possibility of existence of such bodies in other exoplanetary or even binary systems. In ERTBP, the existence of two free parameters, the mass ratio µ and the orbital eccentricity e of the primaries makes the stability analysis much more difficult. Danby[2] employed the Floquet’s theory [3] of differential equation with periodic coefficient and numerically obtained the boundaries separating the stable and unstable region. Bennett [4, 5] employed the same method to analytically define the structure of the unstable region. Alfriend and Rand[6] used a two variable expansion method for treating singular perturbation problems and to obtain algebraic expressions for the transition curves upto O(e3 ). Transition curves, separating stable and unstable domains, were determined by Tschauner [7] using analytical methods, where as Meire[8] used numerical methods for the same. Rajnai etal.[9] determined the stable and unstable domains in µ − e plane by computing the characteristic exponents. They have also presented the frequencies of motion around the triangular point L4 both in the stable and unstable domains, and hence derived the appropriate functions for the frequencies depending on the mass parameter and the eccentricity. In the paper [10], we have explored the effect of the oblateness of the three participating bodies on the pulsating zero velocity surfaces and the fractal basin of the equilibrium points in photogravitational elliptic restricted three body problem. This paper aims at the analysis of the stability of triangular equilibrium point for ERTBP when all the participating bodies are oblate and two massive bodies are radiating as well. The paper is organized as follows: Section 1 gives the introduction. The equations of motion and the variational equations are described in section 2. In section 3, the characteristic exponents for the solution according to Floquet’s theory are presented. In section 4, the results are presented in the form of the expression giving the relation between µ and e and the graphical simulations are discussed. Finally, conclusions are drawn in section 5.
3
2
Variational Equations of Motion
The equation of motion of infinitesimal in ERTBP where all three bodies are oblate and the two primaries are radiating in the barycentric rotating pulsating coordinate system are given as follows [1]: x¯00 − 2¯ y 0 = φUx¯ ; y¯00 + 2¯ x0 = φUy¯;
(1)
where, U=
and
x2 + y 2 , 2 1 q1 q1 A1 + A3 q2 q2 A2 + A3 + 2 (1 − µ) +µ , + + n r¯1 2¯ r13 r¯2 2¯ r23
φ=
1 . 1 + e cos f
(2)
(3)
In this frame of reference, the distance between the primaries and gravitational constant are unity. Also the sum of the masses of the primaries is taken to be unity and mass 2 ratio is given as µ = m1m+m . The prime 0 denotes differentiation with respect to true 2 anomaly f . Since both the primaries are assumed to be luminous bodies, q1 and q2 are assumed to be mass reduction factors of the two primaries. If we assume the oblateness of primary, secondary and infinitesimal bodies are given by the factors A1 , A2 and A3 respectively, where 0 ≤ Ai < 1, i = 1, 2, 3. Also s denotes the semimajor axis of the elliptic orbit followed by the primaries, then the mean motion n is given by[11] 3 2 1 2 (4) n = 3 1 + (e + A1 + A2 ) . s 2
4
Then the coordinates of the triangular equilibrium points are obtained as: 1 δ A1 A2 ∗ x¯ = − µ + + 1−δ − 1+δ , 2 2 2 2 2 1 2/3 ∗ y¯ = ± δ1 − 1 + δ) + A3 4 A1 2/3 2 + 1 + δ − 2δ1 2 1/2 A2 2/3 2 + 1 + δ + 2δ − δ1 , 2 where, 2/3
(5)
2/3
δ = δ1 − δ2 . δi = aqi , i = 1, 2. The linear stability of the Lagrangian equilibrium points are studied using the first varaiational equations: X0 = P X
(6)
where,
P (f, e) =
X =
0 0 1 0 0 0 (0) (0) φUxx φUxy 0 (0) (0) φUyx φUyy −2 ξ η . ξ0 η0
0 1 , 2 0
(7)
where ξ and η are the infinitesimal displacement in the position coordinates of the Lagrangian equilibrium point and the suffix (0) denotes that the partial derivatives are to be computed at the equilibrium point.
3
Determination of characteristic exponent
Exploiting the Floquet’s theory and using the theoretical method developed by Bennett [4], we shall determine the characteristic exponents. For applying the Floquets theory the solution of the system of Eqs.(6) is written in the form: Xk = Yk eλk f 5
(8)
where Yk is the periodic coefficient with period 2π and λk is the characteristic exponent. Dropping the suffixes for the solution in general form, the derivative of the solution with respect to f is given as: Y 0 = (P − λI)Y
(9)
where, I is the unit matrix. Now taking the expansion of the coefficient Y , the characteristic exponent λ and the matrix P in terms of the eccentricity of orbit e as: Y = Y (0) + eY (1) + e2 Y (2) + · · · λ = λ0 + eλ1 + e2 λ2 + · · ·
P (e, f ) = P (0) + eP (1) + e2 P (2) + · · ·
(10) (11) (12)
where,
P (0)
and
0 0 1 0 0 0 = (0) (0) Uxx Uxy 0 (0) (0) Uyx Uyy −2
0 1 , 2 0
P (m) =(− cos f )m C,
(13)
(14)
such that
0 0 0 0 C = (0) (0) Uxx Uxy (0) (0) Uyx Uyy
1 0 0 0
0 1 , 0 0
(15)
Substituting the values of Y and P from Eqs. (10) and (12) in (9) and equating the coefficient of all powers of e, we get the set of equations: Y 0(0) + (Iλ0 − P (0) )Y (0) =0
Y 0(1) + (Iλ0 − P (0) )Y (1) =(−C cos f − Iλ1 )Y (0)
Y 0(2) + (Iλ0 − P (0) )Y (2) =(−C cos f − Iλ1 )Y (1)
Y 0(n) + (Iλ0 − P (0) )Y (n)
+ (C cos2 f − Iλ2 )Y (0) ··· ··· n X = (C(− cos f )m − Iλm )Y (n−m) m=1
6
(16)
The solution of the zeroth order equation will be a constant vector and the nth order n . Then the particular equation with non-homogeneous terms have frequencies upto 2π solution is given as Y
(n)
=
n X
a(n,k) eikf ; n = 0, 1, 2
(17)
k=−n
where,
a
(n,k)
=
(n,k)
a1 (n,k) a2 (n,k) a3 (n,k) a4
.
(18)
Substituting the values from Eqs. (17) and (18) in Eq. (16), we obtain from the first equation of system Eq. (16): (Iλ0 − P (0) )a(0,0) = 0 For the existence of the term a(0,0) , it is necessary that det(Iλ0 − P (0) ) = 0, that is λ0 0 −1 0 0 λ0 0 −1 =0 (0) (0) −Uxx −U λ −2 xy 0 (0) (0) −U −Uyy 2 λ0 yx
or λ40 − Qλ20 + R = 0 where, 0 0 Q = Uxx + Uyy − 4,
0 0 0 2 R = Uxx Uyy − (Uxy ).
Solving Eq. (19) we obtain the value of λ0 as follows: p Q ± Q2 − 4R 2 λ0 = 2 Now, solving the second equation of the system Eqa. (16), we obtain (Iλ0 − P (0) )a(1,0) = −λ1 a(0,0) 7
(19)
That is det(Iλ0 − P (0) ) + λ1 a(0,0) vanishes. Since λ1 appears as a factor in each term, therefore we get λ1 (det(Iλ0 − P (0) ) + a(0,0) ) = 0
That is
λ1 = 0. Next on solving the third equation of the system Eqs. (16), we get the following set of equations: C (Iλ0 + i − P (0) )a(1,1) = − a(0,0) (20) 2 C (21) (Iλ0 − i − P (0) )a(1,−1) = − a(0,0) 2 C (Iλ0 − P (0) )a(2,0) = − λ1 a(1,0) + − − Iλ2 a(0,0) 2 C − (a(1,1) + a(1,−1) ) (22) 2 Substituting the values of a(1,1) and a(1,−1) from Eq.(20) and Eq.(21) in Eq.(22) and solving we get C C (Iλ0 − P (0) )a(2,0) = Re[I(λ0 + i) − P (0) ]−1 + − Iλ2 a(0,0) (23) 2 2 Eliminating a(2,0) by multiplying the adjoint of (Iλ0 − P (0) ) to both sides of the Eq. (23) and simplifying further, the value of λ2 in terms of λ0 is given as: λ2 = Aλ0 ;
(24)
where, A=− and
(Q2 − 4R − 16)λ20 + A0 F0 + A1 F1 + A2 F2 4(Q2 − 4Q − 4R)λ20 + 32R
A0 =((Q + 4)2 (Q − 4) − 4QR)λ20 − R(Q + 4) − 4R2 1 (Q + 1)(λ20 + 1) + 2R F0 = N A1 = − 8λ0 R(2λ20 − (Q + 4)) λ0 F1 = − 2λ20 + Q + 3 N A2 = − λ20 R(Q2 − 4R − 16) 1 2 F2 = − λ0 − Q − 1 N 2 N =λ0 4Q2 + 8Q + 4 − 16R + (Q + 1)2 − 4R 8
(25)
(26)
Hence the characteristic exponents expanded uoto second order of e is given as λ = λ0 (1 + Ae2 )
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(27)
Results and discussions
The transition curves separating the stable and unstable region for the triangular equilibrium points in the µ − e plane can be found by equating the expression for the characteristic exponents to the value for periodic solution in the range of 0 < µ << 12 . The characteristic exponent λj are given by νj = exp(λj T ), j = 1, 2, 3, 4;
(28)
where, νj are the eigen values and T is the period of revolution of the bigger primary. Thus we get λj =
Φj + 2πk ln|νj | + i( ), k = 0, ±1, ±2, · · · T T
(29)
where νj = |νj |exp(iΦj ) and the value of λj is within the multiple of 2πi . The real part T of the characteristic exponent determines whether the solution is bounded or not. So, the value of λ with in the specified range of 0 < µ << 21 is the frequency corresponding to the mean motion resonance λ∗ = ± 2i . Using the value of λ∗ in Eq. (27), we get the relation between µ and e, which gives the transition curves in the µ − e plane. The Figs. 1 to 3 show the transition curves in the µ − e plane varying the values of A3 , A1 and A2 respectively when the oblateness of the other two bodies are assumed to be zero. The shaded regions in the plots depicts the stable region. The Fig. 4 shows the the transition curves in the µ − e plane varying the values of A3 , when A1 = 0.001 and A2 = 0. Fig. 5 shows the the transition curves in the µ − e plane varying the values of A3 , when A1 = 0 and A2 = 0.001 where as Fig. 6 shows the the transition curves in the µ − e plane varying the values of A3 , when A1 = 0.001 and A2 = 0.001.
5
Conclusions
Based on the Floquet’s theory, the stability of oblate infinitesimal’s motions about the triangular equilibrium points in the elliptic restricted three body problem with radiating and oblate primaries has been investigated using the analytical technique. The transition curves in the µ − e plane has been plotted in Fig.1-3. It was found that when all the other perturbing forces (radiation pressure and oblateness) are neglected than both the values of µ, i.e. µa = 0.02859 (the value corresponding to the bifurcation point on the µ-axis) and µb = 0.03852 (limiting the stability domain on the µ-axis) 9
Figure 1: The transition curve in µ−e plane, taking A3 = 0, A2 = 0, q1 = 0.9999, q2 = 0.9999
Figure 2: The transition curve in µ−e plane, taking A1 = 0, A3 = 0, q1 = 0.9999, q2 = 0.9999
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Figure 3: The transition curve in µ−e plane, taking A1 = 0, A2 = 0, q1 = 0.9999, q2 = 0.9999
Figure 4: The transition curve in µ − e plane, taking A1 = 0.001, A2 = 0, q1 = 0.9999, q2 = 0.9999
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Figure 5: The transition curve in µ − e plane, taking A1 = 0, A2 = 0.001, q1 = 0.9999, q2 = 0.9999
Figure 6: The transition curve in µ − e plane, taking A1 = 0.001, A2 = 0.001, q1 = 0.9999, q2 = 0.9999
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agrees exactly with the classical results [2, 6]. For the cases when the oblateness of the second primary or the infinitesimal is increased the values of µa and µb decreases slightly. However it remains same in the case when the oblateness of the heavier primary is increased. It was also observed when the oblateness of the second primary is taken to be non-zero, the values of both µa and µb further decreases for varying values of A3 . However the oblateness is again not found to have much effect on the values.
6
Acknowledgment
The financial assistance from Chattisgarh Council of Science and Technology (Endt. no. 2260/CCOST/MRP/2015) is duly acknowledged with gratitude.
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[10] A. Narayan, A. Chakraborty and A. Dewangan, Pulsating zero velocity surfaces and fractal basin of oblate infinitesimal in the elliptic restricted three body problem Few-Body Syst. 2018 59:43 (2018), doi.org/10.1007/s00601-018-1368-9. [11] A Narayan, A Chakraborty and A Dewangan, Dynamics of oblate test particle under the influence of oblate and radiating primaries in elliptic restricted three body problem J. of Informatics and Mathl. Sci. 10(1) 73-92 (2018)
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