Linear stability and resonances in the generalized photogravitational Chermnykh-like problem with a disc

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1 MNRAS 436, (013) Advance Access publication 013 October 11 doi: /mnras/stt169 Linear stability and resonances in the generalized photogravitational Chermnykh-like problem with a disc Ram Kishor and Badam Singh Kushvah Department of Applied Mathematics, Indian School of Mines, Dhanbad 86004, Jharkhand, India Accepted 013 September 5. Received 013 September ; in original form 013 July 31 ABSTRACT Today, the main concerns in dynamical systems are stability, dynamical processes and the factors causing them, which determine their past evolution. This article presents a systematic study of the linear stability of triangular equilibrium points in the proposed problem, under the influence of perturbations in the form of radiation pressure, oblateness and the presence of a disc. We see that points are stable for mass ratios 0 <μ< μ c = , which is greater than Routh s value μ c = Further, analysis is performed under three main resonance cases and it is observed that the stability region expands the presence of the disc. Perturbed mass ratios μ κ, κ = 1,, 3, for these three resonances are obtained under appropriate approximations. Finally, the effect of these perturbations is analysed with the help of the results obtained and it is noted that they influence the motion of an infinitesimal mass significantly, but the effect of the disc dominates over radiation pressure and oblateness. This theory is applicable not only to four cases examined earlier by researchers but also to a generalized case, as in the present model. Results are limited to a radially symmetric regular disc but can be extended in future. Key words: instabilities radiation mechanisms: general celestial mechanics planets and satellites: dynamical evolution and stability. 1 INTRODUCTION Recent developments in the field of non-linear dynamics, as well as in technology to observe the Solar system, have revived interest in the dynamics of space, especially problems of three bodies. If one of the three bodies is restricted to negligible mass then the resulting problem is known as a restricted three-body problem (RTBP). It comprises an important dynamical system for the study of new investigations regarding motions not only in the Solar system but also in other planetary systems. Motions of small space objects (asteroid, comet, ring, spacecraft, satellite, etc.) in the Solar system, as well as Sun planet systems (Sun Earth, Sun Jupiter, etc.) are the best examples of the RTBP. The problem becomes more interesting when it also includes other types of space structure, such as a belt, disc, ring, etc., that are present in the Solar system or in extrasolar planetary systems. The presence of an asteroid or Kuiper belt (Jiang & Yeh 004b,c) as well as discs of space debris (Rivera & Lissauer 000; Jiang & Yeh 004a; Jiang & Ip 001), either in the Solar system or in an extrasolar planetary system, affects many aspects of the motion of a restricted body such as the equilibrium point, stability properties, periodicity of the orbits, etc. (Jiang & Yeh 006; Yeh & Jiang 006; Kushvah, Kishor & Dolas 01; Kishor & Kushvah 013). At the end of the 0th century, Chermnykh (1987) studied a problem consisting of the motion of a particle under the influ- kishor.ram888@gmail.com (RK); bskush@gmail.com (BSK) ence of the gravity field of a uniform rotating dumbbell; hence this is named as Chermnykh s problem. Later, it was studied by many researchers (Markellos, Papadakis & Perdios 1996; Goździewski 1998; Papadakis 005a,b; Jiang & Yeh 006; Ishwar & Kushvah 006; Kushvah 008) in a different modified form and became a renowned problem of celestial mechanics known as the Chermnykhlike problem. This problem not only generalizes the RTBP but also has a number of important applications in the fields of celestial mechanics, dynamical astronomy and extrasolar planetary systems (Goździewski & Maciejewski 1999; Rivera & Lissauer 000; Jiang & Yeh 004a,b) and moreover in chemistry (Strand & Reinhardt 1979). In view of this, we analyse the linear stability and resonance cases of triangular equilibrium points in the present model, which is formulated in Section, under the gravitational influence of a disc which is rotating about the common centre of mass of the system (Section 3). For a linear stability test, we follow the method described in Moulton (1984) and Murray & Dermott (000). Since, NASA has considered a number of important missions in the basin of equilibrium points of the Sun Earth system: for example, L 1 is home to the SOHO spacecraft and WMAP is at L.Wehaveshown interest in describing the linear stability of L 4,5 points in the Sun Jupiter system under the effect of perturbations such as radiation pressure force, oblateness and a disc (Section 4). It has been seen that the triangular equilibrium point is stable for all values of mass ratio within the linear stability region 0 <μ<μ c = C 013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society Downloaded from on 16 June 018

2 174 R. Kishor and B. S. Kushvah (Szebehely 1967), whereas it is unstable for perturbed values of the mass ratio μ 1, μ and μ 3 for the three main cases of resonance (Bhatnagar, Gupta & Bhardwaj 1994). They have determined perturbed values of the mass ratio under four cases of perturbations: (i) the classical case, (ii) when the larger primary is an oblate spheroid, (iii) when both primaries show oblateness and (iv) when the larger primary is a source of radiation. In our case, we study the problem and obtain the perturbed mass ratio not only in the other case, i.e. when the larger primary is radiating and the smaller primary is an oblate spheroid, but also in the presence of a disc (Section 5). The effect of the perturbations on the motion of an infinitesimal mass in the vicinity of the triangular equilibrium point is described with the help of numerical results in the same section. The numerical computations have been performed with the help of MATHEMATICA (Wolfram 003). In addition to this, great care has been taken over the double precision computation process and accuracy as well the precision goal: both are taken to three decimal places. Finally, conclusions are given in Section 6. EQUATIONS OF MOTION It is supposed that the forces governing the motion of an infinitesimal mass (a mass that does not affect the motion of finite bodies gravitationally) consist of the gravitational attraction of both (larger and smaller) primaries as well as the disc, which is rotating about the common centre of mass of the system. The radiation pressure force of the radiating (larger) primary is also taken into account and its opposing nature to the gravitational attraction results in the mass reduction factor q 1 = (1 F p /F g ) (Ragos & Zagouras 1993), where F p and F g are the radiation pressure and gravitational attraction forces, respectively. The oblateness of the smaller primary comes into the picture in the form of oblateness coefficients A = (Re R p )/(5R ) (McCuskey 1963), where R e and R p are the equatorial and polar radii, respectively, of the oblate body and R is the separation of both primaries. In general, the value of oblate coefficient A (0, 1), whereas A is very small in the Solar system, e.g. A for the Jupiter Io system and for the Saturn Mimas system, the largest value of A (Sharma & Rao 1976). It is supposed that the power-law density profile of the disc having thickness h 10 4 is ρ(r) = c/r p,wherep N (here we take p = 3) and c is a constant that depends on the total mass of the disc. The units of mass, distance and time are normalized in such a way that G = 1, which results in the mean motion of the system n = q A f b (r), where f b (r) is the gravitational force due to the disc, given as (Jiang & Yeh 006) r ρ(r )r 1 f b (r) = r r r E(ξ) + 1 ] r + r F (ξ) dr. (1) Here F(ξ) and E(ξ) are elliptic integrals of the first and second kind, respectively, r is the disc reference radius and ξ = ( rr )/(r + r ). Series expansion of these elliptic integrals over a r b with a suitable approximation provides { (b a) 1 f b (r) = πch ab r + 3 log(b/a) 8 r 3 }, () where a and b are inner and outer radii, respectively, of a radially symmetric disc. Now, let us suppose that ( μ,0)and(1 μ,0)are the coordinates of the larger and smaller primary respectively and P(x, y, 0) is the coordinate of an infinitesimal mass, relative to the synodic frame of reference, where μ = m J /(M S + m J ) is the mass parameter in the Sun Jupiter system (M S and m J are masses of the Sun and Jupiter, respectively). Then the equations of motion of the infinitesimal mass in the xy plane can be written as (Kushvah et al. 01) ẍ nẏ = x, (3) ÿ + nẋ = y, (4) where x = n x q 1(1 μ)(x + μ) μ(x + μ 1) r1 3 r 3 3μA (x + μ 1) r 5 y = n y (1 μ)q 1y r x r f b(r), (5) μy r 3 3μA y r 5 + y r f b(r), (6) with r = x + y, r 1 = (x + μ) + y and r = (x + μ 1) + y the distances of the infinitesimal mass from the common centre of mass of the system, first primary and second primary respectively. The last two terms on the right-hand side in equations (5) and (6) are due to oblateness and the presence of a disc, respectively. 3 TRIANGULAR EQUILIBRIUM POINTS L 4,5 The points at which the motion of a moving particle ceases are known as equilibrium points. The triangular equilibrium point (it forms a triangle with two massive bodies, hence the name) can be obtained from equations (3) and (4) in addition to the vanishing conditions of the velocity as well as acceleration of the infinitesimal mass. In other words, one can evaluate the triangular equilibrium point of the problem by solving the equations x = 0 and y = 0 (7) simultaneously for space variables. Analytically, to solve the above equations for real x and y is a cumbersome task. Therefore, for convenience it is assumed that r 1 = q 1/3 1 (1 + δ 1 )andr = 1 + δ, where δ 1, δ 1. Hence, r1 = (x + μ) + y = q /3 1 (1 + δ 1 ), r = (x + μ 1) + y = (1 + δ ), (8) which yields ( ) x = q/3 1 μ + q /3 1 δ 1 δ, (9) ] ( 1/ y =±q 1/3 1 1 q/ q /3 1 )δ 1 + δ (10) under the linear approximation for δ 1 and δ,where δ 1 = 1 1 n πch(b a) + 3 ab{μ + q /3 1 (1 μ)} 3/ ] 3πch log(b/a) + 8{μ + q /3, (11) 1 (1 μ)} 1 δ = 3(1 + 5 A 1 n + 3A ) + πch(b a) ab{μ + q /3 1 (1 μ)} 3/ ] 3πch log(b/a) + 8{μ + q /3. (1) 1 (1 μ)} Downloaded from on 16 June 018

3 Table 1. Coordinates of L 4 (x 4, y 4 )andl 5 (x 4, y 4 ) at different values of parameters q 1, A and b. Photogravitational Chermnykh-like problem 1743 (x 4, y 4 )at A q 1 b = 1.0 b = 1. b = 1.4 b = ( , ) ( , ) ( , ) ( , ) 0.80 ( , ) ( , ) ( , ) ( , ) 0.70 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 0.80 ( , ) ( , ) ( , ) ( , ) 0.70 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 0.80 ( , ) ( , ) ( , ) ( , ) 0.70 ( , ) ( , ) ( , ) ( , ) The δs are obtained with the help of equations (7) (10) in addition to a suitable approximation in simplification of the expression. The coordinates of the triangular equilibrium points given in (9) (10) agree with the classical results. That is, in the absence of radiation pressure, oblateness and disc simultaneously, the coordinates of the triangular points become x = 1/ μ and y =± 3/. Numerically, we have computed triangular equilibrium points L 4,5 for different values of radiation factor, oblateness coefficient and disc width and these are displayed in Table 1. The mean motion (n) is calculated at the disc s reference radius (r) = Table 1 reflects the fact that the position of the L 4 point is shifted (x-coordinate) towards the origin with either decreasing values of q 1 or increasing values of A, whereas it moves away (y-coordinate) from the line joining the primaries. On the other hand, presence of the disc has the reverse effect. Moreover, in the absence of q 1 and A, changes in the value of b do not influence the x-coordinates of the triangular points (Table 1). 4 ANALYSIS OF STABILITY OF L 4 UNDER THE LINEAR CASE If an infinitesimal body is displaced a very little from the exact equilibrium point and given a small velocity, then it will either oscillate around the respective point, at least for a considerable interval of time, or it will depart rapidly from its original position. The first case implies that the point from which the displacement is made is stable, whereas in the second case it is unstable. Due to the presence of a velocity-dependent term in the expression for effective potential, the criterion to determine the stability of the points by observing the shape of the surface (hills, valley and saddle) of the effective potential is not appropriate. Therefore, we perform a linear stability analysis in the neighbourhood of the triangular equilibrium point by linearizing the equations of motion of an infinitesimal mass in the vicinity of the L 4 point (Moulton 1984; Murray & Dermott 000). Let us take a small displacement and a small velocity, so that the coordinates of the infinitesimal mass and components of its velocity are x = x e + X, y = y e + Y, ẋ = Ẋ and ẏ = Ẋ, wherex, Y, Ẋ and Ẏ are initially very small quantities and (x e, y e ) is the exact position of the infinitesimal body. Substituting these into equations (3) and (4), we obtain the following new differential equations in the neighbourhood of the equilibrium point: Ẍ nẏ = x (x e + X, y e + Y ), (13) Ÿ + nẋ = y (x e + X, y e + Y ), (14) where x (x e, y e ) = y (x e, y e ) = 0. Now we expand the right-hand sides of equations (13) and (14) about (x e, y e ) using Taylor s series up to first-order terms in X and Y.SinceX and Y are very small, the influence of second-order and higher powers in the right-hand side terms is negligible. The above equations therefore change into the following forms: Ẍ nẏ = X 0 xx + Y 0 xy, (15) Ÿ + nẋ = X 0 yx + Y 0 yy, (16) where 0 xx, 0 xy, 0 yx and 0 yy are second-order partial derivatives with respect to x and y, respectively, of the effective potential of the system, which are obtained with the help of equations (5) and (6). Superscript 0 denotes the corresponding value at the equilibrium point. In order to solve equations (15) and (16), let X = P e λt and Y = Qe λt, (17) where P and Q are constants and λ is a parameter characteristic of the solution. On substituting these into equations (15) and (16) and dividing out by common factors, a linear system of equations is obtained as follows: ( ) ( ) λ 0 xx P + nλ 0 xy Q = 0, (18) ( ) ( ) nλ 0 yx P + λ 0 yy Q = 0. (19) For a non-trivial solution, the determinant of the coefficient matrix of these equations must vanish, i.e. λ 0 xx nλ 0 xy nλ 0 yx λ 0 = 0. yy On simplifying, we obtain a biquadratic equation in λ known as the characteristic equation: λ 4 + Aλ + B = 0, (0) where A = 4n ( 0 xx + 0 yy) and B = 0 xx 0 yy ( 0 xy). (1) Substituting the values of 0 xx, 0 xy, 0 yx and 0 yy in equation (1), it is found that ] A = n 3μA 3πch log(b/a) r,0 5 8r0 4 and B = 9μ(1 μ)γ 0, () Downloaded from on 16 June 018

4 1744 R. Kishor and B. S. Kushvah with γ 0 = y 0 + q 1 r1,0 5 r5,0 ( chπ(1 μ) r 5 0 r5, A r,0 ) + q ( 1μ b a r1,0 5 r5 0 ab + log(b/a) ) r 0 ( ) ( 1 + 5A (b a) r,0 ab + log(b/a) ) ], r 0 (3) where subscript 0 indicates the value at the triangular equilibrium point. If we take = λ, then we obtain a quadratic equation in : + A + B = 0. (4) Let λ 1, λ, λ 3 and λ 4 be the characteristic roots of biquadratic equation (0) or quadratic equation (4). Then, a general solution of the system of linear differential equations with constant coefficients (15) and (16) can be expressed in terms of exponentials such as X(t) = P 1 e λ 1t + P e λ t + P 3 e λ 3t + P 4 e λ 4t, (5) Y (t) = Q 1 e λ 1t + Q e λ t + Q 3 e λ 3t + Q 4 e λ 4t, (6) where constants Q 1, Q, Q 3, Q 4 are related to arbitrary constants P 1, P, P 3, P 4 respectively, by means of linear equations (18) and (19). From equations (5) and (6), it is clear that if λ j, j = 1,, 3, 4 are purely imaginary then X and Y are expressible in the form of periodic functions and hence the solutions given by equations (5) and (6) are said to be stable. However, if the roots are of the same magnitude, i.e. multiple pure imaginary, then, due to the presence of a secular term, the solutions are unstable. On the other hand, if λ j, j = 1,, 3, 4 are positive real numbers then the solutions are said to be unstable. In the case of complex roots, if the real part of at least one root is positive, then due to the presence of an exponential term in the solution it becomes unstable, whereas if all the real parts of the complex roots have negative signs then the solution is said to be asymptotically stable (Boccaletti & Pucacco 1996). Now, from the characteristic equation (4), we obtain 1, = A ± A 4B] λ 1, =± 1/ 1 ; λ 3,4 =± 1/. (7) These four roots depend in a simple manner on the parameters μ, A, q 1 and b. In order to ensure stability of the L 4,5 points, the four roots λ j, j = 1,, 3, 4 of the characteristic equation (0) must be purely imaginary. In other words, the sign of the discriminant (A 4B) of equation (4) determines the scenario of stability. Thus, three cases arise on the basis of the value of the discriminant. Now, equating the discriminant of equation (4) with zero, we obtain n 3μA r,0 5 ] 3πch log(b/a) 8r0 4 36μ(1 μ)γ 0 = 0. (8) At classical values of the parameters, i.e. q 1 = 1, A = 0, a = 1and b = 1, equation (8) provides Routh s condition of critical mass ratio μ c = In our case (in the presence of perturbations q 1 = 0.80, A = 0.003, a = 1andb = 1.4), the critical value of the mass ratio is μ c = One can also describe these three cases on the basis of the critical value of mass ratio of the problem μ c.thus,wehave (i) A 4B <0 or μ c <μ 1, (ii) A 4B = 0 or μ = μ c and (iii) A 4B >0 or 0 <μ< μ c. Fig. 1 shows how the nature of the roots varies with the mass parameter. From the figure, it can be seen that roots have real parts for μ c <μ 1/. In this case, the roots are of the form ±α ± iβ and consequently we have at least a positive real part, so the perturbed motion is unstable. At μ = μ c, there are multiple roots of equal magnitude providing secular terms which make the solution unstable. However, in the case of 0 <μ< μ c, the roots are of the form ±iω 1 and ±iω and so the perturbed motion is stable. The nature of the curves shown in Fig. 1 can be understood by considering the analytical solutions to the characteristic equation (0). Case I:whenμ c <μ<1/; A 4B < 0. Let d =+ A 4B. Then, from equation (7), we have 1, = 1 ( A ± id) (9) and hence the four roots λ j, j = 1,, 3, 4 take the form λ 1 = 1 ( A + id) 1/ = α 1 + iβ 1, λ = 1 ( A + id) 1/ = α + iβ, Figure 1. (a) Numerical values of the real (shaded/blue lines) and imaginary (black lines) components of the roots λ j, j = 1,, 3, 4 of the characteristic equation for triangular equilibrium points as a function of μ (0, 0.5). The dashed vertical line represents the value of critical mass μ c = (b) Enlargement of the region about μ c in (a). Downloaded from on 16 June 018

5 Photogravitational Chermnykh-like problem 1745 λ 3 = 1 ( A id) 1/ = α 3 + iβ 3, λ 4 = 1 ( A id) 1/ = α 4 + iβ 4. (30) The lengths of these four roots are equal and given as λ = λ 1,,3,4 = 1 A + d, (31) whereas the principal amplitude of the first root λ 1 is given as A ± ] A θ = θ 1 = arctan + d. (3) d If θ 1, θ, θ 3 and θ 4 are the amplitudes of the four roots λ j, j = 1,, 3, 4, which are related as θ 1 = θ, θ = π + θ, θ 3 = π θ, θ 4 = π θ, (33) then the real and imaginary parts of the roots λ j = α j + iβ j, j = 1,, 3, 4 are related as follows: α = α 1 = α = α 3 = α 4 and β = β 1 = β = β 3 = β 4, (34) where d α = λ +A > 0, λ +A β = > 0. (35) d The real parts of the two roots of the characteristic equation (0) are positive and equal and therefore the triangular equilibrium points are unstable for this range of mass parameter μ. Case II:whenμ = μ c ; A 4B = 0. Since A 4B = 0, from equation (7) we obtain 1, = A A A and λ 1,3 = i ; λ,4 = i, (36) i.e. the characteristic equation (0) has multiple pure imaginary roots. The multiple roots give secular terms in the solution of the equations of motion of an infinitesimal mass and hence the triangular equilibrium points are unstable in this case. Case III:when0<μ< μ c ; A 4B > 0. Since the values of 0 xx and 0 yy are negative for this range of mass parameter, and A > 0. Thus 1 and are both negative and consequently the four roots λ j, j = 1,, 3, 4 of equation (0) are purely imaginary and hence the motion of an infinitesimal mass is stable for A > 4B. The purely imaginary roots occur in the pairs, of the form λ 1, =±i 1/ 1 =±iω 1, λ 3,4 =±i 1/ =±iω (i = 1), (37) A ± A where ω 1, = 4B (38) are real numbers. In this case, the resulting motion of an infinitesimal mass is composed of two periodic motions known as short and long periodic motions, with periods π/ω 1 and π/ω respectively. The short periodic motion with period π/ω 1 π is very close to the orbital period of the second primary with mass parameter μ = , whereas the longer period motion with period π/ω corresponds to liberation in the vicinity of triangular points L 4,5 (Murray & Dermott 000). In equations (5) and (6), if we write P j = ā j + i b j and Q j = c j + i d j,j= 1,, 3, 4, where ā j, b j, c j and d j R, then solutions X(t)andY(t) take the form X(t) = (ā 1 + i b 1 )e λ1t + (ā + i b )e λ t + (ā 3 + i b 3 )e λ 3t + (ā 4 + i b 4 )e λ 4t, (39) Y (t) = ( c 1 + i d 1 )e λ 1t + ( c + i d )e λ t + ( c 3 + i d 3 )e λ 3t + ( c 4 + i d 4 )e λ 4t. (40) The fact that, for the motion of an infinitesimal mass, X, Y in equations (39 and 40) as well as velocity components Ẋ, Ẏ must be real, is used to show that ā 1 = ā = a 1, ā 3 = ā 4 = a, b 1 = b = b 1, b 3 = b 4 = b, c 1 = c = c 1, c 3 = c 4 = c, d 1 = d = d 1, d 3 = d 4 = d. Thus, the coefficients of the exponential terms in equations (39) and (40) consist of complex conjugate pairs and so, with the help of Euler s equation e iθ = cos θ + isin θ, the real solution X, Y can be written as X(t) = a 1 cos ω 1 t + a cos ω t b 1 sin ω 1 t b sin ω t, (41) Y (t) = c 1 cos ω 1 t + c cos ω t d 1 sin ω 1 t d sin ω t. (4) Thus, in the case of purely imaginary eigenvalues, the motion of an infinitesimal mass in the neighbourhood of triangular equilibrium points is oscillatory and hence stable. If we take parametric values q 1 = 0.80, A = , a = 1, b = 1.4 in the vicinity of the point L 4 : ( , ), then eigenvalues ±ω 1, are ±1.15i and ±0.47i, respectively. Let us suppose that X(0) = 10 5, Y(0) = 10 5, Ẋ(0) = 0andẎ (0) = 0. Therefore, the perturbed motion given by equations (41) and (4) of an infinitesimal mass in the neighbourhood of L 4 is X(t) = 10 6 (14.08 cos ω 1 t 4.08 cos ω t sin ω 1 t.85 sin ω t), (43) Y (t) = 10 6 (13.30 cos ω 1 t 3.30 cos ω t 5.85 sin ω 1 t +.39 sin ω t). (44) Further, characteristic frequencies 0 <ω <ω 1 are computed at different values of parameters q 1, A, b and given in Table, which shows that the frequencies are increasing functions of radiation factor q 1, oblateness A and disc s outer radius b. It is also noted that the effect of oblateness of the second primary on ω 1, is negligible. Variation of frequencies is also displayed in the framework of q 1 ω 1, plots (Figs and 3) for different parameter values. The q 1 ω 1, curves in panels (a) and (b) of Figs and 3 show that the effect of A is very minor, whereas the curves in panels (c) and (d) illustrate that there is a rapid change in frequencies with small values of q 1 and it seems to increase with constant growth rate for Downloaded from on 16 June 018

6 1746 R. Kishor and B. S. Kushvah Table. Characteristic frequencies ω 1, at different values of parameters q 1, A and b. ω 1 ω A q 1 b = 1.0 b = 1. b = 1.4 b = 1.6 b = 1.0 b = 1. b = 1.4 b = Figure. Variation of frequency in ω 1 q 1 space at a = 1andμ = :(a)b = 1, A = 0.0; (b) b = 1, A = 0.003; (c) b = 1.4, A = 0.0; (d) b = 1.4, A = larger values of q 1. We note that there is a gap in the curve close to q 1 = 0.07, because the frequency changes its sign (from negative to positive). It is also observed that the presence of a disc in the problem is responsible for the fast growth rate of frequencies ω 1, with q 1. 5 PERTURBED MASS RATIO IN RESONANCE CASES Three main cases of resonance (Marchal 1991, 01; Markellos et al. 1996) of the problem are obtained as follows: ω 1 κω = 0, κ = 1,, 3. (45) From equations (38) and (45), we obtain a quadratic equation in μ, under a suitable approximation regarding its order, which provides the critical mass ratio μ κ for these three main resonance cases for a range of mass parameters 0 <μ<1/. The critical mass ratio depends significantly on the perturbation factors (radiation factor, oblateness and disc width). Therefore, ( ) κ ω1 = = A A 4B ω A + A 4B. (46) Simplifying the above, we obtain ( κ μ(1 μ)γ A ) + 1 = 0, where γ = 9γ 0. (47) κ Downloaded from on 16 June 018

7 Photogravitational Chermnykh-like problem 1747 Figure 3. Variation of frequency in ω q 1 space at a = 1andμ = :(a)b = 1, A = 0.0; (b) b = 1, A = 0.003; (c) b = 1.4, A = 0.0; (d) b = 1.4, A = Again, q 1 (0, 1]; A 1 and let us suppose that q 1 = 1 ɛ 1, b = 1 + ɛ ; ɛ 1, ɛ 1. In order to obtain a quadratic equation in μ, we expand the above expression using Taylor s series and neglect second- and higher order terms. Finally, the critical values of mass ratio μ κ, κ = 1,, 3 are obtained and given as follows: μ 1 = ɛ A ɛ, (48) μ = ɛ A ɛ, (49) μ 3 = ɛ A ɛ. (50) These expressions of critical value of mass ratio agree with that of Deprit & Deprit-Bartholome (1967) for classical values of parameters, i.e. at ɛ 1 = ɛ = A = 0. Moreover, these are similar to the values of Markellos et al. (1996) for ɛ 1 = ɛ = 0and0 A 0.1 up to first order in A and of Kushvah (008). Similar but not exact expressions have also been obtained by Subbarao & Sharma (1975) and Bhatnagar et al. (1994). The resonant values of mass ratio μ for several values of parameters q 1, A and b are computed up to seven decimal places (Table 3). We note that the critical value of mass ratio is a decreasing function of oblateness A and κ, whereas it is an increasing function of q 1 and b. It provides information about the linear stability region, i.e. it shows the upper bound of the stability regions on the μ-axis at different values of parameters q 1, A and b with κ = 1,, 3. Figs 4 6 represent the linear stability regions and corresponding main resonance curves in μ ɛ 1, μ A and μ ɛ spaces for 0 ɛ 1 = 1 q 1 1, 0 A 0.1 and 0 ɛ = b 1 0.4, respectively. From Fig. 4, it is observed that the stability region decreases with the value of ɛ 1,i.e.whenradiationpressure increases (or q 1 = 1 ɛ 1 decreases from 1 to 0) the stability region also decreases (panel a) for all three main resonance cases. However, in the presence of oblateness as well as a disc, the stability region expands (panel b). Fig. 5(a) shows that the stability region decreases with oblateness whereas in the presence of a disc, in addition to radiation pressure force, the rate of decrease of the stability region increases (Fig. 5b). Similarly to the case of the radiation factor, the stability also decreases with the disc s outer radius b = 1 + ɛ (Fig. 6a), whereas, in the presence of radiation pressure and oblateness together, regions expand (Fig. 6b). On the basis of numerical as well as graphical results, we note that the perturbation factors have a significant effect on the motion of a spacecraft or satellite in space. It is observed that the nature of the motion is unaffected but the stability region of the motion varies with radiation pressure and oblateness as well as the width of the disc. The influence of oblateness of the smaller primary is much lower, but still considerable, whereas the presence of a disc affects the motion significantly. In the perturbed case, the positions of the L 4,5 frequencies of stable motion of an infinitesimal mass and stability regions of the three main resonances deviate from the classical results. The stability regions in the resonance case increase with disc s outer radius. 6 CONCLUSIONS The generalized photogravitational Chermnykh-like problem is studied in the context of the linear stability of triangular equilibrium Downloaded from on 16 June 018

8 1748 R. Kishor and B. S. Kushvah Table 3. Mass ratio μ κ (A, b), κ = 1,, 3, 4, 5 at different values of q 1, A and b. q 1 k μ k (0.0, 1.0) μ k (0.0, 1.) μ k (0.0, 1.4) μ k (0.0015, 1.0) μ k (0.0015, 1.) q 1 k μ k ( , 1.4) μ k ( , 1.0) μ k ( , 1.) μ k ( , 1.4) Figure 4. Linear stability region of triangular equilibrium points in the μ ɛ 1 parameter space and the resonance curves ω 1 κω = 0, κ = 1,, 3, (a) without perturbations and (b) with perturbations. points in the Sun Jupiter system under the influence of radiation pressure and oblateness of the massive bodies, respectively, in the presence of a disc rotating about the common centre of the system. On the basis of analysis and numerical results as well as graphical observations, we conclude that the motion of a spacecraft in the vicinity of the triangular points is affected. The opposing natures of radiation pressure and the gravitational force of the Sun reduce the strength of its gravity field, which causes an increment in the stability regions for the motion of a spacecraft. Moreover, due to gravitational attraction of the disc, stability regions expand in the sphere of influence. As regards the oblateness effect of an oblate body, its gravity field increases due to the equatorial bulge, whereas the presence of a disc minimizes its influence and hence its effect is much smaller. Thus, we can say that the presence of a disc in the Sun Jupiter system affects the motion of a spacecraft in the vicinity of the triangular equilibrium points. The perturbed mass ratio of the system is obtained for the three main resonances, both analytically and numerically. It is seen that results agree with those Downloaded from on 16 June 018

9 Photogravitational Chermnykh-like problem 1749 Figure 5. Linear stability region of triangular equilibrium points in the μ A parameter space and the resonance curves ω 1 κω = 0, κ = 1,, 3, (a) without perturbations and (b) with perturbations. Figure 6. Linear stability region of triangular equilibrium points in the μ ɛ parameter space and the resonance curves ω 1 κω = 0, κ = 1,, 3, (a) without perturbations and (b) with perturbations. of the classical case (Deprit & Deprit-Bartholome 1967) and those of generalized cases (Markellos et al. 1996; Kushvah 008). The results may be applied to study the perturbed motion in a generalized dynamical system and will help to analyse the same model in the case of drag forces (P R drag, solar wind drag, etc.) in future. ACKNOWLEDGEMENTS This work is supported by the Department of Science and Technology, Government of India through the SERC-Fast Track Scheme for Young Scientists SR/FTP/PS-11/009]. Some of the references used in the article were collected from the Library of Inter- University Centre for Astronomy and Astrophysics (IUCAA), Pune (India). REFERENCES Bhatnagar K. B., Gupta U., Bhardwaj R., 1994, Celest. Mech. Dyn. Astron., 59, 345 Boccaletti D., Pucacco G., 1996, Theory of Orbits. Springer, Berlin Chermnykh S. V., 1987, Vest. Leningrad Univ.,, 10 Deprit A., Deprit-Bartholome A., 1967, AJ, 7, 173 Goździewski K., 1998, Celest. Mech. Dyn. Astron., 70, 41 Goździewski K., Maciejewski A. J., 1999, Celest. Mech. Dyn. Astron., 75, 51 Ishwar B., Kushvah B., 006, J. Dyn. Syst. Geometric Theor., 4, 79 Jiang I.-G., Ip W.-H., 001, A&A, 367, 943 Jiang I.-G., Yeh L.-C., 004a, Int. J. Bifurcation Chaos, 14, 3153 Jiang I.-G., Yeh L.-C., 004b, AJ, 18, 93 Jiang I.-G., Yeh L.-C., 004c, MNRAS, 355, L9 Jiang I.-G., Yeh L.-C., 006, Ap&SS, 305, 341 Kishor R., Kushvah B. S., 013, Ap&SS, 344, 333 Kushvah B. S., 008, Ap&SS, 318, 41 Kushvah B. S., Kishor R., Dolas U., 01, Ap&SS, 337, 115 Marchal C., 1991, in Roeser S., Bastian U., eds, Predictability, Stability, and Chaos in N-Body Dynamical Systems. Springer, Berlin, p. 73 Marchal C., 01, Studies in Astronautics 4. The Three-Body Problem. Elsevier, Amsterdam Markellos V. V., Papadakis K. E., Perdios E. A., 1996, Ap&SS, 45, 157 McCuskey S. W., 1963, Introduction to Celestial Mechanics. Addison- Wesley, Boston Moulton F. R., 1984, An Introduction to Celestial Mechanics. Dover Press, New York Murray C. D., Dermott S. F., 000, Solar System Dynamics. Cambridge Univ. Press, Cambridge Papadakis K. E., 005a, Ap&SS, 99, 19 Papadakis K. E., 005b, Ap&SS, 99, 67 Ragos O., Zagouras C. G., 1993, Ap&SS, 09, 67 Rivera E. J., Lissauer J. J., 000, ApJ, 530, 454 Sharma R. K., Rao P. V. S., 1976, Celest. Mech., 13, 137 Strand M. P., Reinhardt W. P., 1979, J. Chem. Phys., 70, 381 Subbarao P. V., Sharma R. K., 1975, A&A, 43, 381 Szebehely V., 1967, Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, New York Wolfram S., 003, The Mathematica Book. Wolfram Media, Champaign Yeh L.-C., Jiang I.-G., 006, Ap&SS, 306, 189 This paper has been typeset from a TEX/LATEX file prepared by the author. Downloaded from on 16 June 018