Existence and stability of collinear equilibrium points in elliptic restricted three body problem with radiating primary and triaxial secondary

Transcription

1 Modelling, Measurement and Control A Vol. 9, No., March, 08, pp. -8 Journal homepage: Existence and stability of collinear equilibrium points in elliptic restricted three body problem with radiating primary and triaxial secondary Usha Tadepalli *, Narayan Ashutosh Department of Mathematics, Bhilai Institute of Technology, Chhattisgarh , India Corresponding Author usha6@gmail.com Received: August 07 Accepted: 0 December 07 Keywords: celestial mechanics, collinear points, elliptical restricted three body problem, stability, oblateness, rigid body ABSTRACT The present paper studies the motion of an infinitesimal mass around a stellar primary and triaxial secondary moving around each other in the elliptic orbits about their common center of mass in the neighbourhood of Collinear equilibrium points. The location and stability of the collinear points are found to be affected by the radiation pressure and triaxility parameters. The nature of stability, however, remains unchanged. The collinear points L and L are unstable in the Lyapunov sense. But, the collinear point L shows a stable behavior for small values of radiation and triaxiality parameters.. INTRODUCTION The restricted three body problem has been widely studied by many researchers. This is because of its application to the celestial and stellar bodies. The system consists of two finite bodies, known as primaries, moving about their common center of mass, being attracted by the gravitational attraction of each other. The motion of the third body is influenced by both the primaries. The primaries may describe either a circular or an elliptic path. If the system performs motion about a circular orbit, then it is a circular restricted three body problem otherwise it is an elliptic restricted three body problem. The elliptic restricted three body problem generalizes the circular restricted three body problem. The orbit of the Jupiter around the Sun is a fixed ellipse and the Trojan asteroids are influenced by the gravitational attraction of the Sun and Jupiter. The stability of such systems (ERBP) moving in elliptic orbits was investigated by many authors, [- 8] and many others. The bodies in the classical model of the problem were considered as spherical, but many celestial bodies are either oblate spheroids or triaxial or both, and not spheres. For instance, the Mars, Jupiter, Saturn, Neutron stars, Regulus and white dwarfs are oblate spheroids whereas, the Moon and Pluto and its moon Charon are triaxial. The Earth is also oblate and triaxial as well. This oblateness and triaxiality of the primaries cause perturbations in the system. This inspired many authors to include these characterizations in their study of Restricted/Elliptic Resricted three body problem. The stability of the collinear equilibrium points in the Photogravitational/Generalised Photogravitational Elliptic Restricted three body problem was studied in [9-]. The linear stability of periodic orbits of the Lagrangian equilibrium points of the ERTBP, was studied in [] based on some value of the mass ratio. The dynamical properties of the solar sail in Elliptic Restricted three body problem were also analysed and studied in [-]. The effect of the solar radiation pressure on the location and stability of the equilibrium points was studied in [6] taking bigger primary as radiating and taking both primaries as radiating in [7]. In both studies the result was that the radiation pressures of the primaries affect the location of the collinear points. The positions and stability of the collinear equilibrium points, L,, of an infinitesimal body in the elliptic restricted threebody problem (ERBP) when both primaries of the system are luminous and oblate spheroids moving in elliptic orbits around their common center of mass was studied in [8]. The stability of the Lagrangian equilibrium points in the elliptic restricted three body problem with radiating and triaxial primaries was studied based on Averaging method and Floquet's theory respectively in [9] and [0]. There are five equilibrium points in the Elliptic Restricted three body problem. The equilibrium points are the points at which the particle has zero velocity and zero acceleration and are very important for astronautical applications [6]. Three of the equilibrium points are called collinear points as they lie on the x-axis (axis joining the two primaries) and are denoted by L, L, L. The remaining two are called the Lagrangian points and are denoted by L 4, L. In the present work, we study the effects of solar radiation pressure and the triaxiality of the primaries. The motion of the Collinear equilibrium points has been studied in the frame work of the Elliptic Restricted three body problem. To the best of our knowledge propensities due to radiation of the primary and the triaxiality of the secondary has not been previously studied. The study of the collinear points is important. This is because the orbits near these points are useful for spacecraft missions. These are suitable to set permanent observatories of the Sun, the magnetosphere of the Earth, links with the hidden part of the Moon, and others []. The locations of the collinear points have been found analytically in terms of the mass ratio, solar radiation pressure and the triaxiality of the primaries. The method proposed in [] and [6] has been used. The stability of the collinear points has also been studied by averaging the system. The numerical calculations and the graphs have been plotted using the Wolfram Mathematica 0. and Matlab software, respectively. The locations of the

2 collinear points and the stability has been analysed for the Sun- Earth system. The present paper is organised as follows: Section, which is introduction; Section provides the equations of motion; Section gives the location of the Collinear points; Section 4 focuses on the stability of the different Collinear points. The conclusions of the work are drawn in Section.. EQUATIONS OF MOTION The differential equations of motion of the infinitesimal mass in the elliptic restricted three body problem under radiating and triaxial primaries in the barycentric, pulsating and rotating, non-dimensional coordinates are derived in [9] and given in equation (). The notations in principle are taken from [4] with some minor modifications in the notation being done for adapting to the present problem, presented as : x y = ( + e cos v) x y + x = (+e cos v) y where denotes differentiation with respect to v, and Ω = x +y Let, () + n [( μ)q r + μ r + μ(σ σ ) r μ(σ σ ) r ] () k = n [( μ)q r + μ r + μ(4σ σ ) r μ(σ σ )y r7 ] () Then, () can be written as, x y = [x ( k + μ(σ σ ) ) + μ( μ) ( q (+e cos v) r y + x = where n r r (σ σ ) r + (σ σ )y r7 )] (+e cos v) [ k]y (4) n = + (σ σ ); () and, σ = a b R ; σ = b c R ; r = (x + μ) + y, r = (x + μ) + y (6) The dimensionless variables are introduced using the distance between the primaries given by: r = a( e ) ( + e cos v) Here, m and m are the masses of the bigger and smaller primaries positioned at (x i,0), i=,, q=-δ the radiation pressure; σ, σ are triaxiality parameters, σ i, (i=,)[]; and a, b,c are semi axes and R is the distance between the n primaries; r i (i=,) are the distances of the infinitesimal mass from the bigger and smaller primaries respectively; while e is the eccentricity of the either primary around the other and v is the true anomaly.. LOCATION OF COLLINEAR EQUILIBRIUM POINTS The equilibrium points of the system are the points where the consumption of the resources is the minimum, hence, are given by the equations: x = 0; = 0. (7) y where, Ω is given by equation (). But, the collinear points lie on the x-axis; hence, are given by the conditions: x = 0; = 0; y = 0. (8) y Hence, using equations (7) and (8), we get: f(x) = [x n {( μ)(x+μ)q r + μ(x +μ) r + μ(x +μ)(σ σ ) r }] = 0 (9) There are three collinear equilibrium points. These are denoted by L, lying between the bigger and the smaller primary (-μ<x<-μ); L, lying to the right of smaller primary (x>-μ) and L, lying to the left of the bigger primary (x<-μ).. Location of L To find the solution for L, substituting x = x ρ, such that r = ρ and r = ρ into equation (9), we have: μ ρ n {( μ)q ( ρ) + μ ρ + μ(σ σ ) ρ 4 } = 0 (0) Now, rearranging the terms, and simplifying we have: ρ [ { (σ σ ) δ} {[ 4 { (σ σ ) μ(+(σ σ )) ( μ){ (σ σ ) ρ + { 4δ } (σ σ ) 4δ } { (σ σ ) 4δ } ρ ] = 4δ }] ( ρ) [ 0(σ σ )ρ + (σ σ )ρ {n + 6(σ σ )}ρ ]} () Now, let, μ( + (σ σ )) [ ( μ) { (σ σ ) 4δ ] } Then using series expansion given as: = λ ρ = λ( + c λ + c λ + ) () The simplified equation can be written as:

3 δ (σ σ ) 4δ ρ = λ [ σ ) {+9(σ } λ σ ) δ 9 {+87(σ { (σ σ ) 4δ } } λ + ] () Hence, the solution for L is given by: δ (σ σ ) 4δ x = μ λ [ σ ) {+9(σ 87(σ σ ) δ 9 {+ { (σ σ ). Location of L } λ 4δ } } λ + ] (4) For finding the location of L, substituting x = x + ρ such that r = ρ, r = + ρ. Then, substituting, the values in equation (9), we have: ρ[(+n )(+ρ+ ρ )+(n q) ρ ] = μ [ nρ+ (σ σ ) ρ (+ρ) ( μ) On simplification, we have: ρ [ + { (σ σ ) δ} {[ {+ 7(σ σ ) μ(+(σ σ )) ( μ){+ 7(σ σ ) ρ + { + 4δ } + (σ σ )+ 4δ } {+ 7(σ σ ) ρ ] () + 4δ } ρ ] = + 4δ }] ( ρ) [ 0(σ σ )ρ + 4 (σ σ )ρ { + (σ σ )} ρ ]} (6) Using the series as in equation (), the value of ρ is given as: ρ = λ [ σ ) { 4(σ + 4δ + 7(σ σ ) + 4δ { + 67(σ σ ) + 6δ 9 {+ 7(σ σ ) } λ + 4δ } } λ + ] (7) Hence, the solution for L is given as: x = μ + λ [ σ ) { 4(σ { + 67(σ σ ) + 6δ 9 {+ 7(σ σ ). Location of L + 4δ + 7(σ σ ) + 4δ } λ + 4δ } } λ + ] (8) For the point L substituting x = x ρ such that r = ρ and r = + ρ in equation (9), we have: μ = (n ρ q)(+ρ) μ ρ [+ (σ σ ) (+ρ) n (+ρ) ] (9) taking, ρ = + α, and using the elementary algorithm for division upto O[α 4 ], we have: μ = [( 9 (σ μ 7 σ ) 4 δ) + ( (σ 7 4 σ ) 9 8 δ) ( α) + ( (σ 7 67 σ ) 989 δ) ( α) + ( (σ σ ) 46 δ) ( α) +. ] (0) Now, using the method of successive approximations and Lagrange inversion formula [4], and retaining only linear terms in δ, σ, σ we get: ρ = [( (σ 4 σ ) δ) 7 ( (σ 4 σ ) μ δ) ( ) ( (σ μ 44 σ ) 7 μ δ) ( 04 μ ) + ( + 07 (σ σ ) δ) ( μ μ ) + O [ μ μ ]4 ] () Hence, the solution for L is given as: x = μ [( (σ 4 σ ) δ) 7 ( (σ 4 σ ) μ δ) ( ) ( (σ μ 44 σ ) 7 μ δ) ( 04 μ ) + ( + 07 (σ σ ) δ) ( μ μ ) + O [ μ μ ]4 ] () 4. LINEAR STABILITY OF COLLINEAR POINTS The stability of motion of the collinear points is decided by the following lemma whose formulation is taken from [], with some modifications in notations being done for adapting to the present problem. Lemma: At the collinear points: k = n [( μ)q r + μ r + μ(4σ σ ) r μ(σ σ )y r7 ] > () Proof: For an equilibrium point, we have the condition: x n {( μ)(x+μ)q r + μ(x +μ) μ(x +μ)(σ σ )y r + μ(x +μ)(σ σ ) r r 7 } = 0 (4) The condition for a collinear equilibrium point is y=0, so the equation (4) gets simplified as: x n {( μ)(x+μ)q r + μ(x +μ) μ(x +μ)(σ σ )y r + μ(x +μ)(σ σ ) r r 7 } = 0 () Rearranging the terms, the equation () can be written as: n {( μ)(x+μ)(r r q) + μ(x +μ)(r r ) + r r μ(x +μ)(σ σ )(r r 4 ) + μ( μ)(x+μ)(σ σ ) r } = 0 (6) Next, to prove equation () we analyze each collinear equilibrium point separately.

4 4. Stability at collinear point L Now, at the point L, r + r =, where, r = x + μ, and r = x μ. Substituting the values in equation (6) and simplifying using equation (), we have: [ n [( k + μ( μ)(σ σ ) ) r + μ(σ σ ) μ ( r (σ σ ) r )]] = 0 Since, n 0 and r <, we have: k = + [ μ r ( r + (σ σ ) r (σ σ )) + μ( μ)(σ σ ) r ] (7) Hence, k > for collinear point L. 4. Stability at Collinear point L At L, r r =, r = x + μ and r = x + μ. Inserting, the values in equation (6), and using equation (), and proceeding in the same way as for L, also for collinear point L, we have, k = + [ μ r ( r + (σ σ ) r (σ σ )) + μ( μ)(σ σ ) r ] (8) Hence, for collinear point L, we have, k >. 4. Stability at collinear point L At L, r r =, r = x μ, r = x μ +. Proceeding in the same manner as in L and L, substituting values in equation (6) and using equation () we have: k = + [ μ r { ( r + (σ σ ) r (σ σ ))} + μ( μ)(σ σ ) r ] (9) Hence, k >, for collinear point L also. Thus, for all collinear points L, L and L, we have k >. This completes the proof of lemma. To analyses the stability of motion around the primaries near the collinear points, investigating the roots of the characteristic equations. For this, assuming, that the particle receives a small displacement from the equilibrium position. Then finding the variational equations of motion by substituting the coordinates of displaced point in the equation of motion equation () and expanding by Taylor's series about the collinear points and taking only the linear terms, we get the equation following [0] as: ξ η = φ[ω 0 xx ξ + Ω 0 xy η]; η + ξ = φ[ω 0 yx ξ + Ω 0 yy η].(0) where φ = and (x (+e cos v) 0, y 0 ) are the coordinates of the collinear points respectively. The subscript of Ω denotes the second order partial derivatives of Ω with respect to x and y, as it appears, respectively. Since, all the collinear points lie on the x-axis, hence y=0, resulting, Ω xy = 0. Introducing new variables given by, x = ξ, x = η, x = dξ dv, x 4 = dη dv Substituting these values in equation (0), the system of equations can be written as: dx i dv = P ix + P i x + P i x + P i4 x 4 ; i =,,,4 () where, P = P = P 4 = P = P = P = P = P 44 = 0; P = P 4 = ; P 4 = ; P 4 = ; P = φω 0 0 xx ; P 4 = φω yy where, superscript '0' indicates the values evaluated at respective collinear points. The coefficients of equation () are π periodic functions of v. Hence, taking the averaged system, given by: dx i (0) dv = P (0) i x (0) + P (0) i x (0) + P (0) i x (0) + P (0) i4 x (0) 4 ; i =,,,4 () where, P (0) is = π π P is(v)dv ; i, s =,,,4 0 So, we get: P (0) = Ω e xx 0 ; P (0) 4 = Ω 0 e yy where, superscript '0' indicates the values evaluated at respective collinear points L, L, L. Thus, the characteristic equation for the system of equations () can be given as: λ 4 + Qλ + R = 0 () where, Q = (4 P 0 P 0 4 ); R = P 0 0. P 4 (4) The motion of the infinitesimal particle will be stable near the collinear point when given a small displacement and small velocity, the particle oscillates for a considerable time about the points [9]. That is, the system will be stable if the roots of the characteristic equation are purely imaginary. Hence, the condition for stable roots can be given as: Q < 0; R > 0. Taking the latter inequality, the condition of stability can be written as: 4

5 4 (σ σ ) μ(σ σ ) < k r < + 4 (σ σ ) μ(σ σ ) r If the values of σ, σ are negligibly small. Then, the condition of stability can be written as: < k < () But, from the above subsections, it is clear that for all the collinear equilibrium points, L, L, L we have k>. Thus, the collinear points are unstable, using the above condition given by equation (). The roots of the characteristic equation (6) are given by: λ, = [k ( e ) μ(σ σ ) r ]±[(9k 8k 8μk(σ σ ) r )+8e (+k μ(σ σ ) r )] e (6) Let, λ i = s i, i =,. Hence, the above equation (6) can be written as: s = [k ( e ) μ(σ σ ) r ]+[(9k 8k 8μk(σ σ ) r )+8e (+k μ(σ σ ) r )] s = e [k ( e ) μ(σ σ ) r ] [(9k 8k 8μk(σ σ ) r )+8e (+k μ(σ σ ) r )] e As, k > for collinear points, we have: [(9k 8k 8μk(σ σ ) ) + 8e ( + k μ(σ σ ) r )] r (7) > (8) for the values of σ, σ <, e <. Thus, from the above equations (7) and (8), we have s > 0, s < 0. As λ = s, s > 0 gives two real roots of opposite signs, and s < 0, results into two imaginary roots. Hence, the solution of equation (6) can be written in the form: λ i = C i ε p v + C i ε p v + C i cos(p v C i4 ), i =,. (9) where, p, p, p are the roots of equation (). The first and second terms of equation (9) cause an exponential growth in the values of the roots λ i and dominates the third term. Thus, from the values of the two characteristic roots, it is also clear that the motion is unstable near a collinear equilibrium point. 4. CONCLUSION The formulas derived in the paper can be applied to the Sun and the Earth as primaries and the particle as space craft}. Hence, for the system, the mass parameter μ = m = m +m , the eccentricity of the elliptic orbit of the primaries, e= The location of collinear points for the Sun-Earth system are given in table: [], for different values of δ, σ, σ. The nature of motion around the collinear points can be analysed as: (i) the motion around the point L is unstable for all the values of δ, σ and σ as k > and λ > 0, λ < 0. This can be seen from table: [] and is also evident from graph (figure:) (ii) the point L also exhibits the instability of motion in its vicinity as λ > 0 and λ < 0 and also k> for all values of δ, σ and σ.this is evident from table: []. The instability can also be seen from the graph(figure:) (iii) For L, the motion appears to be stable for some values of δ, σ and σ because the values of k < as well as λ, < 0. As, the value of σ and σ, increases the system becomes unstable, for a fixed value of δ, the roots of the system λ, < 0. That is, the roots will be imaginary implying the stability of the system which is evident from the table: [4] and also from the graph(figure:) When σ and σ are negligibly small, the locations and stability of our results are in confirmation with [6] and [7]. The collinear point L was found to be stable for the value of radiation pressure β 0. in [6] and [7]. This result is also in confirmation with [6] and [7] as the collinear point L is found to be stable in our result also, clearly evident from table: [4]. The collinear points L and L are found unstable, as is given in [6-7] and [8]. The existence and stability of collinear equilibrium points of the elliptic restricted three body problem with radiating primary and triaxial secondary has been analysed by the method given by [6, ]. The rotating coordinate system and dimensionless pulsating variables are used to make the system independent of time. The numerical calculations are performed using the software Mathematica 0. and the graphs are plotted using Matlab software. The location of the collinear points L, L and L are given by equations (4), (8) and (), respectively. The increment in the values of δ, σ and σ affects the locations of the collinear points. The stability of the system can be analysed by the equations (), (7) and (8) using the condition of stability given by equation (). The points L, L exhibit the unstable motion in their vicinity, which is evident from graphs (figure:, figure:). But, the motion around the point L is stable for a particular value of δ, σ and σ (hypothetical values), which can be seen from the figure. Also, the stability has been numerically analysed for the Sun-Earth system. Thus, it can be concluded that the stability of the collinear point L is dependent on the parameters, radiation pressure and the triaxiality parameters, respectively. Increase in the value of the radiation parameter makes motion around the collinear point L stable. On the other hand, increase in the value of triaxiality parameters causes instability of motion around the collinear point L. Table. Location of collinear equilibrium points for Sun-Earth system for different values of δ, σ and σ Collinear Point L L L For σ = , σ =

6 δ = 0.0 ( ,0) (.00999,0) ( ,0) δ = 0.00 ( ,0) (.000,0) ( ,0) δ = 0. ( ,0) (.0099,0) (-0.98,0) δ = 0. (0.9889,0) (.00899,0) ( ,0) For δ = 0.00 For σ = 0.0, σ = 0.0 ( ,0) (.06,0) ( ,0) For σ = 0.00, σ = 0.00 ( ,0) (.009,0) (-0.999,0) For σ = 0.000, σ = ( ,0) (.0006,0) ( ,0) Table. Values of λ,, k for different values of δ, σ, σ and e for L k, λ, k λ λ δ For σ = 0, σ = 0, e = 0 δ = δ = δ = δ = δ = δ = For δ = 0.00, e=0 For σ = 0.0, σ = For σ = 0.00, σ = For σ = 0.000, σ = For σ = 0, σ = 0, δ = 0.00 e= e= e= e= For δ = 0.00 σ = 0.0, σ = 0.04, e = σ = 0.00, σ = 0.004, e = σ = 0.000, σ = , e = Table. Values of λ,, k for different values of δ, σ, σ and e for L k, λ, k λ λ δ For σ = 0, σ = 0, e = 0 δ = δ = δ = δ = δ = δ = For δ = 0.00, e=0 For σ = 0.0, σ = For σ = 0.00, σ = For σ = 0.000, σ = For δ = 0.00 σ = 0.0, σ = 0.04, e = σ = 0.00, σ = 0.004, e = Table 4. Values of λ,, k for different values of δ, σ, σ and e for L k, λ, k λ λ δ For σ = 0, σ = 0, e = 0 δ = δ = δ = δ = δ = For e=0.0 For σ = 0.00, σ = 0.00, δ = For σ = 0.000, σ = 0.000, δ =

7 For σ = 0.000, σ = 0.000, δ = σ = 0.00, σ = 0.00, δ = σ = , σ = , δ = σ = 0.000, σ = 0.000, δ = σ = 0.00, σ = 0.00, δ = REFERENCES Figure. Correlation of characteristic root λ and δ for L Figure. Correlation of characteristic root λ and δ for L Figure. Correlation of characteristic root λ and δ for L [] Arnold VI. (96). Stability of Equilibrium position of a Hamiltonian system of ordinary differential equations in general elliptic case. DOKLADY AKADEMII NAUK SSSR 7():. [] Danby JMA. (964). Stability of the triangular points in the elliptic restricted problem of three bodies. The Astronomical Journal 69: 6. [] Bennett A. (96). Characteristic exponents of the five equilibrium solutions in the elliptically restricted problem. Icarus 4(): [4] Szebehely V. (967). Theory of orbits: The restricted three body problem. [] Broucke R. (969). Stability of periodic orbits in the elliptic restricted three-body problem. AIAA Journal 7(6): [6] Katsiaris G. (97). The three-dimensional elliptic problem, In Recent Advances in Dynamical Astronomy. Springer: 8 4. [7] Markellos VV. (974). Numerical investigation of the planar restricted three-body problem. Celestial Mechanics 0(): [8] Selaru D, Cucu-Dumitrescu C, (99). Infinitesimal orbits around Lagrange points in the elliptic restricted three-body problem. Celestial Mechanics and Dynamical Astronomy 6(4): 46. [9] Ishwar B, Sahoo SK. (006). Stability of collinear equilibrium points in the generalized photogravitational elliptic restricted three body problem. Celestial Mechanics: Recent Trends 99. [0] Zimovshchikov S, Tkhai VN. (004). Instability of libration points and resonance phenomena in the photogravitational elliptic restricted three-body problem. Solar system Research 8(): 64. [] Vijay K, Choudhry RK. (990). Nonlinear stability of the triangular libration points for the photo gravitational elliptic restricted problem of three bodies. Celestial Mechanics and Dynamical Astronomy 48(4): [] Roberts GE. (00). Linear stability of the elliptic Lagrangian triangle solutions in the three-body problem. Journal of Differential Equations 8(): 9 8. [] Baoyin H, McInnes CR. (006). Solar sail equilibria in the elliptical restricted three-body problem. Journal of Guidance, Control, and Dynamics 9(): 8 4. [4] Biggs JD, McInnes C. (009). Solar sail formation flying for deep-space remote sensing. Journal of Spacecraft and Rockets 46(): [] Biggs JD, McInnes CR, Waters T. (009). Control of solar sail periodic orbits in the elliptic three-body problem. Journal of Guidance, Control and Dynamics (): 8 0. [6] Ammar MK. (008). The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three-body problem. Astrophysics and Space Science (4):

8 [7] Narayan, Singh N. (04). Existence and stability of collinear equilibrium points in elliptical restricted three body problem under the radiating primaries. Astrophysics and Space Science 4(): 68. [8] Singh J, Umar A. (0). Collinear equilibrium points in the elliptic rbp with oblateness and radiation. Advances in Space Research (8): [9] Usha T, Narayan A, Ishwar B. (04). Effects of radiation and triaxiality of primaries on triangular equilibrium points in elliptic restricted three body problem. Astrophysics and Space Science 49(): 64. [0] Narayan, Usha T. (04). Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiating and triaxial primaries. Astrophysics and Space Science (): 4. [] G ómez G, Mondelo JM. (00). The dynamics around the collinear equilibrium points of the RTBP. Physica D: Nonlinear Phenomena 7(4): 8. [] Vinti JP. (998). Orbital and celestial mechanics. IAA 77. 8