Orbital Dynamics of an Ellipsoidal Body

Transcription

1 Orbital Dynamics of an Ellipsoidal Body Akash Gupta Indian Institute of Technology Kanpur The purpose of this article is to understand the dynamics about an irregular body like an asteroid or a comet by modelling it as an ellipsoidal body rotating uniformly about it s axis of maximum inertia. To start with, this report discusses the model characterization as to how it serves as a good approximation. Then, the analytic formulation of gravitational potential for the ellipsoidal body under this model is defined and derived. The equations of the system are, then, non-dimensionalized. Fixed points are evaluated and the eigenvectors in their vicinity are calculated to get a localised phase potrait of the system near the same. Bifurcations have also been dealt with. The dynamical equations are,also, numerically solved and plotted in the configuration space, to get an overall idea about the system. After the mid-semester examinations, the stability conditions of the fixed points have been studied. Also, the 4-D poincare map has been constructed for the general case. Kindly note that this system s formulation is quite complex, for it s a six-dimensional system and moreover, there are non-linear integrals involved which themselves are variables throughout the evolution of the dynamical system. These factors cause difficulties in the construction of algorithms for the system and thus, in interpretation of the results and further analysis. 1. Introction Investigation into the orbital dynamics of a spacecraft about planets have been extensively carried out in the last half century. But in recent times, with the advent of space exploration missions about asteroids, and comets, like Rosetta, Hayabusa etc. the need to study the orbital conditions around them has become of utmost importance. Establishing an understanding of the orbital dynamics about such small bodies like asteroids or comets can greatly facilitate in knowing about the origin and evolution of our solar system, and the universe, in general, and so, it has been the prime centre of scientific investigation ring the rendezvous missions. Moreover, in recent times the prospect of asteroid mining has seen a furore in the space instry, for such bodies have rich reserves of unexploited resources. Also, there is a consensus among scientists that in the upcoming future these could serve as temporary bases for long ration space flights. And then amongst all these, one of the detrimental motivation for this is to understand as to how to mitigate the danger of colliding into such asteroids or similar bodies in the future. The primary distinction between a planet and an asteroid like body comes from the fact that the latter has an irregular shape and structure, and are distinctly small in size, though recently distinction has blurred down a bit. This report serves as an attempt to understand the dynamics about such small bodies, by modelling them as ellipsoids and then, using the laws of classical mechanics for necessary investigation and analysis. One interesting thing to note here is that this whole model i.e. an ellipsoid rotating uniformly about one of it s principal axis is based upon the classical definition of gravity, which was proposed by Newton. This model of gravitational pull is not the most accurate, for now, the perfect model that exists is the Einstein s one which takes into consideration the relativistic effects in which gravity is explained as ripples in the space-time curvature. This, though, is an active area of research and so, we do not consider them for reasons of simplistic calculations. 2.1 Model Characterization As discussed above, we model the irregular body as a triaxial ellipsoid of mass M. There are a lot of celestial bodies like Phobos, Mars satellite etc. which have near ellipsoidal shape. The principal axes of the body are denoted by a 1,a 2,a such that a 1 a 2 a. It is assumed to have a uniform density ρ to simplify our calculations. Further, the body is assumed to be rotating around the axis of maximum moment of inertia with uniform angular velocity ω, this defines our system. This assumption in our model too is a good consideration since a vast majority of asteroids are found to be in a uniform rotation state about their maximum moment of inertia, which e to energy dissipations caused by tides raised by complex rotation, is the minimum energy rotation state of the body.

2 φ (x 1, x 2, x ; λ(x 1, x 2, x )) = 0 (6) 2 FIG. 1: Ellipsoidal Figure Further, let s denote the body fixed position vector as r and the body fixed velocity vector as ṙ = v. The Hamiltonian expressed in Lagrangian variables is: J(ṙ, r, t) = 1 2ṙ r (Ω r) (Ω r) U(r) (1) Taking the time derivative yields: J = Ω [(ṙ + Ω r) r] (2) If the asteroid is rotating uniformly, then ω is constant and the Hamiltonian is conserved. Thus, it is possible to define zero velocity surfaces,compute equilibrium points and periodic orbit families and to compute stability parameters. But if the asteroid is not rotating uniformly, then ω is a time periodic function in a body fixed frame. In this case, the equations of motion are time periodic and the Hamiltonian is not conserved. Analysis of this problem is more difficult than the uniformly rotating case. Hence, we shall assume the asteroid to be uniformly rotating, as this adds to the cause of this assumption. 2.2 GRAVITATIONAL POTENTIAL The gravitational force potential for a triaxial ellipsoidal body at a point x 1,x 2,x exterior to the body,using Maclaurin s and Ivory s theorem, is V (x 1, x 2, x ) = µ 4 where µ = 4πGρa1a2a (u) = φ (x 1, x 2, x ; u) =, λ(x 1,x 2,x ) φ (x 1, x 2, x, u) (u) () (a u)(a2 2 + u)(a2 + u) (4) [ ] x 2 1 a u + x2 2 a u + x2 a 2 + u 1 (5) Herein λ is the ellipsoidal coordinate of x 1,x 2,x defined by the ellipsoid which passes through x 1, x 2,x and is confocal to the ellipsoid characterized by a 1,a 2,a and is the largest root of (6). Thereby the attraction of a homogeneous ellipsoid at an exterior point x 1,x 2,x is ẍ = 2πρGB x (7) Where B is a symmetric tidal shaped tensor and captures the effect of ellipsoid s tri-axial nature on it s gravitational field and is defined as B i = λ a 1 a 2 a (a 2 i + u) (8) 2. DYNAMICAL SYSTEM The Equations of Motion in the Body Fixed Reference Frame (BFRF;of the ellipsoidal body) of a point mass, about an ellipsoidal body rotating uniformly with angular velocity ω, about its axis of maximum Moment of Inertia (x ), with the BFRF axes x i, defined along the directions of principal axes a i, and centred at ellipsoid s centre can be expressed as: ẍ 1 = ω 2 x 1 + 2ω 2 2πρGB 1 x 1 (9) ẍ 2 = ω 2 x 2 2ω 1 2πρGB 2 x 2 (10) ẍ = 2πρGB x (11) 2.4 DYNAMICAL SYSTEM The above equations are now nondimensionalized, with Non-Dimesionalized Time: t = t t c, and the Non-Dimensionalized Distance: x i = xi x c. Thereby also, Non-Dimesionalized Semi- Principal Axes a i = ai x c.thus, the Dimensionless quantities: Π 1 = t 2 cω 2, Π 2 = 2t c ω, Π = 2πρGt 2 c, and Π 4,5,6 = ai x c. Under the above considerations, the most appropriate choice of scaling constants seem to be the following, which can be seen evidently from the subsequently defined equations: T ime Scale Constant : t c 1 ω (12)

3 and Length Scale Constant : x c a 1 (1) 2 = (a 2 i + u), (27) i=1 Under the above defined scheme, let Also defining, b i = Π = 2πρGa 1a 2a ω 2 (14) B i a 1 a 2 a = λ (a 2 i + u) (15) Which under non-dimensionalization becomes, b i = λ (a 2 i + u ) (16) where * corresponds to non-dimensionalized counterparts. From here on, for the sake of brevity, we will omit * notation for non-dimensionalized terms. The non-dimensionalised equations of motion in the body fixed reference frame are, henceforth, written as: ẍ 1 = x 1 b 1 x 1 (17) ẍ 2 = x 2 b 2 x 2 (18) ẍ = b x (19) The Six-Dimensional Dynamical System can be expressed as, defined against the previously stated system of equations of motion: 1 = 2y 2 + (1 b 1 )x 1, (20) 2 = 2y 2 + (1 b 2 )x 2, (21) = b x, (22) 1 = y 1, (2) and, a 2 i=1 i x 2 i + λ = 1, (28) a 1 a 2 a ; a 1 = 1. (29) = 2πρGa 1a 2 a ω 2. (0) t c = ω 1 ; x c = a 1 (1) An algorithm has been constructed for evaluating the evolution of the above stated dynamical system, whereby an ellipsoidal body is rotating with a uniform angular velocity or zero angular velocity. For the case of zero angular velocity the dynamical system (Non-Dimensionalized) has been defined a bit differently and is as follows (It can be followed upon from a similar analysis as for the case stated): 1 = b 1 x 1, (2) 2 = b 2 x 2, () = b x, (4) 1 = y 1, (5) 2 = y 2, (6) = y. (7) where, b i = 2 = y 2, (24) = y. (25) λ (a 2 i (26) + u), where, b i = λ (a 2 i (8) + u), 2 = (a 2 i + u), (9) i=1

4 4 and, t c = a 2 i=1 i x 2 i + λ = 1, (40) a 1 a 2 a ; a 1 = 1. (41) = 1 (42) 1 2πρGa1 a 2 a ; x c = a 1. (4) Certain Results are on the following page. 2.5 FIXED POINTS AND BIFURCATION ANALYSIS The conditions for different fixed points are: (y i = 0, x i = 0) (44) (y i = 0, x = 0, x 1 = 0, b 2 = 1) (45) (y i = 0, x = 0, x 2 = 0, b 1 = 1) (46) (y i = 0, x = 0, b 1 = 1, b 2 = 1) (47) becomes satisfied, i.e. in this case, two more solutions exist since x 2 2 = a λ (owing to the just preceding condition mentioned, (46) may not be satisfied at this instant unless a 1 = a 2, in which case as such another case follows), further as is increased, one would witness condition (49) also becoming satisfied, thus total fixed points for the given dynamical system increases to five, i.e. in this case too, two more solutions start to exist since x 2 1 = a λ. Further, solution to the condition (47) exists only when a 1 = a 2 and real solutions exist i.e. 0 Here, yet another kind of bifurcation is observed if, whereby we get infinite solutions or fixed points on a circle in the plane (x 1, x 2 ) since x 2 1 +x 2 2 = a 2 1 +λ, i.e. upon change of parameters a 1 and a 2 such that they become equal, infinite fixed points suddenly exist, up from previously five fixed points. To note is that these fixed points in the body fixed reference frame are actually synchronous periodic orbits in the inertial frame of reference. Also, hereby an algorithm was constructed using MATLAB, which, given the defining characteristics of the considered system, namely, ω, a 1, a 2 and a, outputs the corresponding fixed points and points of bifurcation (a 12 + u) 1 (limiti (50) where i = 1, 2, The condition (58) provides the trivial solution of the dynamical system. It is physically not possible since it lies inside the ellipsoid. The solution for the condition (45) exists only when 0 (a 22 + u) 1 (48) Similarly, the solution for the condition (46) exists only when 0 (a 12 + u) 1 (49) Also, since 0 (a 2 2 +u) 0 (a 2 1 +u), thus as we change the parameters defining this system, namely, a 1, a 2 and a, Bifurcations are observed, since upon changing these parameters, say for instance, as it s increased in magnitude, then initially, from one fixed point, the total number of fixed points increases to three as condition (48) FIG. 2: Bifurcation analysis 2.6 LINEARIZATION OF THE SYSTEM Under the linear analysis of the system, an algorithm has been devised, which computes the Jacobian corresponding to the dynamical system s equations, a 6X6 Matrix, which then is used by the program to compute Eigenvalue and Eigenvectors, thereby characterizing the local behaviour of the system about the fixed points.

5 5 Owing to the complexity of the system, not much information can be derived of the Eigenvalues and the Eigenvectors except the system s nature about these directions. An example is presented below: Example: Given, ρ = 1, a 1 = 1, a 2 = 0.8, a = 0.6, and ω = F P 1 = (11.99, 0, 0), Eigenvalues = -0.77, 1.08i, -1.08i, 0.77, 0.651i, i F P 2 = ( 11.99, 0, 0), Eigenvalues = -0.77, 1.08i, -1.08i, 0.77, 0.651i, i F P = (0, , 0), Eigenvalues = i, i, i, i, 0.408i, i F P 4 = (0, , 0), Eigenvalues = i, i, i, i, 0.408i, i F P 5 = (0, 0, 0): (Existence physically not plausible) y = 0 The expression for hamiltonian is: H = 0.5( x x x 2 + y y y 2) + 2 λ 1 Σ x 2 i i=1 a 2 i +u The value of the same, as per the chosen initial conditions, comes out to be H = We eliminate x using this constant of motion. Then, we take the hypersurface y = 0 as our poincare surface. Finally, we get the following poincare map discussed here in 2 D plots( Fig to 8). 2.7 POINCARE MAP For a uniformly rotating body, the hamiltonian is conserved which.imposes a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint (which would require extra constraint forces). Thus, in effect, conservation of the hamiltonian does not provide any extra information if the equations of motion are already known. But, they are useful because they allow properties of the motion to be derived without solving the equations of motion. So, using the above property, a 6-dimensional system can be reced to a 5-dimensional one. Now, a 4-D poincare surface is defined and it s attendant 4 4 monodromy matrix can be computed. In the vicinity of a periodic orbit, we get a pair of unity eigenvalues of the 6x6 state transition matrix. In regards to our problem, we were able to develop an alogorithm for the formulation of the map on a 4-D poincare surface. Since, a 4 D plot can not be visualised, we plot 6 2-D plots which essentialy contain the same information. (see fig.) We consider the following initial conditions, x 1 = 5 x 2 = 1.25 x = 0 y 1 = 5 y 2 = 7.5 FIG. : x 1 x 2 FIG. 4: x 1 y STABILITY BEHAVIOUR ABOUT THE FIXED POINTS In this section we shall discuss in regard to the stability about the equilibrium point. For it required to characterize the stability characteristic

6 6 FIG. 5: x 1 y 2 FIG. 8: y 1 y 2 Also let, {x i0, y i0 } be the particular solutions of this system then if we have system with x i = x i0 + X i, y i = y i0 + Y i, z i = Z i (5) Then if we evaluate the system to this first order of Taylor series, then X 1 = Y 1 (54) FIG. 6: x 2 y 1 X 2 = Y 2 (55) Y 1 = 2Y 2 + U x1x1 0 x 1 + U x1x 2 0 x 2 (56) Y 2 = 2Y 1 + U x1x2 0 x 2 + U x2x 2 0 x 2 (57) Kindly note that we have not considered X,Y. This is because U zz 0 < 0, thereby out of plane oscillation about all planar points(hyperplane) are stable. Proceeding with the above. If we sought eigenvalues for the system the, we obtain FIG. 7: x 2 y 2 of the phase space surrounding. The stability can be inferred from evaluation of the solutions to the variational equations expanded about this point; which basically the linear stability analysis. Let, where, U = 1 2 (x2 1 + x 2 2) V (51) V = 2 λ φ (u (52) σ 4 +σ 2 (4 U x1x 1 U x2x 2 )+U x1x 1 U x2x 2 = 0 (58) Further note that the following are the second derivatives of the modified potential (U), U x1x 1 = 1 λ ( 1 2x 2 1 (a 2 1 +λ)2 (λ)( ) x 2 j j=1 (a 2 ) j +λ)2 U y1y 1 = 1 λ ( 1 2x 2 2 (a 2 2 +λ)2 δ(λ)( ) x 2 j j=1 (a 2 ) j +λ)2 U x1y 1 = 2x 1 x 2 (a 2 2 +λ)(a2 1 +λ)δ(λ)( x 2 j j=1 (a 2 ) j +λ)2 (a 2 1 +u) (u) + (a 2 2 +u) (u) +

7 7 Now we define certain Stability Parameters {α, β, θ} on whose determination one can effectively determine the stability characteristics about the fixed points: α = 4 U x1x 1 U x2x 2 (59) β = (4 U x1x 1 U x2x 2 ) 2 4U x1x 1 U x2x 2 (60) θ = U x1x 1 U x2x 2 (61) Through further analysis of the roots, one comes to the following categorization of the possible results/stability behaviour about the fixed points upon the four manifolds: Type 1 Type 2 α, β < 0, θ > 0 Eigen Values: Four Complex roots (±p ± iq) Behaviour on Manifolds: Two Stable Spirals, Two Unstable Spirals α, β > 0, θ < 0 Eigen Values: Two Imaginary, One Positive, One Negative Behaviour on Manifolds: One Stable, One Unstable, Neutrally stable oscillations about two whereby U x1x 2 = 0. Also b 1 < b 2 (x 1, y 1 ). Now we shall examine the sub-cases: Case A : If F.P is (±x 1, 0, 0) Then, U x1x 1 > 0, U x2x 2 < 0 thus, θ < 0, β > 0, α is uncertain. From the above we can say that there exists one stable, one unstable and two neutrally stable manifolds(type 2). The general behaviour is of hyperbolic instability. But, it s also possible to find periodic orbits close to the points. Case B : If F.P is (0, ±x 2, 0) Then, U x1x 1 > 0, U x2x 2 > 0 thus, θ > 0 and α, β is uncertain β, α depend on system parameters. Let s say α, β > 0,then the system corresponds to type4, that is, there exists neutrally stable oscillations about all four manifolds. Now, when α > 0, β < 0 (upon changing the parameters), then the system is of type 1. As a result, we have two stable spirals and two unstable spirals along the four manifolds. Further, when parameters change in the same direction, we get α < 0, β < 0. Surprisingly, regardless of this change the qualitative picture near the fixed points remain the same. Finally, when α < 0, β > 0, the stability behaviour is of type 2. Thus, we get one stable manifold, one unstable manifold and neutrally stable oscillations about the other two manifolds. Type Type 4 α < 0, β > 0, θ > 0 Eigen Values: Two Positive, Two Negative Behaviour on Manifolds: Two Stable, Two Unstable α, β < 0, θ > 0 Eigen Values: Four Imaginary Behaviour on Manifolds: Neutrally stable oscillations about all Now following up on our investigation, by examining into the particular case whereby there exist four fixed points of the form (±x 1, 0, 0), (0, ±x 2, 0) 2.9 Determination of Lyapunov Exponent We have developed an algorithm for the numerical determination of the Maximal Lyapunov Exponent, for our Six-Dimensional Dynamical System, whereby we utilise variable time intervals, i.e. the reference orbit and the test orbit evolution is examined, upon the basis distance between them, which when exceeds a threshold limit say, D, the test orbit is renormalized and then the system is re-simulated. Though there have been issues regarding the efficiency of these simulations for owing to the complexities of the dynamical system itself, along with the added intricacies e to constant evaluation of the system ring it s evolution.

8 dynamical system, End Sem Report, Mid sem Presentation 8 FIG. 9: Neutrally stable oscillations near fixed point(type 4). ACKNOWLEDGEMENTS We would like to thank Dr. Sagar Chakraborty and Dr. Ishan Sharma for their invaluable support and guidance throughout the semester regarding the project. Further, we would like to state each of the group member s contributions in the project as per the request of our guide. Pragnya Jha- Mid Sem report, Mid- Sem presentation, End Sem report, End -Sem presentation, Computer program for plotting poincare map: 4-Dimensional (using conservation of hamiltonian) and 5-Dimensional, Studied using the eigenvalues of the 6x6 state transition matrix to predict the existence of a periodic orbit for any general case, Theoretical investigation into interpretation of poincare maps (4- Dimensional and 5-Dimensional System) Chitrasen Singh- Mid Sem Report, End Sem Report, End Sem Presentation, Computer program for plotting poincare map: 5- Dimensional, Theoretical investigation into evaluation of poincare maps (4-Dimensional and 5-Dimensional System) Akash Gupta: Fixed Point and Bifurcation Analysis, Linear Analysis of the Dynamical System, Computer Program for determination and plotting of trajectories numerically, using Ivory s Approach of gravitational potential for ellipsoid (For both the cases of non-rotating, and rotating ellipsoid body), Program for determination of fixed points of the system, and respective eigen information relating to it, and also other stability characterization regarding orbits close to it(stability parameters), given the system parameters, Program for determination of Maximal Lyapunov exponent numerically for the system, Program for numerical determination of 5-Dimensional Poincare Map, System Non-Dimensionalization of the 4. REFERENCES 1.Dirk Brouwer,G Clemence -Methods of celestial mechanics, D J Sheeres- Satellite dynamics about asteroids: Computing poincare maps for the general case.d J Sheeres- Satellite dynamics about tri-axial ellipsoids, H. Goldstein- Classical Mechanics 5.S Chandrashekhar - Ellipsoidal figures of equilibrium, Steven strogatz- Non linear dynamics and chaos 7.Argyris, Faust, Haas- An Exploraton of chaos 8.Stephen Lynch- Dynamical systems with applications using mathematica 9. Mark A Murrison- Notes on lyapunov exponent,1995