Analytical & semi-analytical propagation of space orbits: The role of polar-nodal variables

Transcription

1 Analytical & semi-analytical propagation of space orbits: The role of polar-nodal variables Martin Lara GRUCACI - Scientific Computation Group University of La Rioja, Spain Astronet II - Final meeting Tossa de Mar, June 15 19, 2015 IEEC/Marie-Curie Research Training Network on Astrodynamics support

2 Background Propagation of space orbits: analytical & S/A methods last 2 decades: eclipsed by numerical in practical apps. still may play a role in the propagation of space orbits Space debris mitigation guidelines: EoL disposal strategies guarantee some safe condition for hundreds of years S/A + global optimizers: suitable design strategy Armellin et al. Adv. Space Research 2015 Onboard orbit propagation: micro & nano satellite missions reduced power consumption is a critical requirement Efficiency in evaluating analytical expressions depends on variables used in the theory formulation Delaunay (action-angle of the Kepler problem), non-singular (Poincaré), polar-nodal (Whittaker/Hill) 1

3 Outline The two-body problem Newtonian formulation: vectorial reduction (G,e,t 0 ) Lagrangian, Hamiltonian formulation symmetries: Polar-nodal variables complete reduction by H-J: Delaunay variables Main problem of AST intermediaries Perturbed Kepler problems Recall perturbation methods Brouwer solution Normal forms, averaging Conclusions 2

4 The Kepler Problem Two bodies: Newton gravitational theory + 3rd law m 1 d 2 x 1 dt 2 = Gm 1 m 2 x 2 x 1 3(x 2 x 1 ) m 2 d 2 x 2 dt 2 = Gm 1m 2 x 2 x 1 3(x 2 x 1 ) 2nd order differential system, 6-DOF problem Add accelerations + quadratures: m 1 x 1 + m 2 x 2 = A t + B A, B, constant vectors, functions of the iicc preservation of the linear momentum of c.o.m. Relative motion: d2 x dt 2 = µ x { r = x = x2 x 1 r 3 µ = G(m 1 + m 2 ) singularity at the origin (collision orbits) 3-DOF, integrability needs 3 more independent integrals 3-1

5 By simple vector algebra (Danby s 1962 book, for instance) G = x (dx/dt) = const. (orbital plane) µe = (dx/dt) G µx/r = const. (shape, orientation) conic solution r = (G G/µ)/(1 + e cos f), f (e, x) Superintegrability: 5 independent integrals (G e = 0) corresponding to orbital elements: a, e, I, Ω, ω additive integration constant: time of periapsis passage Energy integral: E = T + V (kinetic plus potential energy) E = 1 2 dx dt dx dt µ r = µ2 (1 e e) 2G G bounded motion for E < 0 ( e < 1) 3-2

6 Lagrangian formulation: L = T V (scalar, state function) may use generalized coordinates L L(q, q), q = dq/dt eqs. of motion d ( ) L L = 0 (2nd order diff. system) dt q q Kepler problem: L = 1 dx 2 dt dx dt +µ r Hamiltonian formulation: H = Q q L, where Q = L/ q flow Hamilton equations: dq dt = H Q, dq dt = H q 1st order differential system Kepler problem: Cartesian coordinates Q X = dx/dt H = 1 2 X X µ dx r dt = X, dx dt = µ x r 3 the same as Newtonian equations, no advantage Good point: invariant under canonical transformations 3-3

7 Polar-nodal variables Polar-nodal variables (r, θ, ν, R, Θ, N), N = Θ cos I x and dx/dt define the (instantaneous) orbital plane ν not defined for I = 0 4-1

8 Match orbital (ˆx, Ĝ ˆx, Ĝ) and inertial frames: rotations x y z X Y Z = R 3 ( ν) R 1 ( I) R 3 ( θ) R 1, R 3, usual rotation matrices R 1 (φ) = cos φ sin φ 0 sin φ cos φ R 3 (φ) = r ṙ = R 0 r θ = Θ/r 0 0 cos φ sin φ 0 sin φ cos φ Canonical transformation (Whittaker s book 1904) H = 1 2 ( R 2 + Θ2 r 2 ) µ r N, Θ, ν constant decouples the differential system: 1-DOF in (r, R) dr dt = H R = R, dr dt = H r = Θ2 r 3 µ r 2 quadrature: dθ dt = H Θ = Θ r 2 θ = θ 0 + Θ 1 r(t) dt 4-2

9 Solution by H-J: Complete reduction Find canonical transform T : (x, X) (y, Y ) such that T : H(x, X) H(x(y, Y ), X(y, Y )) = Φ(, Y ) trivial integration: Y = Y 0, y = y 0 + ( Φ/ Y ) t H-J method: Generating function in mixed var. S(x, Y ) transformation: X = S/ x, y = S/ Y S solved from Jacobi eq.: H (x, S/ x) = Φ(, Y ) PDE; apparently this way complicates the problem Kepler problem: T : (r, θ, ν, R, Θ, N) (l, g, h, L, G, H) choose S S(r, θ, ν, L, G, H) = ν H + θ G + W(r, L, G) N = S ν = H, l = S L = W L, Θ = S θ = G, g = S G = θ + W G, R = S r = W r, h = S H = ν 5-1

10 Hamilton-Jacobi eq. 1 2 [ ( W r ) ] 2 + G2 r 2 µ r = Φ(L, G, ) W = 2Φ + 2µ/r G 2 /r 2 dr solved by quadrature Transformation in mixed variables (h = ν, N = H, Θ = G) R = W r = 2Φ + 2µ/r G 2 /r 2 l = W L = Φ L 1 R(r) dr g = θ + W G = θ + G 1 R(1/r) d ( ) 1 r + Φ G 1 R(r) dr Use anomalies: (1/r) = (1 + e cos f)/p, r = a(1 e cos u) p = G 2 /µ, a = µ/(2φ), e = 1 p/a l = Φ µ L ( 2Φ) 3/2(u e sin u), g = θ + f + Φ/ G Φ/ L l family of transformations parametrized by Φ 5-2

11 Standard Hamiltonian Φ Φ(L, G) = µ2 2G 2(1 e2 ) e e(l, G), nondimensional e e(η), η = G/L Φ L = µ2 ( G 2e e η ) Φ, η L G = µ2 G 3(1 e2 ) + µ2 G 2e e η family of transformations parametrized by e Action-angle variables: angle condition dl = 2π eη 2 de leads to: (1 e 2 ) 3/2 dη = 1 e2 = 1 η 2 = 1 G2 L 2 ( ) η Φ = µ2 g = θ f, l = u e sin u (Kepler equation), 2L2 Kepler probl. completely reduced Φ = Φ(L) degenerate! trivial solution g, h, L, G, H const., l = l 0 + (µ 2 /L 3 ) t l = M, g = ω, h = Ω, L = µa, G = Lη, H = G cos I Delaunay var.: singular for circular and equatorial orbits 5-3 G

12 Perturbed Kepler problems: AST Real problems: non-central potential V = µ r µ r α i i i 2 r i j=0 (C i,j cos jϑ + S i,j sin jϑ)p i,j (sin ϕ) (r, ϑ, ϕ) spherical coord., α equatorial radius P j,k associated Legendre func., C i,j, S i,j, harmonic coeff. Earth case J 2 = C 2,0 = O(10 3 ), C n,m, S n,m = O(J 2 ) 2 Main problem of AST ( H = 1 R 2 + Θ2 2 r 2 non-integrable ) [ ) 2 ( ) ] 3 µ ( α 1 J 2 r r 2 sin2 I sin 2 θ 1 2 Danby 1968, Irigoyen & Simó 93, Celleti & Negrini 95 still, a lot of info can be obtained 6-1

13 Axial symmetry: ν cyclic N constant decouple de motion of the orbital plane Invariant manifolds: polar and equatorial orbits Particular solutions: equatorial and rectilinear orbits Poincaré maps, periodic orbits,... Secular dynamics: constant term of H as a Fourier series slowly precessing ellipse regression of the perigee critical inclination: frozen-perigee orbits Short time spans: neglect second order effects intermediary solutions 6-2

14 Truncations of the main problem Expand the disturbing function: H = K + P Q + P R + P Z H = 1 2 ( µ 2r µ 2r µ 2r R 2 + Θ2 r 2 α 2 r 2 J 2 α 2 r 2 J 2 α 2 r 2 J 2 ) µ r ( 3 ) 2 sin2 I Kepler Equatorial main Cid intermediary 3 2 sin2 I cos 2θ Full problem Radial Hamiltonians: 1-DOF integrable term 1/r 3 : requires elliptic functions 6-3

15 Good intermediaries: secular effects O(J 2 ) Keplerian: K = 1 2 ( R 2 + Θ2 r 2 ) µ r, 1 2π Equatorial main problem E = K + P Q E = 1 2 ( R 2 + Θ2 r 2 ) µ r µ 2r α 2 r 2 J 2, 1 2π 2π 0 2π Cid & Lahulla intermediary C = K + P Q + P R C = 1 ( ) R 2 + Θ2 2 r 2 µ r µ α 2 ( 2r r 2 J sin2 I 2π 2π 1 1 (H C)dM = P Z dm= 0 2π 0 2π 0 paradigm of common intermediaries solution depends on elliptic functions 0 (H K)dM 0 (H E)dM 0 Radial, Θ = const: miss long-period effects in e and I ) 6-4

16 Zonal intermediaries Non-separability: related to 1/r 3 = (1 + e cos f)/(r 2 p) H = H 0 + H 1 (s sin I, c cos I, p = Θ 2 /µ) H 0 = 1 ) (R 2 + Θ2 µ 2 r 2 r µ ( α 2 2p r J s2 + 3 ) 2 s2 cos 2θ H 1 = µ ( ) ( r α 2 2r p 1 r J s2 + 3 ) 2 s2 cos 2θ H 0 2-DOF but integrable (Aksnes 1965, elliptic funct.) include long-period effects in e and I Good intermediary for low e: H 1 M = O(e 2 J 2 ) Other intermediaries Sterne 1957, Garfinkel 1958,..., Oberti 2005 Vinti 1959, Aksenov et al (2nd order effects of J 2 ) 6-5

17 Natural Intermediaries Common intermediaries: same vars. as the main problem Cid & Lahulla (1969): natural intermediary in new variables results from an infinitesimal contact transformation x = x + ɛa( x, X), X = X + ɛb( x, X) retains all secular and periodic effects up to O(J 2 ) Deprit (1981): common intermediaries can be naturalized DRI: H 0 = 1 ( ) R 2 + Θ2 2 r 2 µ r 1 Θ 2 α 2 2 r 2 p 2 J 2 (1 32 ) s2 integrable in trigonometric functions quasi-keplerian system, variable angular momentum useful for onboard orbit propagation (Gurfil & Lara 2014) If second order effects are required: perturbation theory 6-6

18 Perturbation theory Effects of non-centralities small, slightly distorted ellipses VOP method: osculating elem. x (a, e, i, Ω, ω), y = M ẋ = ɛ F (y, x, ɛ), ẏ = n(y, x) + ɛ Ψ(y, x, ɛ) Solution of VOP approached by Picard iterations recall that V = V (f), f f(m, e); Kaula expansions single iteration: main problem in closed form Hautesserres 2013 (elliptic integrals) Picard iterations may require expanding trig. functions mixed secular-periodic terms: M cos(k 1 M +... ) limit the solution to short time intervals Different perturbation methods avoid mixed terms issue books by: Giacaglia, Nayfeh, Ferraz-Mello,

19 H-J perspective: reduction by transformation of variables find (x, y) ( x, ỹ, ɛ) close to identity; ɛ small parameter x = x + j 1 ɛj j F j( x, ỹ), y = ỹ + j 1 ɛj j Ψ j( x, ỹ) such that ( x, ỹ) simpler than (ẋ, ẏ) if truncated to O(ɛ m ) Usual simplifications: reduce the number of DOF; v.g. x = 0<j<m ɛ j F j ( x) + ɛm j 0 ɛ j F j+m ( x, ỹ), ỹ = ñ( x) + 0<j<m ɛ j Ψ j ( x) + ɛm j 0 ɛ j Ψ j+m ( x, ỹ) After truncation, flow in original and tildes are different useful sol. constrained to some regions of phase space example: removing M keeps out tesseral resonances Hamiltonian perturbations: H = H 0 + ɛh 1, H 0 integrable select the form of the transformed Hamiltonian compute corresponding transformation equations 7-2

20 e Ω deg days Moon perturbations (GNSS-type orbit a = km) time history reveals a trend in evolution plus noise Averaging: filters the noise (elements const. on one period) simplifies solution of VOP by removing M days orbit evolution in mean variables evolution eqs. result from a transformation of variables Usual criterium for selecting the new H: Averaging 7-3

21 Brouwer solution to AST (AJ 1959) Model: disturbing effects of J 2, J 3, J 4 and J 5 axial symmetry, H integral, 2-DOF problem Analytical solution in Delaunay variables separates secular, long-, and short-period effects double averaging (based on von Zeipel s method) inclination resonances excluded because of g averaging trouble for low e: short-period corrections l, g Closed form solution using relation: a 2 η dm = r 2 df expansions of f = f(l, e) low convergence for high e Secular terms up to O(J 2 2 ), periodic terms up to O(J 2) Here, results for H = µ 2a + µ r m=2,3 α m r m J mp m (s sin θ) only 8-1

22 H l = µ 2a µ 1 2a 4 J 2 p η(4 2 6s2 ) + µ 3 2a 2 J 3 p (4 3 5s2 )s η e sin g [ ]. µ 3 2a 64 J 2 2 α 4 p η m 4 0,0 + m 0,1 η + m 0,2 η 2 + m 2,2 e 2 cos 2g α 2 m i,j m i,j (s) inclination polynomials s = 1 c 2, c = H/G, e = 1 η 2, η = G/L, a = L 2 /µ Hamiltonian in new (mean) variables; H and L constant Brouwer s 1st order corrections (Delaunay variables) = j k P (e)q(i) trig(jf + kg), trig sin, cos need to evaluate 20 different trigonometric terms! Brouwer s clever arrangement using r, a polar variable... still, evaluate 8 circular functions l, g: e in denominators, deteriorate for low e orbits α 3 8-2

23 H l,g = µ 2a µ 1 2a 2 J 2 p η(2 2 3s2 ) µ 2a J 2 2 α 2 ] α [m 4 p η 4 0,0 + m 0,1 η m 0,2 η 2 completely reduced Ham. in new (action-angle) variables H, L, G constant, secular terms: trivial integration Brouwer s 1st order corrections ( δg = α2 p J m 1 5c 2 0 (s) + m ) 2(s) 1 5c 2e2 sin 2g α p δl =..., δh =..., δg =..., δh = 0, δl = 0 J 3 1 ( s J 2 2 e e ) s c2 cos g trouble for low e (δl, δg), and also for low i orbits (δg, δh) Brouwer s theory does not apply to the critical inclination trouble for orbits very close to it: 1 5c 2 = O(J 2 ) essential singularity, contrary to virtual of e, s = 0 Limited in precision (desirable 2nd order corrections) 8-3

24 Non-singular variables: Lyddane s approach Poincaré canonical elements x 1 = L x 2 = 2 L G cos(g + h) x 3 = 2 G H cos h y 1 = l + g + h y 2 = 2 L G sin(g + h) y 3 = 2 G H sin h No need of recomputing the theory: use Brouwer s results (g + δg) + (h + δh) = g + h + (δg + δh) cancels terms in 1/s 1 c cos(h + δh) = 1 c cos h ( 1 c δh) sin h + O(J 2 2 ) avoid trouble with low I (l + δl) + (g + δg) + (h + δh) = l + g + h + δl + δg 1 + δg 2 + δh cancels terms in 1/s and in 1/e No singularities, but more involved series 9

25 Simplification: Polar-nodal variables Further simplifications to Brouwer s formulas reformulate Brouwer s corrections in polar variables l = l(r, θ, R, Θ, N), g =..., h =... trig. functions reduced to one half! cos 2θ, sin 2θ, r = p/(1 + e cos f), R = (G/p)e sin f still remain singularities for low e and I Izsak (1963) approach: removes singularity for zero e orbits theory correctly computed in action-angle variables, but corrections to polar-nodal variables instead of Delaunay r = r(r, θ, R, Θ, N), θ =..., ν =..., R =... dramatically reduces size of the short-period corrections Extended by Aksnes (1978) to the long-period corrections 10-1

26 [ (2 r = γp ) ( 3s κ 1 + η + 2η ) ] s 2 cos 2θ 1 + κ { [ ] θ = γ 3 ( 4 5s 2) φ s2 + ( 4 6s 2) κ sin 2θ [ ( +2σ 5 6s ) 2 + κ 2 s2 1 + η + ( ]} 1 2s 2) cos 2θ [ ] ν = γc 6(σ + φ) (3 + 4κ) sin 2θ + 2σ cos 2θ R = γ Θ { p (1 + κ)2 2s 2 sin 2θ ( [ ] } 2 3s 2) η + η σ (1 + κ) ] 2 Θ = γθs [( κ) cos 2θ + 2σ sin 2θ N = 0 φ eq. of the center α 2 γ = 1 4 C 2,0 p, σ = pr 2 Θ, κ = p r 1, p = Θ2 µ, 10-2

27 Third-Body perturbations Mass-point approx.: V 3 = µ 3 r 3 Expand in Legendre poly: V 3 = β n2 3 a3 3 r 3 ( r 3 x r 3 x r 3 j 2 r3 2 ( r r 3 ) ) j P j (cos ψ) ψ (x, r 3 ), β = m 3 /(m 3 + m) 3rd-body reduced mass Moon perturbations: slow convergence of the expansion Integral may require up to P 6, SIMOLX up to P 8... very long correction series! Perturbation theory by Lie transforms (Hori 67, Deprit 69) theory properly computed in Delaunay variables transformation to mean variables from generating func. closed form: dl = (1 e sin u)du 11-1

28 Example: P 2 only Delaunay var.: generating function (trouble for low e, i) W 1 = L n2 3 n 2 β a3 3 r j=0 l= 2m= 3 m 0 C j,m (e) Q j,l (I) [ T l (ˆr 3 ) cos(2jω + lω + mu) S l (ˆr 3 ) sin(2jω + lω + mu) u eccentric anomaly, ˆr 3 moon direction (ephemeris) Fourier series with 81 different trig. terms Polar-nodal var.: (no trouble for low e, trouble for low i) V 1 = Θ n2 3 n 2 β a3 3 r 3 3 r 3 p j=0 l= 2 Q j,l [ ] (κ j T l σ j S l ) cos(2jθ + lν) (κ j S l + σ j T l ) sin(2jθ + lν) κ j κ j (r, R, Θ), σ j σ j (r, R, Θ), e-type coefficients polar-nodal: only 14 trigonometric terms! Time in evaluating the corrections reduces to < 1 3!! 11-2 ]

29 The case of low inclinations Polar-nodal: singularities remain for low inclinations Alternative: treat separately that case Aksnes (1972): ephemeris in spherical coord. (r, ϕ, ϑ) sin ϕ = s sin θ, sin(ϑ ν) = c sin θ, cos(ϑ ν) = cos θ s = O(J 2 ) compute ϑ = θ + ν, ϕ = (s sin θ) Alternative: reformulate corrections in nonsingular variables non-canonical elements based on polar-nodal variables ψ = θ + ν, ξ = s sin θ, χ = s cos θ, r, R, Θ evaluation: similar performances as polar-nodal variables may be used only for the case of low I dramatic simplifications of the corrections 11-3

30 Conclusions S/A theories still useful for current engineering problems properly constructed in Delaunay variables virtual singularities for circular and equatorial orbits Reformulate the theory in Poincaré nonsingular variables make the theory as much general as possible, but... evaluation of long Fourier series degrade performance Polar-nodal var.: compact form, straightforward evaluation almost equatorial orbits must be treated separately lack of universality fixed with a conditional statement compensated by the simplif. of the theory for low I New nonsingular variables based on the polar-nodal set similar evaluation performances as polar-nodal 12