Previous Lecture. The Von Zeipel Method. Application 1: The Brouwer model. Application 2: The Cid-Lahulla model. Simplified Brouwer transformation.

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2 2 / 36 Previous Lecture The Von Zeipel Method. Application 1: The Brouwer model. Application 2: The Cid-Lahulla model. Simplified Brouwer transformation.

3 Review of Analytic Models 3 / 36

4 4 / 36 Review: Brouwer Model The J 2 Hamiltonian: H J2 = µ2 2L 2 µj 2req 2 r 3 [B 20 + B 22 cos (2θ)] The Brouwer 1st order Hamiltonian: H BRO = µ2 2L µ 4 J 2 req 2 4 L 3 G 3 ) (1 3 H2 The generating function (polar-nodal variables): W = k [ 2 (2 3s 2 ) ( f l + σ) s 2 σ cos 2θ κ Θp 2 G 2 ] sin 2θ

5 5 / 36 Review: Brouwer Model The Contact transformation: r = r W R θ = θ W Θ R = R + W r Θ = Θ + W θ ν = ν W N N = N + W ν Useful partial derivatives: ( κ r 1 + η + η ) ( 1 + κ ( f l) (r, R, Θ) = σ 1 κ R 1 + η + 2η ) 1 + κ κ Θ 1 + η

6 6 / 36 Analytic Propagation Algorithm 1 Input: osculating initial conditions (a 0, e 0, i 0, ω 0, Ω 0, M 0 ).

7 6 / 36 Analytic Propagation Algorithm 1 Input: osculating initial conditions (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 2 Transform to osculating polar-nodal: (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ).

8 Analytic Propagation Algorithm 1 Input: osculating initial conditions (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 2 Transform to osculating polar-nodal: (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ). 3 Transform to averaged initial polar-nodal (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ) through the inverse CT. 6 / 36

9 Analytic Propagation Algorithm 1 Input: osculating initial conditions (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 2 Transform to osculating polar-nodal: (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ). 3 Transform to averaged initial polar-nodal (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ) through the inverse CT. 4 Transform to averaged initial Delaunay / Classical OEs: (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 6 / 36

10 Analytic Propagation Algorithm 1 Input: osculating initial conditions (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 2 Transform to osculating polar-nodal: (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ). 3 Transform to averaged initial polar-nodal (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ) through the inverse CT. 4 Transform to averaged initial Delaunay / Classical OEs: (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 5 Propagate the averaged (explicit expressions) to get (a (t), e (t), i (t), ω (t), Ω (t), M (t)). 6 / 36

11 Analytic Propagation Algorithm 1 Input: osculating initial conditions (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 2 Transform to osculating polar-nodal: (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ). 3 Transform to averaged initial polar-nodal (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ) through the inverse CT. 4 Transform to averaged initial Delaunay / Classical OEs: (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 5 Propagate the averaged (explicit expressions) to get (a (t), e (t), i (t), ω (t), Ω (t), M (t)). 6 Transform to averaged polar-nodal (r (t), θ (t), ν (t), R (t), Θ (t), N (t)) 6 / 36

12 Analytic Propagation Algorithm 1 Input: osculating initial conditions (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 2 Transform to osculating polar-nodal: (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ). 3 Transform to averaged initial polar-nodal (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ) through the inverse CT. 4 Transform to averaged initial Delaunay / Classical OEs: (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 5 Propagate the averaged (explicit expressions) to get (a (t), e (t), i (t), ω (t), Ω (t), M (t)). 6 Transform to averaged polar-nodal (r (t), θ (t), ν (t), R (t), Θ (t), N (t)) 7 Transform to osculating polar-nodal (r (t), θ (t), ν (t), R (t), Θ (t), N (t)) through the direct CT. 6 / 36

13 Analytic Propagation Algorithm 1 Input: osculating initial conditions (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 2 Transform to osculating polar-nodal: (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ). 3 Transform to averaged initial polar-nodal (r 0, θ 0, ν 0, R 0, Θ 0, N 0 ) through the inverse CT. 4 Transform to averaged initial Delaunay / Classical OEs: (a 0, e 0, i 0, ω 0, Ω 0, M 0 ). 5 Propagate the averaged (explicit expressions) to get (a (t), e (t), i (t), ω (t), Ω (t), M (t)). 6 Transform to averaged polar-nodal (r (t), θ (t), ν (t), R (t), Θ (t), N (t)) 7 Transform to osculating polar-nodal (r (t), θ (t), ν (t), R (t), Θ (t), N (t)) through the direct CT. 8 Transform to any coordinates you like. Propagation is finished. 6 / 36

14 7 / 36 Review: Cid-Lahulla Model The J 2 Hamiltonian: H J2 = µ2 2L 2 µj 2req 2 r 3 [B 20 + B 22 cos (2θ)] The Cid-Lahulla 1st order Hamiltonian: H CID = 1 ) (R 2 + Θ2 2 r 2 µ r k 2 ( 2 3s 2 ) r 3 The generating function (polar-nodal variables): W = (H CID H J2 ) dt

15 The Radial Intermediary of Deprit The J 2 Hamiltonian: H J2 = µ2 2L 2 µj 2req 2 r 3 [B 20 + B 22 cos (2θ)] The Deprit 1st order Hamiltonian: H DEP = 1 ) (R 2 + Θ2 2 r 2 µ r k 2 ( 2 3s 2 ) pr 2 The generating function (polar-nodal variables): W = (H DEP H J2 ) dt 8 / 36

16 Analytic Orbit Propagation Paradigm 9 / 36

17 10 / 36 Analytic Orbit Propagation Paradigm INPUT Initial conditions (6 constants) T 1 INPUT Initial conditions (6 constants) Algebraic operations or Fast numerical integration OUTPUT Real state (6 vectors of length N) T OUTPUT Fictitious state (6 vectors of length N) INPUT Time t (user-defined vector, length( t) = N)

18 11 / 36 Analytic Orbit Propagation Paradigm INPUT Initial conditions (6 constants) T 1 INPUT Initial conditions (6 constants) ESSENTIAL! Algebraic operations or Fast numerical integration OUTPUT Real state (6 vectors of length ) N T OUTPUT Fictitious state (6 vectors of length N) INPUT Time t (user-defined vector, length( t) = N)

19 Final Comments 12 / 36

20 13 / 36 Classic Solution: Brouwer Advantages 1st order: time-explicit linear model. Improved contact transformation (Lyddane, Poincaré). Action-angle variables (long-term predictions at higher orders).

21 Classic Solution: Brouwer Advantages 1st order: time-explicit linear model. Improved contact transformation (Lyddane, Poincaré). Action-angle variables (long-term predictions at higher orders). Drawbacks Non-explicit CT (mixed Delaunay / Poincaré): [ ] q T 1 [ ] q p T p [ ] q = p W = W (q, p) [ q p ] + J 2 W (q, p ) 13 / 36

22 14 / 36 Transitional Solution: Cid-Lahulla Advantages 1st order: closed-form solution; short-periodic terms. Easier to go to higher orders.

23 14 / 36 Transitional Solution: Cid-Lahulla Advantages 1st order: closed-form solution; short-periodic terms. Easier to go to higher orders. Drawbacks Non-explicit CT (polar-nodal) (mixed variables). Elliptic integrals involved at first order. Singular at low eccentricities (but fixable...). 1st order drifts faster than Brouwer at higher inclinations. Not in action-angle variables. Original paper published only in Spanish!

24 15 / 36 Modern Solution: Deprit Advantages 1st order: closed-form solution; short-periodic terms. Explicit CT (Lie transforms, polar-nodal). Easier to go to higher orders.

25 15 / 36 Modern Solution: Deprit Advantages 1st order: closed-form solution; short-periodic terms. Explicit CT (Lie transforms, polar-nodal). Easier to go to higher orders. Drawbacks Singular at low eccentricities (but fixable...). 1st order drifts faster than Brouwer at lower inclinations. Not in action-angle variables.

26 16 / 36 Global View on Available Solutions Formulas (easy-to-read-variables) Brouwer: H (1) BRO = J µ 2 4 Cid-Lahulla: H (1) CID = J µ 2 4 Deprit: H (1) DEP = J µ 2 4 ( req ( req ) 2 η 3 ( 1 + 3c 2 ) p p ( req r r ) 2 ( 1 r ) ( 1 + 3c 2 ) ) 2 ( 1 p ) ( 1 + 3c 2 ) H (1) BRO = 1 2π Essential distinctive features: 2π 0 H (0) 1 dl; 1 2π 2π 0 [ H (1) CID,DEP H(0) 1 ] dl = 0

27 Ongoing Research at ULg Raising The Stake: Analytic Approach to J 2 & Atmospheric Drag 17 / 36

28 Physical Models 18 / 36

29 19 / 36 The Main Challenges Two or more non-integrable problems are mixed.

30 19 / 36 The Main Challenges Two or more non-integrable problems are mixed. How to define the best mathematical model?

31 19 / 36 The Main Challenges Two or more non-integrable problems are mixed. How to define the best mathematical model? Which atmospheric model to use?

32 19 / 36 The Main Challenges Two or more non-integrable problems are mixed. How to define the best mathematical model? Which atmospheric model to use? Which tools to add to the canonical formalism?

33 19 / 36 The Main Challenges Two or more non-integrable problems are mixed. How to define the best mathematical model? Which atmospheric model to use? Which tools to add to the canonical formalism? Is it possible to add other NC perturbations?

34 19 / 36 The Main Challenges Two or more non-integrable problems are mixed. How to define the best mathematical model? Which atmospheric model to use? Which tools to add to the canonical formalism? Is it possible to add other NC perturbations? Presence of uncertainties.

35 19 / 36 The Main Challenges Two or more non-integrable problems are mixed. How to define the best mathematical model? Which atmospheric model to use? Which tools to add to the canonical formalism? Is it possible to add other NC perturbations? Presence of uncertainties. Is accurate analytic propagation reachable?

36 20 / 36 Breaking the Ice: J 2 + drag Pick an atmospheric model.

37 20 / 36 Breaking the Ice: J 2 + drag Pick an atmospheric model. Variational equations: 5 constants + 1 varying Keplerian elements.

38 20 / 36 Breaking the Ice: J 2 + drag Pick an atmospheric model. Variational equations: 5 constants + 1 varying Keplerian elements. Average the whole perturbation effect (J 2 + drag).

39 20 / 36 Breaking the Ice: J 2 + drag Pick an atmospheric model. Variational equations: 5 constants + 1 varying Keplerian elements. Average the whole perturbation effect (J 2 + drag). Solve the simplified equations.

40 20 / 36 Breaking the Ice: J 2 + drag Pick an atmospheric model. Variational equations: 5 constants + 1 varying Keplerian elements. Average the whole perturbation effect (J 2 + drag). Solve the simplified equations. Use the classic Brouwer-Lyddane CT (direct + inverse).

41 20 / 36 Breaking the Ice: J 2 + drag Pick an atmospheric model. Variational equations: 5 constants + 1 varying Keplerian elements. Average the whole perturbation effect (J 2 + drag). Solve the simplified equations. Use the classic Brouwer-Lyddane CT (direct + inverse). Expect rapid drift, highly sensitive to the ICs.

42 21 / 36 Numerical Simulations Setup Orbital Element Initial value Units Semimajor axis (a 0 ) r e km Eccentricity (e 0 ) Inclination (i 0 ) 55 deg Argument of perigee (ω 0 ) 25 deg RAAN (Ω 0 ) 0 deg True anomaly ( f 0 ) 20 deg Reference: Numerical integration (GVE, J 2 + drag). Output: Errors in position and velocity. Propagation time: 2 days.

43 22 / 36 Numerical Simulations Constant Density Position and velocity errors Position errors [m] J 2 + Drag J 2 only (Brouwer) Velocity errors [m/s] J 2 + Drag J 2 only (Brouwer) Time [days]

44 23 / 36 Numerical Simulations Constant Density Differences Brouwer / J 2 +drag rj2, [m] rdrag+j vj2, [m/sec] vdrag+j2 5 x Time [days]

45 24 / 36 Numerical Simulations Exponential Density Position and velocity errors Position errors [m] J 2 + Drag J 2 only (Brouwer) Velocity errors [m/s] J 2 + Drag J 2 only (Brouwer) Time [days]

46 25 / 36 Numerical Simulations Exponential Density Differences Brouwer / J 2 +drag rj2, [m] rdrag+j vj2, [m/sec] vdrag+j Time [days]

47 Relative Motion: Initial Conditions 26 / 36

48 350 km Altitude: Relative Distance & Error Rel dist [km] Error [m] Time [days] / 36

49 350 km Altitude: Errors in X, Y, Z X error [m] Y error [m] Z error [m] Time [days] / 36

50 650 km Altitude: Relative Distance & Error Rel dist [km] Error [m] Time [days] 29 / 36

51 650 km Altitude: Errors in X, Y, Z X error [m] Y error [m] Z error [m] Time [days] / 36

52 31 / 36 Applications Accurate analytic orbit propagator, implementable onboard.

53 31 / 36 Applications Accurate analytic orbit propagator, implementable onboard. Relative motion: quick debris impact risk assessment.

54 31 / 36 Applications Accurate analytic orbit propagator, implementable onboard. Relative motion: quick debris impact risk assessment. Fast and robust collision avoidance maneuvers.

55 31 / 36 Applications Accurate analytic orbit propagator, implementable onboard. Relative motion: quick debris impact risk assessment. Fast and robust collision avoidance maneuvers. Station-keeping / formation-keeping maneuvers.

56 31 / 36 Applications Accurate analytic orbit propagator, implementable onboard. Relative motion: quick debris impact risk assessment. Fast and robust collision avoidance maneuvers. Station-keeping / formation-keeping maneuvers. Fast orbit estimation for large populations ( 200,000).

57 Onboard Propagation: What For? Propelantless rendezvous between two QB50 CubeSats. Low altitude: highly perturbed orbits (drag, J2 ). Nanosatellite: limited onboard computational capabilities. 32 / 36

58 33 / 36 Onboard Propagation: What If? GPS outage: need for orbit prediction. Short-term: relaxed accuracy restrictions. Limited onboard resources: simple, efficient propagator. April 2-3, 2014: GLONASS satellites reported illegal or failure status for 11 hours.

59 34 / 36 Space Debris Orbit Prediction Mitigate impact risk by onboard space debris orbit propagation. Paradigm: relative motion of debris wrt the satellite. Avoidance method: change of attitude / propulsion.

60 Further Reading 35 / 36

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