1. Integration of Perturbed Motion
John L. Junkins
03/05/2006
Integration of Perturbed Orbits

2. Integration of Perturbed Orbits
Outline
Integration of Perturbed Motion
INTRODUCTION COWELL AND ENCKE METHODS
VARIATION OF PARAMETERS
GRAVITY MODELING & OBLATENESS PERTURBATIONS
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3. Integration of Perturbed Motion
Three Quasi-Independent Sets of Issues Must be Addressed:
What physical effects will be considered?
Which set of coordinates will be integrated?
What integration method will be used?
Gravitational perturbation due to non spherical earth
Gravitational perturbation due to attraction of non-central bodies
Aerodynamic forces
Thrust
Solar radiation pressure
Relativistic effects
Rectangular coord. in nonrotating ref. Frame (Cowell’s Method)
Departure motion in rectangular coordinates (Encke’s Method)
Variation-of-Parameters; slowly varying elements of two-body motion: - classical elements other elements
Regularized Variables
K.S. transformed oscillators
Burdet transformed oscillators
Canonical Coordinates
Delunay Variables
Numerical (“special”) Methods:
Single Step Methods:
Analytical continuation
Runge-Kutta methods
Multi Step Methods:
Adams-Moulton method
Adams-Bashford method
Gaussian second sum method
Symplectic Integrators
Analytical (“general”) Methods:
Pedestrian asymptotic expan.
Lindstedt-Poincare methods
Methods of averaging
Multiple time scale methods
Transformation methods
Questions: What is the solution needed for? How precise must the solution be? What software is available?
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4. Integration of Perturbed Orbits
Relative Strengths of Forces Acting on a Typical Satellite
(“Junkins with 10 m2 solar panels” at 350 km above earth)
inverse square attraction
dominant oblateness (J2)
in-track drag (B = 0.35)
higher harmonics of gravity field
cross-track aerodynamic force
attraction of the Moon
attraction of the Sun
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5. Integration of Perturbed Orbits
Gravity Modeling Overview
Potential of a “Potato”:
“associated Legendre functions”
Acceleration:
“spherical harmonic gravity coefficients”
Rectangular
Spherical
Problems: (1) “The more you learn, the more it costs!”
(2) ∞ is a painful upper limit
(3) For n > 3, convergence is very slow.
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6. Integration of Perturbed Orbits
Gravity Overview…
During 1975 – 76, J. Junkins et al developed a (“finite element”) gravity
model based upon the starting observation “horse-sense”: +
Dominate terms
“Everything Else”
. . . Use global model
for these . . .
. . . Use global family of
local, piecewise continuous
functions to model these. . .
Thesis: It takes a >1000 term spherical harmonic series to model U
globally, but UREF can be modeled using 2 or 3 terms and ΔU can be locally modeled with ~ 10 terms computational efficiency results This is the genesis of earliest version of the “GLO-MAP piecewise continuous approximation methods” published by JLJ et al during the mid 1970s. => Gravity model for Polaris submarine-launched ICBMs.
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7. Investigation of Finite-Element Representation of the Geopotential
RADIAL DISTRUBANCE ACCELERATION ON THE EARTH’S SURFACE
(contour interval is 5 x 10-5 m/sec2)
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8. Integration of Perturbed Orbits
FINITE ELEMENT MODELING OF THE GRAVITY FIELD: THE BOTTOM LINES
Basic tradeoff is storage versus runtime
Factors of ~ 50 possible increased speed to calculate local acceleration
In one example, a global 23rd degree and order spherical harmonic expansion
has been “replaced” by 1500 finite elements
RMS of acceleration residuals ≈ , 002 m/sec2
Max acceleration error ≈ , 008 m/sec2
Mean acceleration error ≈ , 000, 03 m/sec2
1500 local functions coefficients each ,000 coefficients total
See: Junkins, J.L., “Investigation of Finite Element Representations of the Geopotential”, AIAA, J., Vol. 14, No. 6, June
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9. Integration of Perturbed Orbits
Cowell’s Method.
Simply the name given to straight-forward numerical integration (e.g., ODE45) of the acceleration differential equations of motion … most usually, using the inertial rectangular coordinate versions of the equations of motion.
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10. Encke’s Method: Integrate Departure Motion from an Osculating Reference Orbit
Osculation Condition at t0
Note that:
Also note:
From which it follows that:
The parenthetic term is a small difference of large numbers,
It is profitable to re-arrange it to avoid numerical difficulties...
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11. Integration of Perturbed Orbits
Encke’s Method: Re-arrangement of Departure Motion
Differential Equation to Avoid SDOLN
(small differences of large numbers!)
On the previous chart we developed the departure differential equation:
This equation can be arranged into a more computationally attractive form:
[note, no small differences of large #’s!]
where
The development of the above form is given on the following 3 pages.
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Integration of Perturbed Orbits

12. Integration of Perturbed Orbits
Encke Manipulations ….
The actual motion is governed by
The osculating orbit satisfies
So the departure (“pertubative”) acceleration is
Making use of
I get
Introduce some useful alternatives since
From which
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13. Integration of Perturbed Orbits
Encke Manipulations ….
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14. Integration of Perturbed Orbits
Encke Manipulations ….
So, finally, we get the (exact!) departure motion differential equation
which lies at the heart of Encke’s Method.
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15. Integration of Perturbed Orbits
Rectification of the Reference Orbit in Encke’s Method
Original osculating
reference orbit (kisses
actual motion at time t0)
Original reference orbit
osculates at time t0:
“Rectified” (new) osculating
Reference orbit (kisses the
actual motion at time t1).
Whenever exceeds some preset tolerance,
The position and velocity at time t1 are used to calcualte a
New “rectified” reference two-body orbit. Note that this
Has the effect of re-setting the “initial” departure position
and velocity to zero. Since rectification can be done as
often as we please (as long as we pay the “overhead”!),
the departure motion can be kept as small as we please.
Updated reference orbit
Osculates at time t1
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16. Integration of Perturbed Orbits
Continuous limit of osculating orbits: Variation-of-Parameters
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17. Effects of Earth Oblateness on the Osculating Orbit Elements
Eight Revolutions of a J2 – Perturbed Orbit*
These results were computed by Harold Black of the Johns Hopkins Applied Physics Lab using
Least square fit of Ω &  above gives
The first order (EQS 10.94, 10.95) secular terms give
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18. Variation of Parameters Tutoring
Consider the two problems
The forced linear oscillator
(1)
The perturbed two-body problem
(2)
We’ll look first at the linear oscillator to illustrate the essential ideas.
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19. Integration of Perturbed Orbits
(1)`
(3)
(4)
(5)
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20. Integration of Perturbed Orbits
(16)
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21. Integration of Perturbed Orbits
(1)``
(6)`
(1)```
(7)
(8)
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22. Integration of Perturbed Orbits
(9)
(8)`
(10)
(11)
(12)
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23. Integration of Perturbed Orbits
(12)`
(13)
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24. Integration of Perturbed Orbits
Of course, the justification for variation-of-parameters “runs deeper”
Than solving linear ODE’s! However, the essence of the ideas is
easy to illustrate for this case.
The inversion for is typically “more significant” for the higher dimensioned case. Lagrange developed an elegant process “Lagrange’s Brackets” and applied it to the perturbed 2-body
problem (Ch. 10 of RHB). We now consider this material.
One notational challenge, RHB does not distinguish between the position and velocity vectors r , v and the functional form of the solution, e.g.
Also, RHB denotes vectors and column matrices with the same symbol, e.g., r.
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25. Integration of Perturbed Orbits
The method of the variation of parameters, as originally developed by
Lagrange, was to study the disturbed motion of two bodies in the form
(10.13)
Where R is the disturbing function defined in Sect The solution
of the undisturbed or two-body motion is known and may be expressed
functionally in the form
(10.14)
Where the components of the vector are the six constants of integration
(orbital elements). As in the previous section, we allow to be a time dependent quantity and require that the two-body solution (10.14) exactly
satisfy the equations (10.13) for the disturbed motion.
A set of differential equations for will result as before; however, they
will not be solvable by quadrature. The new set of equations will in fact,
we transformation of the dependent variables of the problem from the
original position and velocity vectors and to the time-varying
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26. Integration of Perturbed Orbits
To obtain the variational equations, we substitute Eqs. (10.14) into Eqs. (10.13) and use the fact that unperturbed motion satisfies
(10.15)
Here, the partial derivatives serve to emphasize that when the vector of elements is considered to be constant, then Eqs. (10.14) are solutions of the equations which describe the undisturbed motion.
For the actual (disturbed) motion
and, paralleling the arguments used in the previous section, we have the osculation constraint:
(10.16)
As the condition to be imposed on that guarantees the first of Eqs. (10.15). Physically, this means we are requiring the velocity vectors of both the disturbed and undisturbed motion to be identical and consistent with the same osculating two body orbit.
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27. Integration of Perturbed Orbits
Similarly, differentiation of v gives
Eq. of motion
and, substituting A into B, we find that
(10.17)
must result if Eqs. (10.13) are to be satisfied. Equations (10.16) and (10.17) are the required six scalar differential equations to be satisfied by the vector of orbital elements
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28. Integration of Perturbed Orbits
The two Eqs. (10.16) & (10.17):
6 x 6
3 x 6
3 x 6
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29. Integration of Perturbed Orbits
Lagrange’s Immortal Manipulations
6 x x 6
6 x x 6
This gives:
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30. Integration of Perturbed Orbits
The Lagrange Matrix and Lagrangian Brackets
The two vector-matrix variational equations can be combined to produce a more convenient and compact form. For this purpose, we first multiply Eq. (10.16) by Then, multiply Eq. (10.17) by and subtract the two. The result is expressed as
(10.18)
where the matrix
(10.19)
is 6 x 6 and skew-symmetric. The form of the right-hand side of Eq. (10.18) follows from the chain rule of partial differentiation
The element in the ith row and jth column of the Lagrange matrix L is denoted by
and will be referred to as a Lagrangian bracket. From Eq. (10.19) we have
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31. Integration of Perturbed Orbits
An important property of the Lagrange bracket matrix L is displayed when we calculate the partial derivative of the Lagrangian bracket with respect to t. Thus,
And, clearly, the second and fourth terms cancel immediately. Using the gravitational potential function , the second one of Eqs. (10.15) becomes
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32. Integration of Perturbed Orbits
so that
In view of this discussion, we can summarize the properties of the Lagrangian brackets as
(1)
(2)
(3)
or, equivalently,
These properties hold for any choice of elements. The brackets then have additional
Special properties for each particular choice of elements. RHB develops these
Properties for the Classical Elements
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33. Integration of Perturbed Orbits
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34. Integration of Perturbed Orbits
(10.31)
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35. Integration of Perturbed Orbits
Summary of Gauss’ Equations
Finally, we are ready to summarize the complete set of variational equations.
By substituting Eqs. (10.36) and (10.38) into Lagrange’s planetary equations
(noting that ), we obtain
(10.41)
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36. Oblateness Perturbations
Legendre
Polynomials
Legendre
Polynomials
Associated
Legendre
Functions
“Zonal”
Harmonics
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37. Integration of Perturbed Orbits
Z z dm s P r y Y X x
Earth
(For example)
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38. Integration of Perturbed Orbits
If we take origin to be
mass center of earth
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39. Integration of Perturbed Orbits
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40. Variation – of Parameters Battin’s Development – p. 476-489
“Gaussian Form”
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41. Integration of Perturbed Orbits
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42. The Classical Elements Have Singularities at e = 0 …
Roger Broucke and Paul Cefola introduced an attractive alternative… The Equinoctial Orbit Elements
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Integration of Perturbed Orbits

43. Integration of Perturbed Orbits
Equinoctial Orbit Elements => Classical Orbit Elements
Classical Orbit Elements => Equinoctial Orbit Elements
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44. Integration of Perturbed Orbits
Equinoctial Orbit Elements => Rectangular Position and Velocity
Check this messy integral, I did it using Mathematica on line & had to re-arrange it a bit
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45. Lagrange/Gaussian Variation of Parameters for Equinoctial Elements
Notice: no singularities at e = 0 or i = 0, still singularity at e = 1
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46. Universal Variation of Parameters
Check these Variation of parameters equations, I derived them w/o checking references
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47. REGULARIZED INTEGRATION OF GRAVITY PERTURBED TRAJECTORIES
Prepared by:
John L. Junkins
L. Glenn Kraige
L.D. Ziems
R.C. Engels
Department of Engineering Science and Mechanics
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061
May 1980
Prepared for:
U.S. Naval Surface Weapons Center
Dahlgren, Virginia
Final Report
Contract No. N C-A214
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48. Integration of Perturbed Orbits
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49. Integration of Perturbed Orbits
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50. Integration of Perturbed Orbits
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