Integration of Perturbed Motion - School of Engineering ...psingla/Teaching...Continuous limit of...
Transcript of Integration of Perturbed Motion - School of Engineering ...psingla/Teaching...Continuous limit of...
03/05/2006 Integration of Perturbed Orbits Slide 1
Integration of Perturbed Motion
John L. Junkins
03/05/2006 Integration of Perturbed Orbits Slide 2
OutlineIntegration of Perturbed Motion
COWELL AND ENCKE METHODSVARIATION OF PARAMETERSGRAVITY MODELING & OBLATENESS PERTURBATIONS
State Transition & Related Sensitivity Matrices for Perturbed Motion
The Three Body ProblemEQUATIONS OF MOTIONTHE RESTRICTED THREE BODY PROBLEM
Jacobi’s integral & other miraclesThe libration points, stability & the zero velocity surfaces
∂(current state) / ∂(initial state)∂(current state) / ∂(force model parmeters)
Initial Value and Two-Point Boundary Value Problems
03/05/2006 Slide 3
Integration of Perturbed MotionThree 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 nonrotatingref. 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 VariablesK.S. transformed oscillatorsBurdet transformed oscillators
Canonical CoordinatesDelunay Variables
Numerical (“special”) Methods:Single Step Methods:
Analytical continuationRunge-Kutta methods
Multi Step Methods:Adams-Moulton methodAdams-Bashford methodGaussian second sum methodSymplectic Integrators
Analytical (“general”) Methods:Pedestrian asymptotic expan.Lindstedt-Poincare methodsMethods of averagingMultiple time scale methodsTransformation methods
Questions: What is the solution needed for? How precise must the solution be? What software is available?
03/05/2006 Integration of Perturbed Orbits Slide 4
Relative Strengths of Forces Acting on a Typical Satellite(“Junkins with 10 m2 solar panels” at 350 km above earth)
1.0.0010.000 070.000 0050.000 000 20.000 000 080.000 000 04
Source of Perturbing Force 2
| perturbing force || / |GMm r
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
03/05/2006 Integration of Perturbed Orbits Slide 5
Gravity Modeling OverviewPotential of a “Potatoe”:
( )0 0
sin cos sinnn
m m mn n n
n m
RGMU P C m S mr r
φ λ λ∞
⊕
= =
⎛ ⎞ ⎡ ⎤= +⎜ ⎟ ⎣ ⎦⎝ ⎠∑ ∑
Acceleration:
Problems: (1) “The more you learn, the more it costs!”(2) ∞ is a painful upper limit(3) For n > 3, convergence is very slow.
1South:
1East: cos
Radial:
S
E
R
UGr
UGrUGr
φ
φ λ
∂= −
∂∂
= −∂
∂=∂
SphericalRectangular
x
y
z
UGxUGxUGx
∂=
∂∂
=∂∂
=∂
03/05/2006 Integration of Perturbed Orbits Slide 6
During 1975 – 76, J. Junkins et al developed a (“finite element”) gravitymodel based upon the starting observation “horse-sense”:
( ), ,REFU U r θ λ= ( ), ,U r θ λΔ
“Everything Else”Dominate terms
. . . Use global modelfor these . . .
. . . Use global family of local, piecewise continuousfunctions to model these. . .
+
Thesis: It may take a >500 term spherical harmonic series to model Uglobally, but URεF 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 Overview…
03/05/2006 Integration of Perturbed Orbits Slide 7
Investigation of Finite-ElementRepresentation of the Geopotential
RADIAL DISTRUBANCE ACCELERATION ON THE EARTH’S SURFACE(contour interval is 5 x 10-5 m/sec2)
Gravity Potential GM Ur
= + Δ ( )2Radial Acceleration - GM Ur r
∂= + Δ
∂
03/05/2006 Integration of Perturbed Orbits Slide 8
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 ≈ 0.000, 002 m/sec2
• Max acceleration error ≈ 0.000, 008 m/sec2
• Mean acceleration error ≈ 0.000, 000, 03 m/sec2
• 1500 local functions 20 coefficients each 30,000 coefficients total
See: Junkins, J.L., “Investigation of Finite Element Representations of the Geopotential”, AIAA, J., Vol. 14, No. 6, June. 1976.
03/05/2006 Integration of Perturbed Orbits Slide 9
Encke’s Method: Integrate Departure Motion from an Osculating Reference Orbit
The parenthetic term is a small difference of large numbers,It is profitable to re-arrange it to avoid numerical difficulties...
From which it follows that:
3 3osc
osc osc doscr r
δ δ μ⎛ ⎞
= + → = − = − +⎜ ⎟⎝ ⎠
r rr r r r r r a
( ) ( )( ) ( )
osc o o
osc o o
t t
t t
=
=
r r
r r
Osculation Condition at t0
Note that: Also note:
osc
osc
δδ
= +
= +r r rr r r
3
3osc
d
oscosc
r
r
μ
μ
= − +
= −
rr a
rr
( )tδ r
03/05/2006 Integration of Perturbed Orbits Slide 10
Encke’s Method: Re-arrangement of Departure MotionDifferential Equation to Avoid SDOLN
(small differences of large numbers!)
On the previous chart we developed the departure differential equation:
3 3 , oscosc d osc
oscr rδ μ δ
⎛ ⎞= − = − + = +⎜ ⎟
⎝ ⎠
r rr r r a r r r
This equation can be arranged into a more computationally attractive form:
( )3 3 dosc osc
f qr rδδ μ μ= − − +
r rr a [note, no small differences of large #’s!]
where
The development of the above form is given on the following 3 pages.
( )( )
2
3/ 22
2 3 3, , 1 1
oscq qq f q q
r qδ δ δδ
⎛ ⎞⋅ − ⋅ + += + = = ⎜ ⎟
⎜ ⎟+ +⎝ ⎠
r r r rr r r
03/05/2006 Integration of Perturbed Orbits Slide 11
The actual motion is governed by
The osculating orbit satisfies
So the departure (“pertubative”) acceleration is
Making use of
Introduce some useful alternatives since
From which
3 drμ
= − +r r a
3osc oscoscrμ−
=r r
3 3osc
osc doscr r
δ μ⎛ ⎞
= − = − +⎜ ⎟⎝ ⎠
r rr r r a
,osc δ= −r r r I get
( )
3
3 3 3 1 oscd
osc osc
f q
rr r rμ μδ δ
−
⎛ ⎞−= + − +⎜ ⎟
⎝ ⎠r r r a
2 2 2 osc osc osc oscr rδ δ δ δ= − → = ⋅ = − ⋅ + ⋅r r r r r r r r r
2
2 2
21 , oscr q qr r
δ δ δ⋅ − ⋅= + ≡
r r r r
Encke Manipulations ….
03/05/2006 Integration of Perturbed Orbits Slide 12
( ) ( )
( ) ( )
2
2 2
3 312 2
3
3 32
3
2 1 ,
1 1
thus
1 1 1
this can be further manipulated to more attractive forms --here's one of the
osc
osc osc
osc
r q qr rr rq qr r
rf q qr
δ δ δ⋅ − ⋅= + =
= + → = +
⎛ ⎞ ⎡ ⎤= − − = − − +⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠
r r r r
( ) ( )( )
( )
( )( )
( ) ( )
32
32
32
32
33 22
m:
1 1 1 1
1 1
1 1 3 3 1 11 1
qf q q
q
q q qf q qqq
⎡ ⎤+ +⎢ ⎥⎣ ⎦⎡ ⎤= − − +⎢ ⎥⎣ ⎦ ⎡ ⎤+ +⎢ ⎥⎣ ⎦⎡ ⎤ ⎛ ⎞− + + +⎣ ⎦ ⎜ ⎟= =
⎜ ⎟⎡ ⎤ + ++ + ⎝ ⎠⎢ ⎥⎣ ⎦
Encke Manipulations ….
03/05/2006 Integration of Perturbed Orbits Slide 13
So, finally, we get the (exact!) departure motion differential equationwhich lies at the heart of Encke’s Method.
( )
( ) ( )
( )( )
3 3
0 0
2
2
3/ 2
0where
2
3+3q+q 1+ 1+q
when , gro
dosc osc
osc
f qr r
t t
qr
f q q
μ μδ δ
δ δ
δδ δ δ
δ δ
= − − +
= =
− +⋅ − ⋅
=
⎛ ⎞= ⎜ ⎟
⎜ ⎟⎝ ⎠
r r r a
r r
r r rr r r r
r r w too large "rectify the orbit"!→
is computed from a 2-body solution(e.g. the & functions), isusually done by numerical methods (e.g., Runge-Kutta).
osc
F G δ δ δ→ →
rr r r
Encke Manipulations ….
03/05/2006 Integration of Perturbed Orbits Slide 14
Rectification of the Reference Orbit in Encke’s MethodOriginal osculatingreference orbit (kissesactual motion at time t0)
“Rectified” (new) osculatingReference orbit (kisses the actual motion at time t1).
Whenever exceeds some preset tolerance,The position and velocity at time t1 are used to calcualte aNew “rectified” reference two-body orbit. Note that thisHas the effect of re-setting the “initial” departure positionand 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 orbitOsculates at time t1
Original reference orbitosculates at time t0:
( ) ( )( ) ( )
0 0
0 0
osc
osc
t t
t t
=
=
r r
r r
δ δε+ =
r rr r 1( )tδ r
03/05/2006 Integration of Perturbed Orbits Slide 15
Continuous limit of osculating orbits: Variation-of-Parameters
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 6
It is evident that given (t) and (t), I can compute the transformationto determine the elements of the instantaneous osculating orbit:
, , , , , ,t t e t e t e t e t e t e t⇔
The Essence of Variation
r r
r r
3Knowing the equations of motion and the above transformmation,
drcan I determine differential equations for the element
μ= − +
- of - Parameters lies in the affirmative answerto the following question :
rr a
( )
( )1 2 3 4 5 6
, , , , , , , , 1, 2, …,6?
Note that the elements are "slow variables" (since they are constants ofunperturbed motion).
i
ii d
s e t in the formde f t e e e e e e idt
= =a
03/05/2006 Integration of Perturbed Orbits Slide 16
Effects of Earth Oblateness on the Osculating Orbit ElementsEight Revolutions of a J2 – Perturbed Orbit*
These results were computed by Harold Black of the Johns Hopkins Applied Physics Lab using
0 0 0 0 0 30 27378 , 0.01, 30 , 45 , 270 , 90 , 1.0827 10a km e i M Jω −= = = Ω = = = = − ×
Least square fit of Ω & ω above gives 5.207 deg/day, 8.449 deg/dayωΩ = − =
The first order (EQS 10.94, 10.95) secular terms give 5.184 deg/day, 8.230 deg/dayd ddt dt
ωΩ= − =
03/05/2006 Integration of Perturbed Orbits Slide 17
Variation of Parameters Tutoring
Consider the two problems
• The forced linear oscillator
( )2 , , ,dx x a t x xω= − +• The perturbed two-body problem
3
3
a
d
dx
r
x xr
μ
μ
−
−=
= +
+
r r a∼ ,x y z→
We’ll look first at the linear oscillator to illustrate the essential ideas.
(1)
(2)
03/05/2006 Integration of Perturbed Orbits Slide 18
( )
( )
( )
2
00
0 0
0
The solution of
For the unperturbed 0 case is well-known -- I write it in two forms FORM 1:
cos sin
sin cos
d
d
x x aa
xx t x
x t x xt t
ω
ωτ ωτω
ω ωτ ωττ
= − +
→ = ←
⎫= +
= + ⎬≡ −
( ) ( )( ) ( )
( )2
2 2 0 00
0
FORM 2:
cos
sin
where
= + , tan =
x t Ax t A
x xA xx
ωτ φω ωτ φ
ωφω
⎪⎪
⎪⎪⎭
= + ⎫⎪⎬= + ⎪⎭
⎛ ⎞⎜ ⎟⎝ ⎠
(1)`
(3)
(4)
(5)
03/05/2006 Integration of Perturbed Orbits Slide 19
( ) ( )
( ) ( )1 2
1 2
In general, the un-perturbed solution can be written
, , , , "elements"
, , ,
The element are constants of the un-perturbed motion.
The essence of the variation-of-p
i
i
x t f t e e efx t t e et
e
= = ⎫⎪⎬∂
= ⎪∂ ⎭
( )( )
arameters idea is to considerEqs. (6) to be a coordinate transformation for the perturbed problem
and ask the question: How can we "vary the constants" i.e.,
in Eq. (6) so that the homogenous soi ie e t=
lution form of Eq. (6) becomes the solution for the perturbed motion?
(16)
03/05/2006 Integration of Perturbed Orbits Slide 20
( ) ( ) ( ) ( )
22
2
1 2 1 2
22
2
Developments:The unperturbed motion satisfies
and the solution is
, , , , ,
we seek to solve
with a solution of the form
d
d xx xdt
fx t f t e e x t t e et
d x x adt
ω
ω
≡ = −
∂= =
∂
= − +
( ) ( ) ( )( )( ) ( ) ( )( )
( ) ( ) ( )( )
1 2
1 2
1 2
2
1
, ,
, ,
For , , , the chain rule gives the velocity expression
i
i i
x t f t e t e t
dx t f t e t e tdt t
x t f t e t e t
dedx f fdt t e dt=
⎫=⎪⎬∂
= ⎪∂ ⎭
=
∂ ∂= +∂ ∂∑
(1)``
(6)`
(1)```
(7)
(8)
03/05/2006 Integration of Perturbed Orbits Slide 21
( ) ( ) ( )( )
1 2
1 2
1 2
Comparing (7) & (8), we obtain the "osculation" contraint
0
So the velocity solution for the perturbed case is
, ,
Taking the time deriv
de dedx f f fdt t e dt e dt
f t e t e tdx tdt t
∂ ∂ ∂= ⇒ + =∂ ∂ ∂
∂=
∂
2 2 22
2 21
2
2
ative of (8)`, the acceleration is
Substituting (7) & (10) (1) gives
i
i i
ded x f fdt t t e dt
ft
=
∂ ∂= +∂ ∂ ∂
⇒
∂∂
∑
222
1
i
i i
def ft e dt
ω=
∂+ = −
∂ ∂∑
1
1 22 2
2
1 2
(cancellation due to Eq. (1)``)
Equations (9) & (11) can be combined as
0
d
d
a
f f dee e dt
adef fdtt e t e
+
∂ ∂⎡ ⎤ ⎧ ⎫⎢ ⎥ ⎪ ⎪∂ ∂ ⎧ ⎫⎪ ⎪⎢ ⎥ =⎨ ⎬ ⎨ ⎬⎢ ⎥∂ ∂ ⎩ ⎭⎪ ⎪⎢ ⎥ ⎪ ⎪⎩ ⎭∂ ∂ ∂ ∂⎣ ⎦
(9)
(8)`
(10)
(11)
(12)
03/05/2006 Integration of Perturbed Orbits Slide 22
21
1 22 2
1 2
1
2
Now, consider FORM 1:
cos sin ,
1 cos , sin
sin , cos
Equation (12) is then
1cos sin
sin cos
oef e t t
f fe e
f ft e t e
dedtdedt
ωτ ωτ τω
ωτ ωτω
ω ωτ ωτ
ωτ ωτω
ω ωτ ωτ
= + ≡ −
∂ ∂= =
∂ ∂
∂ ∂= − =
∂ ∂ ∂ ∂
⎧⎡ ⎤ ⎪⎪⎢ ⎥ ⎨⎢ ⎥−⎢ ⎥⎣ ⎦ ⎩
( )1 2
0
This is easy to invert for
1 sin , cos
d
i
d d
a
dedt
de dea adt dt
ωτ ωτω
⎫⎪ ⎧ ⎫⎪ =⎬ ⎨ ⎬
⎩ ⎭⎪ ⎪⎪ ⎪⎭
⎛ ⎞= − =⎜ ⎟⎝ ⎠
(12)`
(13)
03/05/2006 Integration of Perturbed Orbits Slide 23
idedt
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.