03/05/2006integration of perturbed orbitsslide 1 integration of perturbed motion john l. junkins
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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
Outline
Integration of Perturbed Motion
INTRODUCTION
COWELL AND ENCKE METHODS
VARIATION OF PARAMETERS
GRAVITY MODELING & OBLATENESS PERTURBATIONS
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 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?
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.
0.001
0.000 07
0.000 005
0.000 000 2
0.000 000 08
0.000 000 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 “Potato”:
0 0
sin cos sinnn
m m mn n n
n m
RGMU P C m S m
r r
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
UG
r
UG
r
UG
r
SphericalRectangular
x
y
z
UG
xU
G
Gz
U
y
“spherical harmonic gravity coefficients”
“associated Legendre functions”
03/05/2006 Integration of Perturbed Orbits Slide 6
During 1975 – 76, J. Junkins et al developed a (“finite element”) gravity model based upon the starting observation “horse-sense”:
, ,REFU U r , ,U r
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.
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
2Radial Acceleration
GMU
r r
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
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.
03/05/2006 Integration of Perturbed Orbits Slide 10
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
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
r r r
r r r
3
3osc
d
oscosc
r
r
rr a
rr
( )t r
03/05/2006 Integration of Perturbed Orbits Slide 11
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:
3 3, osc
osc d oscoscr r
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 r
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 1osc
q qq f q q
r q
r r r rr r r
03/05/2006 Integration of Perturbed Orbits Slide 12
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 dr
r r a
3osc oscoscr
r r
3 3osc
osc doscr r
rr
rr r a
,osc r r r I get
3
3 3 3 1 osc
dosc osc
f q
r
r r r
r r r a
2 2 2 osc osc osc oscr r r r r r r r r r r
2
2 2
21 , oscr
q qr r
r r r r
Encke Manipulations ….
03/05/2006 Integration of Perturbed Orbits Slide 13
2
2 2
331
2 23
33
23
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
rq q
r r
r rq q
r r
rf q q
r
r r r r
32
32
32
32
3322
m:
1 1 1 1
1 1
1 1 3 3
1 11 1
qf q q
q
q q qf q q
Encke Manipulations ….
03/05/2006 Integration of Perturbed Orbits Slide 14
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
0
where
2
3+3q+q
1+ 1+q
when , gro
dosc osc
osc
f qr r
t t
qr
f q q
r r r a
r r
r r r
r r r r
r r w too large "rectify the orbit"!
is computed from a 2-body solution
(e.g. the & functions), is
usually done by numerical methods
(e.g., Runge-Kutta).
osc
F G r
r r r
Encke Manipulations ….
03/05/2006 Integration of Perturbed Orbits Slide 15
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 a
New “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 orbit osculates at time t0:
0 0
0 0
osc
osc
t t
t t
r r
r r
r r
r r 1( )t r
03/05/2006 Integration of Perturbed Orbits Slide 16
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 transformation
to 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
- of - Parameters lies in the affirmative answer
to 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 of
unperturbed motion).
i
ii d
s e t in the form
def t e e e e e e i
dt a
03/05/2006 Integration of Perturbed Orbits Slide 17
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 J
Least square fit of Ω & above gives 5.207 deg/day, 8.449 deg/day
The first order (EQS 10.94, 10.95) secular terms give 5.184 deg/day, 8.230 deg/dayd d
dt dt
03/05/2006 Integration of Perturbed Orbits Slide 18
Variation of Parameters Tutoring
Consider the two problems
• The forced linear oscillator
2 , , ,dx x a t x x • The perturbed two-body problem
3
3
a
d
dx
r
x xr
= +
+
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 19
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 a
a
xx t x
x t x x
t t
2 0
0
22 00
FORM 2:
cos
sin
where
+ , tan
x t A
x t A
xA x
x
x
(1)`
(3)
(4)
(5)
03/05/2006 Integration of Perturbed Orbits Slide 20
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 e
fx t t e e
t
e
arameters idea is to consider
Eqs. (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 so
i ie e t
lution form of Eq. (6) becomes the
solution for the perturbed motion?
(16)
03/05/2006 Integration of Perturbed Orbits Slide 21
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 x
dt
fx t f t e e x t t e e
t
d xx a
dt
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 ft e t e t
dt t
x t f t e t e t
dedx f f
dt t e dt
(1)``
(6)`
(1)```
(7)
(8)
03/05/2006 Integration of Perturbed Orbits Slide 22
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 f
dt t e dt e dt
f t e t e tdx t
dt 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 f
dt t t e dt
f
t
222
1
i
i i
deff
t e dt
1
1 2
2 22
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 23
21
1 2
2 2
1 2
1
2
Now, consider FORM 1:
cos sin ,
1 cos , sin
sin , cos
Equation (12) is then
1cos sin
sin cos
o
ef e t t
f f
e e
f f
t e t e
de
dtde
dt
1 2
0
This is easy to invert for
1 sin , cos
d
i
d d
a
de
dtde de
a adt dt
(12)`
(13)
03/05/2006 Integration of Perturbed Orbits Slide 24
ide
dt
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.
( , )( , ), (osculation constraint)
tt
t
f e
r f e v
03/05/2006 Integration of Perturbed Orbits Slide 25
3 3, more genera ll y:
T
d
d d R
d
d
r rt dtdt
r v vv rr a
r
, ,
d tt t
dt
,rr r v v
The method of the variation of parameters, as originally developed byLagrange, was to study the disturbed motion of two bodies in the form
Where R is the disturbing function defined in Sect. 8.4. The solutionof the undisturbed or two-body motion is known and may be expressedfunctionally in the form
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) exactlysatisfy the equations (10.13) for the disturbed motion.
A set of differential equations for will result as before; however, theywill 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 .
t
tr tv
t
(10.13)
(10.14)
03/05/2006 Integration of Perturbed Orbits Slide 26
To obtain the variational equations, we substitute Eqs. (10.14) into Eqs. (10.13) and use the fact that unperturbed motion satisfies
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:
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.
3 0
t t r
r v
v, r
d d
dt t dt
r r rv =
0d d
dt t dt
r r r
v
t
(10.15)
(10.16)
03/05/2006 Integration of Perturbed Orbits Slide 27
Similarly, differentiation of v gives
and, substituting A into B, we find that
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 .
3
BA
T
d d d R
dt t dt dt r
v v v v
rr
t
v
3
d
dt r
vr =
T TR d R
dt
v
r r
t
(10.17)
Eq. of motion
03/05/2006 Integration of Perturbed Orbits Slide 28
1 2 6
, , , , , are
symbolic for the 2-body
analytical soln , , ,
etc. the ( ) matrix.
x y z x y z
x t
A
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1
x x x x x x
y y y y y y
z z z z z z
x x x x x x
y y y y y y
z
1
2
3
4
5
6
2 3 1 1 6
0
0
0
d
dtd
dtd
dt
RdxdtRdydtRz z z z z dzdt
0
T
d
dt
d R
dt
r
v
r
The two Eqs. (10.16) & (10.17):6 x 6
3 x 6
3 x 6
( ) 0 0 0T
R R RA
x y z
03/05/2006 Integration of Perturbed Orbits Slide 29
subtract 1st from
0
2nd:
TT
T
T T T
d
dt
d R R
dt
R
v
r r r
r
v
r r
Lagrange’s Immortal Manipulations6 x 3 3 x 6
6 x 3 3 x 6
This gives:
T T
( ) Typical element: = =[ , ]=
T T T
T
ij ij i ji j i j
d R
dt
or
d RL L L
dt
r v v r
r v v r
03/05/2006 Integration of Perturbed Orbits Slide 30
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
where the matrix
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
Td R
dt
L
T T
r v v r
L
= TT T T
i i i i
R R R R R x R y R z
x y z
r r
r r
,i j
1,2, ,6i
(10.18)
(10.19)
03/05/2006 Integration of Perturbed Orbits Slide 31
,
i ji j j i
T T T T
i j j i j i i j
r v r v
r v r v v r v r
,T T T T
i jj i j i i j i jt t t
v v v v r v vr
T V
t
v
r
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) becomesV r
03/05/2006 Integration of Perturbed Orbits Slide 32
2 2
,
0
i jj i i j
j i i j
j i i j
V V
t
V V
V V
r r
r r
r r
r r
so that
These properties hold for any choice of elements. The brackets then have additionalSpecial properties for each particular choice of elements. RHB develops theseProperties for the Classical Elements
In view of this discussion, we can summarize the properties of the Lagrangian bracketsas , 0
, ,
, 0
i i
i j j i
i j
elements of
t
L
(1)
(2)
(3)
or, equivalently, and 0T
t
L
L L
i a e
3 2n
a
ptime of perigee
n t
03/05/2006 Integration of Perturbed Orbits Slide 33
3 3
, 0 , 0
1 1, cos , 0 ,
2 2
, cos , 0 , , 0
1, 0 , 0 , 0 , , 0
2 With the elements of the Lagrange matrix determined, Eq. (10.18) may be
written in component form a
i
a nb i a i a nb
na e na ee i e i e e a
b b
i a na e
3
3
3 3
s
sin cos cos2
sin
2
cos2 2 2
cos
2
di nb da na e de Rnab i i i
dt dt b dtd R
nab idt i
nb da na e de R
dt b dtnb d nb d na d R
idt dt dt a
na e d na e d Ri
b dt b dt ena da R
dt
03/05/2006 Integration of Perturbed Orbits Slide 34
These are easily solved for the derivatives of the orbital elements to produce
the classical form of :
1
sin1
sin
Lagrange's planetary equations
d R
dt nab i idi R
dt nab i
3
3 4
2
4
cos
sincos
sin2
2
Equation (10.31) demonstrate explicitly that the matrix is nonsingular so
long as the ecce
i R
nab id i R b R
dt nab i i na e eda R
dt nade b R b R
dt na e na e
d R b R
dt na a na e e
L
ntricity is neither zero nor one and the inclination angle
is not zero. It should be remarked that a different choice of orbital elements
will alleviate these annoying singularities as seen in a la
e i
ter section of this chapter.
(10.31)
03/05/2006 Integration of Perturbed Orbits Slide 35
Summary of Gauss’ Equations
2
sin
sincos
1 sin coscos sin
sin
2sin
1sin cos
cos 2 sin
dh
dh
dr d dh
dr d
dr d
dr d
d ra
dt h idi r
adt h
d r ip f a p r f a a
dt he h i
da a pe f a a
dt h r
dep f a p r f re a
dt hdM b
n p f re a p r f adt ahe
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 obtain2 and p b a h nab
(10.41)
03/05/2006 Integration of Perturbed Orbits Slide 36
Oblateness Perturbations
0 0
1 1
For the general gravitational potential of a finite arbitrary body, we have the
potential energy function
P P cos sin
where the associ
nN nm m m
n n n n nn m
rGM GMV C w w C m S m
r r r
0 0 0 2 0 30 1 2 3
1 21
1 22
2 2 22
1 2 2 23
23
ated Legendre function are
1 1 =1, w = =sin , w = 3 1 , = 5 3 , . . . .
2 2
P 1 cos
P 3 1 3sin cos
P 3 1 3cos
3 3P 1 5 1 cos 5sin 1
2 2
P 15
P P w P w P w w
w w
w w w
w w
w w w
w
2 2
33 2 323
1 15sin cos
P 15 1 15cos
. . . . . . .use recursions for computing higher functions . . .
The coefficients can be related to the mass distribution by
w w
w w w
0 mn
0
mmnnm
n
1 P , sin
cos!C 2 P
sin !S
nn n n n
nn
C C J p w dm wm r
mn mp w dm
mm r n m
AssociatedLegendreFunctions
LegendrePolynomials
“Zonal”Harmonics
0n nP P Legendre
Polynomials
03/05/2006 Integration of Perturbed Orbits Slide 37
Earth(For example)
X
x
Z z
Y
y
s P
r
Space-Fixed
, , , ,
Earth-Fixed
, , , ,
X Y Z r
x y z r
Total potential at
earth
P
G dmV
s
dm
0GR GR 0
Note: and
,
if space-fixed
axes have .
e t t
z
GR
03/05/2006 Integration of Perturbed Orbits Slide 38
1
For rotational symmetry about the z-axis, you can verify that
0 for 0, ,
Thus the potential function reduces to
sin
n mn n
nn
n nn
C S m
rGM GMV J P
r r r
41 2
22
213
22
2
2
2 3
0, 0.001082616, 10 , for > 3
or
13sin 1
2 sin
1 3sin 1
2
nJ J J n
GMV V
r
rGMV J z
r r Cr
rGMV J
r r
Vd GM
dt r
r
r
ˆ ˆ ˆ
d
V V
x y z
a
I J K
If we take origin to be mass center of earth
03/05/2006 Integration of Perturbed Orbits Slide 39
22
2 13
2you verify2
2 13 112
you verify
2 2
1 3 1
2
Perturbing acceleration rectangular components:
31 5
2
3
2
dx
dy
rV J C
r r
V ra J C C
x r r
Va J
y r
22
13 12
2you verify2
2 13 132
11 12 13
3 2
1 5
33 5
2
where
ˆ ˆ ˆ + , cos cos , cos sin , sin
398601.2 sec
dz
12 13 n
rC C
r
V ra J C C
z r r
x y zr C +C C C C C
r r r
km
r =
I J K
2
6378.165
0.001082616
r km
J
03/05/2006 Integration of Perturbed Orbits Slide 40
Variation – of ParametersBattin’s Development – p. 476-
489 v.o.p. of , , , , , a e i M f
“Gaussian Form”
2
2
sin
sincos
, 1
1 sin coscos sin
sin
2sin
1sin cos
cos
dh
dh
dr d dh
dr d
dr d
d ra
dt h idi r
a h p e a edt hd r i
p f a p r f a adt he h i
da a pe f a a
dt h r
dep f a p r f re a
dt hdM b
n pdt ahe
2 sin dr df re a p r f a
ˆ ˆ ˆ d dr r d dh ha a a a e e e
03/05/2006 Integration of Perturbed Orbits Slide 41
22
2 13
2
2 3 2
22
2 3
2
22 3 2
1For = 3 C 1
2
these lead to
3 cos terms periodic in
2
3 sin cos sin 2
2
3 5cos 1 terms periodic
4
M rV J
r r
rdJ i
dt p a
rdiJ i i
dt r p
rdJ i
dt p a
in
f
2
2 3/ 2
2
22 3/ 2
As a first approximation, consider right sides constant, except
for + average over one orbit to obtain
3 cos , 0
2
3 5cos 1
4
rd diJ i
dt p a dt
rdJ i
dt p a
03/05/2006 Integration of Perturbed Orbits Slide 42
The Classical Elements Have Singularities at e = 0 …Roger Broucke and Paul Cefola introduced an attractive alternative…
The Equinoctial Orbit Elements
21
2
3
4
5
6
(1 )
cos( )
sin( )
tan( / 2)cos( )
tan( / 2)sin( )
e p a e semi latus rectum
e e
e e
e i
e i
e L f true longitude
03/05/2006 Integration of Perturbed Orbits Slide 43
Equinoctial Orbit Elements => Classical Orbit Elements2 22 3
2 2 1/22 3
1 2 2 1/24 5
1 13 2 5 4
15 4
13 2
/ (1 )
( )
tan [( ) ]
tan ( / ) tan ( / )
tan ( / )
tan ( / )
a p e e semi major axis
e e e ecentricity
i e e inclination
e e e e argument of perigee
e e longitude of ascending node
f L e e true anoma
ly
Classical Orbit Elements => Equinoctial Orbit Elements 2
1
2
3
4
5
6
(1 )
cos( )
sin( )
tan( / 2)cos( )
tan( / 2)sin( )
e p a e semi latus rectum
e e
e e
e i
e i
e L f true longitude
03/05/2006 Integration of Perturbed Orbits Slide 44
Equinoctial Orbit Elements => Rectangular Position and Velocity
2 24 5
2 24 5
24 5
2 2 24 5 2 3
2 2 24 5 2 2 3
4 5 2 4 4
(cos cos 2 sin ) /
(sin sin 2 cos ) /
( sin cos ) /
/ [sin sin 2 (cos ) (1 ) ] /
/ [ cos cos 2 (sin ) (1 ) ] /
2 / [ cos sin
r L L e e L s
r L L e e L s
r e L e L s
p L L e e L e e s
p L L e e L e e e s
p e L e L e e e
r
v
2 2
1 1 2 1 2
2 2 3/ 2 2 21 2 1 2
0
25
2 2 2 2 2 24 5 4 5 2 3
03/2 21 2
(1 ) tan( / 2) (2tan
(1 ) 1
] /
, 1 , / 1 cos sin
( )(1 cos sin )
L
L
e e e e
e e e e
e s
where
e e s e e r p w w e L e L
dt t
p e e
2
2 2
1 1 2 1 20
) sin
(1 )(1 cos sin )
L
L
e
e e e e e
Check this messy integral, I did it using Mathematica on line & had to re-arrange it a bit
03/05/2006 Integration of Perturbed Orbits Slide 45
Lagrange/Gaussian Variation of Parameters for Equinoctial Elements
1 1 1
32 1 1 12 4 5 2
3 1 1 1 23 4 5 2
24 1
2
1sin (1 )cos ( sin cos
1cos (1 )sin ( sin cos
2
r h
r h
d
d d d
d d d
de e ea
dt w
ede e e eL a w L e a e L e L e a
dt w w
de e e e eL a w L e a e L e L e a
dt w w
de es
dt w
25 1
26 14 53/2
1
2 2 2 2 2 24 5 4 5 2 3
cos
cos2
1( sin cos )
, 1 , / 1 cos sin
h
h
h
d
d
d
L a
de esL a
dt w
de edLw e L e L a
dt dt e w
where
e e s e e r p w w e L e L
Notice: no singularities at
e = 0 or i = 0, still singularity at
e = 1
Universal Variation of Parameters
03/05/2006 Integration of Perturbed Orbits Slide 46
0 0 0 0 0 0 0 0 0 1 2 1 0 1 2 2
0 0 0 0 0 0 0 0 0 1 2 1 0 1 2 2
0 0 0
0 0
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , )
( , , )
F t t G t t F t t G t t
F t t G t t F t t G t t
F t t G t t
F t t
r v r r v v e e e e e er
r v r r v v e e e e e ev
r r v r r v v
v r v r
0
0 01
0 02
1
2
( , , )
( , , ) ( , , )
( , , ) ( , , )T T
d d d
TT
d d d
G t t
so
F t t G t t
F t t G t t
d F GG
dt
d F GG
dt
r v v
r v r r v vee
r v r r v ve
ea a r a v
v v
ea a r a
v v
the , , , functions are computed in terms of the universal functions,
and therefore the entire formulation is valid for all species of conic sections.
F G F G
v
Check these Variation of parameters
equations, I derived them w/o checking
references
03/05/2006 Integration of Perturbed Orbits Slide 47
REGULARIZED INTEGRATION OFGRAVITY PERTURBED TRAJECTORIES
Prepared by:
John L. JunkinsL. Glenn Kraige
L.D. ZiemsR.C. Engels
Department of Engineering Science and MechanicsVirginia Polytechnic Institute and State University
Blacksburg, Virginia 24061
May 1980
Prepared for:
U.S. Naval Surface Weapons CenterDahlgren, Virginia
Final ReportContract No. N60921-78-C-A214
03/05/2006 Integration of Perturbed Orbits Slide 48
03/05/2006 Integration of Perturbed Orbits Slide 49
03/05/2006 Integration of Perturbed Orbits Slide 50