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

03/05/2006 Integration of Perturbed Orbits Slide 1

Integration of Perturbed Motion

John L. Junkins


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?


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


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”


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…


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


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.


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.


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


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


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 ….


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

qq



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



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


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


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


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)


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)


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)


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)


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)


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)


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


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)


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)


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


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


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


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)


,

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


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


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


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)


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)


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


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


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


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


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


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


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


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


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


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


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


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




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

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