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}R-64-533

SPECIAL PERTURBATION TECHNIQUES APPLICABLE TO SPACETRACK

TECHNICAL DOCUMENTARY REPORT NO. ESD-TDR-64-533

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P. Chambliss, Jr. J. Stanfield

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ESD-TDR-64-533

SPECIAL PERTURBATION TECHNIQUES APPLICABLE TO SPACETRACK

TECHNICAL DOCUMENTARY REPORT NO. ESD-TDR-64-533

JULY 1964

P. Chambliss, Jr. J. Stanfield

496L SYSTEM PROGRAM OFFICE ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND

UNITED STATES AIR FORCE L. G. Hanscom Field, Bedford, Massachusetts

Project 496L

(Prepared under Contract No. AF 19 (628)-1648 by the System Development Corporation, Santa Monica, California.)

FOREWORD

This Technical Documentary Report was originally published

as System Development Corporation Technical Memorandum

TM-LX-l45/000/00, dated 13 July 1964. It was prepared

under Contract AF 19(628)-l648, System 496L, Spacetrack,

for Electronic Systems Division, Air Force Systems Command.

ii

ESD-TDR-64-533

ABSTRACT

This report describes in detail three special techniques which

can be specifically applied in the SPACETRACK system to deter-

mine the motion of an artificial satellite under the influence

of perturbing accelerations. These are Variation of Parameters,

Crowell's Method, and Encke's Method.

REVIEW AND APPROVAL

This technical documentary report has been reviewed and is approved.

11 CHARD A. JED], 1st Lt, USAF Contract Technical Monitor I4.96L System Program Office Deputy for Systems Management

111

TABLE OF CONTENTS

Page

1. INTRODUCTION 1

2. VARIATION OF PARAMETERS METHOD 2

a. Description ' 2 b. Analysis 2

3- COWELL'S METHOD 8

a. Description 8 b. Analysis 8

k. ENCKE'S METHOD 12

a. Description 12 b. Analysis 12

BIBLIOGRAPHY 15

iv

1. INTRODUCTION

The purpose of this report is to describe in some detail three special

perturbation techniques which find specific application in the SPACETRACK

system for determining the motion of an artificial satellite under the

influence of perturbing accelerations. The methods, and a brief comment on

each, are:

Variation of Parameters, also commonly called the variation of elements,

expresses the perturbations in the orbit as a function of the elements,

that is to say, the difference between the elements of the orbit at epoch

and those of the osculating orbit at any time (t). Variation of parameters

is used in SPWDC and SPIRDEC.

Cowell's method integrates the equations of motion in rectangular coordinates

directly, giving the rectangular coordinates of the perturbed body. This

method finds application in ESPOD and the final phase of SPIRDEC.

Encke's method differs from Cowell's in that it is the difference between

where the body actually is in rectangular coordinates at some instant and

where it would have been if no perturbative accelerations were present

(two body reference orbit) that is calculated. This method is used in

MUNENDC.

The remainder of this report is devoted to a development of these three methods.

2. VARIATION OF PARAMETERS METHOD

a. Description

In Figure 1, let A be the position of the satellite at some time t . Suppose

at this instant all the perturbing forces acting on it ceased to exist. The

satellite would then continue to move

in an elliptical orbit (AB) with the

constant elements C,, Cp...CV* of the

classical two body problem. Further,

if the position and velocity of the /A (t ) FIGURE 1 * v o

satellite is known at t , we have the requisite number of equations for o

determining the six elements of the ellipse. The ellipse AB is called the

osculating ellipse at time t and the constants C , CU, •••Cg are the

osculating elements at time t . The actual path of the satellite when the

effect of the perturbations is taken into account is not along the elliptical

path AB but rather along some other path represented by AC in Figure 1. The

satellite at the instant t has the same coordinates and, by definition,

the same velocity components in the unperturbed as in the perturbed orbit.

Stated another way, the satellite has the position and is moving instant-

aneously as it would in purely two-body motion. Obviously one could com-

pute a set of elements to define an osculating orbit at any point of the

actual orbit. At time t , just subsequent to t , there could be defined 11 1

a new set of osculating elements C , C , •••Cg, associated with the cor-

responding position and velocity of the satellite at point C in its actual

path. The satellite's position would have been at B at time t , in the absence

of all perturbations. It is these differences between the elements,

C - C , C - C , etc. that are the perturbations of the elements in the

interval tn - t . 1 o

* It should be noted that C , C , ...C^- are identical in the Keplerian case

to a, e, i, u), fl and T or some combination of these elements.

The major perturbation encountered in dealing with the motion of artificial

satellites is the acceleration caused by the oblateness of the earth which

is much greater than the perturbing accelerations produced on it by other

bodies in the solar system. The effect is that the elements of the

Keplerian orbit vary as a function of time. Here the line of nodes and the

perigee point move very rapidly under the noncentral force field. When

the rates of change of the elements are known, the future orbital characteristics

of the satellite can be predicted.

b. Analysis

The analysis known as the variation of parameters begins with the assumption

that the cartesian coordinates defining the position of the satellite are

known. In vector notation these are

r = r (t, Ci; C2, C3, Ch, Cy C6), (1)

where r=xi+yj+zk and i, j and k are unit vectors along the

X, Y and Z axes respectively. The equations of motion of a satellite

of mass M under the central attraction of the earth (mass M ) and acted

upon by a disturbing function R can be written as

r+^H- = VR (2)

dRidRjdRk . .2 ,„ , „ \ / 0\ where V R = ^-y- + g-^ §~~z p, = k (M + MQ). (3)

From equation (l) with the C 's (k = 1, 2, 3...,6) as functions of time, 6 k

~ 5 r , > 9 r C, /(|\

1. Additional perturbations associated with earth satellites arise from

atmospheric drag, solar radiation, etc.

But d r _ d r which implies b t " d t

6

U c* = ° (5) k k=l

Invoking (5) and differentiating equation (k) with respect to t, we obtain

- d2r , \ o2 r C (6) r = —2 +Z^kaa k o t k=l k

Substituting this result back into equation (2) yields

,2 _ - V^- ^2 -

M •*4-*2_ark c-» (T)

d t r k=l k

For the osculating orbit, V R = 0 and the C 's are constants so that

d2 * + k_L _ 0 (8)

a +2 3

0 t r

hence from equation (7)

J\ = V R. ^ S2 r

It is common to rewrite equation (9)> making use of the fact that

d r S I d r 1 d r at ack " acR let rack'

as

6

jU- Ck=VR: <10> B Ck

The time derivatives of the orbital elements can now be found by solving

equations (5) and (10) simultaneously for the C, . However, the solution

of these equations can be performed more readily by a rearrangement which

introduced new functions of the parameters C called Lagrangian brackets.

d r If we take the dot product of equation (10) with ^—~ ; and equation (5) vith

or J

^-z and subtract the two, the resulting six equations may be written as

i —6. a r 3 r or dr

k=l a c. ' a c, _ J k a c, e c,

ck:7R _a_i (J-ii 2,...,6). (ii) o C .

The quantity in brackets in equation (ll) is Lagrange's bracket and is

commonly denoted by

where

IT c~l _ a (x,x) | (y, y) | (zt

5 (x, McT

, z) c,) ' (12)

*1 5 x d x o c. a c,

a x a x a c a c,

with similar expressions for Tc~cJ

, a (z, z) and a (c , cj

The right hand side of equation (ll) is the partial derivative of R with

respect to C. which is J

a R a x a R a^ a R ajz a R a x a c. a 5c. a~7 a c a c.

j y j J j

(13)

Using equations (12) and (l3).> equation (ll) may be written very simply as

k=l

R (j-1, 2,...,6). (Ik) a c.

It is these six equations that are to be solved for C, . Equation (l4)

contains thirty-six Lagrangian brackets but from the definition of these

brackets in equation (12) it is noted that

[> cU: ° [\ cr] • -&. c2 (15)

Therefore, the number of distinct Lagrangian brackets to be evaluated is

only fifteen instead of the original thirty-six.

An example of the differential equations representing the variation of

orbital elements which results from evaluating the Lagrangian brackets (2) are:

da 2 3R dt " na 3M: '

de 1 - e2 aR Jl - e2 dR dt " 2 aM 2 du> na e na e

I

dou cos i aR , Vl - e2 aR dt na y/1 - e sin i ai

di cos i aR 2 r•—? * dt na Jl - e sin i a«J

dQ 1 aR o i ' rr~ >

2 j. na e oe

dM 1 - e2 3R 2 aR — = n- 2 dt na e he na ba

The equations above differ from the differential equations solved for in

SPWDC and SPIRDEC which are in terms of an N-M element set. In terms of the

N-M element set and considering perturbations due to the earth's bulge, (3) radiation-pressure and drag, the variation of parameter equations become

(2) Kozai, Y., "The Motion of a Close Earth Satellite", The Astronomical

Journal, 6k, No. 9, Page 369, November 1959-

(3) Aeronutronic, Special Perturbations Weighted Differential Correction

Program Document, Technical Documentary Report No. ESD-TDR-63-645, 11 Dec. 63.

s dL k L n , dt= e +

d^ k a^ dT= e_ >

dh u u\ — k h dt = e_

•

In SPWDC, these equations are solved using a Runge-Kutta numeric integration

technique to determine the satellite's position and velocity.

3- COWELL'S METHOD

a. Description

Cowell's method of numerical integration makes no explicit use of a conic

section as the first approximation to the orbit, but rather, the equations

of motion in rectangular coordinates are integrated directly, giving the

rectangular coordinates of the disturbed body. The origin is usually taken

at the primary body, but this restriction is not necessary since the center

of mass of the system or of any of the disturbing bodies may be used. The

only restriction is that the motion of all bodies exerting appreciable

effects are known relative to the chosen origin at some time. The only

practical disadvantage of the method is that the integrals contain many

significant figures and change rapidly with time. In consequence, the

integration tables are slowly convergent which compels the use of a small Z

tabular interval.

b. Analysis

Consider two point masses, m and ni ,

with coordinates § , T| Q and § , 71 , a a a bo Q relative to a Cartesian coordinate

system X, Y and Z as illustrated in

Figure 2.

Let r be a vector from the origin of

the coordinate system to m and r, be a b

a vector to m, .

In addition, let s be a unit vector in the direction m HL and i, j, and k

unit vectors from the origin along X, Y and Z.

V &

i(5i> \> <i> 1, a

*.<«.' \> c„) FIGURE 2

From Newton's law, the gravitational attraction between m and m, can be

expressed by:

F = k m m.

(16)

8

where r is the distance between m and m, and k is a constant of proportionality-

depending on

due to ni is

depending on the units of mass, time and distance chosen. The force on m

2 t r

F = m r = §—2- , (l7) k m ni s

a a a r

and similarly, that on m, due to m is

k m m, s

\ - \ ^ • —P (18) r

To determine the components of the force acting on m in the X, Y, Z s directions, i.e., the §-, r|-, and Q- components, it is necessary to obtain

the dot product of F with the unit vectors i, j and k. Thus, the §- com-

ponent of the force on m is

— - — — "2 ma% — — V Fa • ' = ma ra • i = ma ?a = k ~ C0S (Fa> i}' °r>

^a = k ma '"b 3 ' U9) m a r

where r = (?a " Sb>2 * '"a " \'2 + <C. " Cb7] * Similarly, the E,- component of the force on rn due to

1

m is

" .2 ^a " *V \h = K % ma 3 (20)

r

Similar expressions can be written for the r\- and Q- components.

If additional point masses m , m , m , ... are introduced into this system

and denoting any one of these point masses by m., equations similar to (l9)

and (20) expressing the total accelerations of m and UL may be obtained by

summing all these attractions.

In figure 2; m,, represents one such mass whose distance from m and m, is

pl. and

'i,b respectively.

The expression then for the 5- components would be

2 m £ = k m m, a 'a a 0

2 (*b " ?a) (?« - 5a) /*- m m. —^

a j .3 3 J,a

(21)

"b 5b = k V. (s. - SJ

a 3 r + 1 7 (LLL!^

-"b'j T3 J, b

where

J,a

PJ,b

<5a " «/ * (\ " V2 + «a " Cj

(?b " «/ * <\ " V * <Cb " C,

3* 3*

(22)

(23)

(210

Again, similar expressions can be written for the r\- and Q- components. Let

the origin of coordinates be taken at m which is equivalent to the linear 8

transformati on

£-u -? = x; ! . -? = x.. bb 'a *j *a j

It then follows directly from (25) that

(25)

§j "5b = xj " X (26)

Finally, put

2 2 2 2 ' pj= B -x)2+(yj -y)2+(ZJ - z3 * (27)

10

Divide equation (2l) by m and equation (22) by m, and then subtract (2l) from

(22). The result is the equation of motion of m, relative to m and can be

expressed as

\ X \ X ~X

x = - k2 (m + rrL ) £r-) k2 m. -4 + > k2 m. -f- (28)

r 3 J J J

A more familiar expression for equation (28) is

* • • * <\ *">>T + L*'Ait- -i-j <29) X

r"3

This equation and similar equations in y and z are the fundamental equations

in Covell's method. Other accelerations can be combined in the right hand

side of equation (29) such as drag, zonal harmonics of the earth, radiation

pressure, etc.

11

k. ENCKE'S METHOD

a. Description

Encke's method differs from Cowell's in that the coordinates of the disturbed

body are not obtained directly but rather from the difference between the

position the body would have in an osculating orbit referenced to some time t,

called the epoch of osculation, and its true position that is calculated.

The departures from the osculating orbit are the perturbations. The advantage

of this method is that for times near the epoch of osculation, the perturbations

are small and can be expressed by a few significant figures permitting a

larger tabular interval than with Cowell's method. Against this is the

disadvantage that each step of Encke's method takes longer and the perturbations

increase with time requiring an occasional redetermination of the osculating

orbit.

b. Analysis

Let m and m, be two point masses with x y z the coordinates of m, relative

to m where m, is moving under the attraction of m alone. The equations of

motion are known to be

(r^ + ma) -2 , (30) x =-k (m, + m ) — o b a' 3

2 , , yo *o = -k K+maH > (31)

r

z zo- • K2 (r^ + ma) -2 ^ (rr^ + m ) -2 , (32)

ro

where

r = (x + y + z )^ o o o o

12

Let a, P, Y represent the perturbations produced by the presence of additional

masses m. such that the true position (x, y, and z) of m, at any time can be

represented by

x = XQ + a, y yo + p, z = zQ + Y,

and the actual equations of motion are

x = k (n^ + mQ) 7-D k m.

.. L

PJ3

with similar expressions for y and z". Note that

(33)

(3*0

( d d , ds f r=^x +y + z ) .

Subtracting (30) from (3V) yields

x x o x-xo - tf - k (^ + mj — - - a-s k m.

J

x .

\P. r . J 3

(35)

again with similar expressions for P and Y*

Equation (35) could be solved for a, P, Y ^Y direct integration by calculating

x x 0 for each step by the laws of elliptic motion and the term — at each step 3 r^

r o

by extrapolating a and adding it to x to give x, etc., but this approach would

not be convenient in practice for, since a is a small quantity, o is nearly

r3

x o equal —— , and these two terms would have to be calculated to many more

r^ significant figures than are needed in their difference. Encke developed

the following transformation to overcome this difficulty. Treating only

the equation for a, since those for "£ and Y are exactly similar,

3 3 3 r J rJ r J o o

o -

3 —1

x - a (36)

1. Refer to equation (29)

13

But

r2 = x2 + y2 + z2 = (XQ + af + (yQ + p)2 + (ZQ + v)2

= r 2 + 2x a + 2y P + 2z y + a2+ 02 + y2

o o o o '

Dividing (37) by r yields

= 1 + 2 (XQ + \ a)a+ (yQ + &)0 + (ZQ + |Y)V

ro —-•

and putting (x + &*)« + (y„ + &)P + (•„ +*Y)\ >

q = O 2

(37)

(38)

(39

equation (38) may be rewritten as

2 ^5 = 1 + 2 q. (40)

From this,

(l + 2 q) •3/2

1 - (1 + 2 q) r3/J

(41)

(42)

f =

If we now define a function f by

1 - (1 +2 qr3/'2

q

equation (35) can be rewritten in the form

a = k2 £b j ma) (f q x - a) + - 3

I k2m/Xi - x X.i \ (W) P_3

Similar equations for 0 and Y can be expressed; it is these equations that

are solved in Encke's method. Again, as in the discussion of Cowell's method,

additional perturbing accelerations can be added to the right half of equation

(44).

14

BIBLIOGRAPHY

Aeronutronic, Special Perturbations Weighted Differential Correction Program Document, Technical Documentary Report No. ESD-TDR-63-61+5> 11 Dec. 63

Brouwer, D. and Clemence, G. M., Methods of Celestial Mechanics. New York and London: Academic Press, 196T

Kozai, Y., "The Motion of a Close Earth Satellite", The Astronomical Journal, Gk, No. 9, Page 369, November 1959

McCuskey, S. W., Introduction to Celestial Mechanics. Reading, Massachusetts: Addison - Wesley Publishing Co., Inc.

Plummer, H. C., An Introductory Treatise on Dynamical Astronomy. New York: Dover Publications, Inc., 196*0

Smart, W. M., Celestial Mechanics - New York: Longmans, Green and Co., 1953

15

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16

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DOCUMENT CONTROL DATA - R&D (Security classification ol title, body ol abstract and indexing annotation must be entered when the overall report ia classitied)

I. ORIGINATIN G ACTIVITY (Corporate author)

System Development Corporation Santa Monica, California

2a. REPORT SECURITY CLASSIFICATION

Unclassified 2b GROUP/

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3 REPORT TITLE

Special Perturbation Techniques Applicable to Spacetrack

« DESCRIPTIVE NOTES (Type ol report and inclusive dates)

None 5 AUTHOWSJfLasinams, Itrst name, initial)

Chambliss, P., Jr. and Stanfield, J.

6 REPORT DATE July 1964

7« TOTAL NO. OF PACES

15 7b NO. OF REFS

10 il CONTRACT OR GRANT NO.

AF 19(628)-1648 b. PROJECT NO.

496L

9» ORIGINATOR'S REPORT NUMBERfSJ

TM-LX-145/000/00

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ESD-TDR-64-533

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Qualified Requestors May Obtain from DDC. Available from OTS

11. SUPPLEMENTARY NOTES

None

12. SPONSORING MILITARY ACTIVITY

496L Systems Program Office, Hq ESD, L. G. Hanscom Field, Bedford, Massachusetts

13 ABSTRACT

This report describes in detail three special techniques

which can he specifically applied in the SPACETRACK system

to determine the motion of an artificial satellite under the

influence of perturbing accelerations. These are

Variation of Parameters, Crowell's Method, and Encke's

Method.

DD *'JZi< 1473 Unclassified

Security Classification

Unclassified

Security Classification

KEY WORDS LINK A LINK B

BOLf

LINK C

Space Surveillance Systems Satellites (Artificial) Tracking Orbit Equations Mathematical Analysis Variation of Parameters Method Cowell's Method Encke's Method

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