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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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496L SYSTEM PROGRAM OFFICE ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND
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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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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
9 b. OTHER REPORT N O(S) (A ny other numbers that may be assigned this report)
ESD-TDR-64-533
10. A VA IL ABILITY/LIMITATION NOTICES
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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