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EXTENSION OF GAUSS' METHOD FOR THE SOLUTION OF KEPLER'S EQUATION
by
THOMAS J. FILL
B.S., Pennsylvania State University
(1975)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 1976
Signature of AuthorDepartment oflAeronaut'ics and Astronautics
May 21, 1976
Certified byThesis Supervisor
Accepted byChairman, Departmental Graduate Committee
UL1aItA ILJUL 161976
'
F
EXTENSION OF GAUSS' METHOD FOR THE SOLUTION OF KEPLER'S EQUATION
by
THOMAS J. FILL
Submitted to the Department of Aeronautics and Astronautics on
May 21, 1976, in partial fulfillment of the requirements for the degree of
Master of Science.
ABSTRACT
The solution of a transcendental equation known as "Kepler'sequation," which relates position in orbit with time, requires an iterativeprocedure for solution. A method is developed based on one presented byGauss in his Theoria Motus dealing with the problem of position determinationfor time since pericenter passage in cases where elliptic and hyperbolicorbits approached very near unity. The problem of interest here is themore general one of determining final position arid velocity from giveninitial conditions and a specified time interval. Kepler's equation istransformed to an equation which is of the form of a cubic and whichprovides the nucleus of an efficient iteration algorithm. The finalalgorithm is a general form valid for any orbit of any eccentricity andrequires no knowledge of the nature of tKo orbit for application. Universalformulae are developed relating final position and velocity to initialvalues in terms of variables defined in the transformation. Finally, themethod is tested over a wide range of orbits to observe its performanceand comparison is made with the proposed Kepler subroutine for the NASASpace Shuttle orbiter vehicle.
Thesis Supervisor: Richard H. BattinTitle: Lecturer in Aeronautics and
Astronautics
ii
ACKNOWLEDGMENTS
The author is, first of all, indebted to Karl F. Gauss, whose work
in and contributions to the field of astronomy have provided the foundation
for the work presented in this thesis.
Special thanks to Dr. Richard H. Battin, thesis advisor, who
suggested the topic presented here and whose advice, comments and insight
have contributed greatly to its solution. Also, a note of appreciation is
extended to Stanley W. Shepperd whose assistance and suggestions were so
important during the course of the work.
iii
TABLE OF CONTENTS
Chapter Page
I INTRODUCTION 1
2 EXTENDED FORM OF GAUSS' METHOD 10
2.1 The Ellipse 10
2.2 The Hyperbola 13
2.3 The Parabola 15
2.4 Final Position and Velocity Vectors 17
2.5 General Summary of the Iterative Method 21
3 SERIES EXPANSIONS 23
3.1 Selection of the Constants a and 3 23
3.2 Methods of Derivation 24
3.3 Limits on Y 30
3.4 Series Economization 36
3.5 Series about an Arbitrary Point Y0 38
4 ROOTS OF THE CUBIC 44
4.1 Method A 44
4.2 Method B 51
5 FINAL ALGORITHM 55
5.1 Procedure 57
5.2 Tests and Data 58
5.3 Discussion of Results 63
5.4 Comparison with the Proposed NASA Shuttle
Kepler Subroutine 63
iv
Chapter Paye
6 CONCLUSIONS 66
Appendicies
A SERIES REVERSION 68
B SERIES EXPANSION ABOUT AN ARBITRARY POINT Y0 75
Figures
1.1 Parabola 2
1.2 Ellipse 2
1.3 Hyperbola 3
3.1 B vs. Y, Ellipse 25
3.2 B vs. Y, Hyperbola 26
3.3 - C0 vs. Y , Ellipse 27
3.4 C0 vs. Y, Hyperbola 28
3.5 Y vs. AE 34
3.6 Y vs. AH 35
4.1 Graph of Three Real Roots, b vs. x 47
5.1 Flow Diagram of Final Algorithm 59
v
Tables Pay%
3.1 Coefficients of B and C0 Series for a = 931
33.2 Coefficients of B and C0 Series for - T32
3.3 Coefficients of B and C0 Series for a = - 33
3.4 Economized Coefficients of B and C0 Series for
lYl < 2 and =
3.5 Ecomomized Coefficients of B and C0 Series for
jYj < 2 and = 40
3.6 Ecomomized Coefficients of B and C0 Series for
IYI <1 and a = 41
3.7 Economized Coefficients of B and C0 Series for
jYj <1 and ( = 42
4.1 Coefficients of Asymptotic Series 52
5.2 Characteristics of Test Orbits in Apollo Test
Package 60
5.2 Number of iterations for Apollo Test Package 61
5.3 High Eccentricity Test Orbits Resulting from use
of Point Masses for a Circular Orbit around Earth 62
B.1 B and C0 Series Coefficients for Y > 0 82
B.2 B and CO Series Coefficients for Y < 0 86
References 90
vi
SYMBOLS
a Semi-major axis
Constants whose sum is unity
e Eccentricity
E Eccentric anomaly, angle used as position parameter in ellipticorbits
AE E - E0
f Angle referred to as the true anomaly between the radius vectorand eccentricity vector
F, Ft Lagrange Functions relating final position and velocity toG, Gt initial position and velocity
1 ( c- (1-e))
h Massless angular momentum
H Position parameter in hyperbolic orbits
AH H - H 0
M Mean anomaly
p Semi-latus rectum or parameter
q Magnitude of radius vector at pericenter or point of closestapproach
r Magnitude of radius vector
r Position vector
vii
T Time of pericenter passage
t tf - t0 , Time interval from initial to final position
tmax Time interval corresponding to Ymax
tT Sum of time steps
T Time interval computed in the iteration procedure for convergencetest
p Product of the universal gravitational constant and the sum ofthe masses of the two bodies
v magnitude of velocity vector
v velocity vector
viii
CHAPTER 1
INTRODUCTION
The determination of the position and velocity in two-body orbits
leads to the solution of a transcendental equation commonly referred to
as "Kepler's equation" which relates the dependence of position in orbit
with time.
In classical analysis, the shape of these two-body orbits is
described through the use of conics and correspcnding to each conic
Kepler's equation has a different form. A useful quantity in classifying
conics is a constant e called the eccentricity. For a circle e = 0, for
an ellipse e is between 0 and 1, e equals 1 for a parabola, and is greater
than 1 for the hyperbola. Also obtainable from elementary considerations,
is the general polar equation for the conic which can be stated as
r = h2/p P
1+ e cos f 1 + e cos f
where h is the massless angular momentum, p is the product of the
universal gravitational constant and the sum of the masses of the two
bodies, p is the semi-latus rectum or parameter, and f, called the true
anomaly, is the angle between the radius vector and the direction of
pericenter or point of closest approach of the two bodies.
1
For the parabola, Kepler's equation is simply
6 (t - T) = tan3(f/2) + 3 tan(f/2) (1.2)
where T is the time of pericenter passage. Although Eq. (1.2) is a
special form of Kepler's equation it is more commonly known as "Barker's
formula." A graph of the parabola is shown in Fig. 1.1.
GN
F
GF=GN
Figure 1.1 Parabola q
2q-+
In the case of the ellipse, use is made of an angle, denoted by
E, called the eccentric anomaly, which is based on a reference circle
referred to as the "auxiliary circle," and whose geometrical significance
can be seen in Fig. 1.2.
a EC F
Figure 1.2 Ellipse
2
In terms of E, Kepler's equation may be expressed as
M = E - e sin E (1.3)
M = (t -T) (1.4)\a
M is the mean anomaly and a is the semi-major axis.
For the hyperbola, instead of an angle an area is employed as the
auxiliary variable and is also based on a reference geometric shape
referred to as the "equilateral hyperbola" as shown in Fig. 1.3.
equilateral
hyperbola
actualorbit
C AF
Figure 1.3 Hyperbola ae
Then the appropriate variable H is defined as
2Area CAQ =a H
3
so that Kepler's equation may be written as
-- (t - T) = e sinh H - H (1.5)
a3
Kepler's equation is transcendental, so that for a given time it
cannot be solved algebraically for the position parameter. However, there
is one and only one solution and for an analytic solution an iterative
process must be employed.
In his Theoria Motus Gauss addressed the problem of determining
the true anomaly from the time, for elliptic and hyperbolic orbits which
are nearly parabolic. In such cases the conventional methods of solution
could not give the precision required. As Gauss expressed it, "The methods
above treated, both for the determination of the true anomaly from the
time and for the determination of the time from the true anomaly, do not
admit of all the precision that might be required in those conic sections
of which the eccentricity differs but little from unity, that is, in
ellipses and hyperbolas which approach very near to the parabola; indeed,
unavoidable errors, increasing as the orbit tends to resemble the parabola,
may at length exceed all limits. Larger tables, constructed to more than
seven figures would undoubtedly diminish this uncertainty, but they would
not remove it, nor would they prevent its surpassing all limits as soon
as the orbit approached too near the parabola. Moreover, the methods given
above become in this case very troublesome, since a part of them require
the use of indirect trials frequently repeated, of which the tediousness
4
is even greater if we work with the larger tables. It certainly, therefore,
will not be superflous, to furnish a peculiar method by means of which
the uncertainty in this case may be avoided, and sufficient precision may
be obtained with the help of the common tables."
Gauss' method of solution is applicable to orbits of any eccentric-
ity. The required iterative scheme is a "Picard" type iteration, i.e.
successive substitution, there being no need for trials or tests which are
so characteristic of many iterative schemes. Furthermore the method is
applicable to all conic orbits, the advantage here being that the type of
conic encountered need not be known in order to apply thie formulae. Also,
continuity is maintained during transition from one conic to another while
at the same time being free from ambiguities or indeterminant forms. As
will be seen, the speed of convergence is quite rapid. Gauss' method is
briefly outlined here for the elliptic orbit.
Rewriting Eq. (1.4) as
M =3 (t - T) (1.6)
q
q being the pericenter distance, Gauss then chose to replace E arid sin E
by the quantities
P = E - sin E(1.7)
0 10
5
With these, Eq. (1.3) takes the form
(1- e) P + 1 + )% + fbe)Q =M
then as long as E is a quantity of first order,
IE3 1 E5+ Es6 120 5040 E '0'
is a quantity of the third order, while
Q = E- E3+ 1 I E560 1200
is a quantity of the first order.
Q P
Then defining
6 P
2_=_Y1
Y = E2 - 1IE4 - 514E630 5040
is a quantity of second order, while
B = 1+ 8 E4 + 10 E6 +.2800 16800
is a quantity which differs from unity by a quantity of the fourth order.
Finally, Eq. (1.8) becomes
6
(1.8)
0 a a
a 0 0
. .
B {(i e) Y 1 / 2 + -L(1 + 9 e) Y3/2} = M (1.9)
Hence it is readily seen that the choice of and in the definition10 TUof Q was to obtain B, which multiplies the entire left side of Eq. (1.9),
as nearly constant as possible. Eq. (1.9) is essentially an algebraic
equation of third order.
It is easy to see that B can be considered as a function of Y.
Furthermore, in the words of Gauss, "Now, although B may be finally known
from Y by means of our auxiliary table, nevertheless it can be foreseen,
owing to its differing so little from unity, that if the divisor B were
wholly neglected from the beginning, Y would be affected with a slight
error only. Therefore, we will first determine roughly Y, putting B = 1;
with the approximate value of Y, we will find B in our auxiliary table,
with which we will repeat more exactly the same calculations; most
frequently, precisely the same value of B that had been found from the
approximate value of Y will correspond to the value of Y thus corrected,
so that a second repetition of the operation would be superfluous, those
cases excepted in which the value of E may have been very considerable."
The tables referred to by Gauss were constructed to further simplify the
iteration process by reducing the amount of computation even more. Here
corresponding values of B are listed for values of Y, which are in incre-
ments of .004 from 0 to 1.2. In this manner the tables provide a simple
means of obtaining values of E up to 640 7'.
7
To further illustrate the speed of convergence, consider an
elliptic orbit in the x-y plane where pericenter is given by
7mr = (2 x10 m) i V
-p3
(6.2 x 10 m/sec) i'
then for the Earth as the central force and an arbitrary value of 5 hours
for the time since pericenter passage
e = .928735 M = .0764383
For an initial guess of 1 for B, Y is obtained from Eq. (1.9) and is
found to be
y = .59995
Then from Gauss' table,
log B = .0001734 or B = 1.000399
With this value of B, Eq. (1.9) gives
Y = .59998
and again from the table
B = 1.000399
Hence we have converged to the solution after just one correction at
which point E may be calculated from
8
E = B # (1 + 1Y)
with a value of 33.70390.
It is the purpose of this study to extend Gauss' method of solution
of Kepler's equation in standard form to the general problem of determining
final position and velocity vectors for a specified time interval from
given initial conditions at any point in the orbit while at the same time
preserving all or most of the qualities which were inherent in Gauss'
method. Finally, to see just how practical this solution process is,
comparison is made with the algorithm proposed for the on-board computer
in the NASA Space Shuttle orbital vehicle.
9
CHAPTER 2
EXTENDED FORM OF GAUSS' METHOD
The extension of Gauss' method to the solution of Kepler's equation
for some arbitrary interval of time to obtain the final position and
velocity is presented here for the ellipse and hyperbola, respectively.
The parabola is shown to be the limiting form of both the elliptic and
hyperbolic solutions as the eccentricity tends to unity. Furthermore,
universal formulae are derived which permit calculation of final position
and velocity using the initial position and velocity without knowledge of
the type of orbit encountered. Finally a generalized procedure of the
iterative process is presented, the details being left as the topic for
a later chapter.
2.1 The Ellipse
Kepler's equation for a time interval t = tf - t0 corresponding to
an eccentric anomaly difference AE = E - E0 , may be written as(2)
ao (1 ro) sin AEt = AE + (1 - cos AE) - (1 - a (2.1)
where the quantity 0 is defined as
7-o
10
Since q = a (1 - e) we have
= (1 - e)-3/2 {AE - e)1/2 (1 - cos AE )
- (1 -2(1q - e)) sin AE }
Defining the variables
P = AE - sin AE
Q = a AE + B sin AE
R = 1 - cos AE
y = 0(- a-(1 - e))2 q
where a and B are constants to be specified such that a + B
Eq. (2.2) becomes
= (1- e)-3 /2 { ( - e) (a
= 1. Then
AE + B sin AE )
- e)112 (1 - cos AE )
r0+ (1 - a (1 - e))
q (AE - sin AE ) }
3
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
'I 3jq
a
11
+- (
Tip'3q
t = (1-e)3 2 (e(1)-e)Q+q (1 - e)112 R + 2 y P)
(2.7)
Then, as did Gauss, defining the variables
Y 6 P
and also a new variable
c0C vrf0 C h
it is easily verified that
Q = B/tY
Hence, we have in Eq.
23B 2=-t
6P
CoR
flpV/
P = B Y3/26
(2.7)
1tN q
= (1-e)31 2 B {-Q(1 - e) y1/2 +q
WEqw(
+ 3/2
or, defining the variable D such that
12
or
(2.8)
(2.9)2 RY B
C Y-e) 1/2
Y= -0 (1 - e) D2 (2.10)q
then
t = B (D + C D2 +1y D3) (2.11)r0
Hence, we have succeeded in reducing Kepler's equation to a form which
resembles a cubic equation, the solution of which will be discussed in
a subsequent chapter. Notice also that B, C0 and Y are all functions of
AE, so that we may regard B and C0 as functions of Y. This fact, as will
be seen, is of great importance in the solution of Eq. (2.11). Also, in
the definition of Q we leave a and 1 unspecified for the moment instead of
using Gauss' choice of - and 1. The functions B and C0 are both10 1
dependent on Y and their sensitivity to particular values of a and r will
be discussed in a later chapter.
2.2 The Hyperbola
Kepler's equation for the hyperbola may be treated in an analagous
fashion. To the time interval t = tf - t0 corresponds the difference
AH = H - H0 and the relevant form of Kepler's equation is
- 0 (cosh AH - 1) + (1 + -) sinh AHF t -AH +7a a
13
since q = a (e - 1)
t = (e- i)-3/ 2 {-AH
r0+ (1+22
1)1/2 (cosh
(e - 1)) sinh AH }
Again defining the variables
P = sinh AH - AH
Q = cAH +S0 sinh AH
R = cosh AH - 1
q
+ ~ 22 (e - 1))q
gives
'iv
t = (e - 1)-3/2 {(e - 1) Q + (e - 1)1/2 R + 2 y P}
or, using Eqs. (2.8) and (2.9)
= (e- i)-3/2 B {j(e - 1) y/ 2 +] (e - 1)1/2 C Y
+ 3/23 (2.17)
14
N'4- i)
(2.12)
(2.13)
(2.14)
(2.15)
.JI'
Nq3
(2.16)
+ ( -
By letting
Y = (e - 1) D2 (2.18)q
Eq. (2.17) becomes
-L t = B (D + C D 2 +ly D3) (2.19)
The universality of this method becomes apparent here since, for
different type orbits, the same resulting equation is obtained.
Furthermore, from Eqs. (2.18) and (2.10), it is readily seen that Y may be
used to classify conics in a similar fashion to the eccentricity. If
Eq. (2.10) is accepted as the definition of D then for the ellipse Y is
greater than 0, Y is equal to 0 for the parabola, and is less than 0 for
the hyperbola. Hence we now possess a general form for the solution of
Kepler's equation for hyperbolic and elliptic orbits and will show
subsequently that this general form is indeed also valid for parabolic
orbits.
2.3 The Parabola
The parabola can be shown to be the limiting form of both the
ellipse and hyperbola as e tends to unity. Here Kepler's equation is
2 t = {tan(f/2) - tan(f0/2)} + - {tan3(f/2) - tan (f0/2)}
(2.20)
15
As e approaches unity from either the hyperbola or ellipse a
approaches infinity and from the definition of D, Y tends to 0. It is
easily verified that both B and C0 approach unity. Hence we have from
either Eq. (2.11) or Eq. (2.19)
t = 6D + C-O D2 + D3(2.21)32/ 0
Using the fact that for parabolic motion
sin=fV -0 cosf = 1 (2.22)sn 0 r 0 0 rO
and that the root of Eq. (2.21) is
& sin (- - foD= 2 (2.23)
cos(f/2)
then substitution of Eq. (2.22) and Eq. (2.23) does indeed lead to
Eq. (2.20). As a case in point; at pericenter f0 = 0; hence
D = Wtan(f/2) 00=-0 = 20 r 0
Therefore Eq. (2.21) becomes
2) 3P (t -r) =vi tan(f/2) + L- tan (f/2)
16
17
or
2 (t - ) = tan(f/2) + tan3(f/2)p33\p
which is Barker's formula. Therefore as Y approaches 0 (ecr1), the
hyperbolic and elliptic forms reduce to the parabolic form.
2.4 Final Position and Velocity Vectors
Determination of the final position and velocity from given initial
position and velocity vectors and a time interval may be done through the
use of universal formulae expressed in terms of the variables Y, B, D,
and C0.
In general, the final position and velocity vectors may be written
in terms of the Lagrange F and G functions as
r_ = F ro+Gmy
(2.24)
v_ = Ft -o+ Gtm
For elliptic orbits F and G are found to be
F = 1 - (I - cos AE)r0
Ft = - sin AErir 0
G = t-i AE n6 = -~( -csAE
G =1-a-( cos AE)r
making use of Eq. (2.9) and Eq. (2.5)
1 - cos AE 1
lY
Also from Eq. (2.3) and Eq. (2.4)
sin E = Q-caP = B v YB /26
= 7 B vY(-U
but sincer0
Y =D
sin AE = BD{0
P = AE - sin AE
( -tY)
= 1 B VY
Therefore, for an ellipse
F = 1- D B C0BD.3 r0
(1 - ) Gt = 1 --- (1D2 B C)r 20
18
and
D B6
3r0
a 3
FtBrD
(2.25)
- sin AE )
For hyperbolic orbits F and G are
F = 1 - (cosh AH - 1)r0
F = rLiisinh AH
G = t-ji(sinhAH -AH)
G = 1 - (cosh AH - 1)t r
making use of Eq. (2.9) arid Eq. (2.15)
coshAHY -B1 = 1 C U D2B C)
Also from Eq. (2.13) and Eq. (2.14)
sinh AH = Q + a P = B AY (1 + Y)
= B D t(1 + 2 Y)a 6
and
P = sinh AH - AH = 6BY3 /2 _ BD3 r0
Therefore, for the hyperbola
19
F = 1 -D2 B C0
Ft B I (l + Y)tr
G = -BD3 r0
Gt = 1 D2 B CO)
The only difference between Eq. (2.25) and Eq. (2.26) is in the
expression for Ft. This sign difference may be avoided by the convention
that Y<O for the hyperbola, Y=O for the parabola, and Y>O for the ellipse.
Thus, the final position and velocity vectors may be determined through
the use of Eq. (2.24) and Eq. (2.25), where Y determines the conic. Notice
that nowhere in Eq. (2.25) is there any need to know the nature of the
conic.
One further point to be made here is that another means of
calculating y and also Y will be needed if the new form of Kepler's
equation is to be valid for rectilinear orbits as well. This may be
accomplished using the fact that
(1 + e)
2
2 = (2_12 +pj0ro r0 (2.27)
Recall that
(n x )0-(1% g(f~oXY ( LOp 2
Using the identity
(A x B) (CxD) = (A -B) (B *D) - (A -D) (B *C)
20
(2.26)
we have 2
P C 0
Now 2
(1-e)2 _ p (p + )r0 r0
and 2 2(1 -e) _ 1- 2) _2 +0 2 V0
q p r0 r02 r0 p
Hence Eq. (2.6) and Eq. (2.16) become
Y =(1 -c(2- 00 (2.28)
while 2
Y = (2- 0 0) D2 (2.29)
2.5 General Summary of the Iterative Method
The purpose for the following general discussion of the method of
solution is to summarize the work presented in this chapter and also
to give purpose to the material presented in the following chapters.
Given an initial position, velocity and a time interval one may find
the final position and velocity by performing the following sequence of
steps:
1. Calculate r0, V0 , 2and y.
21
2. For an initial guess set B and C0 equal to unity.
3. Substitute these values into the cubic
1'= B (D + C D2 + y D3
and solve for D.
4. With this value of D, calculate Y using Eq. (2.29).
5. From B and C0 expressed as functions of Y, calculate new
values for B and C0.
6. Test for convergence by using the present values of B, C0 and
D in the cubic and recomputing t. Check the error between this
value and the given time interval. If the error is larger than
some specified value, return to step 3. If the error is smaller,
then calculate the final position and velocity using Eq. (2.24)
and Eq. (2.25).
As in Gauss' solution there must be a limit to the magnitude of
AE or AH in the ellipse and hyperbola and hence on the time interval or
else the number of iterations does not remain small. Therefore it is
necessary to account for larger intervals of time. This aspect along with
determining B and C0 as functions of Y, and determining the root to the
cubic, which in this case may not always possess one real root as was the
case with Gauss' solution since C and y are not constant, will be dis-
cussed in the subsequent chapters.
22
CHAPTER 3
SERIES EXPANSIONS
Probably the most convenient method of obtaining B and C0 as
functions of Y is by power series representation. The means of obtaining
these series is discussed along with the selection of the constants (I and
. A procedure to economize the series is then presented and a potential
means by which further reduction in the amount of computation is
obtained.
3.1 Selection of the Constants a and B.
Consideration in the selection of the constants a and 3, in this
instance, must be given not only to B but also to C 0 . For elliptic
motion, if B and C0 are expanded as powers of AE in terms of cx and 3 the
resulting first terms are
B = 1 + TU( - s) (AE)2 +
C0 = 1 +TZ(T- -)(AE) +. .
Clearly selection of 9= , = makes B a value which differs from1- 10 ' 10
3 7unity by a quantity of fourth order in AE. But if a= , 3= are
selected then C0 will differ from unity by a quantity of fourth order
23
3 1in AE. On the other hand if aX= T , =- are chosen, both B and C0
differ from unity by the same quantity of second order in AE.
It is possible to observe both B and C0 as functions of Y for these
selections by evaluating B, C0 and Y for different values of AE (AH) and
then plotting B vs. Y and C0 vs. Y. The resulting curves are shown in
Figs. 3.1 through 3.4. It is easy to see, for both the hyperbola and
ellipse, that to obtain either B or C0 as nearly flat as possible the other
varies significantly. Also notice that for a= -, C remains much10 C09
flatter than B does for a= . These two choices of a will be compared
in tests to see which gives better results.
The following procedure to obtain B and C0 as series in powers of
3 1Y will be explained with a= but results for both a= and 4 are10 1
also presented.
3.2 Methods of Derivation
Two methods of obtaining B and C0 as a power series in Y were
tried. The first uses the technique of series reversion. The procedure
is to obtain Y as a series in AE (AH) and revert the series such that now
AE (AH) is a series in powers of Y. A convenient expression can be
obtained relating B to AE (AH) and Y in which case substitution of the
reverted series gives the final result. The series for C0 can then be
obtained by making use of the reverted series and the B power series.
This method may prove more applicable if the computation is to be done by
24
S1.03
1.02
1.01
0.935
0.99 051.0 1.5 2..5
0.98
253. 0 3.3
097
0.96
FigUre 3.-1 B VS. Y
0-95el7is
0.94
0.93
0.92
0.91
4
B
1.18
1.16
1.14 c =T0
1.12
3
1.10
Ch 1.08
1.06
1.04
1.02
1.00.001.5 2.0 2. 53 -.03.5
Figure 3.2 B vs. Y, hyperbola
-A
Cnc0-
-0.1
-0.2
-0.3
-0.9
10
-. 4Figure 3.3 - C vs. y 3=
-0
4
-0.5 ellipse
-0.6
-0.7
-0.8
-0.9
-1.0
t 3
2 46 6 8 10Y
C0 Figure 3.4 C0 vs. Y
1.4 - hyperbola
1.3
1.2DO
1.1-
2.0 4.0 6.0 8 01.0
- 10
Y~po
hand, but with the aid of a computer the second method proved to be much
quicker and easier. For this reason the reversion method is explained in
detail in Appendix A.
The second method -vas performed on MACSYMA (Project MAC's SYmbolic
MAnipulation system), a large computer programming system used for
performing symbolic as well as numerical mathematical manipulations and
developed by the Mathlab Group Project MAC, of MIT. The procedure was to
obtain the power series in AE, in the case of the ellipse, about AE 0,
for Y, B and C0 from their definitions using the algebra for power series.
Hence,
E2+AE2 AE AE6 79 AE8 6469 AE1030 5040 +36000 ~ 498960000 34054020000
7 3 AE4 AE 6 71 AE8 527 AE10B = 1+ 2800 ~ 84000 S+T258720 - 100900800000 +
/xE2 /xE4 11 AE6 43 AE8 8747 AE1+C0 = 20 4200 + 504000 - 194040000 - 908107200000 +
Now, if a series for B is assumed in the form
CO
B = IAn Yn (3.1)0
substitution of the power series for Y then gives B as a power series in
AE which must be equal to the one above. Therefore, the coefficients of
like powers must be equivalent. The end result is a system of simultaneous
29
equations in the coefficients which may be solved for the coefficients of
Eq. (3.1). The same procedure can be used to obtain C0 as a power series
in Y. Values for the coefficients of the B and C0 power series in Y for
9 33the cases when a = T-o' 10 and - are listed in tables 3.1 through 3.3.
Expansions for the hyperbola were found to differ from those for
the ellipse only in the sign of the coefficients of the odd powers of Y.
Therefore if the convention that Y<O for the hyperbola is invoked, then the
coefficients for the B and C0 power series in Y for the hyperbola become
identical to those listed in tables 3.1, 3.2, and 3.3.
In the remainder of this report this convention will be assumed
unless otherwise stated.
3.3 Limits on Y
Before presenting a procedure to determine the number of terms
needed in the series presented above, some discussion is needed on the
range of Y. Clearly before determining the number of terms needed in the
series, it is important to know the magnitude of Y.
Figs. 3.5 and 3.6 illustrate graphs of Y versus AE and AH,
respectively. Clearly, for the ellipse, Y is not single-valued;therefore
AE maximum must be limited such that no ambiguities arise in the series.
Furthermore, from consideration of the figures presented earlier, Y must
be limited in such a way that the maximum difference of B or C0 from
unity is fairly small.
30
Table 3.1
Coefficients of B and C0 series for c
00 100an B = n+AnY0 I1C = i+ A yn
n
1
2
3
4
5
6
7
8
9
10
11
12
13
0
32800
116800
47186240000
13632802800000
4347419417408000000
2408477533653120000000
403919063892268016640000000
214507273146329300864000000000
7547165962983715687142454151168000000000
132411358246191261452374235852800000000000
557997989924547671035351401973977088000000000000
124142357439709972144656475517523968000000000000
I q____________________________a_______________________________
31
1700
S12000
538624000
223436800000
412990552000000
31117772770880000000
3772832990841766400000000
1050331432535919626240000000
54716100445331297649651538240000000000
106817118047244754666956800000000000
11851702685582548258837850493494272000000000000
190279531951729733916329216162435072000000000000
Table 3.2
Coefficients of B and Co series for
B = 1 n1o Yn
3
20189600
147121504000
17661335323904000
2714513748659200000
8488006389617179250688000000
1309850284955957585395712000000
588032925530971069269517504348160000000
94205546971148533233471020933148835840000000
38835961084332648051671.2632955714051083625037824E34
230981228746584997830291.01063645712408669000302592E36
64417567779201888372016373.138917937420692778362339328E38
7006713442959367729144985491m4.126630781729070772620355436544E42
Co = i+ {An Yn
3
7989600
87121504000
252481105971712000
1235127178721843200000
6865898111617179250688000000
25250217313052231930675200000
3301734880307335181844584828764160000000
162885532294232273233471020933148835840000000
302094967565576376649- 7.43115042003004919119872E32
- 53217485590557684626271.59574744064526684258304E35
- 87042907556862489536050033.138917937420692778362339328E39
- 1255133409038017992997674049I5.38256188921183144254828969984E42
1 I I__ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _. _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
32
-r
34x
n
1
2
3
4
5
6
7
8
9
10
11
12
13
f i
I
Table 3.3
Coefficients of B and C 0 series for 3
B = 1 + An01 yn
-+ 4
1
2
3
4
5
6
7
8
9
10
11
12
13
-. I ____________________________________________________________________________________________
*1 --LU
Co 1+ A yn
3
1115600
841336000
42657137984000
1352179-35875840000
13731694330135705600000
935213025717076899840000000
534645366582981578790092800000000
48211240543188761673565310156800000000
5602284660995720899760238627023940485120000000000
133025677788520291001- 1204772540478809702400000000000
1809728048880088249343138286935081045983232000000000000
732639844176505845864534.730122343541419016192E34
33
n
0
15600
1168000
157413952000
712745600000
5956930135705600000
411725589760000000
5437831417395343367372800000000
855068051700835969433600000000
43869829127363393716516496343040000000000
43558105279941808185326043136UU000000
- 31592487182856854633180599506864057614336000000000000
1754618814144204791.82287323515924250624E33
y
45 -310
Figure 3.5 Y vs. AE40
35
30
25
20
15
3
104
5 9
2 4 6 8 10 12 14 16 18
AE34
y 9 310 / 4
8
Ct=10
7
6
5Figure 3.6 Y vs. AH
4
3
2
1
00 2 4 6 8
AH
35
For comparison purposes, two ranges of Y were selected on the basis
of the figures, to illustrate both the number of terms needed in the B and
C0 series and the number of iterations required in the basic algorithm
for certain test cases to be presented subsequently. These two ranges are
-2<Y<2 and -1 Y.9
In the case of an ellipse; for a = , a Y of 2 corresponds
to a AE of approximately 84' while for a Y of 1, AE is approximately 580.
3These values are roughly the same for a =
3.4 Series Economization
A simple procedure to predict the effect of a term in a power series
is through the use of Chebyshev polynomials 1)If a function is expanded in
tha series of Chebyshev polynomials~then the contribution of the n--
Chebyshev (Tn) term will never be greater than the magnitude of its coef-
ficient since the magnitude of any Chebyshev polynomial does not exceed
unity. Specifically, for a given function
m
f(x) = I an xn where -1:x il (3.2)0
let it be required to find a function
Kg(x) = I bn X (3.3)
0
with K as small as possible, such that
36
jf(x) - g(x)j < p
where p is some maximum permissable error. This may be done by expressing
f(x) in terms of Chebyshev polynomials
mf(x) = Cn Tn(X)
0
and since ITn(x)I < 1, for -1 xs1, then
Kg(x) = Xcn Tn(X) (3.4)
0
within the desired accuracy provided that
mX (cn j< p
n=K+1
After Eq. (3.4) is obtained, then by expressing the Chebyshev polynomials
in terms of the powers of x, we obtain Eq. (3.3).
Economizations were performed to obtain 10 decimal place accuracy,
for reasons which will become apparent later, for the two ranges of Y
discussed earlier. For Y between 2, when a = -, the B series required103
9 terms and the C0 series 9 terms. When 0 = , the B series required
911 terms and the C0 series 7 terms. For Y between 1; when a =
3the B series required 6 terms and the C0 series 7 terms. Whena =- ,
the B series required 8 terms and the C0 series 5 terms. The new
37
coefficients for the four cases noted here are listed in Tables 3.4 through
3.7. Out of curiosity, the B series was economized for jYJ c + and92
a =10. In this case only 4 terms were required.
3.5 Series About an Arbitrary Point Y
If a greater accuracy is required for the values of B and C0 and
the resulting number of terms required in the above series, expanded about
Y = 0, is too large or still further reduction in the amount of computation
in the B and C0 series is desired, this may be had by expanding B and C0
about some arbitrary point Y0. In this way the range of Y may be divided
into smaller ranges. For example, B and C0 might be expanded about the
points Y0=0, 1, 2. Then, if Y varies from -2 to +2, this range may be
subdivided, where the different expansions would cover the ranges
- 5 (Y + 2) -
- 3 (Y + 1) < -
- < (Y -0) <
2=< (Y -1) <
2=3 (Y -2) <s
Clearly the symmetry of the expansions is lost here in that different
series are needed for negative values of Y, (i.e. for hyperbolic motion,
38
Table 3.4
Economized coefficients of B and C0 series for Y < 2 and a =
n B = X bn YC0 = XbYn0 0
784 357 1227 1510556596298377843571227075584000000000
0
672306029706247570163627485698166046720000000
116800
1713582915051373837313742849083023360000000
13632802800000
6871740017219921149401356706201600000000
2408477533653120000000
7856098426713971568714245415116800000000
214507273146239300864000000000
1-
175
11500
53539000
22313650000
41291414875000
311177568522500000
37728329354850650000000
1050331434952968020000000
I I__ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _L _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _
39
0
1
2
3
4
5
6
7
8
9
Table 3.5
3Economized coefficients of B and C0 series for |Y < 2 and a =
n B =X bn y nCO =ybnYn0 0
0
1
2
3
4
5
6
7
8
9
10
11
197671683691837831417197671683686400000000
85506805177870663270400000000
441231980583141724708960460800000000
13905561090805123361198981120000000
749167640858319767168368640000000
201113565984777870663270400000000
1340583750421642432971980800000000
1338326373177870663270400000000
40
69143467538713263567119911750657_69143467540522991616000000000000
3
761401297561372956000882493433841303752251277312000000000000
841336000
4071472448036137631117506571317018429343295078400000000000
- 135217935875840000
56278895619014381682493431234704777509339136000000000000
935213025717076899840000000
1660227236984777491906572560869168167518208000000000000
48211240543188761673565310156800000000
288124270208053749321689265049958905338134528000000000000
- 1330256777885202910011204772540478809702400000000000
I I - -
Table 3.6
Economized coefficients of B and C 0 series for IY< 1 and 9 =10
B = b0
n-I -t2677272312165437747913889689
26772723121654377479138896892677272312175132672000000000
14373266859032855257653248000000000
286850635854381151775093267727231217513267200000000
23898422653221547401520607488000000000
365509625837213538436693180780437831680000000
1017806702893120590800384000000000
236348556031576357350198855853283737600000000
cn
CO = bn0
Yn
1162774609923772832911627746099200000000
360664124610296685772132824924160000000
519095845728329363367065600000000
4508302608091435409961869312000000
44658691167172673413120000000
23013132141714508301557760000000
210883472945420883200000000
12627032128899368960000000
41
0
1
2
3
4
5
6
7
~
yn
Table 3.7
Economized coefficients of B and C0 series for Yj < 1 and a =10
0
yn
3084217703631355122804995720899730842177036257528381440000000000
2631405448412232745681131754270302155571200000000
12226720126207816426970791003616843540725150567628800000000
329317741959020471887131570272661667840000000
2383678248815485925000899777105442590643820953600000000
-1377004486107208113
0
1
2
3
4
5
6
8
27537658068087078912000000000
1549340372354772727410473471180800000000
163517364259646049763724095450809576194048000000000
42
0
657851363267207815197053657851363308339200000000
1779599436739967444898611200000000
7342044897399429053~ 41115710206771200000000
2781028488704629467223979622400000000
785844846547494720557855103385600000000
8143355893811311482653081600000000
n
36547297961574400000000
125422606175071338151003
B and C0 must be expanded about 1Y01 using the hyperbolic definitions and
then the sign of the coefficients for the odd powers of Y chanvd.).
The methods presented earlier are no longer of practical use in
this case. The method of obtaining these coefficients and their values
are presented in Appendix B.
43
CHAPTER 4
ROOTS OF THE CUBIC
The cubic equation in Gauss' solution is monotonic in that the
coefficients are such that only one real root exists. This is not the case
here because of the existence of C and y which are functions of initial
position and velocity. It is now possible that three real roots may exist
and clearly only one is valid. Two methods are presented here for
obtaining the one correct solution of the cubic.
4.1 Method A
Equation (2.11) is
1 3 2 t Vy D + C D + 0 =-
which may be written in the form
1(yD) yD2 + y (yD) = B t
Then by defining
y D = x -C (4.1)
F = (4.2)F 03
44
we have
x3-3s x = 2 b (4.3)
where
E = C2-
2 b = 3y2 F + C3 - 3 c C (4.5)
From elementry algebra, the criterion for one real root in Eq. (4.3) is
b2 _ 3 > 0
In this case it is easily shown that
x = (b + bfT_ 3)1/3 + (b - br 3 1/3 (4.6)
is the real root. This may be written in the form
2 b m 0x = 2-) M0 (4.7)
(in0 - )2 + m
where
mo = (Ibi +{b2 _C3)i/3 (4.8)
The absolute value of b results from taking the positive square root of
b2 Eqs. (4.7) and (4.8) are more desirable than Eq. (4.6) in computing
x due to the fact that only one and not two cube roots is required and
45
that m0 is obtained by adding two positive numbers in contrast to the
subtraction required in Eq. (4.6).
When b2 - E< 0, three real roots exist, two of which are equal
when the equality holds. Here the three roots are obtained by calculating
3 6 = arccos(-) (4.9)
so that
x1 = 2 / cos @
+niF7cos(e++if b > 0x2 =-2 cs( 3 - if b < 0
3 =v2 cos(O +T)
are the three real roots.
Since E must be greater than zero and b lies between 4) for
for these three real roots to exist, selection of the proper root can be
deduced, and easily verified, by plotting the three roots as functions of
b and c as is shown in Fig. 4.1. Using Eq. (4.1), if C is positive then
this equation is equivalent to translation of the y D axis to the right
of the origin whereas if C is negative, translation is to the left.
Rewriting Eq. (4.4) in the form
IC! = +
46
b'C
I x3
WEv2 IE
II
11IIIII
J
Vt
/B
III
x1
I1I
1II
-I
fII
I/
I
Graph of three real roots, b vs. x
x
D
.1 0 i IN 0 &1 i
Figure 4.1l
it can be seen that if y is negative then absolute C will always be less
than / . Hence translation of the axis will be such that x3 is the
correct choice of the root. If y is positive then selection of the roots
x1 or x2 will be based on the signs of b and C.
A simple criterion for selection of the roots can be obtained by
letting 6 run from 0 to R, starting at pt. A in Fig. 4.1 and moving
along the curveinstead of oscillating between 00 and 300 as is seen in
Eq. (4.9). Therefore the roots may be computed using the following
criteria: for b2 3
b3 6 = arccos(-)
then for y < 0 , 120 0- r+
for y > 0 and C < 0 , 0 + 1200 -O
otherwise, 0 remains as calculated (always in the first quadrant)
and
x = 2 /Fcos 6
From a continuity standpoint, since small changes in t and hence in
F and b , must correspond to small changes in the root D, then depending
on the values of y and C there are only certain ranges of t for which
this solution method is valid. For example, consider C positive with b
and e being such that three real roots exist. If b is decreased (by
changing t), some value will be reached such that b = at which point
the value of x = vE is the solution (pt. B in Fig. 4.1). Any further
48
decrease in b results in the existence of one real root and Fig. 4.1
illustrates that a jump from pt. B to point D exists. Hence a discontinuity
in x exists at the point where b = when c > 0. Since this is not
physically possible, t must be constrained, in this case, such that b> -/.
(we know that requiring that b be less than -/7 is incorrect from the
simple fact that three real roots exist when t = 0 and hence continuity
must be maintained from this point). In a similar fashion it is seen that
when C is negative, t must be constrained such that b < /3 . Hence
several tests must be incorporated in the algorithm to maintain these
requirements. This is, of course, not desirable.
1Depending on the selection of a , in particular if a > , it is
possible for y to be equal to or less than zero. If y equals zero then
Eq. (2.11) reduces to a quadratic with solution
D = -1 + (4.10)2 C
Clearly the (+) sign is the correct choice from the simple fact that D
must equal zero when F is zero.
When y is in the vicinity of zero, computation of the root using
Eq. (4.1) leads to an indeterminant form. In this case, an asymptotic
expansion of the form
2D = D + D1 () +2 () + . . . (4.11)
49
is appropriate. Substitution into Eq. (2.11) and equating powers of y gives
a set of expressions for the coefficients of Eq. (4.11) which may be
written in the form
A = /1 + 4 c F
S = CD + 3 D02
A2 = C D2 + 3 Do D2
A3 = C D3 + 3 D0 D2 + D1
= 2FD -
D2A
D2 A1 + D2A2D3 A
n =3-A31 D2 A2 + DA3D A-
A
A more convenient form which eliminates all of the indirect
computation may be arrived at after some inspection of the above set of
coefficients. Rewriting Eq. (4.11) in the form
oo D 0yD = D (1 + Wk ( 0 )k )
1 A
and expressing C and F in the forms
= D0 H (4.12)
(1 + A) D 0A 2 -1 A - 1F = - 2 C = =4F 2D
we substitute into Eq. (2.11) to obtain
50
u4
2
T A ( 2)H3 + (A--!1)H2 +H =A+1
A 2 2
The series definition of H then provides expressions for the coefficients
Wk in terms of A. Table 4.1 provides a list of several of these
coefficients.
The efficiency and practical use of the asymptotic series both
in the amount of computation required and in the accuracy leaves much to be
desired. The coefficients do alternate in sign so that the truncation
error is smaller in absolute value than the first term omitted. On the
other hand, no precise criterion is apparent to decide when the asymptotic
series must be used or when obtaining the root using Eq. (4.1) is no longer
valid. Also, when C is negative (or t is negative in which case F is
also negative), then for y equal zero, t must be such that the radical in
Eq (4.10) is nonnegative. Furthermore, when y is small, t must be such
that A is not in the vicinity of zero, since A is present in the denominator
of the asymptotic series.
4.2 Method B
The second method involves a change in variable from D to x
according to
D = 3+F (4.13)1 +x
51
Table 4.1
Coefficients of asympotic series
1 + [ Wn ( 2 YfA
A3
A + 5 A218
A + 7 A2 + 16 A354
5 P + 45 A2 + 159 A 23 + 31 A4648
7 A + 77
7 A + 91
11 A + 165
(429 A +
A2 + 357 A3 + 847 A4 + 896 A1944
A2 + 518 A3 + 1638 A4 + 2931 A5 + 2431 A6
3888
A2 + 1110 A3 + 4330 A4 + 10455 A5 + 15033 A6 + 10240 A111664
7293 A2 + 56528 A3 + 260865 A4 + 780615 A5 + 1529847 A6
1839739 A7 + 1062347 A8)/839808
52
r-i
n
-I
1
2
3
4
5
6
7
8
Substituting into Eq. (2.11) gives
3x -3Ex = 2b (4.14)
where now
E = 1+3 C F (4.15)
2 b = 2+9 C F + 9 y F2 (4.16)
Note that Eq. (4.14) is identical to Eq. (4.3) with different definitions
of the coefficients. Thus, if b _3 is positive, then the root is
computed using Eq. (4.7).
When b2 - Ec3< 0 , selection of the proper root can be made by
plotting the three roots as was previously done in Fig. 4.1. When F = 0
2 3(t = 0), the roots of Eq. (4.14) are x1 = 2, x2 = x3 = -1 and b -
Hence point A is the solution point for all values of C and y. From a
continuity standpoint, small changes in F, and hence in b, correspond to
small changes in the root. Therefore if b decreases then the correct root
will always be located on that portion of the curve from points A to B.
Hence for three real roots, the correct root is simply calculated from
3 0 = arccos( b (4.17)/E3
and
x = 2/ cos e (4.18)
53
To avoid any discontinuities, the requirement that b > -VE must be met.
This requirement avoids obtaining a value of x in the vicinity of x -1
for which case Eq. (4.13) has a singularity.
The most important characteristic of this method is the elimination
of division by y and hence the need for an asymptotic expansion is avoided.
Also computation of the real root has been simplified along with the number
of criteria to maintain continuity. Another quality evident is the effect
of errors resulting in the calculation of x. Small errors in x using
1method A are magnified by a factor of - which is not the case in this
Y
method.
54
CHAPTER 5
FINAL ALGORITHM
Incorporating any iterative procedure into an algorithm requires the
use of tests along with certain logic operations to account for cases that
may arise which could pose problems if precautions are not taken. In this
solution of Kepler's equation only two such cases arise.
The first deals with time intervals which require a larger transfer
angle than that which is permitted by limitations on the range of Y. This
can be handled simply by computing the time interval corresponding to
the maximum value of Y and comparing with the desired time interval. If
the time interval corresponding to Ymax is greater than the desired time
interval then the iteration process is initiated immediately to obtain the
final position and velocity using the desired time interval. If, on the
other hand, the opposite is true then a transfer angle step corresponding
to Ymax is taken. This is done by computing a position and velocity by
means of the universal formulae derived previously using the values of
B, CO, D, t and Y corresponding to Ymax . Then the time corresponding to
Ymax is subtracted from the desired transfer time interval. This is
continued until the time interval corresponding to Ymax is greater than
the present desired time interval. This procedure will be referred to
subsequently as "time stepping."
For the second case, recall that one requirement in the solution of
55
Eq. (4.14) was that b > -/ (when E is positive), to maintain con-
tinuity and also to avoid the singularity at x = -1 in computing D. Note,
b and c are both functions of y, C, and F while y and C, for the most part,
are functions of the initial position and velocity. Thus, b and E_ vary
with F and hence t. When e is positive and a value of b which is less
than is encountered, a simple relation to determine that value of
F such that b = -/ does not seem to exist.
The value of F where b = -47 is determined from the quadratic
2 3F 2 91 2 2 (y-42 )23 2)) =b2 _ -9F2(9 F2+ C (y - C2) F + (y- 4 C2))
and, unfortunately, division by y is required. If y is very nearly zero,
problems will arise. Furthermore, when c is negative, there being always
only one real root in this case, there is continuity for all values of b
and s but now b must be restricted to only positive values to positively
avoid the singularity at x = -1. Hence the first test is to see if b
is positive or negative. If positive, there is no problem. If negative,
then E must be checked to see if it is positive or negative. If negative,
then a value of F must be determined which will make b at least zero.
Here b is a quadratic itself in F and also requires division by y. If C
is positive then b >_-/ must be tested. If it is true, there is no
problem. If it is false then the above quadratic must be solved for that
value of F where b = -5.
56
Clearly all this testing is very tedious and detracts from the
appeal and potential of this method of solution. It was found that for
all orbits tested that a root less than zero was never encountered and for
the most part was in fact quite large. Thus, the need for the tests
described above can be avoided by the simple expediency of assuring only
positive roots. If a negative root is encountered then a prescribed
fraction of the time interval is subtracted and the iteration is
reinitiated. This fraction would be preferred to be near unity ( say .95)
since fairly small changes in t have a significant effect on the root.
Also, a decrease, rather than an increase, in F (t) is taken due to the
fact that as F approaches zero both b and c approach unity while the
root, x, approaches a value of 2. Maintaining x positive not only
satisfies the continuity requirement when c is positive but avoids the
possibility of having to subtract a small quantity from unity when
computing D.
5.1 Procedure
Before presenting the final algorithm the following quantities are
defined:
tmax = time interval corresponding to Ymax
t = input time interval
t T = sum of time steps (if any)
T = time interval computed in the iteration for convergence test
57
Figure 5.1 illustrates the flow chart diagram used in the tests.
In the following discussion, an iteration will be referred to as executing
blocks 12 to 20 in the flow diagram.
5.2 Tests and Data
All tests described below were made on a Hewlett Packard model
9820A calculator which employs 12 significant figures internally and
displays 10.
The first set of 28 tests consist of a series of orbits which
comprise the test package for the Kepler subroutine in the Apollo project.
The characteristics of these orbits are listed in Table 5.1. These tests
were first run to examine different ranges of Y. Two ranges were examined;
-1 < Y < 1 and -2 < <Y<2, both for a = . The results of the number10
of time steps and iterations needed along with the results of the Kepler
subroutine proposed for the NASA Space Shuttle orbiter vehicle are listed
in Table 5.2.
The second set of tests was used to examine the performance of this
method for orbits of very high eccentricities. These cases evolved from
modelling the Earth as a series of point masses, resulting in orbits of
very high eccentricities. The orbits and results are listed in Table 5.3.
Finally, we note that several cases were run for different values
58
Enter r v , t
itialize t = 0
2
B-- B
Cr0e 77p axzt2a = B ( 0+ 2
+ t=a (1 - tT)
08
3 + 4
y =Ymax y = Y-mY x
B =B max-ellipse B =B max-hyper.
co =C O mxe11ipse oCo Cmax-hyper.
65
tm = B (D +-2CfD yD6--7-
+ tmax - (t - tT)
8 t9t = t - t, T=tc A L v = I
Initial guess; B = C 1
11
0C C CI2O
-0
= 1 + 3 C F1
b = 1 + 4.5(C F1+ y F1)
- b2 -93 +
13 0 14
e = arccos(- --) m0 = [Ib + - E3 cb
x = 2 r'E cos 8 X (2 b m 0)/[(ill0 - C)+
17_3 F1
y z a2
18fff-n N
c0 = + A y B = 1+ A Y
19C C
20
Ji-ID + C D2 + -yD3
2 2 j1 t 5 - T I < E C
.tT tT +
12F 1 +7D2 B C0
G =T+ B D3JI
1=F + G
Ft= D
G = I - ro (I D2 B CO)
K_ = F t ! + Gt Y
27
t - tT (
29
m o
16=E t1 x<
Figure 5.1 Flow Diagram of Final Algorithm
59
La.L
Table 5.1
Characteristics of test orbits in Apollo test package
Test Transfer Angle, degfCase (TrunsferAnomly)deEccentricity Transfer Time, secCase-, (True Anomaly)a
1
2
3
4
5
6
7
8
9
10
10A
10B
lOC
11
12
13
14
15
16
17
18
19
20
21
22
23
2425
7.99998424968
1.00048154767
4.99987664764
1.20000175146
1.80000041363
2.40000321827
3.10000511619
3.59990041015
3.00001706630
3.00000052396
1.51788202155
1.51804097745
1.79577137679
4.99999925706
1.60650689794
3.21302199029
2.00001935050
1.60651090924
1.99999947324
2.38668401433
1.99994918900
1.08575407348
1.51672933663
1.51672339021
1.79675677848
1.40807640275
1.153449786921.09425122755
E01
E-02
EOO
E02
E02
E02
E02
E02
E01
E02
E02
E02
E02
E00
E02
E02
E01
E02
E01
E02
E01
E02
E02
E02
E02
E02
E02E02
9.08472055853
9.08473428889
9.08473428889
9.08473428889
9.08473428889
9.08473428889
9.08473428889
9.08472106471
4.74644650401
4.74644650401
9.99992103557
9.99922247179
9.99922247179
9.99999997122
9.99999997122
9.99999997122
1.00000008927
1.00000008927
2.12962970416
1.87546300185
2.12914297903
2.82215957165
9.99999876956
9.99999998429
9.99999998429
8.96059501396
2.129629704162 70365025705
E-02
E-02
E-02
E-02
E-02
E-02
E-02
E-02
E-02
E-02
E-01
E-01
E-01
E-01
E-01
E-01
E00
E00
E00
[00
E00
E00
E-01
E-01
E-01
E-01
E00Fo
1.14836999999
1.49999999999
7.53999999999
2.13718999999
3.33458999999
4.37899999999
5.40949999999
6.13027999999
5.93839999999
6.40665999999
2.49214659999
2.49386009999
2.49733099999
5.58423099999
1.15658179999
2.31316369999
2.81689999999
1.15658209999
3.15733599999
7.75691699999
2.26909999999
2.99999999999
2.49192869999
2.49192869999
2.49542759999
1.80147099999
3.472936999992-qqqQ qqQqQ
I I
E03
E-01
E01
E03
E03
E03
E03
E03
E02
E03
E05
E05
EC5
E04
E05
E05
E02
E05
E04
E04
E02
E04
E05
E05
E05
E04
E04FM4
60
.II-I
Table 5.2
Number of iterations for Apollo test package
Proposed -1 < Y <1 -2c< Y < 2Test Shuttle number of iterations number of iterationsCase version time steps _-time steps-1
2
3
4
5
6
7
8
9
10
iOA
lOB
10c
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25a__I _ __ _I
4
3
4
5
5
5
5
3
4
5
14
11
10
6
9
10
5
13
8
12
5
7
14
10
10
6
10
7
1
0
0
2
3
4
5
6
0
5
0
0
0
0
0
0
0
0
2
70
30
0
0
1
33
4
1
1
2
2
3
1
1
1
2
2
2
2
2
2
2
7
71
2
2
1
1I
2
32
5
2
2
2
44
5
61
0
0
0
1
2
2
3
4
0
3
0
0
0
0
0
0
0
0
1
5
0
2
0
0
0
1
2
2
5
4
2
2
3
2
7
7
1
2
2
1
1
S
3
2
6
2
2
2
2
6
6
Table 5.3
High Eccintricity Test Orbits resulting from use of
Point Masses for a Circular Orbit around Earth
Characteristics
Time interval equals a quarter orbit
Point masses scaled to (1/iPe)
Test case
1
2
3
4
Point mass
4.99999999999 E-6
2o49999999999 E-6
2.49999999999 E-6
2.49999999999 E-6
Eccentricity
2.50374472748 E5
1.00220726562 E5
6.96929475557 E5
4.93695513720 E5
Results
number o
time ste
1
2
0
1
-1 < Y < 1f number of
ps iterations
2
3
2
2
-2 < Y < 2
number of number of
time steps iterations
0 3
1 4
0 2
0 3
62
Test
Case
1
2
3
4
of a . The value of 9 was superior to an a of for reasonable values
of transfer angle although they were comparable for very small transfer
9angles. Hence, it was concluded that the value of -Is was the proper
choice, just as Gauss had determined for the more elementary problem.
5.3 Discussion of Results
The preceeding results clearly illustrate the application of this
method to the solution of transfer problems for all types of orbits and
for a wide range of eccentricity.
In the comparison of the two ranges of Y, the smaller range did
tend to reduce the number of iterations in several instances. And, as
would be expected, the smaller range increases the number of time steps
but the amount of computation required in taking a time step is small. Also,
the smaller range of Y requires fewer terms in the B and C0 power series.
Selection of the range of Y is largely up to the user although evidence
points towards a smaller range being more benificial. Clearly there is a
point of diminishing returns in reducing the allowable range of Y.
Furthermore reduction of the range of Y below yields very little in
the further reduction of the B and C0 power series. It is felt that a range
of Y between 1 or possibly -would be the best selection.
5.4 Comparison with the
The comparison with the proposed NASA Shuttle Shuttle subroutine
shows a considerable decrease in the number of iterations in most cases.
63
It should be kept in mind that the amount of computation in an iteration
for both methods is different. Interesting enough, this method shows a
substantial decrease in both the amount of logic and computation in one
iteration.
The convergence test in the Kepler subroutine was based on a
relative error of 10-13 between the computed time in the iteration and
the desired time. This error criteria was unobtainable due to the
limitations of the HP 9820A model. Hence a convergence test was selected
based on an absolute difference of 10~4 seconds. Cases were run where
this error was tightened whenever possible and little or no increase was
evidenced in the number of iterations. Hence the comparison of these
results could be made with little or no hesitation that these results
would differ significantly if the same criteria were used.
Of equal importance to the number of iterations is the accuracy
of the final position and velocity vectors. Here again the calculator
posed limitations. For instance, the desired time interval could not be
posed to full accuracy stice only 10 significant figures could be used
without calculator round off. Nevertheless, with a convergence criteria
3f 1O~4 seconds , accuracy of the final position and velocity vectors
was at least 8 significant figures and sometimes even 10. Hence, this
method is not only quicker but also quite accurate. The reason for this
accuracy is due to an added feature in this algorithm. Referring to
the flow chart of Fig. 5.1, the use of T, the computed time in the iteration
for the convergence test, in the statement numbers 22 and 23 has the
64
net effect, upon reaching statement 27 , of taking all the errors which
remain in statement 21, on the final iteration of any number of steps,
and iterating on that time interval error. Almost always this error is so
small that only one iteration is required to obtain a good approximation
of Y, B, and C0 . Here again, in most cases, the limits of the calculator
prevented the effectiveness of this feature from being realized since Y
was quite small and the calculator set B and C0 exactly to 1. In several
cases, though, this was the reason for the increased accuracy of the final
position and velocity.
65
CHAPTER 6
CONCLUSIONS
The method developed here for the solution of Kepler's equation for
the general problem of determining final position and velocity vectoru from
given initial conditions for a specified time interval through the
extension of Gauss' method in the standard form of position determination
for' time since pericenter passage also has resulted in a Picard type
iteration, requiring only successive substitution. Furthernore, the form
is general, thereby being applicable to all conics without knowledge of the
conic encountered and at the same time is continuous during transition from
one conic to another and is free from ambiguities or indeterminant forms.
Also, it has proven itself to be applicable to both rectilinear motion and
to all orbits of any eccentricity even in the cases where the eccentricity
is very large in which case motion is nearly rectilinear. And, the resulting
universal formulae, relating final position and velocity to initial values,
are not only simple but are also expressed in terms of variables which
have already been computed in the iteration process.
The final algorithm exhibits both simplicity and strong convergence
and at the same time has very good accuracy in determining final position
and velocity which results from internal correction of errors in the
time interval accumulated in the iteration process.
66
Finally, comparison with the Apollo version of solving Kepler's
equation showed not only a decrease in the number of iterations but also
a reduction in the amount of logic and computation in any one iteration
thereby further illustrating its simplicity and potential of finding a wide
range of application in computer oriented problems and also presents itself
as a simple method for hand or calculator computation.
Most important is the basic concept behind the method which was to
transform Keplers equation to an equation which is nearly cubic and hence
is solvable through algebraic methods, the result of which is a simple,
straightforward and expedious means of obtaining the final position and
velocity.
It is recommended that if increased accuracy is desired, say to
16 significant figures in which case better accuracy will be required
for B and CO , then serious consideration should be given to using the
B and C0 power series about points other than Y = 0, since the number of
terms needed in the series about Y = 0 could get quite large depending on
the range of Y selected.
67
APPENDIX A
SERIES REVERSION
A procedure for obtaining the coefficients of the expansions of B
and C0 in powers of Y, which is useful if the coefficients are to be
calculated by hand, is through the use of series reversion and the algebra
(1)of power series. The algebraic relations used here are:
If S = 1 +a x + ax2 X2 + a3 X3 + ...
s2= 1 + b x + b.2x2 + b3 X3 +
S3 = 1 + c x + c2 x2 + c3 x3 +.
then for
S3 = S1 92
S 3 = 1 2
= s 3
(n-1)Cn = bn + an + ak bn-k
(n-1)cn = an - bn - X ckbfn-k0
1 1(n-1)an 2 C n Z ~ X ak an-k
0
where c0 = b0-= a0
The variable Y may be expanded in powers of AE using Eq. (2.8)
68
6P _ 60 (AE - sin AE)
Q 9 AE + sin AE
or Y 1 + a AE2 + a 2 AE 4 + a 3 AE 6 + . . .-~r 2 4 6
AE 2 1+b1 AE + b2 AE + b3 AE +...
where
a = 6 (-)nb = )n(2 n + 3)! 10 (2 n + 1)!
Then dividing the two series yields
= 1 + C AE 2 + C2AE4 + c AE6 + . . (2)AE
where(n-1)
Cn = an - bn - Ck bn-k (3)
0
The reversion theorem for a power series states that given an
expansion of the form of Eq. (2) which is convergent in some interval, then
if c1 f 0 there exists one and only one function which can be expanded in
the form
AE2 = 1+d + d Y2+d2 +d3Y3 + d Y4 + (4)
Y
Clearly one method of obtaining the coefficients to Eq. (4) is by
substituting Eq. (2) into Eq. (4) and equating powers. Unfortunately, no
general expression for the n coefficient is obtainable through this
69
process. An alternate procedure is based on the fact that
- (AE 2dn =
dn-1 n!
Differentiating Eq. (2) with respect to AE2 and inverting gives
d 2TY-AE) = 1+ a AE2 + a2 AE + .
where in general
a = -(n + 1) cn
(n-1)- y (k+l)ck a n-k
0 (7)
Now the n = coefficient in Eq. (4) may be obtained as follows:
1. Compute k coefficients in Eq. (6).
2. Starting with n = 2, compute
dn 2-(AE2) 2
dYn
t' 2) d d (n-1)E2d(AE2' d d- (AE2
dY d(AE 2 dY(n-1)
For example, with n = 2,
_d (AE 2dY
d d (AE2
d(AE 2 ) LdY
= (1 + AE +*be2 + ak- k-2)(a 1 + ... + k ak AE2k-2
70
( 5)
(6)
d (AE2)
dY2
2 2 2 4 2 21= a2 (1 + AE + b2 AE + ... + b. AE
1 2 1
Where
= (ji+1) ad+ + j a a + (i-i) a a2 + . + a aa1
1 < i < (k-I)
With n = 3,
AE2 + ... + ak- AE2k )(b2 + 2 b2 2AE2+...k-2AE +
+ (k-i) bk- AE 2k-4)
= a b (1 + b1 AE2 b3 AE2i+ *.. + b AE1 +...
2
b+1I
+1[
2 -
m m a.-+1 1 < i < (k-2)
and in general
2 3 4 2 2i= b 1b1...b1 (1 + b AE + ... + b AE1 + a..)
1 < i < (k + 1 - n)
where
= - t (i+1) b 4 +a. m
b
71
2
b
d3AE2
dYa1(1+a
where3
dnAE2
dYn
ft
b.1
b1
It can be seen that 1 I n < (k+1). At the same time
d nAE2
dY n Y=
2 3 F--
- a b b b
dn-IAE2
dY n Y=-ibL
Hence, if k coefficients are calculated in Eq. (6), then k coefficients
are obtainable in Eq. (4). With the series in Eq. (4) determined, B as
a power series in powers of Y may be obtained since
2 AE 2 -2B2 = (1 + )
Using the binomial expansion
(1 + )- = 1 + a Y + a2Y2 + ... lY0 2 2
where
an - n (n+1)n (60) n
then
B2 = (1 + d Y + d2 y + ... ) (1 + a + a2 y)
- 1 +b Y+b2 2 +b 3 y +.
72
wheren-1
b = a + d + X a. d .n 0 1 fl-0
Hence
B =1+A 1 Y + A2 y2 + A3 y3 +*...
withn-1
A n = bn - bnbn-i0
To obtain the C0 power series in Y;
2R = 2(1 - cos AE) = AE2 ( 1 + *. + 2 1n + ... )
and after substitution of Eq. (4) for AE2 into the above series and expanding
the result is
2R = AE2 1 + b Y + b2 y2 + b3 y 3 + ... )
where the b's are the result of raising Eq. (4) to the corresponding
power of AE2 in the above series and adding the coefficients of similar
powers in Y. Now using the definition of C0
C = (1) (E 2 ) (1 + b Y + b2Y2 +b 3 + .m)(-B y )C 1 V+ 2 V
or
CO = (1 + a1 Y + a2 y + ... ) (1 + d Y + ... ) (1 + b1 Y + ... )
wheren-1
an = -An- - akA n-k
73
Hence, performing the required multiplication
C0 = (1 + w1 Y + w2 Y2 + ... ) (1 + b Y + b2 Y2 + ... )
wheren-1
w = an+ d + 1 akdWn = n n akdn-k0
and finally
Co =0 1+A Y + A2 Y2 + A2 Y3 +...
wheren-1
A = wn +b + wk bnk0
74
APPENDIX B
SERIES EXPANSION ABOUT AN ARBITRARY POINT Y0
The methods presented earlier for obtaining B and C0 as power
series in Y about Y = 0 are no longer practical if the expansion is required
about some arbitrary point Y0 . An alternate procedure can be used in this
instance to obtain numerical values of the coefficients for these two
series which makes use of the simplicity of the functions P, Q, and R
and their derivatives and Leibnitz's formula for the differentiation of a
product, which states
d (u v) = (n) un-k vk where (n n!
dxn 0 k! (n-k)!
For convenience, let
P(i) _dP
d E E= E0
be the convention for all functions, other than B and C0 , in which case
B0i53) - d jB
d E 1 dY E= E0
will be the convention
thLeibnitz's formula for computing the n- derivative of a function
assumes that the values of the (n-1) derivatives are known. To compute
75
y(n); from Eq. ( 2.8)
Q Y = 6 P
Differentiating both sides n times gives
X(n) (n-k) y(k)0 s
or solving for Y(fl)
y(n)
For dnB IdAE n
= 6 p(f)
= 6 P(n) n- (f) Q(n-k)0 (
(n,0) ; If we let
fl= 0, 1, . .
Y (k) n = 1, 2,
AE0
x = z2 = B2 2Y2 = 6 P Q
Then differentiating x = z2 gives
= (n) z(n-k) z(k)
0n = 0, 1, ...
but x = 6 P Q, hence
(n) - 6 n(n) (n-k) (k)
0
Equating Eqs. (3) and (4) gives, when n = 1
2 z z(1)
n = 0, 1, ..
= 6 (P~l Q + p Q l)
and for n > 2,
76
(1)
(2)
(3)
(4)
(5)
2 z z(n) = 6 () p(n-k) Q(k)0(
Q (n-k) z(k)1
And since z = B Y, differentiating with respect to E gives
( n) =
nn) B(n-k,0) y (k)
0
( n) B(n-k,0) Y(k) + 8(n,0)
or
V B(n,0) = z) -n ) B(n-k,o) (k)I
Now B(0,n) = d BdY n AE0
B(1,j) _ ddBdAE dYi
n = 1, 2, ..
be determined using the chain rule
- (= ) B(0,j+1)AE
Differentiating i times with respect to AE gives
8(j = i = 2, 3,0 (i-k) B(k,j+1)
or02 k1)Y(i-k B(i j+')0 k
If j is replaced by n-1 in Eq. (8) then
B(1,n-1) = Y(1) B(On)
77
(6)
V
(7)
(8)
(9)
(10)
Y B~-1,j1) = B('2j
hence, B(1,n-1) must
seen by expanding Eq.
For n = 1, B(120)
from Eq. (10)
For n = 2, B(2,0)
In Eq. (9)
For i=2; j=0
From Eq. (10)
For n = 3, B(3,0)
be determined. The procedure for doing this can be
(9) for several values of n:
is determined from Eq. (7)
B -10) y ()B(01
is determined from Eq. (7)
Y(i) B(1'11 = B(2, 0 ) _ .y(2) B0'1)
y(1) B(0,2)
is determined from Eq. (7)
In Eq. (9)
For i=3; j=0
For i=2; j=1
From Eq. (10)
Y B(2 ,1)
Y() B(1,2 )
B(1,2 )
= B(3,0) _ y(3 ) B(0,1)
= B(2, 1 ) _ y(2) B(0,2)
= (1) B(0,2)
- 2 (2) B 1)
Hence in general for n > 2, in Eq. (9)
Y () B(n-j-1, j+1) = B(n-j,J)_ n- -2 nj-l (n-i-k) B(k, j+1)
j = 0, 1, ... , n-2 (11)
78
dn C0To determine ,
dAE n AE0
using Eq. (2.9) and Eq. (2)
z C0 = 2 R (12)
Differentiating n times with respect to AE gives
n (n) C k,O) z(n-k) = 2 R(n)
0
or
ZCn,0) 2 R(n) _ n (n) C k,0) z(n-k) n = 1, 2,... (13)
dnCand now -0 is obtained using Eqs. (8), (10) and (11) where B
dYn AE0
is simply replaced by CO.
To summarize the procedure of determining the coefficients of
the B and C0 power series in Y about some arbitrary point YO; To calculate
m coefficients to these series the sequence is as follows
1. Evaluate the variables P, Q and R and their m derivatives at the
point AE0 (AHQ.
2. Evaluate
y(O) = 6 P(0) B(00) (0) C(0,0) 2R(0)(0)7 Y() 0 Y (0) B(U,)
79
3. For n = 1, evaluate
y(1) = (6 - Q () (O)) (0)
(P Q(O) + P5) Q() )/z()z() = 3
C(1,0)0
(0,1)0o
4. For n = 2, 3, 4, ..
= (z - B(0'0 ) Y(i))/Y(O)
- B(t1) /Y(l)
= (2 R1) - c1 0 '0) z )/z (0)0
=C(10 /Y~lu
., in
Q(0) y(n) =
2 z(0) z(n)
y(0) B(n,0)
Z(0) C no)
6 p(f) _ n-1 n
0
= 6 () p(n-k)0
(n-k) y(k)
(k) _ n-1
1(n) z(n-k) z(k)
= (n) _ (n) B(n-k,O) Y(k)
1
-2 R(n) _ n-i (nck,0) z(n-k)
0
For j = 0, 1, ... , (n-2)
0- 2 (n-k-1 (n-j-k)B(kj+) o0
80
Y B =-1j 1 B (n-jj)_
For i = 0, 1, ..., (n-2)
y(1) C(n-j-1) C(n-j,) n-j- 2 (n-j-1) y(n-i-k) C(k,j+1)
o 0 0 k 0
Y() B(0,n) - B(1,n-1)
(O~n) C(1,n-1)
Y(1) C(O,) -0C
an=0n/n) n =C0,n
wherem m
B = i+ a n(Y 0)fl C0 = 1 +Ybn (Y 0 f1 1
Table B.1 lists the coefficients of the B and C0 power series
expanded about the points .5, 1.0, 1.5, 2.0. Table B.2 lists the
coefficients for the points -.5, -1.0, -1.5, -2.0. In both tables
the number of significant figures drops from 29 for the zeroth coefficient
to 15 for the 15- . It will be hence safe to assume that the coefficients
from 1 to 5 are correct to 25 figures; coefficients 6 through 10 are
correct to 20 figures and those from 11 through 15 correct to 15 figures.
Finally, it is important to take note of the fact that the sign of Y has not
been accounted for in Table B.2. Hence for Y < 0, the sign of the odd powered
coefficients must be changed.
81
Table B.1
B and C 0 series coefficients for Y > 0
y = 4.99920793653689037165097540983p-10
n
0 1 .0002755662965972664443o2s9588o
I 1.11877805591153866712861977927B-32 1.16954380813962351712854294901B-33 7.1786525665411241568329493859B-54 6.87203163064854247174723199308B-65 6.52028433859334612908012003632B-76 6.56822195689472748648596925367B-87 6.82430353790162025933937716942B-98 7.27487389418031529408518296883B-109 7.90986039151248332596825027154B-1110 8.73854370396782131647310642643B-1211 9.78131643809698547504807697806B-1312 l.1069116458611583976786505865B-1313 1.26434527871678208599139086472B-1414 1.45575149944960705844895462873B-1515 1,68780150807908525717728459312B-16
n ~ bno >9.74636118338569407231225165762B-1
1 - 5. 1 4 9406 5 1621667810950204247743B-22 - 1.56345074645680626916714147211a-33 - 9.70226638716167223960822394225B-54 - 7 .6 1 3 3 97 4 6 813898399539955168998B-65 - 6 .73014 7 610363272866239S37o0243B--76 - 6 .3 9 6 5 4 905195183275317675747415B-87 - 6.38327063905106551650921467205B-98 - 6 .597 2811162099405615336448216B-109 - 7 .00096514595267222395201086224B-1110 - 7 .58 4 0 8 8 12 770680620509108354011B-1211 - 8.3527454764979165253679453711B-1312 - 9 .3 2 4 84 4 02318727311063977333134B-1413 - 1.052850 71016817904803180005289B-1414 - 1. 2 0 0 20 1 31572573363999654808812B-1515 - 1. 3 7947026856651413829240253971B-16
82
Table B.1 Continued
y = 1.05631489201046008637152731699(0
.n.... an0 1.001 2 7 3 1689412471J653016167973BO
1 2.4919720882695221.13421'3252t2B-32 1.30335884274550136593427291 279D-33 6.93508182056833183246520896056B-t54 9.03757154796369830595426307523B-65 9.2376959466027251 $8653309773S4B-76 9.991500563112667655523946951B-67 1.11609566524772285064902702864B-88 1.o 2793806837829632124:14416920 18B-99 1.49619266525570159622529580109B-1010 i.77817384248164148134169809988B-1111 2.'.1.414389357680896372004591775B--1212 2. 60756448726320558370280983698B-.1313 3.2051 106340735 1690963263294498B--1414 3, 9714089105200111 7892837494838B--1515 4.95543517221269066567566830471B1-16
n b-. n0 9.4548 3 64 0826002146714229205519E-1
1 - .*3329551176554104061 39391 3399622 - 1. 74079966745741804312355909275B-33 - 1.16294270187351996292626555513B-44 - 9.82616640226647326 6955920723 123B-65 9.35376604257074116845604273967B-76 - 9.57344128597197692284716116376B-87 - 1.02677828012135613333092218354B-8S 1-* 14496626535256265305616531547B--99 - 1.*30836048419997795077262094159B-1010 -1.526167778239016912829272615B3-1111 - 1.30993163746772264552166926718B-1212- 2:175692662064073143168750565243-1313-2.64509309774744925325001476404B-1414 - 3.2467004391449955031420424195B--1515 - 4. 0100716304420496814271986482B-16
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REFERENCES
1. Abramowitz, M., and I. A. Segun, Handbook of Mathematical Functions,
Dover Pub., Inc., 1965.
2. Battin, R. H., Astronautical Guidance, McGraw-Hill, 1964.
3. Gauss, K. F., Theory of the Motion of the Heavenly Bodies Moving
about the Sun in conic Sections, a translation of Theoria Motus,
Dover Pub., Inc., 1963.
4. Knopp, K., Theory and Application of Infinite Series, Blackie and
Son, 1951.
90