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