Implementation of Gauss-Jackson Integration for Orbit

Implementation ofGauss-Jackson Integration

for Orbit Propagation1

Matthew M. Berry2 and Liam M. Healy3

Abstract

The Gauss-Jackson multi-step predictor-corrector method is widely used in numerical in-tegration problems for astrodynamics and dynamical astronomy. The U.S. space surveil-lance centers have used an eighth-order Gauss-Jackson algorithm since the 1960s. In thispaper, we explain the algorithm including a derivation from first principals and its relationto other multi-step integration methods. We also study its applicability to satellite orbits in-cluding its accuracy and stability.

Introduction

Determination and prediction of orbits requires an orbit propagator that finds thephase space state of a satellite at one time based on its state at another time. Thisfunction has traditionally been performed by an analytical computation such asHamiltonian normalization (general perturbations). While numerical integration(special perturbations) can provide a wider range of force models, until recent yearsthe computation time required made it prohibitive for routine use in space surveil-lance. With greatly increased computation speeds now widely available, the dailyprocessing of 10,000 or more satellites routinely tracked by space surveillance cen-ters can conceivably be based on numerical integration, and a significant fractionof this number are processed numerically now.

The Gauss-Jackson integrator [1] is a fixed-step predictor-corrector method. TheGauss-Jackson software used in space surveillance originated in the 1960s as part

1Based on paper AAS 01-426 presented at the AAS/AIAA Astrodynamics Specialists Conference, QuebecCity, Canada, July 30–August 2, 2001.2Astrodynamics Engineer, Analytical Graphics Inc., 220 Valley Creek Blvd., Exton, PA 19341. E-mail:[email protected]. This work was completed while the first author was a Graduate Assistant in the De-partment of Aerospace and Ocean Engineering at Virginia Tech and an employee of the Naval Research Laboratory.3Research Physicist, Naval Research Laboratory, Code 8233, Washington, DC 20375-5355. E-mail:[email protected].

The Journal of the Astronautical Sciences, Vol. 52, No. 3, July–September 2004, pp. 331–357

331

of the Advanced Orbit/Estimation Subsystem (AOES) [2]. Over the years, AOEShas given rise to a family of astrodynamics applications used in the various spacesurveillance centers whose integrators are essentially unchanged, including a sys-tem known as SpecialK [3]. SpecialK has been used by Naval Network and SpaceOperations Command for special perturbations catalog maintenance for the lastsix years. Another member of the family is Astrodynamics Support Workstation(ASW) [4], used in Cheyenne Mountain to support the special perturbations cata-log and in support of conjunction assessment for NASA. We have chosen to useSpecialK as the implementation on which to focus, but our analysis has broad ap-plicability to all Gauss-Jackson integrators, including the one in ASW. This paperpresents a study of the theory upon which the integrator is based, how it is imple-mented, and its accuracy and stability.

After an overview of multi-step integrators, we start with an explanation of dif-ference tables and operators that form the core of the derivation of all the multi-stepmethods. We apply these to differentiation in the next section, followed by single integration where we derive the nonsummed Adams methods. The Adams methodhas a summed form which is presented in the succeeding section. The next section,on the Störmer-Cowell integrator for second-order differential equations, makesuse of the Adams results. The summed form of the Störmer-Cowell integrator,called the Gauss-Jackson integrator, is presented next. Having avoided the issue ofhow to start the integrator when backpoints are unavailable, we next address thisissue, in which the concept of a mid-corrector is introduced. To this point, the for-mulation is in terms of the difference operators; for computational efficiency, theordinate form sums like terms but makes the result order-specific. The details areincluded in the next section.

With the derivation of the integrators complete, we finish with two sections. Thefirst is a detailed description of implementation complete with pseudo-code thatshould suffice for a reasonably skilled programmer to implement in any general-purpose programming language. The second shows the effect of step size and orderof the Gauss-Jackson integrator on accuracy and stability.

Multi-Step Integrators

An ordinary differential equation (ODE) is an equation of the form

(1)

in which the derivative of a dependent variable y is a function f of that variable andthe independent variable t. In order to find a specific solution of a differential equa-tion the initial conditions must be given. A given differential equationalong with initial conditions is known as an initial value problem. The solution toan initial value problem, , can be found analytically for some differential equa-tions. However, most differential equations must be solved numerically. Algorithmsthat numerically solve initial value problems are known as numerical integrators.Numerical integrators give an approximate solution at distinct values of t, known asmesh points. The numerical solution at a mesh point is denoted , and due toerror in the numerical method will likely be different from the exact solution, .

The differential equation in (1) is called a first-order differential equation be-cause the equation is for the first derivative. Differential equations may have ahigher order

y�tn�yntn

y�t�

�t0, y�t0��

dy

dt� f�t, y�

332 Berry and Healy

(2)

where n is the order of the differential equation. Initial value problems involvinghigher-order ODEs require initial conditions for each derivative through .Though higher-order ODEs may be rewritten as a system of first order equationsand solved with a standard numerical integrator, some numerical integrators are de-signed to directly solve higher-order ODEs. Integrators that solve second-orderODEs are called double-integration methods, while integrators that solve first-orderODEs are called single-integration methods.

Multi-step integrators integrate forward from a particular time to the next meshpoint using function values at the current point as well as several previous meshpoints. The set of the previous points used as well as the current point is called theset of backpoints. One disadvantage of multi-step integrators is that the initial setof backpoints must be found through some startup procedure, which complicatesthe method. The methods develop a Taylor series in the time separation betweenmesh points, and this series must be truncated after some number of terms, whichthereby gives the order of the method. The order is one less than the number ofbackpoints, and is not related to the order of the differential equation.

Multi-step integrators are known as predictor-corrector methods. The algorithmsfirst predict a y value for the next mesh point and the function is evaluated atthe predicted point. That predicted function value is added to the set of backpoints,and a corrector formula is used with this revised set of backpoints to refine the pre-dicted y value.

This procedure offers several variations on the implementation. First, whenadding the predicted value to the set of backpoints, the farthest backpoint from thecurrent point may or may not be dropped. If it is not dropped, the corrector is ofhigher order than the predictor because the corrector uses one more backpoint thanthe predictor. Otherwise, the predictor and corrector are of the same order. Also, asecond evaluation may or may not be performed after the corrector. A second eval-uation improves the accuracy of the method, because improved function values areused in the set of backpoints in the subsequent steps. However, the additional eval-uation causes an increase in run time. Methods that use one evaluation per step areknown as Predict-Evaluate-Correct, or PEC, methods, and methods that perform asecond evaluation are called PECE methods. Some implementations perform addi-tional iterations of the corrector to meet some tolerance, and are called or

methods.The methods can be derived in a variety of ways, but the fixed-step methods can

be derived in a simplified form by using backward difference operators, which aredescribed in the next section. The derivations of the Adams, summed-Adams,Störmer-Cowell, and Gauss-Jackson methods follow. These derivations follow thederivations presented by Maury and Segal [5], and the NORAD document knownas TP008 [6].

The function must be continuous and smooth through the set of back-points. If there are any discontinuities in f, for example going through eclipse whensolar radiation pressure is considered, the integration must either be restarted, ormodified to handle the discontinuity [7].

In this paper we do not address variable step or error control, including s-integration, in numerical integration. The interested reader is referred to [8]and [9] for information on this topic.

f�t, y�

PE�CE�nP�EC�n

f�t, y�

y �n�1�

dny

dtn � f�t, y, y�, . . . , y��n�1�

Implementation of Gauss-Jackson Integration for Orbit Propagation 333

Tables and Operators

Predictor-corrector integrators can be defined in terms of difference tables. For afixed-step method, assume that solution values are known on a discrete set ofequally-spaced mesh points, where h isthe step. As shorthand, these values are referred to as , so this set offive values is , , , , . The corresponding function values are ,where is the numerical solution at the mesh point (not the exact solution ).

The differential equations are solved by taking differences of the function values.There are three kinds of differences, represented by different operators:

forward difference

backward difference

central difference

Note that first central differences cannot be calculated directly, as the half-step pointsare not in the sequence of points, though the second central differences (see below fordefinition of second differences) can be calculated. As previous authors ([10], [11])have recognized, conversion between series in these operators is straightforward.The backward difference derivation is used here, because when one is predicting orcorrecting, i.e., any time other than startup, it is the most natural. Derivations usingother difference operators give algebraically equivalent methods, though differencesin their implementation may give slightly different results due to round-off error.

Not only can the differences of the function f be computed, the differences of thedifferences, or second differences, can be computed, and so on. For instance,the square of the backward difference operator is the operator applied to the firstdifference

(3)

The relationships of the backward differences are illustrated in Table 1. The arrowsin the table point towards the difference; the upper component is always subtractedfrom the lower. For example, .

In addition to the three difference operators, there is also a displacement opera-tor E. The displacement operator is defined as

(4)

Powers of this operator act as expected; e.g., . The backward differenceoperator can be defined in terms of the displacement operator

(5)

and alternatively the displacement operator can be defined in terms of the backwarddifference operator

(6)

The summed integration formulas (see Summed Adams Method section), con-tain negative powers of the operators; e.g., . The backward difference table��2

E �1

1 � �

� � 1 � E�1

E2fi � fi�2

Efi � fi�1

�2f1 � �f1 � �f0

� fi � fi�1 � � fi�1 � fi�2� � fi � 2fi�1 � fi�2

�2fi � ��fi � �� fi � fi�1�

�fi � fi�1/2 � fi�1/2

�fi � fi � fi�1

�fi � fi�1 � fi

y�tn�tnyn

fn � f�tn, yn�t2t1t0t�1t�2

tn � t0 � nh. . . , t0 � 2h, t0 � h, t0, t0 � h, t0 � 2h, . . . ,

yn

334 Berry and Healy

(Table 1) can be extended to the left to get negative powers of the backward difference operator, as in Table 2. Extending the differencing to the left gives

. This relation can be changed into a recursion formula thatdefines the inverse operator

(7)

Note that the initial ( ) term in the sum is arbitrary; the difference relationholds no matter what this value is. This value is effectively an integration constant

. Thus, if the inverse operator is defined as a sum

(8)��1f1 � �C1 � �i

j�1fj,

C1 � �0j�i�1

fj,

if i � 1

if i �1

C1

i � 0

��1fi � fi � ��1fi�1

fi � ��1fi � ��1fi�1


TABLE 1. Backward Difference Table

0

1

2

3 �3f3�2f3�f3f3

�3f2�2f2�f2f2

�3f1�2f1�f1f1

�3f0�2f0�f0f0

�3f�1�2f�1�f�1f�1�1

�3f�2�2f�2�f�2f�2�2

�3f�3�2f�3�f�3f�3�3

�3fi�2fi�fifii

TABLE 2. Inverse Backward Differences

0

1

2 f2 � f1f2C1 � f1 � f2C2 � 2C1 � 2f1 � f2

f1 � f0f1C1 � f1C2 � C1 � f1

f0 � f�1f0C1C2

f�1 � f�2f�1C1 � f0C2 � C1�1

f�2 � f�3f�2C1 � f0 � f�1C2 � 2C1 � f0�2

�fifi��1fi��2fii

the difference relation holds. The constant is discussed further in the Imple-mentation section.

Powers of the summation are understood to be multiple sums, analogous tomultiple differences. The presence of the operator in the Gauss-Jackson for-mulas sometimes gives rise to the name second sum integration formula. The sec-ond sum has an integration constant , analogous to .

Differentiation

Differentiation is the operator that turns a function into its deriva-tive function. The differentiation operator D can be represented in terms of E bynoting that

(9)

for any real number p, using a Taylor series expansion. The exponential of an op-erator is to be interpreted as its ordinary Taylor expansion

(10)

with powers of the operator well-defined. Thus we may identify the two operators

(11)

or taking the logarithm

(12)

This expression can be written in terms of the backward difference operator (6)

(13)

The integration methods can now be derived by taking the inverse of the differen-tiation operator. The Adams method is derived first, and the other methods arederived from it.

Adams Method

Indefinite integration is the operator inverse of differentiation

(14)

so its action on an arbitrary function is to give the antiderivative

(15)

The single-integration operator is computed as the inverse of the differentia-tion operator (13)

(16)

With this operator Adams integration can be developed. The focus here is satelliteorbits, so the function f is replaced with acceleration r.

D�1 � �h

log�1 � ��

D�1

F�t� � D�1f�t�

� dt � D�1

hD � log E � �log�1 � ��

phD � p log E

Ep � ephD

et � 1 � t �t2

2!�

t3

3!� . . .

� ephDf�t0�

� f�t0� � phDf�t0� �� ph�2

2!D2f�t0� �

� ph�3

3!D3f�t0� � . . .

Epf�t0� � f�t0 � ph�

D � d�dt

C1C2

��2��1

C1

336 Berry and Healy

The definite integral is computed by using the displacement operator on theindefinite integral

(17)

For one step, p is 1. This integration corresponds to a predictor in a multi-stepnumerical integration, because the equation finds a value of r at a future time. If

, the predictor operator J can be written in terms of the backward differ-ence operator as

(18)

To integrate from to , shift backward using the displacement operator (6)

(19)

This operation corresponds to the corrector in a multi-step numerical integration,because it assumes that the velocity is already known and uses the correspon-ding acceleration to recalculate a new, better, velocity. It is on this expression thatwe focus initially because the operator has the simplest expansion.

The coefficients of the operators may be developed using a recursion relation[12]. For convenience in developing expansions, we define an operator L and itsTaylor expansion with coefficients as

(20)

We shall develop the expansion of L recursively through the standard Taylor seriesof the logarithm

(21)

which can be substituted in the definition of L

(22)

and then expanded in the unknowns

(23)

Expanding and grouping by powers of �

(24) � � 1

4c0 �

1

3c1 �

1

2c2 � c3�3 � . . . � 1

c0 � � 1

2c0 � c1� � � 1

3c0 �

1

2c1 � c2�2

�c0 � c1� � c2�2 � c3�

3 � . . .��1 �1

2� �

1

3�2 �

1

4�3 � . . . � 1

ci

L � ��

log�1 � ��

1

1 �12 � �

13 �2 �

14 �3 � . . .

� �i�0

ci�i

log�1 � �� 1

2�2 �

1

3�3 �

1

4�4 . . .

L � E�1J � ��

log�1 � �� c0 � c1� � c2�

2 � c3�3 � . . .

ci

nth

E�1J � �1 � ��J � ��

log�1 � ��

tntn � h

� ��

�1 � �� log�1 � ��

J �1

h�E � 1�D�1 �

1

h�1 � ��1 � 1�D�1 �

1

h

�

�1 � ��D�1

p � 1

�Ep � 1�D�1rn � �Ep � 1�rn � �tn�ph

tn

rdt


allows for the development of a recursion relation for the coefficients

(25a)

(25b)

(25c)

or, in general

(26)

for with . The first few coefficients are

(27)

With the coefficients known, integration from to may be written in termsof L as

(28)

A formula for the corrected value can be found by performing the integration andusing the coefficients

(29)

This formula is the Adams-Moulton corrector formula in difference form. Thisform is called the difference form because it uses the differences . An alternateform, called the ordinate form, in which the differences are rewritten in terms of thefunction values, is described in the Ordinate Forms section.

Returning to the predictor (18), the expansion of J can be computed by noting that

(30)

Each coefficient can be expressed as the sum of the coefficients ,

(31)

so they may be computed directly from (27)

(32)

i

�i

0

1

1

12

2

512

3

38

4

251720

5

95288

6

1908760480

7

525717280

8

10700173628800

�i � �i

k�0ck

k ick�i

J � �1 � ��1L � �1 � � � �2 � . . . � �c0 � c1� � c2�2 � . . . � � �

i�0�i�

i

�i

rn � rn�1 � h�1 �1

2� �

1

12�2 �

1

24�3 �

19

720�4 �

3

160�5 � . . .rn

ci

�tn

tn�hrdt � hE�1J rn � hLrn

tntn � hci

i

ci

0

1

1

�12

2

�1

12

3

�1

24

4

�19

720

5

�3

160

6

�863

60480

7

�275

24192

8

�33953

3628800

c0 � 1n � 1

cn � ��n�1

i�0

ci

n � 1 � i

0 �1

3c0 �

1

2c1 � c2

0 �1

2c0 � c1

1 � c0

338 Berry and Healy

and the integral is then

(33)

or

(34)

This formula is the Adams-Bashforth predictor formula in difference form, so calledbecause when applied to the velocity and acceleration at it finds the velocity at thenext mesh point .

In practice, given m known accelerations

(35)

the backward difference table (Table 1) for the derivatives can be computed through, a predicted value for is made based on (34), then that value is corrected

with (29).Although our purpose in deriving the nonsummed Adams-Bashforth and Adams-

Moulton coefficients is to use these results in the Gauss-Jackson method, theseformulae stand on their own as a widely-used numerical integration method offirst-order differential equations. For greater accuracy, summed methods may bepreferable, and we treat the summed form of the Adams method next.

Summed Adams Method

An alternative form of single integration, called the summed form, can be devel-oped using the summation operator . The Adams-Moulton corrector formula (29)may be written so both velocity terms are on the left side

(36)

The difference of the two velocity terms may be written using the backward differ-ence operator

(37)

The corrected value of the velocity can be found by applying to both sidesof (37), [5]

(38)

This expression is the summed Adams corrector formula. The summed form usesthe same coefficients as the Adams-Moulton corrector, but they have been shiftedby one place. The formula is called the summed form because of the presence ofthe summation operator.

rn � h��1 �1

2�

1

12� �

1

24�2 �

19

720�3 �

3

160�4 � . . .rn

��1

�rn � h�1 �1

2� �

1

12�2 �

1

24�3 �

19

720�4 �

3

160�5 � . . .rn

rn � rn�1 � h�1 �1

2� �

1

12�2 �

1

24�3 �

19

720�4 �

3

160�5 � . . .rn

��1

rn�1�m�1

rn�m�1, . . . , rn

tn�1

tn

rn�1 � rn � h�1 �1

2� �

5

12�2 �

3

8�3 �

251

720�4 �

95

288�5 � . . .rn

�tn�h

tn

rdt � hJ rn


A similar process can be performed on the Adams-Bashforth predictor for-mula (34)

(39)

giving the summed Adams predictor. The summed form gives the predicted or cor-rected value by integrating directly from epoch, while the nonsummed form inte-grates from point to point. According to Henrici ([13], p. 327) the summed form ispreferred for reducing the propagation of roundoff error.

Störmer-Cowell

To find a corrector formula for position, the corrector may be applied to theAdams-Moulton corrector (28)

(40)

Application of the corrector to both velocity terms, e.g., , gives anexpression for position

(41)

To find the coefficients in the expansion of

(42)

in terms of the Adams coefficients c (26), note that (42) is the square of (20)

(43)

so the first nine coefficients are

(44)

These are the Cowell corrector coefficients. The corrector formula is given by usingthese coefficients in (41)

(45)

This formula is the Cowell corrector formula [5]. It is a double-integration formula,since it computes the position given acceleration.

As with single integration, the predictor formula is found by shifting the step atwhich the operator is applied, in other words applying the displacement operator Eto both sides of (41)

(46)

Writing the displacement operator E in terms of �, (6), and applying to

(47)

EL2 � �1 � ��1L2 � �1 � � � �2 � . . . � �q 0 � q1� � q 2�2 � . . . � � �

i�1�i�

i

L2

rn�1 � 2rn � rn�1 � h2EL2rn

rn � 2rn�1 � rn�2 � h2�1 � � �1

12�2 �

1

240�4 �

1

240�5 � . . .rn

i

qi

0

1

1

�1

2

112

3

0

4

�1

240

5

�1

240

6

�221

60480

7

�19

6048

8

�9829

3628800

qi � �i

k�0ckci�k

L2 � q0 � q1� � q2�2 � . . .

L2

rn � 2rn�1 � rn�2 � h2L2rn

hLrn � rn � rn�1

hL�tn

tn�hrdt � hL�rn � rn�1� � h2L2rn

rn�1 � h��1 �1

2�

5

12� �

3

8�2 �

251

720�3 �

95

288�4 � . . .rn

340 Berry and Healy

shows that the predictor coefficients can be written as a sum of the correctorcoefficients, just as with single integration. The predictor coefficients can be com-puted from (44)

(48)

Using these coefficients in (46)

(49)

gives the Störmer predictor formula [5].

Gauss-Jackson

The Gauss-Jackson method is the summed form of the Störmer-Cowell method.According to Herrick ([10], p. 12), the method was named Gauss-Jackson becauseof its appearance in a 1924 paper by Jackson [1]. The Gauss-Jackson method is alsoreferred to as the second-sum method. Sometimes, the name is associated with acentral difference derivation, because that is how Jackson showed it. However, aswe mentioned, the derivation by any of the difference operators are equivalent;what is unique about Gauss-Jackson is the presence of the second sum.

The Gauss-Jackson corrector can be derived from the Cowell corrector by not-ing that the position terms in (45) may be combined using a second backwarddifference

(50)

Applying the second sum to both sides gives an equation for position [5]

(51)

Note that , so the first two terms can be simplified to

(52)

This expression is the Gauss-Jackson corrector formula.A similar process on the Störmer predictor (49) gives a summed predictor

(53)

which is the Gauss-Jackson predictor formula. Note that for the predictor the sec-ond sum operators acts on , while for the corrector it acts on .

Startup Formulas

In order to calculate the differences , the difference table (Table 1) must be cal-culated, which depends on having values of acceleration at the backpoints. Tocalculate the difference points must be known. The initial value problemN � 1N th

�i

rn�1rn

rn�1 � h2��2rn � � 1

12�

1

12� �

19

240�2 �

3

40�3 � . . .rn

rn � h2��2rn�1 � � 1

12�

1

240�2 �

1

240�3 �

221

60480�4 � . . .rn

�1 � ��rn � E�1rn � rn�1

rn � h2��2�1 � � �1

12�2 �

1

240�4 �

1

240�5 �

221

60480�6 � . . .rn

��2

�2rn � h2�1 � � �1

12�2 �

1

240�4 �

1

240�5 �

221

60480�6 � . . .rn

rn�1 � 2rn � rn�1 � h2�1 �1

12�2 �

1

12�3 �

19

240�4 �

3

40�5 � . . .rn

i

�i

0

1

1

0

2

112

3

112

4

19240

5

340

6

86312096

7

2754032

8

33953518400

�i


only gives the initial conditions at epoch, so a startup procedure is required tofind position and velocity, and then acceleration, at the other backpoints. Onepossible method is to use a single-step integrator, such as Runge-Kutta, to findthe values at the initial set of backpoints. Alternatively, one may use the two-bodymotion solution, which may be computed analytically. In the latter case particu-larly, we use an iterative method that takes an initial guess of the backpoints andrefines them with corrector formulas. These corrector formulas, which correct avalue using not only previous points, but also future known points, may be calledmid-corrector formulas.

Where two-body motion is close to the actual motion modeled, the initial set ofguess values can be found by using the analytic two-body solution. In other cases,a good analytic approximation of the full motion would be necessary. For a methodthat uses the difference, total points are needed, so N points in additionto epoch must be found. To reduce the error that comes from the two-body solution,instead of propagating N steps forward, N�2 steps are taken both forward and back-ward. These points are numbered using a symmetric index, from �N�2 to N�2,with epoch numbered 0. This system requires that N be even, as it is for the inte-grators in this study. If N were odd an additional point would be needed either be-fore or after epoch.

After the set of guess values is found, each value other than epoch is correctedwith a mid-corrector formula, except for the last value, which uses the correctorformula. The formulas are numbered using the letter j as an index with symmetricindex numbers. So the mid-correctors are numbered to ,and the corrector is numbered . Similarly the predictor can be consideredthe formula.

The mid-correctors are found from the corrector by applying the inverse dis-placement operator repeatedly. Using the corrector operator L as anexample, the relation of operators to coefficient sets can be seen on the diagram foran eighth-order method ( ):

mid-correctors

corrector L

predictor

The coefficients for each mid-corrector may be computed from the next one by application of the operator . For an arbitrary power series in � with coeffi-cients d, multiply the series

(54)

In other words, the coefficients for a particular value of j are just the differences ofcoefficients of adjacent values of powers of � for the next higher value of j. Asshown in the derivation of the Adams and Störmer-Cowell predictors from their

�d0 � d1� � d2�2 � . . . � �1 � �� d0 � �d1 � d0�� d2 � d1��2 � . . .

1 � �

L

1 � �j � 5

j � 4

�1 � ��Lj � 3

bapply �1 � ��1�1 � ��2Lj � 2

aapply 1 � �

�1 � ��7Lj � �3

�1 � ��8Lj � �4

N � 8

E�1 � 1 � �

j � N�2 � 1j � N�2

j � N�2 � 1j � �N�2

N � 1N th

342 Berry and Healy

correctors, the coefficients are also the sums of all the coefficients of the next lowerj, through the particular power of �

(55)

All the coefficients, mid-correctors, corrector, and predictor, taken together maybe viewed as an array, with the rows indexed by j, and the columns by the expo-nent i of �. For an -order integrator, there are terms in the series expan-sion of the operator for each j, and there are values of j; therefore, the fullarray of coefficients has elements. For example, the eighth-orderintegrator has coefficients.

The summed Adams corrector formula (38) can be written using the coefficientarray

(56)

From this, the last mid-corrector formula is obtained from the coeffi-cient differences (54) of the corrector

(57)

The general mid-corrector formulas can be written as

(58)

where . The coefficients for each value of j are obtained bytaking differences of the coefficients for . The mid-corrector coefficients arethus computed recursively

(59)

for .

The predictor formula (39) may be rewritten in terms of the coefficient array

(60)

The eighth-order coefficients are presented in Table 3.�ji

rn�1 � h��1rn � �Ni�0

�N2�1, i�

irn�

�N2 j

N2 � 1

�ji � ��j�1, i � �j�1, i�1

�j�1, i

if i � 0

if i � 0

j � 1�ji�

N2 n

N2 � 1

rn � h��1rn � �Ni�0

�ni�irN

2

� h��1rN2 �1 � ��

1

2�

5

12� �

1

24�2 �

11

720�3 �

11

1440�4 � . . .rN

2 � E�1h��1rN

2� ��

1

2�

1

12� �

1

24�2 �

19

720�3 �

3

160�4 � . . .rN

2 rN

2�1 � E�1rN2

� j � N2 � 1�

rn � h��1rn � �Ni�0

�N2

, i�irn

�

9 � 10 � 90�N � 1� �N � 2�

N � 2N � 1N th

� �k��k

i�0mi�k

� �m0 � m1� � m2�2 � . . . � �1 � � � �2 � �3 � . . . �

�m0 � m1� � m2�2 � . . . �

1

1 � �


Analogously, the Gauss-Jackson corrector (52) can be written using a coefficientarray

(61)

The mid-correctors are then written

(62)

where . The coefficients of the mid-correctors are computed asthe difference (54) of the corrector coefficients; the entire set of coefficients arethus computed recursively

(63)

for . Finally the predictor is included in the array

(64)

The eighth-order coefficients are presented in Table 4.Note that the sum introduced by the term in the mid-correctors or corrector

is accumulated through the point before the point being corrected; this is given bythe term in (52).rn�1

��2�ji

rn�1 � h2��2rn � �Ni�0

�N2�1, i�

irn�

N2 j

N2 � 1

�ji � ��j�1, i � �j�1, i�1

�j�1, i

if i � 0

if i � 0

�N2 n

N2 � 1

rn � h2��2rn�1 � �Ni�0

�ni�irN

2

rn � h2��2rn�1 � �Ni�0

�N2 , i�

irn�

344 Berry and Healy

TABLE 3. Eighth-Order Summed-Adams Difference Coefficients, �ji

i

0 1 2 3 4 5 6 7 8

2571389600

10700173628800

525717280

1908760480

95288

251720

38

512

125

�8183

1036800�33953

3628800�275

24192�863

60480�3

160�19

720�1

24�112�

124

425290304

72973628800

134480

27160480

111440

11720

124

512�

123

�7

12800�3233

3628800�191

120960�191

60480�11

1440�19

720�38

1112�

122

24977257600

24973628800

191120960

27160480

3160

251720�

3124

1712�

121

�2497

7257600�3233

3628800�13

4480�863

60480�95

2881181720�

6524

2312�

120

712800

72973628800

27524192

1908760480�

28371440

3131720�

378

2912�

12�1

�425

290304�33953

3628800�5257

1728013824160480�

1011160

6461720�

16924

3512�

12�2

81831036800

10700173628800�

116034480

52039960480�

220211440

11531720�

23924

4112�

12�3

�2571389600

104684473628800�

1354079120960

144528160480�

450831440

18701720�

1078

4712�

12�4

j

Ordinate Forms

The formulas given above all involve multiple differences and sums. In order tocompute these formulas to a particular order, all the elements of the appropriatedifference tables need to be computed. Calculating the differences can be time-consuming and is unnecessary because the formulas can be re-expressed in termsof the function values themselves.

As shown above in equation (3), powers of the difference operators can be re-duced to linear combinations of the original function values. The backward differ-ence operator raised to any power can be expressed as a linear combination of thefunction value at the mesh points using binomial coefficients

(65)

for . A sum over arbitrary coefficients of the backward difference operatorcan thus be written in terms of the values of the function

(66)

� �Nm�0

zNmrn�m

� �Nm�0

��1�mrn�m �Ni�m

�i� i

m �Ni�0

�i�irn � �N

i�0�i �i

m�0� i

m��1�mrn�m

�ii � 0

�irn � �i

m�0� i

m��1�mrn�m


TABLE 4. Eighth-Order Gauss-Jackson Difference Coefficients, �ji

i

0 1 2 3 4 5 6 7 8

325043353222400

8183129600

33953518400

2754032

86312096

340

19240

112

1125

�330157

159667200�407

172800�9829

3628800�19

6048�221

60480�1

240�1

24001124

45911159667200

6411814400

15713628800

3160480

31604800�

1240�

112

1123

�3499

53222400�289

3628800�289

362880003160480

1240

19240�

16

1122

317228096000�

2893628800�

3160480�

22160480�

340

59240�

14

1121

31722809600

2893628800

15713628800

196048

86312096�

77240

119240�

13

1120

�3499

53222400�641

1814400�9829

3628800�2754032

2371960480�

4960

199240�

512

112�1

45911159667200

407172800

33953518400�

696115120

7311160480�

7948

299240�

12

112�2

�330157

159667200�8183

12960019083113628800�

2019112096

17265160480�

347120

419240�

712

112�3

325043353222400�

427487725760

79656113628800�

4560110080

34753960480�

37180

559240�

23

112�4

j

with the ordinate coefficients defined as

(67)

Notice that these coefficients, unlike the difference coefficients , depend on N, theorder of the integration method. Thus a variation of order to control errors while in-tegrating is feasible in difference form, but is more difficult in ordinate form. Herethe index m numbers the backpoints so the current point is and each previ-ous point has a higher positive value of m.

As an example, consider the summed Adams corrector (38). For a fourth-ordermethod, , the difference coefficients and the ordinate coefficients are

(68)

An alternative formulation includes the term with the sum rather than withthe coefficients. The example summed Adams-Moulton corrector (38) is now written

(69)

With this formulation the coefficients are different from (68)

(70)

All the single-integration corrector and mid-corrector coefficients are computedthis way; however, the term is included in the coefficients for the predictor. Theeighth-order summed Adams coefficients in ordinate form are shown in Table 5,and the Gauss-Jackson coefficients are shown in Table 6. For Gauss-Jackson, the

term , is included with the coefficients and not the sum . Although thisalternate formulation is not necessary, it makes the programming easier, as describedin the next section. Both the Gauss-Jackson and the summed Adams coefficientsmay be compared with [6], Table III (pages 25–26) and [12] Appendix C.3. Ordi-nate forms for the Adams, summed Adams, Störmer-Cowell, and Gauss-Jacksonpredictors and correctors for orders between 1 and 14 may also be found in [5].

In this section, the are generic difference coefficients such as and , and thez are the corresponding ordinate coefficients, which have an extra subscript to in-dicate the order. The actual eighth-order ordinate coefficients a or b do not have anorder subscript even though they are dependent on the order; the two subscripts arethe point of integration j and the point of reference k.

��

��2112 rn�0

ajk

bjk

�0

m

�i

z 4m

0

0

�49

288

1

�112

77240

2

�124

�730

3

�19720

73720

4

�3

160

�3

160

m � 0

rn � h��1rn �1

2rn � ��

1

12� �

1

24�2 �

19

720�3 �

3

160�4 � . . .rn

�12 rn�0

m

�i

z 4m

0

�12

�193288

1

�112

77240

2

�124

�730

3

�19720

73720

4

�3

160

�3

160

z 4m�iN � 4

m � 0

�

zNm � ��1�m �Ni�m

�i� i

m

346 Berry and Healy


TA

BL

E 5

.E

ight

h-O

rder

Sum

med

-Ada

ms

Coe

ffic

ient

s in

Ord

inat

e F

orm

,bjk

k

j0

12

34

3288

521

1036

800

�40

9877

7136

2880

010

2198

4140

3200

�13

5352

319

3628

800

1672

8745

36�

9839

609

4032

0053

9323

351

8400

�94

0102

936

2880

025

713

8960

05

�19

087

8960

042

7487

7257

60�

3498

217

3628

800

5003

2740

3200

�64

6756

7026

1616

136

2880

0�

2401

980

640

2630

7736

2880

0�

8183

1036

800

4

�81

8310

3680

0�

5725

140

3200

1106

377

3628

800

�21

8483

7257

6069 28

0�

5301

7736

2880

021

0359

3628

800

�55

3340

3200

425

2903

043

425

2903

04�

7645

336

2880

0�

5143

5760

066

0127

3628

800

�66

156

7049

9780

640

�83

927

3628

800

1910

936

2880

0�

712

800

2

�7

1280

023

173

3628

800

�29

579

7257

60�

2497

5760

025

6322

680

�17

2993

3628

800

6463

4032

00�

2497

7257

6024

9772

5760

01

2497

7257

600

�14

6940

3200

6811

936

2880

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2527

6936

2880

00

2527

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0�

6811

936

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6940

3200

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9772

5760

00

�24

9772

5760

024

9772

5760

�64

6340

3200

1729

9336

2880

0�

2563

2268

024

9757

600

2957

972

5760

�23

173

3628

800

712

800

�1

712

800

�19

109

3628

800

8392

736

2880

0�

4997

8064

066

156

70�

6601

2736

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4357

600

7645

336

2880

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425

2903

04�

2

�42

529

0304

5533

4032

00�

2103

5936

2880

053

0177

3628

800

�69 28

021

8483

7257

60�

1106

377

3628

800

5725

140

3200

8183

1036

800

�3

8183

1036

800

�26

3077

3628

800

2401

980

640

�26

1616

136

2880

064

6756

70�

5003

2740

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217

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800

�42

7487

7257

6019

087

8960

0�

4

�1

�2

�3

�4

348 Berry and Healy

TA

BL

E 6

.E

ight

h-O

rder

Gau

ss-J

acks

on C

oeff

icie

nts

in O

rdin

ate

For

m,a

jk

k

j0

12

34

1037

9843

915

9667

200

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1158

4399

7920

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0713

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2640

0�

1593

1445

319

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0025

1629

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4�

8660

609

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6322

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8119

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5043

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2224

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3250

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5322

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8701

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3991

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4026

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1330

5600

�91

7039

3193

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7370

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2577

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4331

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5188

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Implementation

To find both position and velocity given a function for acceleration, either a single-integration method must be used once for velocity and the method used again for position, or a single-integration method must be used along witha double-integration method. Assuming that summed and nonsummed forms arenot used together, the four integrators derived above offer four implementations:two Adams, two summed Adams, Adams and Störmer-Cowell, or summedAdams and Gauss-Jackson integration. For convenience, Gauss-Jackson integra-tion is understood to mean Gauss-Jackson and summed Adams integration, andStörmer-Cowell integration is understood to mean Störmer-Cowell and Adamsintegration. Because Gauss-Jackson integration involves sum terms, its imple-mentation is somewhat complicated, and so is described in detail in this section.

Overview

In order to use the predictor-corrector formulas to integrate forward in time withan order method, points must already be known. To preserve the orderof the method, these points must have also been found using an ordermethod. Since initially only one point, epoch, is known, a startup procedure is nec-essary. If initial estimated points are given, the mid-corrector and correctorformulas can be used to refine them. By applying the mid-correctors iterativelyuntil the points converge, the resulting points are accurate through order.

For the eighth-order methods used in this study, a Taylor expansion of the two-body solution to fifth order ( f and g series, [14] equation (4–68)) is used for theinitial estimate, though a single-step integrator, such as Runge-Kutta could be used.The analytic solution generates eight positions and velocities in addition to epoch:four before and four after epoch. The accelerations at all nine points are then foundfrom the positions and velocities with full perturbations, and corrector and mid-corrector formulas are applied to the eight non-epoch accelerations, which gener-ates corrected positions and velocities. These corrected positions and velocities areused to find corrected accelerations, and the cycle repeats, until the accelerationsconverge, i.e., on two successive iterations, the absolute difference is less than aspecified tolerance. This process is denoted SECECE...CE, where “S” is startupestimate, “E” is evaluate, and “C” is correct. This process is done for the eight ini-tial points other than epoch.

Once the startup points have been corrected to satisfactory accuracy, the regularpredictor-corrector process can start. First the predictor equation is used to find theposition and velocity of the first point after startup, and the force model is evaluatedto find the acceleration of that point. This predicted acceleration is then used in thecorrector formula to find a corrected position and velocity. The corrected positionand velocity may be compared to the predicted values, and if they do not match tosome tolerance, another acceleration is evaluated, which is then used in the correc-tor. Note that the procedure is different from startup, where convergence of acceler-ation is checked. This cycle would then be repeated until the position and velocityconverge, or until some specified maximum number of iterations is reached. Thisprocess gives a implementation, with a maximum value of n specified.

Single Integration

The first step in startup of single integration is to use the mid-corrector formu-las to correct the eight estimated velocities around epoch. In difference form, these

P�EC�n

N th

N � 1

N thN � 1N � 1N th


formulas are (58); using the ordinate coefficients b, the formulas are

(71)

with . Note the term is excluded from the coefficient sum in an-ticipation of its inclusion in the acceleration term, as in (69). The integration con-stant (see Tables and Operators) may be determined by making sure that theinitial conditions ( )

(72)

are satisfied; solving for

(73)

The acceleration term may be included with the sum to make software imple-mentation easier. Define the term , which replaces the first two terms in (71), and define a new integration constant Then (71) may be rewritten

(74)

where n ranges from �4 to 4, with

(75)

Table 7 shows the relationship between the summed accelerations and near epoch.For a given value of n, notice the symmetry of the formulas in the last column;changing the sign of n merely changes the sign of the term added to . This for-mulation makes programming simpler and is the reason that the term ismoved from the coefficients to the sum. The are computed successively in orderto compute the startup velocities. While the formulas are correct for , no cor-rection is performed on epoch.

n � 0sn

�r0�2C�1�

sn

sn �

sn�1 �rn�1 � rn

2

C�1�

sn�1 �rn�1 � rn

2

if n � 0

if n � 0

if n � 0

rn � h�sn � �4k��4

bnkrkC�1� � C1 �

r0

2 .sn � ��1rn �

12 rn

C1 �r0

h� �4

k��4b0krk �

r0

2

C1 � ��1r0

r0 � h��1r0 �r0

2� �4

k��4b0krk

n � 0C1

�rn2�4 n 4

rn � h��1rn �rn

2� �4

k��4bnkrk

350 Berry and Healy

TABLE 7. Relationship Between Summation Term and sn

0

1

2 C�1 �12 r0 � r1 �

12 r2C1 � r1 � r2r2

C�1 �12 r0 �

12 r1C1 � r1r1

C�1C1r0

C�1 �12 r0 �

12 r�1C1 � r0r�1�1

C�1 �12 r0 � r�1 �

12 r�2C1 � r0 � r�1r�2�2

sn � ��1rn �12 rn��1rnrnn

Once the startup points have been corrected, integration may proceed forwardwith application of the predictor, followed by one or more applications of the cor-rector. The predictor finds the velocity from the coefficients in row 5 of theb array, and the nine most recent accelerations . The formula also includesa term which is the sum of all the accelerations since epoch and

(76)

with . This expression differs from the difference form given in (60). Forprogramming, the predictor can be written using

(77)

The Adams-Moulton corrector formula in difference form (69), for any pointafter startup, can be rewritten using the eighth-order ordinate coefficients b.

These coefficients are applied to the eight previous accelerations and the currentpredicted acceleration

(78)

with . Note that (78) is the same as (71) for . The corrector can be writ-ten using

(79)

Depending on the implementation, this equation may be applied through severalevaluate and correct (EC) iterations, but only the last acceleration, , changes,so the summation for the first eight need only be done once, then saved throughthe iterations.

Note that when applying the predictor, the first formula in (75) uses one halfof the corrected acceleration and one half of the predicted acceleration in thesum. If the acceleration term is not shifted, then the entire acceleration comesfrom the predicted value. Therefore, the shift has a presumed advantage of usinghalf the corrected acceleration and half the predicted, rather than all predictedacceleration. A possible further improvement would be to use instead of

in (78) with b redefined appropriately. Then this summation would haveonly corrected values.

Double Integration

The mid-correctors, predictor, and corrector for the second integration ordinateform are respectively

(80)

(81)

(82)n � 4 rn � h2��2rn�1 � �4k��4

a4krn�k�4n � 4 rn�1 � h2��2rn � �4

k��4a5krn�k�4

�4 n 4 rn � h2��2rn�1 � �4k��4

ankrk

��1rn

��1rn�1

k � 4

rn � h�sn � �4k��4

b4krn�k�4sn

n � 4n � 4

rn � h��1rn �rn

2� �4

k��4b4krn�k�4

n � 4

rn�1 � h�sn �rn

2� �4

k��4b5krn�k�4

sn

n � 4

rn�1 � h��1rn � �4k��4

b5krn�k�4C1��1

rn�8 . . . rn

rn�1


corresponding to the difference form (62), (64), (61). To find the integration con-stant for double integration, , use in (80)

(83)

Table 2 shows that the second sum on the point immediately before epoch is the dif-ference of the two constants of integration

(84)

which than allows us to solve for , because is known (73)

(85)

For software implementation, a term is defined recursively

(86)

The mid-correctors, predictor, and corrector can now be written in terms of

(87)

(88)

(89)

As with first integration, the summation in the corrector may be split, because thefirst eight accelerations do not change during the EC iteration.

Procedure

A step-by-step procedure for the operation of the integrator can now be written:

Startup11. Use f and g series to calculate eight positions and velocities surrounding epoch

labelled with epoch.12. Evaluate nine accelerations from these positions and velocities, and those of

epoch labelled 13. While the accelerations have not converged:

(a) Calculate (75) and (86).(b) For each point , :

i. Calculate (75) and (86).

ii. Calculate using Table 5.

iii. Calculate using Table 6.�4k��4

ankrk

�4k��4

bnkrk

Snsn

n � 0n � �4 . . . 4S0s0

r�4 . . . r4 .

r0r�4 . . . r4

n � 4 rn � h2�Sn � �4k��4

a4krn�k�4n � 4 rn�1 � h2�Sn�1 � �4

k��4a5krn�k�4

�4 n 4 rn � h2�Sn � �4k��4

ankrkSn

Sn �

Sn�1 � sn�1 �rn�1

2

C2 � C1

Sn�1 � sn�1 �rn�1

2

if n � 0

if n � 0

if n � 0

Sn � ��2rn�1

C2 �r0

h2 � �4k��4

a0krk � C1

C1C2

��2r�1 � C2 � C1

r0 � h2��2r�1 � �4k��4

a0krkn � 0C2

352 Berry and Healy

iv. Calculate (74) and (87).v. Evaluate the updated acceleration using the appropriate force model.

(c) Test convergence of the accelerations.

Predict14. Calculate (86).

15. Calculate .

16. Calculate .

17. Calculate (77) and (88).

Evaluate—Correct18. Evaluate the acceleration .19. Increment n.10. While and have not converged and the maximum number of corrector it-

erations is not exceeded:(a) Calculate (75).(b) If first iteration:

i. Calculate .

ii. Calculate .

(c) Calculate and .(d) Sum the results of 10(a), 10(b), and 10(c) to obtain (79) and (89).(e) Test convergence of and ; if not converged, and this is not the last

allowable corrector iteration, evaluate .11. Go to Step 4 unless this is the final time desired.

Similar procedures are used for implementing two Adams, two summed Adams,or Adams with Störmer-Cowell.

Effects of Step Size and Order on Accuracy and Stability

In general, the accuracy of a numerical integrator improves as the step size isreduced and the order of the method is increased. One exception to this rule isthat round-off error can occur if the step size becomes too small. Another ex-ception occurs when the integrator becomes unstable. Increasing the order maycause instability because higher-order methods are susceptible to instability atlarge step sizes [15] (pp. 139, 146).

To assess the effect of step size and order on accuracy, we compare ephemerisgenerated using step sizes of 30, 60, 120, and 240 seconds, with 6th, 8th, 10th, 12th,and 14th order Gauss-Jackson integrators. The integrators follow the proceduregiven in the Procedure section, modified for the order of the integrator. The testscompare ephemeris generated at one minute intervals over 72 hours. For step sizesgreater than one minute, a fifth-order Hermitian interpolator ([16], p. 28) is used tofind the intermediate points. The perturbation forces include WGS-84geopotential [17], Jacchia 70 [18] drag, and lunar and solar forces. Note that all ofthese perturbations are continuous forces. To simplify this study, discontinuousforces such as solar radiation pressure are not considered. The two test cases usedare the International Space Station (ISS, catalog number 25544), with a period

24 � 24

rn

rnrn

rnrn

a4krnb4krn

�3k��4

a4krn�k�4

�3k��4

b4krn�k�4

sn

rnrn

rn�1

rn�1rn�1

�4k��4

a5krn�k�4

�4k��4

b5krn�k�4

Sn�1

rn

rnrn


of 92.05 min. and eccentricity of 0.001, and CRRES (20712), with a period of607.28 min. and eccentricity of 0.716.

In the tests, a metric for integration accuracy is defined using an error ratio defined in terms of the RMS error of the integration [19]. First define positionerrors as

(90)

The RMS position error can be calculated

(91)

The RMS position error is normalized by the apogee distance and the number oforbits to find the position error ratio

(92)

The error ratios for ISS are given in Table 8, and the error ratios for CRRES aregiven in Table 9. In the comparison the ephemeris generated by the 14th ordermethod with 30 second step size is considered to be the reference. The tables showthat step size affects accuracy more than the order of the method does. The trend issimilar for both the circular and the eccentric objects. In Table 8 the asterisk de-notes that the integration has become unstable. It is obvious when instability hasoccurred because the eccentricity of the integrated ephemeris starts to grow to thepoint of the orbit becoming hyperbolic. The instability at a large step size for highorders shows that if computation time is a higher priority than accuracy, a lowerorder method is preferable, because a larger step size can be used without the threatof instability.

Herrick [10] claims that the corrector is not usually required in Gauss-Jacksonintegration. This saves computation time, because only one evaluation is required(PE) per step. A predictor-only form of Gauss-Jackson can be created by skip-ping steps 10(b)–10(e) of the procedure above. Tables 10 and 11 list error ratios forpredictor-only results for ISS and CRRES, respectively. Once again, the 14th order,30 second step size results, with the corrector, are considered to be the reference.Comparing Table 10 to Table 8, we see that for the circular satellite the resultswith and without the corrector are similar for smaller step sizes and lower orders.

�r ��rRMS

rANorbits

�rRMS � � 1

N�Ni�1

��ri�2

�r � �rcomputed � rreference�

354 Berry and Healy

TABLE 8. Error Ratios for ISS (e � 0.001)

Step Size

Order

(sec) 6 8 10 12 14

—

* *1.2 � 10�41.3 � 10�41.1 � 10�4240

1.1 � 10�78.8 � 10�81.1 � 10�71.1 � 10�79.7 � 10�8120

9.0 � 10�109.7 � 10�101.1 � 10�91.5 � 10�92.4 � 10�960

1.5 � 10�133.8 � 10�131.5 � 10�121.0 � 10�1130

However, when the corrector is not used instability occurs at lower orders andsmaller step sizes than when the corrector is used. For the eccentric satellite, shownin Table 11, there is a somewhat more noticeable loss in accuracy when the correc-tor is not applied. However, the integration of the eccentric satellite remains stablein more cases than the circular case.

In general, when the corrector is applied at least two evaluations are performed(PECE). This means that a predictor-only implementation (PE) has roughly half thecomputation time as a predictor-corrector implementation, because evaluations ac-count for most of the computation time when using a realistic force model. Compu-tation time is also related to the step size; when the step size is doubled thecomputation time is halved. Comparing Table 8 to Table 10, and Table 9 to Table 11,we see that 30 second predictor-only results are more accurate than 60 secondpredictor-corrector results, though these have roughly the same computation time.This justifies Herrick’s claim that the corrector is not necessary; however, if it is notused, care must be taken to avoid instability. The inclusion of the corrector (PEC)with no final evaluation gives an improvement in accuracy over the predictor-onlymethod, with little cost in computation time, since there are no extra evaluations.However this method has the same stability problems as the predictor-only method.

An alternate method performs a second evaluation in which only the two-bodyforce is re-evaluated; the perturbation force from the first evaluation is re-used. Be-cause the two-body evaluation is a simple calculation, this method has essentiallythe same run-time as performing only one evaluation per step. However, in most


TABLE 9. Error Ratios for CRRES (e � 0.716)

Step Size

Order

(sec) 6 8 10 12 14

—

9.0 � 10�52.0 � 10�44.0 � 10�41.9 � 10�59.3 � 10�4240

8.7 � 10�82.6 � 10�82.2 � 10�77.6 � 10�71.1 � 10�7120

1.6 � 10�101.6 � 10�101.2 � 10�103.9 � 10�115.3 � 10�960

1.1 � 10�149.4 � 10�142.5 � 10�131.4 � 10�1130

TABLE 10. Error Ratios for ISS, Predictor only (e � 0.001)

Step Size

Order

(sec) 6 8 10 12 14

*

* *

* * *

* * * *4.9 � 10�4240

1.2 � 10�71.3 � 10�6120

9.2 � 10�101.6 � 10�94.7 � 10�960

1.5 � 10�123.5 � 10�131.9 � 10�121.4 � 10�1130

cases this method has the same stability advantage as performing two full evalua-tions, though the partial evaluation is not as accurate as a full evaluation. Resultsusing a partial evaluation are available in [9].

Summary

This paper derives the Gauss-Jackson integration method widely used in specialperturbations software for space surveillance. Both single and double integrationare derived from difference tables. The ordinate form simplifies the calculations be-cause it avoids the need to calculate tables of differences; however, it requires newsets of coefficients. We show how the theory is actually implemented in the soft-ware. Finally, there is a discussion of stability and accuracy which shows that stepsize has more of an effect on accuracy than the order of the method does, and thathigher orders are subject to instability at large step sizes.

Coefficients may be computed to any order with code that is available at thepermanent link http://hdl.handle.net/1903/2202.

Acknowledgments

We thank Keith Atkins, Shannon Coffey, Felix Hoots, and Alan Segerman for helpful com-ments on the original paper. Fred Lutze provided comments on this version of the paper, and wededicate it to his memory.

References[1] JACKSON, J. “Note on the Numerical Integration of ,” Monthly Notes,

Volume 84, Royal Astronomy Society, 1924, pp. 602–606.[2] KOSKELA, P. E. “Astrodynamic Analysis for the Advanced Orbit/Ephemeris Subsystem,”

Technical Report U-4180, Philco-Ford Corporation, Aeronutronic Division, Newport Beach,California, September 1967.

[3] NEAL, H. L., COFFEY, S. L., and KNOWLES, S. “Maintaining the Space Object Catalog withSpecial Perturbations,” in F. Hoots, B. Kaufman, P. Cefola, and D. Spencer, editors, Astro-dynamics 1997 Part II, Volume 97 of Advances in the Astronautical Sciences, pp. 1349–1360.

[4] CASALI, S. “Conjunction Analysis Approach for the Astrodynamics Support Workstation(ASW),” presented at the Core Technologies for Space Systems Conference, Colorado Springs,Colorado, November 19–21, 2002.

[5] MAURY, J. L. and SEGAL, G. P. “Cowell Type Numerical Integration as Applied to SatelliteOrbit Computation,” Technical Report X-553-69-46, NASA, 1969, NTIS #N6926703.

d2x�dt2 � f �x, t�

356 Berry and Healy

TABLE 11. Error Ratios for CRRES, Predictor only (e � 0.716)

Step Size

Order

(sec) 6 8 10 12 14

*

* *1.4 � 10�31.0 � 10�21.3 � 10�2240

4.3 � 10�62.0 � 10�62.3 � 10�54.2 � 10�5120

5.8 � 10�71.4 � 10�91.9 � 10�92.0 � 10�98.8 � 10�860

5.4 � 10�132.1 � 10�131.8 � 10�136.6 � 10�123.5 � 10�1030

[6] NORAD “Mathematical Foundation for SCC Astrodynamic Theory,” Technical Report TPSCC 008, Headquarters North American Aerospace Defense Command, 1982. Obtainablefrom Defense Technical Information Center, Cameron Station, Alexandria, VA 22304-6145,as AD #B081394.

[7] WOODBURN, J. “Mitigation of the Effects of Eclipse Boundary Crossings on the NumericalIntegration of Orbit Trajectories Using an Encke Type Correction Algorithm,” presented aspaper AAS 01-223 at the AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA,February 11–14, 2001.

[8] BERRY, M. and HEALY, L. “The Generalized Sundman Transformation for Propagation ofHigh-Eccentricity Elliptical Orbits,” Advances in Astronautics, San Diego, CA, February 2002.

[9] BERRY, M. M. A Variable-Step Double-Integration Multi-Step Integrator, Ph.D. thesis,Virginia Polytechnic Institute and State University, Blacksburg, Virginia, April 2004. Availableat permanent link http://scholar.lib.vt.edu/theses/available/etd-04282004-071227/.

[10] HERRICK, S. Astrodynamics: Orbit Correction, Perturbation Theory, Integration, Volume 2,Van Nostrand Reinhold Company, New York, 1972.

[11] FRANKENA, J. F. “Störmer-Cowell: Straight, Summed and Split. An Overview,” Journal ofComputational and Applied Mathematics, 62: 129–154, 1995.

[12] WALLNER, R. N. “Earth Gravitational Error Budget Initial Report,” Technical report, KamanSciences Corporation, 1500 Garden of the Gods Road, Colorado Springs, CO 80907, November1994, System Memorandum KSPACE 95-18.

[13] HENRICI, P. Discrete Variable Methods in Ordinary Differential Equations, John Wiley andSons, New York, 1962.

[14] VALLADO, D. A. Fundamentals of Astrodynamics and Applications, Volume 12 of The SpaceTechnology Library, Microcosm Press and Kluwer Academic Publishers, El Segundo, Califor-nia and Dordrecht, The Netherlands, second edition, 2001.

[15] MONTENBRUCK, O. and GILL, E. Satellite Orbits, Springer, New York, 2000.[16] DAVIS, P. J. Interpolation and Approximation, Dover, New York, 1975.[17] National Imaging and Mapping Agency “Department of Defense World Geodetic System 1984,

its Definition and Relationships with Local Geodetic Systems,” Technical report, National Imag-ing and Mapping Agency, 1997. Available at permanent link http://earth-info.nga.mil/GandG/tr8350_2.html. This publication covers changes made to the standard since 1984 which are notincorporated into this software.

[18] JACCHIA, L. G. “New Static Models of the Thermosphere and Exosphere with Empirical Tem-perature Models,” Technical Report 313, Smithsonian Astrophysical Observatory, 1970.

[19] MERSON, R. H. “Numerical Integration of the Differential Equations of Celestial Mechanics,”Technical Report TR 74184, Royal Aircraft Establishment, Farnborough, Hants, UK, January1975, Defense Technical Information Center number AD B004645.


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Implementation of Gauss-Jackson Integration for Orbit