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fco -v! AFUOD TN 60 34

SPACE TRACK

fu, (o^O

SPACE VEHICLE EPHEMERIS AND

IFFERENTIAL CORRECTION PROGRAM

UNIFIED THEORY

C9

ASTRODYNAMICS DEPARTMENT,

AERONUTRONIC

A DIVISION OF FORD MOTOR COMPANY FORD ROAD, NEWPORT BEACH, CAL,

CONTRACT AF19(604)-5885

14 JUNE 1960

'^^

496L SPO

AIR FORCE COMMAND AND CONTROL DEVELOPMENT DIVISION

AIR RESEARCH AND DEVELOPMENT COMMAND

UNITED STATES AIR FORCE Ä S T I A

JAN 9 1*1

DPOB

AFCCDO-TN-60-34

SPACE TRACK

SPACE VEHICLE EPHEMERIS AND DIFFERENTIAL CORRECTION PROGRAM

UNIFIED THEORY

ASTRODYNAMICS DEPARTMENT

AERONUTRONIC

A DIVISION OF FORD MOTOR COMPANY

FORD ROAD, NEWPORT BEACH, CALIF.

CONTRACT AF19(604)-5885

14 JUNE I960

496L SPO

AIR FORCE COMMAND AND CONTROL DEVELOPMENT DIVISION

AIR RESEARCH AND DEVELOPMENT COMMAND

UNITED STATES AIR FORCE

LAURENCE G. HANSCOM FIELD, BEDFORD, MASS.

Air Force Command and Control Development Division

A96L SPO

SPACE VEHICLE EPHEHERIS and

DIFFERENTIAL CORRECTION PROGRAM UNIFIED THEORY

Astrodynatnics Department Electronic Systems Operation

Aeronutronic A Division of

Ford Motor Company Ford Road, Newport Beach, California

ABSTRACT

This report traces the development of a differential correction theory for lunar and space probes. To overcome the singularities inherent in differential processes with conventional two-body descriptions of the psth, as the eccentricity passes through unity in either the parabolic or rectilinear sense, a uni- fied formulation of the two-body equations was developed. The differential correction procedure is based upon an ephemeris, inte- grated in Encke form by special perturbations, and upon differential expressions derived from the unified two-body equations. The report includes experimentation with'simulated range, range-rate, azimuth, altitude, right ascension and/or declination data to evaluate the performance of the resulting IBM 709 program.

TABLE OF CONTENTS

SECTION

1 \.

TITLE

Introduction

PAGE

1

Technical Approach

Experimentation 31

ii

SECTION 1

INTRODUCTION

Under an amendment to contract AF19(604)-5885, Aeronutronic has undertaken the development of a differential correction program for geocentric orbits of medium to high eccentricity, including lunar probes. Orbits of interest in- clude the parabolic, elliptic, hyperbolic, and rectilinear forms, and an orbit theory which was applicable at once to all of these forms, without the singularities normally associated with the transition between these, was sought. This report discloses the development of a unified orbit theory especially adapted to machine computation; included are the necessary differential relationships and details of the Encke perturbatioi method.*

* The proposal to employ a unified formulation of the two-body equations, and the major portion of the development thereof,

are due to Samuel Herrick. Others on the staff of Aeronutronic who participated in this development include Jeannine Arsenault, R. H. Gersten, G. Matlin, C. Tross, C. T. Van Sant, G. B. Westrom, and L. G. Walters.

SECTION 2

TECHNICAL APPROACH

The Keplerian or two-body orbit is the basis for the perturbation methods of orbit integration and for the differential expressions required for orbit correction. In the perturbation method, the Keplerian path is employed as a reference path, and only the non two-body accelerations (or "perturbations") need be integrated to define the motion. Differential expressions involve the determination of linear cause-and-effeet relationships between observation and orbit, and require a description of the orbit free of singularities in the region of interest. In both the perturbation method and differential correction use of other than two-body (or nearly two-body) form, such as position and velocity, severely limits the computational efficiency.

A review of the two-body orbit description betrays two serious singularities in the region of interest. First is the transition from ellipse to parabola to hyperbola as the eccentricity passes through unity and the semi-major axis becomes infinite. This region is described by three different sets of formulae, none of which is valid in this entire region. The situation is similar to the engineer's description of damped harmonic motion, wherein three mathematical descriptions cover the undamped, critically damped, and overdamped cases. A more subtle singularity occurs when the eccentricity passes through unity but the semi-major axis remains finite. The two-body orbits are degenerate forms of the ellipse and hyperbola, but the path is confined to a rectilinear path and a description which involves the orientation of the orbit plane fails,

The most familiar description of two-body motion is in terms of the six elements:

a semi-major axis

e eccentricity

T time of perigee passage

i inclination

ft longitude of the ascending node

argument of perigee 0)

This set of elements embodies the following singularities:

1. For zero eccentricity (i.e., the circular orbit) cu is indeterminate, and the anomalies used to describe position

in the orbit plane have an indeterminate origin.

2. For zero inclination (i.e., the equatorial orbit), the longitude of the node is indeterminate, and the argument of perigee has an indeterminate origin.

3. For rectilinear motion, the orientation of the orbit plane is undefined*, consequently, both the longitude of the node and the inclination are indeterminate

4. For nearly parabolic motion, the equations for ellipse and hyperbola are singular for unit eccentricity, whereas the para- bolic equations are invalid for non-unit eccentricity.

For the latter two reasons a description of the two-body motion by these familiar elements must be abandoned, and an alternate set of parameters chosen for this study. Incidentally, Aeronutronic has developed descriptions which also overcome singularities of the first two types*, results of this research are published elsewhere *

* See, for example, Aeronutronic publication U-880,

The development of the orbit theory is given in the following sections, These include the basis for parameter selection, the Encke epbemeris formulation in terms of these parameters, the differential expressions for orbit correction, and the geometric transformations required for the utilization of topocentric obser- vation residuals.

2.1 SELECTION OF PARAMETERS

Singularities of the types noted above arise from the parameters employed to describe the two-body orbit. This section will introduce a set of six parameters which removes all singularities save that for zero eccentricity from the theory, thereby unifying the separate descriptions heretofore required in the vicinity of unity eccentricity •

To provide continuity from the elliptic to the hyperbolic region, including the parabolic (e = 1, infinite a) and rectilinear (e ■ 1, finite a) cases, it is necessary to adopt a uniform concept of mean motion n, ordinarily defined by:

ne

r K' N/~M (a)"3/2 elliptic case

n ■ k' «y~/i parabolic case

n, s k' <Jfi (-a)" ' hyperbolic case

In the following, the parabolic definition for mean daily motion will be adopted for all three cases, and the remaining two-body equations will be adjusted accordingly. From this procedure evolves a set of two-body equations identical for the elliptic, parabolic, rectilinear, and hyperbolic cases, with no singularities as the eccentricity passes through unity.

Parallel research undertaken by Or, Samuel Herrick of Aeronutronic has lead to an alternate development in terms of closed form f and g series expressions which is valid for all eccentricities, including zero, and all inclinations. Further development of this completely unified theory is continuing", the development of both a variation-of-parameters theory and of differential expressions for orbit correction will be pursued

Beginning with the parabolic definition of mean daily motion, designated n,* the mean anomaly is related to time by

M - M = n (t-t ) o v o

-3/2 ~ where the usual elliptic definition of M is a M. Kepler's equation, in elliptic form, is written

M a E - e sin E > • (E - sin E) -t- (1-e) E

or

M - a?'1 h r e U + q X

where, by definition,

<\, A X - •>/~ä E

u « X X X * 2

3V a -51 a • 7!

The function U is derived from the (E-sin E) series} for negative a (hyperbolic case) this series may be derived from (sinh F - F) series. For infinite a, (parabola), the M form for Kepler's equation reduces to the familiar Barker's equation for the parabola. Thus the single form is adequate for all conies.

In this unified development, the tilde ( *» ) symbol will be used to denote quantities whose definitions differ by virtue of the unification process, from familiar quantities in the conventional two-body theory. This comment does not apply to the usage in Section 2,4, where the tilde symbol is used to distinguish between equatorial and horizon coordinates of the unit vector triad L, A and D,

The position vector r is readily related to the above quantities through

r > la £ + vui ft«

To avoid the indeterminate nature of <} for tht rectilinear case, where the semi-latus rectum p ■ a (1-e^) is zero and the orbit degenerates to a single line along P, a vector (} *vp £ is introduced such that

yco «/p

The coefficients are

x^ = a (cos E - e) = a (1-e) - a (1-cos E)

= q - C

K ? x2 x4 x6 where C ■ - + —s—

21 a • 41 a «6!

VCü _ 1 %

—- ■•/• sin E ■ X - * U

Note that the elliptic expressions above, expressed in terms of circular functions, become hyperbolic, function! for negative a, typical of the formulae for the hyperbola. The velocity expression r is

£ - fei +~ fi

I where

^=-^sinE=-^-(x-iu)

_ mZJL co8 E = 2JB- (l - - c) X r r a

The unified theory presented here lends itself to automatic machine calculation, where transcendental functions are routinely computed from their T^aylor series representations; on the other hand, lack of tables for U and C in terms of X and a"l will discourage hand computa- tion.

The parameters which suggest themselves for representation of all orbits (save zero eccentricity; note the usage of P) are

M0, a"1, P, ä .

The previous development indicates the procedure for translating from parameters to position and velocity. The inverse transformation from position and velocity to these parameters follows:

~ 1 ~ Given r, r, and t, to obtain M0, —, P and (£:

r2 = r • r

rr ■ r • r

s2 = r • r

1 12 s2

Compute — and q: —= — -■-— (vis - viva integral)

P = (r2 a2 - r2 f2) /M

e2 = 1 - p/a

q = p/(l + e)

~ 1

Compute P and 2_: P. = [-^C1-^C)J r -L /=■ S J r

Compute X and M: S = X - —T-, + n , - 3 ■ , + ... ^.X

U = 31 " a-5'. + I2^"'.

/\y /\, /-w

M = q X + e U

»\< Compute MQ: n = kWji

fW *%* 1i M0 = M - n (t - tc)

2.2 ENCKE EPHEMERIS PROGRAM

The numerical integration of the perturbative accelerations, (and of the total acceleration in Cowell's method), is designated as "special perturbations". Factors to be considered in an orbit theory based on special perturbations include: what is to be integrated numer- ically, the choice of reference orbit, and the choice of the independent variable. In addition, the type of numerical integration procedure, e. c;.,Adams-Bashforth second-difference, second-sum Runge-Kutta, etc., must also be considered.

The special perturbations approach is particularly well suited to space trajectories which depart only slightly from orbits for which an analytical solution exists, e.g., a lunar trajectory.

* Alternatively, series for U directly in terms of S may be employed, i.e.,

3 . 32S5 If I 7 u - - + — + V^ + 3! a.5: a*-7!

and

M - U + qS

In connection with what is to be integrated, a variety of methods for special perturbations have been developed. In effect, however, all of these methods use some modification or combination of the basic concepts characterized by (1) the integration of the total- acceleration, as in Cowell's method, (2) integration of the departures from a fixed reference orbit, as in Encke's method, and (3) integration of the parameters of a varying reference orbit.

The Encke ephemeris program has been shown to be highly efficient for lunar trajectories. The program has demonstrated its ability to integrate trajectories to the moon with 15 integration steps with less than five miles accumulated truncation error at impact. Similar calculation with Cowell's method required almost 200 integra- tion steps.

In the Encke orbit determination method only the departures from a two-body reference orbit are integrated. The position at some epoch (t) on the reference orbit is given by r . The deviatiottsof the vehicle from the reference orbit can then be given as (see Fig.l):

j0 = £ - £ 2.2.1 e.

Considering one component we have the derivatives:

where

Finally

d2e A2 A1 d x d x

d^ dT* dT2

d2x

d r

M*. = X = -

e

d2x dT

X + X

Mx 3

x* = perturbations

2 A ■» K x

^ x -* I y» z 2.2,2

VEHICLE •REFERENCE ORBIT.

> *

FIG. 1 DEVIATION FROM REFERENCE ORBIT

10

For small departures from the reference orbit, as in the case of lunar trajectories, the first term on the right side of the above equation is small and hence one integrates only slightly more than the x term. The first term on the right side is normally developed into a series to avoid obtaining small differences of large quantities. This series is developed in the following manner:

First, x ■ x + %, etc., are substituted into the defining equation for r2

2 2 2 2 r = x + y + z ,

yielding

2 = (x + i ) + (y0 +n) + (z + C ) i

= rg + 2 [ £(xe + ^ £)+ T) (ye + % n.) + C (ze + % C ) ]

2.2.3

and q is taken to be

[i (*.♦% t> + i (y0+%n) + C(«. +\ C) ] , 2.2.4

so that equation 2.2.3 becomes

2

e 1 + 2 q 2,2.5

The first term on the right-hand side of Equation 2,2.2 may be expressed as

e x 1 3 " 3 = 3

r r r e e

[ 1 - Cf) ] x - ( } 2.2.6

where, applying Equation -2.2.5,

(i,3- (1 ♦ 2,)-3/2 2.2.7

11

2.2.8

Tven, using a binomial expansion, Encke's series is defined by

i-(-£) - i- (i + 24 -3a- ^ q+-~-1 «3 •

. fq

Finally, the substitution of Equations 2.2.6 and 2.2.8 into Equation 2.2.2 yields Encke's formula:

ti . (^L_ <fqx-«) + k2x\ «4T,,< 2.2.9 2 3

dv

The perturbative term x typically includes the second, third, and fourth harmonics of the earth's gravitational potential and the perturbations due to the sun and moon.

The procedure employed to calculate the position and veloc- ity on the reference orbit from the unified parameters follows the pat- tern developed in Section 2.1.

12

2.3 DIFFERENTIAL EXPRESSIONS

This section deals with the development of linear differential expressions relating position and velocity to the unified orbit para- meters introduced in Section 2.1. In Section 2.4, these relationships will be extended to include observation residuals referred to topocentric coordinates. In each of these operations, the algebraic manipulations are tedious, but straightforward. The object of this present section is vector relationships relating position and velocity increments Ar, and Ar to increments in the parameters $10, Ai, AP and A^« In order to remove redundancy, the latter two parameters will be developed in terms of related quantities which recognize the unit property of the vector P and the orthogonality of P and §f.

Differential expressions provide the link between observation residuals and corrections to the parameters. The development starts with the vectors of position and velocity:

y r =xQ)P + — £ 2.3.1

<v>

y r = x l+^P $ 2.3.2

where, for convenience, the coefficients of ?£ will be denoted as S and S, respectively. Differentiation leads to:

A r ■ P A* + ^ A? + x Ap + ? A<£ 2.3.3

At = PAx +2_A?+ J^AP + I Aa 2.3.4

13

2.3.1 Differential Expressions for.Position and Velocity Components In the Orbit Plane

In order to facilitate understanding of this differential procedure, those derivatives in equations 2.3.3 and 2.3.4 which, pertain to the position and.velocity components in the orbit plane, i.e., AXQ, AS, AxjjD, As, will be derived first in terms of corrections to the parameters. Subsequently, the differentials associated with the expressions for the vectors defining the orientation of the orbit plane will be discussed. The P and 2 vectors both define the orienta- tion and serve as the chosen axes of the orbit plane.

The position and velocity components in the orbit plane, Xyj , y^ and x^ , y^ , respectively, are defined,

xa) = q <V 'Y 1 S# S = X - - u

a

2.3.5

x . -"fr XCD " r s

r ft (i -1 C) 2.3.6

The differential expressions for x^ , S, x^ and $ are ob- tained from the differentiation of equations 2.3.5 and 2.3.6:

Ax = Aq - AC

AS" = AX - i tu - U A (i) a v a7

»Vf [}*»•**]

2.3.7

2.3.8

2.3.9

r a 2.3.10

where the various terms are defined below. The determination of AC and A^J follows from the series:

14

IT £ XL .H U - 31 " a-51 + T2^?:

X2 X4 X* C " 21 " a-41 + az-61

Differentiating, we obtain

AÜ = C A)C-iy A (-)

&.? A* - ft AC1) a a'

where

v!r 1 t i'V a 71 + 2 9:

* St a = 41

I Ifi , 3 I* a 61 "P- 81

From the formulae for e and r

e = 1 q a '

r = q + e t,

differentiation yields

Ae = -qA (j) - J Aq

and Ar = Aq+eAC + (?Ae

or Ar = elS Afr- (q&+ e£ ) A C1) + (1 - J t)A<

15

Finally, the Z& term is derived from the expressions for ft:

flU # + n (t - t ) o o

I. q fc + e ft

and their derivatives which follow

A ft ■ A Öf since A n * 0 o

A ft » q A^ +e AD+ ?Aq + Ü A e

or AX'=-AM'+-(qÜ'+efr) A(-) - - If A q r o r v^ a' va' r '

At this point it is convenient to express Eq. 2.3,7-10 in terms of A ft" , A(—) and A q; o' a n

A xn = 1 A ft + 3£ A(-) + X Aq 2.3.11 «> m o aq a q *

A If ■ 3 AM + S A(i) + S Aq 2.3.12 m o aq a q '

where

X - - S/r; X -X (eU + q Ü + C : X ~ 1 - X S m aq m a ^ aq ■

3" . (1 - i &Wri S « S (eft + q U) - (U" - - Ü ) m v a " ' aq m a T M v a a'

S = -S S 2.3.13 q m

and A i„-4 £■ + £ A(-) +X Aq 2.3.14 CD m o aq a q *

£.1 Aft" + f A(i) +1' Aq 2.3.15 m o aq a q '

16

where 3» tye If _ %

m 3 r r

i . .tf£i + 2Ä

X« -1 A + # -^2 c<# + ■ *J ♦^ (V - i *.) aq m a a r a a'

m r* a ' a m

>2.3.16

«/p~

i = i (etT + qty + I (qC-+ e fr ) +v^ (J * - V) aq mvan/ rH a' r va a aq ma

To more directly relate the A.i^,.. A1>, etc., to the chosen parameters, the quantity £q is replaced by 9 \p. Note that:

and § • A § = j A P

Then, from the expressions relating p, q, e and

q = p/(l + e)

one obtains

e = 1 - p/a

1/1 \ . q /lv Aq =" (JA P) + ft A (-)

17

The substitution of the last expression will alter equations 2.3.11-16 in the following manner:

Ax *1t Aft + £ A(-) + ? tk&p) co H o ap va' p v2 '' 2.3.17

A S ■ S AM + S A (i) + S (^A p) m o ap a' p N2 r/ 2.3.18

where X and S remain as before, and m m

x = X + *-x ap aq 2e q p e q

ap aq 2e q p e q

2.3.19

and

Ax, = K AH + X A (i) +1 (^Ap) o ap va' p v K/ co m 2.3.20

AS =^ Aft +i A (i) +^ (^Ap) m o ap va' p v2 2.3.21

where again X and S remain unchanged and ö m m ■

i\, nj 2 . a 'o

X = X + *- X ap aq 2e c

S = S ap aq + f » 2e q

p e q

TL-i 4 p e q

2.3.22

Equations^ .3.17, 18, 20, 21 thus give express ions for Ax , A's', Ai , As, which sre different for each observation date, in terms of rrections to the elements, AM , A (i) and ^Ap.

o a 2 co

18

2.3.2 Differential Relationships Extended to the Orientation of the Orbit Plane

The orientation of the orbit plane is defined by P, a unit vector directed toward perigee, and 0^, a vector normal to P in a sense determined by the direction of motion. To avoid the indeterminate behavior of Q for rectilinear motion, the vector

K - Jpä. has been considered in this development.

The reduction of the six components of A p and Ag to fewer quantities leads to a definition of P in terms of its right ascension a* and declination ö*. From these the unit vectors A* and D*, ortho- gonal to P (see Fig. 2) are calculated.

' P = cos 5* cos ct*

P 4. P = cos 5* sin a*

P = sin ö* z

A* <

A * = - sin a* x u

A * = cos n* y a

A * = 0 z

r D * = - sin 5* cos a*

D* I D * = - sin 6* sin a*

D * = cos ö*

19

FIGURE 2

The differentiation of P, A* and D* yields

AP = A* cos 6* Act* + D* AÖ * 2.3.23

AA* = -P cos 6* Aa* + D* sin 6* Aa*

AD* = -A* sin 6 * Aa* - L A6 *

The angle 0*, «• shown in Fig. 2, allows us to derive the following relationships.

2 = A* cog 0* + D* sin 0*

W =-A* sin 0* + D* cos 0*

£ = A* c* + D* s* 2.3.24

where c* = ,/p co» 0*

s* = Jp sin 0*

The differential expression for <£ becomes

Aa" = A* A c* + D* A s* + W* Aa * + P* A6 * 2.3 ?5

where W* = c* (D* sin 6* " F_ cos 6*) - s* A* sin §■

p* . -p s*

21

A further reduction of unknowns in the expressions for Ax , Ala, Ax and A? is carried out by noting that CD ' ' (B

«v *v "k *k it "it 2. • <} = f A p - c Ac+sAs 2.3.26

Incorporating equation 2.3.26 into equations 2.3.17-22 and substituting these along with equations 2.3.23 and 25 into equations 2.3.3 and 4 gives the following.formulae for Ar and A£ io terms of six unknowns, namely, Aft , A(—)t Aa . AÖ , Ac ana As •

Ar= [PX +Kl A»+[P* +§3 ] A(~) 2.3.27 - L - m • m ' o L— ap * ap va'

c (PX + Q S ) + A S J Ac ~ P P ~

+ Ls (P X + Q S ) + D S ] As _ P P _

Ax cosö+WS] Aa+[Dx + P £> ] Aö

and

Ar = [PX'+'b3']Afi' + [pi +QS J A A 2.3.28 — — m am o L— ap -*■ ap va'

+ Lc (PX+QS)+AS]Ac - P " P

s (PX + Q S ) + D S J As - P • P

A x^ cos o + W S J Act + I D x^ + PS JAO

The explicit derivation of ■ and 6 is avoided by expressing the involved quantities A and D* in terms of the given components of P:

22

(P2+P2)1/2 V x T y /

-P /D

P /D x' z

* ■A P

y z

* * D = A P y x z

* * It is obvious from Fig. 2 and Eq. 2.3.24 that c and s may

be obtained from the following dot products:

ff-A'

2 • 2 - S

This completes the derivation of the differential expressions. In the following section, the topocentric coordinates of the observer are introduced, and scalar differential expressions between parameters and residuals in the observed quantities are derived. Details of the structure of the IBM-709 program are reserved to Sect. 3.

2.4 TOPOCENTRIC RESIDUALS

The position of the vehicle referred to the observer is related to the positions of the observer, R, and of the vehicle, r_, both referred to the dynamical center by the fundamental expression.

L + R " PL 2.4.1

23

where L is a unit vector. (See Fig. 3)

The differential relationships relating residuals in the observed position to improvements of the orbital elements are embodied in the derivative of equation 2.4.1,

A£ ■ A (pL) » Ap L + pAL - Ar 2.4.2

It is assumed that there is no requirement for correcting the station vector, R. The previous section developed an expression for Ap_ in terms of corrections to the orbital parameters. In this section A£ is translated into differential expressions involving residuals in the observations. In the discussion to follow, let the subscript c signify that the particular quantity is computed, that is, obtained from the representation, whereas no subscript denotes an observed quantity. The residuals will be taken in the sense "observed-minus-computed."

The observations considered in this study are p, A and h, a and 5, and p . The differential correction program processes any

combination and number of these observed quantities taken one at a time. This implies that although a complete set of observations may be available, for example, A, h and p, such that the components of Ao_ could be obtained directly, the more general method employed in the differential correction program considers each piece of information without regard to the others. This procedure allows incomplete data to be processed and, furthermore, permits independent weighting or rejection of the data components.

First, if only one or two of the a, b, and/or p are available it will be useful to define the auxiliary unit vectors A and D (Fig.4 ) which form an orthogonal set with L. D is the north-pointing tangent to the celestial meridian at the point toward which L is directed. A is also tangent to the celestial sphere at the same point and parallel to the equator plane so that it completes the right-handed orthogonal set L, A, and D. The components of L, A, and D are

= cos 6 cos a

L = cos 6 sin a

L ■ sin 6, z •

f i 2.4.3

24

r North Celestial Pole

Observer

Observer's Meridian

Greenwich Meridian

Equator Plane

Satellite

r /«—Satellite Orbit

Sub-Satellite Point

Figure 3

POSITION RELATIONSHIP OF OBSERVER, SATELLITE AND DYNAMICAL CENTER

25

Celestial Sphere, Centered At Observer

Figure 4

VECTOR RELATIONSHIPS

26

Ax » -sin a

A = cos a

Az ■ o,

> A 2.4.4

and D ■ -sin 6 cos a

Dy = -sin 8 sin a

Dz ■ cos 6,

2.4.5

where a and 5 are the topocentric right ascension and declination of the object. In addition, L is determined from

*J 2.4.6

where

and

P =» r + R c — -

PC ■ £c ' £c

For angular observations, the residuals in the direction cosines, £L, are obtained from

4L • 1 • L

Also it can be seen from Fig. 4 that

27

AL ■ A A (Xcos & + D A 5 2.4.7

Successive dot products of A L with A and D will lead to scalar differ- ential expressions for A acos 5, and A5, respectively.

AL • A » A acos &, 2.4.8

AL Aö 2.4.9

Similarly, the dot product of A£ with L relates a slant-range residual Ap to improvements of the orbit parameters.

4ft" L Ap. 2.4.10

Differential expressions involving observations in a horizon- oriented coordinate system may be obtained in a similar manner by defining the horizon system unit vector triad L, A, and D, whose components are:

and

L , »-cos A cos h

L . = sin A cos h yh 2.4.11

Lzh * sin h,

xh ■ sin A

yh = cos A 1 2.4.12

zh = o,

28

D xh

yh

= cos A sin h

s -sin A sin h 7- £ 2.4.13

fr, ■ cos h,

where h is the altitude or elevation angle and A is the azimuth, measured in the positive sense to the east from north. These horizon system components are rotated into the equatorial system, to which the components of £Q are referred by

L «L.S,L,E+L,Z N

x xh x+ yh x+ zh x

Ax =A,S+A.EJA,Z I (x—.y, z.) 2.4.14 x xh x yh x zh x

D sD.S+D.E+D.Z, x xh x yh x zh x'

The components of the auxiliary unit vectors S, E_ and Z, directed to the south, east, and zenith, respectively, are

Sx = sin 0 cos 6 *

S ■ sin 0 sin 9 I S y

S > -cos 0 , z *

2.4.15

E = -sin 0 \ x

E = cos e y

Ez " o,

2.4.16

29

and

Z = cos 0 cos 6 X

Z r cos ^ sin Q I Z y ' -

sin <P

2.A.17

where <P is the observer's astronomical latitude and 0 is the local hour angle of the vernal equinox. Then

AL ■ A A A cos h + D Ah. 2.4.18

Scalar differential expressions for A A co^s h, and Ah can now be obtained by forming the dot products of AL with A and D, respectively:

AL ' A ■ AA cos h 2.4.19

and

AL D Ah. 2.4.20

Thus, residuals in the observations of angular position in the her izon coordinate system can be related to corrections to the orbit parameters through A£.

through Range-rate residuals Ap can be related to the parameters

p_ • A£ = p A p 2.4.21

or

p_-A£+£-A£-pAp+pAp

pAp-£-A£ + [£ - * pj- Aß

2.4.22

2.4.23

The corrections to the orbit parameters are introduced through A£ and A£ as developed in Sect. 2.3.

30

SECTION 3

EXPERIMENTATION

The unified ephemeris and differential correction theory has been subjected to experimental verification to determine the

t ^curacy and rate of convergence with various types of simulates observations.

EPHEMERIS PROGRAM

The numerical integration program was checked against the Encke Moonshot program. The major differences between the two methods are the choice of parameters used to define the two-body reference orbit and the argument used for the numerical integration, i.e., time for the unified Encke program and eccentric anomaly, E, for the Encke Moonshot program. The results of the comparison are listed in Table 1.

Further experimentation is being conducted to determine integration step-size, time for integration, etc.

DIFFERENTIAL CORRECTION PROGRAM

The program has been verified by correcting a hyperbolic orbit to one that is elliptical with various combinations of range and angle data. Tables 2 through 4 give the sizes of the r.m.s. residuals and list the values of the parameters M , L, P and Q. after each of four correitions. The results of Table 2 were based on 32 range observations spaced over 7 hours and 18 minutes. Over the same time-span, Table 3 involves 15 observations of aand 6 and 17 observations of A and h, and the same number of angular observations were used with 32 range observations in Table 4. Table 5 is a reproduction of the program listing of the residuals in the range and angular observations after one correction.

31

*

Further verification of the program was carried out by correcting from an elliptic to a parabolic orbit and from a hyperbolic to a parabolic orbit. The success of this experimentation, orbit correction through unit eccentricity, satisfies the definition of the unified theory. Tables 6, 7 and 8 list the values of the parameters for the elliptic case and the r.m.s. residuals in the observations after each correction for both the elliptic and hyperbolic runs. In all cases the observations were obtained over a 6 hour and 42 min. period. Thirty-two range observations were used in Table 6, 13 sets of aand 5 and 15 sets of A and h for Table 7, and a combination of 32 range, 13 a and & and 14 A and h observations are included in Table 8. Tables 9 through 12 demonstrate the. use of the program with range rate simulated observations.

32

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