1*7

CALCULUS OP PERTURBATIONS APPLIED TO LUNAR MISSION ANALYSIS

William C. Marshall

Minneapolis-Honeywell Regulator Co.

Minneapolis, Minnesota ABSTRACT

The perturbation or linear prediction theory is applied to a realistic four-body model of the oblate Earth, triaxial moon, sun, and vehicle. The disturbed motion of the vehicle may be caused by combinations of three major error sources, namely:

1) a disturbance in the state coordinates at an initial time, e.g., an error in position and velocity at burnout or in- jection time; 2) small disturbing forces arising from ignored

force centers or slightly inaccurate representation of non- ignored force centers, e.g., the planetary perturbations of the outer planets may be ignored, or the Earth's gravitational field may be imperfectly represented; and 3) fictitious dis- turbing forces which result in truncation and round-off errors propagating in time about the true (unknown) reference

traj ectory.

MOTIVATION

In performing a trajectory or guidance analysis of the motion

of a space vehicle during a lunar mission, a precise mathematical model of all significant external forces that affect the trajectory is of fundamental interest. In many cases, the forces which act are very unpredictable and must be described statistically. Also, some forces may exist which

are only known approximately or are even unknown at the present level of knowledge of the universe.

Presented at ARS Guidance, Control, and Navigation Con- ference, Stanford, Calif., Aug. 7-9, 1961.

iResearch Engineer/Scientist, Military Products Group Research Dept.

2

T h e concept of generalized state coordinates is implied here. Thus, there is no distinction between kinematic or dynamic coordinates or other quantities such as the mass of the vehicle.

1*7

Even though it may not be possible to formulate exact dif- ferential equations of motion, approximate equations of motion can be formulated. In certain problems such as the restricted two-body problem, the equations of motion of a negligible point-mass in a central inverse-squared force field have been analyzed for over 400 years. The results of these studies have been applied fairly extensively to feasibility studies of satellite, lunar, and interplanetary missions over the past five years and up to the present time.

However, with the need for extremely accurate prediction of trajectories, a precision analysis must be contemplated to determine not only error sensitivities to initial and terminal conditions (on position, velocity, and time, for example), but also to determine the effects of numerical integration errors, the effects of stochastic perturbations (such as meteorite bombardment accelerations on the vehicle or roundoff error in numerical computations), the effects of ignoring planetary perturbations of flight path (or light pressure effects), and in general any of the effects of nature that have been omitted in the approximate differential equations of motion.

The science of exterior ballistics is one field where

analysts have for some time been confronted with the necessity of performing precision analysis to describe the effects of wind uncertainties, gravitational uncertainties, initial condition errors, and uncertainties in drag and lift co- efficients on the flight of missiles. This area is described in Refs. 1-5.

Numerical analysts have also been concerned with the pro- gagation of numerical integration errors through the numerical solution of the equations of motion of ballistic missile and celestial bodies as is evidenced in both the western world literature and the Russian literature, Refs. 6-9, 11-19, and 22-25.

It is the purpose of this paper to present what the author believes is the underlying theory, using the calculus of first- order perturbations or linear prediction theory, which applies to the problem areas of precision analysis mentioned pre-

viously. In particular, an analysis technique (which has been called "the adjoint method") will be derived and applied to the equations of motion of a lunar mission space vehicle. The approximate model selected is the most realistic one known to the author, embodying the oblate Earth, triaxial moon, sun, and atmospheric reaction forces.

The section on the derivation of basic relationships pre-

sents a review and derivation of the basic linear prediction

theory already referred to. An attempt is made throughout to interpret the mathematics in the language of astronautics.

The section on ignored forces describes the treatment of disturbing forces (including numerical integration error propagation). Two distinct problems are considered: 1) the inclusion of specified disturbing forces into an analysis;

and 2) the determination of unknown disturbing forces through observations of the disturbed motion of the space vehicle.

The section on the Newtonian Earth-moon equations of motion contains the specific application of the linear prediction theory developed in the preceding sections to a practical set of equations of motion of a space vehicle in Earth-moon three- dimensional space.

DERIVATION OP BASIC RELATIONSHIPS

In this section, the basic relationships or calculus of first-order perturbations are derived in canonical form.

Subsequently, application will be made to the particular equations of motion of a lunar mission vehicle.

Nonlinear Equations of Motion

A generalized system of first-order nonlinear ordinary differential equations are first defined as the true, or pre- cise equations of motion of a mathematical point in configu- ration or phase space with the laws of motion containing m physical constants. The precise equations of motion are

d

— X dt

x

d dt

f (Χ; a, t) [la]

[lb]

where the precise laws of motion are

f(X,a,t)

Χ 2 , * 9 x

n»

A

« 1 , 9

A

X2, * / S Χ ·

>

n '

A

« 1 ,

• S

X 2 , ' Χ · « 1 ,

a

, ·

[2]

1 4 9

and the physical constants are

« 1

[3]

In a particular example, the physical constants may be true constants (whose magnitudes have a small uncertainty); or they may be dependent on time, position, velocity, attitude, or, in general, any combination of the generalized coordinates of the problem.

A solution vector X(t) to the precise equations of motion from precise initial conditions, £(t

) = X , would describe at every instant of time the exact position, velocity, mass, jattitude, etc., of a space vehicle without error. The vector X(t) is sometimes referred to as the state vector of the

system. Between two instants of time, initial time t

and terminal time T, the state vector describes a trajectory or path in (n + 1) dimensional (n state variables and time) con- figuration space, as depicted in Fig. 1.

Obviously, the definition of the precise equations of motion, î(x;8,t), and determination of precise solutions is an im- possible task. Not only is the form of the equations of motion in doubt for some problems in astronautics, but, until recently, there was not general agreement as to the magnitude and error bounds on some fairly standard existant physical constants (26).

Fortunately for the analyst, in almost all problems in astronautics it is possible to formulate approximate equations of motion through the use of Newtonian (or non-Newtonian) mechanics. The approximate formulation of the equations of motion of a space vehicle are then defined by

d ^

— X = dt

d dt

x i X 2

f(X; a, t) [4

]

X ( 0 = X. [4

]

3

Numbers in parentheses indicate References at end of paper.

where the approximate laws of motion are

f i ( x i , X 2 , * · • > χ η; « ι , Α2>· .

f( X ; A , T ) = f a ( x i » X 2 , * · • > xn = « ι , Α2 )· ·

[5]

, « 2 , · · , « 2 , · ·

and the approximate physical constants are

[6]

where it is now tacitly assumed that as many physical con- stants m as exist in nature have been formulated.

The difference between the unknown laws of nature and the formulated laws are next defined as

(X,t) F ( X;A , T ) - F(X;OT,T)

[7]

where 17 may be, in a particular application, as large or as small as the ineptness of the analyst in specifying the approximate laws of motion.

_^ Subsequent to a specification of initial conditions, X ( t

) = X

, Eq. 4 may be solved by analytical or numerical methods to yield a reference solution X(t). Denoting the difference between the exact solution of the (unknown) equations of motion and the reference solution to the approximate equa- tions of motion b y ? ( t ) , then

c ^(t)

€ l (t) f

( t )

X(t) - X(t) [8]

which represents the propagated error in the state vector due to both an error in formulation of the laws of motion and an error ?(t ) in initial conditions from which the reference

ο

trajectory was calculated.

Equations of Disturbed Motion

Differentiating Eq. 8 and substituting from Eqs. 1 and 4,

1 5 1

then

— ht) = f(X; a,t) - f(X; a,t)

dt

which represents the true or precise equation of disturbed motion with respect to the reference solution X(t) as depicted in Pig. 2.

Using the definition of η, Eq. 9 may be rewritten as

— ht) = f(X; a,t) - f(X; a,t) + i?(X,t) [10]

dt

with no loss of generality.

First Variational Equations

Expanding the right-hand terms in Eq. 10 by Taylor's series about the point X(t) on the reference solution gives

i(x,t) = i?(X,t) + ( — \ (x - x) + σ[(Χ - x )

] [11]

f(X

a,t) - f(X

o,t) =(-—\ ( X - X ) + β[(Χ - X)3 ^

d

f

_

t ) = | -\ (X - X) + ij(X,t) + Cf[(X -. Χ)]) [13]

dt

Denoting the difference (X-X) by ΔΧ and ignoring the second- order and higher terms, the first (free) variational equations or first-order perterbation equations are obtained.

— Δ Χ = ( — - \ AX+i,(X,t) [14a]

AX(t„) = i(t ) [14b]

where

d

and AX(t) = f (t) since

5fi df ^t

dx ¹ ' dx ² '

a ^{f g} a ^f

( 9 χ ι d x2

'X2

5fi

— ΔΧ( t ) = — e ( t) dt dt

The solution of Eqs. 14 approximates the true disturbed motion of the space vehicle with respect to the reference solution

Denoting 'the coefficient matrix [df^/dx ) by B(t) it is noted that

— ΔΧ = Β ( t ) · ΔΧ [15]

dt

represents the homogeneous form of the approximate equations of disturbed motion. The approach is now to find the general solution to the homogeneous equations and also a particular solution to the nonhomogeneous Eqs. 14. The sum of these solutions then represents the general solution of the linear approximate differential equations of disturbed motion.

Fundamental Solution Matrix

Noting that the first variational equations represent a nonhomogeneous first-order system of linear ordinary-dif- ferential equations with time varying coefficients B(t) and forcing function 17 ( t ), it is known (from elementary theorems on multiplication and superposition of solutions of linear equations) that if any η linearly independent solution vectors can be obtained to the homogeneous equations and arrayed as column vectors of a time-dependent matrix, then that matrix may be called a fundamental solution matrix.

155

A fundamental solution matrix (or matrix of linearly inde- pendent solutions to the homogeneous variational equations) which at time t = t takes on the numerical value of I , the

Ο 7

identity matrix, has the property that given any arbitrary set of initial conditions A X

the solution vector A^(t) to the homogeneous perturbation equations can be written as

where TI ( t )

to

at t is

AX(t) = 7c(t) - AX(t

) [16]

I and the fundamental solution matrix normalized

TT(

t)

^ 1 1 ,

^ 2 1 , 7 l^{1 2} Π _{2 2 ,}

1 y

η

In short, Eq. 1 6 represents the general solution to the homogeneous first variational equations. It should be noted that Eq. 1 6 contains η arbitrary constants which are the η elements of ), the initial disturbance vector. T o be

Ο 7

specific, the kth column vector of M t ) satisfies the follow- ing linear differential equation:

kth

coordinate

T TLK ~ ^ K ~ " ^ K' V ⁰

d

*

( t

)

= B(t)

dt

:

NK Π .

nk

j

ό

[17]

where k

1 , 2 , 3 , n.

Since TC(t) is composed of η solution vectors of Eq. 1 7 , it may be more easily considered as being the solution matrix generated by (numeric or analytic) integration of the follow- ing matrix differential equation from initial conditions given at time t

:

— π( t)

dt = B( t) π( t) [18

]

M t„ ) [18b]

7l(

t

1 0 0 0 1 0

1 0 0 1 Adjoint Solution Matrix

The purpose of this section is to demonstrate a method (the adjoint method) to obtain the inverse matrix of the funda- mental solution matrix M t ) without resorting to numerical matrix inversion techniques. The usefulness of this method is obvious in later portions of this paper.

The matrix differential equation adjoint (or adjunct) to Eq. 18 is

— A(t) = -A ( t ) · B(t) dt

Λ( t = I

[19 a]

[19

] The solution matrix A(t) to the differential equation, Eq.

19a, may be determined either by numerical or analytic integration and is defined as the adjoint matrix. It is asserted that the unique property of the adjoint matrix A(t) is that it is the inverse fundamental solution matrix. That is

A(t) · 7l(t) = I [20]

This can be proved by first differentiating Eq. 20, then

. π) -

dt

-1 Λ

dt

κ + A

dt Prom Eqs. 18 and 19

- (Λ · π) dt

Integrating

A(t) ·

= -Λ - Β . π + Λ . Β - π

=

0, the null matrix

= Λ( t ) · π( t ) = I · I =

155

Therefore Eq. 20 holds.

It is obvious then from Eq. 20 that A(t) is the inverse fundamental solution matrix of the homogeneous variational equations. From the theory of matrices, it is also known that

A(t) '

= M t ) · A(t) = I [21]

General Solution of the First Order Perturbation Equations To recapitulate, the general solution of the homogeneous equations, Eqs. 15, and a particular solution to the non- homogeneous equations, Eqs. 14, are sought. The general

solution to the homogeneous equations has been demonstrated by Eq. 16.

Now, consider the time derivative of the product A(t) * AX(t).

dt

A(t) · AX(t)J = — A( t)

dt AX(t) + A(t) — AX(t) dt

[22]

Substituting from Eqs. 14 and 19

— A(t) * AX(t) = -A(t) · B(t) ' AX(t)

dt

-I

+ A(t) ' B(t) · AX(t) + A(t) · i)(t)

= A(t) ' rç(t) [23]

Integrating both sides

A(t) · AX(t) = A(t

) · ΔΧ ( t

) + /

A(r) .

[24]

Multiplying both sides of Eq. 24 by 7l(t)=A

( t ) and recalling that A( t )=I

AX(t) = M t ) · ΔΧ( t

) + J

· A(r) .

[25]

Where ft(t)

A(r) in the integral is recognized as the

Green's function, K(t,r) of the nonhomogeneous equations,

Eqs. 14.

Eq. 25 then represents the general solution to the approxi- mate equations of disturbed motion. The integral term in Eq. 25 is recognized as a particular solution of the nonhomo- geneous equations, as can be seen as follows

Let

T(t) =

I\(t) r

(t)

r

(t)

/ Kit) *

Mr) -

[26]

t

Then

d -+ d t -> ->

— T(t) = 7l(t) ' / Λ(Γ)

K(t) * A(t) * n(t)

dt dt t

Substituting from Eq. 18

— Γ(t) = B(t) · Γ(t) +

(t)

dt ^[27]

where F{t ) = 0, the null vector. Eq. 27 is recognized as the nonhomogeneous variational equations, Eqs. 14, where ) = ?( t) = 0 merely generates a particular solution and hence, Eq. 25 is shown to be the general solution to the nonhomogeneous approximate differen- tial equations of disturbed motion. The general solution Eq. 25 may be more easily interpreted when written in the form

' Δ

( t )

AX(t) =

Ax

(t)

Ax

(t)

=

7t(t) · AX(t T(t) ^[28]

The first term in Eq. 28 represents the propagated disturb- ances due to initial disturbances Δ^(t ) in the generalized

—• ο °

coordinates X(t

) at time t

. The second term represents a bias error introduced by ignored disturbing forces acting upon the particle in configuration space over the time interval [t , t] .

Change of Independent Variable

The previous derivation determined the general solution to

157

the approximate e q u a t i o n s of d i s t u r b e d motion i n terms of d i s t u r b a n c e s in i n i t i a l c o n d i t i o n s and a b i a s term i n t r o d u c e d by d i s t u r b i n g f o r c e s a c t i n g over the time of f l i g h t from i n i t i a l time t . I t i s sometimes n e c e s s a r y to o b t a i n the general s o l u t i o n of the nonhomogeneous e q u a t i o n s which

r e l a t e s a term inal e r r o r ΔΧ(Τ) in terms o f I n i t i a l c o n d i t i o n s s p e c i f i e d at any a r b i t r a r y time A X ( t ) .

S p e c i f i c a t i o n of t ermi nal d i s t u r b e d motion ΔΧ(Τ) i s most e a s i l y accomplished by transforming the f i r s t v a r i a t i o n a l e q u a t i o n s under a change of independent v a r i a b l e from time t to time to go t g , i . e . , l e t

tg = Τ - t = time to go [29]

where Τ = t + t^ = t e r m i n a l time

t f = time of f l i g h t of the p a r t i c l e along the r e f e r e n c e t r a j e c t o r y from i n i t i a l time t

to t a r g e t time or terminal time.

From the above d e f i n i t i o n , i t f o l l o w s t h a t the approximate e q u a t i o n s of d i s t u r b e d motion with r e s p e c t to r e v e r s e time are

d -* d -> dL Γ -» ->

— A X

( t . ) = — ΔΧ · = - B ( t

) · AX(t,<) + r / ( t

d t . 8 dt d t

L

[30]

The matrix d i f f e r e n t i a l equation which generates the fundamental s o l u t i o n matrix with respect to reverse time i s

— M t , ) = - B( t^{r f}) " M t . ) [31a]

dt* β g g

M o )

[31b]

The a d j o i n t matrix d i f f e r e n t i a l equation t h a t g e n e r a t e s the i n v e r s e s o l u t i o n matrix to the transformed system of l i n e a r homogeneous d i f f e r e n t i a l e q u a t i o n s i s

— A

( t

) = + A ( t

) · B ( t

) [32a]

d t

g g g

Λ( ο

[32b]

The general solution to the nonhomogeneous equations is (as before)

AX(t

) = 7i(t

) · ΔΧ(ο) + Ti(t

) JT ^é A(t'

) · i /(t'g)dt'g [33]

Multiplying both sides by TX (tg) = A(tg) and rearranging terms,

AX(tg=o) = A(t

) - AX(tg) - A(t'g) . ^(t

g)dt'g [34]

where tg = Τ - t

It is noted that A(tg) represents the solution matrix of Eq. 32 and not Eq. 19. If the independent variable is trans- formed from tg back to real time t again, the desired pre- diction equation is obtained as

ΔΧ(Τ) = A*(t) · AX(t) + / A*(r) . ï(r)dr [35]

T-t

where, in order to distinguish between the two sets of funda- mental solution matrices and their inverses, the following notation has been adopted

rt*(t) = Mtg) = 71 (T-t) ; solution of Eq. 31 [36a]

A*(t) = A(t

) = A(T-t) ; solution of Eq. 32 [36b]

Examining the integral term above, recalling that in the previous derivation it was shown to be a simple bias term i(tg) is again defined as

T U g ) = -/

^ A(t'g) . ^(t'g)dfg Then

- I f ( tg) = + A(tg) - ^(tg)

a]

f(t^o) = 0 [ 3 7 b ]

Denoting the solution to Eq. 37, which may be obtained by numerical or analytic methods, by Γ (t), the prediction equation for the total disturbance at terminal time Τ is

159

ΔΧ(Τ)

Δ

Δ χ

( Τ )

= A*(t) · Δ Χ Η ) + f*(t) [38]

where Λ (t) = A ( t 32 and 37, respectively.

) and Γ (t) = ^it

) are obtained from Eqs.

Variation of Independent Variable

One of the chief sources of error which must be considered in any analysis is the effect of timing errors. The deri- vations of basic variational relationships above implicitly

assumed that the time of flight along the reference path was constant. The manner in which initial, ascent path, or mid- course timing errors propagate along a trajectory to result in terminal errors in position, velocity, mass, attitude, etc., may be critical for some missions. For example, if a lunar

impact vehicle is injected into an astroballistic trajectory, a timing error of one minute at injection is approximately equivalent to a target displacement (a point on the moon's surface) of 38 statute miles.

The basic relationships derived previously need not be modified if the original nonlinear equations of motion are defined by transforming them into an autonomous system through introduction of a new variable called system time, τ . This new variable simply replaces the old independent variable

absolute time t wherever it appears in f t ) , and thus forces the right-hand side of the equations of motion to represent an autonomous system, i.e., time independent.

In order to be able to perform this substitution, the relationship between absolute time t and system time τ must also be defined by means of a new differential relationship as follows:

d

Let 1 [39 a]

r ( t

)

ο

[39b]

r ( t)

Ο

+ t [39c]

If the new variable τ is considered to represent an χ

n+ 1

coordinate in configuration space, the equations of motion may now be written as

d - d

— X(t) = —

dt dt

X i

( t ) χ (t)

2

η

χ (t) n+i

f(X) [40]

where f(X) is of (n+1) dimensions and replaces f(X,t) in Eq.

4 and

— àt

n+i (t) = f ( χ , χ , · · π+ι i' 2

' n

, χ , χ ) = 1

^{n+i [41]}

For convenience of notation, the vector X(t) is henceforth considered as an η-dimensional vector rather than n+1 by tacitly assuming that the new differential equation, Eq. 41, is included in the original system of differential equations of motion.

One immediate effect of this assumption is that the last row of the B(t) matrix, Eq. 14, becomes (0, 0, 0) and also, because of the form of Eq. 41, timing errors Δγ ( t ) are propagated as constants along a trajectory in the absence of forcing functions. This does not mean that the time-error induced effects in position, velocity, mass, attitude, etc., are also propagaged as constants.

The forcing function acting on system time 7/

(t) may in one application represent the time-dilation of non-Newtonian mechanics, or in another application it may represent the

frequency drift of mechanical origin such as that from a vehicle-borne clock.

IGNORED FORCES

The treatment of ignored forces in the analysis of the motion of a body or vehicle under the force laws of celestial mechanics (astrodynamics) usually results in one of two basic types of investigation.

In one form of investigation, it is assumed that the force laws are adequately known to a specified degree of accuracy, and it is desired to determine the effects of uncertainties in the force laws upon the precomputed trajectory and, in particular, the induced terminal error or miss components

161

of the state vector X(t). The second form of investigation is concerned with the problem of determining the unknown disturb- ing forces themselves which cause observed disturbed motion of a space vehicle.

Both types of investigation will be described; the first type, the treatment of assumed ignored forces, will be con- sidered in some detail, whereas the second type, the deter- mination of the disturbing forces, will be considered only briefly.

Treatment of Assumed Ignored Forces

The problem at hand is the prediction of effects propagated in time along a reference trajectory which are induced by uncertainties or inaccuracies in the assumed force laws.

Implicitly in this statement, it is assumed that the adopted force laws are coincident with the unknown laws of nature.

Practically, this discussion is concerned with the effects of ignoring certain terms in the equations of motion (such as planetary perturbations), the effects of numerical integration algorithm (or machine induced) truncation and roundoff errors (which may be thought of as fictitious disturbing forces), and the effects of uncertainties in the physical constants in the adopted equations of motion. It is noted that variations in the force laws may be considered as conceptual variations, such as an inverse-cubed law of gravitational attraction, or they may pertain to a variation in the physical constants only, leaving the mathematical form of the equations intact;

or they may be a combination of the two possibilities mentioned above.

Eq. 7 defines the difference between the unknown laws of nature and the formulated or adopted laws as

η (t) = f(X;Gt, t) - ~f(X;Ct,t)

where η it) = i/(X, t) since X = X(t) = the reference trajectory.

For consistency, ^(t) is now defined as

ï ?(t) = f(X;2,t) - f(X;«,t) + Aiy(t) [42]

where Δ77 ( t ) represents the difference between human knowl-

edge of the force laws and the actual laws and assume that

this difference is of second order in effect and, hence,

ignorable. The quantity f(^, t) then represents the adopted

laws, and t) represents the practical laws used for

precomputation of reference paths in lunar mission problems.

In studying the effects of uncertainties in the physical constants a, as defined by Eq. 6, it is convenient to expand f(X;tt,t) about the point t) in Taylor's series so that

- » /df.\

ηit) = 7/(X

a,t) s / - ^

Δα [43]

where the disturbance in the physical constants vector is

a ^x - a ¹

Δα =

α - α m m

and the generalized gradient of f with respect to the α coordinates is

df/

(9a,

d

f !

df/ df²

df² df²

da ¹ da ²

d?r,

<?f η η

da

m

da

m

da

Numerical Integration Algorithm Errors

The problem of predicting or estimating numerical integration algorithm errors can be treated under the category of ignored forces. Conceptually it is easy to visualize truncation and roundoff errors, which propagate in time through the closed loop system, as representing a disturbed motion about a reference (analytic) trajectory and induced by a fictitious disturbing force η(t). Eq. 14 is used to estimate the error as a function of time once η it) is specified. Specifically, rj it) is the vector whose η coordinates are the η stepwise

truncation and roundoff error estimates. Care must be used in specifying the stepwise truncation error for a coordinate which represents a double integration.

16J

R e f s . 1 1 - 1 6 in t h e E n g l i s h l i t e r a t u r e d e m o n s t r a t e t h e d e t a i l s o f t h i s t y p e o f n u m e r i c a l i n t e g r a t i o n e r r o r a n a l y s i s , w h e r e a s t h e R u s s i a n s V . P. M y a c h i n a n d A. S . S o c h i l i n a i n R e f s . 1 7 a n d 1 8 h a v e d e m o n s t r a t e d a m o r e d e t a i l e d a n a l y s i s , w i t h n u m e r i c a l e x a m p l e s , u s i n g C o w e l l ' s i n t e g r a t i o n a l g o r i t h m a p p l i e d t o t h e e q u a t i o n s o f m o t i o n o f J u p i t e r , U r a n u s , a n d S a t u r n .

D e t e r m i n a t i o n o f D i s t u r b i n g F o r c e s

T h e d e t e r m i n a t i o n o f u n k n o w n d i s t u r b i n g f o r c e s w h i c h c a u s e a s p a c e v e h i c l e t o d e p a r t f r o m i t s r e f e r e n c e t r a j e c t o r y i n t o d i s t u r b e d m o t i o n i m p l i e s t h a t o b s e r v a t i o n s o f t h e d i s t u r b e d p a t h X ( t ) at a r b i t r a r y i n s t a n t s o f t i m e t i , t² , t^m a r e a v a i l a b l e . T h e s e o b s e r v a t i o n s m a y b e c o m p l e t e at a n y t i m e i n s t a n t o r t h e y m a y b e i n c o m p l e t e , i . e . , s o m e o r a l l o b s e r - v a t i o n s o n t h e η - c o o r d i n a t e s o f )?(t) m a y b e a v a i l a b l e . T h e n u m e r i c a l d i f f e r e n c e b e t w e e n t h e o b s e r v e d s t a t e v e c t o r Xi t ) a n d t h e r e f e r e n c e s t a t e v e c t o r at t h e s a m e t i m e i n s t a n t X ( t ) r e p r e s e n t s t h e o b s e r v e d d i s t u r b e d m o t i o n , i . e .

^ o b s ^ ' - ^ o b s ' t ' - * r e f( t)

I f o b s e r v a t i o n s a r e c o m m e n c e d at a p a r t i c u l a r i n s t a n t o f t i m e t f r o m w h i c h a c o m p l e t e r e f e r e n c e t r a j e c t o r y i s d e t e r m i n e d , t h e n it m a y b e a s s u m e d t h a t ks( to) i-s

i d e n t i c a l l y z e r o a n d r e c a l l i n g E q . 2 5 , t h e n

A* o b s( t)

S J

( t , r )

l(r)dr ^[44]

%

w h e r e t h e G r e e n ' s f u n c t i o n o f E q . 1 4 is K(t,r ) = K{t) m

Mr )

T h e m a t r i c e s K i t ) a n d A( t ) a r e g e n e r a t e d b y s o l u t i o n o f t h e m a t r i x d i f f e r e n t i a l E q s . 1 8 a n d 1 9 . T h e d i s t u r b i n g v e c t o r η i t ) a b o v e i s d e f i n e d b y E q . 4 2 a n d i s t h e o b j e c t o f t h e

i n v e s t i g a t i o n .

E q . 4 4 i s a h o m o g e n e o u s l i n e a r m a t r i x i n t e g r a l e q u a t i o n o f V o l t e r r a t y p e . T h e e q u a t i o n m a y b e s o l v e d n u m e r i c a l l y w i t h o u t u n d u e d i f f i c u l t y f o r η i t ) u s i n g f i n i t e d i f f e r e n c e m e t h o d s .

T h i s a p p l i c a t i o n o f t h e c a l c u l u s o f p e r t u r b a t i o n s h a s b e e n r e p o r t e d o n i n d e t a i l b y t h e R u s s i a n A. A. D e b e r d e e v i n b o t h t h e s i s w o r k ( 1 9 5 5 ) a n d m o r e r e c e n t l y in R e f . 2 0 . M e t h o d s o f s o l u t i o n o f t h e i n t e g r a l e q u a t i o n , E q . 4 4 , a r e s u m m a r i z e d a n d c o m p a r e d in R e f . 2 1 .

N E W T O N I A N E Q U A T I O N S O P M O T I O N IN T H E E A R T H - M O O N S Y S T E M N - b o d y P r o b l e m

T h e d i f f e r e n t i a l e q u a t i o n s o f m o t i o n f o r t h e N - b o d y p r o b l e m w i l l f i r s t b e d e f i n e d i n b o t h a b s o l u t e a n d r e l a t i v e f o r m a n d t h e n m o d i f i e d to i n c l u d e a t m o s p h e r i c a n d n o n p o i n t m a s s g r a v i t a t i o n a l f i e l d f o r c e s . T h e d e r i v a t i o n e s s e n t i a l l y f o l l o w s t h e e x c e l l e n t t r e a t m e n t o f t h e s u b j e c t f o u n d i n C h a p . 7 o f R e f . 1 0 .

T h e a b s o l u t e f o r m i s w i t h r e s p e c t to an a r b i t r a r y i n e r t i a l o r s p a c e s t a b l e c a r t e s i a n c o o r d i n a t e f r a m e o f r e f e r e n c e e i t h e r in u n i f o r m m o t i o n o f p u r e t r a n s l a t i o n o r at r e s t in i n e r t i a l s p a c e . T h e r e l a t i v e f o r m o f t h e e q u a t i o n s o f m o t i o n i s r e l a t i v e t o a n i n e r t i a l o r s p a c e s t a b l e c a r t e s i a n c o o r d i n a t e f r a m e o f r e f e r e n c e w h o s e o r i g i n c o i n c i d e s w i t h t h e c e n t e r o f m a s s o f o n e o f t h e N - b o d i e s . T h e r e l a t i v e f o r m o f t h e e q u a t i o n s o f m o t i o n w i l l b e t h e m o s t u s e f u l s i n c e t h e p o s i t i o n s o f t h e s u n , p l a n e t s , m o o n s , a s t e r o i d s , a n d o t h e r m a s s i v e b o d i e s i n t h e s o l a r s y s t e m a r e k n o w n t a b u l a r q u a n - t i t i e s i n o n e o r m o r e c o o r d i n a t e f r a m e s w h o s e o r i g i n i s

u s u a l l y a s s u m e d t o b e l o c a t e d at a d y n a m i c a l c e n t e r , i . e . , t h e c e n t e r o f m a s s o f t h e m o o n o r t h e p l a n e t E a r t h .

A b s o l u t e P o r m o f E q u a t i o n s o f M o t i o n

A N e w t o n i a n f r a m e o f r e f e r e n c e is a s s u m e d w h i c h i s c o m p o s e d o f a r i g h t - h a n d e d a n d o r t h o g o n a l s e t o f a x i s w i t h o r i g i n l o c a t e d i n i t i a l l y at a n y c o n v e n i e n t p o i n t i n t h r e e - d i m e n s i o n a l s p a c e a s i n d i c a t e d in P i g . 3.

I t i s a s s u m e d t h a t w i t h i n t h e N - b o d y s y s t e m , i . e . , t h e v o l u m e o f s p a c e o c c u p i e d b y t h e Ν b o d i e s , t h e r e e x i s t f o r c e s w h i c h a c t o n e a c h o f t h e b o d i e s . T h e s e f o r c e s a r e o f t w o t y p e s , i n t e r n a l a n d e x t e r n a l f o r c e s . T h e i n t e r n a l f o r c e s a r e d e f i n e d t o b e t h e N e w t o n i a n g r a v i t a t i o n a l f o r c e s c a u s e d b y t h e m u t u a l g r a v i t a t i o n a l a t t r a c t i o n o f e a c h o f t h e b o d i e s f o r e a c h o f t h e r e m a i n i n g b o d i e s . N e w t o n ' s T h i r d L a w i s a l s o a s s u m e d so t h a t t h e a t t r a c t i o n o f o n e b o d y f o r a n y o t h e r b o d y is e q u a l a n d o p p o s i t e t o t h e a t t r a c t i o n o f t h e s e c o n d b o d y f o r t h e f i r s t . N e w t o n ' s L a w o f G r a v i t a t i o n t h e n s t a t e s t h a t t h e f o r c e a c t i n g o n t h e i t h b o d y d u e t o t h e g r a v i t a t i o n a l a t t r a c t i o n o f t h e j t h b o d y i s

i n t

[45]

3

I65

w h e r e "r^j i s t h e p o s i t i o n v e c t o r d i r e c t e d f r o m t h e i t h b o d y t o t h e j t h b o d y

k2 = t h e u n i v e r s a l o r G a u s s i a n g r a v i t a t i o n a l c o n s t a n t mi = m a s s o f t h e i t h b o d y

m- = m a s s o f t h e j t h b o d y J

-»

rH = r — = m a g n i t u d e o f t h e v e c t o r ri i

It s h o u l d b e n o t e d t h a t t h e u n i t s o f m a s s , f o r c e , a n d d i s - p l a c e m e n t m u s t b e c h o s e n t o b e c o n s i s t e n t in t h e a b o v e e q u a t i o n .

-> i n t

T h e t o t a l i n t e r n a l g r a v i t a t i o n a l f o r c e F^ o n t h e i t h m a s s p o i n t is t h e s u m .of t h e i n d i v i d u a l g r a v i t a t i o n a l f o r c e s

a c t i n g o n t h e i t h m a s s c a u s e d b y t h e (N_l) r e m a i n i n g m a s s e s ; i . e .

Ν k2mimk

(F\ )

- Σ - *

[46]

ri k k ^ i

w h e r e k is a d u m m y v a r i a b l e o f s u m m a t i o n a n d s h o u l d n o t b e c o n f u s e d w i t h k2.

e x t

T h e t o t a l e x t e r n a l f o r c e (F^) a c t i n g o n t h e i t h b o d y m a y b e t h e s u m o f s u c h f o r c e s o t h e r t h a n t h e g r a v i t a t i o n a l f o r c e s , s u c h as a t m o s p h e r i c d r a g o r l i f t f o r c e s , e l e c t r o m a g n e t i c r a d i a t i o n f o r c e s , p o w e r e d r o c k e t t h r u s t f o r c e s . T h e t o t a l f o r c e a c t i n g o n t h e i t h b o d y r ^ is t h e n

-* -* i n t -> e x t

F

= (F ^L ) ^{+ (F}

⁾

o r t h e d i f f e r e n t i a l e q u a t i o n s o f m o t i o n f o r t h e N - b o d y p r o b l e m in a b s o l u t e f o r m i s

N

d2r; ^ k m

k2

"m, Σ ; .

( F . )

[47]

1 l , — 1 - , 3 I K 1

d t2 =1 Fi k k

k ^ i

R e l a t i v e F o r m o f E q u a t i o n s o f M o t i o n

T h e r e l a t i v e f o r m o f t h e e q u a t i o n s o f m o t i o n s o f a s p a c e v e h i c l e u n d e r e x t e r n a l f o r c e s a n d i n t h e g r a v i t a t i o n a l f i e l d

of N-bodies is more commonly used in trajectory and guidance sensitivity studies, since the equations are usually defined relative to a coordinate or reference frame whose origin coincides with the center of mass of a planet or the sun or moon. These coordinate frames are especially convenient,

since ephemerides or tables of position coordinates of the bodies in the solar system, as well as stars, are referenced to either a heliocentric or geocentric coordinate frame. A standard set of coordinate frames will be defined and dis- cussed later.

Prom Pig. 1 it is evident that

[48]

Differentiating Eq. 48 twice d

dt dt

d dt

[49]

d

d

d

d t

d t

d t

[50]

Using Eq. 47 then

d

i i = 2 k

d t

^

Ν

Ν _m.

k=l r

J

k=l r *

ext

ext

[51]

167

R e w r i t i n g E q . 5 1 g i v e s

d2 _ d t2

i k

•ik

z * e x t -» e x t

(F.) (F.)

[52]

B u t

->

τ . . = — r · · : r · · -

J i i j * J i 1J

T h e r e f o r e , E q . 5 2 m a y b e w r i t t e n a s

->

d2

d t2

^ - k2( m . + m ·

ri j =

Ν

ri j " k = l

i k

[53]

ζ * e x t -* e x t

(Fj) (F.)

[54]

w h i c h is t h e d i f f e r e n t i a l e q u a t i o n o f r e l a t i v e m o t i o n f o r t h e N - b o d y p r o b l e m o f t h e j t h b o d y ( o r s p a c e v e h i c l e ) r e l a t i v e t o a N e w t o n i a n f r a m e at t h e i t h b o d y .

It i s c o n v e n i e n t , at t i m e s , t o s e p a r a t e t h e a c c e l e r a t i o n s c o n t a i n e d in t h e r i g h t - h a n d m e m b e r o f E q . 5 4 i n t o t w o p a r t s : o n e p a r t c o r r e s p o n d i n g to t h e a c c e l e r a t i o n p r o d u c i n g K e p - l e r i a n o r t w o - b o d y m o t i o n , a n d t h e s e c o n d p a r t c o n s i d e r e d as K e p l e r i a n p e r t u r b i n g a c c e l e r a t i o n s a c t i n g o n t h e t w o - b o d y r e f e r e n c e m o t i o n ; i . e .

d2

d t2

r · · + r · ·

d

d t

Ν

Σ

k=l

l

jk

i k

?L

ext -» ext

[56]

and the two-body acceleration r and

£he Keplerian perturbing accelerations of the N-body problem r are

r = r

[57]

Ν

Ί

= k* Σ -"kl^

k=l

-» ext -» ext

+ - [58]

•P 3 -p 3

i

r

j k ^

•ik m-

where k

= Gaussian or universal gravitational constant

^ij = (m^ + mj) = reduced mass of the two-body problem

m

k = mass of any one of the (N-2) remaining bodies The first term under the summation sign in Eq. 52 is commonly referred to in celestrial mechanics as the direct term or direct acceleration of the kth body on the space vehicle, whereas the second term is called the indirect term or indirect acceleration of the kth body on the ith body. In numerical computation of the solution to Eq. 56, the indirect term oftentimes is a tabular quantity, since it may be pre- computed regardless of the motion of the space vehicle, and stored in the memory unit of a digital computer or in tables for hand calculations.

Modified Relative Form Due to Non-N-body Gravitational Forces The treatment of the N-body problem described so far is to consider each of the N-bodies as a gravitating point-mass in the possible presence of external forces acting on the system.

169

It is now assumed that each of the massive bodies, i.e., planets or sun, has the shape and gravitational field of an oblate ellipsoid and the necessary added terms are developed in the relative form of the generalized equations of motion.

The only exceptions to the above assumptions will be the shape and gravitational field of the moon and a space vehicle.

The vehicle will be considered as a mass point of negligible mass so that μ — Ξ m. + = m^, and the moon will be treated as a rigid body with triaxial ellipsoid symmetry with a resulting triaxial gravitational field vector intensity function. The assumption is made that the gravitational accelerations caused by one massive body upon any other massive body are simply as defined by Eq. 45; i.e., the moon, planets, and sun attract one another as if they were point masses, by reason of the distances involved. The negligible mass space vehicle, however, is assumed to be acted on by nonpoint mass gravitational forces arising from the oblate ellipsoidal gravitational field of the massive body (planet or sun) at the dynamical center or origin of the coordinate frame in use. The single exception will be the lunar non- point mass forces when the geocentric coordinate frame is in use or when the center of mass of the moon is at the dynamical center.

The standard form for the gravitational potential of an oblate ellipsoid (planet) is

Φ1( γ )

2

k m^

5 —

„ 3

30 r

30 35 [59]

where i = the planet

a^ = equatorial radius of the ith planet

JN, FL, = second, third, and fourth spherical harmonic coef fi ci ents

r = r- · = xi + yi + zi , position vector of the j-j χ y ζ

jth body or space vehicle of negligible mass

relative to the ith body

T h e g r a v i t a t i o n a l f i e l d i n t e n s i t y v e c t o r g ^ i r ) o f t h e i t h o b l a t e p l a n e t i s d e f i n e d i n c a r t e s i a n i n e r t i a l c o o r d i n a t e s a s

(9Φ. <9Φ. (9Φ.

V r ) = V<*.(x,y,z) = — i

+ — i

+ — - i

[60]

d ^χ dy d ^ζ

T h e s t a n d a r d o b l a t e p l a n e t p o t e n t i a l f u n c t i o n a s g i v e n b y E q . 6 0 m a y t h e n b e u s e d t o o b t a i n

w h e r e c a r e m u s t b e t a k e n i n t h e n u m e r i c a l e v a l u a t i o n o f t h e a b o v e e x p r e s s i o n s f o r t h e c a s e ζ = 0 s o a s t o a v o i d n u m e r i c a l i n s t a b i l i t y .

T h e s t a n d a r d r e l a t i v e f o r m o f t h e g e n e r a l i z e d - e q u a t i o n s o f m o t i o n o f a p o i n t m a s s r e l a t i v e t o a N e w t o n i a n c a r t e s i a n r e f e r e n c e f r a m e w h o s e o r i g i n c o i n c i d e s w i t h t h e c e n t e r o f

1 7 1

m a s s o f t h e s u n o r a n y p l a n e t o r p l a n e t o i d in t h e s o l a r s y s t e m m a y t h e n b e w r i t t e n as

a* _

h

e x t

(F.) e x t

[62]

w h e r e g^ i s g i v e n b y E q . 6 1 .

It s h o u l d b e n o t e d t h a t t h e a b o v e e q u a t i o n s E q s . 6 2 d o n o t c o n t a i n t e r m s c o r r e s p o n d i n g t o t h e p e r t u r b i n g g r a v i t a t i o n a l a c c e l e r a t i o n s i m p o s e d u p o n t h e s p a c e v e h i c l e ( t h e j t h b o d y ) b y E a r t h ' s t r i a x i a l m o o n w h e n t h e i t h b o d y is t a k e n to b e E a r t h o r w h e n t h e i t h b o d y ( a n d h e n c e , t h e d y n a m i c a l c e n t e r ) is t a k e n to b e t h e m o o n . T h e s e c a s e s a r e c o n s i d e r e d as e x c e p t i o n s to t h e s t a n d a r d r e l a t i v e f o r m , E q . 6 2 , a n d t h e l a s t c a s e is t r e a t e d n e x t .

T h e s t a n d a r d f o r m f o r t h e g r a v i t a t i o n a l p o t e n t i a l f u n c t i o n f o r t h e t r i a x i a l e l l i p s o i d a l m o o n i s

( (Δ

k2m .

(Δ

J.

l +

«Δ

+

V

3

V

V

1 - 3

V

[63]

w h e r e r

«Δ

m ^ = m o o n ' s m a s s

= s p h e r i c a l h a r m o n i c c o e f f i c i e n t s

T h e c o r r e s p o n d i n g g r a v i t a t i o n i n t e n s i t y v e c t o r , gy

r e l a t i v e t o t h e c a r t e s i a n ( n o n i n e r t i a l ) s e l e n o - g r a p h i c c o o r d i n a t e f r a m e i s

:

( [ Δ

ί Δ

'

(9Φ

V

<[Δ ' ( Δ (Δ

W h e r e t h e u n i t v e c t o r t r i a d o f t h e s e l e n o g r a p h i c f r a m e i s d e n o t e d b y x ^ , y ^ , a n d z ^ a n d

[ 6 5 ]

C a r r y i n g o u t t h e i n d i c a t e d p a r t i a l d i f f e r e n t i a t i o n

5 Χ

"(Δ

«Δ

<?Φ„

*<Δ

V V

V

-

V«A

V

.

< -

V

V W

- 5

V

-"Vita

W V \ _ ⁵ V

1 - 5

r 2

(Δ

r 2

(Δ

[ 6 6 A]

[ 6 6B]

[ 6 6C]

175

I t i s a p p a r e n t f r o m t h e a b o v e e q u a t i o n s t h a t t h e p o i n t m a s s f i e l d e x p r e s s i o n s m a y b e s e p a r a t e d f r o m t h e r e m a i n d e r o f E q . 6 6 so t h a t

V W - t ^ - V & ' V

da

S i n c e t h e f o r e g o i n g e x p r e s s i o n f o r g^ r e f e r s to t h e n o n - i n e r t i a l s e l e n o g r a p h i c c o o r d i n a t e f r a m e t h a t is d e f i n e d b y t h e r o t a t i n g and l i b r a t i n g p r i n c i p a l a x i s o f s y m m e t r y o f t h e m o o n ' s f i g u r e , t h e r e l a t i v e e q u a t i o n s o f m o t i o n o f a s p a c e v e h i c l e a r e w r i t t e n r e l a t i v e t o a N e w t o n i a n f r a m e w i t h c o m m o n o r i g i n to t h e s e l e n o g r a p h i c f r a m e . T h i s c h o i c e s i m p l i f i e s t h e r e s u l t i n g e x p r e s s i o n s a n d c a l c u l a t i o n s b u t r e q u i r e s t h e i n t r o d u c t i o n o f t h e s e l e n o g r a p h i c r o t a t i o n m a t r i x 0( t ) , w h i c h is d i s c u s s e d l a t e r . T h e e q u a t i o n s o f m o t i o n o f a s p a c e v e h i c l e r e l a t i v e to an i n e r t i a l f r a m e w i t h o r i g i n at t h e m o o n ' s c e n t e r o f m a s s a n d w i t h i , a n d i c o p l a n a r t o E a r t h ' s e q u a t o r i a l p l a n e a r e

k [68]

r A

k

A \

w h e r e r ^ = p o s i t i o n v e c t o r f r o m m o o n c e n t e r t o v e h i c l e r_ = p o s i t i o n v e c t o r f r o m e a r t h c e n t e r t o v e h i c l e r k = p o s i t i o n v e c t o r f r o m v e h i c l e t o t h e k t h b o d y

( p l a n e t o r s u n )

r ^k = p o s i t i o n v e c t o r f r o m t h e m o o n c e n t e r t o t h e k t h b o d y ( p l a n e t o r s u n )

T h e e q u a t i o n s o f m o t i o n o f a s p a c e v e h i c l e r e l a t i v e t o an i n e r t i a l f r a m e w i t h o r i g i n at E a r t h ' s c e n t e r o f m a s s u n d e r t h e p e r t u r b i n g a c c e l e r a t i o n s o f b o t h t h e o b l a t e e a r t h g r a v i t a t i o n a l f i e l d a n d t h e t r i a x i a l m o o n g r a v i t a t i o n f i e l d a r e

-k

d t2 Δ Φ

% Δ³

- ^Γ Θ Δ ⁺ V W +0 () ' W -t

Ν k=l

?. e x t -» e x t

FA F0

r

ûk

*$k

i

[69]

w h e r e

Δ ( « Δ

\ k

% k - ^ φ

A t m o s p h e r i c R e a c t i o n

A t m o s p h e r i c r e a c t i o n f o r c e s , i . e . , d r a g , s i d e , a n d l i f t f o r c e s a r e c o n s i d e r e d as e x t e r n a l f o r c e s a c t i n g o n l y o n t h e v e h i c l e a n d n o t o n a n y o f t h e r e m a i n i n g b o d i e s i n t h e N - b o d y p r o b l e m . H e n c e , t h e t e r m F ^e xV m ^ i s a s s u m e d t o b e

i d e n t i c a l l y z e r o i n t h e e q u a t i o n s o f m o t i o n d e v e l o p e d u p t o t h i s p o i n t . I f t h e s p a c e v e h i c l e , w h i c h w a s c o n s i d e r e d a s a p o i n t m a s s i n t h e p r e c e d i n g d i s c u s s i o n , i s i n r e a l i t y a

" w i n g e d " v e h i c l e , t h e n a r e l a t i v e w i n d - f i x e d c o o r d i n a t e f r a m e m u s t b e d e f i n e d a s w e l l a s t h e f o r c e l a w s .

B y c o n v e n t i o n , d r a g f o r c e s a r e a s s u m e d t o b e a c t i n g i n t h e d i r e c t i o n o f t h e r e l a t i v e w i n d w h e r e t h e r e l a t i v e w i n d v e l o c i t y v e c t o r i s s i m p l y t h e a p p a r e n t v e l o c i t y o f t h e

a t m o s p h e r e w i t h r e s p e c t t o a n a s s u m e d m o t i o n l e s s v e h i c l e . I t s h o u l d b e n o t e d t h a t o t h e r c o n v e n t i o n s e x i s t i n w h i c h d r a g f o r c e s a r e d e f i n e d a s t h e v e c t o r c o m p o n e n t o f t h e t o t a l a t m o s p h e r i c r e a c t i o n f o r c e r e s o l v e d a l o n g t h e l o n g i t u d i n a l a x i s o f a v e h i c l e . U s i n g t h e a d o p t e d c o n v e n t i o n , t h e r e m a i n - ing t w o o r t h o g o n a l c o m p o n e n t s o f r e a c t i o n f o r c e m u s t l i e b y n e c e s s i t y i n t h e p l a n e n o r m a l t o t h e r e l a t i v e w i n d v e c t o r . T h e s e t w o c o m p o n e n t s , w h i c h w i l l b e d e f i n e d , a r e t h e l i f t f o r c e a n d s i d e f o r c e v e c t o r c o m p o n e n t s o f t h e t o t a l r e a c t i o n f o r c e .

T h e a t m o s p h e r e o f a p l a n e t i s a s s u m e d to^ b e r o t a t i n g w i t h t h e p l a n e t1s r o t a t i o n a l v e l o c i t y v e c t o r , Ω ι . H e n c e , t h e

175

v e l o c i t y o f t h e a t m o s p h e r e w i t h r e s p e c t to i n e r t i a l s p a c e at t h e p o s i t i o n o f a v e h i c l e is

VA(R I) = Ω . Χ Γ - J J

[70]

T h e v e l o c i t y o f t h e a t m o s p h e r e V ^ , r e l a t i v e to w h i c h h a s an i n e r t i a l v e l o c i t y ( d / d t ) r -

t h e r e l a t i v e w i n d

a v e h i c l e i s t h e v e l o c i t y o f

VR =

d t

[71]

In i n e r t i a l g e o c e n t r i c - e q u a t o r i a l c o o r d i n a t e s , a s s u m i n g Üi

to be d i r e c t e d a l o n g iz, E q . 7 1 m a y b e w r i t t e n

w h e r e

VR

= - i J

+

y) +

Ωχ) +

Ω;

y y [72]

d t

V i + V i + V i x x y y ζ ζ

T h e p r o j e c t i o n s o f t h e t o t a l a t m o s p h e r i c r e a c t i o n f o r c e v e c t o r o n t o t h e p l a n e n o r m a l t o t h e r e l a t i v e w i n d v e l o c i t y v e c t o r i n t h e d i r e c t i o n s o f VDx ~ r a n d VDx ( VDx r ) a r e d e f i n e d

as t h e s i d e - f o r c e S a n d t h e l i f t - f o r c e v e c t o r s , L , r e s p e c - t i v e l y . T h e r e m a i n i n g c o m p o n e n t o f t h e t o t a l a t m o s p h e r i c r e a c t i o n v e c t o r is i n t h e d i r e c t i o n o f t h e r e l a t i v e w i n d v e c t o r a n d i s d e f i n e d a s t h e d r a g - f o r c e D . T h e t r i a d D , S,

a n d L f o r m s a m u t u a l l y o r t h o g o n a l s e t w i t h u n i t v e c t o r s

iD ' a n d w h e r e

R- VX - Q y >

f-V

^{+ Ωχ^}

ⁱ

^I — ^{1 [73]}

VRx r VRr

y Vz - z Vy + Ωχζ + i y

z Vx " x Vz + ß z y

VD * r

+ ir

x V^y - y V^x - ^Ω( χ 2 + y2

VR * r

[74]

XL = iDx iS =

- ( Vy- O x ) x Vy- y Vx ) - Ω ( x2+ y2 jj

+ Vz [( z Vx- x Vz +O y z ]

"Vz ((yVz-zVy)+OxzJ

+ ( Vx+ ny) [ u Vy- y Vx) - n(x2 + y 2 jj

- ( Vx+ O y ) ^ z Vx- x Vz) + nyz j

+ (Vy-Ωχ) £ ( y Vz- z Vy) + O x z

/ith vR = + £ νχ+ Ω7) 3 + ( V y - Ω χ )2 +

v ^z2 J

r = + [ x2 + y2 + z2]

1/2 1 / 2

[75]

[76a]

[76b]

D r a g , s i d e , a n d l i f t f o r c e l a w s a r e t h e n d e f i n e d i n f o l l o w i n g m a n n e r

D » 1/2 C u V V j ,

S Ξ 1/2 C g A g p VR 2

L *

1/2

[77 a]

[77 b]

[77c]

w i t h D = D i p , S = S is, L = L iL [78]

177