Injection Conditions for Lunar Trajectories

8/3/2019 Injection Conditions for Lunar Trajectories

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X- 643-65-430

INJECTION CONDITIONSFOR

LUNAR TRAJECTORIES

B YR ON A LD K OLE N K I E W I C ZWILLIAM PUTNEYi tI ! ! N 66 - 17 26 2

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NOVEMBER 1965Microfiche (MF) .5-fie53 July65

I GODDARD SPACE FLIGHT CENTERGREENBELT, MARYLAND


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// X-643 -65 -430

INJECTION CONDITIONS FOR LUNAR TRAJECTORIES

Ronald KolenkiewiczWilliam Putney

November 1965

Goddard Space Flight CenterGreenbelt, Maryland


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ABSTRACTThis paper reviews the mechanics of Earth to Moon trajectories as

affected by geo metric considerations and boos ter capabilities. It formu-lates and de scr ibe s the equations fo r a computing procedure which, usingthe two body equations of motion, prov ides approximate initi al injectionconditions ne ar the ear th for either a fixed time of ar ri va l at the Moon o ra fixed tim e of flight to the Moon. Considera tion of multiple c i rc ularparking orb its, a rb itr ar y injection elevation angles, ar bi tr ar y launch s ite ,and booster burning cha rac ter isti cs a r e al so taken into account. Thetraj ect ory followed is an elliptical Earth to Moon tra ns fer tr ajecto ry, withrespect to the Ea rth , which in ter sec ts the Moon before apo-apsis. Adigital computer program having the equations programmed in FORTRANI1 is available.

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CONTENTSPage

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1II. NOMENCLATURE . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . 2

111. GENERAL DISCUSSION OF THE PROBLEM . . . . . . . . . . . . . . . . 6A . Requirement fo r a Translunar Trajectory . . . . . . . . . . . . . . . . 6B. Launch Considerations. ............................ 6C Boost Co nsiderations 6..............................D. Parking Or bit Considerations ........................ 8IV. SOLUTION O F THE PROBLEM ......................... 8

8. Time of Launch into Pro pe r Plane .....................B. Motion in the Plane ............................... 14

1.2.3.4 .5.6.78.

First Boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Parking Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Second Boost ................................. 16Translunar Trajectory .......................... 16Range of Velocity R atios and Flight Tim es . . . . . . . . . . . . .Obtaining Successive Velocity Ratios . . . . . . . . . . . . . . . . .Initial Injection Conditions ........................ 21Solutions f or Successive Orbits .....................

182024

V. REFERENCES ..................................... 26APPENDIX A . ERIVATIONS............................. 27APPENDIX B. ALCULATING PROCEDURE . . . . . . . . . . . . . . . . . . 34

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Figure1234

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67

LIST OF ILLUSTRATIONSPage

79

Trajectories fo r a Lunar Mission . . . . . . . . . . . . . . . . . . . .Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Geometry of the Launch Planes ..................... 11Position of the Launch Site Relative to Greenwich . . . . . . . . .Geometry of the Transfer Ellipse ....................Regula Falsi Iteration ............................ 22Successive Parking Orbit Solutions . . . . . . . . . . . . . . . . . . .

1315

25

V


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I. INTRODUCTIONA calculating procedure, using two body equations of motion, has been de-

veloped which provides approximate injection conditions for earth to moon traj-ectories. The resulting injection conditions a r e intended fo r use as a first guessin generating precision tr aje cto rie s. When the r esu lts obtained are used in aprecision digital computer progr am (for example ITEM, see Reference 1) theytypically produce a lunar impact. They can be furt he r adjusted by itera tin g inthe precision program to provide a specified miss distance for such missionsas a lunar orbiter.

The procedure p resen ted builds on and extends the work of References 2 to4. In Reference 2 a study w a s made to determine the effects on lunar tra je cto rie sof so me typical geometri c constraints. This study concludes that application ofthe various constraints seriously res tric ts th e allowable launch tim es during themonth and day fo r direct-ascent launch; whereas, less serious restrictions re-sult fo r the parking orbit launches. This reference al so presents some of theequations fo r the calculation of geometrical pa ra me te rs involved. Reference 3fu rth er justifies the parking orbi t type of traj ectory . It also presents some ofthe equations neces sar y fo r matching the powered phases of the tra jec tor y to thegeometrical constraints.t rajectories . Reference 4 gives some general discussion on lunar

The prese nt pa per adds the consideration of multiple ci rc ul ar parking orbi ts ,ar bi tr ar y injection elevation (or flight path) angle, and booster burning charac-te ri st ic s. Injection conditions can be deter mined fo r ei th er fixed time of arr iva lat the moon o r a fixed flight tim e to the moon. The mechanics of e ar th to moontrajectories as affected by geometrical co nsiderations and booster capabilitiesare reviewed, and the working equations which are used in a computer prog ramare formulated and described.

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11. NOMENCLATURE

i

A

L

Components of geocentric equa torial coordinate system(Figure 2 ) .

Unit vec tors along geocentric equatorial coordinate sys tem(Figure 2 ) .Unit vector to launch site.Launch time, hours.Number of days from January 0.0 U.T. 1960 to 0.0 U.T. onthe day of launch.Unit vector to the Moon.Components of the unit position vector to the Moon.Time of impact at the Moon, hours.Number of da ys from January 0.0 U.T. 1960 to 0.0 U.T. onthe day of arr iva l at Moon.Inclination.

Azimuth (geocentric).Right ascension.Declination.Unit vecto r no rmal to the o rbi t plane.Components of the unit vector normal to the orbit plane(Figure 2).Angle between the and w x l vectors, see Figure 2.Mean longitude of Suns apparent motion around the Ear thdefined by Equation (14), adians.Longitude (terrestrial).

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.w Absolute rate of spin of the Earth.

R l

Tb2

t i

t 2

T

Total tim e of flight , defined by Equation (33), hours.Distance fro m the center of the Ea rth to th e Moon.First boost burning time , ( t , - t l ), hours.First boost angle, Earth centered angle between Rr and El .Tim e of injection into parking orb it, h ours.Tim e in parking orbit, ( t , - t , ), hours.Recipr ocal of the parking orb it ra te , defined i n Equation (18).Parking or bit angle, Earth centered angle between R , and E , .Earth's gravitational parameter.Distance from the center of the Ear th to the parking orbitinjection.Angular distance (Figure 5) .Second boost angle, Ear th centere d angle between R2 and Ri .Translu nar t raje ctor y angle, Earth centered angle between Riand EmSecond boost burning time, ( t i - t , ), hours.Time of injection into tra nsl un ar trajec tory , hours.Time at beginning of second boost, hours.Time in translunar trajectory, ( tm - t i ), hours.

a Semimajor axis.M Mean anomaly.e Eccentricity.

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EPv

R i

YiV i

vP

Y m

Rl

h P O

hi

V C

V m

Vc m

TtA

Eccentric anomaly.Semilatus rec tum of an ell ips e, a (1 - e ' ) .True anomaly.Distance from the center of the E art h to injection into tran s-lunar trajectory.Ratio of injection distance, R i , to moon's distance, R, .Injection elevation angle (See Figure 5).Injection velocity.Para bolic velocity.Ratio of injection velocity, V i , to parabolic velocity, Vp .Rotation matrix, defined by Equation (42).Rotation matrix, defined by Equation (43).Rotation matrix, defined by Equation (44).Year of lunar impact.Distance fro m the center of the E art h to launch site.Parking orbit altitude.Injection altitude.Circu lar velocity.Luna r impact velocity.

Earth's radius (taken as 6378.165 km).Circu lar velocity at lunar distance.Equivalent to T, defined by Equation (15).

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Equivalent to Tt defined by Equation ( 3 3 ) .Tt BAT, Total tim e difference, defined by Equation ( 3 8 ) .

Minimum value of the velocity ratio, defined by Equation (34).Tl- Maximum velocity ratio, defined by Equation ( 3 5 ) .v2(ATt) 1(AT, ) 2

Total time difference,Total time difference,

Equation (38) , fo r T1 , see Figure 6.Equation ( 3 8 ) , fo r c2 see Figure 6.

% Second and succeeding assu med velocity ra tio s, defined byEquation (39).V k + l

Final value of assumed velocity ratio, see Figure 6.TfS Par ame ter defined by Equation (37).

x component of the moon position vector.m xR y component of the moon position vector .m y

z component of the moon posit ion vector.R m z

SUBSCRIPTS1 Launch.1 Start of parking orbit.2 End of parking orbit.i Injection.

m Moon.n The orbit number.

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,111. GENERAL DISCUSSION OF THE PROBLEM

A . Requirements for Translunar TrajectoryIn general, th ere a r e two requ iremen ts that must be met fo r launching a

vehicle from the sur face of the Earth to the vicinity of the Moon with a minimalamount of energy.

The first is that the vehicle be launched into a plane that is common to th elaunch site at launch time and the Moon at impact time. To launch in a planeother than this necessitates changing the plane of the trajectory after launch.This entails a reduction in the payload that will a r r ive in the vicinity of the Moonsince additional f u e l is required to accomplish this maneuver.

The second requirement is that a specific angular relationship mus t existin the plane between t h e center of Earth and launch site and the line of centersof Ear th and Moon. This comes about from the fact that there ar e a number ofrest rict ion s on various lunar missions such as booster capab ilities, allowabletime in parking orb it, and the time requir ements on the mission.

The following sections will dis cu ss the type of trajectory that must be fol-lowed to satisfy requirements on both launch plane and angular travel in the plane.

B. Launch Considera tionsIn general, for any given launch azimuth there are two time s a day a vehicle

may be launched from the Earth into a tra je cto ry that would take it to t he Moon.The exceptions oc cur only when the absolute value of the Moon's declination isgr ea te r o r equal to the inclination of the traj ec to ry plane; then, eit he r none o rat most one launch pe r day will be possible.

C. Boost ConsiderationsThe re ar e two possible methods fo r leaving th e surface of the Ear th and

going to the vicinity of th e Moon. The first is lunar injection from a direc tlaunch; the second is lunar injection fro m a parking orbit. These trajectoriesa r e shown in Figure 1. Both the lunar injection fr om a dire ct launch as well asfrom a parking or bit have been discu ssed in previous p ap er s (References 2 and3).

In this paper attention will be focused on tr aj ec tor ies that a r e launched fro mthe surface of the E arth and employ a parking or bi t before being injected into atranslunar trajectory. It will also be re stri cted to an elliptical E arth to Moon

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TO MOON

BOOSTCOAST

BOOSTER BURNOUT (99% of energy impar tedto veh ic le ) AND I NJECTI ON INTO TRANS-LUNAR TRAJECTORY

( a ) Lunar injection from a direct launch

TO MOON

I NJECTI ONINTO CIRCULARPARK ING ORBIT

LAST STAGE BURNOUT (99% o fenergy impar ted to veh ic le) A N DTRAJECTORYIGNITED INJECTION INTO TRANSLUNAR

(b) Lunar in jection from a p ar k in g o r b i tFigure 1-Traiector ies fo r a lunar mission

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tra nsf er trajectory, with respect to the Earth, which inters ects the Moon beforethe apo-apsis.

D. Parking Orbit ConsiderationIt is envisioned t h a t capability must be provided for the parking orbit to in-

clude m o re than one revolution around the Earth. The purp ose of thi s is to pro-vide time to check out the variou s s yst em s on-board the spa cecr aft and to estab-lish the orbit elements so the position, time and velocity neces sar y fo r sendingthe spacecraft to the Moon can be calculated. In general, the sp acec raft may beinjected to its translunar trajectory once per revolution from its parking orbit.

IV . SOLUTION O F THE PROBLEMA . Time of Launch into Proper PlaneA s stated previously, one of the requirement s that must be satisfied for avehicle leaving the surface of the Earth and arriving in the vicinity of the Moon

with a minimum amount of energy expended is that the vehicle be launched intoa plane that is common to the launch site a t the time of launch and the Moon atthe time of impact. This plane may be defined if the declination of the launchsite, the launch azimuth, the position of the Moon at the des ired day and time ofimpact are given. In addition, the ti mes of launch pe r day into th is plane may befound if the longitude of the launch site and the des ired day of launch are known.In this section the equations f or finding the ti me of launch will be presented.

Consider the geocentric equatorial coordinate system, x , y, and z shown- -in Figure 2. Associated with these coordinates are the unit vector s I , J , and K.The unit vector is directed toward the launch site at time of launch, tz

on the day of launch, D l , and the unit vector ? is direc ted toward th e Moon attime of impact, t , on the day of impact, D . The plane that contains both unitvectors has an inclination angle, i , at the equator and an azimuth angle, A1 , atthe launch site. The launch site is defined by the right ascen sion, a , , and thedeclination, 6 , . The unit vector W is norm al to the plane defined by Tr and 'fm .

m

The following equations may be wr i t t en

wz = co s i = c o s 6, s i n A,

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UNIT VECTOR

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which is derived in Appendix A- -w * r m = O

where

Equations (2) and (3) may be solved simultaneously (Appendix A) to yield

Y 2r i x + my

-w ry m y - w z m zw = - ( 7 )1 m xExamining Equations (l), ( 6 ) , and (7 ) , i t is see n that the launch declination,

launch azimuth and posi tion of the Moon a t impact define two planes tha t arefixed in space. Launch into the se planes is possible when the launch site on therotating earth pa ss es through the planes and the launch azimuth is in the plane.This occ urs , in gener al, twice each day. Figu re 3 is included to furt her c larifythe geometry of the problem. Launch will be possible once a day o r not at alldepending on whether the value of the radical in Equation ( 6 ) is zero or imaginary,respectively.

Referring to Figure 2 , additional equations that may be wr itten are:

and-( W x r , ) - K = c o s 7 = c o s 6, C O S A,

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z LAUNCH SITE ATFIRST LAUNCH

SECOND LAUNCH PLANEx ( M E A N EQUINOX OF DATE)

Figure 3-Geometry of the launch planes

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which is derived in Appendix A where--r, = I c o s a , c o s b + J s i n a , c o s 6, + K s i n 6,I

Equations (8) and (9) may be solved simultaneously (Appendix A ) to yield(11)wz wx s i n 6, + w c o s 8, c o s A,YC O S a , = (w; - 1) c o s 6,

(12)w wz s i n 8, - wx c o s 6, c o s A ,s i n a l =(w2 - 1) cos 3 ,

and the right ascens ion, n , , a t the time of launch may be found. Figure 4 showsthe relationships that e xis t between the time and angles on the day of launch.The following equation may be wr itten f or the ti me of launch on the launch day.

a , - Y - L,t, = wwhere o is the Earth 's absolute ra te of spin per mean so la r hour, the angle Yis the mean sidereal time at Greenwich, given by the expression

2Y = 1.72218633 + 1.720279168 D, t .67558729 (A)14)

whereY is in radians and D, is the number of days fro m Jan uar y 0.0 U.T. 1960to 0.0 U.T. on t h e day of launch.

Equation (14) is Newcomb's expression given in Reference (5) with the unitschanged to radian m eas ure and the refe ren ce tim e changed fr om Noon 1900 Jan -uary 0 at the Greenwich meridian to Midnight 1960 January 0 at the Greenwichmeridian (Appendix A).

Since th e time of lunar impact wa s assu med and the time of launch calculated,the total t ime of the flight in hour s may be obtained

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W ICH)

(ZEROGREENWICHJ (ZERO HRS. UT)

XHRS. UT) al = Y + utl +L,Figure 4-Posi t ion of the launch site relative to Greenwich

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In summary, the time of launch for a vehicle going fr om the su rface of theEarth to the Moon without a plane change may be found i f the launch declination,longitude and azimuth at the time of launch as well as the position of the Moonat the time of impact are known.

B. Motion in the PlaneOnce the launch time is established, the vehicle can be launched into the

proper plane that will intersect the Moon at the tim e of impact. Launch into theproper plane is a nec ess ary but not a sufficient requirement fo r the vehicle toimpac t th e Moon at the preselected time of impact. Figure 5 shows the geometryin the space fixed plane in which the vehicle is moving after launch. At the pr e-selected time of impact with the Moon, t , the Moon passes through the plane atthe position denoted by the vector Em . In order to impact the Moon, the vehiclemust arr ive at t h e pro per position in the plane at exactly the right moment.Since a parking o rbit approach is to be used i n this p aper, the problem of thein plane motion may be broken into four phases: the fi rs t boost, the parkingorb it, the second boost, and the tra ns lu na r traj ecto ry. The conditions fo r a lunarimpact a r e then that the sum of the ti mes of each of these phases must be equalto the t ot al time given by Equation (15) and the vehicle must be at the positiondenoted by t h e vector E,.

1. Firs t Boost - The first boost phase of the problem begins at the time oflaunch and termina tes at injection into a cir cu lar parking orbit. Both the totalboost time, Tbl and the boost angle, C b l , will be assume d known constants. Thetotal first boost tim e may be exp ressed by the following equation:

2. Parking Orbit - The parking orb it phase of the proble m begins at tim e ofinjection into the parking orbi t (end of first boost) and terminate s at the beginningof the second boost. Since a ci rcu la r parking orbit will be assumed, the totaltime in the parking orbit can be exp ressed by the equation

T = t, - t, = C,


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/

Figu re 5-Geometry of the t ransfer e l l ips e

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In Equation (18) p is the Earth's gravitational parameter, R, is the knownmagnitude of the parking orbit vector, and < is the parking orbit angle, ex-pressed by (See Figure 5)

in which < b l and cb2 a r e known constants, and the angular distance, 6, romlaunch to impact may be obtained from the following equation

which may be rewritten

( = c0s-l (rmxcos a l cos 8, + r s i n a l c o s 8, + rm z i n b l ) (21)myThe translunar trajectory angle, C f , is obtained fro m the exp res sio ns given

in Section (4) following.3. Second Boost - The second boost begins at the t ime the vehicle leave s

the circular parking orbit, t, , and termin ate s at injection into the tran sluna rtrajectory, t i . The total boost time , Tb2 , and the boost angle, < b 2 , and injectionradius, R i , will be as sum ed known constants. The total boost tim e may be ex-pr es se d by the following equation.

Tb = t i - t,2

4. Translunar Trajectory - The translunar trajectory is the portion of theflight between the injection time, t i , and the tim e of lunar impact, tm . The totaltim e of flight during the translunar tra jec tor y, T, , is given by the e xpression

T = tm t ifIn orde r to study this portion of the flight, it was assumed that the translunar

tra jec tor y could be approximated by the two-body equations which neglect theeffect of the Moon gravity on the vehicle. Refer ences (2 ) through (4) ndicate thatthis assumption is adequate in obtaining fi rs t ord e r est im at es of injection condi-tions. Utilizing the above assum ption , Equations (24) through (31)may be written(Reference 6). These lead to a solution of Equation (23).

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where the mean anomaly, M , is given by Kepler's equationM. = E . - e s i n E . ; j = m , iJ J J

the eccentric anomaly, E , may be obtained fro m

where

andPa =-1 - e2 (27)

p = R j ( l t e c o s v . ) ; j = m o r i (28)J

The expressions neces sary to f ind the tru e anomaly, v, and the e ccentricity,e , are given by

and

l ) I= cos-1 [k ( 2 W os2 yi -v i = cos-'[;(2v"2 cos2 yi - I)]

The partic ula r fo rm of these equations are derived from their more familiarfo rm in Appendix A.

Upon assuming an injection velocity rat io , v", fo r an injection elevation angle,yi , Equations (29), (30), and (31) a r e solved which in tu rn y ields a solution f orEquation (23).

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4

The value of the transl una r t raje cto ry angle, C f , nec essa ry t o solve Equation(19) may be found from

i f = urn- uiThe total tim e of the flight, T, , can be expressed by the equation

Thi s equation can be so lved sinc e Tbl , Tpo , Tb2 and Tf are known fro m Equations(16), ( 1 7 ) , (22), and (23) respectively.

It will be noted that the total time of flight, T f , had been previously calcula tedby Equation (15). Therefore, fo r a solution of the problem to exist, the total timeof flight, Tf in Equations (15) and (33) must be equal. If they are not equal, thenanother velocity ratio, ?, is assume d; and the total flight tim es calculated again.This iteration proces s is continued until the tota l flight tim es in Equations (15)and (33) are equivalent. At this time the velocity ratio which yields a solution tothe problem, qf , has been found.

5. Range of Velocity Ratios and Flight Times - As previously stated, thetranslunar trajectory will be an ellipse, with respect to the Earth, which intersectsthe Moon before the apo-apsis. This type of traj ec to ry yields a specific upperand lower limit between which the velocity ratio, q , is to be assumed.

The lower limit, , may be calculated by the following equation (derivedin Appendix A).1 - R (34)

-The upper limit, V, , is taken to be that of parabolic flight.(35)-v , = 1.0

The velocity ratio which yields a solution to the proble m, Tf , lies betweenthese two limits. In or de r to get an idea of the values of th ese limi ts, Equation(34) may be solved fo r its minimum anticipated value. To do this some numbersconsis tent with the ideas previously proposed must be assumed.

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r

Assume:a. The earth to moon distance, I$,, varies between a maximum of 63.8

ea rth ra dii to a minimum of 55.8 earth radii. These values we re ob-tained from Reference 7.

b. The ea rth injection distance, Ri , varies between 1.0 earth radii and 1.1ear th radii. These values seem adequate to cove r the injection range ofcurrent boosters.

c. The injection elevation angle, Yi , varies between zero degr ees and 20degrees. This se em s t o be a reasonable assumption.

Using these assumptions, Equation (34) has been evaluated and the results tabu-lated in Table I. Fro m this table it is se en that the minimum anticipated velocityratio, min V l , is 0.990264. The range of velocity ra tios that will yield a solutionis, therefore, seen to lie between 0.990264 and 1.0.

-u

Since one of the given pieces of data w i l l be the day of launch, D, , it is im-portant to find wha t the range of flight ti me s fro m injection to lunar impac t, T, ,are. This may be done by utilizing the previous assumptions and correspondingvelocity ratios. To calculate the maximum anticipated flight tim e, max T, ,Ecpations (23) to (31) must be evaluated using vl .

This has been done and the results appear in Table I. From this table it isseen that the maximum anticipated flight time, max T, , is 130.1477 hours. Tocalculate the minimum anticipated flight time, min Tf the value of V, must beused. Since this is a parabolic flight, Equations (24) to (28) a r e no longer valid.Instead they are replaced by the following equations (Reference 6 ) .

,-I,

which holds for parabolic flight when V, - vi < 7r and where1 / 2S = [Rf t R; - 2 R i R m c o s (vm - vi)] (37)

The calculation is then made using Equation (35), Equations (29) to (31), andEquations (36) and (37). Results appear in Table I where it is seen that theminimum anticipated flight time, min T, , is 44.9697 hours. The range of flight

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Table IRange of Velocity Ratios and Flight Times

I CalculatedIssumedR mear th radii

55.8II6T5.863.81

Ri?arth radii

1.01.11.01.11.01.11.01.1

Yictegrees0

20020

0200

200

200

200

200

20

I********1 o

.991158

.991139

.990287

.990264

.992253

.992240

.991489

.9914 72********

Tfhours106.5595106.1230106.84111 6.3 54 2129.8469129.3886130.1477129.637 04 5.2 04644.969745.319645.056255.089654.844955.213154.9389

times that will yield solutions is , therefore, seen to lie between 130.1477 and44.9697 hours. Knowing th is , along with the de si re d day of lu nar imp act D, , andthe de sire d number of parking orb it revolutions, it is se en that solutions of theproblem will ex ist anywhere between seven and one days bef ore lunar impact.Therefore, a good choice for D, to obtain all solutions would be in the range f ro mD m - 7 t 0 Dm -1 .

6. Obtaining Successive Velocity Ratios - Having establish ed a lower limitvelocity ratio, v, , and an upper limit velocity rat io, v2 there remains a pro-cedure to be followed in obtaining successive assumed velocity ratios that willultimately lead to a solution. The re are many methods by which this iterationmay be done, the one desc ribed below is known as the Regula F al si method ofiteration. Calling T, from Equation (15) T tA and T, from Equation (33) TtB,the quantity AT, may be calculated by

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.QThe object of the iteration is to yield a AT,, which is a function of V , equal to

zero. Figure 6 shows the procedure when AT, is plotted as a function of ?.After Equation (34) is used in calculating v l , his value is used along with ot herknowns in calculating Tt B, called ( T t B ) l since was used in its calculation, byuse of Equations (23) to (33). Since Tt, is already assumed known, and constantfor this procedure; the maximum value of AT, viz. (AT,), may be calculatedby Equation (38). The upper lim it, q 2 , is known and another value of T tB , viz.(TtB)2, is calculated by use of Equations (29) to (33) along with Equations (36)and (37), since the tra jec to ry is parabolic. This quantity along with the constantand known T,, is used in evaluating Equation (38) for the minimum value of AT, ,viz . (AT,),. Through the two points on the solution curve, [(AT,),, TI] and[(AT, ) 2 , q,] , a straigh t line (straigh t line 1) is constructed. The next value ofthe velocity ra tio to be assumed, T3 , is located at the intersection of stra igh tline 1 and AT, = 0. Analytically the expression yielding thi s new assume d velocityratio and successive assumed velocity ratios is found to be

The value of v3 is then used to calculate ( T t B ) , by use of Equations (23) to (33).A new minimum value of AT,(38). A new st raig ht line (str aig ht line 2) is const ructed through the two points,[ AT,)1 , v"1] and [(AT,), G3], on th e solution curve. The next velocity ratio tobe assumed, V4 is the intersec tion of stra igh t line 2 and AT, = 0. Analyticallyth is a ls o may be found by Equation (39). The iteration continues in this m anneryielding values of ATt closer and closer to zero as it proceeds. When (AT,),equals zero, the point where the solution curve intersects ATt = 0 has beenfound, and the velocity ratio yielding a solution to the problem, ?f is known.

viz. (AT, ) 3 , may then be calculated by Equation

Q

7. Init ial Injection Conditions - f a solution to the problem exists , a valuef o r vf is found; and it is then possible to calculate the initial injection conditionsfor a tra ns lu na r traj ecto ry. These conditions can be obtained in eit he r Cartesiancoordinates (xi y i , z i , X i , Y i , Z i ) o r polar coordinates (Li 6 i Ri , V i , A, ,Yi .

The Cartesian coordinates may be found by the following equations.

R i00


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F ig u re &Reg u la F a ls i i te r a t io n

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and

x i

jriZ .

where the rotations R, (e),R, (e), ndR3 (8) re defined as

0 - s i n e c o s 8

c o s 6 0 - s i n eR2(B) = [ 0 1 0 ]

s i n e o C O S B

Vi00

(43)

The injection velocity, Vi , may be obtained since the final velocity rat io,V,, is known, and the parabolic velocity at injection, Vp , is given (Reference 6) as%

vP =&FTherefore

b

Pvi 5 v, v

23

(45)

(46)


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In polar coordinates three of the quantities R i , V i , and yi are known byproducts of the solution, and the rema ining quantities may be found by the equations -

andRi - y i < i ) (49)

yi(yizi - Zi+) - x i (ZiXiA , = t a n - '18. Solutions f o r Successive Orbits - The problem as outlined thus far will

yield a solution fo r the fi rs t revolution of the parking orb it around the eart h. Toobtain a solution for s ucce ssive orb its , the following technique is used.

Figure 7 i s a sketch of the geom etry fo r the solutions in the plane of thetraj ector y. The position of injection fo r the fi rs t parking orb it solution is indi-cate d by the vector (Fi) and the posi tion of injection fo r the second parkingorb it solution by vector (R i ) 2 . Note that the second position oc cur s af te r re -maining in the parking orbit somewhat more than one full revolution in inertialspace. This comes about fr om a combination of facts.

a. The time of launch fr om the s urf ace of the e ar th to the time of luna rimpact remains the same.

b. The injection elevation ang le, yi , is the same.c. The moon is essenti ally in the sam e position sinc e the time fo r one

parking orbi t ( 2 1.5 hrs .) does not allow fo r the moon to move ve rymuch (moon's motion % 12O/day).

d. Since mor e time is spent in the ear th parking orbit as the number ofparking orbits is increased, it is evident that the time in the tran slun artrajectory must decrease in order for a. to be true.

The above fac ts neces sita te a higher energy translunar trajectory whichgoes through essentially the s am e point indica ted by the ve ctor g m . The only


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PARKING ORBITALT ITUDE

------BOOST

-R m

F IRST PARKINGORBIT SOLUTION

SECOND PARKINGORBIT SOLUTION

IMPACTFigure 7-Successive par bng orbit solutions

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way this can be accomplished is by rotating the position injection vector from(R i ) , to (Ei)*n a counter-clockwise sen se a s shown in Figure 7.To find solutions f o r successive orb its, the procedure is to subtract ( n - 1)

2 77 C, fro m the to tal time given by Equation (15) or

where n is the number of the orbit in which the solution is to be found. In allother respects the problem remains the same.

V.1 .

2.

3.

4.

5.

6 .

7.

REFERENCESShaffer, F., Squire s, R. K., Wolf, H., "Interplanetary Trajectory EnckeMethod (ITEM) Program," Goddard Space Flight Cen ter X-640-63-71 , May1963.Tolson, Robert H., "Effects of s ome typical geome trical cons traints of lunartra jecto rie s,' ' NASA Technica l Note D-938, Augu s t 1961.Clarke, Victor Jr., "Design of Lunar and Interplanetary Ascent Trajectories,''J P L Technical Report No. 32-30, March 1962.Sedov, L. I., "Orbits of Cosmic Rockets Toward the Moon," ARS Journal,Vol. 30 , No. 1, January 1960."Explanatory Supplement to the Astronomical Ephemeris and The AmericanEphemeris and Nautical Almanac," London: H e r Majest y's Stationery Office,1961.Ehricke, K. A., "Space Flight I. Environment and Celestial Mechanics,"Princeton, New Jersey: D. Van Nostrand Company, Inc., 1960.Woolston, Donald S. , "Declination, Radial Distance, and P ha se s of the Moonfo r the Years 1961 to 1971 for use in Traj ecto ry Considerations,'' NASATechn ical Note D-911, August 1961.

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Appendix ADERIVATIONS

1. Derivation of Equation (1): Refer to Figur e Al , and u s e the law of sines

s i n (90" - i ) - s i n (90" S i )s i n A, s i n 90"-

: wz = c o s i = c o s 6, s i n A, (1)2. Simultaneous solution of Equations (2 ) and ( 3 ) to yield Equations ( 6 ) and ( 7 ) :

- -w - w = 1

w 2 = 1 - w ; - w ; = 1 -Y

Which upon simp lifica tion beco mes

(3)


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Solving this quadratic equation in wy yields

-wz r m y r m z k r m u J1 - r,22 - w,2w =Y 2r,2, + r m y

3 . Derivation of Equation (9): Refer to Figure A2, it is seen that

( W X f , ) ' I( = c o s 77The direction co sines of TT yield

Substituting in Equation (1) or the fir st t erm , transposing, and simplifying

cos2 7) = 1 - c o s 2 6, s i n 2 A , - s i n 2 6, = c o s 2 6, c o s 2 A ,Upon taking the square root of both sides

c o s 7) = co s 6, c o s A ,

where the positive value is considered only if launch is in the northern hemisphere@/2 > 6 , > 0) and eastward ( 7~ > A , > 0)

(9): (W x ; I I ) K = C O S 77 = C O S 6, C O S A ,4. Simultaneous solution of Equations (8) and (9) yield Equations (11)and (12).

Since(10)-r I = I C O S a , c o s 6 , -+ J s i n a, c o s 6, t K s i n 6,

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Performing the vector operations yields-r, * W = c o s al co s 6, wx t s i n a l co s 6,w t s i n 6, wz = 0Y

-(W x f , ) K = s i n a , c o s 6, wx - c o s a , c o s 6, w = c o s 6, c o s A, (A2)YUpon substituting (A2) into the second te r m of ( A l )

Which upon transposing and solving f o r cos a , yieldsw, wx s i n 6, t w c o s 6, co s A ,Yc o s a l = (w; - 1) cos 6 ,

Upon substituting (Al ) into the second te r m of (A2)Ws i n z I c o s 6, wx - 2 - s i n a , c o s 6, w - s i n 6, wz) = co s 6, c o s A ,wx Y

which upon transposin g and solving for si n a , yields

w wz s i n 6, - wx cos 6, co s A ,s i n a l = Y (12)( W i - 1) co s 6,

5. Changing the reference time and units of measure in Newcomb's expression.Greenwich mean siderial time at zero hours universal time on successive

dates are computed from Newcomb's expression (Reference 5) .

Yo = 6h 8"45?836 t 8640184342 T + 0?0929 T2 (A31where for any date, T denotes the number of Ju lian centu rie s of 36525 days whichhave elapsed sin ce noon on 1900 January 0.0 at the Greenwich meridian.

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This equation may be written as

Yo = Y l g o , t AT t B T 2

To s h i f t the reference time fr om January 0.5, 1900 to Jan uary 0.0, 1960

yo = y 1 9 0 0 t A (T t T 1 ) t B (T t T , ) * = Y l g o o t AT t B T 2 t (A t 2BT)T1 t BT:where T is the time in Julian centuries between January 0.5, 1900 and J anua ry0.0, 1960 and Tl is the time in Julian centuries after January 0.0, 1960. Thevalue of T in this case is

T = 6 5 6 o 1 3 * = .599958932 J u l i a n c e n t u r i e s36525

: Yo = Yl 6 o t [ (8640184.542 + 2 ( . 0 9 2 9 ) ( . 5 99 9 5 8 9 3 2 ) 1 T, t .(I929 T iand the Equation (A3) becomes

Yo = 6 h 3 4 m 4 1 : 7 6 2 +8 6 4 0 1 8 4 5 6 5 3 T 1 + OSO929Ti ( A 4 )

where T, denotes the number of Jul ian cen turie s which have elapsed sinc e 1960January 0.0.

The units of Equation (A4) may be changed to yield

Y = 1.72218633 t 1.720279168 x D, t 0.67558729 x lo- -36?25) (14)where Y is in radians and D, is the number of days f rom Jan uary 0.0 U.T. 1960.6. Equations (29), (30), and (31) may be obtained in the following manner. Start-

ing with the basic equations (Reference 6).

P = c o s 2 y

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p = a ( I - e 2 )

Equating P in Equations (As) and (A6) and solve for e 2

Substitute for l /a using Equation (A7)

Equation (AT) also implies for parabolic velocity (Le. a =a).

vp = d v cSubstitute this relationship in and solve for e yields

31


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Combining Equations (A5), (A9), and (A8) yields

(30)-e

o s vi = - 2 v 2 c o s 2 y; - 1)

Applying Equation (A5) fo r ea rt h injection dis tance s and lun ar dis tan ces yieldsthe results

2 2p = Rm($) cos2 y, = Xi(?) co s 2 yi

Substitutingp /R, fr om above r es ul ts along with (A9) into Equation (A8) yield s

urn = c o s - f [+ 2 i w c o s2y ; -7. Derivation of t h e lower limit for assume d velocity ratios.

Equation (35) for the lower limit, T1, s obtained in the following manner.Upon substi tut ion of Equation (31) into Equation (29) and squa ring yi elds

- 4 - - 24 R V C O S Y ~ - 4 R V + 1cos2 urn= 4 T 2 ( T - l ) cos2 yi + 1

The limiting ca se occur s when c o s 2 urn = 1 which corresponds to a lunarimpact at the apo-apsis of the tra nsl una r ellipse. Making thi s substitution andsolving for the velocity ratio Equation (A10) becomes

1 - 21 - ( R c o s yi (34)

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Figure A-1-Geometry of the orbi t

F x 7,-UNIT VECTOR

Figure A-2-Geometry descr ibing the ang l e -q

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APPENDIX BCALCULATING PROCEDURE

A. Data RequiredIn ord er t o solve the problem of a lunar impact from a given launch site,

various data a r e required before the calculation may begin. A summary of theseinput data and i ts sou rce a r e given below.

1. The firs t boost a re C b l : Refer to Figure 5. This is a characterist ic ofthe booster to be used in th e mission, It is usually available o r may beobtained fro m data in reports giving the booster characteristics.

2. The first boost time, Tb l : Refer to Figure 5. The same r ema rks inItem 1 apply here.3 . Parking orbit altitude, h,, = R, - I?, . The altitude at which the first

boost injects its payload into a circular orbit . The same rema rks inItem 1 apply here.

4. The second boost a rc , c b 2 : Refer to Figure 5. This is a characterist icof the booster u sed to boost the vehicle fro m cir cu lar parking velocityto translunar injection velocities. This information may be obtained fromdata in reports giving the booster characteristics.

5. The second boost time , Tbp : Refer to Figure 5. The same remarks i nItem 4 apply here.

6 . Injection altitude, hi = Ri - R, . The altitude at which the second boostinjects the vehicle into its translunar trajectory. The sam e rem arks inItem 4 apply here.

7. Injection elevation angle, yi : Refer to Figure 5. This angle is a functionof the particular guidance system employed during the second boost.

8. Launch azimuth, A i . The desir ed launch azimuth at launch site. Dependsupon range safety and tracking requirements.

9. Parking orbit revolutions. The numb er of parking orb it revolutions fo rwhich solutions are desired.

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10. Launch days(s), D, . The number of d ays bef ore impac t that the launchis to take place. Fo r the elliptical traj ect ori es described this is f rom7 to 1 days before D,.

Flight time , T , the time of the flight fr om injection to impac t may begiven instead of D, .11. Lunar impact time, Y,,, D, , t,. The yea r, day and time it is desired

to imp act the moon.

B. Calculating StepsA flow diag ram indicating the solution of t he pro blem is given in Figure B1.

This figure is a guide t o the solution of the problem and indicates the gene ralflow fr om equation to equation in t h e text. Steps in the calculating proce durewhich may be u sed in conjunction with this fig ure a r e given as follows:

1. Given a year , Y, , day, D, , and time, t, , it is de si re d to impact the Moon.The position of the Moon in geoce ntric e quato rial coo rdina tes, K X ,Gy ,R,, may be obtained fro m an ephe me ri s of the Moon. The result ant ofthe se coordinates, I?,,, as w e l l as the components of the unit ve cto r, r m x ,r m y , r,, , are then calculated.

2. Given the launch site declination, 6 1 , and a launch azimu th, A i , compon-ents , w, , wy , w, of the unit vector % normal to and defining the plane ofmotion fro m the su rface of th e Earth to the Moon are obtained from Equa-tions (l), ( 6 ) , and (7). Depending upon the value unde r the radic al inEquation (6) the unit vector w wi l l have two values, one value, o r be imag-inary. These re su lts indicate for the launch azimuth, A i , launch in to theplane is possible twice a day, once a day, o r not at all. If a re al valueof W exi sts the calculating procedure is continued. If two real values ofZ exist, us e one of th em at a time.

3. The right ascension at launch, a1 , may be calcul ated by Equations (11)an d (12). A des ir ed day of launch, DI , is given (i.e. D, is an integer inthe range of D,,, 7 to D;, - 1) and the angle Y is calculated by Equation(14). Since the launch longitude, L i , and the Eart h 's spin rate, W , areknowns, the time of launch after z er o hour U.T. on the day of launch, D l ,can be calculated f ro m Equation (13). The day of launch, DI and the timeof launch, t,, , as well as the day of impact, 0, , and time of impact, t, ,are now all knowns; the total time of the flight, T, , can be calculated fro mEquation (15). This time is designated as TtA.

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4 . A value of the injection angle, y; , is given, and the injection radius, R; ,can be calculated f ro m the given injection altitude, h i , and Earth's radiusR, . A velocity ratio, , calculated by Equation ( 3 4 ) is now obtained.

5. Using this velocity ratio, the eccentricity, e , as well as the true anomaliesto the Moon, v, , and to the injection point, v i , a r e calculated by Equations( 3 1 ) , ( 2 9 ) , and ( 3 0 ) respectively. The tra nsl una r tra jec tor y angle, [ f ,may now be ca lculated f ro m Equation ( 3 2 ) ; and the angular d istance, E , iscalculated by Equation (21). Since the boost er burning angles, [ b , andi b 2 , a r e given, the parking orbit angle, < , can be calculated fro m Equa-tion (19).

A cir cul ar parking orbit w a s considered in this analysis; and the ref ore ,the parking orbit radius, R 1 , is constant and is obtained fro m the givenparking orbit altitude, h,, , and Ear th' s radius. The inverse orbital rat eof the parking orbit, C1 , is defined by Equation ( 1 8 ) . This is used incalculating the time in parking orbit, Tpo , given by Equation ( 1 7 ) .

6. Since the velocity ratio yields an elliptical flight path, the calculationcontinues as follows: The semilatus re ctum, p , and the semimajoraxis, a , may be calculated by Equations (28) and ( 2 7 ) respectively. Hav-ing thes e quantities the ec cent ric anomalies to the Moon, E, , and to theinjection point, E i , can be calculated by Equation (2G); and the meananom alie s to the Moon, M,, and to the injection point, M , ca n be calculatedby Kepler's equation, viz. Equation (25). The time difference on the tra ns -lunar tra je ct or y between injection and lun ar impact, t, - t i , can be cal-culated by Equation ( 2 4 ) . This tim e difference, called T f, is the same asthat defined by Equation ( 2 3 ) .

7 . The boost time s, T b l and T b 2 , are both given and are used with T,, andTf to calculate the total tim e of the flight, T, , from Equation ( 3 3 ) . Thistime is designated as T t B . In step ( 3 ) above, it is se en that the total timeof flight, T, = T,, , has a lre ady been calculated by another method, viz.Equation (15). The difference in these ti mes , AT, , is given by Equation( 3 8 ) -

8. The particular AT, as calculated by Equation (38) for fr om Equation( 3 4 ) is designated, since k = 1, as (AT,) , on Figure 6 . I (AT , ) , is less

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9.

10.

11.

than ze ro , the solution of the problem req ui res that the transfe r trajectoryintersect the Moon after apo-apsis and the calculation is concluded. Anew value of the launch day, D, , gre ate r than before is used, and the cal-culation is reinitiated at step (3), Equation (14). If (AT,) l is gre ate r thanzero, continue.

T-uThe next velocity ratio used, V, , is given by Equation (35). Step (5) isthen repeated. Since the velocity ratio corre spond s to a parabolic traje c-tory, the param eter, s , s calculated by Equation (37) and is used in cal-culating the total time f ro m injection to lunar impact, Tf , given by Equa-tion (36). Step (7 ) is then repeated. The par ticula r AT, as calculated byEquation (38) fo r ?, fr om Equation (35) is designated, sinc e k = 2 , as(AT,), on Figure 6. If (AT,), is gr ea ter than ze ro, the solution of theproblem requires a velocity ra tio gre at er than parabolic; and the calcu-lation is concluded. A new value of the launch day, Dl , le ss than beforeis used, and the calculation is reinitiated at step ( 3 ) , Equation (14). If(AT,), is le ss than zero, a new velocity ratio, T 3 , must be calculated.The next velocity ratio to be assumed may be calculated by Equation (39).Steps (5), (6), (7), and (10) are repeated in that ord er and are continuedto be repea ted until the value of AT, given by Equation (38) is as close tozero as desire d. When this occurs, a velocity ra tio which ha s yielded asolution to the problem, ?f , has been obtained.A t this point se ver al things m a y be calculated depending upon th e partic-ula r purpose of making the calculation. Two possibilitie s come to mind,and to distinguish them one will be called Option A and the other Option B.

Option A: In this case it is assumed the purpose for making the cal-culation is to obtain all the possible injection conditions that will yield alunar impact at a given des ire d time f or given launch azimuth, boos terconditions, and injection elevation angle if the parking orbit is allowed toorbit the earth several ( sayn ) times. For this option n w a s assumed tobe equal to one through st ep s 10. The value of n is now inc rea sed to two,and Equation (50) is used in place of Equation (15), the e ntir e calculationbeing repeated from step 4. This value of n is increased as many time sas desired repeating the calculation from step 4. Having done this, asolution has been found fo r each revolution of the parking orbit (i.e. nof them). Assuming two values of wy were obtained when Equation (6)was solved (see step 2 ) , th er e remain s now to repeat the calculation fr omsteps (3) to (10) fo r the second possible value of wy . Having done this(for n = l ) , he values of n ar e increased again one at a t ime as w aspreviously done. Now the two possible solutions pe r day f o r each parking

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orbit revolution have been found, yielding 2n possible solutions. Theresul ts thus fa r have been fo r only one assumed launch day, D, . Sincelunar impact at the desired time is possible, as previously explained,for launch time s between 44.9697 and 130.1477 hour s befor e l una r im -pact, mo re launch days must be investigated. To be su r e to include al lpossible ca se s, the launch days to be investigated a r e between D, - 7and D, - 1, or a total of 6 days. Doing this for n revolutions in theparking orbit f or each day is seen t o yield a possible 12 11 solutions perlaunch azimuth per elevation angle that will impact the moon at the sa m etime.

Option B: In this case it is assumed the purpose for making the cal-culation is to obtain initial injection conditions that will have a givendesired flight tim e, Tf , fro m injection to lunar impact. Lunar impactwill be ne ar the de si re d time, but not at it. Solutions will be obtainedin the fi rs t parking orbit f or a given launch azimuth, booster conditions,and injection elevation angle. The calculation is made as indicated insteps 1 through 10 , and a solution is obtained. A t this point the desiredflight time ( T f ) D , s compared with flight time, Tf , that was the solutionto the problem, and the ir difference, ATf = (AT f )D - Tf , is obtained.This time difference is algebraically added to the original l unar impacttime (thus having the effec t of moving the moon), and s te ps 1 hrough 10a r e again repeated, yielding a new solution. Again a tim e dif fer enc eATf , is obtained, and the pr oc es s is repeated until this time differenceis zero. A solution durin g the first revolution of the parkin g o rb it havinga given tim e of flight fr om injection to impact has thus been found. Thissolution will yield a lunar impac t nea r (s ay within a day) the originalimpact time desired. The original impact time is then utilized with thesecond value of wy obtained when Equation (6) w a s solved, and the entireprocess of Option B is repeated. The final re su lt s will yield a total oftwo solutions, having the s am e desir ed flight time, in the fir st parkingorbit for a given launch azimuth, injection elevation angle, and boostconditions that impact the moon close to the de si red impa ct time.

12 . Having found the final velocity ratio, ? f , all the by-products n ec es sar yf o r finding the injection vector co mponents in both Cartesian co ordin atesmay be found. Par abo lic velocity, V, , is found using Equation (45) whichthen enables the calculation of V i f r o m Tf , Equation (46). A l l the neces-sary data is now known to ca lculate rotation m at ri ce s, Equations (42),(43), and (44), which are used in obtaining the C artesian position com-ponents and velocity components f r o m Equations (40) and (41) respec-tively. Using the se components the remain ing unknown pol ar coordinatecomponents, viz. L , 6 , an d A i , may be calculated by use of Equation(47), (48), and (49) respectively.

( X i , Yi, Zi, G i , Yi , i i ) or polar coordinates (Li, S i , X i , Vi , A i ,Yi)

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' N a c : N umber s in pueorhesis referto equations im the texr.

k = k t l

LUNARGE OC E N T R KEQUlTORlALCOORDINATES

twz = cos i = co s SI sin A, (1

1wz W si n 8, t w, cos 8, cos A,COS^, = (11

( W t - 1) co s SIw, wx si n 8, - W cos 6 , cos A,s i n a, = (12(w3- 1) co s SI

T = 1.72218633 t 1.720279168. x lo- ' 4

1T, = t m- tl t 24 (D - D, ) Tt r (15I ,+,

1

e = [ 4 i 2 (V'- 1) Cos' yi + 11'" (31)I

1

h, = urn- u .P =IL

I M, = Et - - t i

Tb


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, L l . o & i = l . o ,j, = m. i (25) l

minitiate ati p c c after a p a p i s .Let DI = DI t I andreinitiate at equation (14).(AT,) + 0

> oE-'"'I3I I

1-- , = V I+,P = #A,, = VI vp~1 - s i n 6 c o s 6[ s i n 6

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Injection Conditions for Lunar Trajectories

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