The motion of ballistic missiles

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TG 1164JULY 1971Copy No. 8

Technical MemorandumTHE MOTION OFBALLISTIC MISSILESby L.S. GLOVER and J. C. HAGAN

THE JOHNS HOPKINS UNIVERS!TY i APPLIED PHYSICS LABORATORY

Approved for public release; dltrlbutlion uilimited.

R11roduk(d byNATIONAL TECHNICALINFORMATION SERVICESrnlitoll~d, Vn 72151


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The Johns Hopkins University Applied Physics Lab. Unclassified8621 Georgia Ave.Silver Spring, Md. 20910 I NAThe Motion of Ballistic Missiles

A [It ,C r4lI I IVE! NOT F 5 7'p of ripurt oand IfcitcI ivJ datit,)

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L. S. Glover and J. C. IHagen6 EPrORT 0DTE 7I. T O T A L NO. OF PAGE S Tb, N0 O REPSJuly 1971 390 20

Ha. CONT4ACT COR GRANT NO. 90 , ORIGINATOR'S 4EPIORT N MIBER(SI)

N00017-62-C-0604o ECTN0. G 1164Wb. 0 TH ER HEPORT NOIB5 (Aniy a(her 'urnbe ti Ihetf rniy be asslaripi(tthin report)

d,II DISTIRIBUTION STATEMENT

Approved for public release; distribution unlimited.11. 5UPPLEII4ENTAIRY NOTES 12. SPONSORING MI LI TA RY ACTIVITY

Strategic Systems Project Officeil. ABST14ACT

This handbook on the motion of ballistic missiles has been preparedprimarily for use in training new personnel charged with the responsibility ofevaluating the performance of reentry bodies. Aerodynamic fundamentals andcharacteristics of the atmosphere are reviewed briefly, followed by discussionsof the effucts on ballistic missile motion of gravity, earth rotation, the atmo-sphere, and initial body rates. The anomalous motion resulting from varioustypes of mass and/or aerodynamic asymmetries is discussed in detail. Simplesolutions are given for the impact dispersion resulting from asymmetries andnonstandard meteorological characteristics at the impact location. Various tra-jectory simulations and flight test analyses are also discussed. Numerical ex-amples illustrate the material.

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UNCLASSIFIEDSecurity Claunification

14. KEY WORDSBallistic missileBallistic flight test dataTrajectory simulationAerodynamic characteristicsTraining manualImpact dispersionMissile trajectoryNumerical solution

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ThEl JOHNS HOP~i(NSUNIVENIiVIAPPLIED PHYSICS LABORATORY

IlLVR $PRIM* MAVI.6AND

ABSTRACT

This handbook on the motion of ballistic missileshas been prepared primarily for upe in training new per-sonnel charged with the responsibility of evaluating theperformance of reentry bodies. Aerodynamic fundamen-tals and characteristics of the atmosphere are reviewedbriefly, followed by discusEions of the effects on ballisticmissile motion of gravity, earth rotation, the atmosphere,and initial body rates. The anomalous motion resultingfrom various types of mass and/or aerodynamic asym-metries is discussed in detail. Simple solutions aregiven fo r the impact dispersion resulting from asnme-tries and nonstandard meteorological characteristics atthe impact location. Various trajectory simulations andflight test analyses are also discussed. Numerical ex-amples illustrate the material.

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IhNOWN1 HOPKI4 UNIolv"ItVAPPSLIE PHYSICS LABORATORY

ABSTRA CT

This handbook on the motion of ballistic missileshas been prepared primarily for use in training new per-sonnel charged with the responsibility of evaluating theperformance of reentry bodies. Aerodynamic fundamen-tals and characteristics of the atmosphrre are reviewedbriefly, followed by discussions of the effects on ballisticmissile motion of gravity, earth rotation, the atmosphere,an d initial body rates, The anomalous inotion resultingfrom various types of mass and/or aerodynamnic asym-metries is discussed in detail. Simple solutions aregiven for the impact dispersion resulting from asymme-tries and nonstandard meteorological characteristics atthe impact location. Various trajectory simulations andflight test analyses are also discussed. Numerical ex-amples illustrate the material.

,Ny

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THE OHNNEOPKINS UNIVERSITYAPPLIED PHYSICS LABORATORYSILVER S04114I MAIyLANO

CONTENTS

List of Illustrations . viiList of Tables . . xi

1 Introduction 1.2 Aerodynamics . . 53 Atmosphere . . 434 Equations of Motion . 714.1 Effects of Gravity . 754. 2 Effects of Earth Rotation . 954.3 Effects of Atmosphere . . 1074.4 Effects of Body Dynamics . 1294.5 Effects of Asymmetries . . 1574.6 Examples . , . 2355 Dispersion a , . 2696 Trajectory Simulation . . 3237 Flight Dynamics Data Analysis . 3338 Constants and Conversion Equations . 365

Index , . . 36 9

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tWIql JONhlO '40lPM1PW6iNIVIUSIITYAPPLIED PH',SICS LABORATORY

ILLUSTIIA TIONS

2-1 Variation of Viscosity withTemperaturc 35

2-2 Variation of Viscwqity withTemperature .36

2-3 Cone Axial Force Coefficient, LaminarFlow, Cold Wall, 8 = 100 37

2-4 Cone Axial Force Coefficient, TurbulentFlow, liot Wall, 6 = 10* 3 8

2-5 Pitch Damping Coefficient for 10 Semiangle Cone 39

2-6 Roll Damping Coefficient, Laminar Flow,Cold Wall, 6 = 10* 40

2-7 Roll Damping Coefficient, TurbulentFlow, Hot Wall, 6 = 10 41

3-1 Typical Atmospheric TemperatureProfile and Altitude Regions . 65

3-2 Standard Atmosphere Characteristics 663-3 Effect of Water Vapor on Density 674-1 Range versus y 0 and V . 894-2 Initial y versus V and Range . 904-3 Flight Time versus Yo, R 3000 nmi 914-4 Flight Time versus Range, Minimum

Energy Trajectory 924-5 Range Sensitivity to V . 9. 34-6 Range Sensitivity to Y 0 944-7 Variation of II(K) with K 1254-8 Variation of I (K ) with K 12(6I'

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IT4g jaid MOR~NIN uNher.3TYAPPLIED PHYSICS LABORATORY

LVIIN 11PRIN41MARYLAND

IL.,L.f RATIONS (cont'd)4-9 Variation of I, (K) with K 1 274-10 g g versus K . 1284-11 Body Axis System .1554-12 Variation of J 0(W with X . . 1554-13 Variation of A and A0 with X . 2274-14 Variation of X with D cot 0 2284-15 Variation of G with X and D 0. 2 * 2294-16 Variation of F(X, D) with X, ID 0. 2 * 2304-17 Roll Lock-In Criteria, Body A . . 2314-18 Roll Lock-In Criteria, Body B . 23124-19 Roll Lock-In Criteria, Body C . * 2334-20 & Decay Characteristics .2594-21 Significant Events . . 26004-22 Required for g L 20'.20 2614-23 pr and w versus Altitude 2624-24 Conditions for Roll Resonance

at h = 1)000Feet 2634-25 Required for Lock-In and Spintdrough Zero Roll Rate .. 264K4-26 Boundaries of~ and for Lock-In

at First Riesonance . . . 2654-27 Variation of r with Initial Roll

Rate mm. . 2665-1 Limiting Values of K . * 309S. L.5-2 Limiting Values of A/E 3105-3 Variations of E with h, 1962

Standard Atmosphere . . 311

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SHC JOMNS HOPKINI UNIV9N*ITYAPPLIED PHYSICS LABORATORY

L ~ILv6A IPUIWO MAPYLSK

ILLUSTRATIONS (cont'd)

5-4 Dispersion Resulting from Wind andDeviation in Density 312

5-5 Dispersion Rcsulting from MeteorologicalCharacteristics at High Altitude 31 3

5-6 Probability of Encountering Zero RollRate 314

5-7 Maximum Altitude for Zero Roll Rate , 3155-8 Maximum Dispersion Resulting from

Se and r/d 3165-9 Dispersion versus 00 3175-10 Dispersion Probability . 3187-1 Bit Presentation Scheme, DITAP

Flight Records . .3537-2 Timing Traces, DITAP Flight

Records 3547-3 Bit Presentation Scheme, DITAR Flight

Records 3557-4 Timing Traces, DITAR Flight

Records . 3567-5 Sample Roll Rate History 3577-6 Sample Lateral Acceleration History . 3587-7 Sample Longitudinal Acceleration

History .. 3597-8 Sample Pitch Rate History 3607-9 Angle of Attack Work Sheet . . 3617-10 Aerodynamic Pitch Frequency (Weather-

cock Frequency) Work Sheet * 3627-11 Amplification Factor Work Sheet . 3637-1 2 Rotation of Maneuver Plane WorkSheet . . . 364

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I* 'M[ JOMNIl H'OPKIN3NUNIV[IRIYI't

APPLIED PHYSICS LABORATORYSILVIS SlPSlf,N MAi~llr,.A'i

TAB1,ES

2-1 Mass and Aerodynamic Characteristics 423-1 1062 Standard Atmosphere 683-2 Constants fo r the 1962 Standard

Atmosphere 694-1 Reentry Characteristics 2675-1 Values of 13 3195-2 Values of F 32 0w5-3 Values of 5 3215-4 Values of F 11322

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5 APPLIED PHYSICS LABORATORY11111102111IMS MARYL.AND

~; K 1.NTRODUCTION


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APPLIED -HYSICSLABOrATORY

The term ballistic missile as used herein describesan!j body in motion that is unpowered and uncontrolled. Themotioa may occur in either the atmosphere or the exnatmo-sphere. However, most of the interesting motion phe-nomena occur wiithin the atmosohere, and most of the dis-cussions concern reentry-type vehicles entering the atmo-sphere at speeds from 10 000 to 25 000 ft/sec.

'l l( p'iimaryv purpose_) oft thiL, rclp()rt is to pre sentmaterial suitable for' training new personnel charged withthe responsibility of evaluating the performance of reentrybodies; therefore, the subject matter is treated as simplyas possible while retaining enough detail to explain themissile behavior. Although the material is written pri-marily for personnel unfamiliar with the detai's of missilemotion, it provides experienced personnel with readilyavailable equations suitable for estimating order-of-mag-nitude effects of phenomena frequently encountered in flighttest analysis, performance, and preliminary design work.

A brief discussion ot the aerodynamic character-istics of missiles and the nature of the atmosphere is fol-lowed by a discussion of vehicle motion as affected bygravity, earth rotatior., the atmosphere, and initial bodyrates. Also included is a detailed discussion of the anoma-lois motion resulting from various types of mass and aero-dynamic asymmetries. Simple solutions are given for theimpact dispersion resulting from asymmetries and non-standard meteorological characteristics at the impactlocation. Simulations required for various types of studiesand flight test analysis complete the subjects discussed.Numerical examples illustrate the material presented.

In some sections (particularly parts of Section 4)the material is new and is given in considerable detail,Since some readers may not be interested in details, asummary of the material covered and the major conclu-sions derived is given at the beginning of each section(each subsection for Section 4).

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TMa JOHNS NMOPKINOUNIV90SITVAPPLIED PHYSICS LABORATORYBILVIS OSPONO MARYLAND

An attempt was made to use symbols and units ofmeasure currently used by specialists in the variousfields covered in this report. As a result, a particularsymbol mayhave different definitions in various sections.A complete list of symbols is given at the beginning of eachsection (each subsection for Section 4), and conversionfactors for units of measure are given in Section 8.

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flu .AewDrsANSurNvmweln- ~APPLIED PH4YSICS LAISCRATOftY

2. AEAODYNAMICS

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THE JOHNS NiOPKINi UNIVE"ISIYAPPLIED PHYSICS LABORATORYt lllILVE.lSPR,NG MAmv.ANO

SYMI3OLS FOR SECTION 2

Symbol I)efinition Typical. UnitsA axial ro''rce poundsCA axial force coefficientCAB base drag coefficientCA friction drag coefficientFCA pressure drag coefficientPc, g. refers to center of gravityC rolling moment damping coefficient

pC pitching moment coefficient -C pitching moment coefficient at 0-m 0C pitching moment damping coefficientm qC dC /d&ata"= 0el mCN normal force coefficientC N dC N/dWat 0c. p. refers to center of pressureD total drag poundsDF drag resulting from skin friction poundsd reference diameter feetK Knudsen number (see Eq. (2-7))L lift poundsor

body length feet

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THt JOHNS NOPING UINIV"mUITAPPLIED PHYSICS LABORATORY

MILVIR SPRING M ANVLANO

Symbol Definition Typical UnitsL(p) rolling moment resulting from roll

rate (roll damping moment) lb-ftI moment arm for asymmetric force feetM Mach number = V/I-or

pitching moment lb-ftM(q) pitching moment resulting from pitchrate (pitch damping moment) lb-ftN normal force poundsN normal force resulting from anasy asymmetry poundsN normal force for a sym.metricals vehicle poundsP ambient pressure lb/ft2P cone surface pressure lb/ft 2PF flap surface pressure lb/ft 2p roll rate rad/secq pitch rate rad/secq dynamic pressure, (y/2)PM 2 01l(1 /2)pV2 lb /ft2

2 21R gas constant = 1716 ft /sec -ORRe Reynolds number, oVL/14 = VL/h -S reference area ft 2S.M. static margin

surface area excluding the base area ft 2wT absolute ambient air temperature ORT reference temperature for viscosity OROV vehicle velocity ft/sec

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TYIOJOMW NO*tINS UIV||glvTlAPPLIED PHIYSICS LABORATORYSILVIft BP*N;% MANU6AN6

Symbol Definition Typical UnitsX distance from nose to a point on thebody centerline feetx g. distance from vehicle nose to the c. g. feetx distance from vehicle nose to the c. p. feetAX X -x feetc. p. c. g.Y distance normal to the body surface feet(w vehicle total angle of attack radianst local angle of attack radiansv ratio of specific heats - 1. 48 cone half-angle radians( asymmetry deflection angle radians

ambient air density slug/ft 3M dynamic viscosity slug/ft-secAo dynamic viscosity at T slug/ft-sec

kinematic viscosity ft 2/secshear stress lb/ft2AO body bend angle degrees

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TMK Jaws MOPIINGI NIVIRSITa*1APPLIED PHYSICS LABIORATORY

SUMMARY AND CONCLUSIONS

The aerodynamic forces of interest for reentrybody studies are discussed, The significance of the aero-dynamic coefficients is reviewed, and equations are pre-sented for estimating the magnitude of the coefficientsfor a sharp-nosed conical hody. Several types of asym-metric cones arc nlso con Hidn(rcd. Numerical examplesare given for a 10 serniangle cone having a length of 10feet.

The major conclusions are:1. The following aerodynamic forces and moments

are important in the analysis of motion of a typical reentrybody:v Force perpendicular to tho body centerline

(normal force) and resulting moment aboutthe body center of gravity (pitching moment)* Force parallel to the body centerline (axial

force)* Moments resulting by virtue of body pitch,yaw, or roll rotational rate (pitch, yaw, orroll damping moments)* Forces and/oi moments resulting from bodyasymmetries.2. For uncontrolled vehicles, it is important thatthe normal force act through a point (center of pressure)aft of the center of gravity. In this case the vehicle issaid to be statically stable. The ratio of the distance be-tween the center of gravity and the center of pressure tosome characteristic body dimension (for example, length)is called the stability nialgin and Is usually expressed asa percentage of the characteristic length.3. The aerodynamic forces and moments are repre-sented by dimensionless coefficients, thereby facilitatingevaluation of the forces and moments over a wide varia-

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Tw x JOHNS MOPKINI UNgVDII4YAPPLIED PHYSICS LABORATORYgiILvIImPlllii6. M&WVkbIIII

4, In general, the coefficients are functions ofvehicle angle of attack and the similarity factors, Machnumber - an indication of the effect of compressibility ofthe air, Reynolds number - an indication of the nature ofthe boundary layer on the vehicle, and the Knudsen num-ber - an indication of the size of the vehicle relative tothe mean free path of the ambient air particles.

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?MI' JOM41S HOP1q.N UNIVR~P91?APPLIED PHYSICS LABORATORYSi VI *PRllhO MAHYLANt,

2.1 Aerodynamic ForcesA body moving through air experiences forces -aerodynamic forces - resulting from the motion. Theaerodynamic forces of interest for a reentry vehicle are:1. Axinl force,2. Norinmal force,A. itch damping force,

' toll damping force.The axial force (for a symmetrical vehicle) acts

along the body centerline as shown in the sketch for a cone.

N

A C. ".

V

This force is produced primarily by relatively high pres-sure acting on the conical surface (pressure drag) andrelatively low pressure acting on the cone base (base drag).In addition, a force produced by air viscosity (skin fric-tion drag) exists on the conical surface.

A normal force (normal to the body centerline)'"exists when the body centerline is not directed along the

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YH I JOHNS HOPKIN8 ,NIVINIITYAPPLIED PHYSICS LABORA'rORY

velocity vector, The angle between the velocity vectorand the body centerline is called the angle of attack, .The normal force results from high pressure acting onthe windward side of the body compared to low pressureacting on the leeward side. The integration of this pres-sure difference over the cone surface is the normal force.The pressure differential across the body alsc produces amornent about the body center of gravity, Poor convenienct-in analysis, this aerodynamic moment is computed as theproduct of the normal force and a moment arm (XX1 . ), where X is the longitudinal location at 1h6chthe fotal normal fo ce acts and is called the center ofpressure and Xc. g. is the longitudinal location of the cen-ter' of gravity. Assume that a vehicle has its center ofpressure forward of the center of gravity and that somedisturbance causes an initial angle of attack. This angleof attack results in a normal force, Since the normalforce is forward of the center of gravity, the aerodynamicmoment produced is in a direction to increase the angle ofattack, This process continues, and the missile eventuallytumbles. Similar reasoning will show that, for a centerof pressure located aft of the center of gravity, the aero-dynamic moment will always tend to cancel a perturbationin &, Therefore, a vehicle having a center of pressureforward of the center of gravity is said to be staticallyunstable; a vehicle having a center of pressure aft of thecenter of gravity is said to be statically stable. The valueof the ratio(Xc, p. - Xc. g,/L w AX/L is called the staticmargin. The margin must be positive for uncontrolledvehicles and for high performance vehicles is usually 3%to 5%.

The use of the word "normal" to define the forceIn the plane of ;T and perpendicular to the body centerlineis used here to be consistent with the usual terminologyfound in the literature. It is hoped that there will be noconfusion with telemetry terminology where "normal"force and 'lateral" force are the two components of thenormal force as defined above. In an attempt to avoidconfusion, in some sections of this report we have used

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TNS JO NOHOPKINS rNIIvuuTYAPLIED PHYUICS LABORATORYBILVIN OP01.14 &iAP.I6ftD

the term 'transverse' force to mnean the force in the planeof a and perpendicular to the body centerline.

The pitch and roll damping forces exist by virtueof the body pitch and roll rates, respectively, and there-fore are called dynamnic forces. Consider a body pitchingabout Its center of gra,-vity with rotational velocity, q, Thepitching motion produces a locail velocity at X equal to1X - X )q. The distribution of velo.zity along the bodyis shon n the sketch,

Mg.The local velocity distribution produces a localanrgle of attack, &I, whose magnitude is given by the equa-

tion:(X - X)q

The local angle of attack results in a. local normalforce which, at all points along the body, tends to resistthe pitching motion. The Integral of the local forces (thedamingforce) is tosmall to be of significance in ballis-tic missile aaye.Hwvr h apn oetwhich is the integral of the local forces multiplied by theirmoment arms, is very important in some portions of thetrajectory as will be shown later.15


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THCO~HN%MOPKIN&INIiV2 1ItYAPPLIED PHYSIC5 LABOnATORYt$11V SIPRING ARI&NIA

Tile example cited above i.s just one type of dampingeffect. Any force that exists by virtue of the pitch rate isa pitch damping force, and these forces do not always re.-sist body pitch motion. The forces that do resist the pitchmotion are said to be stable forces (or moments), andthose that reinforee the pitch motion are said to be un-stable forces (or moments).

The only roll damping force on a symmetrical bodyof revolution is the force produced by the viscosity of theair, (skin friction). This force always tends to retard theroll rate.

Ideally, it would be desirable to calculate the oero-dynamic forces accurately by theoretical means. However,thio is nut always possible, and it is necessary to mea-sure the forces in ground tests by blowing air past a fixedbody (wind tunnel), or by propelling the body through astationary instrumented region (ballistic range), or some-times by using a combination of both techniques. Bconomi-cally it is not possLble to obtain measurements on theactual vehicle for actual reentry conditions of ambientpressure, temperature, velocity, etc. To circumventthis difficulty, the aerodynamicist defines the forces nn dmoments in terms of aerodynamic coefficie'ts which npplyover a wide range of environmental conditions. The follow-ing coefficients are of interest in the study of renritry bodymotion:

A - CA S (2-2)N mCNqS (2-3)

M(q) _C (-. - iSd (2-4)q

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?b41 JOH$1404SHOkpINN'VtN6ITVAPPLI[0 PHYSICS LABORATORY

where q at(1 /2)pV2 (,Yf2)pM 2 an(_ is called dynamic pres-sure. Tile bar over the q is deleted In somic sections ofthis report where there is no possible confusion betweenthe symbols for pitch rate and dynaminic pressure. Thereference diniensions S anrd d niay be iany convenientdimensions. For a cone, they nre usuanlly tnken as theba sec ja0,Rca d h118 (din met.er, and thusu~ referen~ce dimen-

rrom dimensional nnalys Is, it may he shown thatfor a given ;; thle "'rody"-namic coe-fficients CA, CN (andn geometrically similar model (externnal, shape) providedcertain modeal stimilarity f'actors are tile same as thosefor the actual vehicle. The nimajor similarity factors ofIinterest for, the reentry body are:

TieIch numbor. is tile ratio of the vehtclevelocity (V) to the velocity of a sound wave (V-ylT). "'hereare four significant Which number re,,gimes. At subsonicdspeeds (M ft0 to 0. I), the acrdynnai coefficients areweakly dependent upon M. Tho transonic Mach numberregime extends from thle Ma~rch number at which a localMach num-ber over the body fi1rs~t becomes sonic (MI 1)to the Mach nui-nbr, nt which.1 the flowfield over the hodyis predominantly supersonic. This Mach number ra-ngespeed results in the, appetirance of shock waves, Trhestrength and location of the wnveF are highly sensitive tochanges in Mach number. Therefore, in the transonicspeed range, thle aeurodynnmic coefficients are usuallysensitive to change in M. In the supernonic speed range(M. P 1. 5 to (6), thle shock wave stteength Is dependent uponMach number but the shock location is somewhat insensitiveto changes In Mach number so that the aerodynamic coeffi-cients are moderately dependent upon M. The effect of

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114l JOamNm OPRIN9 UNIVSII6TWAPPLIED PHYSICS LABORATORY

*IhVOR @POllS 14A.y lkO

aerodynamic heating 12 moderate in the supersonic speedrange. At hypersonic speeds (M > 6) , the aerodynamiccoefficients tend to be relatively insensitive to M andaerodynamic heating is severe.

The Reynolds number, pVL/M = VL/v, is indica-tive of the importance of the dynamic viscosity of the air,M4(or kinernntic viscosity t). As a result of viscousforces, a region of low velocity is produced in the vicinityof the body. In fact, for continuum flow (discussed below)the velocity of the air relative to the vehicle is zero at thevehicle surface. The region of degraded velocity is calledthe boundary layer. Since the layer of air adjacent to thebody "sticks" to the surface, a shear stress, r, or anelement of friction drag, dD, results, tending to reducethe velocity of the vehicle,

Y (NORMAL TO S'URACE)j --------- LAMINAR-TURBULENT

LAMINAR BOUNDARYLAYER THICK(NESSTURBULENT BOUNDARY LAYER,' "THICKNINI

BODY SURFACE

The total friction drag is given by:dr,dS . (2-6).,DF = .II" dSw dyA ("

Therefore, the viscous drag is proportional to the dynamicviscosity and the velocity gradient at the wall. The vis-*cosity is very nearly a function of air temperature only

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?WEl JO~kuU MO,UOD~til.MIIKftIJAPPLIED PHYSIC LAMOHATORY

and is given by Sutherland's equation plotted in Figs. 2-1and 2-2. The velocity gradient is a strong function of theReynolds number. At very high Reynolds numbers, theflow in the boundary layer contains much eddy or vorticalmotion. As a result of mixing, the high energy air nearthe outer porLion of the flow is transported in toward thesurface. Since the layer of air adjacent to the wall sticksto the surface, this type of boundary layer has a verylarge velocity gradient at the surface and the skin frictionis relatively high. This type of boundary layer is calleda turbulent boundary layer. At sufficiently low Reynoldsnumbers, the eddy motion within the boundary layer dis-appeari, and the velocity gradient at the wall and skinfriction become much less than those for Lhu turbulentcase. This type of boundary layer is called a laminarbounda,'y layer. At low Reynolds numbers the flow tendsto be laminar; at high Reynolds numbers the flow tends tobe turbulent. The Reynolds number at whinh transitionfrom laminar to turbulent flow occurs varies with bodyshape and type of trajectory. A typical value of Re atwhich transition begin- to occur on a slender conical bodyis 107.

The surface heating rate is proportional to the skinfrictioi and therefore is greater for turbulent flow thanfor laminar flow. In fact, the heating rate in turbulentflow may be nearly an order of magnitude greater thanthat for laminar flow.

The Knudsen number is the ratio of the mean freepath of the ambient air particles to a characteristic dimen-sion of the reentry vehicle. This ratio may be expressedin terms of Re and M by the relation

K = 1.49 M (2-7)R eFor the very low values of density that exist early

in the reentry phase, the particles of air are spaced atdistances that are large compared to, say, the body length

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tHR JOHN$I NHOPINf UN1VCSIItyAPPLIED PHSICS LABORATORY*ILVlI* *pegINO MAPYLANO

or diameter. An air particle bouncing off the body is no tlikely to encounter another air particle. This region offlow is called the free molecular flow regime. At a some-what lowtr altitude, the air particles have frequent colli-sions with other air particles in the vicinity of the reentrybody. However, there are insufficient particles in thefilm of air adjacent to the body surface to cause it tostick to the surface. Therefore, although a boundarylayer exists, the film of air adjacent to the wall "slides"along the body surface, This flow regime is called theslip flow regime, At still higher densities, the air par-ticles at the surface stick to the surface, This flow regimeis called the continuum flow regime,

The characteristics of the aerodynamic forces andsurface heating depend upon the type of flow that exiotson the body. There are no sharp boundaries between thevarious types of flow, but approximate boundaries are asfollows:

Continuum to slip -- 1i0 2

Slip to transition -111m 10

Transition to free molecular M 10R eThe transition regime considered here is transition inbasic flow regimes which occurs at Knudsen numbers ofapproximately unity. This transition should not be con-fused with transition from laminar to turbulent flow in theboundary layer.

The Mach number, boundary layer, and basic fluidflow regimes are shown for a typical reentry body in Fig.4-21, page 260. Note that practically all the reentry

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THI ..OWNS IIOPKIN, U vI vAPPLIED PHYSICS LABORATORY

phenomena for which aerodynamic data are required occurin the continuum flow regime; only this regime will be con-sidered in the remairnder of this report.

Sometimes the lift nnd drrnf4 components of thenerody.iamic for'ce neroqutired for nnnlysis. The reln-tionship between the liift-dmtn g systclin of forces nnd thenormal force-axinl force system is shown in the sketch.

SNA L

a4V

From this force diagram, the lift and drag coeffi-cients may be obtained from the following equations:C C cos - C sin a (2-8)L, N ACD =CA cos e + CN sin (2-y)

Differentiating Eq. (2-8) with respect to c, and notingthat the variation of CA with ey is small, the slope of thelift curve at a = 0 is given by the equation:

CL =CN -CA (2-10)

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tHil JOHNS HOpKININ jNIV IIvAM-t.ILD PHYSICS LABORFATORYGI-10N SPIIN. M.S-tAN.,

2. 2 G(enural ChnaincteristicsIt is beyond the scope of this r'eport to provide pro-ce(duI'cr5lo'f estilmting the naerodynalmlic coefficuients for n

body of arbhira'ry shape, Ilowever, a few re mareks on thegenernl behavioi' of tie coefficients nae given in this see-tio)n, and :ppi' xinim te equa tions applicanle to n shn rp nosedCWne :i 1,fiVe.ii ill SCct ea 2. 3.Axial 'rcee. For most bodies of interest, the

axial foorce coefficient is weakly dependent upon c forsmall values of ' v(10 to 2*), strongly dependent upon M,andt moderately dependent upon R.0 . For a sharp slendercone, the order of magnitude of CA is 0- 1.

Normal Porce. The normal force coefficient isstrongly dependent upon rv and M and nearly independentof R1. For small values of fy, CN varies linearly with nso that CN = CNrY. For a sharp slender cone, CN a. 2per radinn.

Center of Pressure. In general, the center ofpressure is strongly dependent upon o' and M and weaklydependent upon Re. However, for small vwlues of rY,Xcp is independent of ry . Fo r a sharp slender cone,Xe p:, (2/3)L,

Pitch lDnn-ipng. The pitch damping coefficient isstrongly dependent upon M, weakly dependent upon &for small values of ry, and nearly independent of Re. Foraa sharp slender cone, a typical value of Cmq is -1 perradian. The negative value of CnMq indicates a stable damp-ing coefficient.

1oll Damping, The roll damping coeffici.nt for asymmetrical vehicle is very small, negative, hltiily de-pendent upon Re and M, and weakly dependent upon rfor small values of IFor a cone, a typical vwlue of theroll clamping coefficient is -0. 003 per radian,

- 22 -

tk " "'4


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r?HU[ JO94NS NOP4dINg) UHIVEMITYI

APPLIgED PHYSICS LABORATORY

2. 3 Approximate Equations for a Sharp ConeFor a major portion of the reentry trajectory, the

Mach number is in the hypersonLc regime. The aerodynamiccoefficients are nearly Independent of Mach number, andgood approximations to the aerodynamic coefficients forsharp nosed cones are given by relatively simple equations.

The skin friction drag coefficient, CAF, is verysensitive to the cone surface temperature, to the type ofboundary layer, and to surface roughness. To providesome idea of the order of magnitude of the coefficient,CAF, without becoming involved in details, equations aregiven below which apply to laminar flow with a cold wall(typical of early reentry conditions) and to a turbulentflow with a hot wall (typical of late reentry conditions), Asmooth wall was assumed.

An estimate of C, is given in which it is assumedthat the shear stress Is irected along the vector+ pr where pr is the circumferential velocity of a pointon the cone surface. The circumferential component ofIr is proportional to C1 pThe following equations are applicable to a cone ofhalf angle 8:

2CN - 2 cos 8 (2-11)N2

Xc.p. 2sec 8 L (2-12)

C = C + C + C (2-13)A AP AB A (2132C 2 sin 8 (2-14)

1,CAB M. (2-15)

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?UK JOHNS HOPKINS NIVINSITYAPPLIED PHYSICS LABORATORY

1. 9 ( 2+sin 2)Cosl (2-16)CA tai 8 RLaminar flow, cold wall

tan 2Co 8A 0.___ (1 + M _-;n 6

C cc)1. CA 01

2. 4. 2mer i Bodies.Boie inee to b+ syterin (2r1lay

Considymtri aBodieos hc teaymtrcfrei

sroduce byevr thsmall proubrncescate neavte iastruend of the body, as shown in the sketch.

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TIMiR OHMN NO$ MO NIS|V9IT?APPLIED PHYSICS LABORATORYSILVIO Se SIN4 APVLANO

At a 0, the protuberance results in a downwardasymmetric normal force, Nasy. This force causes thenose to pitch up, producing an angle of attack. The angleof attack produces a normal force, Ns. The vehicle con-tinues to pitch up until the moment about the center ofgravity is zero. The zero moment condition is called thetrim condition, and the values N and re that exist at thiscondition are called the trim normal force and trim angleof attack, respectively,It is convenient to define another dimensionlessaerodynamic coefficient called the pitching moment coef-

ficient, Cm, such that the moment about the center ofgravity is given by!M=CM q Sd. (2-20)

From the above sketch (using a nose-up moment as posi-tive):asy 1 (s c.p. c.g.

The total normal force is given by:N Ns - Nasy (2-22)

Writing Eqs, (2-21) and (2-22) in coefficient form:N s i N (Xc - X )

C asy s c.p. c.g. * (2-23)m - - dq Sd qSNs NasyC IS--(- , 2-24)IR S S

The term, Ns/lS, is the CN for a symmetrical vehicle.The Nasy term is usually negligible compared to Ns/qS

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THl JOHN*41 iJPKINi UNIVRIENITYAPPILIED PHYSIC,, LABORATORY216YINSPmIN4 MARh.At.O

and may be ignored in Eq. (2-24). However, the effect ofthis term on Cm cannot be ignored. LetasylaC mqSd o

Then (X xC C C C. . (2-25)N d0If the body has a linear variation of CN with , then Eq,(2-25) may be written:

C C " C d. , (2-26)mIf Cmo and X., p, are independent of 9, then Cm varieslinearly with a.

Within the limitations imposed by the assumptionslisted above, typical Cm characteristics for a stable sym-metrical and a stable asymmetrical vehicle are shown inthe sketch, page 27 ,Note that C is the value of Cm at o 0, and

aTR is the value of ey at Cm = 0. The value of TR isobtained from Eq . (2-26) as:Cm Cm i0 0 (2-27)~TR (X -x ) Z2-7

CN d

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TH JIOHISi "O INs UNIVOWAIIYAPPLIED PHYPICS LABORATORYS*I4.VlURkHU UAUI h.A

m CL ASYMMETRICAL VEHICLE

, -. SYMMETRICAL VEHICLE

In terms of the static margin defined previously:Cm 0

-R 0 o(2-28)TR (L/d)CN S,'M.In some reports the term Cn, is used. An equa-

tion for Crma may be obtained by differentiating Eq. (2-26)with respect to . Therefore:

Cm - CN I (2-29)To provide a feel for the magnitude of Cm . con-

sider the flap-type protuberance discussed at the beginningof this section.

- 27 -

iN


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Thi JOINI 4OPKINS UNIVURSIIVAPPLIED PHYSICS LANORATORY&ILvIR SPRINGi MARflV6AND

1 ley1a

5qA

The pressure acting on the flop and cone surfaces may beestimated using Newtonian theory which, in the simplestform, states that (for zero angle of attack):PF "2 2 sin 2 ( + c) (2-30)

q

= 2 sin 28 (2-31)

The force resulting from the asymmetry may be dividedinto normal and axial force components. For small valuesof 6 and (, the axial component Is small compared tothe normal component. Neglecting the axial component,the total increment in normal force is:

Nasy-SF(PF P (2-32)Then:

N I S *IPM qSd qSd IS d . (2q )28 1

- 238-

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Tr" JOHN$ MOPKINN UNIV910ITYAPPL.OID PHYSICS LAgORATORY

6%.Vt4t 1 ING M&AYI.A4ND

Using Eqs. (2-30) and (2-31), and assuming 6 and ( to besmall:

Cm (2T- (2-34)0As an example, let8 -100, 1/d 1, ,SF/S 0.1,

and I 20. For this asymmetry, Eq. (2-34) shows thatCmo 0 0027.

Another type of asymmetry for which Cmo can bereadily evalUL:Lcd Is one in which the body centerline is thearc of a circ*le, This type of asymmetry, for example, isan approximation to a distortion resulting from lateralloads or heating effects. For this asymmetry the momentis noarly a pure couple and Cmo may be expressed inturms of the change in slope of the body centerline fromnose to base.

007

At high hypersonic Mach numbers:C O 0.008 Le (2-35)mo

where A is in degrees.

-29 -

.. . ,. ........ ...,..


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fTl JIOHNSIHOPKIPN UNIVERIIIvAPPLIED PHYSICS LABORATORY

SILVIN SPMI2N MANYt.AND

2. 15 ExamplesA typical reentry body will be used in many ex-

amples In later sections of this report. To illustrate thematerial in the preceding subsections, the aerodynamicdata required for subsequent examples will be computed.rh typical body is a 10-foot-long cone with a basediameter of 3. 5 feet. These dimensions correspond to acone semiangle of about 100. The mass characteristicsare assumed to be as follows:

Weight 1000 poundsX = 6. 27 feet aft of the nosePitch moment of inertia about thecenter of gravity t: 300 slug-ft

2

Roll moment of inertia - 30 slug-ft 2

We will use a reference length d of 3. 5 feet and a refer-ence area S of 9. 65 ft 2 .At hypersonic speeds and for angles of attack closeto zero, the aerodynamic coefficients are as follows:Xc , sec2 1oo0 10 -6,87 feet (2-12)C. p. 3

2C = 2 cos 8 1.94 per radian (2-11)N-It is observed from Eqs. (2-13) to (2-17) that CAis a function of M, Re, wall temperature, and type ofboundary layer. The valves of CA for two combinations

of wall temperature and boundary layer type are shown inFigs. 2-3 and 2-4. The laminar flow, cold wall case istypical of conditions that would exist early in the reentryflight, say above 100 000 feet; the turbulent flow, hot wallcase is typical of conditions which would exist loter in


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THE JOHN@ OPKINS UNIVR"4IYAPPLIED PHYSICS LABORATORYSIIA1 SPRImNSlMANYIANS

flight, say below 100 000 feet. As will be shown later,most of the phenomena of Interest occur for values of Hefrom about 10 6 to 108. For these values of it~ ftA 0.iA

At M = 10 approximately (60% Ot ' CA results frompresrure drag (CAP), 30% from skin friction (CAF), andabout 10% from base drag (CAn). The percentages varysomewhat with Re, M, and type of boundary layer, butthe given values are typical of the relative importance ofthe three sources of axial force for low-drag reentrybodies. For blunted bodies, the pressure drag mayaccount for nearly 100% of the total drag.

Values of Cmq (Eq. (2-18)) are shown as a func-tion of X , /L i'n Fig. 2-5, and values of C,(Eq. (2-.19))are shown inFigs. 2-6 and 2-7. p

In the examples included in subsequent sections wewill use typical values of the aerodynamic coefficients.The values of the rmass and aerodynamic characteristicsthat will be used are shown in Table 2-1.

31i -31-


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Til, JOHNN* "O"'B aN .'aNIV~qiTYAPLIEDl PYSICS LABORATORYDILtY1 SPRIh MAMILAND

BIBLIOGRA PHIY

2-1 S. A. Schaaf :tlid L. Talbot, tl-ndbook of Super-[onic Aei 'odynaIAC8, Sectiun 16i, Mechanics ofRarefied Gnses, NAVORDSYSCOM Report 1488(Vol. 5), 1"',n!o,

2-2 M. Tobak an, \ 1, Wel'rend, Stability Deriva-tives of Cones nt Supersonic Speeds, NACA TNT776-8, Se.pte,.,L.,.t_' ,

Preceding page blank-33-

_"_'_ _-,_.., =,


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A12P IED PHYSICS LABORATORY

SILVIp-Tp I I IA

LU JI( I-

u...I0 2L

z

I II z

0rfu

Preceding page blank*35


43/351

?"t JOHNO HOPKINOS N141VCSIIYAPPLIED PHYSICS LAtORATORYISkLYS *eSh~ U&*$..SN.A

4 -

33,1 X10- SLUGFTSE

!iQ2 - -To 619 o R

1 -- J ~ T AND To IN 0 R"-,UTHERLAND'S FORMULA M (rT~ 191oi 1Qf- 12 l96.7/TO0I 4+ 198.71

0 400 R00 12DO 1(oo 2000 2400 2800 3200 3600 4000TEMPERATURE (CR)

FIg, 2.2 VARIATION OF VISCOSITY WITH TEMPERATURE

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I... __ _


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,i YMI jhWId NOl.w6 UWi'4aus'1 'V

APPLIED PHYSICS LABORA-r)H"

0.42

0.3

0.210104 106

Sig, 2.3 CONE AXIAL FORCE COEFF ICIENT, LAMINAR FLOW, GOLD WALL,8 - 10.

-37-


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TMK iJOmNs MOPMKINIUNIVIrS1IITYAPPLIED PHYSICS LABORATORY

SLVIO 5,0paN MAR6A~ND

',2 -....... . .....- I..... ,. ...... r .... r-i T- 'r ....... - ... . ----i ... '7M" 10 20

< 0.1 _____n_____

10 5 10 7 10u 10t

Fig, 2.4 CONE AXIAL FORCE COEFFICIENT, TURBULENT FLOW, HOT WALL, 6 10038

4 - 38-I...

S,'.,________,_,_._,___.


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TII JOHNS MOPKINS UNIVUIIYTYAPPLIED PHYSICS LABORATORY

.SLVIA 2P..IQ MARYLAND

1

.1.2

-0.4 -

0 , I I I I0.60 0.54 0.58 0.62 0.66 0.70X cg./L

FIg, 2"5 PITCH DAMPING COEFFICIENT FOR 100 SEMIANGLE CONE

-39S...... . . .. . .... .... , . -. ,','-;-:... ", ',',s..,' t".. ,


47/351

S IJHN$p'AIToVYA' *La, _P- ,,e',.A TO

.0,04

-0.04 M - 20o.030

.o.02 0l

iOoo0~

10 0

9 F,2-6 ROLL DAMPING COr LAmiaR FLO, COLO NALL, 8 1I

40'


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THE. OHN$lOMrKNI NIV9M01161TYJAPPLIED PHYSICS LABORATORYSIhlYSS *pPIMIS MAU~hANS

0.0003 T' "

-0.002

.0,00`1

lo' 10 g

FIg. 2-7 ROLL DAMPING COEFFICIENT, TURBULENT FLOW, HOT WALL, 6 100

-41-

i . .. ...S " -' "" "'.g, :.:. r ', m .I,, h." ' "- .: r...l, "r. , 1,- ,


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THi JOHNS HOPKINS UNIVliUMITAPPLIED PHYSICS LAUONATOV

Oiwvlm NPimI MARVLANI,

Table 2-1Mass and Aerodynamic Characterlistics

Masi3 CharacteristicsL 10 feetS 9, 65 ft 2d m 3. 5 feet

X g, .6,27 feetWeight (W) - 1000 pounds

Mass (m) - 31. 1 slugsRAI moment of Inertia (Iv)

30 slug-ft 2Pitch moment of inertia (Iy I) 300 slug-ft

lya2 0. 99

Ix .2 0. 79

Aerodynamic Characteristics Combined ParametersC W = l01b/ft

2C 2 per radian u 100Sa 0. 03 5 per degree A

. 6. 67 feet Al~c'P. S.M. =- " /4%CA -0.104Cm -1per radian AX 0q 14,

C, -0 . 003 per radianp

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?H l JOHNlS UOmIms UNlIVIryIYAPPLItD PHYSICS LABORATORYSILVli OPPING MhiV6A-0

3. ATMOSPHERE

AA

!: [ - 43 -


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lHIr JIONNSI '40pINSI UNIVSNS,?YlTAPPLIED PHYSICS LABORATORYSINKO HOPfINS "NIVIPSI.Y

SYMBOLS FO R SECTION 3

Symbol Dinlittion T_. c nlitsa speed oL sound 'it/sec

Napierian baseCoriolis force poundsc horizontal force resulting from apressutre gradient pounds2

g acceleration of gravity ft/secH scale height, RT/g 0 feeth geometric altitude feeth geopotential altitude feetCT 2P static pressure of dry or moist air ib'ft2P partial pressure of dry air 1b/ftp saturation pressure at T lb/ft2slb/ft 2p partial pressure of water vapor l/tR gas constant for dry air ft2/sec -

radius of earth feet0 ~2 2 OR gas constant for water vapor ft /sec -RAOB's weather measurements in which aballoon is used as instrumentation

carrierROCOB3's weather measurements in which a

rocket is used as instrumentationcarrier

S. L. sea levelT temperature of dry or moist air RT dew point temperature OR

- 45-Preceding page blank


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TH~r JONNI 9IOpMINII UNIV|ftI1IT

APPLIED PHYSICS LADORATORY

Symbol Definition Typical Untist time secondsV missile velocity ftsec,V wind speed ft/,ccqCW weight of nn elermentary volume of nir poundsdx, dy, dh dimensions of an elementary volume

of air feetZ refers to time at the zero meridianc change in density defined by Eq. (3-22) percentv ratio of specific heats0 relative humidity

orlatitude degrees

X mean free path feetkinematic viscosity ft2 /sec

P density of dry air x P/RT slug/ft 3pr density of moist air slug/ft 3

earth rotation rate rad/secSubscriptsIn refers to a reference altitude or to altitudes at which

weather measurements are obtainedo refers to sea level conditions


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APl'.tD PHYSICS LABORATORY3ILVUN SPANQ~ MARYLA"D

SUMMAARY ANF) CONCLUJSIONS

The atnisphore from se,- level to urhout 40 0 00 0feet Plays '1n imlpoi't:uit vole ill the moition it'~ umiir Vvehicles. In this section the generna mi tur1 , of ti ntill -sphere Is considered, and equations required for studiesconcerning vehicle motion Lire discusised. The vnria.blvsof most concern are atmospheric, density, temperature,wind, and moisture profiles (vtitrat ions with altitude),The 19~62 Standard Ati-osphere i~s used in many anialyses.Some of the charactvristics of this atmosphere are pro-sented, and differences from- othow types of sttandalrd daysare noted.

Tho major conciusionis iru-1 . Most phenomena of interest in the analysis ofreentry body motion occur in an altitudo region cnfled thehomosphere (son level to approxiimately 300 000 feet), Inthis region the atmosphere obeys two laws of considerableimportance. The perfect gas law Provides a r'elationshipbetween pressure, temperature, and density; the perfectgas law in conjunction with the hydrostatic equilibriumn-

law provides a relationship between density, temperature,and altitude, The hydrostatic law 1is frequently used asan interpolation equation for tabulnr values of densityversus altitude, permitting relatively few volues of the in-dependent variable for a given aiccuracy. Trhis considera -tion is of particular importance in trajectory simulationwork.2, Winds are obtained from direct measurements(b,,alloon observations, or PKA01Vs, for the lower aitm-osphere

and rocket observationsor lROCOl3's, for the uipper atnmo-sphere), from forecasts based on ohservations~or froml'orecasts based on statistical analyses of many previousobservations (climatological data). Thiese sources arelisted in the order of decreasing accuracy, and all sources

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THICJOHN$ NOPKiNS UNiVItPS'VAPPLIED PHYSICS LABORATORYSILViE lPIRING MARTVLAND

are frequently employed in some phase of annlysts of bodymotion, A law used extensively in the forecast procedureis the geostrophic wind law, which is derived by equatingthe Coriolis and pressure forces, However, this law be-comes invalid at low latitudes.3. If the mo.."txirc content of the air is high, threperfect gas law n-ust be applied using partial, pressures

of air and water mixtures with appropriate gas constants.Sometimes moisture is accounted for by using an effective(or virtual) temperature. The effect of water vapor is todecrease the density when it is compax'ed with a computeddensity in which the moisture content is neglected. Neglectof the moisture effect results in a density error of a fewpercent at most, and the error decreases rapidly with de-creasing altitude and decreasing temperature. For mostanalyses this effect may be ignored.

4BI

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T ~tH9 JO~HN$HOPKINS w~.I~(RSITvAPPL ILD PHYSIC93 LABORAI CRY

SMtVIN PPINU M~qILANV.

T1he ~ m~p ie s; dlivided rinto thi'wee br)oad roltitudebands cal led tile haro sphiere, uxc ~pher e, :11idnterplaneta ryg,18 (tPIg. 3i-1). T110' hrosp13)IM.'O (xisti froml S(e:ilvl uCC1LPoabout 500 km 1 (1. (65 . 10 feet or 275 rim i). lih.operCxi~tS fV01m 500 to ;1h(oot (GO 00 0 kmI (331 000 Mi110. III thee1x0Hphere'L, theC iij1 ruIit; jso klow' tjiot. iton lii idlyIever collide with on , ainother. Molecules enltering nt thlebaise of this region dlescribe ballistic trajectories, andsome, tra vel beoynd the gin Vi taltLIonal 1'i(-ld of theL planet,III thilik region, temlper~ature -ins rico significnr ice sincethere is rio Mnxwollalan distribution of vel'ocity. Int er -planetary gas u-cists at tittitudos above about GO 0 000 km1.I

The barosphere is subdivided into the troposphere,stratosphere, niesosipherc, and thermosphere with thletropopnuse, it~ratopnuse, mresopause, and thlermlopauselocnted between theose regions, Lis shown in FPig. 3-1,These altitude bands are, baised upofl thle temperaturevariation with altItude,. Tile region from sea lev~el to thetop of tile mesophere is also called the homosphere.

The behavior of each altitude band varios some-what with cartint latitude, Many types of standard atmo-spheres are in commion use; only three will be consideredin this section. Thle 1062 Standard Atmosphere is typicalof middle latitude locations; the Tropical Atmosphere is* typical of low latitude locations (0 1 to 20' N. latitude), andthe Polar Atmosphere is typical of high latitude locations* ~(north of 60' N. latit~ude).

In thle troposphere the variation of temperaturewith altitude is strongly influenced by the earthi acting asa heat source. The temperature, in general, decreaseswith increasing altitude. The top of thle troposphereoccurs at an altitude of 36 000 feet for th-e Standard Atmo-sphere, at 55 000 feet for tile Tr'opical Atmosphere, anid at83 000 feet for the Polar Atmosphere. Essentially allweather phenomena occur in the troposphiere.


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APPLIED PH.Y$iCS LA1HORA1CR'v

The.( tropopause is the top of' the troposphere Indis -an altitude of minimi-umn temperature. i'or the Standu rdAtmosphere the temperatlure is constant from 3(3 000 to66i 000 feet, where it begins to rise; for the tropicail ntmno-sphere the temprnlnture begins to rise rit 55 00 0 feet; forthe polar' atmiosphecre the temiperatureVL is (mnstant to Wi 000feet, Where it b. girls to rise. The teniperatLre rise is aresult of ;I)5Iovploiiliot iltI8\'i(1i'ht sulni, rtidiations byozone. The temper'ature Conitinlues to riisc with mecca singaltitude to about 1.563 00 0 feet for the Standard Atmosphere.Characteristics to suifficiently high a.ltitudes --re not avaU.-aible for the tropical and polar, ati-nosphereSg. Above- thestratosphere the tei-perature again decreases with in-creasing altitude., reaching ;i mninimum temperature atthe inesopause. Pox' the Standard Atmosphere, this 1lti-tud.a is 263 09J0 feet. Above the m-esopause the tempera-I ture Again Increases with increasing altitude up to thetherniopause--, which occurs at approximately 1 65 0 000feet (275 nmi). Thu temperature rise is caused by theabsorption of ultravio~let rays with a wavelength of 100k.

The homosphere (sea level to 300 00 0 feet) is thealtitude band of primary interest for analysis of reontrybody motion. In this altitue region, the atmosphere Ischaracterized by constant proportions of N2, 021 andargon, and the atmosphere obeys the hydrostatic equilib-riumn and perfect gas equations.For analysi s of reentry vehicles, tile atmosphericproJerty of most interest is air density; of secondaryinterest is air temperature. The ratios of density f'w thetropical arid polar atmospheres to the density for the10)62 Standard Atmo)sphere are shown in Fig. 3-2. Itmay be expected that the tropical. atmosphere would havea lower density and the polir atmosphere a higher den-I ~sity than those of the 1962 Standard Atmosphere. Ho0w-Iover, this is true only for altitudes b~elow about 20 000to 30 000 feet. Therefore, depending upon thQ vehicledesign and trajectory, the tropical day may be equivalent

to an atmosphere of greater density than tha~t of the 1962

'50LRW.....


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THN JOH04%"OPM04% kINIVIIUIISITVAPPLIED PHYSICS [.ABcM4ATnF1Y

Starviard Dlay, and the polar day may be, equivalent to Anatmosphere of lessor density than thnt of the 1062 StandardDay.

The toemperature, in addition to the effect on den-sity, qffects Cie sp.)cd of' sound, and thas Mach numberVI/a). rhe spceed of sounfd ( = i W T ) is1 proportional to thes qu a rc VMut of' tiw nvnibi eat ternipe 1'tu re aad tli o'e c harane-teristics are aLso shown in Fiig. 3-2, Notu that the sp,udof sound at a given altitude does not vary from. one stan-dard atmosphure to anothev by more than about 7. 5l%.This amount of variation is r'elati~vely unimportant. How-*ever, extreme variations in temperature from the I J62Standard Atmosphere, such as those of the winter months-it northern latitudes, may have a significant effect on areentry body trajectory by affecting the Mach numberhistory.

Some, characteristics of the 1962 Standard Atmo-sphere are given as a function of altitude in Table 3-1.More elaborate tables Lire given In R~ef. 3-1. Similardata for the tropical and polar days are given in Reis.3- and 3,3.3. 2 Variation of Density and Temperaturc with Altitude

Consider an elementary volumne of air (page .52) ofbase area dxdy and height dh, If the elemnent: of volumois in static equilibrium (the acceleration of the elume--ntof volume is zero), then the following equation applies:

dxdy(P + dp dh ) -Pdxdy + W -The weight of the volume of air is given by:

W = gdxdydhTherefore,

dP - gd h (3-1)

-li1t


58/351

AHIE JOHN$ OPKII4 UNIVCRNITVAPPLIED PHYSICS LABORATORY

SILVIA SPRING MHATLAND

p+dPRdh

dx P

This equation Is called the hydrostatic equilibrium equa-tion, and all three standard atmospheres dlicussed in thissection obey this equation, A second equation that is validin the homosphere is the perfect gas law which states!P = RT , (3-2)

Differentiating Eq. (3-2):dP - R(Tdo + pdT) . (3-3)

Substituting Eq. (3-3) into Eq. (3-1) and expressing g interms of go:p T RTg

By definition:dh dh (3-5)G go0

- 52

......................................................................


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THKI OHPN5 HOPKINN UNIVI"GItyAPPLIED PHYSICS LABORATORY

OILVI1mNPRING MAR.TLANU

where hG is called the geopotential Rltitude and h is thegeometric altitude. If g is assumed to vary inverselywith the square of the distance from the center of theearth:

R0 0 ('-6)

Substituting Eq. (3-6) into Eq. (3-5) and integrating:hG +h- h (3-"7 )

Horh

h G (3-8)hG1 "-

In the remainder of this section, the solution toEq. (3-4) will be given in terms of the geopotential altitude.However, the results may be obtained in terms of h byusing Eq. (3-8). It should be noted that, since Ro f 2. 09 x107 feet, h f hG for low altitudes. Even at h 100 000feet, hG and h differ by only 477 feet.

In terms of hG , Eq. (3-4) becomes:dT = go

d_ + -- dh (3-9)p -T RT GThe closed form solution to Eq. (3-9) depends upon thenature of the variation of T with hG . For practical ap-plications it is necessary to consider only two cases, aconstant temperature (isothermal) relation and a linearrelation.

; - 53 -


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hic JOHNi HO4KINO UNIVUEROISYAPPLI rJ PHYSICS LABORATORY

IILVEE1S'PlNG M&I.yLA..D

Consider an altitude interval from hGn to hG. Inte-grating over this altitude band, Eq. (3-9) becomes:

go hG Idh (3-10)n n h GnIf T is constant, Eq. (3-10) maybe integrated to give:

h -hGn 13-1,l)j -"P exp (311

where RT'H - - (3-12)90If T varies linearly over the altitude band, then:

+dTT nT - (h - ) (3-13)n d h-- G Gn

In this case, Eq. (3-10) may be integrated to give:R dT ,(3-14)

For calculations, it is frequently desirable to de-fine the density and temperature variations with hG by anequation rather than to use tables, If the atmosphereobeys the hydrostatic equation, it may be divided intoaltitude bands for which the temperature is accuratelydefined by linear variations of T with hG . Then thedensity and temperature within a given band are definedusing Eqs. (3-11), (3-13), and (3-14) and the conditionsat either the lower or upper boundary of the band.

- 54 -

,- , .. . ,. . ,. , '. .,, .,,,,C ,;: k,., .- : ... . .'.': . . . .. l' . -, . .. v


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T atOHNII HOPKINS UNIV1 IrPPLIED PHYSICS LABORATORY.vian sPlUme MARYLAND

The characteristics of the 1962 Standard Atmo-sphere may be defined to within 0. 5% accuracy using Eqs.(3-11), (3-13), and (3-14) and the data in Table 3.-2.

For simple calculations, it is frequently assumedthat the entire atmosphere is at a constant tempernture.Such an atmosphere is called an isothermal or exponentialatmosphere. In this case, Eq. (3-11) becomes

Values of and H are selected to provide the best accu-racy -for pin the altitude region of most interest for theparticular study. The quantity H is called the scaleheight. From Eq. (3-11), the scale height is the change inaltitude corresponding to a change in density (or pressure,since T is constant) by a factor of 1/e.3.3 Nonstandard Atmosphere

Frequently it is necessary to consider atmospheresother than the Standard Atmospheres. The quantities ofprimary interest for studies of reentry body motion (ne-glecting such specialized studies as the effects of windshear and wind gust) are density, temperature, horizontalwind direction, and horizontal wind speed. These dataare available from observations, forecasts, or' climato-logical characteristics. The latter data are character-istics based on statistical analyses of many observations.

Observations are made in two altitude regions.Balloons are used to obtain temperature, pressure, dewpoint, and wind data at altitudes from the surface up to thehighest altitude possible. Frequently, the upper limit inaltitude is about 30 000 feet. Sometimes the data extendto observations at about 100 000 feet. These observationsare called RAOB's. In the northern hemisphere, RAOB'sare obtained daily at OOZ and I 2Z hours at many observa-tion points. Rockets are used to make observations of

- 15 -

"- -U--


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?41 JO1N4H HOPKINS WNINY 1RI17YVAPPLIED PHYSICS LABORATORYSILVIN SPRING. Mh*YAftCA

temperature, density (sometimes), and winds as a functionof geometric altituie for altitudes from about 60 00nf to200 000 feet. Data from these soundings are calledROCOB's and are made only at a very few locations,usually near 12Z hours Mondays, Wednesdaye, and Fridays.3. 3. 1 RAOB )at-DryAir

The RAOB pressure and temperature datta are con-verted to density data by Eq. (3-2), The geopotential alti-tude corresporiding to the observed P and T are calcu-lated using the hydrostatic equation. Since data aro ob-tained at a large number of altitudes, the temperaturesfrom one measurement to the next may be averaged andused as a constant so that Eq. (3-i1) applies and may bewritten in the form:

h uhG + R(Tn + Tn+I) n (3-16)Gn+l hn 2 gn+nwhere n corresponds to the measurement at one altitudeand n+1 corresponds to the measurement at tne succeedingaltitude,

Equation (3-16) is evaluated in stepwise fashionstarting at the surface conditions where hGn is the surfaceelevation of the observation location.3. 3. 2 RAOB Data - Moist Air

The values of density computed uaing Eq . (3-2) arevalid provided the effect of water vapor may be neglected.The effect of water vapor on density may be evaluated bywriting the equation for the density of the air-water mix-ture as the sum of the partial densities:PA + "w (3-17)

RT R Tw

I I,


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IN JH iOMl oPHNOkl UNIVIrNSIIIAPPLIrD PHYSICS LABORATORYi lilL~S111 B.I...l MUY&lli CN

whereP " A + Pw

In terms of p and using R/Rwfl 0. 622:Pw

0( 1 - o 378 ,)(3-19)If the moisture content is given in terms of relativehumidity, 0 - Pw/Ps, rather than Pw, then,

,' 1 - 0. 37 8 (3-20)The water vapor pressure, Pwo is the saturation pressureat TD, and P 5 is the saturation pressure at T. Empiricalequations fo r the saturation vapor pressures are:

P.in. exp 2. 65

Ps exp whc2.i5 g-ne5TrThe effect of water vapor is to decrease the den-i s~ity compared to a computed density which ignores the

effect of moisture. The percent difference in density isgiven by:

37,. oD (3-22)37 exp 22. 65 - 9P Tx

A plot of the error as a function of altitude anid dew pointis given in Wig. 3-3. Also shown are the standard

57-

-- -


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T'lE JOHNIS HON INS UNIVERSIYVAPPLIED PHYSICS LABORATORYil-VIM SPRING MANiNhANG

temperatures for the 1962 Standard Day and the StandardTropical Day. For these standard days, the effect of 100%relative humidity (TD = T) on percent change in density ismaximum at sea level and is about 0. 7% for the 1962 Stan-dard Day and about 1. 8% for the Standard Tropical Day.Only for very hot humid days is water vapor likely to haveany significant effect on density, and even this effect issignificant only for very low altitudes.3. 3.3 1OCOB Data

The effect of moisture is negligible at altitudeswhere ROCOB data are obtained; therefore Eq. (3-2) isvalid. When density and temperature both are availablefrom an observation, the only data reduction required isthe conversion from geometric altitude to geopotentialaltitude by Eq. (3-7). However the density data are usuallynot nieasured directly. In this case, Eq. (3-16) may beused in the following form:r

0n+1 n [R(Tn + Tn+l) (hGn - hn+l) 3-23)where p+l is obtained in stepwise fashion using a knowninitial value of density (from RAOB data, for example),measured T, and hG computed from Eq. (3-7) using mea-sured h.3.3.4 Winds

A wind is defined in magnitude (speed) and direc-tion from which it is blowing, measured clockwise fromthe north. In some current forecast procedures, the pre-dicted winds are not based on measured winds but uponmeasured pressures and temperature, using the geo-strophic wind equations,The geostrophic wind is obtained by balancing thehorizontal pressure force acting on an elementary mass

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THR JOI, NI HOPKINS UNIV1PSITVA4PLIED PHYIICS LABORATORYMiMV1M0MINi. MARLAP4

of air by the Coriolis force, The Coriolis force is per-pendicular to the horizontal velocity of the elementarymass and, in the Northern Hemisphere, is directed to theright (to the left in the Southern Hemisphere). The magni-tude of the force acting on an elementary mass of air,pdxdydh, is given by the equation:

F c 2V wto sin 0 dxdydh , (3-24)A horizontal pressure force acts on the elementarymass whenever a horizontal pressure gradient exists.Consider the following horizontal high pressure system.Let x be the horizontal distance perpendicular to theisobars, y the horizontal distance parallel to the isobars,and h the vertical dimension,

Vw ISOBAR

Fi

FC HIGHX PRESSURE

dx

The pressure on the two surfaces of area dxdh areequal, so that the net pressure force acting on the massin a direction parallel to the isobar is zero. The pres-sure on the two dxdy surfaces are unequal, but the weightof air is just balanced by the pressure force (hydrostatic

59


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?He JOWNGOPKINS UNi~aRSITVAPPLIED PHY4SICS LABORATORYSILVIA $PRIM*S MAPVLAN13

equilibrium is assumed), so that no net vertical forceexists. Let the pressure acting on the dydh surface thatL~ s farthest from the high pressure center be P. Then thepressure oui the dydh surface nearest the center isFr + aF dx. The corresponding forces are Pdydh and(P + dx)dydh. Therefore, the net force is,

F ~ dxdydh,(2)The total horizontal pressure force is erpendicu-

lar to the isobar and directed away from the high pres-sure, Since the geostrophic wind is obtained by a balanceof the pressure and Coriolis forces, the Coriolis force isperpendicular to the isolbar and acts toward the high pres-sure center, am ehown in the sketch, For a Coriolis forceIn this direction, the wind maust be directed along the iso-bar with high pressure to the right (for the Northerm Hemi-Isphere), inope F. cFp5 from Eqs. (3-24) and (3-25):2V siW (3-26)

Equation (3-26), the geostrophic wind equation, Is a,curateexcept for the first few thousand feet above the surfacewhere friction effects cause the wind to be inclined awayfrom the Isobar toward low pressure and the magnitude islower than that given by Eq. (3-26), Typically, the mnaxi-mum deviation in direction is 250, and the maximum devia.tIon in speed is a factor of one-third to one-*half the geo-strophic wind, depending upon surface roughness. In addi-tion, the equation is not valid at latitudes near the equator,The equation is valid for the Southern Hemisphere, bu t thedirection of the velocity along the isobar is opposite to thatdiscussed for the Northern Hemisphere,N

Equation (3-26) is applied at various altitude levelsto obtair. the variation of wind speed with altitude. The

~~6.


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?me JO4N$OKINI UNIVgNSI1'YAPPLIED PHYSICS LABORATORYIfIlbulll SIDII MARYLANIIwind direction is obtained from the slope of the isobar atthe location of interest. Therefore, accuracy in windcharacteristics ,'equires accurate estimates of density,horizontal pressure gradient, and shape of the isobars.

tV

.i.

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1!T4e JOMNS 400IWO uI1N'1I0I6TVAPPLIAO PHYSICS LMORATORY

SILVIA bFA'N6 M&AIM~ANf

REFERENCES

"3-1 U. S, Standard Atmosphere, 19G2, prepared undersponsorshlp of National Aeronautics and SpoceAdminiistration, United States Air Force, andUnited States Weather Bureau, Washington, D. C.,December 1962,

3-2 Climatic Extremes for Militar.j_,utpment, Mil.Std. 210A, 2 August 1967,3-3 L. F. Fehlner and E. V. Nice, Tabulation ofStandard Atmospheres at 100-Foot Intervaos of. A1lt~ude, APL/JItU TO313-1; AugM 1058.

PI

1'__ __ _______ --. 4 .,.%i1 -


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T1,9 JOHN% HOPKINs UNIVtP5Ilyo;APIILf PHYSICS LABORATORY

SILVIO S*poIh MARVI.AND

INTERPLANE-rARY GA S

000 FEET33 000 rirnrNOTE: NOT TO SCALE EXOSPHIFR,

ABOVIE 320 0HEFEETLBEnm/ I tl 275 nrmil -- 'r~HE=RMVOP'AUSE t -

320T"HEAMOSPHEE

-/MESOPAUSE--

240-

200 -

-- STRATOPAUSE--160

120 STRATOSPHERE

40 ---- TROPOPAUSE - "TROPOSPHERE

3W0 34 0 38 0 430 480 500 !40TEMPERATURE ( RI

Fig, 3.1 TYPICAL ATMOSPHERIC TEMPERATURE PROFILE AN D ALTITUDE REGIONSPreceding page blank - -

~


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TM9 JOHNSNOPRIN48 UNIVWKNSIYAPPLIED PHYSICS LABORATORY

06 C

wI-L iIccJ

LL.

-66 -


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THE JOWI.S MHOPKINSUNIvUlI?vAPPLIED PHYSICS LASOAtOmY

0,5 1.0 2.0 3.0 4.0 5,07 0 - -- - - .. . . .

e- 0.1%50

:40

S30 -STAi",ARD

TROPICAL TE PERATUR20 1962 STANDARDTEMPERA ,

10 N

: o ,I , ,I, ,\i-60 .40 .20 0 20 40 60 so 100 12 0

DEW POINT (0 F)Fig, 3.3 EFFECT OF WATER VAPOR ON DENSITY

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Tilt JOH4N%9OPKII4S UNVIEtNSITYAPPLIED PHYSICS LABORATO~RY

SIVI SPRING MAQVI.ANfl

1 -L0n ca (0 f(0 M0 M0.tr 1 1 0f -~lMCIS m cc t - in In al M U M

"II('I CV M 1 )to o 4 l

4'4

M- to -zte

+- +-- - , +++

t, C;. ic 6cl 0

NNL0

0) ell co c mC - ~ o-

-4 -4 -4 I 4 In0 W

a. (0 to M..t4N..i-.4.-~a m

. . i .. NI M '10 4 0 r'-M ~in M) tHol"0in 04 WInN C4

-4.

-. 00 -u


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IfWI JOINS 4OPRINS UNIVBPSIBTAPPLIED PHYSICS LABORATORY

SI1VIN SPRIN'k MA*VLA.D

Table 3-2Constants for the1962 Standard Atmosphere

dT *Td(feet) (0 R) (fR/ft) (slug]ft3)

0.000 518. 670 -3, 56619991E-0:3 2, 37680982E-0336089.238 389,970 0, OOOOOOOOE+00 7. 06119929E-0465616. 750 389. 970 5. 48639B00E-04 1. 7082000OE-04

104985. 875 411. 570 1. 536199B2E-03 2. 56609928E-05154199. 375 487, 170 0. OOOOOOOOE+00 2. 769 79 972 E-06170603. 625 487. 170 -1, 09729986E--03 1. 47349965E-06200131.188 454. 770 -2. 19459995E-03 4.871 89993E-072591B6. 313 325. 170 0. OOOOOOOOE+00 3. 88259984E-08291153. 188 325. 170 1. 69529999E-03 6. 1507989BE-09323002. 625 379. 170 2. 83429981 E-03 9. 65109992E-10354753. 250 469. 170 5. 68669662E-03 1. 907099B)E-10386406.125 649. 170 1. 14439987E-02 4, 72659967E-1I480780. 813 1729. 170 8. 63499939E-03 3, 56239951E-12512045. 938 1999. 170 5. 77439740E-03 2. 24879924E-12543215, 188 2179.170 4. 06099856E-03 1. 55919982E-12605268. 375 2431. 170 2. 92699994E-03 8. 43449DN4E-13728243. 750 2791. 170 2. 38109985E-03 3. 03459948Eh-13939894. 688 3295. 170 2. 01519998E-03 6. 95599719E1-14

1234645. 000 38B9.170 1. 63550000E-03 1. 26079973E-141520799. 000 4357. 168 1. 10109989E-03 3. 05989996Z-151798726, 000 4663.168 7. 32979970E-04 9. 00309B37E-162068776. 000 4861.168 0. OOOOOOOOE+00 3. 04629894E-16

'The temperature gradient is applicable to the altitude handbeginning at the altitude for which the gradient is listed andending at the next highest altitude.

I' , -69-

........................................ .....*~..-- ,,-.


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THE JOHNSHOPKtINSNIVERSITYAPPLIED PHYSICS LABORATORY6ILYRR SPRING6.MANYLAND

4, EQUATIONS OF MOTION

Precdingpageblan1


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21TI JOHNS HOPKINS UNIVRIrIIIITYAPPLIID PHYSICS LABORATORYBIN-VIP SPRING. MAVL6AND

The motion of a reentry body during ballistic flightis affected primarily by five factors:

1. Gravity,2. Earth rotation,3. Atmosphere,4. Body dynamics,5, Asymmetries,

Each factor is considered in simplified form to show thegeneral effects on the trajectory.

',.,

73.

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IYNI J@44NM@9WN1UNIVSNUIYVAPPL ID PHYuICS LAIOATOY m&wse.m MAN'i.ANS

IIII

4,1 EFFECTS OF GRAVITYIIWA

I

5 Preceding page blank -75-


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?HI JOHN$ HOPKINS UNI1[eI?.yAPPLIED PHYSICS LABORATORYI ," llSIVIN SPillO MAIWILhI.D,

SYMBOLS FOR SECTION 4.1

Symbol Definition Typical Unitsg acceleration of gravity ft/sec2I constant defined by Eq. (4.1-15)K constant defined by Eq. (4. 1-5)R range feet or nmir distance from the center of the earthto a point on the trajectory feetr r/rt time secondsV velocity ft/seeV /VI -rSflight path angle degrees6 central angle, measured from apogee radians ordegreesSubscriptsa refers to conditions at apogeee refers to conditions on the surfaceof the eartho refers to initial conditions

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ThE JOHNS ,OPKINS UNIvIIITVAPPLIED PHYSICS LABORATORY

MILVE SPRINdSMARYLA0d1


The effect of gravity only on the trajectory of aballistic missile is considered. Equations that define thetrajectory (velocity, time, flight path angle, altitude, nn drange) in terms of initial conditions are presented,Plots are given which show the relationship ofrange and flight time as a function of initial velocity and

flight path angle for minimum and nonminimum energytrajectories. Also shown are plots of range sensitivityto initial velocity and flight path angles, An example ofthe uses of the equations is given in Example 1, Section4.6.

The major conclusions are:1. A ballistic missile follows an elliptical flightpath. For a given initial velocity, there is an initialflight path angle that results in maximum range, and thetrajectory is called a minimum energy trajectory. Forminimum energy trajectories, the flight path angle de -creases from 450 for very short range to approximately

320 for a range of 3000 nmi; the velocity increases from0 ft/sec at zero range to approximately 20 000 ft/sec at3000-nmi range; the total flight time increases from 0seconds at zero range to about 1300 seconds for 3000-timirange; the trajectory apogee increases from 0 nmi at0 nmi range to about 550 nmi at 3000-nmi range (a maxi-mum apogee of about 70 0 nmi occurs for a range of 5500nmi and a flight path angle of 22. 5O).

2. Trajectories for which the initial flight pathangles are greater than those for the minimum energytrajectory are called lofted trajectories; trajectories forwhich the initial flight path angles are less than those forthe minimum energy trajectory are called delofted, ordepressed, trajectories. For a given range, the totalflight time increases with Increasing flight path angle.Preceding page blank


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weINSP4 HONPKINS UNIVUNUITYAPPLIED PHYSICS LAUORATORYBILVUR PPIN MAm1mIaNIS

3. The range for a minimum energy trajectoryis extremely sensitive to initial velocity (about 2800ft/ft/sec at 3000-nmi range) but is very insensitive toinitial flight path angle. With Increasing loft at a rangeof 3000 nmi, the sensitivity to initial velocity decreasesbut the sensitivity to Initial flight path angle increases,For delofted trajectories at 3000-nmi range, the sensi-tivity to both initial velocity and initial flight path angleis greater than that for a minimum energy trajectory,

80

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Tl~ JOHIN@ OOKINNINIVR"lyAPPLIED PHYSICS LABORATORYGILVEISSOpINSMAVLA&ND

"Consider a nonrotating, spherical earth without anatmosphere. The geometry of the trajectory is shown inthe sketch.

VO0V

The trajectory of a body with initial velocity Vo atro and fired with a flight pathyo may be obtained by apply-Ing the laws of gravity, conservation of energy, and con-servation of angular momentum as defined by Eqs. (4. 1 -1),(4, 1-2), and (4,1-3), respectively.

2 2goro gr (4. 1-1)2Vo V2

t gr - gr (4.1-2)2 ~ 0 0r V cos0 rV cos (4. 1-3)00 0It may be shown that the trajectory is ellipticalV2 V2

for 0' < 1, parabolic for -- 1, and hyperbolic for2g r 2g r0 0 0 0

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THE JOHN$ HOPKINS UNIVENS8tYAPPLIED PHYS!CI LABORATORYSILVIR SPuINs MASVLAND

V2 V20 00 > 1. For ballistic vehicles, 0 is always lessgoro 2goro,

00 0 0than I so that the trajectory is elliptical. Using Eqs.(4. 1-1) through (4. 1-3) and the properties of an ellipse,it may be shown that the trajectory is defined by theequation:- 2 cos20 0 (4.-4)I -K cose ,

whererKr r 0

- 2 ( -

K - I - VE ( 14 byosti (4.-5)IThe range is the arc length OB and is given by the

equation:R 2roe (4.1-6)

where 80 can be obtained from Eq. (4. 1-4) by settingr I and solving for cos eo. Thus:

I - V- Cos2_ o,Cos 19 = K . . .or 2V0 cov 0 sin 7 0

tan 1V C2 2 (4. 1-7)" I-V 2cos8O20 0"-82 -

ii


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?"t JOHNS HOPKINS UNIVISII 'APPLIE D PHYSICS LABORATORY

It should be noted that the equations derived aboveare general insofar as the reference nltitude, re , Is (-on-cerned, That is, ro is not necessarily a surface coridi-tion. H-lowever, if rv is not n surface condition, care mustbe exorcised in evnlunting the icference vLlodty, V-79g0 ,and It . In terms of surface conditions:

'r (r 0 0 ) , 26 000 (4.1-8)0 O0R~ 2r.6 (4-ne ,0From Eqs. (4, 1 -6) and (4 , 1-7), it is observed that

a given range may be obtained for varioua combinations ofVo andy , However, there is one combination of Vo atid*o that resuilts in minimum Vo. This trajectory, calledthe minimum energy trajectory (MET), represents theminimum booster impulse forx a given range or, alter-natively, represents the maximum range that may be ob-tained from a given booster. The minimum energy tr,-Jectory is obtained by operating on Eq. (4. 1.-7) usingstandard procedures for obtaining maximum or minimumvalues, The results are the following:

yo~ :2+-"tr "-o}or o (4. 1-10)

* y070=45 -T+: T3-2 2V -tan y (4. 1-11)0 minIt is noted thnt '/ decreases from 45' at very shortrange to 00 at maximum range (R/ro ft). The velocity

increases from 0 st zero range to Vo -gr. at maxinmumrange. The latter velocity is the velocity for a circular

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?11JOWNI NO NINOS ONIV906tV~1APPLIED PHYSICB LABORATCORIYlSvIm 3pmlh,2 kAII?&.AND

orbit. For ranges that will be considered in this report (R11000 to 3000 nmi) the values of y and V0 computed fromr.Jqs.(4, 1-6) and (4.1-7) are shown in Figs. 4-1 and 4-2. Forall examples given in this section, ro = re 2.09 x 107feet,The time of flight may be derived in the followingmanner. The rate of change in 6 is given by the f'ollowingexpression:

- Cos (4,1-12)rThe negative sign is required since, for convenience inderiving the equation of the ellipse, 8 is measured fromapogee, Using Eqs, (4. 1-3) and (4 , 1-12) the equation fortime, becomes:

rt r d0 , (4,1-13)VCos

Substituting for r from Eq, (4, 1-4) and integrating givesthe result:ts (V 0 co s '0) (4.1-14)

where I is given by the equation:

In I + 2 tan- ( V - -I (9/ 2 )\ (4.]1 -15)Vo 2 - V a Cosn o, ' -Ko I - Kand is evaluated between the limits e and So,

If one is interested in just the total trajectory time,Eqs. (4 . 1-14) and (4. 1-15) may be simplified by noting- 84 -'Ii

I t !il ' l ll l ' l...... ,l


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TM9 JOHNS HOPKINS WNIV3R1ITVAPPLIED PHYSICS LABORA'ORYSILVt SPlRING MARYLANO

that the trajectory is symmetrical about E = 0, and theequation for the time becomes:

2V Co K sLn8 j -1K, tan (6c,/2)g ... + - Ka (4.1-16)

A plot of Eq. (4, 1-16) is shown for a range of 3000nmi in Fig. 4-3. For a given range, time increases withincreasing initial loft angle, -/,. The minimum time for arange from 0 to B is that computed for a circular orbittrajectory from 0 to B; that is yo = 0, Vo - gC-ro, andt 692 seconds for R = 3000 nmi. For a minimumenergy trajectory, K tanyo and Eq. (4 . 1-16) maybewritten as!

tar tar -&t 2 -2 Cos 3 atan 2y2 tan' 0 (4.1-17)J go 1/-n)ool ta~n V'

The time for a minimum energy trajectory variesfrom 0 seconds at zero range to 1rYTO7"- = 2520 secondsfo r maximum range (R = ffr 0 ). For ranges of 1000 to 3000nmi, the flight time is shown in F!g. 4-4.

The local flight path angle may be expressed in theform: dr

tan " (4.1-18)


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?HL JOH4N$ HOPKINS UNIVIRI[ITYAPPLIED PHYSICS LABORATORY

SILvE1 SPINGl MARYLANO

Solving for dr/d8 from Eq. (4. 1-4) and substituting intoEq. (4 . 1-1 B):

K sin 84l1tan = 1- K cos 9Differentiating Eq. (4. 1-19) with respect to t and

using Eq . (4. 1-1 2), it may be shown that j is given bythe following equation:3K(K - cos 9) V cos (4.1-20)

(1 - K cos 6)2 rEvaluating Eq. (4. 1-20) for ccnditions near impact,

it may be shown that # is very small for 1000 to 3000 nmitrajectories. Therefore, in the absence of an atmosphere,the flight path during reentry is nearly a straight line.This fact will be useful. later when we consider the effectsof the atmosphere.

The velocity at any point along the trajectory maybe obtained by solving Eqs. (4. 1-1), (4. 1-2), and (4, 1-4)to obtain the following equation:-= 2 2K cos 8 + 1'V (4,.1-21)

V cosy 0At apogee 6 =y 0, and from Eqs. (4.1-4) and(4.1-21): -

2 2- VCos 700a I -K

SV I -KV cosy 0 "8 -

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TH4 OHN$ HOPKINS UNIVEREITYA4PLIED PH'ISICS LABORATORY

SILVSP SP~ilNO MARYLA~N

orl~t. For a minimum energy trajectory, K = tan yo ando- -=1 - tan 2 y0, so that the altitude (r - ro) and velocity

at apogee are given by the equations:h =- (si-n + cos )0 ) cos 'Yo- 1, (4.1-22)V 9 1o+tan (4.1-23)" Cos 1 +tan )I0 0

or"a o cos ^/ + siny (471-24)

For a minimum energy trajectory, the altitude atapogee is maximum (ha %; 00 nmi for ro = re) for Y.22. 50 (R o 5500 nmi) and varies from about 250 nmi forR = 1000 nmi to about 550 nml for R = 3000 nmi. Sincethe sensible atmosphere extends to about 50 nmi, mostof the trajectory occurs in th e exoatmosphere. The ratioof velocity at apogee to initial velocity decreases from 1at zero range to 1/V-for maximum range. For rangesof 1000 to 3000 nmi, Va/Vo varies from 0. 71 to 0.72.

Another consideration of interest is the sensitivityof range to the initial values of VD and 7 0 . These sensi-tivity values may be obtained in the following manner:

- ~ (4.1-25)Wv ae avo 0 0aRi/6.6 0 and~60 /av may be obtained fromn Eqs. (4. 1-6)and (4. 1-'t) so that:

(F V sin 2oy2R4)0 K (4.1-26)0o

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1i'il JOHN& HOPKINSI UNIVIIAITYAPP LIED PHYSICS LABORATORYSI.VglR@PRNQ MANyRLAN.

Similarly, aR/ayo mrnybe obtained from Eq. (4.1-7):-2 -- 2 22r V (cos 2Y -V cos vY)S T7 . 3 K2 (4.1-27)

Plots of Eqs. (4. 1-26) and (4. 1-27) are shown inFigs. 4-5 and 4-6, respectively. Note that the range of aminimum energy trajectory is sensitive to initial speedbut not to initial flight path angle, With increasing loftangle, the range sensitivity to Vo decreases but rangesensitivity to Yo increases.

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Twi JOwNi Pi~0pKlIlg UNIV5UiII"APPLIED PHYSICS LABORATORY

"3600 --- p

p 3003200 450

600

2800

2400

12000

1600

1200

800

400 I ,[ I I I___ __ ___ ..10 12 14 16 18 20 22 24 26V0 (thousands of ft/tao)

FIg.4-1 RANGE VERSUS 7'0 AND Vo

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R 1O0nnil 1500 2000 2500 3000S650

-- MINIMUM40 * *ENERGY0 -'

10 12 14 16 le 20 22 24 26Vo (thousondi of ft/iecl

Fig. 4.2 INITIAL 7'o VERSUS V, AN D RANGE

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1200

:1I

1800

1400MINIMULM ENERGY

1000 - ~ _30 40 so 70FIg. 4.3 FLIGHT TIME VERSUS Y,, R - 3000 NM I

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SYMBOLS FOR SECTION 4, 2Symbol Definition Typical UnitsCA axial force coefficienth altitude feet or nmLI refers to the impact locationL refers to the launch locationR range from L to I nmir re + h feetr earth radius = 2.09 x 10x feet

e= 3441 nmiS reference area ft 2t total flight time seconds

itncrement in reentry time secondsV velocity in inertial coordinates ft/secW vehicle weight pounds

vehicle ballistic coefficient, W 2lb/f"y flight path angle in inertial coordinates

(positive for launch angles) degrees0 latitude, positive for N. latitude degreesA longitude, positive for E. longitude degreesAA earth rotation angle during time t degrees$ azimuth angle, positive direction is

clockwise from north degreesazimuth angle from L to I (Eq. (4 . 2-19)) degrees

60 half the range angle for nonrotatingearth (Eq. (4. 1-7)) radian

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Symbol Definition Typical Units

Se half the range angle for rotating0 earth (Eq. (4 , 2-17)) radians 10~~-4raseearth rotation rate = 0. 729 x 1 rad/secSubscriptse refers to quantities measured relative

to rotating earthI refers to impact conditionL refers to launch condition

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The equations of motion given in the precedingsection ire acpplic('ble to the rotating earth case providedthe trajectory is dlefined in the inertial axis system. There.-fore, initial conditions m

The motion of ballistic missiles

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