BALLISTIC TRAJECTORIES

BALLISTIC TRAJECTORIES 

It’s time we face reality, my friends…we’re not exactly rocket scientists

Hour 1: 

LECTURE ISSUES 

• The Two Body Problem 

• EOB parameters for a given range 

Hour 2: 

• The Hit Equation 

• Sensitivity parameters 

• Lofted and Depressed trajectories 

Hour 3: 

• Effect of earth rotation 

• Approximate method for estimating ballistic missile parameters

THE TWO BODY PROBLEM (1)

THE TWO BODY PROBLEM (2)

CONSTANTS OF MOTION (1) 

SPECIFIC ANGULAR MOMENTUM


SPECIFIC ENERGY


SPECIFIC ENERGY

Trajectory Equation (1)




THE FOUR TYPES OF ORBITS

SPACE VEHICLE TRAJECTORIES 

Example: if e1 (hyperbola) E must be positive

ORBIT 

CHARACTERIZATION

SOLUTIONS OF THE TWO BODY PROBLEM (1)

SOLUTIONS OF THE TWO BODY PROBLEM (2)

THE ORBITAL VELOCITY (1) 

= 0

THE ORBITAL VELOCITY (2) 

GEOSYNCHRONOUS ORBIT

GEOSYNCHRONOUS MISSION DESIGN

THE ESCAPE VELOCITY 

• A vehicle will escape earth if it has a parabolic 

(e=1) or hyperbolic (e>1) trajectory 

• In the case of a parabolic trajectory, the kinetic 

and potential energies are equal 

• The escape velocity is larger than orbital 

velocity by a factor of 2 1/2 

• If at burnout V b Km/sec the vehicle will 

escape earth independent of the direction of 

motion

Ballistic Missile Trajectory 

Satellite Orbit 

THE ELLIPTICAL ORBIT 

GEOMETRICAL RELATIONS

BALLISTIC MISSILE TRAJECTORY

THE LINK BETWEEN GEOMETRICAL AND 

PHYSICAL PARAMETERS

APOGEE PARAMETERS

• At the end of a rocket launch 

of a space vehicle, the 

burnout velocity is 9 km/sec 

in a direction due north and 3 0 

above the local horizon 

• The altitude above sea level is 

500 mi 

• The burnout point is located 

at the 27 th parallel above the 

equator 

• Calculate and plot the 

trajectory of the space vehicle 

Anderson/ 

Introduction to Flight P.312 

EXAMPLE 1

SOLUTION OF EXAMPLE 1

SOLUTION OF EXAMPLE 1

References 

• Thomson: Introduction to Space Dynamics 

• Bate: Fundamentals of Astrodynamics 

• Anderson: Introduction to Flight 

• Regan: Re-Enrty Vehicles Dynamics 

• Madonna: Orbital Mechanics 

• Battin: An Introduction to the Mathematics and 

Methods of Astrodynamics

BALLISTIC MISSILE S 

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Trajectory Equation (5) 

The Keplerian Trajectory

THE HIT EQUATION 

• Hit equation calculates optimal 

burnout angle as well as burnout 

velocity for minimum Energy 

trajectories 

• Initial data required: 

(1) Burnout altitude 

(2) Burnout range 

Minimum Energy

BURNOUT VELOCITY AS FUNCTION OF RANGE AND 

BURNOUT TRAJECTORY ANGLE

MINIMUM ENERGY TRAJECTORIES

OPTIMAL BURNOUT ANGLE 

(MINIMUM ENERGY)

ERROR ANALYSIS

SENSITIVITY COEFFICIENTS

COORDINATES FOR ANALYSIS OF PLANAR 

BALLISTIC MISSILE TRAJECTORY 

ALBERT WHEELON: FREE FLIGHT OF A BALLISTIC MISSILE 

ARS JOURNAL DECEMBER 1959 PP.915-926

RANGE SENSITIVITY TO BURNOUT VELOCITY

RANGE SENSITIVITY TO BURNOUT TRAJECTORY ANGLE

SENSITIVITY COEFFICIENTS

RANGE SENSITIVITY TO BURNOUT VELOCITY

TIME OF FLIGHT

TIME OF FLIGHT (AND OTHER PARAMETERS) 

MINIMUM ENERGY TRAJECTORIES

Typical Trajectories of Theatre Ballistic Missiles (TBM) 

Range 

(km) 

120 

500 

1,000 

2,000 

3,000 

Burn-out 

velocity (km/s) 

1.0 

2.0 

2.9 

3.9 

4.7 

Boost Phase 

(s) 

16 

36 

55 

85 

122 

Flight Time 

(min) 

2.7 

6.1 

8.4 

11.8 

14.8

BALLISTIC MISSILE DISPERSION 

RANGE

ACCELERATION PROFILES FOR MEDIUM RANGE BALLISTIC MISSILES 

(1) dv = Gdt 

G – End of boost acceleration 

dt – Time delay due to thrust termination

BALLISTIC MISSILE DISPERSION 

SIDE DEVIATION

THE CEP CONCEPT

# % 

# " 

$ 

# 

# % # 

Example: Sigma_R=2600 m Sigma_Y=3400 m CEP=3550 m

TRANSFORMING BALLISTIC TRAJECTORY INTO A 

SATELLITE TRAJECTORY

TRAJECTORY TYPES Lofted 

Minimum Energy 

Radar Beam 

Depressed 

Local 

Horizon

LOFTED AND DEPRESSED TRAJECTORIES 

• For each range, there are 

infinite combinations of 

burnout velocity and 

burnout trajectory angle 

• If: 

(1) Burnout velocity is 

known (i.e. rocket motor 

is given) 

(2) A range (less than minimum 

energy range) has been 

specified 

There are two trajectories 

leading to this range: 

(a) Lofted trajectory 

(b) Depressed trajectory

LOFTED AND 

DEPRESSED TRAJ. 

• RANGE OF INTERCONTINENTAL 

BALLISTIC MISSILES IS ABOUT: 

10000 KM 

• RANGE OF SUBMARINE BALLISTIC 

MISSILES IS ABOUT: 

5000 KM

TYPICAL TRAJECTORIES AND VELOCITY VS ALTITUDE 

ALTITUDE (KM) 

ALTITUDE (KM) 

2000 KM 

10000 KM 

RANGE (KM) 

VELOCITY (KM/SEC)

DEPRESSED TRAJECTORIES 

• Late detection and relatively low interception 

altitudes (attacker’s advantage) 

• RV appears without its decoys (defense advantage) 

• Larger dispersion of impact points 

(as compared with Minimum Energy trajectories) 

• RV needs special design 

(aerodynamic heating problems) 

• Prediction of PIP less accurate 

(as compared with Minimum Energy trajectories)

LOFTED TRAJECTORIES

ICBM 9000 KM TRAJECTORIES FOR: 

15 deg,24 deg,35 deg and 40 deg

AN APPROXIMATE METHOD FOR ESTIMATING THE 

BASIC PARAMETERS OF A BALLISTIC MISSILE (1)

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BALLISTIC MISSILE LENTH/DIAMETER RATIO 

• Full configuration L/D = 12-15 (Upper limit set by aeroelasticity constraints 

• Ballistic Cone L/D = 2.5-3.0 

• Stabilizing Unit (including engine) L/D = 2.0 

• Equipment Section L/D = 1.5-2.0 

• Motor (or: Fuel/Oxidizer tanks) L/D = 5.0-9.0 

Compressed Air Bottles

BURN-OUT 

TIME VS RANGE

END OF BOOST 

ALTITUDE VS RANGE


BASIC PARAMETERS OF A BALLISTIC MISSILE (2) 

)







EXAMPLE: 1360 KM TBM 

• Design a TBM to deliver 850 Kg (W/H) to a range of 1300 Km 

• Preliminary estimates: 

Burnout altitude: 72 Km 

Burnout range: 64 Km 

Burnout time: 107 Sec 

• Results: 

Burnout velocity: 3207 m/s 

Flight time (total): 646 sec 

Maximum altitude: 333 Km 

Velocity at apogee: 2342 m/s 

Central angle: 11.1 Deg

EXAMPLE: 1360 KM TBM

LAUNCH MASS(KG) 

60000 

50000 

40000 

30000 

20000 

10000 

LAUNCH MASS OF A SINGLE STAGE 

SINGLE STAGE TBM RANGE=1360 KM 

ISP=240 SEC 

ISP=250 SEC 

ISP=260 SEC 

0 

0.90 0.92 0.94 0.96 0.98 1.00 

MASS RATIO = MP/(MP+MS)

LAUNCH MASS(KG) 

45000 

40000 

35000 

30000 

25000 

20000 

15000 

10000 

LAUNCH MASS OF A SINGLE STAGE 

SINGLE STAGE TBM RANGE=2000 KM 

ISP=240 SEC 

ISP=250 SEC 

ISP=260 SEC 

0.95 0.96 0.97 0.98 0.99 1.00 

MASS RATIO = MP/(MP+MS)

DOUBLE STAGE TBM

DOUBLE STAGE TBM 

R=2000 KM

TWO IDENTICAL STAGES TBM 

RANGE: 2000 KM

THREE STAGES TBM 

RANGE: 2000 KM

EFFECTS OF EARTH ROTATION 

Example : 

• A ballistic missile with a range 

of 4750 Km and flight time of 

1386 seconds has been launched 

towards a target traveling on 

latitude 60 0 

• While flying in space, the 

target has changed its position 

(relative to an inertial frame of 

reference) by 321 Km

CORIOLIS ACCELERATION

HITING A MOVING TARGET 

(EARTH ROTATION EFFECTS) 

DATA: launch site and target Longitude and Latitude at the minute of launch 

PURPOSE: finding Elevation and Azimuth required to hit the target

HITTING A MOVING TARGET 

FLAT TRAJECTORY (1)

HITTING A MOVING TARGET 

FLAT TRAJECTORY (2) 

EQATIONS TO SOLVE: (3),(6)

HITTING A MOVING TARGET (3) 

SPACE TRAJECTORY


SPACE TRAJECTORY


SPACE TRAJECTORY


SPACE TRAJECTORY

EARTH ROTATION EFFECT ON IMPACT POINT

EARTH ROTATION EFFECT: VELOCITY RELATIVE TO INERTIAL SYSTEM

SOLVING THE ROTATING HIT PROBLEM (1)





IMPORTANT! IMPOTRANT!

HITTING A TARGET WITH BALLISTIC MISSILE

• A missile is fired 

from: 

North Korea (40 0 N,125 0 E) 

To: 

Anchorage, Alaska(62 0 N,150 0 E) 

• Missile Data: 

R=10000 KM 

V B =5900 m/sec 

h b =280 KM 

S b =300 KM 

t b =290 sec 

• Find the required azimuth and 

elevation angle (local values) 

to hit the target

KOREAN MISSILE: LAUNCH DATA

REFERENCES 

• Free flight of a ballistic missile 

Albert D. Wheelon 

ARS Journal, December 1959, pp.915-926 

• Re-Entry Vehicle Dynamics 

Frank J. Regan 

AIAA Education Series 

Careful! several printing mistakes in the text 

• Fundamentals of Astrodynamics 

Bate, Muller, White 

Dover

END OF SLIDES

• The Two Body Problem 3-24 

• Example (Anderson) 25-27 

• References 28 

• Ballistic Missiles Trajectories 29-41 

• Time Of Flight 42-44 

• Dispersions 45-49 

• Satellite Injection 50 

• Lofted and Depressed Traj. 51-57 

• BM Preliminary Design 58-72 

• Earth rotation Effects 73-90 

• Example 91-93 

• References 94

MIDDLE EAST RANGES

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920 

150 

1000 

26 

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BALLISTIC TRAJECTORIES

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