BALLISTIC TRAJECTORIES
BALLISTIC TRAJECTORIES BALLISTIC TRAJECTORIES
BALLISTIC TRAJECTORIES It’s time we face reality, my friends…we’re not exactly rocket scientists
- Page 2 and 3: Hour 1: LECTURE ISSUES • The Two
- Page 4 and 5: THE TWO BODY PROBLEM (2)
- Page 6 and 7: CONSTANTS OF MOTION (2) SPECIFIC EN
- Page 8 and 9: Trajectory Equation (1)
- Page 10 and 11: Trajectory Equation (3)
- Page 12 and 13: THE FOUR TYPES OF ORBITS
- Page 14 and 15: ORBIT CHARACTERIZATION
- Page 16 and 17: SOLUTIONS OF THE TWO BODY PROBLEM (
- Page 18 and 19: THE ORBITAL VELOCITY (2) GEOSYNCHRO
- Page 20 and 21: THE ESCAPE VELOCITY • A vehicle w
- Page 22 and 23: BALLISTIC MISSILE TRAJECTORY
- Page 24 and 25: APOGEE PARAMETERS
- Page 26 and 27: SOLUTION OF EXAMPLE 1
- Page 28 and 29: References • Thomson: Introductio
- Page 30 and 31: Trajectory Equation (5) The Kepleri
- Page 32 and 33: BURNOUT VELOCITY AS FUNCTION OF RAN
- Page 34 and 35: OPTIMAL BURNOUT ANGLE (MINIMUM ENER
- Page 36 and 37: SENSITIVITY COEFFICIENTS
- Page 38 and 39: RANGE SENSITIVITY TO BURNOUT VELOCI
- Page 40 and 41: SENSITIVITY COEFFICIENTS
- Page 42 and 43: TIME OF FLIGHT
- Page 44 and 45: Typical Trajectories of Theatre Bal
- Page 46 and 47: ACCELERATION PROFILES FOR MEDIUM RA
- Page 48 and 49: THE CEP CONCEPT
- Page 50 and 51: TRANSFORMING BALLISTIC TRAJECTORY I
<strong>BALLISTIC</strong> <strong>TRAJECTORIES</strong><br />
It’s time we face reality, my friends…we’re not exactly rocket scientists
Hour 1:<br />
LECTURE ISSUES<br />
• The Two Body Problem<br />
• EOB parameters for a given range<br />
Hour 2:<br />
• The Hit Equation<br />
• Sensitivity parameters<br />
• Lofted and Depressed trajectories<br />
Hour 3:<br />
• Effect of earth rotation<br />
• Approximate method for estimating ballistic missile parameters
THE TWO BODY PROBLEM (1)
THE TWO BODY PROBLEM (2)
CONSTANTS OF MOTION (1)<br />
SPECIFIC ANGULAR MOMENTUM
CONSTANTS OF MOTION (2)<br />
SPECIFIC ENERGY
CONSTANTS OF MOTION (3)<br />
SPECIFIC ENERGY
Trajectory Equation (1)
Trajectory Equation (2)
Trajectory Equation (3)
Trajectory Equation (4)
THE FOUR TYPES OF ORBITS
SPACE VEHICLE <strong>TRAJECTORIES</strong><br />
Example: if e1 (hyperbola) E must be positive
ORBIT<br />
CHARACTERIZATION
SOLUTIONS OF THE TWO BODY PROBLEM (1)
SOLUTIONS OF THE TWO BODY PROBLEM (2)
THE ORBITAL VELOCITY (1)<br />
= 0
THE ORBITAL VELOCITY (2)<br />
GEOSYNCHRONOUS ORBIT
GEOSYNCHRONOUS MISSION DESIGN
THE ESCAPE VELOCITY<br />
• A vehicle will escape earth if it has a parabolic<br />
(e=1) or hyperbolic (e>1) trajectory<br />
• In the case of a parabolic trajectory, the kinetic<br />
and potential energies are equal<br />
• The escape velocity is larger than orbital<br />
velocity by a factor of 2 1/2<br />
• If at burnout V b Km/sec the vehicle will<br />
escape earth independent of the direction of<br />
motion
Ballistic Missile Trajectory<br />
Satellite Orbit<br />
THE ELLIPTICAL ORBIT<br />
GEOMETRICAL RELATIONS
<strong>BALLISTIC</strong> MISSILE TRAJECTORY
THE LINK BETWEEN GEOMETRICAL AND<br />
PHYSICAL PARAMETERS
APOGEE PARAMETERS
• At the end of a rocket launch<br />
of a space vehicle, the<br />
burnout velocity is 9 km/sec<br />
in a direction due north and 3 0<br />
above the local horizon<br />
• The altitude above sea level is<br />
500 mi<br />
• The burnout point is located<br />
at the 27 th parallel above the<br />
equator<br />
• Calculate and plot the<br />
trajectory of the space vehicle<br />
Anderson/<br />
Introduction to Flight P.312<br />
EXAMPLE 1
SOLUTION OF EXAMPLE 1
SOLUTION OF EXAMPLE 1
References<br />
• Thomson: Introduction to Space Dynamics<br />
• Bate: Fundamentals of Astrodynamics<br />
• Anderson: Introduction to Flight<br />
• Regan: Re-Enrty Vehicles Dynamics<br />
• Madonna: Orbital Mechanics<br />
• Battin: An Introduction to the Mathematics and<br />
Methods of Astrodynamics
<strong>BALLISTIC</strong> MISSILE S<br />
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Trajectory Equation (5)<br />
The Keplerian Trajectory
THE HIT EQUATION<br />
• Hit equation calculates optimal<br />
burnout angle as well as burnout<br />
velocity for minimum Energy<br />
trajectories<br />
• Initial data required:<br />
(1) Burnout altitude<br />
(2) Burnout range<br />
Minimum Energy
BURNOUT VELOCITY AS FUNCTION OF RANGE AND<br />
BURNOUT TRAJECTORY ANGLE
MINIMUM ENERGY <strong>TRAJECTORIES</strong>
OPTIMAL BURNOUT ANGLE<br />
(MINIMUM ENERGY)
ERROR ANALYSIS
SENSITIVITY COEFFICIENTS
COORDINATES FOR ANALYSIS OF PLANAR<br />
<strong>BALLISTIC</strong> MISSILE TRAJECTORY<br />
ALBERT WHEELON: FREE FLIGHT OF A <strong>BALLISTIC</strong> MISSILE<br />
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)<br />
MINIMUM ENERGY <strong>TRAJECTORIES</strong>
Typical Trajectories of Theatre Ballistic Missiles (TBM)<br />
Range<br />
(km)<br />
120<br />
500<br />
1,000<br />
2,000<br />
3,000<br />
Burn-out<br />
velocity (km/s)<br />
1.0<br />
2.0<br />
2.9<br />
3.9<br />
4.7<br />
Boost Phase<br />
(s)<br />
16<br />
36<br />
55<br />
85<br />
122<br />
Flight Time<br />
(min)<br />
2.7<br />
6.1<br />
8.4<br />
11.8<br />
14.8
<strong>BALLISTIC</strong> MISSILE DISPERSION<br />
RANGE
ACCELERATION PROFILES FOR MEDIUM RANGE <strong>BALLISTIC</strong> MISSILES<br />
(1) dv = Gdt<br />
G – End of boost acceleration<br />
dt – Time delay due to thrust termination
<strong>BALLISTIC</strong> MISSILE DISPERSION<br />
SIDE DEVIATION
THE CEP CONCEPT
# %<br />
# "<br />
$<br />
#<br />
# % #<br />
Example: Sigma_R=2600 m Sigma_Y=3400 m CEP=3550 m
TRANSFORMING <strong>BALLISTIC</strong> TRAJECTORY INTO A<br />
SATELLITE TRAJECTORY
TRAJECTORY TYPES Lofted<br />
Minimum Energy<br />
Radar Beam<br />
Depressed<br />
Local<br />
Horizon
LOFTED AND DEPRESSED <strong>TRAJECTORIES</strong><br />
• For each range, there are<br />
infinite combinations of<br />
burnout velocity and<br />
burnout trajectory angle<br />
• If:<br />
(1) Burnout velocity is<br />
known (i.e. rocket motor<br />
is given)<br />
(2) A range (less than minimum<br />
energy range) has been<br />
specified<br />
There are two trajectories<br />
leading to this range:<br />
(a) Lofted trajectory<br />
(b) Depressed trajectory
LOFTED AND<br />
DEPRESSED TRAJ.<br />
• RANGE OF INTERCONTINENTAL<br />
<strong>BALLISTIC</strong> MISSILES IS ABOUT:<br />
10000 KM<br />
• RANGE OF SUBMARINE <strong>BALLISTIC</strong><br />
MISSILES IS ABOUT:<br />
5000 KM
TYPICAL <strong>TRAJECTORIES</strong> AND VELOCITY VS ALTITUDE<br />
ALTITUDE (KM)<br />
ALTITUDE (KM)<br />
2000 KM<br />
10000 KM<br />
RANGE (KM)<br />
VELOCITY (KM/SEC)
DEPRESSED <strong>TRAJECTORIES</strong><br />
• Late detection and relatively low interception<br />
altitudes (attacker’s advantage)<br />
• RV appears without its decoys (defense advantage)<br />
• Larger dispersion of impact points<br />
(as compared with Minimum Energy trajectories)<br />
• RV needs special design<br />
(aerodynamic heating problems)<br />
• Prediction of PIP less accurate<br />
(as compared with Minimum Energy trajectories)
LOFTED <strong>TRAJECTORIES</strong>
ICBM 9000 KM <strong>TRAJECTORIES</strong> FOR:<br />
15 deg,24 deg,35 deg and 40 deg
AN APPROXIMATE METHOD FOR ESTIMATING THE<br />
BASIC PARAMETERS OF A <strong>BALLISTIC</strong> MISSILE (1)
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<strong>BALLISTIC</strong> MISSILE LENTH/DIAMETER RATIO<br />
• Full configuration L/D = 12-15 (Upper limit set by aeroelasticity constraints<br />
• Ballistic Cone L/D = 2.5-3.0<br />
• Stabilizing Unit (including engine) L/D = 2.0<br />
• Equipment Section L/D = 1.5-2.0<br />
• Motor (or: Fuel/Oxidizer tanks) L/D = 5.0-9.0<br />
Compressed Air Bottles
BURN-OUT<br />
TIME VS RANGE
END OF BOOST<br />
ALTITUDE VS RANGE
AN APPROXIMATE METHOD FOR ESTIMATING THE<br />
BASIC PARAMETERS OF A <strong>BALLISTIC</strong> MISSILE (2)<br />
)
AN APPROXIMATE METHOD FOR ESTIMATING THE<br />
BASIC PARAMETERS OF A <strong>BALLISTIC</strong> MISSILE (3)
AN APPROXIMATE METHOD FOR ESTIMATING THE<br />
BASIC PARAMETERS OF A <strong>BALLISTIC</strong> MISSILE (4)
AN APPROXIMATE METHOD FOR ESTIMATING THE<br />
BASIC PARAMETERS OF A <strong>BALLISTIC</strong> MISSILE (5)
EXAMPLE: 1360 KM TBM<br />
• Design a TBM to deliver 850 Kg (W/H) to a range of 1300 Km<br />
• Preliminary estimates:<br />
Burnout altitude: 72 Km<br />
Burnout range: 64 Km<br />
Burnout time: 107 Sec<br />
• Results:<br />
Burnout velocity: 3207 m/s<br />
Flight time (total): 646 sec<br />
Maximum altitude: 333 Km<br />
Velocity at apogee: 2342 m/s<br />
Central angle: 11.1 Deg
EXAMPLE: 1360 KM TBM
LAUNCH MASS(KG)<br />
60000<br />
50000<br />
40000<br />
30000<br />
20000<br />
10000<br />
LAUNCH MASS OF A SINGLE STAGE<br />
SINGLE STAGE TBM RANGE=1360 KM<br />
ISP=240 SEC<br />
ISP=250 SEC<br />
ISP=260 SEC<br />
0<br />
0.90 0.92 0.94 0.96 0.98 1.00<br />
MASS RATIO = MP/(MP+MS)
LAUNCH MASS(KG)<br />
45000<br />
40000<br />
35000<br />
30000<br />
25000<br />
20000<br />
15000<br />
10000<br />
LAUNCH MASS OF A SINGLE STAGE<br />
SINGLE STAGE TBM RANGE=2000 KM<br />
ISP=240 SEC<br />
ISP=250 SEC<br />
ISP=260 SEC<br />
0.95 0.96 0.97 0.98 0.99 1.00<br />
MASS RATIO = MP/(MP+MS)
DOUBLE STAGE TBM
DOUBLE STAGE TBM<br />
R=2000 KM
TWO IDENTICAL STAGES TBM<br />
RANGE: 2000 KM
THREE STAGES TBM<br />
RANGE: 2000 KM
EFFECTS OF EARTH ROTATION<br />
Example :<br />
• A ballistic missile with a range<br />
of 4750 Km and flight time of<br />
1386 seconds has been launched<br />
towards a target traveling on<br />
latitude 60 0<br />
• While flying in space, the<br />
target has changed its position<br />
(relative to an inertial frame of<br />
reference) by 321 Km
CORIOLIS ACCELERATION
HITING A MOVING TARGET<br />
(EARTH ROTATION EFFECTS)<br />
DATA: launch site and target Longitude and Latitude at the minute of launch<br />
PURPOSE: finding Elevation and Azimuth required to hit the target
HITTING A MOVING TARGET<br />
FLAT TRAJECTORY (1)
HITTING A MOVING TARGET<br />
FLAT TRAJECTORY (2)<br />
EQATIONS TO SOLVE: (3),(6)
HITTING A MOVING TARGET (3)<br />
SPACE TRAJECTORY
HITTING A MOVING TARGET (4)<br />
SPACE TRAJECTORY
HITTING A MOVING TARGET (5)<br />
SPACE TRAJECTORY
HITTING A MOVING TARGET (6)<br />
SPACE TRAJECTORY
EARTH ROTATION EFFECT ON IMPACT POINT
EARTH ROTATION EFFECT: VELOCITY RELATIVE TO INERTIAL SYSTEM
SOLVING THE ROTATING HIT PROBLEM (1)
SOLVING THE ROTATING HIT PROBLEM (2)
SOLVING THE ROTATING HIT PROBLEM (3)
SOLVING THE ROTATING HIT PROBLEM (4)
SOLVING THE ROTATING HIT PROBLEM (5)
IMPORTANT! IMPOTRANT!
HITTING A TARGET WITH <strong>BALLISTIC</strong> MISSILE
• A missile is fired<br />
from:<br />
North Korea (40 0 N,125 0 E)<br />
To:<br />
Anchorage, Alaska(62 0 N,150 0 E)<br />
• Missile Data:<br />
R=10000 KM<br />
V B =5900 m/sec<br />
h b =280 KM<br />
S b =300 KM<br />
t b =290 sec<br />
• Find the required azimuth and<br />
elevation angle (local values)<br />
to hit the target
KOREAN MISSILE: LAUNCH DATA
REFERENCES<br />
• Free flight of a ballistic missile<br />
Albert D. Wheelon<br />
ARS Journal, December 1959, pp.915-926<br />
• Re-Entry Vehicle Dynamics<br />
Frank J. Regan<br />
AIAA Education Series<br />
Careful! several printing mistakes in the text<br />
• Fundamentals of Astrodynamics<br />
Bate, Muller, White<br />
Dover
END OF SLIDES
• The Two Body Problem 3-24<br />
• Example (Anderson) 25-27<br />
• References 28<br />
• Ballistic Missiles Trajectories 29-41<br />
• Time Of Flight 42-44<br />
• Dispersions 45-49<br />
• Satellite Injection 50<br />
• Lofted and Depressed Traj. 51-57<br />
• BM Preliminary Design 58-72<br />
• Earth rotation Effects 73-90<br />
• Example 91-93<br />
• References 94
MIDDLE EAST RANGES
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