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A BALLISTIC MISSILE PRIMER
Steve Fetter
The Rocket Equation
Consider a single-stage rocket with a lift-off mass Mlo and a burn-out mass Mbo. In the absence of gravity and air resistance, the change in the rocket's velocity from lift-off to burn-out (the "delta-v") is given by
∆ v v M e e M lo bo
= log (1)
where ve is the exhaust velocity of the propellant. If the rocket has n stages, the total
delta-v of the rocket is given by
∆ v ∆ v ∆ v ∆ v v M t n e e M lo i bo^ i
n = + + + = (^) i ^
L log (2)
where vei is the exhaust velocity of the i th stage and ( Mlo/Mbo ) i is the ratio of the rocket mass when the i th stage ignites to its mass when the stage burns out. The mass ratio of the i th stage is given by
M M
M M M m M s M M M m
M s
M m
bo lo (^) i
t t t p p i t t t p
p i t j
i p
i i i
i j
^
^ =^
= −
−
=
1 2 1 2
1 1
1
1
L L
( ) ( ) (3)
where M (^) pi is the propellant mass in the i th stage, si is the fraction of propellant in the
i th stage that remains unburned, M (^) ti is the total mass of the j th stage before it is
ignited, and mp is the mass of the rocket payload. The unburned fraction is very low in modern boosters (e.g., s = 0.0012 in the Minuteman-II first stage) and can therefore usually be ignored. Thus, if we know the total mass, the propellant mass, and the exhaust velocity of each stage, we can estimate the total delta-v (and therefore the range) of the rocket for a given payload mass mp.
Substituting equation 3 into equation 2, we find that the ratio of the payload mass to the total mass of the rocket at liftoff is given by
m M m
M M
v v
p t i
n p
t p
i i e
n
i
i i i =
= − − −
1
1
1 1 exp ∆^ (4)
Unfortunately, this equation cannot be solved for mp as a function of ∆ vt , because the ∆ vi are functions of mp. An approximation to equation 4 can be useful, however, for a quick estimate of the maximum payload capability of a missile. If we assume that the ratio of the initial stage mass to the mass of propellant burned is a constant f for all n stages [ (Mt/Mp)i = f ], that the exhaust velocity of all stages is equal ( vei =
ve ), and that the total delta-v is equally divided among the stages, (∆ vi ≈ ∆ vt/n ), then
m M m
f v n v
p T p
t e
n
'
exp
= − − −
1 1 ∆^ (5)
where MT’ is the total mass of the rocket (without the payload).
The exhaust velocity is usually stated in terms of the specific impulse, or the impulse (force x time) produced per unit weight of propellant consumed. The specific impulse is related to the exhaust velocity by the equation
ve = g Isp (6)
where g is the acceleration due to gravity at sea level (9.81 m/s^2 ). Note that the units of Isp are seconds. The thrust is the exhaust velocity multiplied by the rate at which propellant is consumed:
T v dM dt
g I dM e dt = p = (^) sp p (7)
If we assume that all the propellant is consumed during the burn time of the missile, tbo , then the average thrust is given by
T^ $^ g I $ M sp t
p bo
The specific impulse is a characteristic property of the propellant system, although its exact value varies to some extent with the operating conditions and design of the rocket engine (e.g., the combustion chamber pressure). The theoretical specific impulse for a variety of propellants is given in table 1. The specific impulse at higher altitudes is somewhat greater (due to the lower atmospheric pressure), so
Table 2. Total booster mass Mt , propellant mass Mp , average specific impulse Isp , average thrust T , burn time tbo , and burn-out altitude hbo for several ballistic missiles.
Missile Stage
Mt (kg)
Mp (kg)
Mt Mp
Isp (s)
T
(kN)
tbo (s)
hbo (km) Minuteman IIa^1 23,230 20,780 1.12 267 899 61 34 2 7,270 6,230 1.17 284 266 65 96 3 2,010 1,660 1.21 275 76 59
Minuteman III 3 3,710 3,306 1.12 285 152 61 190
MXb^1 48,700 44,300 1.10 284 2315 53 22 2 27,800 24,900 1.12 304 1365 54 82 3 8,200 6,790 1.21 306 329 62 196
Pershing IIb^1 4,110 3,580 1.15 276 172 58 18 2 2,600 2,250 1.16 279 134 46
Titan IIc^1 116,850 112,720 1.04 288 2170 147 2 26,810 24,150 1.11 313 407 182 315
SS-18d^1 171,000 154,900 1.10 317 5670 85 2 38,500 36,000 1.07 337 992 120
DF-3e^1 65,500 61,400 1.07 241 1020 140 100
V-2f^1 11,800 8,800 1.34 198 252 68 28
Scud-Bg^1 4,900 3,700 1.32 240 120 70 30 aJohn Simpson, Aerojet General, Sacramento, CA, and Larry Hales, Thiokol Chemical Corporation, Brigham City, UT, personal communication, 15 July 1991. Stages 1 and 2 of Minuteman III are identical to those of Minuteman II. b
c"Short burn time ICBM characteristics and considerations," (Denver, CO: Martin Marietta, 20 July 1983). d"Titan II Space Launch Vehicle: Payload Users Guide," (Denver, CO: Martin Marietta Corporation, August 1986). eRolf Engel, "The SS-18 Weapon System,"^ Military Technology , Vol. 13, No. 3 (March 1989), pp. 112-121. Zuwei Huang and Xinmin Ren, "Long March Launch Vehicle Family—Current Status and Future Development," Space Technology f , vol. 8, no. 4 (1988), pp. 371-375.
gGregory P. Kennedy,^ Vengeance Weapon 2: The V-2 Guided Missile^ (Washington: Smithsonian Press, 1983). Steven Zaloga, "Ballistic Missiles in the Third World: SCUD and Beyond," International Defense Review , Vol. 21 (November 1988), pp. 1423-1427.
Table 3. Approximate values of ∆ vag for solid and liquid stages of long-range missiles. ∆ vag (km/s) Stage Solid-fuel Liquid-fuel 1 0.6 1. 2 0.15 0. 3 0.15 --- Source: E.H. Sharkey, "The Rocket Performance Computer," RM-23003-RC (Santa Monica: The RAND Corporation, 1959).
A more precise method of calculating the burn-out velocity is to solve numerically the equations of motion of the missile. If the missile's thrust vector is aligned with its velocity vector (i.e., gravity is used to turn the missile), the equations of motion are:
dv dt
T AC v M
v v
GM x x y
x (^) = − d^ x e
−
12 2 2 2 3 2
ρ / (10a)
dv dt
T AC v M
v v
GM y x y
y (^) = − d y e
−
12 2 2 2 3 2
ρ / (10b)
where vx and vy are the components of the velocity vector v in the x and y directions, T is the thrust, ρ is the atmospheric density (a function of altitude), A is the cross- sectional area of the missile, Cd is the drag coefficient (a function of v ), M is the missile mass (a function of time, as propellant is consumed and empty stages are jettisoned), G is the gravitational constant (6.67× 10 –20^ km^3 s-2kg-1), Me is the mass of the Earth (5.97× 1024 kg), and x and y are measured from the center of the Earth. The atmospheric density as a function of altitude is given in table 4; table 5 gives the drag coefficient of the V-2 missile as a function of velocity.
If the rate of propellant use is constant, all of the propellant is consumed during the burn, and the empty rocket body is jettisoned after burn-out, the mass is equal to
M(t) = Mt – Mp ( t/tbo ) + mp [ t ≤ tbo ] (11)
M(t) = mp [ t > tbo ]
where the tbo = g Mp I $^ sp / T $. This equation is easily generalized to n stages.
Equation 10 is solved by integrating numerically from x = 0 and y = Re until z = Re , adjusting the initial velocity vector so as to achieve the desired (or maximum) range for a given payload mass.
The Range Equation
The range of a missile depends on its velocity, altitude, and angle at burn-out. The burn-out velocity vbo required for a given range, altitude, and angle is given by^1
v G M R R h R R h
bo e^ e e bo e e bo
= −
1 (^2 )
cos sin sin sin
φ θ θ φ θ
where hbo is the burn-out altitude, φ is equal to rb / Re , where rb is the ballistic range of the missile, and θ is the angle of the missile at burn-out with respect to the vertical. The maximum range is attained when θ = (φ + π)/4; this is also known as a “minimum-energy” trajectory. The burn-out altitude varies from 30 km for short- range missiles to 200 km to 400 km for ICBMs (see table 2). Table 6 gives vbo for several values of rb and hbo.
Table 6. The burn-out velocity vbo (km/s) as a function of the maximum ballistic range rb (km) and burn-out altitude hbo (km).
rb Burn-out altitude, hbo (km) (km) 0 30 100 200 300 400 500 2.17 2.11 1.97 --- --- --- 1,000 3.02 2.97 2.85 --- --- --- 2,000 4.11 4.07 3.98 3.86 --- --- 3,000 4.87 4.83 4.75 4.64 4.53 4. 4,000 5.43 5.40 5.32 5.22 5.12 5. 6,000 6.25 6.22 6.15 6.05 5.96 5. 8,000 6.81 6.78 6.71 6.61 6.52 6. 10,000 7.20 7.17 7.10 7.01 6.92 6. 12,000 7.48 7.45 7.39 7.30 7.21 7. 14,000 7.68 7.65 7.59 7.50 7.41 7.
(^1) Paul Zarchan, Tactical and Strategic Missile Guidance (Washington, DC: American Institute of Aeronautics and Astronautics, 1990), p. 232.
Note that rb includes only the ballistic portion of the trajectory. To estimate the total range r , one must add the downrange distance traveled during the rocket burn: r = rb + rbo. As a rough estimate, rbo ≈ hbo tanθ.
For ranges less than 500 km, the curvature of the Earth can be neglected and equation 13 can be approximated by
r v g
gh b v bo bo bo
= + +
2 (^1 1 2 ) sin cos 2 cos
θ θ θ
The maximum range, rmax , occurs when θ = π/4 (45 degrees), in which case
v bo ≈ g r ( max − 2 h bo ) (15)
where rmax includes the distance traveled during boost. The maximum height above the Earth, or apogee, is given by hmax ≈ (rmax/4).
With this background, we can now explore specific missile designs.
China’s DF-3 Missile
The Chinese DF-3 missile is a single-stage liquid-fuel intermediate-range ballistic missile (IRBM). The DF-3 booster is also the first stage of the DF-4 ballistic missile and the CZ-1 space launch vehicle (SLV). Because China is marketing space-launch services, it has made information about their SLVs, including the CZ-1, available to the public.
As noted in table 2, the lift-off thrust of the CZ-1 (and therefore of the DF-3) is 104 tonnes (te). The average specific impulse of the engine is 241 seconds— significantly less than the theoretical maximum of 275 s for nitric acid and UDMH (see table 1). The mass of the first stage is 65.5 te, of which 61.4 te is propellant. With this information we can estimate the payload mass or throwweight of the missile as a function of its range. We do this under two assumptions: first, that the propellant mass is a constant 61.4 te; and second, that the propellant mass is increased or decreased to compensate for changes in the payload mass, so that the total mass of the missile remains constant (in this case, 67.5 te).
Constant Propellant Mass. If the propellant mass is held constant at 61.4 te, then the rocket equation gives
Table 7. The throwweight of the DF-3 missile as a function of the maximum range, for a constant propellant mass Mp = 61.4 te; and for a constant total missile mass MT = 67.5 te. Range Throwweight mp (tonnes) (km) Constant Mp Constant MT 1000 11.3 9. 1500 7.0 6. 2000 4.4 4. 2500 2.9 2. 3000 1.8 1. 3500 1.0 1. 4000 0.4 0. 4500 ---- 0.
It should be emphasized that it is not a simple matter to change the throwweight significantly, since this will change the center of mass and therefore the aerodynamic stability of the missile. Deceases in throwweight (and corresponding increases in range) will also lead to higher accelerations and aerodynamic loads during boost, and to higher velocities and increased aerodynamic heating during reentry.
Note that the above analysis, while analytically simple, does not explicitly include the effects of gravity and air resistance on the booster during launch. These effects were included implicitly through ∆ vag and hbo. It is instructive to check the accuracy of these calculations by solving numerically the equations of motion, since dependable design information is available for the DF-3. The results, which are given in table 8, are in excellent agreement with those obtained with equations 17 and 19.
Table 8. The maximum range of the DF-3, for a constant propellant mass Mp = 61.4 te; and for a constant total missile mass MT = 67.5 te. Throwweight Maximum Range r (km) (te) Constant Mp Constant MT 0.0 4400 4500 0.5 3800 3900 1.0 3400 3400 2.0 2800 2800 5.0 1800 1700 10.0 1000 900
Israel’s Jericho-II/Shavit Missile
Very little is known publicly about the Israeli Jericho II missile. The Shavit space launch vehicle (SLV), which has been used to orbit two Israeli satellites, is widely believed to be based on the Jericho II. From the orbital characteristics and estimated masses of these satellites one can obtain a fairly good idea of the throwweight/range capabilities of the Shavit, and therefore of the Jericho II.
The first satellite, which was launched on 19 September 1988, was placed in an elliptical orbit with a perigee of 250 km, an apogee of 1150 km, and an inclination of 148 degrees; the perigee and apogee of the second satellite were 200 km and 1450 km. (These orbital parameters have been independently verified.) The latitude of the launch site was 32 degrees, and the satellites were launched due west over the Mediterranean Sea (to avoid overflying Arab territory).^4
The velocity needed to put a satellite into orbit is given by^5
v G M c e R (^) e a = − ^
where a , the semi-major axis of the orbit, was 7070 km for the first satellite and 7200 km for the second satellite, which gives vc = 8.29 km/s and 8.35 km/s, respectively.
(^4) Jackson Diehl, "Israel Launches Satellite Into Surveillance Orbit, Washington Post , 4 April 1990, p. A35; Steven E. Gray, Lawrence Livermore National Laboratory, personal communication. 5 Samual Glasstone, Sourcebook on the Space Sciences (Princeton: van Nostrand, 1959).
All that remains is to substitute values for f , ve , and ∆ vag into equation 24, along with the estimates of vc and ∆ vr derived above. For a large, solid-fuel rocket, typical values are f = 1.15, ve = 2.6 km/s, and ∆ vag = 1.0 km/s.^6
The satellite mass given by Israeli was 156 kg for the first satellite and 170 kg for the second satellite; including a guidance and control package would bring the total satellite payload mass to at least 200 kg. Assuming that ms = 200 kg, the throwweight predicted by equation 24 is given in table 9 as a function of the maximum range of the missile, assuming a burn-out altitude and range of 200 and 450 km, respectively. Some analysts have speculated that the Jericho II is simply is the first two stages of the Shavit SLV; table 9 also gives the throwweight of the first two stages of a three-stage Shavit.
Table 9. The throwweight of the Shavit SLVas a function of the maximum range, and the throwweight of the first two stages. Range Throwweight mp (tonnes) (km) First 2 stages All 3 stages 1000 8.3 11. 1500 5.2 6. 2000 3.7 4. 3000 2.2 2. 4000 1.4 2. 5000 1.0 1. 6000 0.8 1. 8000 0.5 0. 10000 0.3 0.
Another—and better—way to estimate the capability of the Israeli missile is to compare it to a missile of similar size whose details are well known, scaling the mass up or down to achieve the orbital capability demonstrated by the Shavit. The total missile mass given by equation 22 is 33 te, which is about the same as the U.S. Minuteman-II missile (32.5 te). Both missiles use solid propellants, and it is
(^6) See table 2. Also see Glasstone, Sourcebook on the Space Sciences , and Sharkey, "The Rocket Performance Computer."
reasonable to assume that Israel could achieve the same level of performance as that of the 1960s-vintage Minuteman II.
Using equations 2 and 3 and the missile parameters given in table 2, one finds that the Minuteman II could deliver a 700-kg payload to a range of 10,000 km, which is in good agreement with the 800-kg throwweight declared by the United States under the START Treaty.^7 Using the same assumptions, the Minuteman-II missile would be capable of launching a 160-kg satellite into the same orbit as the first Israeli satellite. Thus, the Shavit and the Minuteman II are missiles of similar capability. (The Minuteman III, with its more advanced third stage, is capable of launching 300-kg satellite into the same orbit.)
Table 10 gives the throwweight of the Minuteman II as a function of the maximum range, assuming constant propellant mass. The throwweight of the Shavit, if used as a ballistic missile, should be similar. Table 10 also gives the throwweight of the first two stages of the Minuteman II; if the Jericho II is indeed the first two stages of the Shavit, then the throwweight of the first two Minuteman stages should be comparable to that of the Jericho II. Despite the simplicity of the earlier estimates based on equation 24, the correspondence between tables 9 and 10 is remarkably good.
It is interesting to compare the throwweight of the first two stages of the Minuteman II with that of the DF-3. At a range of 1700 km both missiles have roughly equal throwweights (about 5 te). At longer ranges, however, the two-stage missile has the advantage: although the DF-3 is limited to ranges of less than 4000 km, the first two stages of the Minuteman could deliver a 200-kg payload at intercontinental ranges. This clearly demonstrates the importance of multi-stage rocket technology for long-range delivery.
(^7) “Memorandum of Understanding on the Establishment of the Data Base Relating to the Treaty Between the United States of America and the Union of Soviet Socialist Republics on the Reduction and Limitation of Strategic Offensive Arms,” dated 1 September 1990, gives a throwweight of 800 kg for the Minuteman II. According to the Treaty, this may be the greatest throwweight demonstrated in flight tests (excluding the first seven tests, unless the throwweight in one of the these tests exceeds by more than 20 percent the throwweight in subsequent tests), or the throwweight at 11,000 km, whichever is greater. Using the equations presented here, Minuteman II would have a throwweight of 800 kg at a range of 9,000 km.
thrust of 12 te) and use equation 10 to estimate the maximum range for a given payload; these results appear in table 11. The assumptions about the missile’s characteristics appear to be accurate, inasmuch as the throwweight is predicted to be 1.0 te at 300 km.
Iraq claimed that by reducing the throwweight they were able to extend the Scud’s range to 650 km; table 11 shows that the throwweight at this range would be only one-eighth of its throwweight at 300 km if the propellant mass remained constant. If, on the other hand, propellant is added to compensate for the reduced payload mass (i.e., a constant total missile mass of 5.9 te), then the throwweight would be about 300 kg at a range of 650 km. Since Iran claimed that the modified Scud (dubbed the “al Hussein” by Iraq) carried a warhead weighing 160 to 180 kg, constant propellant mass is the best assumption.^11 Iraq fired nearly 200 al Husseins against Iranian cities; many were launched at Teheran, which was over 500 km from the nearest Iraqi launch sites. A throwweight of 200 kg is consistent with available estimates of the damage caused by these attacks.^12
It was widely reported that Iraq had further extended the range of the Scud-B by lengthening the missile to carry additional propellant; apparently two al Abbas missiles were made from three cannibalized Scuds. This can result in a increase in throwweight much greater than 50 percent, since the mass of the engines and tail section stay the same—only the mass of the fuel tanks must increase. In the V-2, for example, the fuel tanks account for only one-quarter of the mass of the empty booster. If we assume the same fraction for the Scud-B, and further assume that the al Abbas modification results in a 50% increase in the fuel-tank and propellant mass only, then f = 1.25 for the modified missile. The throwweight as a function of range under these assumptions is given in table 11. Note that the al Abbas could deliver a 1-te payload to a range of 440 km, and that its throwweight at 900 km would equal that of the Scud-B at only 650 km (assuming a constant propellant mass). These estimates are consistent with reports that the al Abbas had a range of up to 900 km, and that it could deliver the normal 1.0-te Scud warhead to ranges significantly greater than the al Hussein missile could.^13
(^11) W. Seth Carus and Joseph S. Bermudez, Jr, "Iraq's Al-Husayn Missile Programme," Jane's Soviet Intelligence Review , May 1990, pp. 205, states that Iran claims that the al Hussein missile had a payload containing 190 kg of explosives and a maximum range of 600 km. They also state that the fuel tanks were lengthened and 1040 kg of additional propellant added to the missile, but this clearly would not be necessary to achieve such a small throwweight at this range. 12
13 See appendix B [NEED NEW FOOTNOTE HERE]. See Aaron Karp, "Ballistic Missile Proliferation," p. 386; Zagola, "Ballistic Missiles," p. 1425.
Table 11. The range of the Scud-B and al Abbas missiles as a function of throwweight, for constant propellant mass (3.7 te for Scud, 5.55 te for al Abbas ); and constant missile mass (5.9 te for Scud, 7.9 te for al Abbas ). Maximum Range r (km) Throwweight Scud-B / al Hussein al Abbas (tonnes) Constant Mp Constant MT Constant Mp Constant MT 0.0 730 980 1000 1150 0.125 640 840 890 1010 0.25 560 720 800 890 0.5 450 530 640 700 1.0 300 300 440 440
Missile Ranges in Perspective
To put these and other ranges given in this appendix in perspective, table 12 gives the minimum range between several countries that possess ballistic missiles and major cities in the Middle East region. Note that every city listed (and many major cities not listed) is within IRBM range of every emerging missile-capable country. For example, every city is within the range of the DF-3 missile (2,800 km) from Saudi Arabia and Iraq, and every city but Tripoli is within DF-3 range of Iran. The Middle East is a small neighborhood; even missiles with ranges of 1,000 km or less can strike many potential adversaries.
Table 13. The mass (in tonnes) and unit flyaway cost (in million FY 1986 dollars) of several U.S. ballistic missiles.
Missile
Mass (te)
Unit Flyaway Cost (million FY 1986$)
Cost/Mass ($/kg) Minuteman III 35 7.8 220 MX/Peacekeeper 88 22 250 Poseidon C3 29 5.0 170 Trident C4 30 8.1 270 Trident D5 57 28 490 Lance 1.3 0.16 120 Pershing II 7.4 2.5 340 Source: Thomas B. Cochran, William M. Arkin, and Milton M. Hoenig, Nuclear Weapons Databook, Vol. I: U.S. Nuclear Forces and Capabilities (Cambridge, MA: Ballinger, 1984).
For comparison, table 14 gives the mass, payload, combat radius, and unit flyaway cost of several U.S. aircraft. Once again, the flyaway cost per unit takeoff mass varies by only a factor of four from the least expensive aircraft (A-7) to the most expensive (B-2). The average cost for fighter and attack aircraft is roughly $700 per kilogram of takeoff mass.
Referring to table 14, the U.S. A-7 aircraft has a maximum payload of 5.9 te, a combat radius of 880 km, and a unit flyaway cost of $8.5 million dollars. To deliver an equal payload at a range of only 300 km would require at six Scud-B missiles at a cost of roughly $6 million dollars. Given the unreliability and inaccuracy of ballistic missiles, an A-7 would only have to complete an average of one mission (an attrition rate of 50%) to be cost effective compared the Scud, other considerations aside.
As another example, consider the comparison between the U.S. A-6 and the Chinese DF-3 missile. Both delivery systems have (or could have) roughly equal payload/range capabilities; the A-6 costs about $19 million, while the DF-3 costs about $10 million. Once again, attrition rates must be very high to make the DF-3 a cost-effective alternative. Since most air defenses cannot impose such high attrition rates, the popularity of ballistic missiles must be explained by other factors, such as speed, political control, prestige, or psychological impact.
Table 14. The maximum takeoff weight, payload, combat radius, and unit flyaway cost in 1986 dollars of several U.S. aircraft.
Aircraft
Takeoff Mass (te)
Payload (te)
Combat Radius (km)
Flyaway Cost (million 1986$)
Cost/Mass ($/kg) A-4 11.6 4.5 1250 7.7 660 A-6 27.4 8.2 1250 19 690 A-7 19 5.9 880 8.5 450 F-15 31 7.3 1350 22 710 F-16 15 5.4 930 13 870 F-18 20 7.7 850 24 1200 B-1 217 29 4600 228 1000 B-2 168 23 5000 274 1600 Source: Thomas B. Cochran, William M. Arkin, and Milton M. Hoenig, Nuclear Weapons Databook, Vol. I: U.S. Nuclear Forces and Capabilities (Cambridge, MA: Ballinger, 1984), and International Institute for Strategic Studies, The Military Balance 1988-1989 (London: IISS, 1988).
