Computation of the Different Errors in the Ballistic Missiles Range

International Scholarly Research Network ISRN Applied Mathematics Volume 2011, Article ID 349737, 16 pages doi:10.5402/2011/349737

Research Article Computation of the Different Errors in the Ballistic Missiles Range F. A. Abd El-Salam1, 2 and S. E. Abd El-Bar3, 4 1

Department of Math, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia Department of Astronomy, Faculty of Science, Cairo University, Cairo 12613, Egypt 3 Department of Applied Mathematics, Faculty of Applied Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Tanta university, Tanta, Egypt 2

Correspondence should be addressed to S. E. Abd El-Bar, so [email protected] Received 8 June 2011; Accepted 25 July 2011 Academic Editor: Z. Huang Copyright q 2011 F. A. Abd El-Salam and S. E. Abd El-Bar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The ranges of the ballistic missile trajectories are very sensitive to any kind of errors. Most of the missile trajectory is a part of an elliptical orbit. In this work, the missile problem is stated. The variations in the orbital elements are derived using Lagrange planetary equations. Explicit expressions for the errors in the missile range due to the in-orbit plane changes are derived. Explicit expressions for the errors in the missile range due to the out-of-orbit plane changes are derived when the burnout point is assumed on the equator.

1. Introduction The fundamental problem of astrodynamics is the orbit determination and orbit correction. For a spacecraft moving under the influence of gravitational field of Earth in free space no air drag, the trajectory is an ellipse with the center of Earth lying at one of the foci of the ellipse. This constitutes a standard two-body-central-force problem, which has been treated, in detail, in many standard textbooks 1, 2. The trick is to first reduce the problem to two dimensions by showing that the trajectory always lies in a plane perpendicular to the angular momentum vector. Then, the problem is set up in plane-polar coordinates. Angular momentum is conserved, and the problem, effectively, reduces to one-dimensional problem 3. A ballistic missile is a missile that follows a suborbital ballistic flight path with the ballistic missile objective of delivering one or more warheads often nuclear to a predetermined target. The missile is only guided during the relatively brief initial powered phase of

2

ISRN Applied Mathematics vbo φbo φbo Reentry point

Burnout point

Ψ Ω

F′

rm

′ rbo

No

Ho

Γ rbo

al

rizo

φbo

t gen Tan

rbo

nta l

Ψ/2

F

a

b

Figure 1: a Geometry of a typical ICBM trajectory. b Some orbital parameters at the burnout point. where Γ, Ψ, Ω, and Λ are the powered flight, free flight, reentry, and total range angles, respectively, and Rb , Rff , Rre , and Rt are their ground ranges.

flight, and its course is subsequently governed by the laws of orbital mechanics and ballistics. To date, ballistic missiles have been propelled during powered flight by chemical rocket engines of various types. Therefore, the ballistic missile trajectory consists of three parts; see Figure 1a. 1 The powered flight portion, sometimes called boost phase, takes usually from 3 to 5 minutes shorter for a solid rocket than for a liquid-propellant rocket; the altitude of the missile at the end of this phase is typically 150 to 400 km depending on the trajectory chosen, and the typical burnout speed is 7 km/s. 2 The free-flight portion, or the midcourse phase which constitutes most of the flight time, takes approximately 25 minutes. It is a part of an elliptic orbit with a vertical major axis; the apogee is at an altitude of approximately 1,200 km; the semimajor axis is between 3,186 km and 6,372 km; the projection of the orbit on the Earth’s surface is close to a great circle, slightly displaced due to Earth rotation during the time of flight; the missile may release several independent warheads, and penetration aids such as metallic-coated balloons, aluminum chaff, and full-scale warhead decoys. 3 The reentry phase during which energy is dissipated as a result of friction with the atmosphere, starts at an ill-defined point at an altitude of 100 km. It takes about 2 minutes to impact at a speed of up to 4 km/s for early ICBMs less than 1 km/s. Ballistic missiles can be launched from fixed sites or mobile launchers, including vehicles transporter erector launchers, TELs, aircraft, ships, and submarines. The powered flight portion can last from a few tens of seconds to several minutes and can consist of multiple rocket stages. When in space and no more thrust is provided, the missile enters free flight. In order to cover large distances, ballistic missiles are usually launched into a high suborbital spaceflight; for intercontinental missiles, the highest altitude apogee reached during free

ISRN Applied Mathematics

3

flight is about 1200 km. The reentry stage begins at an altitude where atmospheric drag plays a significant role in missile trajectory and lasts until missile impact.

2. Types of Ballistic Missiles Ballistic missiles are categorized according to their range, the maximum distance measured along the surface of the Earth’s ellipsoid from the point of launch of a ballistic missile to the point of impact of the last element of its payload. Various schemes are used by different countries to categorize the ranges of ballistic missiles as follows: tactical ballistic missile: range between about 150 km and 300 km, battlefield range ballistic missile BRBM, range less than 200 km, theatre ballistic missile TBM: range between 300 km and 3500 km, short-range ballistic missile SRBM: range 1000 km or less, medium-range ballistic missile MRBM: range between 1000 km and 3500 km, intermediate-range ballistic missile IRBM or long-range ballistic missile LRBM: range between 3500 km and 5500 km, intercontinental ballistic missile ICBM: range greater than 5500 km, and submarine-launched ballistic missile SLBM: launched from ballistic missile submarines SSBNs, and all current designs have intercontinental range. Short and medium-range missiles are often collectively referred to as theater or tactical ballistic missiles TBMs. Long- and medium-range ballistic missiles are generally designed to deliver nuclear weapons, because their payload is too limited for conventional explosives to be efficient though the U.S. may be evaluating the idea of a conventionally armed ICBM for near-instant global air strike capability despite the high costs. The flight phases are like those for ICBMs except with no exoatmospheric phase for missiles with ranges less than about 350 km. Sometimes, the designers of the ballistic missiles need to perform maneuvers in flight or make unexpected changes in direction and range; this type is known as a quasi-ballistic missile or a semi ballistic missile. At a lower trajectory than a ballistic missile, a quasi-ballistic missile can maintain higher speed, thus allowing its target less time to react to the attack, at the cost of reduced range.

3. Literature Survey McFarland 4 treated ballistic missile problem by modeling spherical Earth, Earth rotation, and addition of atmospheric drag using the state transition matrix. Forden 5 described the integration of the three degrees of freedom equations of motion, and approximations are made to the aerodynamic for simulating ballistic missiles. Bao and Murray 6 improved the ground range calculation of a ballistic missile trajectory on a nonrotating oblate Earth. Isaacson and Vaughan 7 described a method of estimating and predicting ballistic missile trajectories using a Kalman filter over a spherical, nonrotating Earth. They determined uncertainties in the missile launch point and missile position during flight. Harlin and Cicci 8 developed a method for the determination of the trajectory of a ballistic missile over a rotating, spherical Earth given only the launch position and impact point. The iterative solution presented uses a state transition matrix to correct the initial conditions of the ballistic missile state vector based upon deviations from a desired set of final conditions. Akgul ¨ and Karasoy 9 developed a trajectory prediction program to predict the full trajectory of a tactical ballistic missile. Vinh et al. 10 obtained a minimum-fuel interception of a satellite, or a ballistic missile, in elliptic trajectory in a Newtonian central force field, via Lawden’s

4

ISRN Applied Mathematics

theory of primer vector. Kamal 11 developed an algorithm includes detection of cross-range error using Lambert scheme in free space in the absence of atmospheric drags. Bhowmik and Sadhukhan 12 investigated the advantages and performance of extended Kalman filter for the estimation of nonlinear system, where linearization takes place about a trajectory that was continually updated with the state estimates resulting from the measurement. They took tactile ballistic missile reentry problem as a nonlinear system model and extended Kalman filter technique is used to estimate the positions and velocities at the X and Y direction at different values of ballistic coefficients. Kamal 13 presented an innovative adaptive scheme which was called “the multistage Lambert scheme”. Liu and Chen 14 presented a novel tracking algorithm by integrating input estimation and modified probabilistic data association filter to identify warhead among objects separation from the reentry vehicle in a clear environment.

4. Statement of the Problem Our ICBM problem concerns with the determination of the free-flight range angle taking into account the perturbation in the orbital elements. Let us define the dimensionless parameter as Q v/vc 2 rv2 /μ, where r is the magnitude of the position vector of the missile relative to the Earth, v is the missile speed at any point in its orbit, vc is the corresponding missile circular speed at this point, and μ Gm1 m2  μ Gm1 , m2  m1 , where G is the gravitational constant, m1 is the mass of the Earth, and m2 is the missile mass. From the orbital mechanics of two body and the symmetry of the free-flight portion shown in Figures 1a and 1b, we have 15  cos

Ψ 2



1 − Qbo cos2 φbo , − cos fbo  1 Qbo Qbo − 2 cos2 φbo

    2 − Qbo Ψ Ψ , sin sin 2φbo 2 Qbo 2

4.1 4.2

where φbo is the flight path angle, fbo the true anomaly, and Qbo the dimensionless parameter at the burnout point. Equation 4.1 gives the free-flight range by which the free-flight range angle can be computed for any given combination of burnout conditions rbo , vbo , and φbo . The problem can now be specified as “given a particular launch point and target, it is required to calculate Λ, and knowing Γ, Ω, Ψ can be calculated”. Equation 4.2 gives the flight-path angle, which provides two trajectories to the missile. The trajectory corresponding to the larger value of φbo is called the high trajectory and to the smaller value is the low trajectory. The nature of the trajectory, high or low, depends primarily on the value of Qbo . This is obvious from 4.2, where if 1 Qbo < 1, Ψ is always less than 180◦ ; otherwise, the right side of 4.2 exceeds 1, and both high and low trajectories are possible, 2 Qbo 1, one trajectory is circular, and for Ψ < 180◦ , both high and low trajectories are possible φbo 0 for low, while high trajectory only is possible for Ψ > 180◦ , and low trajectory skims Earth φbo 0 for high,

ISRN Applied Mathematics

5

3 Qbo > 1, 4.2 yields one positive and one negative value for φbo regardless of range. The low trajectory, corresponding to the negative value, is not practical, since it would penetrate the Earth. When the right hand side of 4.2 equals unity, we obtain a single trajectory called the maximum range trajectory, sinΨ/2 Qbo /2 − Qbo , and the flight path angle is φbo 1/4180◦ − Ψ.

5. In-Orbit-Plane Changes The variations in the parameters {φbo , rbo , vbo } due to changes in the orbital elements {a, e} take place in the plane of the orbit. Therefore, in what follows, we will compute the errors in Ψ due to changes in the mentioned orbital elements. To do this, we need first the following partial derivatives:     ∂Ψ 2 sin Ψ 2φbo csc2φbo − 1 , ∂φbo 4μ Ψ ∂Ψ 2 2 sin2 csc2φbo , ∂rbo rbo 2 vbo

5.1

8μ ∂Ψ Ψ sin2 csc2φbo . 3 ∂vbo rbo vbo 2

5.1. Error in Ψ due to the Change in the Semimajor Axis We can write the change in the free flight range angle due to the change in the semimajor axis Δa Ψ as follows: Δa Ψ

∂Ψ ∂vbo ∂Ψ ∂rbo Δa Δa. ∂rbo ∂a ∂vbo ∂a

5.2

The required derivatives are given by ∞

 ∂rbo −1n en − en 2 cosn f, ∂a n 0

−1/2 1/2

1 μ ∂vbo 2 2 1

e 1 − e −

2e cos f . ∂a 2 a3

5.3

The involved products are ∞

 4μ ∂Ψ ∂rbo Ψ 2 2 sin2 csc2φbo −1n en − en 2 cosn f, ∂rbo ∂a 2 rbo vbo n 0

−1/2 1/2

1 μ3 8 ∂Ψ ∂vbo 2Ψ 2 2 − csc2φ 1

e 1 − e sin

2e cos f . bo 3 ∂vbo ∂a 2 a3 rbo vbo 2

5.4

6

ISRN Applied Mathematics

Using the Lagrange planetary equations, we computed Δa taking into account the oblate model of the Earth retaining the zonal harmonics up to J4 . The integration is performed between fbo  and 2π − fbo . Substitution of the obtained expression into 5.2 yields  Δa Ψ

4μ

 sin2

 ∞

 Ψ csc2φbo −1n en − en 2 cosn fbo 2 n 0

2 2 rbo vbo ⎫

  ⎬ 3



 1/2 −1/2 μ 8 Ψ 1 − e2 − sin2 csc2φbo 1 e2 2e cos fbo 3 3 ⎭ 2 a rbo vbo

× J2

2 5

    αij sin if bo sin jω − J3

i 1 j −2

J4

4 9

3 7

   βkl sin kf bo coslω

k 1 l −3

   γmn sin mf bo sinnω,

m 1 n −4

where nonvanishing coefficients are given by R2 R3 R4   α β γmn , , β , γ ij kl mn kl 2a1 − e2  4a2 1 − e2  32a3 1 − e2   3

3 α1,0 − −4e 6es2 , α1,2 − es2 , α2,2 −6s2 , 2 2  15 2 1

5 es , 36es 45es3 , β2,3 es3 , β0,1 α3,2 2 2 2



  1 1 −60es 75es3 , −24s 30s3 , β1,1 β3,3 −15s3 , β2,1 2 2  35 5

315 4 es , γ5,4 − γ1,0 − 48e − 240es2 210es4 , β4,3 − es3 , 2 2 2   5

5

35 4 γ3,2 − 168es2 − 196es4 , γ3,4 es , γ2,2 − 96s2 − 112s4 , 2 2 2  e3  5

 β1,−3 γ4,4 −140s4 , γ1,2 − −72es2 − 84es4 , η6 β2,3 , 2 8   2 e2 e  4e   4 α1,2 , α1,0 , α1,0 −η 1 α2,0 −η4 α1,0 , α1,−2 η 4 4 2     e2 e2 e2 α3,0 −η4 α 1,0 , α1,2 η4 − 1 α1,2 − eα2,2 − α3,2 , 2 4 12     e2 e2 1 e   4 e   α5,2 −η4 α3,2 , α2,2 −η α1,2 1 α2,2 α3,2 , 2 2 2 2 20     2 e2 e3  e  1  4 e    β3,−1 α1,2 α2,2 1 α3,2 , α3,2 −η η6 β0,1 , 12 3 3 2 24 αij

5.5

ISRN Applied Mathematics

7 

 e2  e3  e   β5,1 α 2,2 α 3,2 , −η η6 β2,1 , 16 4 40    3e 3e3 e3  3e2  e3   6   β7,3

β0,1 β1,1 β2,1 , η6 β4,3 , β1,−1 η 2 8 4 8 56        3e 3e3 e4  3e2 3e 3e3  6     , γ1,−4

β0,1 1 β1,1

β2,1 η8 γ3,4 , β1,1 η 2 8 2 2 8 16    3e 3e3 e4  3e2  e3   6   γ3,−2

β2,3 β3,3 β4,3 , η8 γ1,2 , β1,3 η 2 8 4 8 48   2 e3  e3   6 3e   γ4,0 β0,1 β1,1 , −η8 γ1,0 , β2,−1 η 8 16 8       2 3e2 e4  1 3e 3e3 1  6 3e     , γ5,0 β0,1

β1,1 1 β2,1 −η8 γ1,0 , β2,1 η 8 2 2 8 2 2 80       3e2 e4  1 3e 3e3 3e2   6 1    γ7,2 1 β2,3

β3,3 β4,3 , γ , β2,3 η −η8 2 2 2 2 8 8 112 3,2     3 e4  3e2  1 3e 3e3  6 e    , γ9,4 β0,1 β1,1

γ , −η8 β3,1 η β2,1 24 12 3 2 8 144 5,4         3e2 3e 3e3 1 1 3e 3e3  6 1    β3,3 η ,

β2,3 1 β3,3

β4,3 3 2 8 3 2 3 2 8   3 3e2   6 e  β4,1 η , β β 32 1,1 16 2,1       2 3e2 1 3e 3e3 1  6 3e    , β

β3,3 1 β4,3 β4,3 η 16 2,3 4 2 8 4 2   3 3e2   6 e  , β6,3 η β β 48 3,3 24 4,3     3 3e2  1 3e 3e3  6 e   β5,3 η , β β

β4,3 40 2,3 20 3,3 5 2 8    3e2 e4 e3  e4   8  γ1,−2 η

γ1,2 γ2,2 γ3,2 , 2 4 2 16     3e2 e4 3e4  8 2  γ1,0 η

− 1 3e γ1,0 , 2 4 8        3e4 3e3 3e2 e4  8 2    γ1,2 −η 1 3e , γ1,2 2e γ2,2

γ3,2 8 2 2 4   3 e4   8 e  , γ2,−2 η γ γ 4 1,2 32 2,2 α4,2

4

8

ISRN Applied Mathematics       3 3e2 e4 1 3e3 e3  e4   8   8 e  γ2,0 η

γ3,4 γ4,4 γ5,4 , − 2e γ1,0 , γ1,4 −η 2 4 2 16 4 2 2         3e3 3e3 1 3e4 1  8 1  2   , γ2,2 −η 2e γ1,2 1 3e γ2,2 2e γ3,2 2 2 2 8 2 2       3e3 1 3e2 e4 e3   8 1   γ2,4 −η 2e γ3,4

γ4,4 γ5,4 , 2 2 2 2 4 4    4 1 3e2 e4  8 e  γ3,0 η −

γ1,0 , 48 3 2 4         3 2 e4 3e3 1 1 3e4  8 1   2  , e γ1,2 2e γ2,2 1 3e γ3,2 γ3,2 −η 3 2 4 3 2 3 8         3e3 3e4 1 1 3e2 e4  8 1 2    γ3,4 −η , 1 3e γ3,4 2e γ4,4

γ5,4 3 8 3 2 3 2 4       3 3e3 1 3e2 e4 1  8 e    γ4,2 −η , γ

γ2,2 2e γ3,2 8 1,2 4 2 4 4 2         3e3 3e3 1 3e4 1  8 1  2   γ4,4 −η , 2e γ3,4 1 3e γ4,4 2e γ5,4 4 2 4 8 4 2     4 e3  1 3e2 e4  8 e   γ5,2 −η , γ γ

γ3,2 80 1,2 10 2,2 5 2 4         3e3 3e2 e4 1 1 3e4  8 1   2  γ5,4 −η ,

γ3,4 2e γ4,4 1 3e γ5,4 5 2 4 5 2 5 8       3 3e3 1 3e2 e4 1  8 e    γ6,4 −η , γ

γ4,4 2e γ5,4 12 3,4 6 2 4 6 2   4 e3   8 e  γ6,2 −η , γ γ 96 2,2 12 3,2       4 4 e3  1 3e2 e4 e3   8 e    8 e  γ7,4 −η γ8,4 −η . γ γ

γ5,4 γ γ 112 3,4 14 4,4 7 2 4 128 4,4 16 5,4 5.6

5.2. Error in Ψ due to the Change in the Eccentricity We can write the change in the free flight range angle due to the change in the eccentricity Δe Ψ as follows: Δe Ψ

∂Ψ ∂rbo ∂Ψ ∂vbo ∂Ψ ∂φbo Δe Δe Δe. ∂φbo ∂e ∂rbo ∂e ∂vbo ∂e

5.7

ISRN Applied Mathematics

9

The required derivatives are given by

∂rbo ∂e

 −1 ∂φbo sin f 1 e2 2e cos f , ∂e ∞

 a −1n nen−1 − n 2en 1 cosn f,

∂vbo ∂e

n 0

5.8



−1/2 1/2

μ 1 e2 2e cos f 1 − e2 . a

The bracket 1 e2 2e cos f vanishes when {e cos f ± i sin f, i due to the fact that the eccentricity is a real value. The involved products are

√

−1}, which is impossible

 −1     ∂Ψ ∂φbo 2 sin f 1 e2 2e cos f sin Ψ 2φbo csc2φbo − 1 , ∂φbo ∂e ∞

 4aμ ∂Ψ ∂rbo Ψ 2 2 sin2 csc2φbo −1n nen−1 − n 2en 1 cosn f, ∂rbo ∂e 2 rbo vbo n 0

5.9

−1/2 1/2

8μ3/2

∂Ψ ∂vbo Ψ 2 2 1

e 1 − e

2e cos f sin2 csc2φbo . 3 1/2 ∂vbo ∂e 2 rbo vbo a Using the Lagrange planetary equations, we computed Δe retaining the zonal harmonics up to J4 . The integration is performed between fbo  and 2π − fbo . Substitution of the obtained expression into 5.7 yields

Δe Ψ

⎧ ⎨ ⎩

 −1     2 sin f 1 e2 2e cos fbo sin Ψ 2φbo csc2φbo − 1  ∞

 Ψ n n−1 n 1 ne cosn fbo csc2φ −

2e n −1 bo 2 2 2 rbo vbo n 0 ⎫

⎬ 1/2

−1/2 μ3

Ψ 16 2 2 2 1 − e 1

e csc2φ

2e cos f sin bo bo 3 ⎭ a 2 rbo vbo

4aμ



sin2

⎧ 2 5 ⎨     × J2 δij cos if bo sin jω ⎩ i 1 j −2  − J3 2πX1

7 3







σ mn sin mf bo cosnω

m 1 n −3

⎫ ⎬  

J4 2πX2 , 2Qkl sin kf bo sinlω ⎭ k 1 l −4 

9 4

5.10

10

ISRN Applied Mathematics

where X1 η

4





48s − 60s

3



e2 1 2



 X2 η4 720s2 − 840s4 e sin 2ω,

cos ω,

5.11

where nonvanishing coefficients are given by

δ1,0

  3R2 −8 12s2 , − 16a2 1 − e2  δ1,2

3R2 s2 , 8a2 1 − e2 

15R2 es2 , 16a2 1 − e2    R3 e −72s 90s3 , 32a3 1 − e2  δ4,2

σ1,1

σ1,3

5R3 es3 , 32a3 1 − e2 

δ0,2 −

3R2 es2 , 16a2 1 − e2 

9R2 es2 21R2 s2 , δ , 3,2 4a2 1 − e2  8a2 1 − e2      R3 e 36s − 45s3 R3 48s − 60s3 σ1,−1 , σ0,1 , 32a3 1 − e2  32a3 1 − e2      R3 −144s 180s3 R3 e −60s 75s3 σ2,1 , σ3,1 , 32a3 1 − e2  32a3 1 − e2  δ2,2

σ2,3 −

5R3 s3 , 8a3 1 − e2 

25R3 s3 , 8a3 1 − e2 

σ5,3 −

σ3,3 −

45R3 es3 , 16a3 1 − e2 

35R3 es3 , 32a3 1 − e2      5R4 96 − 480s2 420s4 5R4 e −72s2 84s4 2 Q1,0 Q2,0 − , Q0,2 − , e 256a4 1 − e2  256a4 1 − e2        5R4 −48s2 56s4 5R4 e 288s2 −336s4 5R4 432s2 −504s4 − , Q2,2 − , Q3,2 − , 256a4 1 − e2  256a4 1−e2  256a4 1−e2    5R4 e 168s2 − 196s4 35R4 es4 105R4 s4 , Q2,4 , Q3,4 − , Q4,2 − 4 2 4 2 256a 1 − e  256a 1 − e  128a4 1 − e2  σ4,3 −

Q1,2

δ2,0

  3R2 e −4 6s2 , − 16a2 1 − e2 

445R4 s4 315R4 es4 − , Q , 6,4 128a4 1 − e2  256a4 1 − e2    e e δ1,−2 η2 δ0,2 , δ1,0 η2 −δ1,0 − δ2,0 , 2 2     e e 1 e δ1,2 η2 −δ1,2 − δ2,2 − δ0,2 , δ2,0 η2 − δ1,0 − δ2,0 , 2 2 4 2   e e e 1 e η2 − δ1,2 − δ2,2 − δ3,2 , δ3,0 −η2 δ2,0 , δ5,2 −η2 δ4,2 , 4 2 4 6 10     1 e 1 e e δ3,2 η2 − δ2,2 − δ3,2 − δ4,2 , δ4,2 η2 − δ3,2 − δ4,2 , 6 3 6 8 4     e2 e2 e2 σ1,−1 σ1,1 , σ 1,−1 η4 eσ0,1 1 σ 1,−3 η4 σ1,3 , 2 4 4

Q4,4 −

δ2,2

210R4 es4 , 64a4 1 − e2 

Q5,4 −

ISRN Applied Mathematics 

11

   e2 e2 e2 σ1,1 eσ2,1 σ3,1 , σ 1,1 η eσ0,1 σ1,−1 1 4 2 4       2 2 e2 e 1 e e 4 e 4 e σ0,1 σ1,1 1 σ2,1 σ3,1 , σ0,1 σ1,−1 , η σ 2,−1 η 8 2 2 2 2 8 2 4

σ 2,1

 σ 2,3 η

4

 σ 3,1 η

4

e 1 σ1,3 2 2



e2 1 2

e2 e 1 σ1,1 σ2,1 12 3 3  σ 4,3 η



 e e2 σ2,3 σ3,3 σ4,3 , 2 8



e2 1 2

σ 1,3 η  σ 3,3 η

 σ 5,3 η  Q1,−2 η

6

4

e2 e 1 σ3,3 σ4,3 20 5 5

 

Q1,0

3e2 η − 1 4  

Q1,2 η −

3e 3e3

2 8

  Q2,0 η − 6

  Q1,4 η − 6



Q2,4 

Q3,2

1 η − 2

Q1,0 −

 e2 e σ2,1 σ3,1 , 16 4

 e σ4,3 σ5,3 , 4



 e2 σ1,3 eσ2,3 σ3,3 , 4



e2 1 2

3e 3e3

2 8



 e e2 σ3,3 σ4,3 σ5,3 , 3 12



 σ5,3 ,

σ 5,1 η4

3e 3e3

2 8 

 e2 e σ4,3 σ5,3 , 24 6

3e 3e3

2 8

σ 7,3 η4

 3e2 e3 Q2,2 − Q3,2 − Q4,2 , 4 8





Q2,0 ,

Q2,−2 η

 3e2 e3 Q2,4 − Q3,4 − Q4,4 , 4 8 

1 Q1,2 Q3,2  − 2









3e2 1 4

1 Q2,4 − 2

3e 3e3

2 8

3e 3e3

2 8 1 Q2,2 − 3

e2 σ5,3 , 28



6

 3e2 e3 Q0,2 Q1,2 , 8 16



3e 3e3

2 8







σ 6,3 η

 e3 Q2,0 − Q2,0 , 8

Q1,2 −

3e2 1 2

 4

e2 σ3,1 , 20







e2 σ1,−1 , 12



e2 1 2

e2 1 2

3e2 1 2

1 Q1,0 − 2

e3 3e2 1 Q1,2 − η − Q0,2 − 24 12 3 6

σ 4,1 η







3e2 1 Q0,2 − η − 8 2 6







Q2,2



Q0,2 −

6

σ3,1 ,

 3e2 e2 Q0,2 Q1,2 Q2,2 , 4 8



3e e3

4 8

 4



6

6

e2 1 2

4

e2 e 1 σ1,3 σ2,3 12 3 3

3e 3e3

2 8



e2 e 1 σ2,3 σ3,3 16 4 4

4



4



σ 3,−1 η4

Q7,2 −η6



3e2 1 2

e3 Q4,2 , 56



 3e2 Q2,2 − Q4,2 , 8





 3e2 e3 Q3,4 − Q4,4 − Q5,4 , 8 16

3e2 1 2



1 Q3,2 − 3



3e 3e3

2 8



 Q4,2 ,

12

ISRN Applied Mathematics    3e2 e3 1 3e 3e3 Q1,0 −

Q2,0 , Q3,0 η − Q3,−2 η6 Q0,2 , 12 3 2 8 24   e3 e3 3e2 6 Q2,0 , Q4,0 η − Q1,0 − Q5,0 −η6 Q2,0 , 32 16 40   e3 e3 3e2 Q4,2 , Q6,2 η6 − Q3,2 − Q1,−4 η6 Q2,4 , 48 24 8       e3 3e2 3e2 1 3e 3e3 1 6 Q2,2 −

Q3,2 − 1 Q4,2 , Q4,2 η − Q1,2 − 32 16 4 2 8 4 2       3e2 3e2 1 3e 3e3 3e2 1 6 Q2,4 −

1 Q4,4 − Q5,4 , Q4,4 η − Q3,4 Q5,4  − 16 4 2 8 4 2 16       e3 3e2 3e2 1 3e 3e3 1 6 Q3,4 −

1 Q5,4 , Q5,4 η − Q2,4 − Q4,4 Q6,4  − 40 20 5 2 8 5 2       e3 3e2 3e2 1 3e 3e3 1 6 Q4,4 −

Q5,4 − 1 Q6,4 , Q6,4 η − Q3,4 − 48 24 6 2 8 6 2     e3 e3 3e2 1 3e 3e3 6 Q5,4 −

Q6,4 , Q7,4 η − Q4,4 − Q9,4 −η6 Q6,4 , 56 28 7 2 8 72       e3 e3 3e2 1 3e 3e3 3e2 6 6 Q3,2 −

Q4,2 , Q6,4 . Q5,2 η − Q2,2 − Q8,4 η − Q5,4 − 40 20 5 2 8 64 32 

6

5.12

6. Out-of-Orbit Plane Changes All out-of-orbit plane changes, for example, ΔΩ, Δi will cause a cross-range errors Δψ× .

6.1. Error in Ψ due to the Change in the Ascending Node For the sake of the simplicity, let us take the burnout point on the equator. For some reason, it was displaced by an amount, Δx. This displacement could be interpreted as a change in the longitude of the ascending node ΔΩ if the rest of the orbital elements were kept fixed. Due to this change, a cross-range error, ΔΩ ψ× , at impact occurs, the value of which is obtained, in a similar manner as the previous subsection, by applying the law of cosines. Hence, cos ΔΩ ψ× sin2 Ψ cos2 Ψ cos ΔΩ.

6.1

Since ΔΩ ψ× , ΔΩ are small angles, then we have ΔΩ ψ× ≈ ΔΩ cos Ψ.

6.2

ISRN Applied Mathematics

13

Using the Lagrange planetary equations, we computed ΔΩ retaining the zonal harmonics up to J4 . The integration is performed between fbo  and 2π − fbo . Substitution of the obtained expression into 6.2 yields 

 ΔΩ ψ× ≈

J2

2πX8

J3

2 3 b 1 c 0

⎧ ⎨

2πX9

⎩

     b − ρbc −1 1 sin bf bo cω

3 5

qαβ



α 1 β −1

⎫ ⎬    −1α − 1 cos αfbo βω ⎭

6.3

⎫⎫ ⎬⎬     cos Ψ,

J4 2πX10 − pij −1α 1 sin if bo jω ⎭⎭ ⎩ i 1 j −2 ⎧ ⎨

4 7

where  X8

 η2 ρ0,0 ,

X9 η

4

 eq1,1

sin ω,

X10 η

6

3e2 1 2

  P0,0

 3e2  P cos ω ,

4 2,2

6.4

where nonvanishing coefficients are given by R2 ρbc , 4a2 s1 − e2   −ρ2,2 6s 1 − s2 ,

 ρbc

ρ0,0

 q3,3 −15s2 1 − s2 ,

R3 qαβ , 8a3 s1 − e2   140s3 1 − s2 ,

 qαβ

P4,4

  P0,0 s 1 − s2 −240 420s2 ,

  , ρ1,0 η2 eρ0,0

ρ1,2 η2

e

 , ρ2,2

R4 Pij , 64a4 s1 − e2 

  1 − s2 45s2 − 12 ,

Pij q1,1

  P2,2 s 1 − s2 240 − 560s2 , !

" 1  ρ2,2 , 2     e2 4  , q1,1 η − 1 2

ρ2,2 η2

2   " 2 2 e  4 e  , ρ q ρ3,2 η q1,−1 η q1,1 , 6 2,2 4 1,1    e  e e2  4   , , q1,3 η − q3,3 , q2,1 η4 − q1,1 q2,3 η4 − q3,3 4 2 2        e e2  1 e2 4 4   , , 1 q3,3 q3,1 η − q1,1 , q3,3 η − q4,3 η4 − q3,3 12 3 2 4         3 e2  3e 3e3 4 6 e  6  , P

P0,0 , q5,3 η − q3,3 , P 1,−2 η P 1,0 η 2 20 8 2,2 2 8        3 2 3e 3e3 6  6 e  6 6e  , ,

P2,2 , P P P 1,2 η P 1,4 η P 2,0 η 2 8 8 4,4 8 0,0 !

14 P 2,2

P 3,2

    1 3e2  1 P2,2 , η 2 2

    1 3e 3e3 

P2,2 , η 3 2 8

P 4,4

 P 2,4 η

6

P 3,4

6

    1 3e2  1 P4,4 , η 4 2  P 6,4 η

    1 3e 3e3 

P4,4 , η 3 2 8

P 5,2 η

6

 3e2  , P 8 4,4

 P 4,2 η

6



6

6

ISRN Applied Mathematics   3 6 e  , P P 3,0 η 12 0,0

6

 e3  , P 40 2,2

 3e2  , P 24 4,4

P 5,4 

P 7,4 η

6

6

 3e2  , P 16 2,2

    1 3e 3e3 

P4,4 , η 5 2 8 6

 e3  . P 56 4,4 6.5

6.2. Error in Ψ due to the Change in the Inclination Again, assume that the burnout point is on the equator and that the actual launch azimuth differs from the intended value by an amount Δβ. This amount could be interpreted as a change in the orbital inclination if all other orbital elements were kept fixed. Due to this change, a cross-range error, Δi ψ× , at impact occurs, the value of which is obtained, in a similar manner as the previous subsection, by applying the law of cosines. Hence, cos Δi ψ× cos2 Ψ sin2 Ψ cos Δi.

6.6

Since Δi ψ× , Δi are small angles, then we have Δi ψ× ≈ Δi sin Ψ.

6.7

Using the Lagrange planetary equations, we computed Δi retaining the zonal harmonics up to J4 . The integration is performed between fbo  and 2π − fbo . Substitution of the obtained expression into 6.7 yields  Δi ψ× ≈

J2

3     Zi −1i − 1 cos ifbo 2ω i 1



J3 2πX3

3 5

    − Ckl −1k 1 sin kfbo lω

k 1 l −1

2πX9

5 3

qαβ

α 1 β −1



⎫ ⎬    −1α − 1 cos αfbo βω ⎭

⎫⎫ ⎬⎬    

J4 2πX4 Spq −1p − 1 cos pfbo qω sin Ψ, ⎭⎭ ⎩ p 1 q −2 ⎧ ⎨

7 4

6.8

ISRN Applied Mathematics

15

where 

X3 η4 eC11 cos ω,

X4 η 6

3e2  S sin 2ω, 4 22

6.9

where nonvanishing coefficients are given by C1,1 15s3 − 12s,

C3,3 −15s3 ,

S2,2 −240s2 280s4 ,

S4,4 −140s4 , √ R3 1 − s2 Ckl 3 Ckl , 8a s1 − e2 

 R2 6s 1 − s2 , z 2 2 4a 1 − e 



e Z1 − η2 z, 2 C1,−1

e2 4  η C 1,1 , 4

C1,1



Spq

√ R4 1 − s2 Spq , 64a4 s1 − e2 

1 e Z2 − η2 z, Z3 − η2 z, 2 6   e2 e2 η4 C 1,1 , 1 C1,3 η4 C 3,3 , 2 4

e 4  e e2 4  η C 1,1 , η C 1,1 , C2,3 η4 C 3,3 , C3,1 2 2 12   e2 1 e e2 4  1 η4 C 3,3 , η C 3,3 , C4,3 η4 C 3,3 , C5,3 3 2 4 20 C2,1

C3,3

S1,−2

S2,2

1 − 2

S3,4

1 − 3 S5,2



e3 η6 S 2,2 , 8



3e2 1 2



S1,2 −



3e 3e3

2 8

η 6 S

2,2 ,

S2,4

e3 − η6 S 2,2 , 40

4,4 ,

S5,4

S4,2 1 − 5

 η6 S 2,2 ,

3e2 6  η S 4,4 , − 8

 η 6 S

3e 3e3

2 8

S3,2

3e2 6  η S 2,2 , − 16



3e 3e3

2 8

S7,4 −

e3 6  η S 4,4 , 8

S1,4 − 1 − 3

S4,4



1 − 4

3e 3e3

2 8



3e2 1 2

 η6 S 4,4 ,

e3 6  η S 4,4 , 56

S6,4 −

 η6 S 2,2 ,  η6 S 4,4 ,

3e2 6  η S 4,4 , 24

6.10

7. Conclusions and Future Work Due to the high sensitivity of the ballistic missile range to the different kinds of errors, we computed the explicit expressions for the errors in the missile range due to the in-orbit plane

16

ISRN Applied Mathematics

changes. We derived explicit expressions for the errors in the missile range due to the outof-orbit plane changes when the burnout point is assumed on the equator. In a forthcoming work, we aim to generalize this situation. Also, we aim to do the corresponding algorithms and give numerical examples.

Acknowledgments The authors are deeply indebted to the Professor Dr. M. K. Ahmed, the professor of space dynamics at Cairo University, Faculty of Science, Department of Astronomy, for his valuable discussions and critical comments and ideas that help us to finalize this work. This research work was supported by a Grant no. 627 from the deanship of the scientific research at Taibah university, Al-Madinah Al-Munawwarah, Saudi Arabia.

References 1 H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1981. 2 J. B. Marion, Classical Dynamics of Particles and Systems, Academic Press, New York, NY, USA, 2nd edition, 1970. 3 J. T. Wu, “Orbit determination by solving for gravity parameters with multiple arc data,” Journal of Guidance, Control, and Dynamics, vol. 15, no. 2, pp. 304–313, 1992. 4 J. S. McFarland, “Modeling the ballistic missile problem with the state transition matrix: an analysis of trajectories including a rotating earth and atmospheric drag,” Semester Report, Virginia Tech, Blacksburg, Va, USA, 2004. 5 G. Forden, “GUI missile flyout: a general program for simulating ballistic missiles,” Science and Global Security, vol. 15, no. 2, pp. 133–146, 2007. 6 U. N. Bao and E. D. Murray, “Computation of effective ground range using an oblate earth model,” The Journal of the Astronautically Sciences, vol. 51, no. 3, pp. 291–305, 2003. 7 J. A. Isaacson and D. R. Vaughan, Estimation and Prediction of Ballistic Missile Trajectories, RAND Corporation, Santa Monica, Calif, USA, 1996. 8 W. J. Harlin and D. A. Cicci, “Ballistic missile trajectory prediction using a state transition matrix,” Applied Mathematics and Computation, vol. 188, pp. 1832–1847, 2007. 9 A. Akgul ¨ and S. Karasoy, “Development of a tactical ballistic missile trajectory prediction tool,” Istanbul University Journal of Electrical & Electronic Engineering, vol. 5, no. 2, pp. 1463–1467, 2005. 10 X. Vinh, T. Kabamba, and T. Takehira, “Optimal interception of a maneuvering long-range missile,” Acta Astronautica, vol. 48, no. 1, pp. 1–19, 2001. 11 S. A. Kamal, “Cross range error in the lambert scheme,” in Proceedings of the 10th National Aeronautical Conference, S. R. Sheikh, Ed., pp. 255–263, College of Aeronautical Engineering, PAF Academy, Risalpur, NWFP, Pakistan, April 2006. 12 S. Bhowmik and C. Sadhukhan, “Application of extended kalman filter to tactical ballistic missile re-entry problem,” 2007. 13 S. A. Kamal, “The multi-stage-lambert scheme for steering a satellite-launch vehicle SLV,” in Proceedings of the 12th IEEE International Multitopic Conference, M. K. Anisx, M. K. Khan, and S. J. H. Zaidi, Eds., pp. 294–300, Bahria University, Karachi, Pakistan, December 2008. 14 C. Y. Liu and C. T. Chen, “Tracking the warhead among objects separation from the reentry vehicle in a clear environment,” Defence Science Journal, vol. 59, no. 2, pp. 113–125, 2009. 15 R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics, Dover, New York, NY, USA, 1971.

Advances in

Operations Research Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Advances in

Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Applied Mathematics

Algebra

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences

Mathematical Problems in Engineering

Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Discrete Mathematics

Journal of

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014