Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 349737, 16 pages
http://dx.doi.org/10.5402/2011/349737
Research Article

Computation of the Different Errors in the Ballistic Missiles Range

1Department of Math, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
2Department of Astronomy, Faculty of Science, Cairo University, Cairo 12613, Egypt
3Department of Applied Mathematics, Faculty of Applied Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
4Department of Mathematics, Faculty of Science, Tanta university, Tanta, Egypt

Received 8 June 2011; Accepted 25 July 2011

Academic Editor: Z. Huang

Copyright © 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.

Abstract

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 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 1(a). (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).

fig1
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,and𝑅𝑡 are their ground ranges.

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 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. Akgül 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 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=𝐺(𝑚1+𝑚2)=𝜇=𝐺𝑚1,𝑚2𝑚1, 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 1(a) and 1(b), we have [15]Ψcos2=cos𝑓bo=1𝑄bocos2𝜙bo1+𝑄bo𝑄bo2cos2𝜙bo,(4.1)sin2𝜙bo+Ψ2=2𝑄bo𝑄boΨsin2,(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),(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, sin(Ψ/2)=𝑄bo/(2𝑄bo), and the flight path angle is 𝜙bo=(1/4)(180Ψ).

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𝜕𝜙bo=2sinΨ+2𝜙bocsc2𝜙bo,1𝜕Ψ𝜕𝑟bo=4𝜇𝑟2bo𝑣2bosin2Ψ2csc2𝜙bo,𝜕Ψ𝜕𝑣bo=8𝜇𝑟bo𝑣3bosin2Ψ2csc2𝜙bo.(5.1)

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 Δ𝑎Ψ as follows:Δ𝑎Ψ=𝜕Ψ𝜕𝑟bo𝜕𝑟bo𝜕𝑎Δ𝑎+𝜕Ψ𝜕𝑣bo𝜕𝑣bo𝜕𝑎Δ𝑎.(5.2) The required derivatives are given by𝜕𝑟bo=𝜕𝑎𝑛=0(1)𝑛𝑒𝑛𝑒𝑛+2cos𝑛𝑓,𝜕𝑣bo1𝜕𝑎=2𝜇𝑎31+𝑒2+2𝑒cos𝑓1/21𝑒21/2.(5.3) The involved products are 𝜕Ψ𝜕𝑟bo𝜕𝑟bo=𝜕𝑎4𝜇𝑟2bo𝑣2bosin2Ψ2csc2𝜙bo𝑛=0(1)𝑛𝑒𝑛𝑒𝑛+2cos𝑛𝑓,𝜕Ψ𝜕𝑣bo𝜕𝑣bo1𝜕𝑎=2𝜇3𝑎38𝑟bo𝑣3bosin2Ψ2csc2𝜙bo1+𝑒2+2𝑒cos𝑓1/21𝑒21/2.(5.4) 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 (2𝜋𝑓bo). Substitution of the obtained expression into (5.2) yields Δ𝑎Ψ=4𝜇𝑟2bo𝑣2bosin2Ψ2csc2𝜙bo𝑛=0(1)𝑛𝑒𝑛𝑒𝑛+2cos𝑛𝑓bo𝜇3𝑎38𝑟bo𝑣3bosin2Ψ2csc2𝜙bo1+𝑒2+2𝑒cos𝑓bo1/21𝑒21/2×𝐽252𝑖=1𝑗=2𝛼𝑖𝑗sin𝑖𝑓bosin(𝑗𝜔)𝐽373𝑘=1𝑙=3𝛽𝑘𝑙sin𝑘𝑓bocos(𝑙𝜔)+𝐽494𝑚=1𝑛=4𝛾𝑚𝑛sin𝑚𝑓bosin(𝑛𝜔),(5.5) where nonvanishing coefficients are given by 𝛼𝑖𝑗=𝑅22𝑎1𝑒2𝛼𝑖𝑗,𝛽𝑘𝑙=𝑅34𝑎21𝑒2𝛽𝑘𝑙,𝛾𝑚𝑛=𝑅432𝑎31𝑒2𝛾𝑚𝑛,𝛼1,03=24𝑒+6𝑒𝑠2,𝛼1,23=2𝑒𝑠2,𝛼2,2=6𝑠2,𝛼3,2=152𝑒𝑠2,𝛽0,1=1236𝑒𝑠+45𝑒𝑠3,𝛽2,3=52e𝑠3,𝛽2,1=1260𝑒𝑠+75𝑒𝑠3,𝛽1,1=1224𝑠+30𝑠3,𝛽3,3=15𝑠3,𝛽4,3=352𝑒𝑠3,𝛾1,05=248𝑒240𝑒𝑠2+210𝑒𝑠4,𝛾5,4=3152𝑒𝑠4,𝛾2,25=296𝑠2112𝑠4,𝛾3,25=2168𝑒𝑠2196𝑒𝑠4,𝛾3,4=352𝑒𝑠4,𝛾4,4=140𝑠4,𝛾1,25=272𝑒𝑠284𝑒𝑠4,𝛽1,3=𝜂6𝑒38𝛽2,3,𝛼1,2=𝜂4𝑒24𝛼1,2,𝛼1,0=𝜂4𝑒1+24𝛼1,0,𝛼2,0=𝜂4𝑒2𝛼1,0,𝛼1,2=𝜂4𝑒1+22𝛼1,2𝑒𝛼2,2𝑒24𝛼3,2,𝛼3,0=𝜂4𝑒212𝛼1,0,𝛼2,2=𝜂4𝑒2𝛼1,2+12𝑒1+22𝛼2,2+𝑒2𝛼3,2,𝛼5,2=𝜂4𝑒2𝛼203,2,𝛼3,2=𝜂4𝑒2𝛼121,2+𝑒3𝛼2,2+13𝑒1+22𝛼3,2,𝛽3,1=𝜂6𝑒3𝛽240,1,𝛼4,2=𝜂4𝑒2𝛼162,2+𝑒4𝛼3,2,𝛽5,1=𝜂6𝑒3𝛽402,1,𝛽1,1=𝜂63𝑒2+3𝑒38𝛽0,1+3𝑒24𝛽1,1+𝑒38𝛽2,1,𝛽7,3=𝜂6𝑒3𝛽564,3,𝛽1,1=𝜂63𝑒2+3𝑒38𝛽0,1+1+3𝑒22𝛽1,1+3𝑒2+3𝑒38𝛽2,1,𝛾1,4=𝜂8𝑒4𝛾163,4,𝛽1,3=𝜂63𝑒2+3𝑒38𝛽2,3+3𝑒24𝛽3,3+𝑒38𝛽4,3,𝛾3,2=𝜂8𝑒4𝛾481,2,𝛽2,1=𝜂63𝑒28𝛽0,1+𝑒3𝛽161,1,𝛾4,0=𝜂8𝑒38𝛾1,0,𝛽2,1=𝜂63𝑒28𝛽0,1+123𝑒2+3𝑒38𝛽1,1+121+3𝑒22𝛽2,1,𝛾5,0=𝜂8𝑒4𝛾801,0,𝛽2,3=𝜂6121+3𝑒22𝛽2,3+123𝑒2+3𝑒38𝛽3,3+3𝑒28𝛽4,3,𝛾7,2=𝜂8𝑒4𝛾1123,2,𝛽3,1=𝜂6𝑒3𝛽240,1+3𝑒2𝛽121,1+133𝑒2+3𝑒38𝛽2,1,𝛾9,4=𝜂8𝑒4𝛾1445,4,𝛽3,3=𝜂6133𝑒2+3𝑒38𝛽2,3+131+3𝑒22𝛽3,3+133𝑒2+3𝑒38𝛽4,3,𝛽4,1=𝜂6𝑒3𝛽321,1+3𝑒2𝛽162,1,𝛽4,3=𝜂63𝑒2𝛽162,3+143𝑒2+3𝑒38𝛽3,3+141+3𝑒22𝛽4,3,𝛽6,3=𝜂6𝑒3𝛽483,3+3𝑒2𝛽244,3,𝛽5,3=𝜂6𝑒3𝛽402,3+3𝑒2𝛽203,3+153𝑒2+3𝑒38𝛽4,3,𝛾1,2=𝜂83𝑒22+𝑒44𝛾1,2+𝑒32𝛾2,2+𝑒4𝛾163,2,𝛾1,0=𝜂83𝑒22+𝑒441+3𝑒2+3𝑒48𝛾1,0,𝛾1,2=𝜂81+3𝑒2+3𝑒48𝛾1,2+2𝑒+3𝑒32𝛾2,2+3𝑒22+𝑒44𝛾3,2,𝛾2,2=𝜂8𝑒34𝛾1,2+𝑒4𝛾322,2,𝛾1,4=𝜂83𝑒22+𝑒44𝛾3,4+𝑒32𝛾4,4+𝑒4𝛾165,4,𝛾2,0=𝜂8𝑒34122𝑒+3𝑒32𝛾1,0,𝛾2,2=𝜂8122𝑒+3𝑒32𝛾1,2+121+3𝑒2+3𝑒48𝛾2,2+122𝑒+3𝑒32𝛾3,2,𝛾2,4=𝜂8122𝑒+3𝑒32𝛾3,4+123𝑒22+𝑒44𝛾4,4+𝑒34𝛾5,4,𝛾3,0=𝜂8𝑒414833𝑒22+𝑒44𝛾1,0,𝛾3,2=𝜂81332𝑒2+𝑒44𝛾1,2+132𝑒+3𝑒32𝛾2,2+131+3𝑒2+3𝑒48𝛾3,2,𝛾3,4=𝜂8131+3𝑒2+3𝑒48𝛾3,4+132𝑒+3𝑒32𝛾4,4+133𝑒22+𝑒44𝛾5,4,𝛾4,2=𝜂8𝑒38𝛾1,2+143𝑒22+𝑒44𝛾2,2+142𝑒+3𝑒32𝛾3,2,𝛾4,4=𝜂8142𝑒+3𝑒32𝛾3,4+141+3𝑒2+3𝑒48𝛾4,4+142𝑒+3𝑒32𝛾5,4,𝛾5,2=𝜂8𝑒4𝛾801,2+𝑒3𝛾102,2+153𝑒22+𝑒44𝛾3,2,𝛾5,4=𝜂8153𝑒22+𝑒44𝛾3,4+152𝑒+3𝑒32𝛾4,4+151+3𝑒2+3𝑒48𝛾5,4,𝛾6,4=𝜂8𝑒3𝛾123,4+163𝑒22+𝑒44𝛾4,4+162𝑒+3𝑒32𝛾5,4,𝛾6,2=𝜂8𝑒4𝛾962,2+𝑒3𝛾123,2,𝛾7,4=𝜂8𝑒4𝛾1123,4+𝑒3𝛾144,4+173𝑒22+𝑒44𝛾5,4𝛾8,4=𝜂8𝑒4𝛾1284,4+𝑒3𝛾165,4.(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 Δ𝑒Ψ as follows:Δ𝑒Ψ=𝜕Ψ𝜕𝜙bo𝜕𝜙bo𝜕𝑒Δ𝑒+𝜕Ψ𝜕𝑟bo𝜕𝑟bo𝜕𝑒Δ𝑒+𝜕Ψ𝜕𝑣bo𝜕𝑣bo𝜕𝑒Δ𝑒.(5.7) The required derivatives are given by𝜕𝜙bo𝜕𝑒=sin𝑓1+𝑒2+2𝑒cos𝑓1,𝜕𝑟bo𝜕𝑒=𝑎𝑛=0(1)𝑛𝑛𝑒𝑛1(𝑛+2)𝑒𝑛+1cos𝑛𝑓,𝜕𝑣bo=𝜕𝑒𝜇𝑎1+𝑒2+2𝑒cos𝑓1/21𝑒21/2.(5.8) The bracket [1+𝑒2+2𝑒cos𝑓] vanishes when {𝑒=cos𝑓±𝑖sin𝑓,𝑖=1}, which is impossible due to the fact that the eccentricity is a real value.

The involved products are 𝜕Ψ𝜕𝜙bo𝜕𝜙bo𝜕𝑒=2sin𝑓1+𝑒2+2𝑒cos𝑓1sinΨ+2𝜙bocsc2𝜙bo,1𝜕Ψ𝜕𝑟bo𝜕𝑟bo=𝜕𝑒4𝑎𝜇𝑟2bo𝑣2bosin2Ψ2csc2𝜙bo𝑛=0(1)𝑛𝑛𝑒𝑛1(𝑛+2)𝑒𝑛+1cos𝑛𝑓,𝜕Ψ𝜕𝑣bo𝜕𝑣bo=𝜕𝑒8𝜇3/2𝑟bo𝑣3bo𝑎1/21+𝑒2+2𝑒cos𝑓1/21𝑒21/2sin2Ψ2csc2𝜙bo.(5.9) Using the Lagrange planetary equations, we computed Δ𝑒 (retaining the zonal harmonics up to 𝐽4). The integration is performed between (𝑓bo) and (2𝜋𝑓bo). Substitution of the obtained expression into (5.7) yields Δ𝑒Ψ=2sin𝑓1+𝑒2+2𝑒cos𝑓bo1sinΨ+2𝜙bocsc2𝜙bo+14𝑎𝜇𝑟2bo𝑣2bosin2Ψ2csc2𝜙bo𝑛=0(1)𝑛𝑛𝑒𝑛1(𝑛+2)𝑒𝑛+1cos𝑛𝑓bo+16𝑟bo𝑣3bo𝜇3𝑎1+𝑒2+2𝑒cos𝑓bo1/21𝑒21/2sin2Ψ2csc2𝜙bo×𝐽252𝑖=1𝑗=2𝛿𝑖𝑗cos𝑖𝑓bosin(𝑗𝜔)𝐽32𝜋𝑋1+73𝑚=1𝑛=3𝜎𝑚𝑛sin𝑚𝑓bocos(𝑛𝜔)+𝐽42𝜋𝑋2+94𝑘=1𝑙=42𝑄𝑘𝑙sin𝑘𝑓bo,sin(𝑙𝜔)(5.10) where𝑋1=𝜂448𝑠60𝑠3𝑒1+22cos𝜔,𝑋2=𝜂4720𝑠2840𝑠4𝑒sin2𝜔,(5.11) where nonvanishing coefficients are given by 𝛿1,0=3𝑅28+12𝑠216𝑎21𝑒2,𝛿2,0=3𝑅2𝑒4+6𝑠216𝑎21𝑒2,𝛿0,2=3𝑅2𝑒𝑠216𝑎21𝑒2,𝛿1,2=3𝑅2𝑠28𝑎21𝑒2,𝛿2,2=9𝑅2𝑒𝑠24𝑎21𝑒2,𝛿3,2=21𝑅2𝑠28𝑎21𝑒2,𝛿4,2=15𝑅2𝑒𝑠216𝑎21𝑒2,𝜎1,1=𝑅3𝑒36𝑠45𝑠332𝑎31𝑒2,𝜎0,1=𝑅348𝑠60𝑠332𝑎31𝑒2,𝜎1,1=𝑅3𝑒72𝑠+90𝑠332𝑎31𝑒2,𝜎2,1=𝑅3144𝑠+180𝑠332𝑎31𝑒2,𝜎3,1=𝑅3𝑒60𝑠+75𝑠332𝑎31𝑒2,𝜎1,3=5𝑅3𝑒𝑠332𝑎31𝑒2,𝜎2,3=5𝑅3𝑠38𝑎31𝑒2,𝜎3,3=45𝑅3𝑒𝑠316𝑎31𝑒2,𝜎4,3=25𝑅3𝑠38𝑎31𝑒2,𝜎5,3=35𝑅3𝑒𝑠332𝑎31𝑒2,𝑄1,0=2𝑒𝑄2,0=5𝑅496480𝑠2+420𝑠4256𝑎41𝑒2,𝑄0,2=5𝑅4𝑒72𝑠2+84𝑠4256𝑎41𝑒2,𝑄1,2=5𝑅448𝑠2+56𝑠4256𝑎41𝑒2,𝑄2,2=5𝑅4𝑒288𝑠2336𝑠4256𝑎41𝑒2,𝑄3,2=5𝑅4432𝑠2504𝑠4256𝑎41𝑒2,𝑄4,2=5𝑅4𝑒168𝑠2196𝑠4256𝑎41𝑒2,𝑄2,4=35𝑅4𝑒𝑠4256𝑎41𝑒2,𝑄3,4=105𝑅4𝑠4128𝑎41𝑒2,𝑄4,4=210𝑅4𝑒𝑠464𝑎41𝑒2,𝑄5,4=445𝑅4𝑠4128𝑎41𝑒2,𝑄6,4=315𝑅4𝑒𝑠4256𝑎41𝑒2,𝛿1,2=𝜂2𝑒2𝛿0,2,𝛿1,0=𝜂2𝛿1,0𝑒2𝛿2,0,𝛿1,2=𝜂2𝛿1,2𝑒2𝛿2,2𝑒2𝛿0,2,𝛿2,0=𝜂2𝑒4𝛿1,012𝛿2,0,𝛿2,2=𝜂2𝑒4𝛿1,212𝛿2,2𝑒4𝛿3,2,𝛿3,0=𝜂2𝑒6𝛿2,0,𝛿5,2=𝜂2𝑒𝛿104,2,𝛿3,2=𝜂2𝑒6𝛿2,213𝛿3,2𝑒6𝛿4,2,𝛿4,2=𝜂2𝑒8𝛿3,214𝛿4,2,𝜎1,1=𝜂4𝑒𝜎0,1+𝑒1+22𝜎1,1+𝑒24𝜎1,1,𝜎1,3=𝜂4𝑒24𝜎1,3,𝜎1,1=𝜂4𝑒𝜎0,1+𝑒24𝜎1,1+𝑒1+22𝜎1,1+𝑒𝜎2,1+𝑒24𝜎3,1,𝜎2,1=𝜂4𝑒28𝜎0,1+𝑒2𝜎1,1+12𝑒1+22𝜎2,1+𝑒2𝜎3,1,𝜎2,1=𝜂4𝑒28𝜎0,1+𝑒2𝜎1,1,𝜎2,3=𝜂4𝑒2𝜎1,3+12𝑒1+22𝜎2,3+𝑒2𝜎3,3+𝑒28𝜎4,3,𝜎3,1=𝜂4𝑒2𝜎121,1,𝜎3,1=𝜂4𝑒2𝜎121,1+𝑒3𝜎2,1+13𝑒1+22𝜎3,1,𝜎4,1=𝜂4𝑒2𝜎162,1+𝑒4𝜎3,1,𝜎4,3=𝜂4𝑒2𝜎162,3+𝑒4𝜎3,3+14𝑒1+22𝜎4,3+𝑒4𝜎5,3,𝜎1,3=𝜂4𝑒1+22𝜎1,3+𝑒𝜎2,3+𝑒24𝜎3,3,𝜎3,3=𝜂4𝑒2𝜎121,3+𝑒3𝜎2,3+13𝑒1+22𝜎3,3+𝑒3𝜎4,3+𝑒2𝜎125,3,𝜎5,3=𝜂4𝑒2𝜎203,3+𝑒5𝜎4,3+15𝑒1+22𝜎5,3,𝜎5,1=𝜂4𝑒2𝜎203,1,𝑄1,2=𝜂63𝑒2+3𝑒38𝑄0,2+3𝑒24𝑄1,2+𝑒28𝑄2,2,𝜎6,3=𝜂4𝑒2𝜎244,3+𝑒6𝜎5,3,𝑄1,0=𝜂61+3𝑒24𝑄1,03𝑒2+3𝑒38𝑄2,0𝑒38𝑄2,0,𝜎7,3=𝜂4𝑒2𝜎285,3,𝑄1,2=𝜂63𝑒2+3𝑒38𝑄0,21+3𝑒22𝑄1,23𝑒2+3𝑒38𝑄2,23𝑒24𝑄3,2𝑒38𝑄4,2,𝑄2,0=𝜂63𝑒4+𝑒38𝑄1,0121+3𝑒22𝑄2,0,𝑄2,2=𝜂63𝑒28𝑄0,2+𝑒3𝑄161,2,𝑄1,4=𝜂63𝑒2+3𝑒38𝑄2,43𝑒24𝑄3,4𝑒38𝑄4,4,𝑄7,2=𝜂6𝑒3𝑄564,2,𝑄2,2=𝜂63𝑒28𝑄0,2123𝑒2+3𝑒38𝑄1,2+𝑄3,2121+3𝑒22𝑄2,23𝑒28𝑄4,2,𝑄2,4=𝜂6121+3𝑒24𝑄2,4123𝑒2+3𝑒38𝑄3,43𝑒28𝑄4,4𝑒3𝑄165,4,𝑄3,2=𝜂6𝑒3𝑄240,23𝑒2𝑄121,2133𝑒2+3𝑒38𝑄2,2131+3𝑒22𝑄3,2133𝑒2+3𝑒38𝑄4,2,𝑄3,0=𝜂63𝑒2𝑄121,0133𝑒2+3𝑒38𝑄2,0,𝑄3,2=𝜂6𝑒3𝑄240,2,𝑄4,0=𝜂6𝑒3𝑄321,03𝑒2𝑄162,0,𝑄5,0=𝜂6𝑒3𝑄402,0,𝑄6,2=𝜂6𝑒3𝑄483,23𝑒2𝑄244,2,𝑄1,4=𝜂6𝑒38𝑄2,4,𝑄4,2=𝜂6𝑒3𝑄321,23𝑒2𝑄162,2143𝑒2+3𝑒38𝑄3,2141+3𝑒22𝑄4,2,𝑄4,4=𝜂63𝑒2𝑄162,4143𝑒2+3𝑒38𝑄3,4+𝑄5,4141+3𝑒22𝑄4,43𝑒2𝑄165,4,𝑄5,4=𝜂6𝑒3𝑄402,43𝑒2𝑄203,4153𝑒2+3𝑒38𝑄4,4+𝑄6,4151+3𝑒22𝑄5,4,𝑄6,4=𝜂6𝑒3𝑄483,43𝑒2𝑄244,4163𝑒2+3𝑒38𝑄5,4161+3𝑒22𝑄6,4,𝑄7,4=𝜂6𝑒3𝑄564,43𝑒2𝑄285,4173𝑒2+3𝑒38𝑄6,4,𝑄9,4=𝜂6𝑒3𝑄726,4,𝑄5,2=𝜂6𝑒3𝑄402,23𝑒2𝑄203,2153𝑒2+3𝑒38𝑄4,2,𝑄8,4=𝜂6𝑒3𝑄645,43𝑒2𝑄326,4.(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,  ΔΩ𝜓×, at impact occurs, the value of which is obtained, in a similar manner as the previous subsection, by applying the law of cosines. Hence,cosΔΩ𝜓×=sin2Ψ+cos2ΨcosΔΩ.(6.1) Since ΔΩ𝜓×, ΔΩ are small angles, then we haveΔΩ𝜓×ΔΩcosΨ.(6.2) Using the Lagrange planetary equations, we computed ΔΩ (retaining the zonal harmonics up to 𝐽4). The integration is performed between (𝑓bo) and (2𝜋𝑓bo). Substitution of the obtained expression into (6.2) yields ΔΩ𝜓×𝐽22𝜋𝑋8+32𝑏=1𝑐=0𝜌𝑏𝑐(1)𝑏+1sin𝑏𝑓bo+𝑐𝜔+𝐽32𝜋𝑋9+53𝛼=1𝛽=1𝑞𝛼𝛽(1)𝛼1cos𝛼𝑓bo+𝛽𝜔+𝐽42𝜋𝑋10+74𝑖=1𝑗=2𝑝𝑖𝑗(1)𝛼+1sin𝑖𝑓bo+𝑗𝜔cosΨ,(6.3) where𝑋8=𝜂2𝜌0,0,𝑋9=𝜂4𝑒𝑞1,1sin𝜔,𝑋10=𝜂61+3𝑒22𝑃0,0+3𝑒24𝑃2,2cos𝜔,(6.4) where nonvanishing coefficients are given by 𝜌𝑏𝑐=𝑅24𝑎2𝑠1𝑒2𝜌𝑏𝑐,𝑞𝛼𝛽=𝑅38𝑎3𝑠1𝑒2𝑞𝛼𝛽,𝑃𝑖𝑗=𝑅464𝑎4𝑠1𝑒2𝑃𝑖𝑗,𝜌0,0=𝜌2,2=6𝑠1𝑠2,𝑃4,4=140𝑠31𝑠2,𝑞1,1=1𝑠245𝑠2,𝑞123,3=15𝑠21𝑠2,𝑃0,0=𝑠1𝑠2240+420𝑠2,𝑃2,2=𝑠1𝑠2240560𝑠2,𝜌1,0=𝜂2𝑒𝜌0,0,𝜌1,2=𝜂2𝑒2𝜌2,2,𝜌2,2=𝜂212𝜌2,2,𝜌3,2=𝜂2𝑒6𝜌2,2,𝑞1,1=𝜂4𝑒24𝑞1,1,𝑞1,1=𝜂4𝑒1+22𝑞1,1,𝑞1,3=𝜂4𝑒24𝑞3,3,𝑞2,1=𝜂4𝑒2𝑞1,1,𝑞2,3=𝜂4𝑒2𝑞3,3,𝑞3,1=𝜂4𝑒2𝑞121,1,𝑞3,3=𝜂413𝑒1+22𝑞3,3,𝑞4,3=𝜂4𝑒4𝑞3,3,𝑞5,3=𝜂4𝑒2𝑞203,3,𝑃1,2=𝜂6𝑒38𝑃2,2,𝑃1,0=𝜂623𝑒2+3𝑒38𝑃0,0,𝑃1,2=𝜂63𝑒2+3𝑒38𝑃2,2,𝑃1,4=𝜂6𝑒38𝑃4,4,𝑃2,0=𝜂66𝑒28𝑃0,0,𝑃2,2=𝜂6121+3𝑒22𝑃2,2,𝑃2,4=𝜂63𝑒28𝑃4,4,𝑃3,0=𝜂6𝑒3𝑃120,0,𝑃3,2=𝜂6133𝑒2+3𝑒38𝑃2,2,𝑃3,4=𝜂6133𝑒2+3𝑒38𝑃4,4,𝑃4,2=𝜂63𝑒2𝑃162,2,𝑃4,4=𝜂6141+3𝑒22𝑃4,4,𝑃5,2=𝜂6𝑒3𝑃402,2,𝑃5,4=𝜂6153𝑒2+3𝑒38𝑃4,4,𝑃6,4=𝜂63𝑒2𝑃244,4,𝑃7,4=𝜂6𝑒3𝑃564,4.(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,  Δ𝑖𝜓×, at impact occurs, the value of which is obtained, in a similar manner as the previous subsection, by applying the law of cosines. Hence,cosΔ𝑖𝜓×=cos2Ψ+sin2ΨcosΔ𝑖.(6.6) Since Δ𝑖𝜓×,Δ𝑖 are small angles, then we haveΔ𝑖𝜓×Δ𝑖sinΨ.(6.7) Using the Lagrange planetary equations, we computed Δ𝑖 (retaining the zonal harmonics up to 𝐽4). The integration is performed between (𝑓bo) and (2𝜋𝑓bo). Substitution of the obtained expression into (6.7) yields Δ𝑖𝜓×𝐽23𝑖=1𝑍𝑖(1)𝑖1cos𝑖𝑓bo+2𝜔+𝐽32𝜋𝑋3+53𝑘=1𝑙=1𝐶𝑘𝑙(1)𝑘+1sin𝑘𝑓bo+𝑙𝜔+2𝜋𝑋9+53𝛼=1𝛽=1𝑞𝛼𝛽(1)𝛼1cos𝛼𝑓bo+𝛽𝜔+𝐽42𝜋𝑋4+74𝑝=1𝑞=2𝑆𝑝𝑞(1)𝑝1cos𝑝𝑓bo+𝑞𝜔sinΨ,(6.8) where𝑋3=𝜂4𝑒𝐶11cos𝜔,𝑋4=𝜂63𝑒24𝑆22sin2𝜔,(6.9) where nonvanishing coefficients are given by 𝐶1,1=15𝑠312𝑠,𝐶3,3=15𝑠3,𝑆2,2=240𝑠2+280𝑠4,𝑆4,4=140𝑠4,𝑅𝑧=24𝑎21𝑒26𝑠1𝑠2,𝐶𝑘𝑙=𝑅31𝑠28𝑎3s1𝑒2𝐶𝑘𝑙,𝑆𝑝𝑞=𝑅41𝑠264𝑎4s1𝑒2𝑆𝑝𝑞,𝑍1𝑒=2𝜂2𝑧,𝑍21=2𝜂2𝑧,𝑍3𝑒=6𝜂2𝑧,𝐶1,1=𝑒24𝜂4𝐶1,1,𝐶1,1=𝑒1+22𝜂4𝐶1,1,𝐶1,3=𝑒24𝜂4𝐶3,3,𝐶2,1=𝑒2𝜂4𝐶1,1,𝐶2,3=𝑒2𝜂4𝐶3,3,𝐶3,1=𝑒2𝜂124𝐶1,1,𝐶3,3=13𝑒1+22𝜂4𝐶3,3,𝐶4,3=𝑒4𝜂4𝐶3,3,𝐶5,3=𝑒2𝜂204𝐶3,3,𝑆1,2=𝑒38𝜂6𝑆2,2,𝑆1,2=3𝑒2+3𝑒38𝜂6𝑆2,2,𝑆1,4𝑒=38𝜂6𝑆4,4,𝑆2,21=21+3𝑒22𝜂6𝑆2,2,𝑆2,4=3𝑒28𝜂6𝑆4,4,𝑆3,21=33𝑒2+3𝑒38𝜂6𝑆2,2,𝑆3,41=33𝑒2+3𝑒38𝜂6𝑆4,4,𝑆4,2=3𝑒2𝜂166𝑆2,2,𝑆4,41=41+3𝑒22𝜂6𝑆4,4,𝑆5,2𝑒=3𝜂406𝑆2,2,𝑆5,41=53𝑒2+3𝑒38𝜂6𝑆4,4,𝑆6,4=3𝑒2𝜂246𝑆4,4,𝑆7,4𝑒=3𝜂566𝑆4,4,(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 changes. We derived explicit expressions for the errors in the missile range due to the out-of-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.

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