Computation of the Different Errors in the Ballistic Missiles Range

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).

(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,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/𝑣_𝑐)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, 𝑣_𝑐 is the corresponding missile circular speed at this point, and 𝜇=𝐺(𝑚₁+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 1(a) and 1(b), we have [15]cosΨ2=−cos𝑓^bo=1−𝑄^bocos²𝜙_bo1+𝑄_bo𝑄_bo−2cos²𝜙_bosin2𝜙_bo,(4.1)+Ψ2=2−𝑄𝑄^bo_bosinΨ2,(4.2) where 𝜙_bo is the flight path angle, 𝑓_bo the true anomaly, and 𝑄_bo 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 𝑟_bo, 𝑣_bo, and 𝜙_bo. 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 𝜙_bo 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 𝑄_bo. This is obvious from (4.2), where if(1)𝑄_bo<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)𝑄_bo=1, one trajectory is circular, and for Ψ<180^∘, both high and low trajectories are possible (𝜙_bo=0 for low), while high trajectory only is possible for Ψ>180^∘, and low trajectory skims Earth (𝜙_bo=0 for high),(3)𝑄_bo>1, (4.2) yields one positive and one negative value for 𝜙_bo 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)=𝑄_bo/(2−𝑄_bo), and the flight path angle is 𝜙_bo=(1/4)(180^∘−Ψ).

5. In-Orbit-Plane Changes

The variations in the parameters {𝜙_bo,𝑟_bo,𝑣_bo} 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𝜕𝜙_bo=𝜕Ψ2sinΨ+2𝜙_bocsc2𝜙_bo−1,𝜕𝑟_bo=4𝜇𝑟²_bo𝑣²_bosin²Ψ2csc2𝜙^bo𝜕Ψ,𝜕𝑣_bo=8𝜇𝑟_bo𝑣³_bosin²Ψ2csc2𝜙^bo.(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 Δ_𝑎Ψ as follows:Δ_𝑎Ψ=𝜕Ψ𝜕𝑟_bo𝜕𝑟_bo𝜕𝑎Δ𝑎+𝜕𝜕𝑣Ψ_bo𝜕𝑣_bo𝜕𝑎Δ𝑎.(5.2) The required derivatives are given by𝜕𝑟_bo𝜕𝑎=^∞_𝑛=0𝜕𝑣_bo(−1)^𝑛𝑒^𝑛−𝑒^𝑛+2cos^𝑛𝑓,𝜕𝑎=−21𝜇𝑎³1+𝑒²+2𝑒cos𝑓^1/21−𝑒²^−1/2.(5.3) The involved products are 𝜕Ψ𝜕𝑟_bo𝜕𝑟_bo𝜕𝑎=𝑟4²_bo𝜇𝑣²_bosin²Ψ2csc2𝜙^bo∞_𝑛=0𝜕Ψ(−1)^𝑛𝑒^𝑛−𝑒^𝑛+2cos^𝑛𝑓,𝜕𝑣_bo𝜕𝑣_bo𝜕𝑎=−21𝜇³𝑎³8𝑟_bo𝑣³_bosin²Ψ2csc2𝜙^bo1+𝑒²+2𝑒cos𝑓^1/21−𝑒²^−1/2.(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 (𝑓_bo) and (2𝜋−𝑓_bo). Substitution of the obtained expression into (5.2) yields Δ_𝑎Ψ=4𝜇𝑟²_bo𝑣²_bosin²Ψ2csc2𝜙^bo∞_𝑛=0(−1)^𝑛𝑒^𝑛−𝑒^𝑛+2cos^𝑛𝑓_bo−𝜇³𝑎³8𝑟_bo𝑣³_bosin²Ψ2csc2𝜙^bo1+𝑒²+2𝑒cos𝑓_bo^1/21−𝑒²^−1/2×𝐽₂₅⎫⎪⎬⎪⎭_2𝑖=1_𝑗=−2𝛼^{𝑖𝑗}sin𝑖𝑓_bosin(𝑗𝜔)−𝐽₃₇_3𝑘=1_𝑙=−3+𝐽₄₉𝛽^{𝑘𝑙}sin𝑘𝑓_bocos(𝑙𝜔)₄_𝑚=1_𝑛=−4𝛾^{𝑚𝑛}sin𝑚𝑓_bosin(𝑛𝜔),(5.5) where nonvanishing coefficients are given by 𝛼^𝑖𝑗=𝑅²2𝑎1−𝑒²𝛼^𝑖𝑗,𝛽^𝑘𝑙=𝑅³4𝑎²1−𝑒²𝛽^𝑘𝑙,𝛾^𝑚𝑛=𝑅⁴32𝑎³1−𝑒²𝛾^𝑚𝑛𝛼_1,0,=−32−4𝑒+6𝑒𝑠²,𝛼_1,2=−32𝑒𝑠²𝛼_3,2,𝛼_2,2=−6𝑠²,=152𝑒𝑠²,𝛽_0,1=1236𝑒𝑠+45𝑒𝑠³,𝛽_2,3=52e𝑠³𝛽_2,1,=12−60𝑒𝑠+75𝑒𝑠³,𝛽_1,1=12−24𝑠+30𝑠³,𝛽_3,3𝛽_4,3=−15𝑠³,=−352𝑒𝑠³,𝛾_1,0=−5248𝑒−240𝑒𝑠²+210𝑒𝑠⁴,𝛾_5,4=−3152𝑒𝑠⁴𝛾_2,2,=−5296𝑠²−112𝑠⁴,𝛾_3,2=−52168𝑒𝑠²−196𝑒𝑠⁴,𝛾_3,4=352𝑒𝑠⁴𝛾_4,4,=−140𝑠⁴,𝛾_1,2=−52−72𝑒𝑠²−84𝑒𝑠⁴,𝛽^1,−3=𝜂⁶𝑒³8𝛽^2,3𝛼_1,−2,=𝜂⁴𝑒²4𝛼^1,2,𝛼^1,0=−𝜂⁴1+𝑒4²𝛼^1,0,𝛼^2,0=−𝜂⁴𝑒2𝛼^1,0𝛼_1,2,=𝜂⁴−1+𝑒2²𝛼^1,2−𝑒𝛼^2,2−𝑒²4𝛼^3,2,𝛼^3,0=−𝜂⁴𝑒²12𝛼′^1,0𝛼_2,2,=−𝜂⁴2𝑒𝛼^1,2+121+𝑒2²𝛼^2,2+𝑒2𝛼^3,2,𝛼^5,2=−𝜂⁴𝑒²20𝛼^3,2𝛼_3,2,=−𝜂⁴1𝑒2²𝛼^1,2+𝑒3𝛼^2,2+131+𝑒2²𝛼^3,2,𝛽^3,−1=𝜂⁶𝑒³24𝛽^0,1𝛼_4,2,=−𝜂⁴1𝑒6²𝛼^2,2+𝑒4𝛼^3,2,𝛽^5,1=𝜂⁶𝑒³40𝛽^2,1𝛽_1,−1,=𝜂⁶32𝑒+38𝑒³𝛽^0,1+3𝑒²4𝛽^1,1+𝑒³8𝛽^2,1,𝛽^7,3=𝜂⁶𝑒³56𝛽^4,3𝛽_1,1,=𝜂⁶32𝑒+38𝑒³𝛽^0,1+1+3𝑒²2𝛽^1,1+3𝑒2+38𝑒³𝛽^2,1,𝛾^1,−4=𝜂⁸𝑒⁴16𝛾^3,4𝛽_1,3,=𝜂⁶32𝑒+38𝑒³𝛽^2,3+3𝑒²4𝛽^3,3+𝑒³8𝛽^4,3,𝛾^3,−2=𝜂⁸𝑒⁴48𝛾^1,2𝛽_2,−1,=𝜂⁶38𝑒²𝛽^0,1+𝑒³16𝛽^1,1,𝛾^4,0=−𝜂⁸𝑒³8𝛾^1,0𝛽_2,1,=𝜂⁶38𝑒²𝛽^0,1+1232𝑒+38𝑒³𝛽^1,1+121+32𝑒²𝛽^2,1,𝛾^5,0=−𝜂⁸𝑒⁴80𝛾^1,0𝛽_2,3,=𝜂⁶121+32𝑒²𝛽^2,3+1232𝑒+38𝑒³𝛽^3,3+3𝑒²8𝛽^4,3,𝛾^7,2=−𝜂⁸𝑒⁴112𝛾^3,2𝛽_3,1,=𝜂⁶2𝑒4³𝛽^0,1+3𝑒²12𝛽^1,1+1332𝑒+38𝑒³𝛽^2,1,𝛾^9,4=−𝜂⁸𝑒⁴144𝛾^5,4𝛽_3,3,=𝜂⁶3132𝑒+38𝑒³𝛽^2,3+131+32𝑒²𝛽^3,3+1𝛽_4,1332𝑒+38𝑒³𝛽^4,3,=𝜂⁶3𝑒2³𝛽^1,1+3𝑒²𝛽_4,316𝛽^2,1,=𝜂⁶31𝑒6²𝛽^2,3+1432𝑒+38𝑒³𝛽^3,3+1𝛽_6,341+32𝑒²𝛽^4,3,=𝜂⁶4𝑒8³𝛽^3,3+3𝑒²𝛽_5,324𝛽^4,3,=𝜂⁶4𝑒0³𝛽^2,3+3𝑒²20𝛽^3,3+1𝛾_1,−2532𝑒+38𝑒³𝛽^4,3,=𝜂⁸32𝑒²+𝑒4⁴𝛾^1,2+𝑒³2𝛾^2,2+𝑒⁴𝛾_1,016𝛾^3,2,=𝜂⁸32𝑒²+𝑒4⁴−1+3𝑒²+3𝑒⁴8𝛾^1,0𝛾_1,2,=−𝜂⁸1+3𝑒²+3𝑒⁴8𝛾^1,2+2𝑒+3𝑒³2𝛾^2,2+3𝑒²𝛾_2,−22+𝑒4⁴𝛾^3,2,=𝜂⁸𝑒4³𝛾^1,2+𝑒⁴𝛾_1,432𝛾^2,2,=−𝜂⁸32𝑒²+𝑒4⁴𝛾^3,4+𝑒³2𝛾^4,4+𝑒⁴16𝛾^5,4,𝛾^2,0=𝜂⁸𝑒4³−212𝑒+32𝑒³𝛾^1,0𝛾_2,2,=−𝜂⁸212𝑒+32𝑒³𝛾^1,2+121+3𝑒²+3𝑒⁴8𝛾^2,2+1𝛾_2,422𝑒+32𝑒³𝛾^3,2,=−𝜂⁸212𝑒+32𝑒³𝛾^3,4+1232𝑒²+𝑒4⁴𝛾^4,4+𝑒³𝛾_3,04𝛾^5,4,=𝜂⁸4𝑒8⁴−1332𝑒²+𝑒4⁴𝛾^1,0𝛾_3,2,=−𝜂⁸1323𝑒²+𝑒⁴4𝛾^1,2+132𝑒+32𝑒³𝛾^2,2+131+3𝑒²+3𝑒⁴𝛾_3,48𝛾^3,2,=−𝜂⁸131+3𝑒²+3𝑒⁴8𝛾^3,4+132𝑒+32𝑒³𝛾^4,4+1𝛾_4,2332𝑒²+𝑒4⁴𝛾^5,4,=−𝜂⁸𝑒8³𝛾^1,2+1432𝑒²+𝑒4⁴𝛾^2,2+1𝛾_4,442𝑒+32𝑒³𝛾^3,2,=−𝜂⁸412𝑒+32𝑒³𝛾^3,4+141+3𝑒²+3𝑒⁴8𝛾^4,4+1𝛾_5,242𝑒+32𝑒³𝛾^5,4,=−𝜂⁸8𝑒0⁴𝛾^1,2+𝑒³10𝛾^2,2+1𝛾_5,4532𝑒²+𝑒4⁴𝛾^3,2,=−𝜂⁸1532𝑒²+𝑒4⁴𝛾^3,4+152𝑒+32𝑒³𝛾^4,4+151+3𝑒²+3𝑒⁴𝛾_6,48𝛾^5,4,=−𝜂⁸1𝑒2³𝛾^3,4+1632𝑒²+𝑒4⁴𝛾^4,4+1𝛾_6,262𝑒+32𝑒³𝛾^5,4,=−𝜂⁸9𝑒6⁴𝛾^2,2+𝑒³𝛾_7,412𝛾^3,2,=−𝜂⁸1𝑒1⁴2𝛾^3,4+𝑒³14𝛾^4,4+1732𝑒²+𝑒4⁴𝛾^5,4𝛾^8,4=−𝜂⁸1𝑒2⁴8𝛾^4,4+𝑒³16𝛾^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 Δ_𝑒Ψ as follows:Δ_𝑒Ψ=𝜕Ψ𝜕𝜙_bo𝜕𝜙_bo𝜕𝑒Δ𝑒+𝜕𝜕𝑟Ψ_bo𝜕𝑟_bo𝜕𝑒Δ𝑒+𝜕𝜕𝑣Ψ_bo𝜕𝑣_bo𝜕𝑒Δ𝑒.(5.7) The required derivatives are given by𝜕𝜙_bo𝜕𝑒=sin𝑓1+𝑒²𝜕𝑟_bo+2𝑒cos𝑓⁻¹,𝜕𝑒=𝑎^∞_𝑛=0𝜕𝑣_bo(−1)^𝑛𝑛𝑒^𝑛−1−(𝑛+2)𝑒^𝑛+1cos^𝑛𝑓,𝜕𝑒=𝜇𝑎1+𝑒²+2𝑒cos𝑓^1/21−𝑒²^−1/2.(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𝜕𝜙_bo𝜕𝜙_bo𝜕𝑒=2sin𝑓1+𝑒²𝜕Ψ+2𝑒cos𝑓⁻¹sinΨ+2𝜙_bocsc2𝜙_bo−1,𝜕𝑟_bo𝜕𝑟_bo𝜕𝑒=𝑟4²_bo𝑎𝑣𝜇²_bosin²Ψ2csc2𝜙^bo∞_𝑛=0𝜕Ψ(−1)^𝑛𝑛𝑒^𝑛−1−(𝑛+2)𝑒^𝑛+1cos^𝑛𝑓,𝜕𝑣_bo𝜕𝑣_bo𝜕𝑒=𝑟_bo8𝑣𝜇³_bo𝑎^3/21/21+𝑒²+2𝑒cos𝑓^1/21−𝑒²^−1/2sin²Ψ2csc2𝜙^bo.(5.9) Using the Lagrange planetary equations, we computed Δ𝑒 (retaining the zonal harmonics up to 𝐽₄). The integration is performed between (𝑓_bo) and (2𝜋−𝑓_bo). Substitution of the obtained expression into (5.7) yields Δ_𝑒Ψ=⎧⎪⎨⎪⎩+2si4n𝑎𝑓𝜇1+𝑒²+2𝑒cos𝑓_bo⁻¹sinΨ+2𝜙_bocsc2𝜙_bo−1𝑟²_bo𝑣²_bosin²Ψ2csc2𝜙^bo∞_𝑛=0(−1)^𝑛𝑛𝑒^𝑛−1−(𝑛+2)𝑒^𝑛+1cos^𝑛𝑓_bo+16𝑟_bo𝑣³_bo𝜇³𝑎1+𝑒²+2𝑒cos𝑓_bo^1/21−𝑒²^−1/2sin²Ψ2csc2𝜙^bo×𝐽₂₅⎫⎪⎬⎪⎭_2𝑖=1_𝑗=−2−𝐽₃𝛿_𝑖𝑗cos𝑖𝑓_bosin(𝑗𝜔)2𝜋𝑋¹+⁷₃_𝑚=1_𝑛=−3+𝐽₄𝜎_𝑚𝑛sin𝑚𝑓_bocos(𝑛𝜔)2𝜋𝑋²+⁹_4𝑘=1_𝑙=−42𝑄_𝑘𝑙sin𝑘𝑓_bosin(𝑙𝜔),(5.10) where𝑋₁=𝜂⁴48𝑠−60𝑠³1+𝑒²2cos𝜔,𝑋²=𝜂⁴720𝑠²−840𝑠⁴𝑒sin2𝜔,(5.11) where nonvanishing coefficients are given by 𝛿_1,0=−3𝑅²−8+12𝑠²16𝑎²1−𝑒²,𝛿^2,0=−3𝑅²𝑒−4+6𝑠²16𝑎²1−𝑒²,𝛿^0,2=−3𝑅²𝑒𝑠²𝛿_1,216𝑎²1−𝑒²,=3𝑅²𝑠²8𝑎²1−𝑒²,𝛿^2,2=9𝑅²𝑒𝑠²4𝑎²1−𝑒²,𝛿^3,2=21𝑅²𝑠²𝛿_4,28𝑎²1−𝑒²,=15𝑅²𝑒𝑠²16𝑎²1−𝑒²,𝜎^1,−1=𝑅³𝑒36𝑠−45𝑠³32𝑎³1−𝑒²,𝜎^0,1=𝑅³48𝑠−60𝑠³𝜎_1,132𝑎³1−𝑒²,=𝑅³𝑒−72𝑠+90𝑠³32𝑎³1−𝑒²,𝜎^2,1=𝑅³−144𝑠+180𝑠³32𝑎³1−𝑒²,𝜎^3,1=𝑅³𝑒−60𝑠+75𝑠³𝜎_1,332𝑎³1−𝑒²,=5𝑅³𝑒𝑠³32𝑎³1−𝑒²,𝜎^2,3=−5𝑅³𝑠³8𝑎³1−𝑒²,𝜎^3,3=−45𝑅³𝑒𝑠³𝜎_4,316𝑎³1−𝑒²,=−25𝑅³𝑠³8𝑎³1−𝑒²,𝜎^5,3=−35𝑅³𝑒𝑠³𝑄_1,032𝑎³1−𝑒²,=2𝑒𝑄^2,0=−5𝑅⁴96−480𝑠²+420𝑠⁴256𝑎⁴1−𝑒²,𝑄^0,2=−5𝑅⁴𝑒−72𝑠²+84𝑠⁴𝑄_1,2256𝑎⁴1−𝑒²,=−5𝑅⁴−48𝑠²+56𝑠⁴256𝑎⁴1−𝑒²,𝑄^2,2=−5𝑅⁴𝑒288𝑠²−336𝑠⁴256𝑎⁴1−𝑒²,𝑄^3,2=−5𝑅⁴432𝑠²−504𝑠⁴𝑄_4,2256𝑎⁴1−𝑒²,=−5𝑅⁴𝑒168𝑠²−196𝑠⁴256𝑎⁴1−𝑒²,𝑄^2,4=35𝑅⁴𝑒𝑠⁴256𝑎⁴1−𝑒²,𝑄^3,4=−105𝑅⁴𝑠⁴𝑄_4,4128𝑎⁴1−𝑒²,=−210𝑅⁴𝑒𝑠⁴64𝑎⁴1−𝑒²,𝑄^5,4=−445𝑅⁴𝑠⁴128𝑎⁴1−𝑒²,𝑄^6,4=−315𝑅⁴𝑒𝑠⁴256𝑎⁴1−𝑒²,𝛿_1,−2=𝜂²𝑒2𝛿^0,2,𝛿_1,0=𝜂²−𝛿^1,0−𝑒2𝛿^2,0,𝛿_1,2=𝜂²−𝛿^1,2−𝑒2𝛿^2,2−𝑒2𝛿^0,2,𝛿_2,0=𝜂²−𝑒4𝛿^1,0−12𝛿^2,0,𝛿_2,2=𝜂²−𝑒4𝛿^1,2−12𝛿^2,2−𝑒4𝛿^3,2,𝛿_3,0=−𝜂²𝑒6𝛿^2,0,𝛿_5,2=−𝜂²𝑒10𝛿^4,2,𝛿_3,2=𝜂²−𝑒6𝛿^2,2−13𝛿^3,2−𝑒6𝛿^4,2,𝛿_4,2=𝜂²−𝑒8𝛿^3,2−14𝛿^4,2,𝜎_1,−1=𝜂⁴𝑒𝜎^0,1+1+𝑒²2𝜎^1,−1+𝑒²4𝜎^1,1,𝜎_1,−3=𝜂⁴𝑒²4𝜎^1,3,𝜎_1,1=𝜂⁴𝑒𝜎^0,1+𝑒²4𝜎^1,−1+1+𝑒²2𝜎^1,1+𝑒𝜎_2,1+𝑒²4𝜎^3,1,𝜎_2,1=𝜂⁴𝑒8²𝜎^0,1+𝑒2𝜎^1,1+121+𝑒2²𝜎^2,1+𝑒2𝜎^3,1,𝜎_2,−1=𝜂⁴𝑒8²𝜎^0,1+𝑒2𝜎^1,−1,𝜎_2,3=𝜂⁴2𝑒𝜎^1,3+121+𝑒2²𝜎^2,3+𝑒2𝜎^3,3+𝑒²8𝜎^4,3,𝜎_3,−1=𝜂⁴𝑒²12𝜎^1,−1,𝜎_3,1=𝜂⁴1𝑒2²𝜎^1,1+𝑒3𝜎^2,1+131+𝑒2²𝜎^3,1,𝜎_4,1=𝜂⁴1𝑒6²𝜎^2,1+𝑒4𝜎^3,1,𝜎_4,3=𝜂⁴1𝑒6²𝜎^2,3+𝑒4𝜎^3,3+141+𝑒2²𝜎^4,3+𝑒4𝜎^5,3,𝜎_1,3=𝜂⁴1+𝑒2²𝜎^1,3+𝑒𝜎_2,3+𝑒²4𝜎^3,3,𝜎_3,3=𝜂⁴1𝑒2²𝜎^1,3+𝑒3𝜎^2,3+131+𝑒2²𝜎^3,3+𝑒3𝜎^4,3+𝑒²12𝜎^5,3,𝜎_5,3=𝜂⁴2𝑒0²𝜎^3,3+𝑒5𝜎^4,3+151+𝑒2²𝜎^5,3,𝜎_5,1=𝜂⁴𝑒²20𝜎^3,1,𝑄_1,−2=𝜂⁶32𝑒+38𝑒³𝑄^0,2+3𝑒²4𝑄^1,2+𝑒²8𝑄^2,2,𝜎_6,3=𝜂⁴2𝑒4²𝜎^4,3+𝑒6𝜎^5,3,𝑄_1,0=𝜂⁶−1+34𝑒²𝑄^1,0−3𝑒2+38𝑒³𝑄^2,0−𝑒³8𝑄^2,0,𝜎_7,3=𝜂⁴𝑒²28𝜎^5,3,𝑄_1,2=𝜂⁶−32𝑒+38𝑒³𝑄^0,2−1+3𝑒²2𝑄^1,2−3𝑒2+38𝑒³𝑄^2,2−3𝑒²4𝑄^3,2−𝑒³8𝑄^4,2,𝑄_2,0=𝜂⁶−34𝑒+𝑒8³𝑄^1,0−121+32𝑒²𝑄^2,0,𝑄_2,−2=𝜂⁶38𝑒²𝑄^0,2+𝑒³16𝑄^1,2,𝑄_1,4=𝜂⁶−32𝑒+38𝑒³𝑄^2,4−3𝑒²4𝑄^3,4−𝑒³8𝑄^4,4,𝑄_7,2=−𝜂⁶𝑒³56𝑄^4,2,𝑄_2,2=𝜂⁶−38𝑒²𝑄^0,2−1232𝑒+38𝑒³𝑄^1,2+𝑄_3,2−121+32𝑒²𝑄^2,2−3𝑒²8𝑄^4,2,𝑄_2,4=𝜂⁶−211+34𝑒²𝑄^2,4−1232𝑒+38𝑒³𝑄^3,4−3𝑒²8𝑄^4,4−𝑒³16𝑄^5,4,𝑄_3,2=𝜂⁶−2𝑒4³𝑄^0,2−3𝑒²12𝑄^1,2−1332𝑒+38𝑒³𝑄^2,2−131+32𝑒²𝑄^3,2−1332𝑒+38𝑒³𝑄^4,2,𝑄_3,0=𝜂⁶−31𝑒2²𝑄^1,0−1332𝑒+38𝑒³𝑄^2,0,𝑄_3,−2=𝜂⁶𝑒³24𝑄^0,2,𝑄_4,0=𝜂⁶−3𝑒2³𝑄^1,0−3𝑒²16𝑄^2,0,𝑄_5,0=−𝜂⁶𝑒³40𝑄^2,0,𝑄_6,2=𝜂⁶−4𝑒8³𝑄^3,2−3𝑒²24𝑄^4,2,𝑄_1,−4=𝜂⁶𝑒³8𝑄^2,4,𝑄_4,2=𝜂⁶−3𝑒2³𝑄^1,2−3𝑒²16𝑄^2,2−1432𝑒+38𝑒³𝑄^3,2−141+32𝑒²𝑄^4,2,𝑄_4,4=𝜂⁶−31𝑒6²𝑄^2,4−1432𝑒+38𝑒³𝑄^3,4+𝑄_5,4−141+32𝑒²𝑄^4,4−3𝑒²16𝑄^5,4,𝑄_5,4=𝜂⁶−4𝑒0³𝑄^2,4−3𝑒²20𝑄^3,4−1532𝑒+38𝑒³𝑄^4,4+𝑄_6,4−151+32𝑒²𝑄^5,4,𝑄_6,4=𝜂⁶−4𝑒8³𝑄^3,4−3𝑒²24𝑄^4,4−1632𝑒+38𝑒³𝑄^5,4−161+32𝑒²𝑄^6,4,𝑄_7,4=𝜂⁶−5𝑒6³𝑄^4,4−3𝑒²28𝑄^5,4−1732𝑒+38𝑒³𝑄^6,4,𝑄_9,4=−𝜂⁶𝑒³72𝑄^6,4,𝑄_5,2=𝜂⁶−4𝑒0³𝑄^2,2−3𝑒²20𝑄^3,2−1532𝑒+38𝑒³𝑄^4,2,𝑄_8,4=𝜂⁶−6𝑒4³𝑄^5,4−3𝑒²32𝑄^6,4.(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,  Δ_Ω𝜓_×, at impact occurs, the value of which is obtained, in a similar manner as the previous subsection, by applying the law of cosines. Hence,cosΔ_Ω𝜓_×=sin²Ψ+cos²ΨcosΔΩ.(6.1) Since Δ_Ω𝜓_×, ΔΩ are small angles, then we haveΔ_Ω𝜓_×≈ΔΩcosΨ.(6.2) Using the Lagrange planetary equations, we computed ΔΩ (retaining the zonal harmonics up to 𝐽₄). The integration is performed between (𝑓_bo) and (2𝜋−𝑓_bo). Substitution of the obtained expression into (6.2) yields Δ_Ω𝜓_×≈𝐽₂2𝜋𝑋⁸+³_2𝑏=1_𝑐=0+𝐽₃−𝜌_𝑏𝑐(−1)^𝑏+1sin𝑏𝑓_bo+𝑐𝜔2𝜋𝑋⁹+⁵_3𝛼=1_𝛽=−1+𝐽₄𝑞_𝛼𝛽(−1)^𝛼−1cos𝛼𝑓_bo+𝛽𝜔2𝜋𝑋¹⁰+⁷_4𝑖=1_𝑗=−2−𝑝_𝑖𝑗(−1)^𝛼+1sin𝑖𝑓_bo+𝑗𝜔cosΨ,(6.3) where𝑋₈=𝜂²𝜌^0,0,𝑋₉=𝜂⁴𝑒𝑞^1,1sin𝜔,𝑋₁₀=𝜂⁶1+32𝑒²𝑃^0,0+3𝑒²4𝑃^2,2cos𝜔,(6.4) where nonvanishing coefficients are given by 𝜌^𝑏𝑐=𝑅²4𝑎²𝑠1−𝑒²𝜌^𝑏𝑐,𝑞^𝛼𝛽=𝑅³8𝑎³𝑠1−𝑒²𝑞^𝛼𝛽,𝑃^𝑖𝑗=𝑅⁴64𝑎⁴𝑠1−𝑒²𝑃^𝑖𝑗𝜌_0,0,𝑞=−𝜌_2,2=6𝑠√1−𝑠²,𝑃_4,4=140𝑠³√1−𝑠²,𝑞_1,1=√1−𝑠²45𝑠²−12_3,3,=−15𝑠²√1−𝑠²,𝑃_0,0=𝑠√1−𝑠²−240+420𝑠²,𝑃_2,2=𝑠√1−𝑠²240−560𝑠²,𝜌_1,0=𝜂²𝑒𝜌^0,0,𝜌_1,2=𝜂²𝑒2𝜌^2,2,𝜌_2,2=𝜂²12𝜌^2,2,𝜌_3,2=𝜂²𝑒6𝜌^2,2,𝑞_1,−1=𝜂⁴𝑒4²𝑞^1,1,𝑞_1,1=𝜂⁴−1+𝑒2²𝑞^1,1,𝑞_1,3=𝜂⁴−𝑒4²𝑞^3,3,𝑞_2,1=𝜂⁴−𝑒2𝑞^1,1,𝑞_2,3=𝜂⁴−𝑒2𝑞^3,3,𝑞_3,1=𝜂⁴−1𝑒2²𝑞^1,1,𝑞_3,3=𝜂⁴−131+𝑒2²𝑞^3,3,𝑞_4,3=𝜂⁴−𝑒4𝑞^3,3,𝑞_5,3=𝜂⁴−2𝑒0²𝑞^3,3,𝑃_1,−2=𝜂⁶𝑒8³𝑃^2,2,𝑃_1,0=𝜂⁶232𝑒+38𝑒³𝑃^0,0,𝑃_1,2=𝜂⁶32𝑒+38𝑒³𝑃^2,2,𝑃_1,4=𝜂⁶𝑒8³𝑃^4,4,𝑃_2,0=𝜂⁶68𝑒²𝑃^0,0,𝑃_2,2=𝜂⁶121+32𝑒²𝑃^2,2,𝑃_2,4=𝜂⁶38𝑒²𝑃^4,4,𝑃_3,0=𝜂⁶1𝑒2³𝑃^0,0,𝑃_3,2=𝜂⁶1332𝑒+38𝑒³𝑃^2,2,𝑃_3,4=𝜂⁶1332𝑒+38𝑒³𝑃^4,4,𝑃_4,2=𝜂⁶31𝑒6²𝑃^2,2,𝑃_4,4=𝜂⁶141+32𝑒²𝑃^4,4,𝑃_5,2=𝜂⁶4𝑒0³𝑃^2,2,𝑃_5,4=𝜂⁶1532𝑒+38𝑒³𝑃^4,4,𝑃_6,4=𝜂⁶32𝑒4²𝑃^4,4,𝑃_7,4=𝜂⁶5𝑒6³𝑃^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,  Δ_𝑖𝜓_×, at impact occurs, the value of which is obtained, in a similar manner as the previous subsection, by applying the law of cosines. Hence,cosΔ_𝑖𝜓_×=cos²Ψ+sin²ΨcosΔ𝑖.(6.6) Since Δ_𝑖𝜓_×,Δ𝑖 are small angles, then we haveΔ_𝑖𝜓_×≈Δ𝑖sinΨ.(6.7) Using the Lagrange planetary equations, we computed Δ𝑖 (retaining the zonal harmonics up to 𝐽₄). The integration is performed between (𝑓_bo) and (2𝜋−𝑓_bo). Substitution of the obtained expression into (6.7) yields Δ_𝑖𝜓_×≈𝐽₂₃_𝑖=1+𝐽₃𝑍_𝑖(−1)^𝑖−1cos𝑖𝑓_bo+2𝜔2𝜋𝑋³+⁵_3𝑘=1_𝑙=−1+2𝜋𝑋₉−𝐶_𝑘𝑙(−1)^𝑘+1sin𝑘𝑓_bo+𝑙𝜔+⁵_3𝛼=1_𝛽=−1+𝐽₄𝑞_𝛼𝛽(−1)^𝛼−1cos𝛼𝑓_bo+𝛽𝜔2𝜋𝑋⁴+⁷_4𝑝=1_𝑞=−2𝑆_𝑝𝑞(−1)^𝑝−1cos𝑝𝑓_bo+𝑞𝜔sinΨ,(6.8) where𝑋₃=𝜂⁴𝑒𝐶^11cos𝜔,𝑋₄=𝜂⁶3𝑒²4𝑆^22sin2𝜔,(6.9) where nonvanishing coefficients are given by 𝐶_1,1=15𝑠³−12𝑠,𝐶_3,3=−15𝑠³,𝑆_2,2=−240𝑠²+280𝑠⁴,𝑧=𝑅₂𝑆_4,4=−140𝑠⁴,4𝑎²1−𝑒²6𝑠√1−𝑠²,𝐶^𝑘𝑙=𝑅³√1−𝑠²8𝑎³s1−𝑒²𝐶_𝑘𝑙,𝑆^𝑝𝑞=𝑅⁴√1−𝑠²64𝑎⁴s1−𝑒²𝑍₁𝑆_𝑝𝑞,=−𝑒2𝜂²𝑧,𝑍₂=−12𝜂²𝑧,𝑍₃=−𝑒6𝜂²𝑧,𝐶_1,−1=𝑒²4𝜂⁴𝐶′_1,1,𝐶_1,1=1+𝑒²2𝜂⁴𝐶′_1,1,𝐶_1,3=𝑒²4𝜂⁴𝐶′_3,3,𝐶_2,1=𝑒2𝜂⁴𝐶′_1,1,𝐶_2,3=𝑒2𝜂⁴𝐶′_3,3,𝐶_3,1=𝑒²12𝜂⁴𝐶′_1,1,𝐶_3,3=131+𝑒2²𝜂⁴𝐶′_3,3,𝐶_4,3=𝑒4𝜂⁴𝐶′_3,3,𝐶_5,3=𝑒²20𝜂⁴𝐶′_3,3,𝑆_1,−2=𝑒³8𝜂⁶𝑆′_2,2,𝑆_1,2=−3𝑒2+38𝑒³𝜂⁶𝑆′_2,2,𝑆_1,4=−𝑒³8𝜂⁶𝑆′_4,4,𝑆_2,2=−121+32𝑒²𝜂⁶𝑆′_2,2,𝑆_2,4=−3𝑒²8𝜂⁶𝑆′_4,4,𝑆_3,2=−1332𝑒+38𝑒³𝜂⁶𝑆′_2,2,𝑆_3,4=−1332𝑒+38𝑒³𝜂⁶𝑆^4,4,𝑆_4,2=−3𝑒²16𝜂⁶𝑆^2,2,𝑆_4,4=−141+32𝑒²𝜂⁶𝑆′_4,4,𝑆_5,2=−𝑒³40𝜂⁶𝑆′_2,2,𝑆_5,4=−1532𝑒+38𝑒³𝜂⁶𝑆′_4,4,𝑆_6,4=−3𝑒²24𝜂⁶𝑆′_4,4,𝑆_7,4=−𝑒³56𝜂⁶𝑆′_4,4,(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.