Defeating Missile Defense Radars in the Endo-Atmosphere

S. Frankel and A. Mihalyi

December, 1998

I. Review

Since the invention of radar, simple decoy systems for defeating its ability to recognize and follow targets have been proposed. In 1997, in a series of articles in the I.E.E.E. Forum, the properties of such decoys were reviewed and debated. The simplest countermeasure method is the deployment of chaff, small pieces of aluminum or metalized coatings, which produce too huge a number of reflections for the radar to handle. Modern radar technology has made great strides, and if quick frequency changing methods were to be deployed at the transmitter, the relative frequency dependences of chaff and warhead reflections might, in principal, be used to distinguish the two. Also, on re-entry into the atmosphere the warhead would separate from the chaff. Another well known method to disguise warheads consists of surrounding a warhead with a large radar radiation reflecting balloon and deploying a number of empty balloons along with this warhead. These are simple devices that are expected to be effective above the atmosphere but might fail in the re-entry phase, the light balloons showing much more deceleration inside the atmosphere than the heavy warhead.

Many decades have passed since the introduction of inert decoys. Advances in electronic and microwave circuitry now make possible a whole class of electronic decoys. New light, very wide-band antennas and radar amplifiers are off-the-shelf items and allow for more sophisticated active decoys that cannot be distinguished from the warheads. One can combine such antennas and amplifiers into transponders which return the received radar signal to the radar, overpowering the small reflected signals. Located on both the small decoys and the larger warhead, it was shown some time ago that radar systems, such as those to be incorporated in THAAD, the Theater High Altitude Area Defense system, can be deceived. These large return transponder signals are called ``deceptive jammers'' since they do not saturate the radar but re-radiate a confusing signal immediately at the received radar frequency, anywhere in the range 2-20 gigaherz. These signals obscure the different lengths of decoy and warhead. They can also be provided with signals that can defeat autocorrelation methods which are used to identify objects by storing and processing consecutive received radar pulses from the target.

This present work deals mainly with the question of the ability of such decoys to be distinguished from warheads after they reenter the atmosphere. It is important that the decoys deployed with the warhead be not too heavy so that they adversely affect the maximum range of the warheads launched by the missiles. They can be physically small since the radars cannot distinguish their lengths from the lengths of the warheads: the transponder signals destroy that capability. But if, as a result, they are too light, their deceleration in the atmosphere will be large and distinguishable by the fine time resolution of modern radars which can accurately measure the deceleration.

The effect of the atmosphere is both to cause a deceleration along the missile path and an acceleration normal to the drag force, called the ``lift''. The drag force is defined as the force along the missile trajectory. The deceleration is defined as the (drag) force on the missile divided by the missile mass. To closely match the deceleration and the lift of decoy and warhead turns out to be possible, as this paper will show. This is can be accomplished because the length, the opening angle, the mean density, and the radius of the rounded nose of the cone-like decoy can be chosen to make decoy and warhead indistinguishable after penetration into the atmosphere. Since the two need not be distinguished below 50 km, where the density has become high, one can use Newtonian models for the deceleration. We shall see that because it is relative decelerations and lifts that must be matched, and not the exact magnitudes, the calculations are quite simple. In many cases, analytic expressions are derivable which give basic clues to the design, although simple computer calculations are sometimes used to obtain some of the results tabulated in the appendix.

II. Basic Drag and Lift Considerations

In this section, we consider entrance into the atmosphere for a warhead axis lined up along its velocity axis, since this illustrates many of the qualitative features. (In the Appendix we give the formulas for a symmetric, conical warhead possessing a rounded-nose cone and making an ``access'' angle, $\alpha$, with the airflow direction.)

Figure 1 shows the geometry of a warhead, a solid cone with a rounded nose. The nose radius is R_N and the radius of the rear of the cone is R_C . The cone angle is $\theta_0$ and the rounded nose joins smoothly to the cone. At that point the radius of the sphere section is R_A . In this figure, l is the length of the truncated cone not the overall length. Also shown, for later discussion, is a possible ``fin'' of length L, making an angle $\delta$ with the missile axis.

However, we start with the simplest case, that of pointed conical warhead of length, l. and nose radius zero. The deceleration, D_p , can be calculated using the Newtonian model. Whether one uses elastic scattering of molecules or calculates the deceleration on the assumption that the component of momentum normal to the surface is lost, one finds that the deceleration is given by:

1) $D_p = \frac {k sin^2\theta_0 }{l \rho}$

Here $\theta_0$ is the half-angle of the cone, l is the cone length, and $\rho$ is the average density of the warhead or decoy. k is a constant that is model dependent and is of no import in what follows.

Eq. 1 gives the basic scaling variable describing the deceleration. (Corrections to it due to rounded nose warheads are examined later.)

Since we wish to have small decoys, we must match the deceleration of the warhead by making the decoy mean density higher. Since the average density of a warhead, like the Mark 21, is small, it is possible to add weight to the decoy, e.g., by using titanium inserts (with density = 19.3 ${\rm gms/cm^3}$), to compensate for the smaller decoy length.

The effects of a finite radius nose on a cone warhead, do not scale easily. The equations appear in the appendix, but some simple effects can be seen by examinimg the drag, D_n , on a hemisphere alone, i. e., l = 0. The force on the sphere is proportional to the cross sectional area, $\pi R^2$, so that D= $\vec F/m$, is $\sim \frac{ \pi R^2}{ 2/3 \rho \pi R^3}$. Therefore

2) $D_n = \frac{3k}{2\rho R_N}$

Comparing 2) with 1) we see that the nose makes a very large contribution to D compared with the rest of the cone, both because $sin^2 \theta$ is small and $l \gt\gt R_N$. Thus, using the same radius for the decoy and warhead noses increases the decoy/warhead deceleration ratio. (If one wishes, however, to use identical radii for the noses of the decoy and warhead, another possibility for matching the drags can be employed. One could increase the drag of the warhead to make it slow down and keep pace with a slowing decoy. Thus we examine the possibility of modifying the warhead dynamically, perhaps as soon as it enters the atmosphere, by deploying a thin fin, as shown by the insert in Fig. 1. These can be designed to equalize the decelerations, as shown in the appendix.)

It is also of interest to know whether the lifts, i.e. the accelerations normal to missile direction, are different for warhead and decoy. These differences are difficult for the radars to measure and lift is therefore not a sensitive tool to distinguish decoy from warhead. Nevertheless, we have examined the lift as well.

If the axis of the warhead is not identical with that of the trajectory, the drag forces and transverse forces will vary with the access angle, the angle between the missile axis and its trajectory. To ensure that this variation will not allow one to distinguish decoy from warhead we have calculated the variations with $\alpha$ in the material in the appendix. Meanwhile, it is instructive to examine a simple geometry, the drag and lift for a flat area making an angle, $\alpha$, with the trajectory axis. The momentum transfer along that axis, as shown in Fig. 2., is $mv \times (1- 2 cos\alpha) = mv \times 2 sin^2 \alpha$. while the momentum transfer normal to the trajectory axis is $mv \times sin 2\alpha$.Integrating these momentum transfers over the surfaces of the missiles modifies this dependence somewhat. To a good approximation the lift is linear with $\alpha$ while the deceleration varies slightly more rapidly with $\alpha$.

III. Illustrative Calculations

In this section we present plots of drag and lift as a function of the various geometrical parameters. Because the cone angles are so small we use the approximation $\rm sin \alpha = \rm \alpha$ in all our plots. (For example, at 8 degrees, $\rm \alpha$ = .1396 while $\rm sin \alpha$ = .1392 )

Our illustrative calculations have been carried out for a warhead with the following dimensions, which are the dimensions of the Mark 21 re-entry vehicle: $\theta$ = 8.4 degrees, nose radius = 3.6 cm, warhead length = 1.72 meters. From these dimensions and the warhead weight, the mean density of the warhead is 0.7. We have used a warhead density of 1.0 and a decoy density of 7.2 in our calculations. The formulas in the appendix allow the reader to examine other warhead parameters. Here we wish to give the most important features.

Because the missiles may enter the atmosphere making access angles relative to their path that are not zero, we have calculated the decelerations vs access angle to see if the relative decoy/warhead decelerations, D_d/D_w , varied significantly with access angle. For a spherical object there would be no access angle dependence, so, as the size of the spherical nose of a missile increases, there will be less $\alpha$ dependence.

We first illustrate, in Fig. 3, a worst case scenario: small decoy, one-tenth the length of the warhead and with the same nose radius. The deceleration of the decoy is 16 times that of the warhead at zero access angle, falling to 5 times at an access angle equal to the decoy opening angle. (The reason for the falloff of the deceleration ratio at the largest $\alpha$ is that the circular nose drag force is essentially angle independent, unlike the cone drag force which increases with $\alpha$. Thus the longer warhead increases its deceleration more than the shorter decoy at large $\alpha$.)

We next illustrate how the deceleration and lifts vary with access angle for a specific case of a small decoy, one-tenth the length of the warhead but with a nose radius of one quarter the warhead nose radius. We see again from Fig. 4 that the decelerations for both decay and warhead have essentially the same shape (For this case the deceleration of the decoy is 5 times the deceleration of the warhead at zero access angle dropping to 2.4 at the maximum access angle of 8.4 degrees.

Fig. 5 shows a plot of the lifts for the small nose case, illustrating evidence for our remark that the lift should be linear with $\alpha$.

Because the $\alpha$ dependences are not strong, it is sufficient to summarize the results by plotting only the zero access data. Therefore, shown in Fig 6 are the plots of D_d/D_w as a function of R_d/R_w for different values of the length ratios, l_d/l_w .

IV. Anti-Simulation

It may be desirable to have the warhead and decoy have the same nose radii so that the wakes will be more similar. Yet, because the drag force is dominated by the rounded nose of the light decoy, this requires using longer decoys. Another approach is to increase the warhead drag force, rather than decreasing the decoy drag force. This can be accomplished by adding a small fin at the rear of the warhead as indicated in Fig. 1. This could be a fin making, say, a 45 degree angle with the warhead axis. It could be extended after the warhead enters the exoatmosphere or just before re-entry.

As illustration, Fig 7 shows the decelerations for a 45 degree fin angle and examines the fin length for a decoy body length ratio of l_d/l_w = 0.1 . Note the excellent match at any $\alpha$ for L = 8.49 cm. For a longer length ratio of .15 the fin length drops to 4.24 cm, and for a length ratio of .2 it drops to 2.12 cm.

Other protrusions of different shapes could also be considered, such as a series of small protrusions spaced around the rear of the warhead.

V. Deployment Strategies

Even if a decoy cannot be exactly matched to the warhead at every access angle, $\alpha$, one can confuse the radars by diffusing the decoy parameters. For each warhead the decoys are arranged to have decelerations and lifts which bracket those of the warhead. The decoys are also deployed in the plane of the warhead but at a spread of angles. Around the minimum energy trajectory, (a missile angle of elevation around 45 degrees), all the decoys as well as the warhead have the same eventual target point. These different trajectories, combined with somewhat different decelerations and lifts also give a spread to the nutations of the missiles' noses as they enter the atmosphere.

VI. Masking Warhead and Decoy Wobbles

When warheads and decoys approach their targets they are rarely moving with their axes exactly along the flight path and they may be nutating around the warhead and decoy axes. Their axes may make different angles with their paths. Differences between the warhead and decoy rotational motions could be detected by the radar. If the transponder antenna patterns were completely isotropic, the signals received by the antennas would be independent of the warhead and decoy orientation and the reradiated signal would also be isotropic. Thus there would be no possibility for the radars to distinguish decoy from warhead using differences in nutation. It would not be necessary to use completely isotropic antennas if one knew enough about the level of wobble sensitivity that the defender pessessed. One disadvantage of using completely isotropic antennas is the loss in gain, but for warheads close to the receiving radar antennas, gain is no problem.

In our previous work we examined typical spiral antennas which lie in a plane and which had very large bandwidths, their antenna patterns being identical from 2 to 18 ghz. By accepting narrower frequency bands, say from 8-12 ghz, which bracket a typical 10 ghz radar frequency, one can make the antennas smaller and with a larger angular spread. Conical helix and spiral helix antennas represent other varieties that can easily be designed and manufactured.

However, one can also mask the relative wobbles electronically by using the same technique that was proposed for defeating autocorrelation methods. A simple analogue circuit could modulate the transponder signals with a frequency approximating typical nutation frequencies. In this way even missiles that do not nutate will appear to nutate so, once again, decoy and warhead will be indistinguishable.

VII. Infra-red Countermeasures

Up to now we have treated how our ``smart decoys'' can fool the tracking radars but in this brief section we comment on how the decoys and warheads could be configured to fool an infrared seeker.

The infrared emission from an object is determined by 1) its absolute temperature, 2) its emissivity, and 3) its radiating surface area. The rate at which it cools is determined by its heat capacity. Its heating on re-entry is determined by its surface materials.

Warheads can be cooled with liquid air to reduce the infrared emission. Our decoys are being slightly heated by the transponder amplifier. The designer, knowing the weight and material composition of decoy and warhead as well as the surface areas, can control the 1) surface materials, 2) the materials determining the mean density, and 3) the internal heating to match the infrared output over the missiles path. There are enough parameters to do so. Even here, the decoys would be made to have a variety of emissions, so there would be no decoy ``infra-red signature" that the defender can store and remember.

VIII. Conclusions

We have seen that the use of the small transponder equipped missiles, warhead and decoy, which are indistinguishable from their radar returns above the atmosphere, can be shaped to have the same deceleration when entering the atmosphere. The decoys can also be deployed with a variety of larger and smaller decelerations and slightly different trajectories to make them even more effective. None of the techniques used in our transponder system push any part of the technology. Engineers in any country can easily design and build them. It would be a nice exercise for engineering students to go the next step, using well established microchip circuitry in miniature computers as well as GPS circuitry to make ``intelligent'' decoys, far smarter than the ones the authors have described.

At the present writing, THAAD, the Theater High Altitude Missile Defense and PAC3, the endoatmospheric interceptor system, which have not been tested against traditional decoys, need also to be tested against the new electronic decoys that we have described. They are quite rudimentary, so a ``rogue'' nation, such as North Korea or Iraq, could easily deploy them. There is little merit in the argument that such simple decoys are too difficult to incorporate. In theater warfare the warhead range may often be larger than the distance to target so, throw weight may be not so important.

VIII. Appendices

Equations Describing Drag and Lift for Warhead and Decoy

We provide the basic equations relating to calculations for both a pointed warhead, which has simple scaling laws, and the more complicated rounded nose warhead. The basic theory is that of Newtonian mechanics. Different assumptions about the scattering of molecules from the warhead surfaces lead to different magnitudes for the drag but, fortunately, these cancel when comparing objects with different shapes. Whether one examines the case of elastic molecular scattering or the case of a moving fluid where the momentum normal to the surface is lost to the fluid, the dependence of the drag or lift on the warhead shape and the incident angle are the same although muliplying parameters are different.

A. Calculating the Momentum Change

Figure 1 shows the geometry of a warhead, a solid cone with a rounded nose. The nose radius is R_N and the radius of the rear of the cone is R_C . The cone angle is $\theta_0$ and the rounded nose joins smoothly to the cone. At that point the radius of the sphere section is R_A . Shown in the insert is a possible circular fin of length L.

We will first study zero degree access angle.

The change in momentum along the trajectory direction of a molecule reflected ``optically'' from the conical surface is.

1) $\Delta mv = mv - mv( cos 2\theta_0) = (mv) 2 sin^2\theta_0$

Since the cross sectional area of the cone is ($\pi R_C^2 - \pi R_A^2$), the total momentum transfer to the cone section is:

2) $\Delta mv = (mv)( 2 sin^2\theta_0) ( \pi R_C^2 - \pi R_A^2 )$

or

2') $\Delta mv = (mv)( 2 sin^2\theta_0) ( \pi R_C^2) [ 1 - \gamma{^2} cos^2{\theta_0} ]$

where $\gamma = R_N / R_C$

For $R_N^2 < R_c^2$ we obtain the scaling result of equation 1.

To determine the change in momentum due to reflection off the spherical nose one has to integrate over all the incident angles up to $\theta_0$.

Thus we need to calculate:

3) $\int^{\pi/2}_{\theta_0} (mv)( 2 sin^2\theta) \times 2\pi R_N cos \theta \time... ... \theta d\theta = 4\pi R_N^2 \int^{\pi/2}_{\theta_0} sin^3 \theta d sin\theta$

So the final result for the nose is:

4) $\Delta mv = (mv)( \pi R_N^2) ~~ [1 - sin^4\theta_0]$

Since for real warheads $sin^4\theta < 1$, we used this approximation in obtaining Eq. 2. of section II.

The result for the complete warhead can be rewritten in terms of the ratio, $\gamma$, as:

5) $\Delta mv = (mv) ( \pi R_C^2 ) [ 2sin^2{\theta_0} + \gamma^2( 1 - 2sin^2{\theta_0} cos^2 \theta_0 - sin^4 {\theta_0})]$

This result in this form is well known

B) Calculating the Drag Force, F, and the Deceleration, D

The force is the change in momentum per unit time, so:

6) $\vec F = d\vec p/ dt = \rho_a dA (dl/dt) v (2 sin^2\theta)$

Here dA and dl are infinitesimal regions defining the volume of the gas, $\vec p$ is the momentum, and dl/dt is the velocity of the gas relative to the cone and $\rho_a$ is the air density.

Thus the deceleration, D, is:

7) D = $\vec F /M = \rho_a / \rho_m V\times v^2 \int 2 sin^2 \theta dA = \rho_a / \rho_m V \times v^2 (2 \sin^2 \theta) (\pi R_C^2 - \pi R_N^2 cos^2\theta)$

where V is the volume of the missile and $\rho_m$ is its mean density.

Since the volume of the truncated cone is:

8) $V_c = \pi l/3 ( R_c^2 + R_c R_n cos \theta + R_n^2 cos^2\theta)$

where l is the length of the truncated conical section and the volume of the nose is:

9) $V_n = (2/3) \pi R_N^3 (1 - 3/2 sin \theta + 1/2 sin^3 \theta ) = (2/3) \pi R_N^3 f(\theta)$

The final formula for the deceleration is:

10) $D = [(\rho_a/ \rho_m) v^2]$ $\frac { 2 sin^2 \theta \pi (R_c^2 - R_n^2 cos^2 \theta ) + \pi R_n^2 ( 1 - s... ... R_c^2 + R_c R_n cos \theta + R_n^2 cos^2\theta) +(2/3) \pi R_n^3 f(\theta) }$

$\rho_a$ is the air density and $\rho_m$ is the mean missile density.

C. The Pointed Cone

Setting R_n = 0 in eq. 10, we regain the simple formula:

11) $D = [6\rho_a v^2]$ $\frac { sin^2 \theta}{ l \rho_m}$

This scaling formula tells us the main features of the design of a decoy with the same drag as the warhead. We must compensate for the smaller length of the decoy by using a larger density and a smaller cone angle. It is best to try for the largest density since reducing the cone angle of the decoy reduces the volume into which the transponder must fit.

Arbitrary Access Angle - $\alpha$

We now turn to the case of the warhead entering the atmosphere at an angle $\alpha$ relative to the warhead axis. (The rounded nose is not shown in Fig. 3.) We only consider access angles up to the cone angle $\theta_0$.In this case:

12) D = $[\rho_a v^2] \frac {[ (l^2/2)\frac{sin \theta}{cos^2 \theta} + l \frac{R_n}{... ..._c cos \theta + R_c^2 ) + 2/3 R_n^3 ( 1 - 3/2 sin \theta +1/2 sin^3 \theta )]}$

or, alternately:

12') D = $[\rho_a v^2] \frac { (l/cos \theta \times [R_c/2 + R_n] \times d(\theta ,\al... ..._c cos \theta + R_c^2 ) + 2/3 R_n^3 ( 1 - 3/2 sin \theta +1/2 sin^3 \theta )]}$

13) $d(\theta , \alpha) = \int_0^{2\pi} [ 1- ( cos \theta cos \alpha + sin \theta sin \alpha cos \phi)^2 ]^{3/2} d\phi$

It is useful to note that d is only a function of the relative angles $\theta$ and $\alpha$ and not a function of any of the other parameters of the missile. When the last term in D is negligible, D will then have the same $\alpha$ dependence for any R_n .

Some useful approximations: Setting $cos\theta$ =1 and neglecting $sin\theta$ relative to 1:

14) D = $\frac{ [(l^2/2)sin \theta + l R_n] \times d + \pi R_n^2 ]}{ \pi \rho [(l/3) (R_n^2 +R_n R_c + R_c^2 ) + 2/3 R_n^3 )]}$

Since R_n is small and less than l, we can neglect the last term in the denominator and $\pi R_n^2$ in the last numerator term. defining $R_n/R_c = \gamma$, we obtain:

15) D = $\frac{[(l^2/2)sin \theta + l R_c \gamma] }{ \pi \rho [(l/3) R_c^2 (1 + \gamma ) )]} \times d$

Since $sin\theta = tan\theta$ to a good approximation we can write $R_c = lsin\theta$ to obtain:

16) D = $\frac { 3(1 +2\gamma) }{2\pi \rho l sin\theta (1 +\gamma ) } \times d(\alpha) \cong \frac { 3(1 + \gamma)}{2\pi \rho l sin\theta } \times d(\alpha)$

We can examine $d(\alpha)$ in the limit of $\alpha = 0$ to see whether it reduces to the proper scaling function for $\alpha$ = 0. Thus 13) becomes:

17) $d = 2\pi \sin^3 \theta$

and finally

18) $D(\alpha = 0) = \frac{ 3sin^2\theta}{\rho l} \times \frac { (1 +2 \gamma) }{ (1 + \gamma )} \cong \frac{ 3sin^2\theta}{\rho l} \times (1 + \gamma)$

which is the approximate nose-modified scaling function for D

We have not been able to find any simple approximation so we must evaluate the $d(\alpha)$integral numerically.

We now turn to the formula for the lift, L.

21) L = $[\rho_a v^2]\frac { (l/cos \theta \times [R_c/2 + R_n] \times j(\alpha) }{ \pi... ..._c cos \theta + R_c^2 ) + 2/3 R_n^3 ( 1 - 3/2 sin \theta +1/2 sin^3 \theta )]}$

In this case, for all values, the lift, L, is the product of a missile shape factor and a function, $j(\alpha)$, which is only a function of $sin\theta$ and $sin\alpha$.

22) $j/\pi = 3 sin^3\theta sin^3\alpha - sin\theta sin\alpha + 3 cos^2\theta cos^2\alpha sin\theta sin\alpha$

The first term can be neglected and setting the cos terms equal to unity we have:

23) $j/\pi = 2 sin\theta sin\alpha$

Thus we see that the lift will be linear with $sin\alpha$.

Acknowledgements: The authors would like to thank the Research Foundation, the Science and Technology Wing, and the Physics Department, all of the University of Pennsylvania, for partial support of this effort.

re-entrywash printed April 7, 1999

 

Figure: Parameters of a truncated cone with rounded nose. The insert shows the parameters of a possible circular ``fin''.

 

Figure: Scattering from an Area Element. The component of momentum transfer along the missile direction and that perpendicular to the missile direction are shown. They depend differently on the power of the angular deflections. The drag force is proportional to $mv (1-cos 2\theta) = mv \times 2 sin^2 \theta$ while the lift force is proportional to $mv \times sin 2\theta$.For small angles the ratio of Drag to lift forces increases with $\theta$.

 

Figure: Equal nose radii: smallest decoy, one-tenth the length of the warhead; worst case.

 

Figure: For smaller R_d/R_w = 1/4, the deceleration is reduced to 5. at $\alpha$ = 0, and 2.4 at maximum $\alpha$.

 

Figure: Lift acceleration for the previous figure. The lift difference is smaller for the smaller nose and the decoy lift is,in this case, larger than the lift of the warhead by a ratio of 1.1.

 

Figure: Ratio of decoy to warhead deceleration as a function of decoy nose radius for various length decoys

 

Figure: Anti-Simulation. Use of fin on the warhead to match the deceleration for a small length decoy l_d/l_w = 1/3 and a non-zero nose radius.

 

Figure: Parameters for a tilted warhead of opening angle $\theta$ and access angle $\alpha$. The rounded nose is not shown.