By George Lewis and Theodore Postol

While the recent National Academy of Sciences (NAS) Report, “Making Sense of Ballistic Missile Defense,” doesn’t cite any specific numbers for radar ranges, their figure 5-8, shown below, shows ranges of about 1,500 km for the current AN/TPY-2 X-band radars and 3,000 km for their proposed stacked TPY-2 radars (which they refer to as GBXs).  However, we believe that these ranges are much too large, particularly for discrimination, which is what the proposed GBX radars are for.  To provide a basis for discussion, here we provide our own estimates for the ranges of these radars, with all our parameters and assumptions spelled out.

Figure 5-8 from NAS Report, showing ranges of TPY-2 radars in Turkey and Japan and of stacked TPY-2 (GBX) radars in Britain, Greenland, North Dakota and Cape Cod. This figure shows ranges of about 1,500 km for the TPY-2 radars and 3,000 km for the stacked TPY-2 radars.

Below we first estimate the range of a TPY-2 radar.  We assume perfect coherent integration for a total of 20 ms of radar emitting time, which requires a dwell time of 100 ms.  Specifically, we assume perfect coherent integration of 20 one millisecond pulses at a pulse repetition frequency of 200 Hz.  (However, any other combination of pulses lengths and pulse repetition frequencies consistent with the radar’s average power and a dwell time of 100 ms will give the same result.)     All the other parameters we use are either relatively well established numbers or are estimates we believe to be favorable to the radar.

The maximum range and signal-to-noise ratio are related by the radar equation:[1]

R_max = maximum radar range (m), ρ = antenna aperture efficiency = 0.8, P_av = radar average power (W), A = antenna area (m²), G = antenna gain, n = number of pulses integrated, E_i(n) = integration efficiency, σ = radar cross section of target (m²), k = Boltzmann’s constant (1.38×10^-23 J/K), T₀ = 290 K, F_N = receiver noise figure, B = receiver bandwidth (Hz), τ = pulse length (s), f_P = pulse repetition frequency (Hz), (S/N)₁ = signal-to noise ratio required as if detection were based only on a single pulse, L_f = fluctuation loss (for a Swerling target model), and L_S = system loss.  The terms for propagation effects (such as Earth reflections or refraction) and atmospheric attenuation have been neglected.[2]

We assume perfect integration, neglect fluctuation losses, and make the standard simplifying assumption that (Bτ) = 1.  Then this equation becomes:

We next assess each parameter in this equation:

P_av: The radar average power P_av is taken to be 81,000 W.  This is based on the number of transmit/receive modules times the average power per module, which is taken to 3.2 W.  The number of modules is known to be 25,344.[3]  There is no publicly available official figure for the module average power (that we know of).  However, the modules used in the TPY-2 antenna appear to be third generation modules, where the first generation modules had peak powers of about 6 W and average powers of about 1.2 W, and the second generation modules had peak powers of about 10 W and average powers of about 2 W.  We estimate that the third generation modules have peak and average powers 60% higher than the second generation, or peak powers of about 16 W and average powers of about 3.2 W.[4]

ρ: We take the antenna aperture efficiency to be 0.8.  This may well be high, as the American Physical Society’s Boost Phase Study used a value of about 0.65 for the THAAD (TPY-2) radar.[5]

A: The antenna area is known to be 9.2 m².[6]

G: The antenna gain is estimated to be 103,000, using the relationship G = ρ(4πA/λ²), where the wavelength λ = 0.03 m.

n: We assume perfect integration of n = 20 one-millisecond pulses.

σ: As will be discussed in subsequent post, we take the target to be a conical warhead with a radar cross section at X-band of 0.01 m². 

F_N: We estimate the noise figure F_N to be 1.4, which is equivalent to a system temperature of about 400 K. For comparison, the 2003 American Physical Society Boost-Phase Study assumed a system temperature of 500 K (for a beam which grazed the ground).  An earlier MITRE viewgraph showed a system temperature of 600 K (600K/290K = 2.1) for an X-Band Ground-Based Radar (GBR).[7]   

f_P: We assume a module duty factor of 0.2, which is consistent with the module peak and average powers discussed above.[8] With the 1 millisecond pulses length assumed above, this gives f_P = 200 Hz.  Thus for our baseline case which integrates 20 pulses, the dwell time will be 100 milliseconds.

S/N: For our baseline case, we consider two values of S/N.  First a low value of S/N = 20 (which we refer to as the “detection” value) and a higher figure of S/N = 100 (which we refer to as the “discrimination” value).[9]

L_S: We estimate L_S = 8 dB = 6.3.[10]

Then for our TPY-2 baseline cases we get:

            R = 870 km     detection (S/N = 20)

            R = 580 km     discrimination (S/N = 100)

As discussed in our previous post (September 20, 2012), the stacked TPY-2 (or GBX) has an average power, antenna aperture and gain all greater than that of the TPY-2 by a factor of two.  Thus the ranges for the NAS’s proposed GBX radars should be greater than those of the TPY-2 radars by a factor (2 x 2 x 2)^0.25 = 1.68.

Thus for our GBX baseline cases we get:

            R = 1,460 km   detection (S/N = 20)

            R = 970 km      discrimination (S/N = 100)

Given that the primary purpose of the NAS’s proposed GBX radars is discrimination, we find that the GBX radars fall short by almost two orders of magnitude (3,00/970)⁴ = 91 in terms of the power-aperture-gain product needed to get the 3,000 km range shown in the NAS Report.