Because the circle is one special case: one radar's detection range at one aspect (usually nose-on), at an altitude where the Earth does not block the line of sight, against a missile shot at a target flying straight at the site. Your heading changes the aspect the radar sees at every bearing, your altitude brings in the horizon, and what you do after launch changes the missile's reach.
\(\theta = \mathrm{wrap}(\varphi + 180^\circ - \psi)\), measured clockwise from the nose. From the aircraft the radar lies at bearing \(\varphi + 180^\circ\); subtracting the heading measures that from the nose. \(\theta = 0^\circ\) is nose-on, \(180^\circ\) tail-on.
\(R(\varphi) = R_\text{ref}\,[\sigma_b(\theta(\varphi))/\sigma_\text{ref}]^{1/4}\): the R⁴ law evaluated, bearing by bearing, at the aspect that bearing produces, with \(\sigma_b\) read from the signature pattern in the radar's band.
The heading is the same everywhere; only the bearing from the radar changes, and the bearing alone decides which face the radar sees — nose-on upstream of the site on your track, the beam abeam of it, the tail past it.
A cross: 45.6 km nose-on, needles to 91.0 km at the two beams, 60.8 km at the tail, and pinched to 21.6 km in the quarters between them. At X-band shaping works, and the notional pattern gathers its energy into a small nose spike (−15 dBsm) and large beam spikes (−3 dBsm) over a −28 dBsm floor.
At VHF the airframe's features are comparable to the wavelength (the resonance region), so shaping cannot steer the echo: the pattern moves only about 2 dB (+3 to +5 dBsm) around the compass, and the fourth root turns 2 dB into about a 12% change in range (89.9 to 100.9 km).
It is honest only along the line straight at the site. Passing abeam, you present your beam and the contour spikes far outside the ring; in the quarters the real contour sits well inside it. A route planned to skirt the nose-on ring can be detected much earlier than planned.
\(d = 4.12\left(\sqrt{h_r} + \sqrt{h_t}\right)\) km with \(h_r\) (radar mast) and \(h_t\) (aircraft altitude) in meters, using the 4/3-Earth refraction allowance. L11's one-ended \(4.12\sqrt{h}\) is one of the two terms.
Detection needs both an echo strong enough and a line of sight, so the effective range is the smaller: \(R_\text{eff}(\varphi) = \min(R(\varphi), d)\). The two bounds never add.
Where \(d = R\): \(h^\ast = (R/4.12 - \sqrt{h_r})^2\). Below it the Earth sets detection; above it the signature does. For the TTR on a 10 m mast seeing the B-21 nose-on at 45.6 km, \(h^\ast \approx 62.5\) m; against the VHF EWR's 89.9 km it is about 350 m.
A stealthier signature shortens the R⁴ ranges, which lowers the crossing altitude \(h^\ast\): the quieter you are, the lower you must fly before the horizon adds anything. Terrain-following buys most against the long-range, low-band layers and little against a TTR that already sees you late.
MEZ: the reach against a target flying straight at the site, \(R_\text{MEZ} = (v_m + v_t)\,t_\text{max}\). LEZ (the lethal or no-escape zone): the reach against a target that turns directly away, \(R_\text{LEZ} = (v_m - v_t)\,t_\text{max}\). It is a teaching model — real envelopes come from fly-out models.
Between the rings the turn escapes: the missile must chase at \(v_m - v_t\) and runs out of flight time. Inside the LEZ even a fleeing target is caught — no escape. The turn costs progress toward the target and shows the radar your tail.
\(R(\alpha) \approx (v_m + v_t\cos\alpha)\,t_\text{max}\), the radial approximation, with \(\alpha\) the angle between your velocity and the direction to the site — the same angle as the aspect \(\theta\). It gives the MEZ at \(0^\circ\), the LEZ at \(180^\circ\), and \(v_m t_\text{max}\) abeam.
Inside the tracking radar's effective contour, inside the missile envelope, and outside the minimum range \(R_\text{min}\). Nose-on against the TTR the track starts at 45.6 km inside a 50 km MEZ, so the track — not the missile — sets the first shot.
Overlap: a point inside two or more lethal zones. Gap: inside early-warning coverage but outside every lethal zone — seen, not shot. Seam: the boundary where one site's coverage hands off to the next, where tracks drop and cues arrive late. All three are heading-dependent.
Both halves of each lethal zone depend on heading: the TTR contour through the aspect's \(\sigma\), and the missile envelope through \(\cos\alpha\). In the lesson's laydown the gap is 20 km at 090°, 24 km at 000°, and 57 km at 045°, where both sites see a quiet quarter of the X-band pattern.
In Joint doctrine (JP 3-01; the DoD Dictionary) it is airspace in which engagement responsibility rests with surface-to-air missiles — an airspace-control measure. Threat analysis and mission planning borrow the letters for a site's kinematic reach, which is the sense used here; air-to-air literature calls the inner zone the no-escape zone.