Demo — Threat Laydown

Demo — Threat Laydown#

Put one SAM site on a map and a flying wing in the air around it — the other L26 outlines are a chip away — and draw what the site can actually do: where its radars see the aircraft, where the Earth hides it, and where its missile can reach it. Every one of those boundaries moves when the aircraft changes heading or altitude, or when it turns away after a launch. This demo draws them all on one plan view, then lays several sites together so you can find the gap.

The model#

Detection. Each radar sees the aircraft through the R⁴ law against Project 3’s references — a VHF early-warning radar (EWR) that detects 2 m² at 90 km and an X-band target-tracking radar (TTR) that detects 0.03 m² at 45 km — with the aircraft’s RCS read from its signature pattern at the aspect the radar sees. For an aircraft on heading \(\psi\) at bearing \(\varphi\) from the site (both clockwise from North),

\[ \theta = \mathrm{wrap}\left(\varphi + 180^\circ - \psi\right), \qquad R(\varphi) = R_\text{ref}\left[\frac{\sigma_b\big(\theta(\varphi)\big)}{\sigma_\text{ref}}\right]^{1/4}. \]

Horizon. Detection also needs a line of sight over the 4/3 Earth, with the radar’s mast height \(h_r\) and the aircraft’s altitude \(h_t\) in meters:

\[ d = 4.12\left(\sqrt{h_r} + \sqrt{h_t}\right)\ \text{km}, \qquad R_\text{eff}(\varphi) = \min\big(R(\varphi),\, d\big). \]

Engagement. The missile is a constant-average-speed teaching model: speed \(v_m\), maximum time of flight \(t_\text{max}\), against an aircraft flying at \(v_t\). With \(\alpha\) the angle from the aircraft’s velocity to the site — the same angle as the aspect \(\theta\) —

\[ R_\text{MEZ} = (v_m + v_t)\,t_\text{max}, \qquad R_\text{LEZ} = (v_m - v_t)\,t_\text{max}, \qquad R(\alpha) \approx (v_m + v_t\cos\alpha)\,t_\text{max}. \]

The lethal zone is inside the TTR’s effective contour, inside that heading envelope, and outside a fixed 4 km minimum range. The pattern is interpolated linearly in dB between its 5° rows, exactly as the companion script does it.

Aircraft. The four chips are the L26 Planform Alignment outlines, and the default is the Flying wing. Their X-band rows come from L26’s edge-lobe model at 10 GHz: each edge throws a lobe along its normal and a weaker one behind, over the −25 dBsm speckle floor. Those lobes are under 0.25° wide, far finer than the map can draw, so each 5° row holds the mean RCS over its sector — a spike keeps its energy but is spread across the row it falls in, which is why the spikes read about +4 to +7 dBsm here, not the +16 to +22 dBsm tips on the L26 compass at its default 3 GHz. At VHF (\(\lambda = 2\) m) these outlines are in the resonance region, where shaping loses its grip and the edge-lobe model does not apply, so every aircraft uses the Project 3 VHF column. The reading and the companion script work the same flying wing, so their numbers match the tiles.

Interactive demo#

Open in full screen

Walkthrough#

The map is North-up. The zoom chips above it set the outer range ring — 50, 100 or 200 km, with rings every 25 km (every 50 km at 200) — and the presets reset it. The band chips pick which radar’s contour is drawn and read out, and the aircraft chips under them swap the outline; the heading slider turns the aircraft (the compass in the corner shows it); altitude and mast height set the horizon; the three missile sliders set the MEZ, the LEZ and the orange heading envelope. The nine tiles read the nose-on approach — the aircraft flying straight at the site — except the last one, which reads the farthest the radar sees the aircraft at any aspect (or, laid out, the gap).

  1. Read the default. X TTR, heading 090°, the flying wing at 12,000 m, zoom 50 km. Four copies of the aircraft sit on the contour, all on the same heading: nose-on to the west, a beam to the north and the south, the tail to the east. The tiles read nose 24.4 km, beam 25.5 km, tail 24.3 km — and the ninth, Max \(R_\text{det}\), 164.3 km at ±40° off the nose. The contour is an X: a ragged core about 25 km in radius, which the dashed nose-on ring (the circle a chart would draw) follows, and four thin arms that run straight off the map. Click 200 km to see them whole: they reach 164.3 km at ±40° and ±140°.

  2. Turn the aircraft. Swing the heading slider. The X rotates with the aircraft; the dashed ring does not move. Nothing about the site changed — only which face of the aircraft each bearing sees.

  3. Change bands. Click VHF EWR, then 100 km — at 50 km the whole contour is off the map. It reads 89.9 km nose-on, 100.9 km at the beam and 89.9 km at the tail, with a maximum of 100.9 km at ±90°: at VHF the pattern barely changes with aspect, so the contour is nearly the circle the X-band contour is not. Every aircraft chip gives the same ring, because every aircraft uses the Project 3 VHF column.

  4. Come down. Click Low flyer (50 m). The horizon falls to 42.2 km — still outside the wing’s 24.4 km nose-on range, so Which binds stays the R⁴ law — but the shaded seen region now stops at the horizon: all four arms are cut back to 42.2 km. Nose-on, the horizon would bind only below \(h^* = (24.44/4.12 - \sqrt{10})^2 \approx 7.7\) m on a 10 m mast — under the slider’s 15 m floor. The stealthier the aircraft, the lower the horizon must come. Now pick the 4th-gen fighter: its 66.2 km nose-on range puts the horizon in charge (Which binds Horizon, seen at 42.2 km). Climb it slowly and find where the tile flips back: still the horizon at 150 m (63.5 km), the R⁴ law at 200 m (71.3 km) — the crossing is at \(h^* = (66.2/4.12 - \sqrt{10})^2 \approx 167\) m.

  5. Work the missile. Back at Single site with the flying wing, the MEZ is 50 km and the LEZ 30 km, and the First shot tile reads TTR track at 24.4 km: the radar sees the wing later than the missile could reach it, so the track sets the first shot. Drag the missile sliders down (\(v_m\) = 0.60 km/s, \(t_\text{max}\) = 20 s, \(v_t\) = 0.35 km/s) and the MEZ shrinks to 19 km — now the missile, not the track, limits the shot. The orange envelope always runs from the MEZ on the side you come from to the LEZ on the side you leave.

  6. Lay out a defense. Click Layered (gap) (it zooms to 100 km): a VHF EWR whose coverage spans the first line, two SAMs 100 km apart, and a third behind them. At heading 090° the two front SAMs see the wing’s beams as it passes between them — 25.5 km, inside the 40 km missile envelope abeam — so each lethal zone reaches only 25.5 km toward the other and a 49.0 km gap opens: the green aircraft on the dotted track flies through it, seen by the EWR but not shot. Swing the heading: at 000° the gap is 51.3 km (one SAM sees the nose, the other the tail), and at 045° it is 45.4 km. Now try 040°: each SAM looks straight down a spike — SAM-1 a leading edge, SAM-2 a trailing edge — both lethal zones fill the envelope, and the gap closes to 20.0 km. Five degrees of heading cost 25 km of gap.

  7. Change airframes. Back at Single site (X TTR, 090°, 12,000 m), click through the other three L26 chips; their spikes leave the 50 km map, so click 200 km to see them whole. The 4th-gen fighter reads nose 66.2 km, beam 113.7 km, tail 93.3 km, and its many edge angles throw spikes in more than a dozen directions — the longest 135.1 km at ±80° off the nose, 129.1 km at ±40° and 122.1 km at ±175°. Its 66.2 km nose-on track also lies outside the 50 km MEZ, so First shot now reads MEZ. The 5th-gen fighter is quiet nose-on (24.4 km) and tail-on (24.3 km), with nothing past 34.2 km within ±30° of the nose, but its aligned edges pile their energy into a few spikes: 153.8 km at ±40°, 145.4 km at ±145°, 137.8 km at ±80°, and a 122.9 km beam. The Hopeless Diamond reads the wing’s 24.4, 25.5 and 24.3 km, with four spikes of 153.5 km at ±70° and ±110°, square to its four edges.

Key observations#

  • Heading shapes the ring. The detection contour is the R⁴ law at the aspect each bearing produces. For the flying wing against an X-band TTR it is an X, not a circle; a route planned on the nose-on ring alone can be caught by a spike far outside it.

  • The outline sets the spikes. Swap the airframe and the ring changes shape: it reaches out wherever an edge normal of the L26 outline points at the radar. The flying wing and the diamond line their edges up and put everything into four spikes; the 4th-gen fighter’s many edge angles scatter them all round. The nose-on ring hides all of it — for the wing it reads 24.4 km, under a sixth of the 164.3 km its spikes reach.

  • Altitude picks the bound. Detection is \(\min(R, d)\). Up high the horizon is hundreds of kilometers out and the signature decides; down low the Earth decides. The stealthier the aircraft, the lower the altitude at which the horizon starts to help.

  • Seen is not shot. The missile’s reach runs from the MEZ against a closing target to the LEZ against a fleeing one. Between them, turning away escapes; inside the LEZ it does not. The lethal zone is the overlap of the track and the envelope — and at altitude the flying wing’s track, not the missile, often sets the first shot.

  • Gaps move. Every overlap, gap and seam in a laydown depends on heading, because both the contour and the envelope do. That is why L34 plans routes on a graph whose edges know which way the aircraft is flying.

Source#

MATLAB bundle · L32_ThreatRings.m↓

The companion script reads FlyingWing_SignaturePattern.csv (bundled beside it) — the flying wing’s X-band rows from this demo beside the Project 3 VHF column — interpolates the pattern in dB, and prints each radar’s nose, beam, tail, minimum and maximum detection ranges. It then sweeps the wing’s altitude against the VHF EWR nose-on on a 10 m mast, printing the horizon and which bound binds at each and the crossing altitude, then the X TTR’s crossing for contrast; computes the MEZ, the LEZ and the beam envelope for the default missile; and draws the plan view (the X-band contour, the nose-on ring, the envelope, and the MEZ and LEZ rings). It carries no layered laydown — that part of the demo is geometry you can reproduce by placing the same single-site model at several sites.

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