Demo — Threat Laydown#
Put one SAM site on a map and the B-21 in the air around it, 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 B-21’s RCS read from the Project 3 signature pattern at the aspect the radar sees. For an aircraft on heading \(\psi\) at bearing \(\varphi\) from the site (both clockwise from North),
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:
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 \(\theta\) —
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.
Interactive demo#
Walkthrough#
The map is North-up with range rings every 25 km. The band chips pick which radar’s contour is drawn and read out; 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.
Read the default. X TTR, heading 090°, the B-21 at 12,000 m. 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 45.6 km, beam 91.0 km, tail 60.8 km, and a minimum of 21.6 km at ±55° off the nose. The contour is a cross, and the dashed nose-on ring — the circle a chart would draw — matches it only along one line.
Turn the aircraft. Swing the heading slider. The cross rotates with the aircraft; the dashed ring does not move. Nothing about the site changed — only which face of the aircraft each bearing sees.
Change bands. Click VHF EWR: 89.9 km nose-on, 100.9 km at the beam. At VHF the pattern barely changes with aspect, so the contour is nearly the circle the X-band contour is not.
Come down. Click Low flyer (50 m). The horizon falls to 42.2 km — inside the 45.6 km nose-on range — and the Which binds tile turns to Horizon: the Earth now sets first detection, and the red lethal zone is clipped by the horizon ring. Raise the mast and watch the horizon back off; at 40 m it reaches 55.2 km. Then climb slowly and find where the tile flips back to the R⁴ law (between 50 and 75 m for the TTR on a 10 m mast — the crossing is at 62.5 m).
Work the missile. Back at the default, the MEZ is 50 km and the LEZ 30 km, and the First shot tile reads TTR track at 45.6 km: the radar sees the B-21 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.
Lay out a defense. Click Layered (gap): 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 aircraft’s beams as it passes between them; the missile envelope abeam (40 km) is shorter than the TTR’s beam spike, so each lethal zone reaches 40 km toward the other and a 20 km gap opens — the green aircraft on the dotted track flies through it, seen by the EWR but not shot. Now swing the heading: at 000° the gap is 24.4 km, and at 045°, where both sites see a quiet quarter, it opens to 56.8 km.
Key observations#
Heading shapes the ring. The detection contour is the R⁴ law at the aspect each bearing produces. For the B-21 against an X-band TTR it is a cross, not a circle; a route planned on the nose-on ring alone can be caught by a beam spike far outside it.
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 B-21’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 B21_SignaturePattern.csv (bundled beside it), interpolates the pattern in dB, and prints each radar’s nose, beam, tail and minimum detection ranges — the numbers in the first three tiles. It then sweeps the aircraft’s altitude against a 10 m mast, printing the horizon and which bound binds at each, and the crossing altitude; computes the MEZ, the LEZ and the beam envelope for the default missile; and draws the plan view of step 1 (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.