Demo — Thermal Crossover

Demo — Thermal Crossover#

An LWIR sensor does not measure how hot the target is; it measures how much hotter or colder the target is than whatever sits behind it. That contrast swings across the day and passes through zero twice — once after sunrise and once after sunset. This demo runs a simple diurnal model so you can watch the airframe’s LWIR contrast climb, cross zero, and flip sign, and read off the two crossover times.

The contrast that vanishes#

The LWIR signature lives on thermal contrast between the skin and the background:

\[ \Delta T = T_{\text{skin}} - T_{\text{bg}}. \]

The model gives each surface a sinusoidal diurnal swing about a daily mean of 15 °C. The terrain background swings hard (amplitude 18 °C, peaking mid-afternoon and bottoming out before dawn), while the aircraft skin swings less and lags a touch (amplitude 6 °C):

\[ T_{\text{bg}} = 15 + 18\sin\!\frac{2\pi(t-8)}{24}, \qquad T_{\text{skin}} = 15 + 6\sin\!\frac{2\pi(t-9)}{24}. \]

Because the terrain amplitude is the larger of the two, it overtakes the skin near midday (target reads cold) and falls below it at night (target reads warm). The sign changes of \(\Delta T\) are the crossover times.

Interactive demo#

Open in full screen

Walkthrough#

  1. Read the two temperature curves. The high-amplitude curve is the terrain background; the gentler, slightly lagging one is the aircraft skin. Both ride the same daily mean.

  2. Watch them cross. Twice a day the two curves intersect. Between the crossings the terrain is warmer; outside them the skin is warmer.

  3. Drop to the lower plot. It sits directly under the temperature curves, so you can read both at once: \(\Delta T = T_{\text{skin}} - T_{\text{bg}}\) is negative through midday (target colder than terrain) and positive overnight (target warmer). The dashed line marks \(\Delta T = 0\).

  4. Find the crossover markers. The black filled circles sit where \(\Delta T\) changes sign — about 7.5 h and 19.5 h, roughly one hour after sunrise and one hour after sunset. The two tiles on the right read them off as clock times.

  5. Drive the four sliders. Background swing, skin swing, skin phase lag, and the daily mean — with three presets (baseline, desert day, overcast) as starting points. Widen the background swing and the crossings barely move; add phase lag and they walk across the morning. Then drag the daily mean from one end to the other and watch the crossover times and the peak contrast not move at all: the mean sits in both curves and cancels out of the difference. That is “contrast, not temperature” made visible in one drag.

  6. Reason about the ingress. An LWIR sensor loses the skin at those two instants. Ask what a stern-aspect MWIR seeker sees at the same moment — and note it does not care.

  7. Switch to the “IR signature by aspect” tab. Notional stacked bars break the signature into the lesson’s five components — hot parts, exhaust plume, aero-heated skin, reflected sunlight, reflected sky/earthshine — and re-weight them for the band you pick. In the MWIR the hot parts and plume dominate and the stern is by far the worst aspect; in the LWIR the aero-heated skin carries the signature at every aspect, and reflected sunlight all but vanishes. That skin term is the one the crossover kills.

Key observations#

  • Contrast, not temperature, is what the LWIR sensor sees. A warm target on warmer terrain reads as a cold silhouette.

  • Crossover happens twice daily, about an hour after sunrise and after sunset, because the larger-amplitude terrain curve overtakes and then falls below the skin.

  • Midday is negative, night is positive. Sun-baked terrain beats the skin by day; terrain radiating to cold sky drops below the skin at night.

  • Crossover zeroes only the LWIR skin term. The plume and hot parts still shout in the MWIR — an ingress timed at crossover hides the skin, not the tailpipe. You do not disappear in every band at once.

  • The five components are three emitters and two reflectors. The second tab keeps them in that order: hot parts, plume, and aero-heated skin radiate the aircraft’s own heat; reflected sunlight and reflected sky/earthshine return borrowed light.

  • The daily mean cancels out. Crossover times and peak contrast depend only on the two swing amplitudes and the phase lag between them — never on how warm the day is. Slide the mean and nothing moves.

  • Set the skin swing equal to the background swing with zero lag and the two curves become one: \(\Delta T\) is zero at every hour, so there is no crossover time and the tiles read “—”. That is the degenerate case, not a bug — and it is worth asking why a real airframe never sits there. (Because the skin is convectively tied to the air, so it swings less than sun-baked ground and lags it.)

Source#

MATLAB bundle · L22_ThermalCrossover.m↓

The companion script builds a 24-hour time axis at one-minute resolution, evaluates the two sinusoidal temperature curves, forms \(\Delta T\), and finds the sign changes to locate the two crossover times. It prints them near 7.5 h and 19.5 h and plots both the temperature curves and the contrast curve with the crossover instants marked.

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