# Demo — IR Detection Range
A passive IR sensor detects a target the instant the radiated heat arriving at its aperture clears its own noise floor. This demo puts that condition on a single axis so you can watch two things at once: how far a sensor sees a bright source versus a dim one, and how the exponential atmosphere quietly steals most of the advantage that raw intensity seems to promise.
## The concept
Source intensity $J$ (watts per steradian) spreads over the inverse square of range and is attenuated by atmospheric transmittance $\tau(R)=\exp(-\alpha R)$. The irradiance reaching the sensor is therefore
$$
E(R) \;=\; \frac{J\,\exp(-\alpha R)}{R^{2}},
$$
and detection occurs wherever $E(R)$ stays at or above the noise-equivalent irradiance (NEI). Watch the units: with $J$ in W/sr and NEI in W/m², the $R^2$ in the denominator is in meters while $\alpha R$ uses kilometers (the source script converts via $R_m = 10^3\,R_{km}$). The crossover defines the maximum range, which scales as
$$
R_{\max} \;\propto\; \sqrt{\dfrac{J\,\tau}{\text{NEI}}}.
$$
Contrast the square-root-of-$J$ dependence with radar's $R_{\max}\propto\sigma^{1/4}$: cutting radar cross-section starves the radar, but it leaves $J$ — and therefore IR range — untouched.
## Interactive demo
Open in full screen
## Walkthrough
1. **Read the curve.** The falling curve is $E(R)=J\exp(-\alpha R)/R^2$ on a log-irradiance axis; the horizontal dashed line is the sensor's NEI. Where the curve crosses NEI is $R_{\max}$, marked with a dot.
2. **Slide $J$.** Push source intensity up and the whole curve lifts, moving the crossing outward — but notice how little the crossing moves once the exponential term dominates.
3. **Slide $\alpha$.** Raise the extinction coefficient (haze, humidity) and the curve bends down faster; a clear day ($\alpha\approx 0.15$ /km) and a hazy one give very different reach.
4. **Slide NEI.** Lowering the noise floor (better cooling, better detector) drops the dashed line and pushes $R_{\max}$ out.
5. **Load the two-target preset.** Stern ($J=1000$ W/sr, plume plus hot parts) versus beam ($J=50$ W/sr, skin only). The vacuum rule promises $\sqrt{20}\approx 4.5\times$ more range for the stern; the atmosphere delivers only about $1.4\times$.
## Key observations
- **The atmosphere is the referee.** In vacuum, range would scale as $\sqrt{J}$ and the stern would out-reach the beam by $4.5\times$. With $\tau(R)=\exp(-\alpha R)$ in play, stern reaches about 54 km and beam about 39 km — only $1.4\times$.
- **Diminishing returns on intensity.** Because the crushing term is exponential, every doubling of $J$ buys less added range than the last. Modest signature trimming buys modest range.
- **This is the counter-LO argument, and the setup for L24.** Radar range dies with $\sigma^{1/4}$; IR range lives on $J$, which stealth does not touch. To hide from IR you must attack $J$ by orders of magnitude, component by component — plume, hot parts, skin — which is exactly the signature-suppression fight the next lesson opens.
## Source
MATLAB bundle · L23_IRDetectionRange.m↓
The companion script sweeps range from 0.5 to 80 km, computes $E(R)=J\exp(-\alpha R)/R^2$ for the stern ($J=1000$ W/sr) and beam ($J=50$ W/sr) sources against a clear-air $\alpha=0.15$ /km and NEI $=10^{-10}$ W/m², plots both on a log-irradiance axis with the NEI floor, and prints the two crossing ranges so the class can compare the $4.5\times$ vacuum promise with the $1.4\times$ the atmosphere actually delivers.