Dose, not duration
Counting minutes is convenient and physically meaningless. What changes when you model accumulation as an integral instead.
"Twenty minutes" is not a quantity of exposure. It is a quantity of time during which some exposure happened. The two are only interchangeable when the intensity is constant, and in almost every real system worth modelling, it is not.
This distinction sounds pedantic until you look at the numbers. Under an ultraviolet index of 9, a minute delivers nine times the erythemally weighted energy of a minute under an index of 1. A recommendation expressed in minutes is therefore wrong by up to an order of magnitude depending on the hour, the latitude and the season — while appearing perfectly precise.
Accumulation is an integral
The quantity that actually drives the biological response is the integral of intensity over the exposure window:
D = ∫ₜ₀^t¹ k · I(τ) · f(τ) dτ
Written this way, several things that were previously separate features of the product collapse into parameters of one model. Protection is a factor inside the integrand, not a multiplier applied at the end. Ground reflectivity is another. A cloud passing overhead is a change in I(τ), handled automatically. The recommendation is no longer a number from a table — it is the solution of D(t) = D* for t, where D* is the threshold for this specific person.
That solution is found by stepping the real hourly forecast forward in short increments and stopping when a limit is reached. Because the stopping condition is explicit, the product can also report which limit stopped it — a benefit target, a risk threshold, or simply the sun going down. Users are told not only how long, but why.
Units are the cheapest error detector available
Once accumulation is an integral, everything has dimensions, and dimensions catch mistakes early. Erythemal dose has an established unit — the standard erythema dose, one hundred joules per square metre — and personal thresholds are published in it. Working in real units means our parameters are directly comparable with the dermatological literature instead of being tuning constants with no external referent.
If a constant in your model has no unit, you cannot tell whether it is calibrated or merely convenient.
Where else this applies
The pattern is not specific to ultraviolet light. Any system where a stimulus varies over time and the response depends on accumulated exposure has the same structure, and the same failure mode when it is modelled in units of time:
- Noise exposure — an equal-energy trade between level and duration is the entire basis of occupational limits.
- Pharmacokinetics — area under the concentration curve, not hours since the dose.
- Fatigue and wear — cumulative damage under a varying load, not machine run-time.
- Training load — intensity-weighted volume, which is why athletes stopped counting hours decades ago.
- Cost and rate limits — tokens consumed under a varying price, not requests made.
In each case the same substitution is available: replace the count with the integral, put the modifying factors inside it, and let the threshold be the thing you solve for. The model gets simpler, not more complicated, because the special cases that used to need their own rules become consequences of one equation.