Heat and altitude: how many passengers an aircraft cannot carry

Why heat and altitude cost passengers

Density decides everything. Wing lift and drag are directly proportional to air density, and density depends on pressure, temperature and humidity⁶. That is why pilots work not from the airfield elevation but from “density altitude” — the altitude at which the standard atmosphere would have the same air⁷. To an aircraft, a hot day at sea level is the same as a cool one in the mountains: it behaves as if it were taking off from its density altitude⁷.

Two blows to the take-off. Thin air has two effects at once: a higher lift-off speed is needed and thrust is lower, so acceleration is weaker⁷. The aircraft lifts off at the same indicated airspeed, but its true airspeed is higher — the take-off run is longer⁷. The opposite lever is weight: +10 % weight lengthens the take-off by at least 21 %⁷. So when the runway is not long enough, weight comes off.

Flat rating. A jet engine is set up to hold take-off thrust constant up to a certain temperature — the “kink” (corner) point: usually 30 °C at sea level¹, up to 45 °C for A320 engines, up to 30 °C for the A321⁸, up to 37.4 °C for the Dash 8-400⁹. Below the kink, heat hardly matters; above it, thrust falls together with the allowable weight. Hence the threshold in the calculation.

The limit is not just the runway. Certification rules require take-off performance to be determined for every weight, altitude and temperature¹⁰. The take-off must fit on the runway even with an engine failure¹⁰, and after lift-off a twin-engine aircraft on one engine must still climb at a gradient of at least 2.4 %¹⁰. The allowable weight is the lowest of all limits: runway, climb, obstacles, tyre speed, brakes¹¹. Denver's runway is 4.9 km long, yet the 737-800 runs into its tyre speed limit².

Which temperature to use. US airports are designed for the mean daily maximum of the hottest month¹¹ — we take the same temperature for the flight calculation.

How many passengers are offloaded

An A320 at maximum take-off weight: how many passengers (100 kg each with baggage) would have to be offloaded. Dot — estimated correction, line — spread of the slope; orange — where the basis is weaker (high altitude, short runway).

Denver — with the ISA+12 threshold for high airfields; in reality the weight is limited there even on a cool day (tyre speed), so the estimate is, if anything, too low.

How the limit grows with temperature

Reduction of the 737-800 allowable take-off weight, % of maximum (79 t). Dots — Coffel & Horton (2015), table 2², from Boeing charts; straight lines between the dots.

Each line starts at the threshold and then rises almost linearly. The shorter the runway, the steeper: at LaGuardia (2.1 km) the limit grows from 4.5 to 6.8 t between 31 and 33 °C. The slopes and thresholds were derived by the researcher from the paper's three points — the paper itself does not give them. The authors convert the same 4.5 t into 52 passengers — they assume 86 kg per passenger with carry-on²; the calculator uses 100 kg with checked baggage, so we get fewer passengers. In practice cargo is offloaded first².

What the Airbus and Boeing charts show

For airport planning, manufacturers publish charts showing what weight the aircraft can lift from a runway of a given length at a given elevation, at standard temperature and in the heat. We read the figures off them for the 737-800 (standard, +15, +25 and +35 °C above standard)⁴ and for the A320ceo and A320neo (standard and +15 °C)⁵. The chart below is the 737-800 on a 2.9 km runway: at sea level the weight starts to fall only beyond +25 °C above standard, i.e. around 40 °C; at 4,000 ft already beyond +15 °C, and past +25 °C the drop speeds up; at 8,000 ft the weight is limited even on a standard day.

Horizontal axis — degrees above the standard atmosphere. 737-800, CFM56-7B27, 9,500 ft (2.9 km) dry runway; weight capped at the maximum take-off weight (79 t).

These charts give the slopes for the calculation. For the 737-800 the loss of allowable weight is 0.26 % per degree from standard to +25 °C and 0.53 % (0.32–0.67) from +25 to +35 °C; every 1,000 ft of elevation takes about 2.1 %⁴. The second figure matches the slope the researcher derived from Coffel and Horton for Phoenix — 0.48–0.57 %: two independent bases agree. For the A320 the weight barely changes up to +15 °C above standard — 0.05–0.15 % per degree, because its engines hold thrust up to 45 °C; elevation takes 1.2–1.4 % per 1,000 ft⁵. Airbus has no charts above +15 °C, so for heat beyond the kink point we use the 737-800 slope for the A320 — an assumption.

Table for the calculation

Threshold — how many degrees the hot month is warmer than the standard atmosphere (ISA: 15 °C at sea level, 2 °C less for every 300 m of elevation) before the limit starts.

What we change in the calculator

ΔISA = T_{hot month} − (15 − 1.98 · h)
M_allow = min( MTOW ; M_rwy(L) · [1 − a_h·h − k₁·min(ΔISA, D) − k₂·max(0, ΔISA − D)] )
offload = max(0, M_plan − M_allow); passengers = offload ÷ 100 kg
cost per passenger = flight cost ÷ (N − passengers)

Coefficients from the Airbus and Boeing charts^4,5; if the runway length is unknown, the simplified table above applies: one slope k from the threshold.

1. ◷ Queued for the calculatorOnly when the flight is weight-limited. The calculator already knows the planned take-off weight; if it is below the corrected allowable weight, nothing changes. So heat does not touch short flights.
2. ◷ Queued for the calculatorThe table row is chosen by the runway and elevation of the airfield (both are in the reference data); the temperature is the mean maximum of the flight's month.
3. ◷ Queued for the calculatorOffloaded passengers reduce revenue; the flight costs stay the same — cost per passenger rises. Delhi in May: −9 passengers, at 45 °C — about 26.
4. ◷ Queued for the calculatorEverything carries the orange “estimate” flag. For turboprops, E-Jets and airfields above 2,100 m we do not compute a correction but show a warning.
5. ⚑ Mind when entering dataUntil the correction is in — at a hot or high airfield on a long sector, reduce the load by hand using the table above.

How it was checked

The log has 176 steps: the researcher's search (trace/h1.jsonl) and the theory from the owner's documents (trace/h2.jsonl). Quotes were checked (verify/): found verbatim 74, partly 2, not found 0. A separate agent reproduced the calculation of the thresholds, the slopes and the Delhi example; table 2 of the journal version of Coffel & Horton matched the conference version in every figure.