ATPL aircraft performance · Lesson 6 of 15

Density altitude and take-off distance

A hot day at a high airfield makes the air thin. The aeroplane behaves as if the airfield were higher still. This lesson shows how to put a number on that, and what it does to the runway you need.

The aeroplane in this lesson is a generic light single-engine piston aeroplane with illustrative performance figures. It is not any real type, and its own figures are not from any flight manual. The wind, slope, surface and safety factors are the CAA's figures from CAP 698 and Safety Sense Leaflet 7. The physics of density altitude follows the FAA Pilot's Handbook of Aeronautical Knowledge (see Sources).

1. Why the aeroplane cares about air density

Air density is the mass of air in a given volume. Thin air has low density. Thin air hurts a piston aeroplane three times over at take-off:

• The wings need more true speed to make the same lift. So the ground run is longer.
• The engine breathes less air, so it makes less power.
• The propeller pushes on less air, so it makes less thrust.

Two things make air thin: low pressure and high temperature. Height gives low pressure. Heat makes the air expand. A hot, high airfield has both.

2. Pressure altitude

Performance tables need one standard to compare against. The International Standard Atmosphere (ISA) is that standard. It is a model of an average atmosphere: hPa and °C at sea level, with the temperature falling by about °C for every 1,000 ft of height.

The pressure altitude is the height in the ISA at which the pressure equals today's pressure at the airfield. It is what the altimeter reads when you set hPa. To find it from the airfield elevation and today's QNH (the sea-level pressure the tower gives you), use the ATPL rule of ft per hPa (PA is the pressure altitude):

PA = elevation + (1013 − QNH) × 27

Low pressure means a higher pressure altitude. On a day with QNH 1003 hPa, a 2,000 ft airfield has a pressure altitude of ft. PPL teaching rounds the rule to ft per hPa, which would give ft. The UK ATPL question bank is set with 27 ft per hPa, so use 27 ft in the exam.

The ISA temperature at each pressure altitude, from °C minus °C per 1,000 ft.

3. Density altitude

The pressure altitude fixes the pressure. The temperature still varies. The density altitude is the height in the ISA at which the air has today's density. The FAA handbook defines it as "pressure altitude corrected for nonstandard temperature".

The correction uses the ISA deviation: today's outside air temperature (OAT) minus the ISA temperature at that pressure altitude. Every degree warmer than ISA adds about ft of density altitude. This is the exam rule (DA is the density altitude):

ISA temp  = 15 − 2 × (PA ÷ 1,000)
deviation = OAT − ISA temp
DA        = PA + 120 × deviation

A warm day gives a positive deviation and a higher density altitude. A cold day gives a negative deviation, and the density altitude can fall below the pressure altitude. The aeroplane then performs better than the elevation suggests.

4. Take-off distance against density altitude

The take-off distance is the ground run plus the climb to a screen height of 50 ft. For our illustrative aeroplane it is m at sea level, ISA, maximum weight, on a level dry paved runway with no wind. Two rules then stretch it:

• Every 1,000 ft of density altitude multiplies the distance by . The effect compounds.
• The distance grows with the square of the weight. The handbook says % more weight needs at least % more distance. The square rule gives %.

Move the three sliders. The dot is today's take-off. The dashed line is the take-off distance available (TODA), m at this airfield. Find the hottest day on which the take-off still fits at maximum weight.

At the starting settings the density altitude is ft and the distance is m, which fits. Warm the day to 40 °C and the density altitude rises to ft. The distance becomes m. Now load to maximum weight and it becomes m. That is longer than the runway.

5. Correcting for wind, slope and surface

The chart above assumes a level, dry, paved runway and no wind. Real runways differ. Each difference multiplies the distance by a factor:

• Headwind component: the part of the wind blowing straight down the runway towards you. The aeroplane needs less ground speed to reach lift-off speed. The factor is ((V_lof − wind) ÷ V_lof)². With lift-off at kt, a headwind of 10% of that speed cuts the distance to of its value. A tailwind of the same size raises it to . These match the handbook's 19% and 21%. For planning, CAP 698 does not let you count on all of a reported headwind. Use no more than % of a headwind, and allow for at least % of a tailwind. The figure below and the worked example apply that rule to the wind you set.
• Runway slope: an upslope fights the acceleration. CAP 698 adds % to the distance for each 1% of upslope. A downslope takes the same away.
• Surface: grass drags on the wheels. CAP 698 and Safety Sense Leaflet 7 multiply by for dry grass and for wet grass.
• A safety factor allows for an average pilot, an average aeroplane and an average day. The CAA's Safety Sense Leaflet 7 says to multiply the book distance by before comparing it with the runway.

The first bar is today's distance from the chart above. Each later bar applies one more correction. Try a 10 kt tailwind. Then switch the safety factor on.

6. Worked example

Airfield elevation ft. QNH hPa. OAT °C. Take-off weight kg. Headwind kt, upslope %, . Runway available m.

1. Pressure altitude. − = hPa, times = ft. Pressure altitude = ft.
2. ISA temperature. − × = °C.
3. ISA deviation. − = °C.
4. Density altitude. + × = + = ft.
5. Book distance. Density factor = . Weight factor ( ÷ )² = . Distance = × × = m.
6. Wind. The reported headwind is kt. CAP 698 allows % of it: kt. (( − ) ÷ )² = . Distance = m.
7. Slope. × = m.
8. Surface. × = m. This does not fit the m runway, even before the safety factor. It is m over.
9. Safety factor. × = m. The shortfall grows to m.

7. Step through the calculation

What to remember

• The aeroplane feels air density, not elevation.
• Pressure altitude = elevation + (1013 − QNH) × 27 ft (the ATPL rule; PPL teaching rounds it to 30 ft). Low QNH raises it.
• ISA temperature = 15 °C − 2 °C per 1,000 ft of pressure altitude.
• Density altitude = pressure altitude + 120 ft per °C warmer than ISA. Colder than ISA takes it down.
• Take-off distance compounds with density altitude and grows with the square of weight. 10% more weight is at least 21% more runway.
• Correct for wind with the squared speed ratio. A tailwind of 10% of lift-off speed costs 21%. Plan on no more than 50% of a reported headwind and at least 150% of a tailwind (CAP 698).
• Multiply by the slope, surface and safety factors. Compare the factored distance with the runway available.

Sources

• Federal Aviation Administration, Pilot's Handbook of Aeronautical Knowledge, Chapter 11, Aircraft Performance (public domain). Definitions of pressure altitude and density altitude; the effect of weight (10% more weight, at least 21% more distance); the effect of wind (figure 11-19); the ISA table (figure 11-2).
• National Weather Service, El Paso, Density altitude formula: the 145,366 × (1 − ratio^0.235) form used in the exact step. The NWS version also corrects for humidity; this lesson leaves humidity out, as the FAA handbook says is usual.
• UK Civil Aviation Authority, CAP 698, Aeroplane Performance Manual. Source of the wind rule (no more than 50% of a reported headwind, at least 150% of a tailwind), the surface factors (dry grass 1.2, wet grass 1.3), the slope factor (5% per 1% of upslope) and the 1.33 take-off safety factor.
• UK Civil Aviation Authority, Safety Sense Leaflet 7, Aeroplane Performance. The same surface, slope and safety factors, written for the private pilot.
• The exam rules of thumb (27 ft per hPa, 2 °C per 1,000 ft, 120 ft per °C) are the rounded values used in UK ATPL teaching. PPL teaching rounds the pressure rule to 30 ft per hPa. The aeroplane's own figures (500 m, 1,100 kg, 60 kt, 10% per 1,000 ft) are illustrative values chosen for this lesson, in the range seen in light aeroplane manuals. They are not from any flight manual.