What density is, and the equation behind it
Pressure and temperature are two ways of describing the same air. Density is the number that actually flies the aeroplane. Lift, thrust, climb rate, takeoff run and landing speed all answer to density directly, which is why this short chapter carries more weight in performance planning than either of the two before it.
Density is mass per unit volume. Its unit is grams per cubic metre or kilograms per cubic metre, and Oxford gives three equivalent ways to express it:
- An absolute figure, in g/m³
- A percentage of the standard surface density, called relative density
- The altitude in the standard atmosphere at which the observed density would occur naturally, called density altitude
The gas equation
Joshi derives density from the fundamental gas equation, and it is worth seeing once because everything on the next page falls out of it.
Substituting ISA values, 1013.25 hPa and 288 K, gives the standard sea level density: 1225 g/m³.
Read the equation rather than memorise it. Pressure sits on top, temperature sits on the bottom. Raise the pressure and density rises with it. Raise the temperature and density falls. That single line is the whole of the next page.
Density of moist air
Water vapour obeys the same gas equation, with a gas constant 8/5 times that of dry air. Split the total pressure into the dry air's share and the water vapour's share and work through the algebra, and one result falls out that is worth remembering on its own:
Density altitude, defined precisely
Density altitude is the altitude in the ISA at which a given atmospheric density occurs. Pressure altitude and density altitude are the same number only in ISA conditions. The moment the real temperature departs from standard, they split, and Joshi gives the rule for exactly how far:
That single fact explains the whole of the last page of this chapter. A hot aerodrome behaves, as far as the aircraft is concerned, as though it were sitting at a much higher and thinner altitude than its signboard says.
Density altitude is defined as the altitude above msl at which a given atmospheric density occurs in the ISA. The Pressure altitude and the Density altitude have the same value in the ISA.IC Joshi, Aviation Meteorology, chapter 4
Density may be defined as mass per unit volume and may be expressed as grams per cubic metre, a percentage of the standard surface density, or the altitude in the standard atmosphere to which the observed density corresponds, density altitude.Oxford ATPL Vol 9, 3, Introduction
The three things that change density
Every question in this chapter reduces to one of three proportionalities. Learn the three sentences below word for word, because DGCA asks for them almost verbatim.
Pressure
Squeeze air and it compresses into a smaller volume, so more mass fits in the same space.
In the atmosphere this is why climbing reduces density on its own: less air above you means lower pressure, and lower pressure means lower density, before temperature is even considered.
Temperature
Heat a fixed volume of air and it expands, so the same volume now holds less mass.
Warm air is thin air. This is the one factor a pilot can watch happen in real time, on a hot afternoon at a high aerodrome.
Humidity
A water vapour molecule has less mass than the nitrogen or oxygen molecule it displaces, so a more humid parcel of air is lighter than a dry one at the same pressure and temperature.
Both books agree this is the smallest of the three effects and can usually be ignored for aviation purposes, but Joshi adds a specific exception: at hot, moist tropical airfields such as Bahrain or Singapore, it is enough to matter, and the practical advice is to add 10 per cent to the computed takeoff distance and expect a reduced climb rate.
Density with height, and the strange reversal at the poles
Climbing changes both pressure and temperature, and the two pull density in opposite directions. Falling pressure lowers density. Falling temperature, on its own, would raise it. In the real troposphere, the pressure effect wins by a wide margin, so density still falls steadily with height, just not quite as fast as pressure alone would suggest.
How fast density falls
Near the surface, Joshi's rule of thumb is simple: density falls about 3 per cent for every 1000 ft, and this approximation holds well up to about 20 000 ft. Both books also give the same halving pattern Oxford quotes for the standard atmosphere as a whole:
| Height | Density, per cent of sea level |
|---|---|
| Sea level | 100% |
| 6 km, about 20 000 ft | 50% |
| 11 km, about 36 000 ft | 25% |
| 17 km | 12.5% |
| 40 000 ft | 25%, Oxford's figure |
| 60 000 ft | 10% |
The reversal with latitude
At the surface this is intuitive: the poles are cold, cold air is dense, so density at sea level is lowest at the equator and greatest at the poles. What is not intuitive is that this pattern does not hold all the way up.
The reasoning is the same warm and cold column argument from chapter 2. Because the equator is warm, pressure there falls slowly with height, so at any fixed altitude the equator retains relatively high pressure. Because the poles are cold, pressure there falls quickly, so the same altitude over the poles has relatively low pressure. Follow that through to density and the ranking at altitude flips.
Why this actually matters operationally
The reversal is not a curiosity, it sets which aircraft prefer which routes:
- Low flying, piston engine aircraft sit below the crossover, where density is higher at high latitude. Their operational efficiency is greater in high latitudes than in the tropics.
- High flying jet aircraft cruise above the crossover, where density is higher near the equator. Their operational efficiency is greater in the tropics than at high latitudes.
Density of air at sea level is lowest near the equator and greatest at the poles. This distribution is maintained unto about 8.0 km. Above 8.0 km a reversal occurs and the density becoming more near the equator than at poles.IC Joshi, Aviation Meteorology, chapter 4
At about 26000 ft density remains constant with an increase in latitude. Above 26000 ft density decreases with an increase in latitude. Maximum deviation from standard occurs at about 50000 ft.Oxford ATPL Vol 9, 3.5
What low density does to an aircraft
Everything so far has been about the atmosphere. This page is about what the atmosphere does to the aircraft flying through it, and it is where density stops being an abstract number and starts appearing directly in performance figures.
The lift equation
Oxford gives lift in a form worth knowing exactly, because density sits inside it as plainly as it can:
L is lift, CL is the coefficient of lift, ρ is density, V is true airspeed, and S is wing area.
Density sits alongside the square of true airspeed. Halve the density and, to keep the same lift, the wing needs the same increase in speed it would need if it had lost a quarter of its area. That is why low density aerodromes need longer runways, not just slightly longer ones.
What actually gets worse
Both books list the same consequences, because they are the same physics working through three different systems on the aeroplane:
- Instrument accuracy. Airspeed indicators and Mach meters are calibrated against a standard density, so their readings drift from true values as density changes.
- Aircraft and engine performance. Low density reduces lift at a given speed, so a higher true airspeed is needed to get airborne, which increases the takeoff run and reduces the maximum permitted takeoff weight. Both landing and takeoff speeds are greater, and engine thrust itself is reduced because there is less oxygen to burn.
- Climb performance. Climb outs are slower, and Joshi is direct about it: allowance has to be made for these effects in planning runway length and in calculating the aircraft's all up weight.
Where this bites hardest
Oxford names the two families of airfield that combine to make density altitude a real operational limit rather than a textbook exercise:
| Type | Why | Examples |
|---|---|---|
| High | Lower pressure altitude to begin with | Denver, Nairobi, Sana'a |
| Hot | Warm air raises density altitude further, 120 ft per degree above ISA | Bahrain, Khartoum, Singapore |
The two effects stack. An airfield that is both high and hot, such as Nairobi or Bahrain on a warm day, can see a density altitude many thousands of feet above its actual elevation, which is exactly the calculation the interactive figure on the previous page performs.