# Density altitude

The density altitude is the altitude relative to standard atmospheric conditions at which the air density would be equal to the indicated air density at the place of observation. In other words, the density altitude is the air density given as a height above mean sea level. The density altitude can also be considered to be the pressure altitude adjusted for a non-standard temperature.

Both an increase in the temperature and a decrease in the atmospheric pressure, and, to a much lesser degree, an increase in the humidity, will cause an increase in the density altitude. In hot and humid conditions, the density altitude at a particular location may be significantly higher than the true altitude.

In aviation, the density altitude is used to assess an aircraft's aerodynamic performance under certain weather conditions. The lift generated by the aircraft's airfoils, and the relation between its indicated airspeed (IAS) and its true airspeed (TAS), are also subject to air-density changes. Furthermore, the power delivered by the aircraft's engine is affected by the density and composition of the atmosphere.

## Aircraft Safety

Air density is perhaps the single most important factor affecting aircraft performance. It has a direct bearing on:[2]

• The lift generated by a wing — a reduction in the air density reduces the wing's lift.
• The efficiency of a propeller or rotor — which for a propeller (effectively an airfoil) behaves similarly to lift on a wing.
• The power output of an engine — the power output depends on the oxygen intake, so the engine output is reduced as the equivalent dry-air density decreases, and it produces even less power as moisture displaces oxygen in more humid conditions.

Aircraft taking off from a “hot and high” airport, such as the Quito Airport or Mexico City, are at a significant aerodynamic disadvantage. The following effects result from a density altitude that is higher than the actual physical altitude:[2]

• An aircraft will accelerate more slowly on takeoff as a result of its reduced power production.
• An aircraft will need to achieve a higher true airspeed to attain the same amount of lift — this implies both a longer takeoff roll and a higher true airspeed, which must be maintained while airborne to avoid stalling.
• An aircraft will climb more slowly as a result of its reduced power production and reduced lift.

Due to these performance issues, an aircraft's takeoff weight may need to be lowered, or takeoffs may need to be scheduled for cooler times of the day. The wind direction and the runway slope may need to be taken into account.

## Skydiving

The density altitude is an important factor in skydiving, and one that can be difficult to judge properly, even for experienced skydivers.[3] In addition to the general change in wing efficiency that is common to all aviation, skydiving has additional considerations. There is an increased risk due to the high mobility of jumpers (who will often travel to a drop zone with a completely different density altitude than they are used to, without being made consciously aware of it by the routine of calibrating to QNH/QFE).[4] Another factor is the higher susceptibility to hypoxia at high density altitudes, which, combined especially with the unexpected higher free-fall rate, can create dangerous situations and accidents.[3] Parachutes at higher altitudes fly more aggressively, making their effective area smaller, which is more demanding for a pilot's skill and can be especially dangerous for high-performance landings, which require accurate estimates and have a low margin of error before they become dangerous.[4]

## Calculation

The density altitude can be calculated from the atmospheric pressure and the outside air temperature (assuming dry air) using the following formula:

${\displaystyle {\text{DA}}={\frac {T_{\text{SL}}}{\Gamma }}\left[1-\left({\frac {P/P_{\text{SL}}}{T/T_{\text{SL}}}}\right)^{\frac {\Gamma R}{gM-\Gamma R}}\right].}$

In this formula,

${\displaystyle {\text{DA}}=}$ Density altitude in meters (${\displaystyle \mathrm {m} }$);
${\displaystyle P=}$ (Static) atmospheric pressure;
${\displaystyle P_{\text{SL}}=}$ Standard sea-level atmospheric pressure (${\displaystyle 1013.25}$ hectopascals (${\displaystyle \mathrm {hPa} }$) in the International Standard Atmosphere (ISA), or ${\displaystyle 29.92}$ inches of mercury (${\displaystyle \mathrm {inHg} }$) in the U.S. Standard Atmosphere);
${\displaystyle T=}$ Outside air temperature in kelvins (${\displaystyle \mathrm {K} }$) (add ${\displaystyle 273.15}$ to the temperature in degrees Celsius (${\displaystyle {^{\circ }\mathrm {C} }}$));
${\displaystyle T_{\text{SL}}=}$ ISA sea-level air temperature ${\displaystyle =288.15~\mathrm {K} }$;
${\displaystyle \Gamma =}$ ISA temperature lapse rate ${\displaystyle =0.0065~\mathrm {K} /\mathrm {m} }$;
${\displaystyle R=}$ Ideal gas constant ${\displaystyle =8.3144598~\mathrm {J} /(\mathrm {mol} ~\mathrm {K} )}$;
${\displaystyle g=}$ Gravitational acceleration ${\displaystyle =9.80665~\mathrm {m} /\mathrm {s} ^{2}}$;
${\displaystyle M=}$ Molar mass of dry air ${\displaystyle =0.028964~\mathrm {kg} /\mathrm {mol} }$.

### The National Weather Service (NWS) Formula

The National Weather Service uses the following dry-air approximation to the formula for the density altitude above in its standard:

${\displaystyle {\text{DA}}=(145442.16~\mathrm {ft} )\times \left(1-\left[{\frac {(17.326~{^{\circ }\mathrm {F} }/\mathrm {inHg} )\times P}{459.67~{^{\circ }\mathrm {F} }+T}}\right]^{0.235}\right).}$

In this formula,

${\displaystyle {\text{DA}}=}$ Density altitude in feet (${\displaystyle \mathrm {ft} }$);
${\displaystyle P=}$ Station pressure (static atmospheric pressure) in inches of mercury (${\displaystyle \mathrm {inHg} }$);
${\displaystyle T=}$ Station temperature (outside air temperature) in degrees Fahrenheit (${\displaystyle {^{\circ }\mathrm {F} }}$).

Note that the NWS standard specifies that the density altitude should be rounded to the nearest ${\displaystyle 100~\mathrm {ft} }$.

### Approximation Formula for Calculating the Density Altitude from the Pressure Altitude

This is an easier formula to calculate (with great approximation) the density altitude from the pressure altitude and the ISA temperature deviation:

${\displaystyle {\text{DA}}={\text{PA}}+(118.8~\mathrm {ft} /{^{\circ }\mathrm {C} })\times ({\text{OAT}}-{\text{ISA temperature}}).}$

In this formula,

${\displaystyle {\text{PA}}=}$ Pressure altitude in feet (${\displaystyle \mathrm {ft} }$) ${\displaystyle ={\text{Station elevation in feet}}+(27~\mathrm {ft} /\mathrm {mb} )\times (1013~\mathrm {mb} -{\text{QNH}})}$;
${\displaystyle {\text{QNH}}=}$ Atmospheric pressure in millibars (${\displaystyle \mathrm {mb} }$) adjusted to mean sea level;
${\displaystyle {\text{OAT}}=}$ Outside air temperature in degrees Celsius (${\displaystyle {^{\circ }\mathrm {C} }}$);
${\displaystyle {\text{ISA Temperature}}=15~{^{\circ }\mathrm {C} }-(1.98~{^{\circ }\mathrm {C} })\times \left({\dfrac {\text{PA}}{1000~\mathrm {ft} }}\right)}$, assuming that the outside air temperature falls at the rate of ${\displaystyle 1.98~{^{\circ }\mathrm {C} }}$ per ${\displaystyle 1,000~\mathrm {ft} }$ of altitude until the tropopause (at ${\displaystyle 36,000~\mathrm {ft} }$) is reached.

Rounding up ${\displaystyle 1.98~{^{\circ }\mathrm {C} }}$ to ${\displaystyle 2~{^{\circ }\mathrm {C} }}$, this approximation simplifies to become

{\displaystyle {\begin{aligned}{\text{DA}}&={\text{PA}}+(118.8~\mathrm {ft} /{^{\circ }\mathrm {C} })\times \left[\left({\frac {\text{PA}}{500~\mathrm {ft} }}\right){^{\circ }\mathrm {C} }+{\text{OAT}}-15~{^{\circ }\mathrm {C} }\right]\\&=(1.2376\times {\text{PA}})+[(118.8~\mathrm {ft} /{^{\circ }\mathrm {C} })\times {\text{OAT}}]-1782~\mathrm {ft} .\end{aligned}}}