an introduction to air density and density altitude calculations

8/10/2019 An Introduction to Air Density and Density Altitude Calculations

1/22

An Introduction to Air Density and Density Altitude Calculations

Density Altitude On-Line Calculators:

For your convenience, the following density altitude calculators are available for use on this web site:

-with dew point

-using relative humidity

What is density altitude?

Density altitude is defined as the altitude at which the density of the International Standard Atmosphere

(ISA) is the same as the density of the air being evaluated. (The Standard Atmosphere is simply a

mathematical model of the atmosphere which is standardized so that predictable calculations can be

made.)

So, the basic idea of calculating density altitude is to calculate the actual density of the air, and then find

the altitude at which that same air density occurs in the International Standard Atmosphere.

In the following paragraphs, we'll go step by step through the process of calculating the actual density of

the air, and then determining the corresponding density altitude.

And finally, at the very end of this article, we'll compare the accurate density altitude calculations with

the results of a greatly simplified equation which ignores the effects due to water vapor in the air.

Some different meanings of the word "altitude":

As odd as it may seem, an aircraft altimeter does not actually know anything about altitude, it only

measures pressure. For pilots, it is very important to understand that an aircraft altimeter only measuresair pressure (not true altitude). This point is especially important to understand with the ever-increasing

use of GPS. An aircraft flying at a specific pressure altitude (as indicated by an altimeter set to 29.92 in-

Hg) may note some significantly different altitude displayed on a GPS (which measures actual distance

above mean sea level). In some cases this altitude difference is small... but in other cases it could be

enough to cause a mid-air collision if a pilot was flying on a GPS mean-sea-level (MSL) altitude rather

than the assigned pressure altitude.

Density altitude is yet another sort of altitude, based solely on air density. Density altitude is neither

"pressure altitude" nor "mean sea-level altitude", it is simply the altitude in the International Standard

Atmosphere model at which the air has a certain value of density... hence the name density altitude.

Therefore, it's crucial to always verify what is meant by "altitude".

Now... on to Density Altitude.....

Density and Density Altitude:
http://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da_rh.htmhttp://wahiduddin.net/calc/calc_da_rh.htmhttp://wahiduddin.net/calc/calc_da_rh.htmhttp://wahiduddin.net/calc/calc_da_rh.htmhttp://wahiduddin.net/calc/calc_da.htm


2/22

Although the concept of density altitude is commonly used to describe the effect on aircraft and engine

performance, the underlying property of interest is actually the air density.

For example, the lift of an aircraft wing, the aerodynamic drag of an aircraft, and the thrust of a

propeller blade are all directly proportional to the air density. Similarly, the downforce of a racecar

spoiler is also directly proportional to the air density. Similarly, the horsepower output of an internalcombustion engine is related to the air density, the correct size of a carburetor jet is related to the air

density, and the pulse width command to an electronic fuel injection nozzle is also related to the air

density.

In general, if you really want to be precise and consistent, it will be best to focus attention on the actual

air density, not density altitude.

Density altitude has been a convenient yardstick for pilots to compare the performance of aircraft at

various altitudes, but it is in fact the air density which is the fundamentally important quantity, and

density altitude is simply one way to express the air density. Actually, it would be far more meaningful,

useful and precise if we would simply use the actual air density in kg/m3, and if the data in aircraft pilot's

handbooks were also expressed in actual air density.

Hopefully, someday all of the aircraft performance tables/charts and weather reporting systems will

simply use the actual air density and thereby avoid this arcane concept of density altitude... but, for

now, we're stuck with "density altitude".

Note: If you're just hunting for a simple calculation for density altitude without the effects of moisture,

you will find aSimpler Methods of Calculationsection near the end of this article. But, for those who

want to understand the effects of moisture on density altitude, please read on.

Units:

The 1976 International Standard Atmosphere (which is used as the basis for these Density Altitude

calculations) is mostly described in metric SI units, and I have chosen to use those same units (in

general) throughout this article. Seeref 8andref 9for conversion factors to your favorite units.

Air Density Calculations:

To begin to understand the calculation of air density, consider the ideal gas law:

(1) P*V = n*Rg*T

where: P = pressure

V = volume

n = number of moles
http://wahiduddin.net/calc/density_altitude.htm#simplehttp://wahiduddin.net/calc/density_altitude.htm#simplehttp://wahiduddin.net/calc/density_altitude.htm#simplehttp://www.digitaldutch.com/unitconverter/index.htmhttp://www.digitaldutch.com/unitconverter/index.htmhttp://www.digitaldutch.com/unitconverter/index.htmhttp://physics.nist.gov/Pubs/SP811/appenB8.htmlhttp://physics.nist.gov/Pubs/SP811/appenB8.htmlhttp://physics.nist.gov/Pubs/SP811/appenB8.htmlhttp://physics.nist.gov/Pubs/SP811/appenB8.htmlhttp://www.digitaldutch.com/unitconverter/index.htmhttp://wahiduddin.net/calc/density_altitude.htm#simple


3/22

Rg = universal gas constant

T = temperature

Density is simply the mass of the molecules of the ideal gas in a certain volume, which may be

mathematically expressed as:

(2) D = m / V

where: D = density

m = mass

V = volume

Note that:

m = n * M

where: m = mass

n = number of moles

M = molar mass

And define a specific gas constant for the gas under consideration:

R = Rg / M

where R = specific gas constant

Rg = universal gas constant

M = molar mass

Then, by combining the previous equations, the expression for the density becomes:

(3)

where: D = density, kg/m3

P = pressure, Pascals ( multiply mb by 100 to get Pascals)

R = specific gas constant , J/(kg*degK) = 287.05 for dry air

T = temperature, deg K = deg C + 273.15

As an example, using the ISA standard sea level conditions of P = 101325 Pa and T = 15 deg C, the air

density at sea level, may be calculated as:

D = (101325) / (287.05 * (15 + 273.15)) = 1.2250 kg/m3

This example has been derived for the dry air of the standard conditions. However, for real-world

situations, it is necessary to understand how the density is affected by the moisture in the air.


4/22

Neglecting the small errors due to non-ideal gas compressibility and vapor pressure measurements not

made over liquid water (seeref 14), the density of a mixture of dry air molecules and water vapor

molecules may be simply written as:

(4a)

Which, with some substitutions and rearranging (seeref 15), may also be written as:

(4b)

where: D = density, kg/m3

Pd= pressure of dry air (partial pressure), Pascals

Pv= pressure of water vapor (partial pressure), Pascals

P = Pd+ Pv= total air pressure, Pascals ( multiply mb by 100 to get Pascals)

Rd = gas constant for dry air, J/(kg*degK) = 287.05 = R/Md

Rv = gas constant for water vapor, J/(kg*degK) = 461.495 = R/Mv

R = universal gas constant = 8314.32 (in 1976 Standard Atmosphere)

Md = molecular weight of dry air = 28.964 gm/mol

Mv = molecular weight of water vapor = 18.016 gm/mol

T = temperature, deg K = deg C + 273.15

To use equation 4a or 4b to determine the density of the air, one must know the actual air pressure(which is also called absolute pressure, total air pressure, or station pressure), the water vapor pressure,

and the temperature.

It is possible to obtain a rough approximation of the absolute pressure by adjusting an altimeter to read

zero altitude and reading the value in the Kollsman window as the actual air pressure. Near the end of

this page I'll discuss how to use the altimeter reading to accurately determine the actual pressure.

Alternatively, there are many little electronic gadgets that can measure the actual air pressure and the

vapor pressure directly, and quite accurately.

The water vapor pressure can easily be determined from the dew point or from the relative humidity,

and the ambient temperature can be measured in a well ventilated place out of the direct sunlight.

In the following section, we'll learn to calculate the water vapor pressure.

Vapor Pressure:
http://wahiduddin.net/calc/density_altitude.htm#b14http://wahiduddin.net/calc/density_altitude.htm#b14http://wahiduddin.net/calc/density_altitude.htm#b14http://wahiduddin.net/calc/density_altitude.htm#b15http://wahiduddin.net/calc/density_altitude.htm#b15http://wahiduddin.net/calc/density_altitude.htm#b15http://wahiduddin.net/calc/density_altitude.htm#b15http://wahiduddin.net/calc/density_altitude.htm#b14


5/22

In order to calculate water vapor pressure, we need to first calculate the saturation vapor pressure.

There are many algorithms for determining the saturation vapor pressure, but for simplicity we'll just

look at two algorithms:

A very accurate, albeit quite odd looking, formula for determining the saturation vapor pressure is a

polynomial developed by Herman Wobus (seeref 2) :

(5) Es = eso/ p8

where: Es = saturation pressure of water vapor, mb

eso=6.1078

p = (c0+T*(c1+T*(c2+T*(c3+T*(c4+T*(c5+T*(c6+T*(c7+T*(c8+T*(c9))))))))))

T = temperature, deg C

c0 = 0.99999683

c1 = -0.90826951*10-2

c2 = 0.78736169*10-4

c3 = -0.61117958*10-6

c4 = 0.43884187*10-8

c5 = -0.29883885*10-10

c6 = 0.21874425*10-12

c7 = -0.17892321*10-14

c8 = 0.11112018*10-16

c9 = -0.30994571*10-19

For situations where simplicity is desirable and slightly less accuracy is acceptable, the following

equation offers good results, especially at the higher ambient air temperatures where the saturationpressure becomes significant for the density altitude calculations.

(6)

where: Es = saturation pressure of water vapor, mb

Tc = temperature, deg C

c0= 6.1078

c1= 7.5

c2= 237.3

Seeref 2andref 11for additional vapor pressure formulas.

Here's a calculator that compares the saturation vapor pressure for any given temperature, showing the

results from using equations 5 and 6 given above:
http://wahiduddin.net/calc/density_algorithms.htmhttp://wahiduddin.net/calc/density_algorithms.htmhttp://wahiduddin.net/calc/density_algorithms.htmhttp://wahiduddin.net/calc/density_algorithms.htmhttp://wahiduddin.net/calc/density_algorithms.htmhttp://wahiduddin.net/calc/density_algorithms.htmhttp://atmos.nmsu.edu/education_and_outreach/encyclopedia/sat_vapor_pressure.htmhttp://atmos.nmsu.edu/education_and_outreach/encyclopedia/sat_vapor_pressure.htmhttp://atmos.nmsu.edu/education_and_outreach/encyclopedia/sat_vapor_pressure.htmhttp://atmos.nmsu.edu/education_and_outreach/encyclopedia/sat_vapor_pressure.htmhttp://wahiduddin.net/calc/density_algorithms.htmhttp://wahiduddin.net/calc/density_algorithms.htm


6/22

Saturation Vapor Press Calculator

Air Temperature: degrees C

Reset

Sat vapor press from Eqn 5: mb

Sat vapor press from Eqn 6: mb

The Smithsonian reference tables (see ref 1) give the following values of saturated vapor pressure values

at specified temperatures. Entering these known temperatures into the calculator will allow you to

evaluate the accuracy of the calculated results.

Deg C Es, mb

30 42.430

20 23.373

10 12.272

0 6.1078

-10 2.8627

-30 0.5088

Armed with the value of the saturation vapor pressure, the next step is to determine the actual value of

vapor pressure.

When calculating the vapor pressure, it is often more accurate to use the dew point temperature rather

than the relative humidity. Although relative humidity can be used to determine the vapor pressure, the

value of relative humidity is strongly affected by the ambient temperature, and is therefore constantly

changing during the day as the air is heated and cooled.


7/22

In contrast, the value of the dew point is much more stable and is often nearly constant for a given air

mass regardless of the normal daily temperature changes. Therefore, using the dew point as the

measure of humidity allows for more stable and therefore potentially more accurate results.

Actual Vapor Pressure from the Dew Point:

To determine the actual vapor pressure, simply use the dew point as the value of T in equation 5 or 6.

That is, at the dew point, Pv = Es.

(7a) Pv = Es at the dew point

where Pv= pressure of water vapor (partial pressure)

Es = saturation vapor pressure ( multiply mb by 100 to get Pascals)

Actual Vapor Pressure from Relative Humidity:

Relative humidity is defined as the ratio (expressed as a percentage) of the actual vapor pressure to the

saturation vapor pressure at a given temperature.

To find the actual vapor pressure, simply multiply the saturation vapor pressure by the percentage and

the result is the actual vapor pressure. For example, if the relative humidity is 40% and the temperature

is 30 deg C, then the saturation vapor pressure is 42.43 mb and the actual vapor pressure is 40% of

42.43 mb, which is 16.97 mb.

(7b) Pv = RH * Es

where Pv= pressure of water vapor (partial pressure)

RH = relative humidity (expressed as a decimal value)

Es = saturation vapor pressure ( multiply mb by 100 to get Pascals)

Dry Air Pressure:

Now that the water vapor pressure is known, we are nearly ready to calculate the density of the

combination of dry air and water vapor as described in equation 4a, but first, we need to know the

pressure of the dry air.

The total measured atmospheric pressure (also called actual pressure, absolute pressure, or station

pressure) is the sum of the pressure of the dry air and the vapor pressure:

(8a) P = Pd + Pv

where: P = total pressure


8/22

Pd = pressure due to dry air

Pv = pressure due to water vapor

So, rearranging that equation:

(8b) Pd = P - Pv

where: P = total pressure

Pd = pressure due to dry air

Pv = pressure due to water vapor

Now that we have the pressure due to water vapor and also the pressure due to the dry air, we have all

of the information that is required to calculate the air density using equation 4a.

Calculate the air density:

Now armed with those equations and the actual air pressure, the vapor pressure and the temperature,

the density of the air can be calculated.

Here's a calculator that determines the air density from the actual pressure, dew point and air

temperature using equations 4, 6, 7 and 8 as defined above:

Air Density Calculator

Air Temperature degrees C

Actual Air Pressure mb

Dew Point degrees C

Reset

Air Densitykg/m

3

Moist Air is Less Dense...


9/22

As you may have noticed, moist air is less dense than dry air. It may seem reasonable to try to argue

against that simple fact based on the observation that water is denser than dry air... which is certainly

true, but irrelevant.

Solids, liquids and gasses each have their own unique laws, so it is not possible to equate the behavior of

liquid water with the behavior of water vapor.

The ideal gas law says that a certain volume of air at a certain pressure has a certain number of

molecules. That's just the way this world works, and that simple fact is expressed as the ideal gas law,

which was shown above in equation 1.

Note that this is the gas law... not a liquid law, nor a solid law, but a gas law. Hence, any mental

comparisons to the behavior of a liquid are of little help in understanding what is going on in the air, and

are likely to simply result in greater confusion.

According to the ideal gas law, a cubic meter of air around you, wherever you are right now, has a

certain number of molecules in it, and each of those molecules has a certain weight. The key tounderstanding air density changes due to moisture is grasping the idea that a given volume of air has

only a certain number of molecules in it. That is, whenever a water vapor molecule is added to the air, it

displaces some other molecule in that volume of air.

Most of the air is made up of nitrogen molecules N2 with a somewhat lesser amount of oxygen O2

molecules, and even lesser amounts of other molecules such as water vapor.

Since density is weight divided by volume, we need to consider the weight of each of the molecules in

the air. Nitrogen has an atomic weight of 14, so an N2 molecule has a weight of 28. For oxygen, the

atomic weight is 16, so an O2 molecule has a weight of 32.

Now along comes a water molecule, H2O. Hydrogen has an atomic weight of 1. So the molecule H20 has

a weight of 18. Note that the water molecule is lighter in weight than either a nitrogen molecule (with a

weight of 28) or an oxygen molecule (with a weight of 32).

Therefore, when a given volume of air, which always contains only a certain number of molecules, has

some water molecules in it, it will weigh less than the same volume of air without any water molecules.

That is, moist air is less dense than dry air.

Some examples of calculations using air density:

Example 1)The lift of an aircraft wing may be described mathematically (see ref 8)as:

L = c1* d * v2/2 * a

where: L = lift

c1= lift coefficient
http://www.grc.nasa.gov/WWW/K-12/airplane/lifteq.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/lifteq.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/lifteq.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/lifteq.html


10/22

d = air density

v = velocity

a = wing area

From the lift equation, we see that the lift of a wing is directly proportional to the air density. So if a

certain wing can lift, for example, 3000 pounds at sea level standard conditions where the density is1.2250 kg/m3, then how much can the wing lift on a warm summer day in Denver when the air

temperature is 95 deg (35 deg C), the actual pressure is 24.45 in-Hg (828 mb) and the dew point is 67

deg F (19.4 deg C)? The answer is about 2268 pounds.

Example 2) The engine manufacturer Rotax (seeref 6) advises that their carburetor main jet diameter

should be adjusted according to the air density. Specifically, if the engine is jetted properly at air density

d1, then for operation at air density d2 the new jet diameter j2 is given mathematically as:

j2= j1* (d2/d1)(1/4)

where: j2= diameter of new jetj1 = diameter of jet that was proper at density d1

d1= density at which the original jet j1 was correct

d2= the new air density

That is, Rotax says that the correct jet diameter should be sized according to the fourth root of the ratio

of the air densities. (Note: according to Poiseuille's Law, the volumetric flow rate through a circular cross

section is proportional to the fourth power of the diameter.)

For example, if the correct jet at sea level standard conditions is a number 160 and the jet number is a

measure of the jet diameter, then what jet should be used for operations on the warm summer day in

Denver described in example 1 above? The ideal answer is a jet number 149, and in practice the closest

available jet size is then selected.

Example 3)In the same service bulletin mentioned above, Rotax says that their engine horsepower will

decrease in proportion to the air density.

hp2= hp1* (d2/d1)

where: hp2= the new horsepower at density d2

hp1= the old horsepower at density d1

If a Rotax engine was rated at 38 horsepower at sea level standard conditions, what is the availablehorsepower according to that formula when the engine is operated at a temperature of 30 deg C, a

pressure of 925 mb and a dew point of 25 deg C? The answer is approximately 32 horsepower. (See also

details on theSAE method of correcting horsepower.)

Importance of Air Density:
http://wahiduddin.net/calc/density_altitude.htm#b6http://wahiduddin.net/calc/density_altitude.htm#b6http://wahiduddin.net/calc/density_altitude.htm#b6http://wahiduddin.net/calc/cf.htmhttp://wahiduddin.net/calc/cf.htmhttp://wahiduddin.net/calc/cf.htmhttp://wahiduddin.net/calc/cf.htmhttp://wahiduddin.net/calc/density_altitude.htm#b6


11/22

So far, we've been discussing real physical attributes which can be precisely measured, with air density

being the weight per unit volume of an air mass. The air density, as shown in the previous examples,

affects the lift of a wing, the fuel required by an engine, and the power produced by an engine. When

precision is required, air density is a much better measure than density altitude.

Air density is a physical quality which can be accurately measured and verified. On the other hand,density altitude is a rather conceptual quantity which depends upon a hypothetical "standard

atmosphere" which may or may not accurately correspond to the actual physical conditions at any given

location. Nonetheless, density altitude has a long heritage and remains a common (although rather

hypothetical) representation of air density.

Back on the trail of Density Altitude...

The definition of density altitude is the altitude at which the density of the 1976 International Standard

Atmosphere is the same as the density of the air being evaluated. So, now that we know how to

determine the air density, we can solve for the altitude in the International Standard Atmosphere that

has the same value of density.

The 1976 International Standard Atmosphere (ISA) is a mathematical description of a theoretical

atmospheric column of air which uses the following constants (seeref 16):

Po= 101325 sea level standard pressure, Pa

To= 288.15 sea level standard temperature, deg K ( 15 deg C)

g = 9.80665 gravitational constant, m/sec2

L = 6.5 temperature lapse rate, deg K/kmR = 8.31432 gas constant, J/ mol*deg K

M = 28.9644 molecular weight of dry air, gm/mol

In the ISA, the lowest region is the troposphere which extends from sea level up to 11 km (about 36,000

ft), and the model which will be developed here is only valid in the troposphere.

The following equations describe temperature, pressure and density of the air in the ISA troposphere:

(9) (seeISApg 10, Eqn 23)

(10) (seeISApg 12, Eqn 33a)

(11) (seeISApg 15, Eqn 42)
http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16http://wahiduddin.net/calc/density_altitude.htm#b16


12/22

where: T = ISA temperature in deg K

P = ISA pressure in Pa

D = ISA density in kg/m3

H = ISA geopotential altitude in km

One way to determine the altitude at which a certain density occurs is to rewrite the equations andsolve for the variable H, which is the geopotential altitude.

So, it is now necessary to rewrite equations 9, 10, and 11 in a manner which expresses altitude H as a

function of density D. After a bit of gnashing of teeth and general turmoil using algebraic substitutions of

those three equations, the exact solution for H as a function of D, may be written as:

(12)

Using the numerical values of the ISA constants, that expression may be evaluated as:

where H = geopotential altitude, km

D = air density, kg/m3

Now that H is known as a function of D, it is easy to solve for the Density Altitude of any specified air

density.

It is interesting to note that equations 9, 10 and 11 could also be evaluated to find H as a function of P as

follows:

where H = geopotential altitude, km

P = actual air pressure, Pascals

Now that we can determine the altitude for a given density, it may be useful to consider some of thedefinitions of altitude.

Different Flavors of Altitude:

There are three commonly used varieties of altitude (see ref 4). They are: Geometric altitude,

Geopotential altitude and Pressure altitude.
http://mtp.jpl.nasa.gov/notes/altitude/altitude.htmlhttp://mtp.jpl.nasa.gov/notes/altitude/altitude.htmlhttp://mtp.jpl.nasa.gov/notes/altitude/altitude.htmlhttp://mtp.jpl.nasa.gov/notes/altitude/altitude.html


13/22

Geometric altitude is what you would measure with a tape measure, while the Geopotential altitude is a

mathematical description based on the potential energy of an object in the earth's gravity. Pressure

altitude is what an altimeter displays when set to 29.92.

The ISA equations use geopotential altitude, because that makes the equations much simpler and more

manageable. To convert the result from the geopotential altitude H to the geometric altitude Z, thefollowing formula may be used:

(13)

where E = 6356.766 km, the radius of the earth (for 1976 ISA)

H = geopotential altitude, km

Z = geometric altitude, km

Density Altitude Calculator:

The following calculator uses equation 12 to convert an input value of air density to the corresponding

altitude in the 1976 International Standard Atmosphere. Then, the results are displayed as both

geopotential altitude and geometric altitude, which are very nearly identical at lower altitudes.

Note that since these equations are designed to model the troposphere, this calculator will give an error

message if the calculated value of altitude is beyond the bounds of the troposphere, which extends from

sea level up to a geopotential altitude of 11 km.

Density Altitude Calculator 1

Air Density kg/m3

Reset

Geopotential altitude H m

Geometric altitude Z m


14/22

Here's a calculator that uses the actual pressure, air temperature and dew point to calculate the air

density as well as the corresponding density altitude:

Density Altitude Calculator 2



Dew Point degrees C

Reset

Air Density kg/m3

Geopotential altitude H m

Geometric altitude Z m

Density Altitude calculations using Virtual Temperature:

As an alternative to the use of equations which describe the atmosphere as being made up of a

combination of dry air and water vapor, it is possible to define a virtual temperature for an atmosphere

of only dry air.

The virtual temperature is the temperature that dry air would have if its pressure and specific volume

were equal to those of a given sample of moist air. It's often easier to use virtual temperature in place of

the actual temperature to account for the effect of water vapor while continuing to use the gas constant

for dry air.

The results should be exactly the same as in the previous method, this is just an alternative method.

There are two steps in this scheme: first calculate the virtual temperature and then use that

temperature in the corresponding altitude equation.


15/22

The equation for virtual temperature may be derived by manipulation of the density equation that was

presented earlier as equation 4a:

Recalling that P = Pd + Pv, which means that Pd = P - Pv, the equation may be rewritten as

Finally, a new temperature Tv, the virtual temperature, is defined such that

By evaluating the numerical values of the constants, setting Pv= E, noting that Rd= R*1000/Mdand that

Rv=R*1000/Mv, then the virtual temperature may be expressed as:

(14)

where Tv = virtual temperature, deg K

T = ambient temperature, deg K

c1= ( 1 - (Mv/ Md) ) = 0.37800

E = vapor pressure, mb

P = actual (station) pressure, mb

where Mdis molecular weight of dry air = 28.9644

Mvis molecular weight of water = 18.016

(Note that for convenience, the units in Equation 14 are not purely SI units, but rather are US customary

units for the vapor pressure and station pressure.)

The following calculator uses equation 6 to find the vapor pressure, then calculates the virtual

temperature using equation 14:

Virtual Temperature Calculator



16/22


Dew Point degrees C

Reset

Virtual Temperature degrees C

The virtual temperature Tvmay used in the following formula to calculate the density altitude. This

formula is simply a rearrangement of equations 9, 10 and 11:

(15)

Using the numerical values of the ISA constants, equation 15 may be rewritten using the virtualtemperature as:

where H = geopotential density altitude, km

Tv= virtual temperature, deg K

P = actual (station) pressure, Pascals

Using the Altimeter Setting:

When the actual pressure is not known, the altimeter reading may be used to determine the actual

pressure. (For more information about ambient air pressure measurements see thepressure

measurement page.)
http://wahiduddin.net/calc/pressure.htmhttp://wahiduddin.net/calc/pressure.htmhttp://wahiduddin.net/calc/pressure.htmhttp://wahiduddin.net/calc/pressure.htmhttp://wahiduddin.net/calc/pressure.htmhttp://wahiduddin.net/calc/pressure.htm


17/22

The altimeter setting is the value in the Kollsman window of an altimeter when the altimeter is adjusted

to read the correct altitude. The altimeter setting is generally included inNational Weather Service

reports,and can be used to determine the actual pressure using the following equations:

According to NWS ASOS documentation, the actual pressure Pais related to the altimeter setting AS by

the following equation:

(16)

By numerically evaluating the constants and converting to customary units of altitude and pressure, the

equation may be written as:

Pa= [ASk1- ( k2 * H ) ]1/k1

where Pa= actual (station) pressure, mb

AS = altimeter setting, mb

H = geopotential station elevation, m

k1 = 0.190263

k2 = 8.417286*10-5

When converted to English units, this is the relationship between station pressure and altimeter setting

that is used by the National Weather Service ASOS weather stations (seeref 10) as:

Pa= [AS0.1903- (1.313 x 10-5) x H]5.255

where Pa= actual (station) pressure, inches Hg

AS = altimeter setting, inches Hg

H = station elevation, feet

(Note: several other equations for converting actual pressure to altimeter setting are given inref 12.)

Using these equations, the altimeter setting may be readily converted to actual pressure, then by using

the actual pressure along with the temperature and dew point, the local air density may be calculated,

and finally the density may be used to determine the corresponding density altitude.

Given the values of the altimeter setting (the value in the Kollsman window) and the altimeter reading

(the geometric altitude), the following calculator will convert the altitude to geopotential altitude, and

solve equation 16 for the actual pressure at that altitude.

Altimeter Values to Actual Pressure
http://weather.noaa.gov/http://weather.noaa.gov/http://weather.noaa.gov/http://weather.noaa.gov/http://wahiduddin.net/calc/density_altitude.htm#b10http://wahiduddin.net/calc/density_altitude.htm#b10http://wahiduddin.net/calc/density_altitude.htm#b10http://wahiduddin.net/calc/density_altitude.htm#b12http://wahiduddin.net/calc/density_altitude.htm#b12http://wahiduddin.net/calc/density_altitude.htm#b12http://wahiduddin.net/calc/density_altitude.htm#b12http://wahiduddin.net/calc/density_altitude.htm#b10http://weather.noaa.gov/http://weather.noaa.gov/


18/22

Altimeter Setting hPa (mb)

Altimeter Reading meters

Reset

Geopotential Altitude meters

Actual Pressure hPa (mb)

Using National Weather Service Barometric Pressure:

Now you're probably wondering about converting sea-level corrected barometric pressure, as reported

in a weather forecast, to actual air pressure for use in calculating density altitude. Well the good news is

that yes, sea level barometric pressure can be converted to actual air pressure. The bad news is that the

result may not be very accurate.

If you want accurate density or density altitude calculations, you really need to know the actual air

pressure.

In order to compare surface pressures from various parts of the country, the National Weather Service

converts the actual air pressure reading into a sea level corrected barometric pressure. In that way, the

common reference to sea level pressure readings allows surface features such as pressure changes to be

more easily understood.

But, unfortunately, there really is no fool-proof way to convert the actual air pressure to a sea level

corrected value. There are a number of such algorithms currently in use, but they all suffer from various

problems that can occasionally cause inaccurate results (see ref 7).

It has been estimated that the errors in the sea level pressure reading (in mb) may be on the order of 1.5

times the temperature error for a station like Denver at 1640 meters. So, if the temperature error was10 deg C, then the sea level pressure conversion might occasionally be in error by 15 mb. At the very

highest airports such as Leadville, Colorado at an elevation of 3026 meters (9927 ft), perhaps the error

might be on the order of 30 mb.
http://wahiduddin.net/calc/density_altitude.htm#b7http://wahiduddin.net/calc/density_altitude.htm#b7http://wahiduddin.net/calc/density_altitude.htm#b7http://wahiduddin.net/calc/density_altitude.htm#b7


19/22

And further complicating matters, without knowing the details of the algorithm that was used to

calculate the sea level pressure, it is likely that there will be some additional error introduced in the

process of converting the sea level pressure back to the desired actual station pressure.

These error estimates are probably on the extreme side, but it seems reasonable to say that the density

altitude calculations made using the National Weather Service sea level pressure calculations may havean uncertainty of 10% or more.

When using pressure data from the National Weather Service, be certain to find out if the pressure is

the altimeter setting or the sea-level corrected pressure. They may be quite different in some situations.

Simpler Methods of Calculation...

If you really want to know the actual density altitude, it will need to be calculated in the general manner

that has been described above. However, there are simple approximations which have been developed

over the years.

For example, a particularly convenient form of density altitude approximation is obtained by simply

ignoring the actual moisture content in the air. Here is such an equation which has been used by the

National Weather Service (seeref 13)to calculate the approximate density altitude without any need to

know the humidity, dew point or vapor pressure:

17)

where: DA= density altitude, feet

Pa= actual pressure (station pressure), inches Hg

Tr = temperature, deg R (deg F + 459.67)

This simplified equation (17) is, basically, just equation (12) rewritten in US customary units with no

pressure contribution due to water vapor pressure.

The following calculator can be used to compare the results of the accurate calculations (in geometric

altitude, as described earlier on this web page) with the results from the preceding simplified equation:

Comparison ofActual versus Simplified

Density Altitude

Air Temperature degrees F
http://wahiduddin.net/calc/density_altitude.htm#b13http://wahiduddin.net/calc/density_altitude.htm#b13http://wahiduddin.net/calc/density_altitude.htm#b13http://wahiduddin.net/calc/density_altitude.htm#b13


20/22

Actual Air Pressure inches-Hg

Dew Point degrees F

Reset

Air Density kg/m3

Actual Density Altitude feet

Simplified Density Altitude

feet

The results for dry air (very low dew point) are nearly identical, while the greatest errors in the

simplified equation are when there is a lot of water vapor in the air, i.e. high temperature accompanied

by a high dew point.

To explore the effects of water vapor, consider, for example, a hypothetical ambient temperature of 95

deg F, with a dew point of 95 deg, at an altitude of 5050 feet and an altimeter setting of 29.45 , the

actual air pressure would be 24.445 in-Hg and the actual Density Altitude would be 9753 feet, while the

simplified equation gives a result of 8933 feet.... an error of 820 feet. The actual air density in this case

would be reduced by about 3%, compared to dry air.

Or, for a hypothetical 95 deg F foggy day at sea level, with a dew point of 95 deg F and an altimeter

setting of 29.92, the actual density altitude is 2988 ft, while the simplified equation gives a result of

2294 ft... an error of 694 ft. Similar to the previous example, the actual air density in this would be

reduced by about 3%, compared to dry air.

Those examples are quite extreme, but in actual practice it is quite common to see errors on the order

of 200 to 400 ft along the sea coast and in the sweltering mid-west, which may be inconsequential, or

may be significant, depending upon your specific situation.

So, if you don't mind some error when the air has a lot of water vapor, then the simplified equation,

which is much easier to calculate, may suit your needs.

But if you really want the utmost accuracy in determining the density altitude, then you'll have to deal

with the gory details of vapor pressure and compute the "real" density altitude.


21/22

Based on the reported observations from a variety of US airports, it appears that the ASOS and AWOS-3

automated weather observation systems (which report weather conditions including density altitude at

many airports in the US) use a simplified equation which gives essentially the same results as equation

17 above. That is, it appears that the current ASOS/AWOS density altitude does not account for effects

of moisture in the air.

You can compare the actual Density Altitude with the ASOS/AWOS-3 reported values using the

calculator at:Density Altitude Calculator - with selectable units.

However, before you get too distressed by such seemingly "sloppy" ASOS/AWOS calculations, keep in

mind that the International Standard Atmosphere is merely a conceptual model which may or may not

accurately represent the conditions at any given location on any given day. That is, "density altitude"

and "standard atmosphere" are theoretical concepts which are based upon a number of assumptions

about the atmosphere, and may or may not accurately depict the actual physical conditions at any

actual location, no matter how accurate the calculations may be.

Actually, it would be far more meaningful, useful and precise if ASOS/AWOS reported the actual air

density in kg/m3, and if the performance data in pilot's handbooks was also expressed in terms of actual

air density in kg/m3. But that's not what is currently done. Currently, data in terms of "altitude" and

"density altitude" are generally what we're given. That's a pity.

Hopefully, someday all of the aircraft performance tables/charts and weather reporting systems will be

expressed in terms of the actual air density and thereby avoid this arcane concept of density altitude...

but, for now, we're stuck with "density altitude".

If we really want to be precise and consistent, we should be using the actual air density, not this

theoretical quantity called density altitude.

Density Altitude Calculation Algorithm...

For those who want to do their own density altitude calculations, here's a list of the steps performed by

my on-lineDensity Altitude Calculator:

1. convert ambient temperature to deg C,

2. convert geometric (survey) altitude to geopotential altitude in meters,

3. convert dew point to deg C,

4. convert altimeter setting to mb.5. calculate the saturation vapor pressure, given the ambient temperature

6. calculate the actual vapor pressure given the dew point temperature

7. use geopotential altitude and altimeter setting to calculate the absolute pressure in mb,

8. use absolute pressure, vapor pressure and temp to calculate air density in kg/m3,

9. use the density to find the ISA altitude in meters which has that same density,
http://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htmhttp://wahiduddin.net/calc/calc_da.htm


22/22

10. convert the ISA geopotential altitude to geometric altitude in meters,

11. convert the geometric altitude into the desired units and display the results.

an introduction to air density and density altitude calculations

Documents