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Air Density and Density Altitude Calculations
updated: June 1, 2009
mucky
Density Altitude On-Line Calculators:
The following density altitude calculators are available on this web site:
- using English Units - using Metric Units - with selectable Units - using relative humidity
What is density altitude?
The density altitude is 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 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 that ignores the effects due to water vapor in the air.
Some different meanings of the word "altitude":
An aircraft altimeter measures only air pressure... nothing else. If the air pressure changes, due to temperature or humidity, then an aircraft altimeter will of course change to indicate the actual air pressure. Nonetheless, the aircraft altimeter is simply measuring air pressure.
As odd as it may seem, an aircraft altimeter does not actually measure altitude, it only measures pressure. Hence, the name "pressure altitude" is properly applied to any aircraft altimeter reading.
For pilots, it is very important to understand that an aircraft altimeter only measures air pressure (not 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) may note some other altitude displayed on the GPS (which measures actual distance above mean sea level). In some cases this difference is small... but in some cases it could be enough to cause a mid-air collision if a pilot was flying on a GPS altitude rather than pressure altitude. (To solve that problem, some GPS units do include an air pressure sensor so that they can indicate pressure altitude.)
Therefore, it is crucial to always verify what is meant by "altitude", and differentiate a pressure-based measurement of "pressure altitude" from a distance-based measurement of actual altitude.
Density altitude is a concept based solely on air density, and is neither "pressure altitude" nor "mean sea-level altitude", but is strictly "density altitude" (the altitude at which the air has a certain value of density).
Now... on to Density Altitude.....
Density and Density Altitude:
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 and the thrust of a propeller blade are all directly proportional to the air density. The downforce of a racecar spoiler is also directly proportional to the air density. Similarly, the horsepower output of an internal combustion 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.
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.
(Note: If you're just hunting for a simple, but not very accurate, approximation for density altitude, be sure to study the "Simpler Methods of Calculation" section near the end of this article.)
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). See ref 8 and ref 9 for 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*R*T
Where: P = pressure V = volume n = number of moles R = gas constant T = temperature
Density is simply the number of molecules of the ideal gas in a certain volume, in this case a molar volume, which may be mathematically expressed as:
(2) D = n / V
Where: D = density n = number of molecules V = volume
Then, by combining the previous two equations, the expression for the density becomes:
(3)
Where : D = density, kg/m3 P = pressure, Pascal’s ( multiply mb by 100 to get Pascals) R = 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.
Neglecting the small errors due to non-ideal gas compressibility and vapor pressure measurements not made over liquid water (see ref 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 (see ref 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:
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 (see ref 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 saturation pressure 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
See ref 2 and ref 11 for additional vapor pressure formulas.
Here's a calculator that compares the saturation vapor pressure using equations 5 and 6 given above:
Saturation Vapor Press Calculator
Top of Form
Air Temperature 20degrees C
Top of Form
Sat vapor press from Eqn 5 mb
Sat vapor press from Eqn 6 mb by Richard Shelquist
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.
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 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 CalculatorTop of Form
Air Temperature degrees C
Actual Air Pressure mb
Dew Point degrees C
Top of Form
Air Density kg/m3
by Richard Shelquist
Moist Air is Less Dense...
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 comparisons to a liquid are of little help in understanding what is going on in the air, and may simply result in more 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.
Most of the air is made up of nitrogen molecules N2 with a somewhat lesser amount of oxygen O2 molecules, and then 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. Notice that a water molecule is lighter weight than either a nitrogen molecule or an oxygen molecule.
Therefore, when a given volume of air, which contains only a certain number of molecules, has some water molecules in it (which are very light weight), it will weight less than the same volume of air without any water molecules.
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 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 is 1.2250 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 (see ref 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 jet j1 = 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 available horsepower 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 the SAE method of correcting horsepower.)
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 (ISA) 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 International Standard Atmosphere is a mathematical description of a theoretical column of air. To get the proper results, it is necessary to use the following constants that are specified in the 1976 International Standard Atmosphere document:
Po = 101325 sea level standard pressure, PaTo = 288.15 sea level standard temperature, deg Kg = 9.80665 gravitational constant, m/sec2
L = 6.5 temperature lapse rate, deg K/km
R = 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). The model that will be developed here is only valid in the troposphere. The equations that define the air in the troposphere are:
(9)
(10)
(11)
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 and solve for the variable H, which is the geopotential altitude.
So, it is now necessary to rewrite equations 9, 10, and 11 in a manner that expresses altitude H as a function of density D. After a bit of gnashing of teeth and general turmoil, 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 = geo potential 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 = geo potential 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 the definitions 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.
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, the following formula may be used:
(13)
where E = 6356 km, the radius of the earth (for 1976 ISA) H = geo potential 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
Top of Form
Air Density kg/m3
Top of Form
Geopotential altitude H m
Geometric altitude Z m by Richard Shelquist
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
Top of Form
Air Temperature degrees C
Actual Air Pressure mb
Dew Point degrees C
Top of Form
Air Density kg/m3
Geopotential altitude H m
Geometric altitude Z m by Richard Shelquist
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.
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/Md and 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 Md is molecular weight of dry air = 28.9644 Mv is 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
Top of Form
Air Temperature degrees C
Actual Air Pressure mb
Dew Point degrees C
Top of Form
Virtual Temperature degrees C by Richard Shelquist
The virtual temperature Tv may 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 virtual temperature 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 the pressure measurement page.)
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 in National Weather Service reports, and can be used to determine the actual pressure using the following equations:
According to NWS ASOS documentation, the actual pressure Pa is 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 (see ref 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 in ref 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 Setting to Actual Pressure
Top of Form
Altimeter Setting hPa (mb)
Geometric Altitude meters
Top of Form
Geopotential Altitude meters
Actual Pressure hPa (mb) by Richard Shelquist
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 was 10 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.
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 have an 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.
Density Altitude Algorithm...
Here is a list of the steps performed by my Density 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,10. convert the ISA geopotential altitude to geometric altitude in meters,11. convert the geometric altitude into the desired units and display the results.
My On-Line Density Altitude Calculators:
Click here for Density Altitude Calculator with English units only.
Click here for Density Altitude Calculator with Metric units only.
Click here for Density Altitude Calculator using relative humidity rather than dew point.
Click here for Density Altitude Calculator with both English and Metric units.
Click here for new Engine Tuner's Calculator that includes relative horsepower, air density, density altitude, virtual temperature, absolute pressure, vapor pressure, relative humidity and dyno correction factor!
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 many forms of simpler approximations and generalizations that have been developed over the years, but please note that they are not really the density altitude, they are just numbers that approximate the density altitude.
In some situations, the density altitude approximations can be fairly accurate, but in some real life situations with high moisture content in the air, the approximations can sometimes be quite inaccurate. The simpler form of the approximations is obtained by simply ignoring the actual moisture content in the air.
Nonetheless, for those who really want a simpler equation, here is an equation used by the National Weather Service (see ref 13) to calculate the approximate density altitude without any need to know the humidity, dew point or vapor pressure:
where: DA = density altitude, feet Pa = actual pressure (station pressure), inches Hg Tr = temperature, deg R (deg F + 459.67)
This simplified equation is, basically, just equation (12) above 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 of Actual versus Simplified Density Altitude
Top of Form
Air Temperature degrees F
Actual Air Pressure inches-Hg
Dew Point degrees F
Top of Form
Air Density kg/m3
Actual Density Altitude feet
Simplified Density Altitude feet by Richard Shelquist
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.
For example, on a hot, rainy summer afternoon here in Colorado, 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 pressure is 24.445 in-Hg and the actual Density Altitude is 9753 feet, while the simplified equation gives a result of 8933 feet.... an error of 820 feet.
So, if you don't mind errors approaching 10% when the air is saturated with a lot of water vapor (that is, on a hot day with the dew point approaching the ambient temperature), then the simplified equation, which is much easier to calculate, may suit your needs. But if really want the utmost accuracy, then you'll have to deal with the gory details of vapor pressure.
enjoy....
Richard ShelquistLongmont, Colorado
