The Derivation of the Density Altitude

MJ Mahoney

Created: Oct 26, 2005
Last Revision: Oct 27, 2005

Summary

From 0 to 11 km (0 to 36,000 ft), the Density Altitude is given by:

                     DA11km

where To = 288.15 K, LR= -6.5 K/km, Rho_Gk.PNGo = 1.225 kg/m3, Rd = 287.053 J/kg K,Gamma Gk45 =9.80665 m/s2, and epsilon_a = 0.622004944.

From 11 to 20 km (36,000 to 65,600 ft), the Density Altitude is given by:

                      DA24

where  zo = 11 km, To = 216.65 K, po = 226.3206 hPa, Rd = 287.053 J/kg K, Gamma Gk45 =9.80665 m/s2, and epsilon_a = 0.622004944.


DA from 0 to 11 km

The derivation of the density altitude begins with the equation for the pressure as a function of altitude for the US Standard Atmosphere (1976), where temperature is used as a proxy for altitude, that is, T=To + LR (z-zo). To an altitude of 11 km (36,000 ft):

(1)     p_po.PNG

Using the ideal gas law to eliminate p and po, IGL.PNG and po.PNG, we obtain:

(2)    Rho_Rhoo.PNG

which leads to:

(3)     T_To.PNG

The expression on the right is just T=To + LR (z-zo) divided by To (with zo=0 as it is at MSL). It allows us to solve for the density altitude (z) in terms of the ratio of the actual density to the density at MSL:


(4)     DA

where, using the ideal gas law, the actual density (Rho) is given by:

(5)     Rho_pTRH
Substituting Equation (5) into equation (6), we finally obtain an expression for the density altitude to 11 km:

(6)      DA11km

Equation (6) is only valid to 11 km (36,000 feet). Another expression is derived below for altitudes between 11 and 20 km. In Equation (5) Tv is the virtual temperature. Because water vapor is lighter than dry air, it has a gas constant (Rv) which is larger that that of dry air (Rd). The virtual temperature is defined to allow only the gas constant for dry air to appear in the gas law. Note that if the relative humidity (RH) is zero, the density is just that for dry air (p/(Rd T). What the virtual temperature does is reduce the density by decreasing the actual pressure (p) by the water vapor partial pressure (e) times (1-epsilon_a), where epsilon_a is the ratio of the gas constants for dry air and water vapor, or equivalently, the inverse ratio of their gram molecular weights (Mv/Md). The last expression in equation (5) is obtained by using the definition of the relative humidity (RH): RH_e_es.PNG, as the ratio of the vapor pressure (e) and the saturation vapor pressure (es(T)) expressed as a percentage.

DA from 11 to 20 km

From 11 to 20 km, the ISA atmosphere is isothermal (To = 216.65 K), and therefore has a different relationship from Equation (1) for pressure. In this case,

(7)     DA20

If this equation is solved for the altitude z, we get:

(8)     DA23

We now need to come up with an expression for po/p. To get this, we first note that the density is given by:

(9)      DA21
where the subscripts d and v refer to dry air and water vapor, respectively, e is the partial pressure of water vapor, and epsilon_a is the ratio of the gas constants for dry air and water vapor. The ratio po/p can now be written as:

(10)      DA22

where we have substituted Equation (9) into the middle expression to get the final result. If we now substitute this final result into Equation (7), we get the following result for the density altitude from 10 to 20 km (36,000 - 65,000 feet):

(11)      DA24
In this equation, zo = 11 km, To = 216.65 K, po = 226.3206 hPa, Rd = 287.053 J/kg K, Gamma Gk45 =9.80665 m/s2, and epsilon_a = 0.622004944.