Equations for Computing Temperature and Air Density Ratios at Different Altitudes

When computing aircraft performance at different altitudes, two useful variables are the air density ratio (dr) and temperature (T). Using a ratio is helpful because you can recompute values, such as dynamic pressure (Q), by simply multiplying the seal level value by the ratio. And regardless of what units you are using to represent air density, you can compute that value by multiplying the sea level value of that unit by the ratio.

The International Standard Atmosphere (ISA) tables contain standard values for these variables at different heights (h), stated in both meters and feet. These tables have been adopted by the International Civil Aviation Organization (ICAO) and can be found in ICAO Doc 7488/3. The US Standard Atmosphere is essentially identical.

Are there simple equations that I can use in an Excel spreadsheet to compute the air density ratio and temperature in SI units and US units that will agree with the ISA tables?

Yes, there are equations that can provide you with values that agree with the values in the ISA tables down to the nearest 100,000th (five digits), using the Geopotential altitudes.

Different equations apply to the Troposphere, which extends from Sea Level up to 11,000 meters (36,089 feet rounded), the Tropopause, which extents up to 20,000 meters (65,617 feet rounded) and the portion of the Stratosphere up to 30,000 meters (98,425 feet rounded). This is well above the altitude at which wings lose effectiveness.

Since the pressure ratio is related to the density ratio and the temperature ratio and since the temperature ratio is easily computed, I am adding computations for the temperature ratio and the pressure ratio.

The equations are described using Excel operators, such as * for multiplication and ^ for exponentiation, and Excel functions, such as EXP for e raised to the specified power. The units are described standard notations, such as ft for feet, m for meters, s for second and kg for kilogram. The notations do not use superscript, so that meters squared is m2, etc. I am using P for pressure, p for density, Tr for temperature ratio, pr for density ratio, Pr for pressure ratio and L for lapse rate.

1. TEMPERATURE AND TEMPERATURE RATIO

The equations for computing standard temperature are fairly simple and the results agree with the tables exactly. The temperature ratio can be computed by computing the Kelvin temperature and dividing the result by 288.15K. Thus, we will include the equations for computing T(K) in US units.

a. Troposphere

In the Troposphere, the basic equation is:
T = T0 - L * h
Where T0 is the temperature at sea level, L is the lapse rate and h is the height.

In SI units, T0 is 15 degrees Centigrade (15C) or 288.15 degrees Kelvin (288.15K), X is .0065 and h is in meters. The equations are:
T(C) = 15(C) - (0.0065(C) * h(m))
T(K) = 288.15(K) - (0.0065(K) * h(m))

In US units, T0 is 59 degrees Fahrenheit (59F), X is .00356616F and h is in feet. The equation is:
T(F) = 59(F) - (0.00356616(F) * h(ft))
T(K) = 288.15(K) - (0.0.0019812(K) * h(f))

b. Tropopause

In the Tropopause, the temperatures are constant.

In SI units:
T(C) = -56.5C
TK) = 216.65K

In US units:
T(F) = -69.7F
TK) = 216.65K

c. Stratosphere

In the Stratosphere, the temperature increases with height. The basic equation is:
T = T20 + L * (h - h20)
where T20 is the base temperature, L is the lapse rate and h20 is the base height.

In SI units, T20 is -56.5C or 216.65K, X is 0.001, h0 is 20000 meters. The equations are:
T(C) = -56.5(C) + (0.001(C) * (h(m) - 20000))
T(K) = 216.65(K) + (0.001(K) * (h(m)) - 20000))
which can be simplified to:
T(C) = -76.5(C) + (0.001(C) * h(m))
T(K) = 196.65(K) + (0.001(K) * (h(ft))

In US units, T20 is -67.7F, X is 0.00054864 and h0 is 65,617 feet. The equations are:
T(F) = -69.7F + 0.00054864 *(h(ft) - 65617)
T(K) = 216.65(K) + (0.0003048(K) * (h(m)) - 65617))
which can be shortened to:
T(F) = -105.7F + 0.00054864 * h(ft)
T(K) = 196.65(K) + (0.0003048(K) * (h(ft))

2. DENSITY RATIO

The equations for computing standard Density ratio are more complex, but the results (with added improvements) agree with the tables to the nearest .00001.

a. Troposphere

In the Troposphere, the basic form of the equation is the same in SI and US units:
pr = a ^ b
where A is the temperature index (in Kelvins) and the exponent b equals 4.2561.

Where height is measured in meters, standard the equation is:
pr = (1 - (0.0065 * h(m) / 288.15)) ^ 4.2561
You can improve the results by changing the exponent as follows:
d = (1 - (0.0065 * h(m) / 288.15)) ^ 4.25587971

Where height is measured in feet, the standard equation is:
pr = (1 - (h(ft) * 0.0019812/288.15)) ^ 4.2561
You can improve the results by changing the exponent as follows:
pr = (1 - (h(ft) * 0.0019812/288.15)) ^ 4.255882

b. Tropopause

In the Tropopause, the basic equation is:
pr = dr11 / exp((h - h11) * g / (R * T11)) where dr11 is the density ratio at the start of the Tropopause (.297076), h11 is the base height, g is gravitational acceleration, R is the specific gas constant for dry air and T11 is the base temperature (216.65K). Since g, R and T11 are constant, they can be combined into a single constant (X). The revised equation is:
pr = .297076 / exp(X * (h - h11))

In SI units, g is 9.80665 ft.s2, R is 287.05287 m2/s2 and h11 is 11,000 meters. Thus, X is 0.000157689. The equation is:
pr = .297076 / exp(0.000157689 * (h(m) - 20,000))
which can be shortened to:
pr = .297076 / exp(0.000157689 * h(m) - 1.734579))

In US unites, g is 32.1740486, R is 3089.8113775 ft2/s2 and h11 is 36,089 feet. Thus X is 0.000048063. The equation is:
pr = .297076 / exp(0.000048063 * (h(ft) - 36,089))
which can be shortened to:
pr = .297076 / exp(0.00004806346 * h(ft) - 1.734546))

c. Stratosphere

In the Stratosphere, we must first compute the Kelvin temperature first.

In SI units, the equation is:
T(K) = 196.65(K) + (0.001(K) * (h(m))
In US units, the equation is:
T(K) = 196.65(K) + (0.0003048(K) * (h(ft))

The basic equation for computing density altitude is: pr = (P20/(p0RT(K)*(T(K)/216.65)^34.1632)) where P20 is the base air pressure, p0 is the air density at Sea Level and R is the specific gas constant for dry air. Since these are constants, they can be combined into a single value (X). The modified equation is:
pr = (X / (T(K) * (T(K) / 216.65) ^ 34.1632))

In SI units, P20 is 5474.9 N/m2, p0 is 1.225 kg/m3, and R is 287.05287 m2/s2. In US units, P20 is 114.345664 lb/ft2, p0 is0.0023769 slgs/ft3, and R is 3089.8113775 ft2/s2. Thus, in both cases, X is 15.569627, which means that we can use the same equation for both SI and US purposes:
pr = (15.569627/(T(K)*(T(K)/216.65)^34.1632))

3. PRESSURE RATIO The equation for computing the pressure ratio is:
Pr= Tr*pr.
The equation for computing the temperature ratio is:
Tr=T(K)/288.15
We have included equations for computing T(K) in both SI and US units. Once you have the temperature ratio, you can use the density ratio and the formula above to compute the pressure ratio in both SI and US units.

RECAP

SI Units

Troposphere (0 meters to 11,000 meters):
T(C) = 15(C) - (0.0065(C) * h(m))
T(K) = 288.15(K) - (0.0065(K) * h(m))
pr = (1-(0.0065*h(m)/288.15))^4.25587971

Tropopause (11,000 meters to 20,000 meters):
T(C) = -56.5C
T(K) = 216.65K
pr = .297076/exp(0.000157689*h-1.734579)):

Stratosphere (20,000 meters to 30,000 meters)
T(C) = -76.5(C) + (0.001(C) * h(m))
T(K) = 196.65(K) + (0.001(K) * (h(m))
pr = (15.569627/(T(K)*(T(K)/216.65)^34.1632))

US Units

Troposphere (0 feet to 36,089 feet):
T(F) = 59(F) - (0.00356616(F) * h(ft))
pr = (1-(h(ft)*0.0019812/288.15))^4.255882

Tropopause (36,089 feet to 65,617 feet):
T(F) = -69.7F
T(K) = 216.65K
pr = .297076/exp(0.00004806346*h(ft)-1.734546))

Stratosphere (65,617 feet to 98,425 feet):
T(F) = -105.7F + 0.00054864 * h(ft)
T(K) = 196.65(K) + (0.0003048(K) * (h(ft))
pr = (15.569627/(T(K)*(T(K)/216.65)^34.1632))

SOURCE DOCUMENTS

Here are links to the ISA Tables and the Excel workbooks that I used to test these equations.