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Deriving Equations for Atmospheric Pressure and Density

The levels of air pressure and air density in the atmosphere as used in the Barometric Formula depend on the altitude, the amount and composition of the atmosphere and the temperature profile. Atmospheric pressure, density and temperature are related via the ideal gas law.

(1)

Ideal Gas Law

where'
' =' 'air pressure
' =' 'air density
' =' 'absolute temperature
' =' 'Spezific Gas Constant; dry air = 287.058 J/(kg·K)

We can not measure the amount of atmosphere directly, but we can measure a rough temperature profile with balloons. The overall dependence of atmospheric pressure with altitude can be derived by observing the pressures acting on a small volume. Using calculus we can then derive the pressure curve for the whole atmosphere. Using the temperature profile and the ideal gas law we can also derive the density gradient from the pressure gradient.

  1. determining the pressure on a small volume element
  2. creating an emirical simplified model for the temperature profile of the atmosphere
  3. deriving the atmospheric pressure profile as a function of altitude and temperature profile using calculus
  4. deriving the density profile from the pressure and temperature profile using the ideal gas law

Pressure Differential

To derive the pressure as a function of altitude, we need to calculate how the pressure changes with altitude. Lets regard an ifinitesimal small cube of gas and calculate the pressures acting at the top and bottom area of the cube.

The downward force acting on the area at the top of the cube is the weight of the column of atmosphere above the cube. The downward force acting at the bottom of the cube is the force at the top plus the weight of the cube. So we have the pressures at the top and bottom of the cube respectively:

(2)
(3)

The pressure difference between the top and bottom of the small cube is:

(4)
with
where'
' =' 'weight of the cube
' =' 'gravitational acceleration
' =' 'density of the air in the cube
' =' 'volume of the cube
' =' 'base and top area of the cube
' =' 'height of the cube

Inserting into the equation for we get:

(5)

So the vertical pressure differential acting on a small volume element of density under the gravitational acceleration g is:

(6)
where'
' =' 'air pressure differenctial at altitude h
' =' 'gravitational acceleration
' =' 'air density at altitude h
' =' 'height differential

Using the ideal gas law (1) we can replace the density in (6) by the pressure to get the following differential equation:

(7)
where'
' =' 'air pressure differenctial at altitude h
' =' 'gravitational acceleration
' =' 'air pressure at altitude h
' =' 'absolute air temperature at altitude h in Kelvin
' =' '287.058 J/(kg·K) = Spezific Gas Constant for dry air
' =' 'height differential

Solving Nummerically

The differential equation (7) can be solved for the pressure gradient nummerically. In this case g and T can be arbitrary functions or measurements. Nummerically we solve the following equation iteratively:

(8)
with
and

We begin with some start conditions, which could be measurements, for P0(h0), g0(h0), T0(h0). After each iteration we get the new pressure for the next height increment and we may have different g and T values. There are more complicated ways for nummerical integrations, that yield more accuracy.

Analytical Solutions

If we make the simplifying assumptions that g is constant for the height range we want to solve for and T is constant or a linear function of h, we can solve the differential equation (9) analytically.

(9)

We can bring the pressure P from the right side of the equal sign to the left side, so we have all pressure terms on the left, which simplifies solving the differential equation:

(10)

Note: Pressure and temperature are commonly dependent on altitude. The temperature has to be measured empirically. It can also be constant over some altitude range.

In the International Standard Atmosphere model, used in aviation and other fields, the empirically measured temperature gradient is divided into some ranges and approximated either as constant (isotherm) or linearly dependent on altitude.

Pressure Isotherm

Lets first derive the pressure equation for the simpler case of constant temperature.

We can solve (10) by integrating both sides of the equal sign. The integration 1/P dP = ln(P) is applied to the left side, the integration of the right side is straight forward A · ∫ dh = A·h:

(11)
(12)

Applying the limits:

(13)

To get rid of the logarithms we apply ex to both sides, because eln(X) = X:

(14)

Bringing the constant to the right hand side we get finally the formula for pressure as a function of altitude for constant temperature:

(15)
where'
' =' 'pressure at altitude h above sea level
' =' 'pressure at the reference altitude href above sea level
' =' '9.80665 m/s2 = gravitational acceleration
' =' '287.058 J/(kg·K) = Spezific Gas Constant for dry air
' =' 'constant absolute temperature in the range from href to h
' =' 'altitude above sea level
' =' 'reference altitude above sea level

This formula makes the simplification that the gravitational acceleration g is constant in the altitude range href ... h.

Pressure with Linear Temperature Gradient

We start again with the differential equation derived at (10), repeated here:

(16)

But this time temperature T is not constant, but a linear function of altitude h, so that T(href) = Tref:

(17)

Inserting (17) into (16) we get the new differential equation:

(18)

We can solve (18) by integrating both sides of the equal sign. The general integral ∫ (1/X) dX = ln(X) is applied to both sides:

(19)
(20)

Putting in the limits yields:

(21)

Applying ln(A) − ln(B) = ln(A/B) on both sides yields:

(22)
with

To get rid of the logarithms we apply ex to both sides and because exp(−β · ln(X)) = exp(ln(X)·(−β)) = exp(ln(X))β = Xβ we get:

(23)

Bringing to the right hand side we finally get our formula for pressure as function of altitude in the case of a linear temperature gradient:

(24)
with
where'
' =' 'pressure at altitude h above sea level
' =' 'pressure at reference altitude href above sea level
' =' 'temperature at reference altitude href
' =' 'temperature gradient (negative lapse rate)
' =' 'altitude above sea level
' =' 'reference altitude above sea level
' =' '9.80665 m/s2 = mean gravitational acceleration at sea level
' =' '287.058 J/(kg·K) = Spezific Gas Constant for dry air

Density Isotherm

Using the ideal gas law (1) we can derive the equation for the density in an altitude range of constant air temperature T from the equation for Pressure Isotherm:

(25)

And because is the reference density we get finally:

(26)
where'
' =' 'air density at altitude h
' =' 'air density at reference altitude href, at which the constand temperature range starts
' =' 'constant absolute temperature in the range from href to h
' =' 'altitude above sea level
' =' 'reference altitude above sea level
' =' '9.80665 m/s2 = mean gravitational acceleration at sea level
' =' '287.058 J/(kg·K) = Spezific Gas Constant for dry air

Density with a Linear Temerature Gradient

Using the ideal gas law (1) we can derive the equation for the density in an altitude range with a linear temperture gradient (17) from the equation for Pressure with a Linear Temerature Gradient:

(27)

We can combine the 2 equal expressions in parenthesis to get finally:

(28)
with
where'
' =' 'density at altitude h above sea level in a range with a linear temperature gradient
' =' 'density at reference altitude href above sea level
' =' 'temperature gradient (negative lapse rate)
' =' 'temperature at reference altitude href
' =' 'altitude above sea level
' =' 'reference altitude above sea level
' =' '9.80665 m/s2 = mean gravitational acceleration at sea level
' =' '287.058 J/(kg·K) = Spezific Gas Constant for dry air
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