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Standard Atmosphere Definition & Equations

The standard atmosphere consists of isothermal and gradient regions defined by variations in temperature with altitude. In the isothermal regions, temperature remains constant, so pressure and density decrease exponentially with increasing altitude. In the gradient regions, temperature decreases linearly with altitude based on the lapse rate. Pressure and density in these regions decrease according to equations that incorporate the varying temperature profile. The standard atmosphere table is constructed by integrating equations of hydrostatic balance starting from sea level values and moving through each atmospheric region.

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118 views4 pages

Standard Atmosphere Definition & Equations

The standard atmosphere consists of isothermal and gradient regions defined by variations in temperature with altitude. In the isothermal regions, temperature remains constant, so pressure and density decrease exponentially with increasing altitude. In the gradient regions, temperature decreases linearly with altitude based on the lapse rate. Pressure and density in these regions decrease according to equations that incorporate the varying temperature profile. The standard atmosphere table is constructed by integrating equations of hydrostatic balance starting from sea level values and moving through each atmospheric region.

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Suraj NK
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Definition of standard atmosphere

 The keystone of the standard atmosphere is a defined variation of T with altitude, based
on experimental evidence. This variation is shown in the below figure.

Fig 1: Temperature distribution in the standard atmosphere

 Standard atmosphere consists of series of straight lines , some vertical (called the
constant temperature or isothermal regions) and others inclined( called the gradient
regions).

Consider the hydrostatic equation, dp = - ρg dhG ………………………… (1)


 hG – geometric altitude
 Since in the above equations, g=g(hG) i.e, local gravitational acceleration is varying with
altitude.
 The above equation should be integrated to get variation of pressure with altitude
,p=p(hG).
 To simplify the integration, we make assumption that g is constant throughout the
atmosphere. So eq (1) reduces to dp = - ρgo dh…………….. (2)
 h- geo-potential altitude
 go – constant gravitational acceleration
 Given T=T(h) as defined by the above figure(1), then P=P(h) and ρ=ρ(h) follow from
the laws of physics, as shown in the following.

 First consider, dp = - ρgo dh


 Divide by equation of state, p= pRT

 …………………………………(3)
 Consider first the isothermal (constant temperature ) layers of the standard atmosphere,
as shown by vertical lines in fig(1) and sketched in fig(2).

fig 2: isothermal layer


 The temperature, pressure and density at the base of isothermal layer is T 1 ,P1 and ρ1
respectively. The base is located at a given geo-potential altitude h1 . now consider a
given point in the isothermal layer above the base, where the altitude is h. the pressure p
at h can be obtained by integrating eq(3) between h1 and h.

……………………..(4)
 Note that g0 , R and T are constants that can be taken outside the integral.
………………………….(5)

………………………(6)
Thus equation 5 and 6 gives the variation of p and ρ versus geo-potential altitude for the
isothermal layers of the standard atmosphere.

 Considering the gradient layers as sketched in below fig(3)

Fig(3) :gradient layer


 Temperature variation is linear and is geometrically given as

where a is lapse rate for the gradient layer

We substitute this result in to eq(3)

………..(7)
Integrating between the base of the gradient layer and some point at altitude h, also in gradient
layer, eq(7) yields
…………………(7)
From the equation of state and eq(7) , we have

………… (8)

…………………….(9)
 Eq(9) gives T=T(h) for the gradient layers: when it is plugged into eq(7), we obtain
P=P(h) and similarly from eq(8) gives ρ=ρ(h).
 Looking at fig(1), start at sea level(h=0) where standard sea level values of prssure ,
density and temperature – PS , ρs and Ts are

From eq( 3) and eq( 8), the table of values for the standard atmosphere can be constructed

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