## Relations between height, pressure, density and temperature

**Introduction:**

g = Gravitational acceleration at a certain altitude (g_{0} = 9.81m/s^{2}) (m/s^{2})

r = Earth radius (6378km) (m)

hg = Height above the ground (Geometric height) (m)

ha = Height above the center of the earth (ha = hg r) (m)

h = Geopotential altitude (Geopotential height) (m)

p = Pressure (P a = N/m^{2})

ρ = Air density (kg/m^{3})

ν =1

ρ = Speciﬁc volume (m^{3}/kg)

T = Temperature (K)

R = 287.05J/ (kgK) = Gas constant

Ps = 1.01325 × 10

5N/m2 = Pressure at sea level

ρs = 1.225kg/m^{3} = Air density at sea level

Ts = 288.15K = Temperature at sea level

a =dT

dh = Temperature gradient (a = 0.0065K/m in the troposphere (lowest part) of the earth atmosphere) (K/m)

**Relation between geo potential height and geometric height:**

Newton’s gravitational law implicates:

(1)

The hydrostatic equation is: dp = −ρgdh_{g..
}

However, g is variable here for diﬀerent heights. Since a variable gravitational acceleration is diﬃcult to work with, the geo potential height h has been introduced such that: dp = −ρg_{0}dh

So this means that:

And integration gives the general relationship between geo potential height and geometric height:

(2)

**Relations between pressure, density and height:**

The famous equation of state is: p = ρRT

Dividing the hydrostatic equation (1) by the equation of state (2) gives as results: