Geoscience Reference
In-Depth Information
This is the basic equation for barometric height measurements, and it also holds
for density:
d
ρ
ρ
g
R
d
·
=−
dh
.
(24)
T
v
Strictly speaking, gravity and temperature are also dependent on height, but con-
sidering only small height changes, we can assume them to be constant. Integration
between Earth surface (index
B
) and height
h
yields:
g
R
d
·
e
−
T
v
dh
p
=
p
B
·
,
(25)
g
R
d
·
e
−
T
v
dh
ρ
=
ρ
B
·
.
(26)
The factor of
dh
deserves a closer look. Since the exponent has to be dimension-
less, the factor has the dimension 1/m. Thus, we use
g
R
d
·
1
H
T
v
=
(27)
where
H
is the barometric scale height, and it can be considered as the height of
an atmosphere with constant density. Since in a homogeneous atmosphere
ρ
=
ρ
B
,
Eq.
22
can be integrated easily. The limits for the integration are
p
B
and zero for
p
,
as well as zero and
H
for
h
:
p
B
=
·
ρ
B
·
.
g
H
(28)
With the gas law we get the height of the homogeneous atmosphere:
R
d
·
T
v
,
B
g
H
=
(29)
or
1
3
e
8
p
H
=
H
0
(
1
+
α
·
t
)
+
(30)
where
H
0
is the height of a homogeneous atmosphere of dry air at 0
◦
C. Table
1
shows
the homogeneous heights at various surface temperatures for dry and saturated air.
Note that if the atmosphere would consist of pure oxygen only, then
H
would be
8.250km at 0
◦
C, for helium 57.7km, and for hydrogen 114.6km.
Ta b l e 1
Values of Scale
Height
H
in km for different
surface temperatures and dry
or saturated air
(
g
t
B
in
◦
C
H
in km for dry air
H
for saturated air
−
10
7.686
7.703
0
7.979
7.997
s
−
2
)
=
9
.
806m
/
10
8.272
8.310
20
8.565
8.640
30
8.858
9.000
