Journal
of
Research
of
t
he
Noti
onal
Bureau
of
Standard
s Vol. 50, N
o.1,
Januar
y 1953
Research
Paper
2385
The
Constants
in t
he
Equation for Atmospheric
Refractive
Index
at
Radio Frequencies
Ernest K. Smith, Jr.,
and
Stanley
Weintraub
Re
cent
improv
e
ment
s
in
microwav
e
tec
hniqu
es
hav
e
resulted
in
precise meas
urem
e
nt
s
at
th
e
Nationa
l
Bur
eau
of
Standard
s, the
Nationa
l
Phy
sical
Laboratory
,
and
elsewhere,
which
indicat
e t
ha
t t he conve
ntional
constants
K
\=
79
° K/
mb
and
K
2
'=
4,800° K
in
the
expression for
the
refractivity
of
air
, N = (n - 1) 10
6
= (K d T )
[p
+ K
2
'
(e
/T)] s
hould
be
revi
sed .
Variou
s
laboratori
es
app
ea
r
to
hav
e
arriv
ed
at
this conclus
ion
ind
ep
en
de
ntly,
wi
t.h
the
result
t
ha
t there
are
seve
ral
differ
ent
sets
of
constants
in
c
urr
ent use.
In
much
of
propagation
work
the
absolute
value of
the
refractive
in
dex of the
atmosphere
is of small
moment.
Howev
er,
in
s
om
e
work
it
is
imp
o
rtan
t,
and
it
seems highly desiI
abl
e to decide
upon
a
particular
set
of
con
sta
nt
s.
Through
conside
rat
ion
of the
various
r
ecent
ex
perim
ents
a rel
ation
N = (77.6/ T)
[p
+ 4810
(e/ T)] is deriv
ed,
where p is t he
tota
l
pr
e s
ur
e,
in
millibars, e is t he
par
tia
l
pre
ss
ur
e of
wat
er
vapor,
in
millibars,
an
d T is
ab
solu
te
te
mperature
(O
C+ 273
).
Thi
s
expre
sion is con-
sidere
d
to
be
good
to
0
.5
pe
rc
ent
in
N f
or
fr
eque
ncies
up
to
30,000 megacycles
and
normally
enco
un
tere
d
range
s of
temperature,
pI' ssure,
an
d h
umidity.
R ece
nt
improve
ment
s in microwave techniques
hav
e
re
sulted
in
measureme
nts
at
the
National
Bur
ea
u of
Standard
s [1]/ the National
Phy
sical
Laborator
y
[2],
a
nd
elsewhere
[3
, 4, 5
],
which
hav
e
indi
cate
d
that
the
conventional constants
in
the
ex
pre
ss
ion for the refractive index of air
at
radio fr
e-
qu
encies shou
ld
be
revised. Various laboratories
appear
to
ha
ve arrived
at
this conclusion indepen-
de
ntl
y,
with
the
result
that
there are seve
ral
differe
nt
sets
of
constants in
current
use
[6
, 7, ,
9]
.
Th
e
sources of these recent changes, such as
hav
e been
run
to earth,
have
b
ee
n found to be
ba
ed on indivi-
dual
rather
than
co
ll
ecLive
r
es
ults. Almost all the
proposed c
onstant
s seem to re
pr
ese
nt
a sub
sta
ntial
improvement
over the former values.
Th
e
authors
propose a
set
of
constanLs derived from
what
is fe
lt
to be the
most
reliable 0f
Lh
e rece
nt
microwave
and
optical measure
ment
s of
the
refractive index of
I .
.
dry
air
and
from a rece
nt
s
urv
ey
of
wat
er
vapor
Debye constants.
It
is hoped t
hat
these new con-
sta
n ts will provide a common meeting g
round
for
the
laboratories desiring change
rath
er
than
inje
ct
ju
st
another
set
of
va
lu
es
int
o the field.
For
an
accuracy of 0.5 percent in N, the scaled-
up
refractiv
it
y of moist air
[N
=
(n-
l )
10
6
],
where n
y is
the
refractive index, some simplifying ass
umption
s
m
ay
be
made
if the use of
the
relation is to
be
re-
st
ri
cte
d to certain limits of the variabl
es.
The
limits
in
this case restrict
its
use to
temperat
ur
es
of
-5
0 to + 40° C,
total
pressures
of
200 to 1,100
mb
,
water-vapor
partial
pr
ess
ur
es
of 0 to 30
mb
,
and
a
frequency range of
0 to 30,000
Me
.
Th
e
co
ns
tituent
s
of
dr
y air
and
even
water
vapor
may
be assumed to
obey the ideal gas law
[10]
.
Th
e refractive index
of
water
vapor, a polar molecule
with
an
electric
dipol
e,
may
be
represented
by
a two-term Debye
relation
[ll]
.
The
permeability of air
at
radio fre-
quencies due to oxygen
may
be taken as 1 +
O.4
X
10
-
6
[
J]
.
Di
spersion
ma
y
be
neglected.
Th
e
Lor
e
ntz
polarization term
ma
y
be
ignored.
\.b
sol
ut
e zero
.
Fi
gures
In
brackets in
di
cate the literature references
at
the
end
of this pa.per.
39
for
tempera
Lur
e
ma
y be
taken
as - 273° C
rath
er
than
- 273.16° C [12].
Ther
e
ha
s b
ee
n no proof of
var
iation in
tbe
co
m-
position of the
dry
gases
of
Lhe free atmosphere
e
ith
er
with
l
at
itud
e or
with
heig
ht
up
to
the
iono-
s
ph
ere [13].
Th
e
water
vapor
c
ontent
, of course,
var
i
es
widely. A contributions to
Lh
e
tota
l re-
fra
ct
iv
e
ind
ex obey
an
acldiLive
rul
e,
a
th
r
ee
-t
erm
ex
pression
ma
y be formulated
[ll],
in
which
the
fir
st
te
rm
ex
pr
esses the
um
of the distortion of
el
ectronic c
har
ges of
the
dr
y-gas mol
ec
ules
und
er
the
influence of
an
applied electromagnetic field ;
the
second term,
th
ese di
st
ortions for
water
vapor
;
and
the third term, the e
fT
ect
of
the
orien
tat
ion of
the
electric dipol
es
of
water
vapor
und
er
the
influence of
a
fi
e
ld
.
Thu
s,
us
in
g N for the scaled-up
refractivity
[N
=
(n-
l )106
],
=K
1
(~)+K
2
(
f)+
Ka (;2} (1)
where n i
Lb
e refractive
ind
ex
at
radio fre
qu
encies;
P
a
,
the partial press
ur
e
of
the
dr
y gase ;
e,
the
partial
pr
e sure of
wat
er
vapor;
and
T, the absolute tem-
peratur
e.
In
radio work one is
inte
rest
ed in propagation
through
th
e free atmosphere.
Ther
efore, the com-
position of air should
be
taken
to include
an
average
amount
of carbon dioxide. However,
labora
tory
meaSUl'
e
ment
s usually are
made
on
COrfree
air
because of variable
co
n
ce
ntration
s of CO
2
in
th
e
laborator
y. Hence, those values
of
€- 1 originall
published for
CO
2
-free air
ha
ve been
ad
ju
sted
for
0.03 perce
nt
CO
2
content
by
raising
them
0.02 per-
ce
nt
.
Th
ese values are al
so
g
iv
en on a real
ratb
er
than
an
ideal gas basis.
Three
determinations s
hown
in
table 1 are considered.
Th
e fir
st
shown,
that
of
Barrell
[14], is
an
average
of
the
constant
term
(n
for A=
00)
of
the
optical
Cauchy
dispersion e
qua-
tions for
standard
air used
in
three
of
the principal
metrology laboratories of the
wo
rld.
Theor
et
ical
co
nsiderations indicate th
at
the dielectric
constant
for
dr
y ail' will be the i
sa
me for optical and radio
fre
qu
enc
ie
s. Barrell's value is
co
nverted to dielec-
tric c
on
s
tant
from the relation n=w
with
Il
, the
permeability,
ta
ken as uni
ty
at
optical fre
qu
encies.
Th
e second valu
e,
th
at
of
Birnbaum
,
Kr
yder, a
nd
Lyons
[1]
,was originally pu blished on
an
ideal
ba
sis
but
ha
s here been
co
n
verte
d to the
va
lue per tinent to
real gases.
Th
e
last
deter
min
at
ion,
that
of
Ess
en
a
nd
Froom
e
[2],
ha
s been
adju
sted to include
CO
~
.
Th
e
un
cer
ta
inties li
ste
d are
sta
nd
a
rd
e
rr
ors.
TAB
LE
1. D
Ty
-ai
r Tefractive
index
and
diel
ec
tric constant at
0° C
an
d 1 atmosphere
Frequency
of measure-
Mea
sw'ed N
(.--
1)10
6
Year
ment
-----
--
Barrell
[14J
____
.
__
.
__
.
OpticaL
____
28
7.7
,±
0.0"
fi75
.6± 0.1 3 1951
Birnbaum, Kryder
9,000
Mc
_
__
_
575.8± 0.36 1951
"nd
Lyons
tiJ.
Essen and roomc
24
,000 M c
___
288
.2
1±
0.1 57
6.
1± 0.2 1
951
[2J
.
Mean
'·"
Iu
e
___________
_
_________
• 288.0
.±
0.0, 575.7
,±
0.
10
--._-.--
Ut
ilizing the
tota
l press
Ul"C
, p= P d+
c,
one
ma
y
write
For
use in the limited te
mp
erature
range - 50° to
+ 40° C, negligible e
rror
is incurred
thro
ugh lumping
th
e seco
nd
and third terms
in
(7).
Thi
s m
ay
be
accomplished by dividing these t
er
ms by cfT and
solving for the composite con
st
ant
, K
4
,
in the rel
at
ion
(8) ,
which, for T= 273 ° K, results in
(9) \
Th
e t
wo
-te
rm
formula for dry air is now
N =
77.6
~ +
3.73
X
I0
5
;2'
(10)
a
Derived
Il'om
n=
..;;;.
where
,,
- 1= OAXIO-6 is taken
for
radio frequenci
es
to
accou
nt
for the permeab
ili
ty
. which
ma
y be
wr
i
tten
Th
e
stat
i
st
ical me
an
value of dielect
ri
c
co
n
sta
nt is
then
co
nver ted to
re
fra
ct
ive ind
ex.
Th
e cons
tant
K 1 is eva
lu
ate
d from
(2)
which
ste
ms from (1) when
c=
o.
Sett
ing N = 2S8.04,
p = 1013.25
mb
, T=
273
° C
and
solving for K
1
:
. (3)
A rece
nt
s
urv
ey of dete
rmin
at
ions of the dielectric
cons
tant
of
wat
er
vapor
in
th
e
mi
crowave region
b~
Birnb
a
um
and
C
ha
tte
rjee [
15]
is used to e
valuat
e
the
wat
er-
vap
or con
sta
nts K z
and
K s in (1). As-
s
umin
g ideal gas behavior, which is permissible here
as only low
partial
pre
ss
ur
es are of
int
er
es
t,
th
e
co
nstants
m
ay
be
eva
lu
ate
d from the Dcbye con-
s
tant
s .!l a
nd
B (molar polarization P= .!l
+Hf
T)
dete
rmined
by Bi
rnb
aum a
nd
C
hat
terj
ee.
Th
is
results in
(4)
(5)
where the
un
certa
inti
es are again
sta
ndard errors.
Substituting
the
va
lues of (3), (4),
and
(
5)
in (1),
and
reducing
th
e
va
lu
es
to t
hr
ee
fi
gures
wh
ere
signifi
ca
n t
N = 77.6
~+72
f+3.
75 X
10
5
;2
' (6)
40
(
ll
)
Th
ese
la
st
two rel
at
ions are the ones proposed for
gen
era
l radio meteorological use.
Li
ste
d
in
ta
ble 2 are s
om
e of the values of K
1
,
K
z
,
and
K
a
,
t
ha
t
hav
e been used by
author
s throu
gh
the years. In examples 1 tlu'ough 7 the cons
tant
K I
wa
s
drawn
from
th
e Smithsonian
Ph
ysical Tables
(1933).
Th
e
water-vapor
co
n
st
ants K2 a
nd
K s in
examples 2 to 7
represent
an
averag
e of the deter-
mination
s of several, differe
nt
workers
in
each case.
Essen a
nd
Froom
e in example 8 relied on their own
excellent experime
nt
al work
to
eva
lu
ate
Kl
a
nd
K
s.
Th
eir value of K
t
is seen from table 1
to
be in excel-
lent agreement
with
the
va
lu
es
of
Barr
ell a
nd
of
Birnbamn
,
Kr
yder,
and
Lyons. However, their
value for
Ka is low because of the fact th
at
their
measured
va
lue of the dielect
ri
c c
on
sta
nt
of
wat
er
vapor
is
about
1 perce
nt
lo
we
r
than
those dete
rmin
ed
TABLE
2. Consta
nts
used by diffel'ent authors
N=(n-
l )
10
6
=
Kl
'!!...
+ K
2
!...
+
](
3
~
l ' 'T
']'2
1
_____
A_ut_h_
Ol
_.
____
!I
__
'"_
'ea
_r
1
~
1
__
I
(.
_'_I
___
K
_3
__
1
I.
Sc
be
ll
eng,
BW
Tow
s,
and
Ferrell [
16J
________________ 1
933
2.
En
glund, Crawford,
and
Mumford
[17
J_ ____________ 1935
3. 'Waynick
[1
8]
________
.
_______
1940
4. S
mith
-Rose a
nd
Stickla
nd
[1
9J
__
______________
________
1943
5.
NDRC,
Burrows a
nd
Att·
wood
[201
__________
__
______
1946
6.
Meteorological Factors in
P
ro
pagation [
21J
____
__
_____
1946
7.
Radi
ation
Laboratory, vol.
13 (Kerr)
[lll
______________ 1951
8. Essen and
Fr
oome
[2
1
_______
1
91\
1
9.
Smith a
nd
W eintra
ub
_
______
1952
79
G75
1. 35 x
10
'
79
. 1 68. 3 3. 81
79
68
.5
3.
72
79 68
3.77
79
68
3.8
79
e8 3. 8
79
(
79
) 3. 8
77
. 64
64
. 68 3. 718
77.
6
72
3. 75
by
other
recent workers
in
the
fi
eld.
It
is true
that
their meas
ur
eme
nt
s were performed in a region
ad
ja
cent to the 1.35-cm
water-vapor
absorption
band
,
but
this effect
ha
s been shown to be too small
to acco
unt
for
the
discrepancy. As Essen
and
Froome did
not
hav
e refra
ct
ive-ind
ex
determinations
over the wide
temperature
range
necessary to
eva
l-
u
ate
K2
and
K3 indepe
nd
e
ntl
y,
they
utilized an
opt
ical
va
lue of re
fra
ctive
ind
ex
(K
3
= 0) to d
ete
r-
mine K
2
•
Consequently, their value for K2 is
about
10
pe
rc
e
nt
low
at
radio
frequencies.
How
ever, the
e
ff
ect of this on the e
quation
is negligibl
e.
Th
e
authors
thank
George
Birnbaum
,
Arthur
1ifary
ott
,
Jack
W. H erb
streit
,
and
Kenn
e
th
A.
N orton for their ad
vi
ce
and
assistance
in
this work.
References
[1]
G.
Birnbaum
, S. J .
Kr
yder,
and
H .
Lyon
s, J.
App
lied
Phy
s.
22,
95 (1951
).
[2]
L. Ess
en
and
K.
D .
Froo
me,
Pro
c.
Ph."
R.
Soc. (London)
[B]
64, 862 (1951) .
[3]
C. M.
Cra
in,
Phy
s. R ev. 74, 691 (1948).
[4]
W. F .
Gab
riel,
Pr
oc.
In
st.
Radi
o
Eng
rs. 40, 940 (Aug.
1952
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[5)
W.
E.
Phillip
s,
Pro
c.
In
st.
Radi
o
Eng
rs. 3
8,
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);
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Crain
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Pr
oc.
In
st.
Radi
o E ngrs. 40, 164 (Feb.
1952
).
2
33
150
-
5R
- -4
41
[6]
C.
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As
lak
s
on
and
O. O.
Fickei
ssen,
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s. Am. Ge
op
hys.
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ion 31, 819 (1950
).
[7]
Cornell
E.
E . Research
Int
e
rim
E ngin eering Re
por
t
No
. 13, p. 23 (
Octob
er 27, 1951
).
[
8]
L.
Essen
and
K.
D .
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ome,
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t
ur
e
167,
512 (1951).
[9]
L. J.
And
erson,
NEL
Re
port
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[10]
H.
Barr
ell
and
J. E.
Sea
rs,
Phil
os.
Tran
s. [A] 23
8,
2
(1940) .
[11] P
ropagat
ion of S
hor
t Radio
Wave
s,
Edit
ed
by
D .
E.
Kerr
, R
adiat
ion
Laboratory
Se
ri
es 13, p. 189 (
McGraw-
H ill
Book
Co
.,
I
nc
., New
Yo
rk, N .
Y.,
1951
).
[12]
ibid
. p. 643.
[13] I
onosp
heric
Radi
o
Pr
opagation,
NBS
Circ. 462, p. 35
(J
une
1948
).
[14] H .
Barr
ell, J.
Optical
Soc.
Am.
U,
295 (1951
).
[1
5]
G.
Birnbaum
and
S.
K.
Cha
tte
rjee, J.
App
l
ied
Phy
s. 23,
220 (
Feb
. 1952).
[16]
J. C.
Sche
lleng, C. R.
Burrow
s,
and
E. B. Fe
rr
e
ll
,
Pr
oc.
In
st.
R
adio
Engr
s. 21, 427 (1933).
[17]
C. R.
Eng
lund
, A. B.
Crawford,
and W. W. M
umf
o
rd
,
Be
ll
Syst
em Tech. J.
14,
369 (1935
).
[1
8]
A. H .
Wa
ynick
,
Pr
oc.
In
st. R a
di
o
Eng
rs. 28, 468 (1940
).
[19] R. L . Smit
h-R
ose
and
A. C .
St
ic
kland
, J.
In
st .
El
ec.
Engr
s. (
London
) 90, p
t.
III
, 12 (1943).
[20]
Radio
Wav
e
Propa
gat
ion,
Con
sol
idated
Su
mm
ary Tech-
ni
ca
l Report of
th
e
Commi
ttee
on
Propa
gat
ion,
NDRC
,
Edit
ed
by
C. R .
Bu
rr
ow
s
and
S. S.
Attwood
(A
cademic
Pr
ess,
In
c., New York, N . Y
.,
1949), p. 219.
[21] M
ete
orolog
ica
l
Fa
ctors in
Radi
o
Wav
c
Propa
g
ation
,
Fo
rewo
rd
(
Ph
ys. Soc.,
Londo
n, 1946).
BO
U
LDER
,
CO
LO.
, O
ct
ober 6, ] 952.