Tables of Ordinary and Extraordinary Refractive Indices, Group Refractive Indices and h’o,x(f)-Curves for Standard Ionospheric Layer Models

110 Pages·1960·7.812 MB·English

by  Walter Becker (auth.)

Diese Mitfeilungen setzen eine von Erich Regener begründete Reihe fort, deren Hefte
auf der vorletzten Seite genannt sind.
Das Max-Pianck-lnstitut für Aeronomie vereinigt zwei Institute, das Institut für Strato
sphärenphysik, und das Institut für lonosphärenphysik.
Ein (S) oder (I) beim Titel deutet an, aus welchem Institut die Arbeit stammt.
Anschrift der beiden Institute:
(20 b) Lindau über Northeim (Hann.)
ISBN 978-3-540-02580-1 ISBN 978-3-662-22453-3 (eBook)
DOI 10.1007/978-3-662-22453-3
TABLES OF
ORDINARY AND EXTRAORDINARY REFRACTIVE INDICES,
GROUP REFRACTIVE INDICES
AND h' {f)-CURVES
0 1 X
FOR STANDARD IONOSPHERIC LAYER MODELS
by
Walter Becker
2
Contents
1. Introduction
2. The basic formulae of the tabulated data
2. 1. Formulae for vertical group refractive indices
2. 2. Formulae for numerical calculations of vertical group refractive indices
2. 3. Formulae for the calculation of h' (f)-traces
2. 3. 1. General remarks
2. 3. 2. Special calculations
2.3.2.1. Numerical calculations of standard h' (f)-traces
01 X
2.3.2.2. Numerical virtual path calculations of ionospheric
echoes vertically penetrating a parabolic layer
3. Practical application of the tabulated data
3. 1. General remarks
3. 2. Graphical representation of some tabulated data
Acknowledgement
References
Tabulated data
Tables 1 - 120 Vertical refractive and group refractive indices
Tables 121 - 160 Numerical data of standard h' (f)-traces
O, X
Tables 161 - 170 Numerical virtual path data of ionospheric echoes vertically
penetrating a parabolic layer
Table 171 Numerical data of the standard layer models used here
3
1. Introduction
The N(h)-Working Party, a Group in Commission III of URS! (URS! Information
Bulletin No.112, p.12) suggested these calculations of ordinary and extraordinary
refractive indices n , vertical group refractive indices c/U , and virtual
o,x o,x
heights h1 (f), for an Epstein, cosine and parabolic layer model. c ist the free
01 X
space velocity of light . U and U denote vertical ordinary and extraordinary
0 X
group velocities. The data are intended to facilitate real height (h) computations from
observed, h1 (f)-traces. Especially the h1 (f)-data are intended also to allow
o,x o,x
for tests of existing reduction methods. For ionization minimum investigations an
additional set of tables is presented. These tables represent the virtual paths
6 h1 (f) of sounding pulses which penetrate an ionospheric layer of parabolic
o,x
shape; they can be used together with the abovementioned standard h 1 (f)-curves
o,x
to give ordinary and extraordinary h1 (f)-traces for any combination of a lower
O, X
parabolic layer and an upper Epstein, cosine or parabolic electron density distribution.
Ordinary group refractive index tables have already been published by D. H. SHINN [ 1]
and by Vv. BECKER [ 2] • Their value s of cp , the angle of inclination of the earth 1 s
magnetic field, are slightly different from those used here. These tables may be used
as additional sets for interpolation purposes. D. H. SHINN1 s tables are the least
accurate ("one unit in the third decimal place"). W. BECKER1s tables become
inaccurate in the fifth decimal place. The relative accuracy of the G0- and GR-tables
presented here is better than 10 -B and that of the normalized virtual heights is better
than 10-S.
As for D.H. SHINN1s and W. BECKER1s numerical calculations the data given here
are also based on Appleton-Hartree1 s formula for the ordinary and the extraordinary
refractive in die es of an ionized gas. It is assumed that energy losses due to electron
collisions can be neglected.
2. The basic formulae of the tabulated data
2.1. Formulae for vertical group refractive indices
The following calculations are based on the Appleton-Hartree formula ( 1 ) for the
refractive indices of an ionized gas.
r
2 f2 /f2
n 2 1 0 ( 1 )
01 X f~ /f2 li;r' f2
L
2- + + 4 ?""
2 2 2 2
1-f /f 1-f /f
0 0
The upper sign of the square root holds for the ordinary and the lower for the extra
ordinary wave component. Energy losses by electron collisions are neglected.
In equation ( 1 ) the abbreviations used have the following meaning:
,r;;;;
f plasma frequency
0 y~
4
3
N electron density per cm
e charge of an electron ;
m = mass of an e lectrori ;
Tt 3.14159 ••.
eH
gyrofrequency
2 Ttmc
H total intensi ty of the earth' s magnetic fie ld;
cp angle of inclination of the earth's magnetic field;
fL fH • sin cp
fT fH • cos cp
f frequency of observation.
The ordinary and extraordinary group refractive indices cfU are defined by
O, X
equation ( 2 ); it is :
c d
- = df (fn ) { 2 )
U0 , X O,X
c = free space velocity of light and
U _ordinary, e_xtraordinary group velocity.
01 X
~quation ( 2) can be transformed ( 3) :
2 r2
1
(1 + ;r0 { 3)
n
o,x
o,x
where
N 2- +
o,x
and
{N -2}
O, X
For h'(f)-calculations it is sometimes necessary to have a Taylor approximation of
the group refractive indices for small plasma frequencies ( 4 ).
r f2
y
lim lim ( 1 + A +. • • ) = 1 • ( 4 )
f /f-+o o, x f jf-+ 0 f O, X
0 0
5
The coefficients A are derived from equation ( 3 ) and are ( 5) :
01 X
r'fjr2
A 4 (1 + ) - 1 ( 5 )
o,x N2
f2 No, x
o,x
+ 4 L
7
2. 2 Formulae for numerical calculations of
vertical group refractive indices
Equations ( 1) and ( 3 )are inappropriate for numerical calculations, because n
01 X
becomes zero and c/U infinite for certain values of f and f . For practical
o,x 0
purposes it is useful to calculate
t
for the ordinary component according to D. H. SHINN [ 1] and
for the extraordinary component according to W. BECKER [ 3)
The symbols t and tR have the following meanings :
t is the refractive index with fH = 0.
tH is the refractive index for purely longitudinal propagation, or fH = fL •
f2R f2X - f xfH if f X > fH
( 6 )
f2R f2X + fxfH if f X < fH •
f is the extraordinary frequency of observation.
X
By this procedure values of G and GR are finite even when n or n becomEC
0 0 X
zero, e. g. when
f2 f2 for the ordinary component; or
0
( 7 )
f2 f2 for the extraordinary component.
0 R
6
As t is equal to tR for the ordinary wave component, tR is appropriate as a
common parameter.
Also in order to get as accurate data as possible, equations ( 1 ) and ( 3 ) have to
be transformed because they implicitly contain expressions such as:
which, calculated step by step, give poor results when a is small. However the
accuracy becomes very high when the equivalent expression given by equation ( 8 )
is calculated.
F
=---a==
1- ( 8 )
1+~
Thus useful formulae for numerical calculations of ordinary ray indices can be
obtained:
( 9)
4 ti<P
where y =
2 2
yR cos cp
c tR 'R ti.p [ ( 1 + xR)
G tR n { 1 + - 2 ]} ' ( 10)
0 Uo 0 M2 v'1 + y t4R 1 +V1 +y t~
where
lVl
and
1 tg2 <P
A = {.:... (1+ --) - 1} ( 11)
0 2M1 M1 yT+Y
where 2 tlc:t> 4ts2 c:p
M1 1 + y =
1 +-yr+Y 2 2
yR cos <t>
The limiting cases are :
n
lim 0 1 lim G = lim ( 1 + xRA0+.) = 1 ( 12 )
tR cos <t> 0
tR-o tR-1 xR-o
7
For the numerical calculation of the extraordinary indices one gets the following set
of formulae :
ß
1 - +Jii+OI
2 1
n y~ ( 13 )
X
1-y+il- ßy~
1+yl+Uf
Where
a 4 Sl0 ll2 cp ß
2/1 +/12
y
GR = c t R = tnR { 1+...!.. [1+~{ 2~ _(1+~)(1+x)}J}.(14)
ux X N2 2 1+ y1+a.~ ~
where
N 1 - y +/1- ß --=-Y..,;:,~==
1+y'1+a~
AxR = (1-y){~ (1- ( 15 )
N1
where
y N 1 = 2 - y 2 cos 2 <t> ( 1 + ,V 1 ~+ y ) ;
Again the limiting cases are :
2
And for purely longitudinal propagation, e, g. fH fL , fT o the previous
formulae reduce to:
(17)
8
1 + xR • y ( 18)
2(1-y)
2. 3. Formulae for the calculation of h'(f)-traces
2.3.1. General remarks
Ordinary and extraordinary virtual heights h' are defined by equation ( 19 ).
01 X
JR
h=h
h' (f) (19)
01 X
h=o
hR means the real reflection height, that is that height at which the respective
reflection conditions ( 7) or ( 20) are fulfilled.
r2
0 = 1 or (20)
r2
R
With the abbreviations hitherto used :
( 21 )
t = 1
( 19) becomes: R
f
d~
h' (fR) = 2 <tf-tR) dtR'
01 X t = 0 o, X R
R
or
t R=j a1 .
h' (fR) 2 ( 22)
01 X o, R
t = 0
R
dxR
is proportional to the gradient of the respective ionospheric layer. The ordinary
dh
and extraordinary virtual heights h' (fR) for a constant reduced frequency fR
01 X
belong to the same real reflection height, but do not have the same frequency of Ob-
servation. The frequencies of observation are in fact : fR and fx • It seemed
physically meaningful to publish only such corresponding pairs of virtual heights for
fR and fx in this booklet.