- Email: [email protected]
Oblique echoes from over-dense meteor trails
Oblique echoes from over-dense meteor trails* L. A. Radio
Propagation
Laborat,ory,
&?s?~7h-G Stanford University,
(Receiver1 26 Nay
Stanford,
California
1958)
Abstract-Ray paths are computed for waves refracted by meteor trails having a Gaussian radial distritmtion of ionization densitv. From the spreading of initially parallel rays as they pass through the trail, a ~neasu~ of reflected signaiintensity vs. scattering angle is obtjained; the results are presented in polar scattering diagrams, valid in the limit, of large trail size. Equivalence theorems are derived relating both intensity and scattering angle for rays incident upon the trail at an arbitrary angle to the intensity and ncattering angle for rays incident in the plane normal to the trail. Curves are presented showing the dependence of echo duration on forward-scatter-angle with trail orientation as parameter; it is found that the seca 4 law developed for under-dense trails applies to over-dense trails only if the plane of propagation contains the trail axis. If not, the effective secant exponent may be as small as 0.3. The theory is compared wit,h ~UCKINLEY and h!kNAMARAk durat,ion measurements. It is found that although the general agreement is satisfactory, the details of their experimental results depend on the way that winds change The ray theory is also compared with KEITEL'S wave theory solution. Unforthe trail orientation. tunately, he could not get wave solutions for dense trails of age greater t,han 0,357 per cent of t)he minilnum echo duration. Even so, the ray solutions agree with KEITEL’S results for scat’ter angles up to 1X0, thus including all angles available to gronnd-based stations. 1.
recent years, a great dea.1 of propagate sky wave signals over was shown by ESHLEMAN and t,rails--those having fewer than Is
Fig.
INTR~DTTCTI~S
interest has centered upon the use of meteors to distances of one-thousand or more kilometers. It MANNING (1954) that for under-dense meteor of trail length-echo durat,ion 1014 electrons/m
1. The geometry
of oblique
meteor
reflection.
should vary with the forward scat,ter angle $2~ the secant squared. Fig. 1 illustrates t’he geometry. In the over-dense case, however, there is no adequate theory. HIXES and FORSYTH have deduced a secant-square duration law for oblique reflection from over-dense trails by neglecting the effect of the ionization beyond the critical density radius (HINES and FORSYTH, 1957). However, in a previous paper (MANNING, 1953) the present author had already shown that the error of this neglect may approach 100 per cent for back-scatter as the echo duration is reached. Since interest in the oblique case is greatest when the duration exceeds that pcssible for back reflection, severe error may be anticipated from neglecting the ionization Indeed, as time advances, a larger t;ld larger beyond the “reflecting radius”. fract,ion of the ionization will have diffused beyond the reflecting radius, so that * Jointly supported St1vsl Research).
by the U.S.
Army
Signal
Corps, U.S.
82
_4ir Force,
and the U.S.
Si\v>- (Office
of
Obliqueechoesfrom over-densenleteortrails refraction becomes dominant in fixing the scattering pattern. That important deviation from a secant-square law occurs for over-dense trails has been shown experimentally by MCKINLEY and MCNAMARA (1956). In the present paper the properties of such oblique echoes will be computed using the ray-path method. Precise answers can only be expected in the limiting case of large trails, but t,he approach shares this limit,ation wit,11all other meteoric echo t’heories-none exist t,hnt ra.n in praotice treat more than a limited range of size or density. 2. RAY-PATHS IN OVER-UEXSE METEOR TRAILS &hough the method of approach to be used is quite general. we shall restrict our discussion to a Gaussian radial distribuCon of ionization in the meteor trail.
Fig. %. The co-ordinatesof a ray path. This distribution So let
is perhaps as close to the true distribution
N =
as we can now get’.
4GDT exp (-?/4Dt)
where N is the electron volume density, Q the electron line density, D the coefficient We may then express the of diffusion, r the radial co-ordinate and t is time. dielectric constant ,LA~ = 1 - 81N/f2 of trail as iu2 = 1 -
l/[y exp
(r2/v)l
(2)
where ,Uis the refractive index, y = t/t, is the time t divided by the back-scatter duration t, (at which time ,u = 0 at r = 0), and n2 = 4Dt, is the square of the “trail radius” at the back-scatter duration. The value of t, is given by f, = 81&/4nDf”
(3)
lvhere f is the radiofrequency and rationalized M.K.S. units are assumed. When a ray enters the ionized region about the trail axis, it is bent in two directions. Referring to Fig. 2, let PQ be a differential element of such a ray path. The refractive index at P is determined by the distance r of the ray from the trail axis. The direction of travel of the ray may be described by the two angles, y and 5. The first angle, y, is that between the plane containing P and the trail axis (plane l), and the plane containing PQ and an element parallel to the trail axis (plane 2). 83
1,. A.
MANNING
The secoJJd angle, l> is measured between UQ and a plane yerpendiculaJ3 to the trail axis. If y = 0, the ray pat’h is in the axial plane, and is goverJJed by SJJell’s law, p sin 6 = constant. If 6 = 0, the ray path is in a plane perpendicular to the trail axis, and is governed by Bouger’s rule, ,ur sin v = constant. In general. however, t)he ray path is noJJ-planar. and the following t’wo relat,ioJJs Jnust be used to find the ray patjh: p sill 6 = sin to 2 8 (4) (5)
,UUT siJJy cos ,t = A cos 5, 2 AC
where to is the value of &’outside the ionized region and A is the distalice fJVJl1the trail axis to plane 2.
Fig.3. The
refraction
and divergence
of a narrow
beam.
To find the signal intensity refracted from the trail to a distant receiver. consider the spreading imparted to a pencil-like beam of square cross-sect,ion. Fig. :3 shows such a beam. As is evident from equation (4), the beam is not spread in the &direct,ioJJ, but becomes fan-shaped because of spreading in the v-direction. At the radius of closest approach the beam is highly pinched and there is a caustic. Since t,he trail spreads the beam only in one dimension, it will be sufficient if we compube dAld26 as a measure of the scattering properties of the trail. We shall find dB/dA bv first computing 6, the “half planar-scatter-angle”; this deviation in turn will be ‘found by integrating the change in y along the path. Because [ is of no concerJJ iJJthis integrat,ion, we shall eliminate E between equatiorls (4) and (5). The result, is -
r\/(/2 3. CALCULATION To find t.he planar-scatter-angle
s”) sin ye = AC OF THE
RAY
(6) PATHS
26, we note that
where r0 is the radius of closest approach. have
PuttiJJg equation
84
(6) in equatioJJ (7). we
1’1)011substituting
,LL~ from equation
(2). letting s = 4’ and (T = ;P.
we have
(!‘I But if (’ -= I. and the entering ray has no axial component.
s = I’. G = ;‘. so
an expression of precisely the same form as equation (9). Thus we are led to observe tha~t it is only necessary to compute 6 vs. A and y for to = 0. For any other value of 60. I) ma,,v be found from the following equivalence theorem: B(A. y, [,)
= b(A cos lo, y COS2[“. 0)
(11)
In the actual calculation, equation (10) is not the most, convenient form of the integral for 6, because of the infinite limit. Instead, the variable of integration has been cha,nged to ,u. and the relabion used is
(1”)
Because two parameters must be varied, in addition to the variable of integration. equation (12) is hard to evaluate profusely enough to facilitate numerical differentiat,ion with respect to A. Hence? although critical cases were evaluated from equation (12) using an electronic computer. x simple approxima’te formula
was used whenever possible. Consider Fig. 4, which applies to propagation in the plane normal to the trail. In this special case from equation (5) we have ,~(r,)r, = 3. since 6 = 0 and y = 7712at closest approach. Thus for a given y, or trail age, we may assume T,,and get A. When a ray is travelling tangentially in a radial distribution of refractive index, the radius of curvature p is p = ,u drld,u. Thus at r0 the my has a radius of curvature p,,(r,). Par radii I > rO, p > p,,. If now we draw circles of radius A and p0 as shown in Fig. 4, the ray path extensions must be tan gent to the circle of radius A, and must fall outside the circle of radius pO. Experience shows, however, that for most cases, the deviations are very nearly t)he same 8.5
L.
&A. MANNIHG
It then follows that. approsi-
if t(he ray ext’ensions were tangent to the p0 circle. matelp.
as
sin $ = 4_i-!Yn ro + PO
(13)
where A = r0 [I -
I/(r esp (ro2/a2y)}]
p. = (ylr&
exp (ro2/a2y) -
and 11
Whenever justified on the basis that equations (12) and (13) give the same results. d(A/a)/d6 has been computed by evaluating the derivative of equation (14) 90r
0-
0
Fig.
0*5
A/a
I.0
5
5. Dependence of the half planar-scattering-angle 6 upon normalized radius of closest, approach A/u.
the
directly on t’he comput’er. However, preliminary slide-rule calculations equation (13) were not prohibitive as they were for equation (12).
made with
i. THE SCATTERING DIAGRAMS In view of the equivalence theorem (1 l), it is sufficient to compute reradiation patterns for rays lying in a plane perpendicular to the trail axis. Thus we shall interpret this special case before discussing the changes of scale needed to describe incidence at a general angle. Figure 5 summarizes the behaviour of the half planar-scattering-angle 0 as a function of A/a, where a is the radius of the trail at the back-scatter duration, and it is assumed that [ = 0. Recall that 14 is the radius of closest approach of the undeviated ray path, and y is t/tb, time normalized by the back-scatter duration; thus y > 1 represents enhanced duration at forward scatter. It will be seen t’hat However at the back-scatter duration for y < 1, rays are refracted in all directions. y = 1 the minimum planar deviation 26 is n/2, and regardless of A none of t’he rays have a vector component towards the transmitter. When y > 1. so that no part of the trail has a negative dielectric constant? the rays directed at the asis 86
Oblique echoes from over-dense meteor trails
(A = 0) are neither bent nor reflected; as y increases the remaining rays experience progressively less maximum deviation. In Fig. 6 are plotted the normalized polar scattering diagrams for t,rails of a, variety of normalized ages y, again for the case 6 = 0 (propagation in the plane normal to the t,rail axis.) The radial scale is d(A/a)/& in reciprocal radians. Upmt The plots thus represent, signal intensit,y. rather than signal amplit’ude.
Fig. 6. Normalized
polw~ scattering diagrams for propagdion normal to the trail axis.
in the plane
inspect,ing the plots, note first the way in which the back-scatter intensity decreases to zero at y = 1. This special case has previously been treat’ed (MANXI?;G. 1953). Next, it will be seen that to the extent that ray path calculations are accurat’e, thv duration of the echoes is the same for any transmitter-met’eor-receiver angle up to For y 5 1. 90°. For larger scattering angles, increased duration is apparent’. notice that at the moment when t’he echo drops out the receiver is in a caustic. For such forward-scatter-angles it would be expected that t,he signal int’ensit?should increase gradually with Oime to a sharp maximum before dropping out. A~~,~~~ hns described oblique path T’HF echoes from exceptionally large meteors 87
L. A. MANNING that have bhese properties (ALL~CK1948). Finally, it may be noted that as y increases beyond 1, the whole scale of the reradiation pattern becomes larger. The signal intensity then increases roughly as set 6 for deviations less than those corresponding t,o minimum intensity. Since the plots of Fig. 6 are normalized by a it follows t’hat t’he magnitude of scattered int’ensity increases direct,ly with a. and the scattered amplitSude as the back-scat,ter deviation tlj’-l. In general the incident ray will not lie in the plane normal to the trail axis. It, is t,hen desirable to determine t,, and 6 in tjerms of more easily measurable qua’ntities. Kefer to Fig. 7: it n-ill be seen that + (the ha,lf forward-scatt’er-a’ngle) and /? (t,he
Fig. 7. The co-ordinates of a ray incidrnt~ upon a trail at an arbitrary angl?.
angle between the projection of the original ray direction in the tangent plane, and t,he intersection of the tangent plane with a plane normal to the trail axis) are more suitable as external measures of the reflection geometry. The meaning of + was already made clear in Fig. 1. The advantage in using /J’as a co-ordinate is that for random trail orientations. each value of ,8 is equally probable. Note that p as defined here is the complement of t’hat’ used by ESHLEMAN and MAXSIXC: (1%~). From Fig. 7 it, will now be seen that and
tan to = tan /j sin h cos f$ -= cos lo cos h
(14) (15)
If p and h are assumed given, it is a simple matter to compute to from equaGon (14). then 4 from equation (1.5). Thus we have a procedure for adapting the normalplane results of Fig. 5 to arbitrary oblique geometry. Values of cos lo found using equation (14) are lmt in the equivalence theorem of equation (1 l), and the results apply to a forward-scatter-angle 24 found from equation (15). The polar diagrams of Fig. 6 apply only to the special case of E,, equal to zero. Ali equivalence theorem exists relating intensities scatt,ered in ohher directions. Thus from equatmion (11) it will be seen that’
dd ~
4Ala)
= n’ 88
CO8
(,,
Oblique echoes from over-dense met,eor t,rwils
if 0’ is ~lc?/d(.fI/n ) evaluated for 6, = 0, but with A and y replaced by A cos tu and y cos? :“,,* Hence the intensity I satisfies the equivaIence relation: I(S, ?I, to) =: see t0 I(6. y cos2 lo, 0)
(16)
Equations ( 14) and (15) are required to interpret the intensity patterns in terms of the f~tr~~nrcl-scatter-angle 24, and the trail orientation p. In general, the effect of
05
2.0
2.5
Forward- scatter dumtion y moi = bock -scatter duration IJig. 8. The relatjiom between half forwttld-scatter-angle
and duration normalized by thr heck-smtter duration; @ expresses the Wail orient,ntion, and is zero if the trail is normnl to the plane of the trrtnsmitt~~.rcreiver paths, 90’ if the t,rail if in that plane.
glancing incidence is t’o increase the duration. For p other t,han zero, the polar scattering diagrams will not be confined within as narrow a wedge about the forward direction as for incidence normal to the trail axis. 5. ECHO DURATION AND AMPLITWE It will be seen immediately from equations (11) or (16) that when to is not zero t,he duration is increased by t&hefactor sec2 to3 while the effective t,rail diameter a, and so the intensity, are increased by the factor set f,,. Thus when the trail lies in t’he plane of propagation the see2 d, duration increase with obliquity is still to be expect,ed. However, for /? not 90”. a much smaller increase results. The act,ual variation of duration wit’h 4; and /3 may be found by noting in Fig. 5 that the echo duration for a particular value of 6. with /I = 0, is determined by the value of y for t,hat member of t#hefamily tangent to the horizontal hue ~-equals-a-~onstallt. The duration in the general case is then y sec2 to with lo found from equation (14) for an assumed ii’_ This duration applies to a half forward-scatter-angle 4 found from equation ( 15). Fig. 8 shows the relation between half forward-scatler-angle and normalized clnration computed in the above manner. Note that the curve /3 = 90” is a plot, of ;‘ilMX:ZZsee2 #. Thus the duration increase predicted for over-dense trails is less than is expected for all orientatJions of under-dense trails: t,here the sec7r’$ law with SD
m N 2 derives from the existence of a t, a 2"'law of duration versus vvarelciigth regardless of orientation. Upon inspecting Fig. S. it is interesting to inquire whether the curves for /3 not zero may be approximated by laws of the form y = see”’ 45. It may be concluded that the exponents m must range from 0.3 or 0.4 to represent ,!l = 0. up to 2.0 for /5’ = 90”. If the trails are randomly oriented, it may be shown using equabion (3) of ESHLEMAN and MAXNIKG (l!XS), t’hat the median value of /? is 4.5’. For t,his p a secant law wit.h exponent about 0.8.5 is a representative fit).
The only controlled experimental test of the dependence of duration 011 obliquity available to the writer is due to ,MCKIXLEY and MCNAMARA (1956). They operated a 33 MC/S pulse transmitter at Ottawa and received simultaneously at Ottawa and Scarboro, 335.8 km distant. The echoes studied ranged in duration from roughly 1 t’o 100 sec. From range data the forward-scatter-angle 2’+ was found and the value of the exponent m best fitting the relation yrn = set?” 4 was presented. Separate determination of 7thfor t, less than and greater t,han t3.5 set yielded m = 1.73 and m = 1.13 uit,h probable errors in the means of 0.11 alld 0.05. Some tendency for m to decrease with increase in t’he value of set + was also noted. Because of the geometry of the paths in MCKIXLEY and MCNAJIARA’S esperiment, specular meteor reflect’ions from linear trails not distorted by atmospheric winds should only have been obtained if p were zero. Thus at first glance it might appear that a theoret,ical m of 0.3 or 0.4 should be used in the comparison. Han-ever n number of difficulties arise that we shall now discuss. First note that MCKISLEY and McNA~L~RA break the distribution in two aft t, = 3.5 sec. They call the two groups “under-dense” and “over-dense”. However the theoretical division between under-dense alit1 over-dense t,rails occurs in the bridging region where the under-dense and over-dense duration formulae merge. For back-scatt)er t)hese equations are t ,,<,= L2/J67r’D
(17
and t(,,, = 0.X85 :< 10-14 (1)3L2/4Trw
(18)
where D is the coefficient’ of diffusion. The bridging region occurs approxinintely7 where 0.855 x lo-l4 Q = I, so we see the critical duration is about t,.,.i, = P/14011 Takiilg y = 9.1 m and D = 3 m2/sec, we find trrit is about O-2 sec. Hence WC must assume that most of MCKIXLEY and McNA~~ARA’s meteors are over-dense. However, we expect a gradual transition from the sec2 4 behaviour of under-deiise t,rails t)o the behaviour predicted on t’he basis of ray optics. A density well above t(he crit’ical value, and in the region of MCKIXLEY and NcNaarA~A’s so-called ,‘over-dense” sample, seems appropriate for the comparison with theory. In MCKINLEY and MCNAX~RA’S “over-dense” sample, however. ‘???,was found to be 1.13. This is much greater than t’he theoretical value for @=O. Now another influence on t,he trail must be considered. It was shown some time ago by MCK~SLEY
Oblique echoes from over-dense meteor trails
and &fILLMAN (1947) that in the case of t’hose meteoric echoes enduring over Ti-IO set the aspect sensitivity of detection is lost. The author has developed the detailed theory of the loss of aspect sensit’ivity (MAXXING, 1957) and finds it is a direct result of distortion of the originally-straight meteor trails by winds. We see therefore, that we must no longer assume ,B to be zero, since after 5-10 set some part of the trail will be turned to present a value of p as large as 90’. Since the echo duration will be fixed by that part of the trail t’urned through the most favourable angle. we see that the mean value of m should lie somewhere between O-3 and 2.0. Individual observations greater than m = % can be expected to result when turning of t’he trail occasionally makes back-scat’ter detection impossible. Clearly an accurate check of the present theory is not possible based on MCKINLEY arid MCNAMARA’S data. However the fact’ that their mean m of l-13 is very nearly the average of 0.3and 2.0 is encouraging. And, as they point out, in view of the large percent,age of over-dense trails in their “under-dense” sample, their mean m of 1.73 does not mean a failure of the secant squared law for under-dense trails. 5. COMPARISOS WITH WAVE THEORY SOLUTIONS Attempts to determine the applicability of the present ray t’heory solutions by comparison with exact wave solutions are hampered by the extreme difficulty of obtaining wave solutions for dense trails of appreciable age. KEITEL has carried the wave approach about to t,he limit (KEITEL, 1955) but his most ambitious example, calculated on a large electronic computer. carries the case with line density Q = 1017 electrons/m only t’o a time equal to 0.357 per cent of the backAt this time the trail radius is only l/n wavelengths. Such scatter duration. durations allow much less time for expansion than do the values of normalized time y of one or more for which the ray theory has been devTeloped. Thus no valid comparison of over-dense echo behaviour wit’11 exact calculations is possible in a region where both may be expected to apply. It is none-the-less interesting to note how good the ray results are even for KEITEL’S very small trails. KEITEL expresses his results in terms of a reflection coefficient defined by scat,tered field at R ' ( -~~incident~fieti~ - “1
p = JkL:u)
(19)
where hi = 2rr/n and R, is the dist’ance from t)he meteor to the receiver. Our results are presented in terms of the angle d8 t8hrough which energy confined between parallel rays d(A/a) apart is spread. Jn terms of U/d6 the second bracketed quantity in equation (19) becomes scattered
field
incident, field
= ,~'[(l/:!R,,)dA/dd)
Now a2 = 41X,. so (C = 1[O.SXr, X 10-l” Q/713]1’2 and p = 5.13 +{d(&)/dh)
(%O)
(21)
In Fig. 9, KEITEL'S wave solution for k(ayliz) = P, Q == 10ii and @ = 0 has been compared with the ray-path calculation for y = 0.00357 t’o which it, corresponds. It will be noted t,hat t*he agreement, is excellent for forward-scatter-angles
“6 up to about 155”. In the cross-hatched region. however. the exact polar scattering diagram has a broad peak of finite amplitude; the ray solution has a narrow infinite peak. The discrepancy in this region must be expected in view of the very small y that KEITEL used. For such a small t)rail, the exact ionization distribution is not of great importance, and as KEITEL shows. t’he shape of the true overall scattering diagram is similar to the wave solution for a metal cylinder. However,
Forward scatter
Fig.
!). C’omparisori of ray solution with I
for n
for 6 < SO”, the ray theory gives a better fit’ than the metal cylinder even with 1 only 0.00357. It is expected that for y’s in the pertinent range near unity, the metal cylinder approximation will no longer be valid, and the ray treatment will be useful over an extended range of forward-scatter angles. It may be further noted that values of 6 greater than 80” are not possible for transmission between ground-based equipments. Thus the cross-hat’ched area of Fig. 9 is excluded on practical grounds. Although ray theory makes no distinction between incident waves with electric polarization along or across t’he trail, wave theory shows cross-polarization to be It is therefore expected t’ha,t for parallel-polarization t,he ray more complex. t’heory may be applied to smaller trails than for cross-polarizabion. s. CONCLUSIOTS Experimental studies of the duration of forward-scattered meteor echoes from over-dense trails have shown that the see2 + duration increase with obliquity predicted for under-dense trails does not hold. No theory of long-duration echoes has existed capable of discussing this question. The present theory, applicable in the limiting case of large Gaussian ionizat,ion trails. demonstrat’es that the effective
exponent of set C$for forward-scatter may vary from about 0.3 to 2.0 depending upon the orientation of the t,rail. In estimating practical duration increases, it is necessary to take into account in addition t’he distortion and change of orientation of the trail due to winds. Thus the mean practical exponent may be somewhat above the median expected for random trail orientation: t’he most favonrahl>r oriented part’s of t’he trail will determine the observed duration. Comparison of the theoretical polar scattering diagrams with wave theor? calculations is unsatisfactory because no existing wave solutions do more t,ha,ll dent the over-dense trail problem. However, the ray solutions compare closely wit’11 t,he most ambit’ious wave solutions over the experiment’ally observable range of angles despite the small trail radius of the examples. An equivalence theorem has been found relating the ray solutions for scat-lering at an arbitrary angle with t,he trail axis to the scattering in the plane normal to the axis. If the plane of propagat,ion contains the trail axis, t’he sec2 C$duration la,w holds even for over-dense t,rails. 4s t’he plane of propagation deviates farther ant1 farther from the axis the predicted duration increase becomes smaller and smaller. but never vanishes entirely. REFERENCES :\LLEN
E.
.Jtt.
\1’.,
1948
I’roc. Inst.
Rrrtlio Enyrs.,
X.Y.
36,
346. Esm,~k~~s
HIXES
V.
(‘.
KEITEL
K.
ant1 FORSYTH
0.
(:.
and MAX;NIN(:
L.
A.
1’. A.
H.
Inst. Radio I:‘nyrs., S. Y. 42, 530.
1954
Proc.
1957
Cunad. ,J. I’roc. Inst.
1955
f'hys.
Rdio
35,
1033.
Enyrs.,
.\‘.1’.
43,
1481. JIANNINU
L.
MANNING:
1,. A4.
1953
.k
1959
93
.J. dtmosph. Terr. I’hys. 4, 21!). Proceedings of Fifth Meeting, I.C.S.U. Mixed Commission on the Ionosphere. J. Atmos~h. Terr. Ph,ys. In press. Proc. Inst. Radio Enyrs., S.Y. 37, 364.
Propagation
Laborat,ory,
&?s?~7h-G Stanford University,
(Receiver1 26 Nay
Stanford,
California
1958)
Abstract-Ray paths are computed for waves refracted by meteor trails having a Gaussian radial distritmtion of ionization densitv. From the spreading of initially parallel rays as they pass through the trail, a ~neasu~ of reflected signaiintensity vs. scattering angle is obtjained; the results are presented in polar scattering diagrams, valid in the limit, of large trail size. Equivalence theorems are derived relating both intensity and scattering angle for rays incident upon the trail at an arbitrary angle to the intensity and ncattering angle for rays incident in the plane normal to the trail. Curves are presented showing the dependence of echo duration on forward-scatter-angle with trail orientation as parameter; it is found that the seca 4 law developed for under-dense trails applies to over-dense trails only if the plane of propagation contains the trail axis. If not, the effective secant exponent may be as small as 0.3. The theory is compared wit,h ~UCKINLEY and h!kNAMARAk durat,ion measurements. It is found that although the general agreement is satisfactory, the details of their experimental results depend on the way that winds change The ray theory is also compared with KEITEL'S wave theory solution. Unforthe trail orientation. tunately, he could not get wave solutions for dense trails of age greater t,han 0,357 per cent of t)he minilnum echo duration. Even so, the ray solutions agree with KEITEL’S results for scat’ter angles up to 1X0, thus including all angles available to gronnd-based stations. 1.
recent years, a great dea.1 of propagate sky wave signals over was shown by ESHLEMAN and t,rails--those having fewer than Is
Fig.
INTR~DTTCTI~S
interest has centered upon the use of meteors to distances of one-thousand or more kilometers. It MANNING (1954) that for under-dense meteor of trail length-echo durat,ion 1014 electrons/m
1. The geometry
of oblique
meteor
reflection.
should vary with the forward scat,ter angle $2~ the secant squared. Fig. 1 illustrates t’he geometry. In the over-dense case, however, there is no adequate theory. HIXES and FORSYTH have deduced a secant-square duration law for oblique reflection from over-dense trails by neglecting the effect of the ionization beyond the critical density radius (HINES and FORSYTH, 1957). However, in a previous paper (MANNING, 1953) the present author had already shown that the error of this neglect may approach 100 per cent for back-scatter as the echo duration is reached. Since interest in the oblique case is greatest when the duration exceeds that pcssible for back reflection, severe error may be anticipated from neglecting the ionization Indeed, as time advances, a larger t;ld larger beyond the “reflecting radius”. fract,ion of the ionization will have diffused beyond the reflecting radius, so that * Jointly supported St1vsl Research).
by the U.S.
Army
Signal
Corps, U.S.
82
_4ir Force,
and the U.S.
Si\v>- (Office
of
Obliqueechoesfrom over-densenleteortrails refraction becomes dominant in fixing the scattering pattern. That important deviation from a secant-square law occurs for over-dense trails has been shown experimentally by MCKINLEY and MCNAMARA (1956). In the present paper the properties of such oblique echoes will be computed using the ray-path method. Precise answers can only be expected in the limiting case of large trails, but t,he approach shares this limit,ation wit,11all other meteoric echo t’heories-none exist t,hnt ra.n in praotice treat more than a limited range of size or density. 2. RAY-PATHS IN OVER-UEXSE METEOR TRAILS &hough the method of approach to be used is quite general. we shall restrict our discussion to a Gaussian radial distribuCon of ionization in the meteor trail.
Fig. %. The co-ordinatesof a ray path. This distribution So let
is perhaps as close to the true distribution
N =
as we can now get’.
4GDT exp (-?/4Dt)
where N is the electron volume density, Q the electron line density, D the coefficient We may then express the of diffusion, r the radial co-ordinate and t is time. dielectric constant ,LA~ = 1 - 81N/f2 of trail as iu2 = 1 -
l/[y exp
(r2/v)l
(2)
where ,Uis the refractive index, y = t/t, is the time t divided by the back-scatter duration t, (at which time ,u = 0 at r = 0), and n2 = 4Dt, is the square of the “trail radius” at the back-scatter duration. The value of t, is given by f, = 81&/4nDf”
(3)
lvhere f is the radiofrequency and rationalized M.K.S. units are assumed. When a ray enters the ionized region about the trail axis, it is bent in two directions. Referring to Fig. 2, let PQ be a differential element of such a ray path. The refractive index at P is determined by the distance r of the ray from the trail axis. The direction of travel of the ray may be described by the two angles, y and 5. The first angle, y, is that between the plane containing P and the trail axis (plane l), and the plane containing PQ and an element parallel to the trail axis (plane 2). 83
1,. A.
MANNING
The secoJJd angle, l> is measured between UQ and a plane yerpendiculaJ3 to the trail axis. If y = 0, the ray pat’h is in the axial plane, and is goverJJed by SJJell’s law, p sin 6 = constant. If 6 = 0, the ray path is in a plane perpendicular to the trail axis, and is governed by Bouger’s rule, ,ur sin v = constant. In general. however, t)he ray path is noJJ-planar. and the following t’wo relat,ioJJs Jnust be used to find the ray patjh: p sill 6 = sin to 2 8 (4) (5)
,UUT siJJy cos ,t = A cos 5, 2 AC
where to is the value of &’outside the ionized region and A is the distalice fJVJl1the trail axis to plane 2.
Fig.3. The
refraction
and divergence
of a narrow
beam.
To find the signal intensity refracted from the trail to a distant receiver. consider the spreading imparted to a pencil-like beam of square cross-sect,ion. Fig. :3 shows such a beam. As is evident from equation (4), the beam is not spread in the &direct,ioJJ, but becomes fan-shaped because of spreading in the v-direction. At the radius of closest approach the beam is highly pinched and there is a caustic. Since t,he trail spreads the beam only in one dimension, it will be sufficient if we compube dAld26 as a measure of the scattering properties of the trail. We shall find dB/dA bv first computing 6, the “half planar-scatter-angle”; this deviation in turn will be ‘found by integrating the change in y along the path. Because [ is of no concerJJ iJJthis integrat,ion, we shall eliminate E between equatiorls (4) and (5). The result, is -
r\/(/2 3. CALCULATION To find t.he planar-scatter-angle
s”) sin ye = AC OF THE
RAY
(6) PATHS
26, we note that
where r0 is the radius of closest approach. have
PuttiJJg equation
84
(6) in equatioJJ (7). we
1’1)011substituting
,LL~ from equation
(2). letting s = 4’ and (T = ;P.
we have
(!‘I But if (’ -= I. and the entering ray has no axial component.
s = I’. G = ;‘. so
an expression of precisely the same form as equation (9). Thus we are led to observe tha~t it is only necessary to compute 6 vs. A and y for to = 0. For any other value of 60. I) ma,,v be found from the following equivalence theorem: B(A. y, [,)
= b(A cos lo, y COS2[“. 0)
(11)
In the actual calculation, equation (10) is not the most, convenient form of the integral for 6, because of the infinite limit. Instead, the variable of integration has been cha,nged to ,u. and the relabion used is
(1”)
Because two parameters must be varied, in addition to the variable of integration. equation (12) is hard to evaluate profusely enough to facilitate numerical differentiat,ion with respect to A. Hence? although critical cases were evaluated from equation (12) using an electronic computer. x simple approxima’te formula
was used whenever possible. Consider Fig. 4, which applies to propagation in the plane normal to the trail. In this special case from equation (5) we have ,~(r,)r, = 3. since 6 = 0 and y = 7712at closest approach. Thus for a given y, or trail age, we may assume T,,and get A. When a ray is travelling tangentially in a radial distribution of refractive index, the radius of curvature p is p = ,u drld,u. Thus at r0 the my has a radius of curvature p,,(r,). Par radii I > rO, p > p,,. If now we draw circles of radius A and p0 as shown in Fig. 4, the ray path extensions must be tan gent to the circle of radius A, and must fall outside the circle of radius pO. Experience shows, however, that for most cases, the deviations are very nearly t)he same 8.5
L.
&A. MANNIHG
It then follows that. approsi-
if t(he ray ext’ensions were tangent to the p0 circle. matelp.
as
sin $ = 4_i-!Yn ro + PO
(13)
where A = r0 [I -
I/(r esp (ro2/a2y)}]
p. = (ylr&
exp (ro2/a2y) -
and 11
Whenever justified on the basis that equations (12) and (13) give the same results. d(A/a)/d6 has been computed by evaluating the derivative of equation (14) 90r
0-
0
Fig.
0*5
A/a
I.0
5
5. Dependence of the half planar-scattering-angle 6 upon normalized radius of closest, approach A/u.
the
directly on t’he comput’er. However, preliminary slide-rule calculations equation (13) were not prohibitive as they were for equation (12).
made with
i. THE SCATTERING DIAGRAMS In view of the equivalence theorem (1 l), it is sufficient to compute reradiation patterns for rays lying in a plane perpendicular to the trail axis. Thus we shall interpret this special case before discussing the changes of scale needed to describe incidence at a general angle. Figure 5 summarizes the behaviour of the half planar-scattering-angle 0 as a function of A/a, where a is the radius of the trail at the back-scatter duration, and it is assumed that [ = 0. Recall that 14 is the radius of closest approach of the undeviated ray path, and y is t/tb, time normalized by the back-scatter duration; thus y > 1 represents enhanced duration at forward scatter. It will be seen t’hat However at the back-scatter duration for y < 1, rays are refracted in all directions. y = 1 the minimum planar deviation 26 is n/2, and regardless of A none of t’he rays have a vector component towards the transmitter. When y > 1. so that no part of the trail has a negative dielectric constant? the rays directed at the asis 86
Oblique echoes from over-dense meteor trails
(A = 0) are neither bent nor reflected; as y increases the remaining rays experience progressively less maximum deviation. In Fig. 6 are plotted the normalized polar scattering diagrams for t,rails of a, variety of normalized ages y, again for the case 6 = 0 (propagation in the plane normal to the t,rail axis.) The radial scale is d(A/a)/& in reciprocal radians. Upmt The plots thus represent, signal intensit,y. rather than signal amplit’ude.
Fig. 6. Normalized
polw~ scattering diagrams for propagdion normal to the trail axis.
in the plane
inspect,ing the plots, note first the way in which the back-scatter intensity decreases to zero at y = 1. This special case has previously been treat’ed (MANXI?;G. 1953). Next, it will be seen that to the extent that ray path calculations are accurat’e, thv duration of the echoes is the same for any transmitter-met’eor-receiver angle up to For y 5 1. 90°. For larger scattering angles, increased duration is apparent’. notice that at the moment when t’he echo drops out the receiver is in a caustic. For such forward-scatter-angles it would be expected that t,he signal int’ensit?should increase gradually with Oime to a sharp maximum before dropping out. A~~,~~~ hns described oblique path T’HF echoes from exceptionally large meteors 87
L. A. MANNING that have bhese properties (ALL~CK1948). Finally, it may be noted that as y increases beyond 1, the whole scale of the reradiation pattern becomes larger. The signal intensity then increases roughly as set 6 for deviations less than those corresponding t,o minimum intensity. Since the plots of Fig. 6 are normalized by a it follows t’hat t’he magnitude of scattered int’ensity increases direct,ly with a. and the scattered amplitSude as the back-scat,ter deviation tlj’-l. In general the incident ray will not lie in the plane normal to the trail axis. It, is t,hen desirable to determine t,, and 6 in tjerms of more easily measurable qua’ntities. Kefer to Fig. 7: it n-ill be seen that + (the ha,lf forward-scatt’er-a’ngle) and /? (t,he
Fig. 7. The co-ordinates of a ray incidrnt~ upon a trail at an arbitrary angl?.
angle between the projection of the original ray direction in the tangent plane, and t,he intersection of the tangent plane with a plane normal to the trail axis) are more suitable as external measures of the reflection geometry. The meaning of + was already made clear in Fig. 1. The advantage in using /J’as a co-ordinate is that for random trail orientations. each value of ,8 is equally probable. Note that p as defined here is the complement of t’hat’ used by ESHLEMAN and MAXSIXC: (1%~). From Fig. 7 it, will now be seen that and
tan to = tan /j sin h cos f$ -= cos lo cos h
(14) (15)
If p and h are assumed given, it is a simple matter to compute to from equaGon (14). then 4 from equation (1.5). Thus we have a procedure for adapting the normalplane results of Fig. 5 to arbitrary oblique geometry. Values of cos lo found using equation (14) are lmt in the equivalence theorem of equation (1 l), and the results apply to a forward-scatter-angle 24 found from equation (15). The polar diagrams of Fig. 6 apply only to the special case of E,, equal to zero. Ali equivalence theorem exists relating intensities scatt,ered in ohher directions. Thus from equatmion (11) it will be seen that’
dd ~
4Ala)
= n’ 88
CO8
(,,
Oblique echoes from over-dense met,eor t,rwils
if 0’ is ~lc?/d(.fI/n ) evaluated for 6, = 0, but with A and y replaced by A cos tu and y cos? :“,,* Hence the intensity I satisfies the equivaIence relation: I(S, ?I, to) =: see t0 I(6. y cos2 lo, 0)
(16)
Equations ( 14) and (15) are required to interpret the intensity patterns in terms of the f~tr~~nrcl-scatter-angle 24, and the trail orientation p. In general, the effect of
05
2.0
2.5
Forward- scatter dumtion y moi = bock -scatter duration IJig. 8. The relatjiom between half forwttld-scatter-angle
and duration normalized by thr heck-smtter duration; @ expresses the Wail orient,ntion, and is zero if the trail is normnl to the plane of the trrtnsmitt~~.rcreiver paths, 90’ if the t,rail if in that plane.
glancing incidence is t’o increase the duration. For p other t,han zero, the polar scattering diagrams will not be confined within as narrow a wedge about the forward direction as for incidence normal to the trail axis. 5. ECHO DURATION AND AMPLITWE It will be seen immediately from equations (11) or (16) that when to is not zero t,he duration is increased by t&hefactor sec2 to3 while the effective t,rail diameter a, and so the intensity, are increased by the factor set f,,. Thus when the trail lies in t’he plane of propagation the see2 d, duration increase with obliquity is still to be expect,ed. However, for /? not 90”. a much smaller increase results. The act,ual variation of duration wit’h 4; and /3 may be found by noting in Fig. 5 that the echo duration for a particular value of 6. with /I = 0, is determined by the value of y for t,hat member of t#hefamily tangent to the horizontal hue ~-equals-a-~onstallt. The duration in the general case is then y sec2 to with lo found from equation (14) for an assumed ii’_ This duration applies to a half forward-scatter-angle 4 found from equation ( 15). Fig. 8 shows the relation between half forward-scatler-angle and normalized clnration computed in the above manner. Note that the curve /3 = 90” is a plot, of ;‘ilMX:ZZsee2 #. Thus the duration increase predicted for over-dense trails is less than is expected for all orientatJions of under-dense trails: t,here the sec7r’$ law with SD
m N 2 derives from the existence of a t, a 2"'law of duration versus vvarelciigth regardless of orientation. Upon inspecting Fig. S. it is interesting to inquire whether the curves for /3 not zero may be approximated by laws of the form y = see”’ 45. It may be concluded that the exponents m must range from 0.3 or 0.4 to represent ,!l = 0. up to 2.0 for /5’ = 90”. If the trails are randomly oriented, it may be shown using equabion (3) of ESHLEMAN and MAXNIKG (l!XS), t’hat the median value of /? is 4.5’. For t,his p a secant law wit.h exponent about 0.8.5 is a representative fit).
The only controlled experimental test of the dependence of duration 011 obliquity available to the writer is due to ,MCKIXLEY and MCNAMARA (1956). They operated a 33 MC/S pulse transmitter at Ottawa and received simultaneously at Ottawa and Scarboro, 335.8 km distant. The echoes studied ranged in duration from roughly 1 t’o 100 sec. From range data the forward-scatter-angle 2’+ was found and the value of the exponent m best fitting the relation yrn = set?” 4 was presented. Separate determination of 7thfor t, less than and greater t,han t3.5 set yielded m = 1.73 and m = 1.13 uit,h probable errors in the means of 0.11 alld 0.05. Some tendency for m to decrease with increase in t’he value of set + was also noted. Because of the geometry of the paths in MCKIXLEY and MCNAJIARA’S esperiment, specular meteor reflect’ions from linear trails not distorted by atmospheric winds should only have been obtained if p were zero. Thus at first glance it might appear that a theoret,ical m of 0.3 or 0.4 should be used in the comparison. Han-ever n number of difficulties arise that we shall now discuss. First note that MCKISLEY and McNA~L~RA break the distribution in two aft t, = 3.5 sec. They call the two groups “under-dense” and “over-dense”. However the theoretical division between under-dense alit1 over-dense t,rails occurs in the bridging region where the under-dense and over-dense duration formulae merge. For back-scatt)er t)hese equations are t ,,<,= L2/J67r’D
(17
and t(,,, = 0.X85 :< 10-14 (1)3L2/4Trw
(18)
where D is the coefficient’ of diffusion. The bridging region occurs approxinintely7 where 0.855 x lo-l4 Q = I, so we see the critical duration is about t,.,.i, = P/14011 Takiilg y = 9.1 m and D = 3 m2/sec, we find trrit is about O-2 sec. Hence WC must assume that most of MCKIXLEY and McNA~~ARA’s meteors are over-dense. However, we expect a gradual transition from the sec2 4 behaviour of under-deiise t,rails t)o the behaviour predicted on t’he basis of ray optics. A density well above t(he crit’ical value, and in the region of MCKIXLEY and NcNaarA~A’s so-called ,‘over-dense” sample, seems appropriate for the comparison with theory. In MCKINLEY and MCNAX~RA’S “over-dense” sample, however. ‘???,was found to be 1.13. This is much greater than t’he theoretical value for @=O. Now another influence on t,he trail must be considered. It was shown some time ago by MCK~SLEY
Oblique echoes from over-dense meteor trails
and &fILLMAN (1947) that in the case of t’hose meteoric echoes enduring over Ti-IO set the aspect sensitivity of detection is lost. The author has developed the detailed theory of the loss of aspect sensit’ivity (MAXXING, 1957) and finds it is a direct result of distortion of the originally-straight meteor trails by winds. We see therefore, that we must no longer assume ,B to be zero, since after 5-10 set some part of the trail will be turned to present a value of p as large as 90’. Since the echo duration will be fixed by that part of the trail t’urned through the most favourable angle. we see that the mean value of m should lie somewhere between O-3 and 2.0. Individual observations greater than m = % can be expected to result when turning of t’he trail occasionally makes back-scat’ter detection impossible. Clearly an accurate check of the present theory is not possible based on MCKINLEY arid MCNAMARA’S data. However the fact’ that their mean m of l-13 is very nearly the average of 0.3and 2.0 is encouraging. And, as they point out, in view of the large percent,age of over-dense trails in their “under-dense” sample, their mean m of 1.73 does not mean a failure of the secant squared law for under-dense trails. 5. COMPARISOS WITH WAVE THEORY SOLUTIONS Attempts to determine the applicability of the present ray t’heory solutions by comparison with exact wave solutions are hampered by the extreme difficulty of obtaining wave solutions for dense trails of appreciable age. KEITEL has carried the wave approach about to t,he limit (KEITEL, 1955) but his most ambitious example, calculated on a large electronic computer. carries the case with line density Q = 1017 electrons/m only t’o a time equal to 0.357 per cent of the backAt this time the trail radius is only l/n wavelengths. Such scatter duration. durations allow much less time for expansion than do the values of normalized time y of one or more for which the ray theory has been devTeloped. Thus no valid comparison of over-dense echo behaviour wit’11 exact calculations is possible in a region where both may be expected to apply. It is none-the-less interesting to note how good the ray results are even for KEITEL’S very small trails. KEITEL expresses his results in terms of a reflection coefficient defined by scat,tered field at R ' ( -~~incident~fieti~ - “1
p = JkL:u)
(19)
where hi = 2rr/n and R, is the dist’ance from t)he meteor to the receiver. Our results are presented in terms of the angle d8 t8hrough which energy confined between parallel rays d(A/a) apart is spread. Jn terms of U/d6 the second bracketed quantity in equation (19) becomes scattered
field
incident, field
= ,~'[(l/:!R,,)dA/dd)
Now a2 = 41X,. so (C = 1[O.SXr, X 10-l” Q/713]1’2 and p = 5.13 +{d(&)/dh)
(%O)
(21)
In Fig. 9, KEITEL'S wave solution for k(ayliz) = P, Q == 10ii and @ = 0 has been compared with the ray-path calculation for y = 0.00357 t’o which it, corresponds. It will be noted t,hat t*he agreement, is excellent for forward-scatter-angles
“6 up to about 155”. In the cross-hatched region. however. the exact polar scattering diagram has a broad peak of finite amplitude; the ray solution has a narrow infinite peak. The discrepancy in this region must be expected in view of the very small y that KEITEL used. For such a small t)rail, the exact ionization distribution is not of great importance, and as KEITEL shows. t’he shape of the true overall scattering diagram is similar to the wave solution for a metal cylinder. However,
Forward scatter
Fig.
!). C’omparisori of ray solution with I
for n
for 6 < SO”, the ray theory gives a better fit’ than the metal cylinder even with 1 only 0.00357. It is expected that for y’s in the pertinent range near unity, the metal cylinder approximation will no longer be valid, and the ray treatment will be useful over an extended range of forward-scatter angles. It may be further noted that values of 6 greater than 80” are not possible for transmission between ground-based equipments. Thus the cross-hat’ched area of Fig. 9 is excluded on practical grounds. Although ray theory makes no distinction between incident waves with electric polarization along or across t’he trail, wave theory shows cross-polarization to be It is therefore expected t’ha,t for parallel-polarization t,he ray more complex. t’heory may be applied to smaller trails than for cross-polarizabion. s. CONCLUSIOTS Experimental studies of the duration of forward-scattered meteor echoes from over-dense trails have shown that the see2 + duration increase with obliquity predicted for under-dense trails does not hold. No theory of long-duration echoes has existed capable of discussing this question. The present theory, applicable in the limiting case of large Gaussian ionizat,ion trails. demonstrat’es that the effective
exponent of set C$for forward-scatter may vary from about 0.3 to 2.0 depending upon the orientation of the t,rail. In estimating practical duration increases, it is necessary to take into account in addition t’he distortion and change of orientation of the trail due to winds. Thus the mean practical exponent may be somewhat above the median expected for random trail orientation: t’he most favonrahl>r oriented part’s of t’he trail will determine the observed duration. Comparison of the theoretical polar scattering diagrams with wave theor? calculations is unsatisfactory because no existing wave solutions do more t,ha,ll dent the over-dense trail problem. However, the ray solutions compare closely wit’11 t,he most ambit’ious wave solutions over the experiment’ally observable range of angles despite the small trail radius of the examples. An equivalence theorem has been found relating the ray solutions for scat-lering at an arbitrary angle with t,he trail axis to the scattering in the plane normal to the axis. If the plane of propagat,ion contains the trail axis, t’he sec2 C$duration la,w holds even for over-dense t,rails. 4s t’he plane of propagation deviates farther ant1 farther from the axis the predicted duration increase becomes smaller and smaller. but never vanishes entirely. REFERENCES :\LLEN
E.
.Jtt.
\1’.,
1948
I’roc. Inst.
Rrrtlio Enyrs.,
X.Y.
36,
346. Esm,~k~~s
HIXES
V.
(‘.
KEITEL
K.
ant1 FORSYTH
0.
(:.
and MAX;NIN(:
L.
A.
1’. A.
H.
Inst. Radio I:‘nyrs., S. Y. 42, 530.
1954
Proc.
1957
Cunad. ,J. I’roc. Inst.
1955
f'hys.
Rdio
35,
1033.
Enyrs.,
.\‘.1’.
43,
1481. JIANNINU
L.
MANNING:
1,. A4.
1953
.k
1959
93
.J. dtmosph. Terr. I’hys. 4, 21!). Proceedings of Fifth Meeting, I.C.S.U. Mixed Commission on the Ionosphere. J. Atmos~h. Terr. Ph,ys. In press. Proc. Inst. Radio Enyrs., S.Y. 37, 364.