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(meteorobs) Some Forward-Scatter Notes (long)



Hello Philip and Steve,

i have been following the "Temiscaming Results" and "Waubaushene Results"
threads with interest, and hope that I can add some constructive input into
the discussion.  

In his report, Philip stated the basic geometry requirement for
forward-scatter, which I am repeating here from my March 2, 1998 MeteorObs
post:


"The basic geometry is that in order to cause a forward scatter reflection,
the meteor trail must lie within a plane (called the tangent plane) which
is tangent to an ellipsoid having the transmitter and receiver as its foci.
 The entire reflection path will also lie within a plane (called the plane
of propagation), which contains the transmitter, reflection point, and
receiver.  The plane of propagation will be normal to (at right angles to)
the meteor tangent plane.  

Important note:  the meteor itself can be at any orientation within the
tangent plane -- it need not be normal itself to the propagation path.
There is, however, greater signal loss when the meteor trail is
perpendicular to the propagation plane than when it is parallel to the
propagation plane.

A third useful constraint  is that most meteor reflections will occur
within the narrow altitude band of about 85 to 105 km altitude.  Thus, the
sphere formed by the 95 km altitude band, the meteor tangent plane, and the
ellipsoid having the transmitter and receiver as foci must all meet (or be
tangential) at the reflection point."


Another often quoted set of thumb rules for radiometeor reflections is the
proportionalities concerning the used radio frequency wavelength and echo
power, duration, and echo numbers.  These are:

* The echo power is proportional to lambda^3

* The echo duration is proportional to lambda^2

* The number of echoes is roughly proportional to lambda

where:
lambda = transmitted RF wavelength


But these thumb rules only tell a portion of the story, and care must be
taken not to draw an incorrect conclusion.  Philip Gebhardt wrote(24 Oct
1999 ):

"Beyond the question "Can both the transmitting antenna and the receiving
antenna 'see' the meteor," the main criterion for hearing a meteor burst is
that the signal path obeys the law of specularity (i.e. angle of incidence
equals angle of reflection). That being the case, the only consideration
is: Is there an appropriately oriented meteor within 'sight' of the
antennas to reflect the signal? Any meteor that is tangent to the family of
ellipsoids having the transmit site as one focus and the receive site as
the second focus will provide a signal burst. The location of the meteor
(relative to me) is irrelevant, so it is as likely to be overhead as at 45
deg or at the horizon. (This is another purpose of The Waubaushene
Project--to determine if elevation angle is indeed irrelevant.)"


If Philip will allow, i would like to correct this statement (that is, the
meteor trail elevation angle to the receiving station IS relevant), drawing
heavily upon the radiometeor enthusiast's "Bible" -- "Meteor Science and
Engineering," D.W.R. McKinley, (McGraw-Hill, 1961).  These notes come from
Chapter 8 (on back-scatter) and Chapter 9 (forward-scatter), and those who
have access to this book are strongly encouraged to verify my notes and
inspect the accompanying figures.  The "classical" equations for
forward-scatter from a meteor trail are as follows:


** Underdense trails (electron line density, Q < 1E14 electrons / meter):


* Underdense Echo Power

The echo power received at the receiving station in a forward scatter
underdense echo is given by (Eq. 9-3, page 239), as the product of two
fractions:

P_r = ((P_t * g_t * g_r * lambda^3 * sigma_e) / (64pi^3)) * 
     ((Q^2 * sin^2(gamma)) / ((r1 * r2) * (r1 + r2) * (1 - sin^2(phi) *
cos^2(beta)))),

where:
P_r = power seen by receiver (Watts),
P_t = power produced by transmitter (Watts),
g_t = gain of transmitting antenna,
g_r = gain of receiving antenna,
lambda = RF wavelength  (m),
sigma_e = scattering cross section of the free electron (m^2),
Q = electrons per meter of path,
r1 = distance between meteor trail and transmitter (m),
r2 = distance between meteor trail and receiver (m),
phi = angle between r1 line and normal to meteor path tangent plane, or
phi = 1/2 angle between r1 and r2 lines,
beta = angle between trail and the intersection line of the tangent plane
and plane of propagation,
gamma = angle between the electric vector of the incident wave and the line
of sight to the receiver.

A useful substitute for sigma_e is:
sigma_e = 1.0E-28 * sin^2(gamma) m^2,
which reduces in the back-scattter case to simply:
sigma_e = 1.0E-28 m^2.

* Underdense Echo power decay 

A second useful expression from this chapter for the exponential decay over
time of the underdense echo power is given by (Eq. 9-4, page 239), as an
exponential (e^x) raised to a fraction):

P_r(t)/P_r(0) = exp(- (((32pi^2 * D * t) + (8pi^2 * r0^2)) / (lambda^2 *
sec^2(phi)))),

where:
P_r(t)/P_r(0) = normalized echo power as a function of time (t),
t = in seconds (sec),
D = electron diffusion coefficient (m^2/sec), 
r0 = initial meteor trail radius (m).

The diffusion coefficient, D, will increase roughly exponentially with
height in the meteor region.  An empirical derivation from Greenhow &
Nuefeld (1955) is given for meteor altitudes of h = 80 km to h = 100 km:

log10(D) = (0.067 * h) - 5.6, 

for D in m^2/sec.

The initial meteor trail radius is another empirically derived value, given
in two studies as:

* 1956 & 1959 ARDC data;

log10(r0) = (0.075 * h) - 7.2,

h = meteor altitude (75-120 km)
r0 = trail radius (m)

* Manning (1958);

log10(r0) = (0.075 * h) - 7.9.


* Underdense echo duration

An approximate expression for the duration of an underdense trail is given
by Eq. 9-6, page 240:

t_uv = (lambda^2 * sec^2(phi)) / (16pi^2 * D)


** Overdense trails (electron line density, Q > 1E14 electrons / meter):

The classical expressions for the overdense trails contain many more
assumptions and estimations than for the underdense trails.  Their full
theory is still under development today.  However, the classical equations
can still be used to glean some of the basic characteristics of these
events.  I am showing these here in their final form, skipping some
intermediate steps and approximations.

* Overdense echo power

This is Eq. 9-7 on page 242:

P_r = 3.2E-11 * ((P_t * g_t * g_r * lambda^3 * Q^(1/2) * sin^2(gamma)) / 
     ((r1*r2) * (r1+r2) * (1 -sin^2(phi) * cos^2(beta)))).


* Overdense Echo Duration

An approximate expression for overdense echo duration is given by Eq. 9-8
on page 242:

t_ov = 7E-17 * ((Q * lambda^2 * sec^2(phi)) / D).



** General Notes

A few of the more important relationships from these equations:

* Note that the thumb rules initially given concerning wavelength, lambda,
are verified in these equations, at least for echo power and duration.

* The electron line density, Q, is a function of the meteor mass ,
velocity, and composition, much as is meteor magnitude.  Some important
relationships from the above equations can be gleaned:

-- for underdense trails;

     Echo power is proportional to Q^2
     Echo duration is independent of Q (!)

--  for overdense trails;

     Echo power is proportional to Q^(1/2)
     Echo duration is proportional to Q


These correlations were used as one of the criteria for statistically
separating underdense from overdense echoes recorded at Poplar Springs,
Florida.

* The diffusion coefficient, D, and initial trail radius, r0, are the
primary reasons for the well known "height-ceiling" effect in
forward-scatter systems.  Most systems are limited to an effective ceiling
of about 105-110 km above which echoes cannot normally be detected.  The
trail radius becomes a limiting factor due to electron density decrease and
destructive interference between the reflections from different portions of
the trail at the first Fresnel zone -- front to back and side to side.  The
diffusion coefficient, D, decreases the amount of time it takes for the
trail to reach these poor reflection conditions.

Additionally, there is also a "hight-floor" effect seen in slow, overdense
trails, which begins to seriously decrease their durations at about the
80-85 km altitude level.  This is also currently under investigation, and
is thought to be due to the more rapid free electron recombinations and
attachments at this lower altitude (higher air density) region.

The upshot of these two effects is that most forward-scatter systems tend
to be more sensitive to meteors which occur in the 85-105 km altitude band,
with an average of about 95 km.  This makes the systems most responsive to
medium-speed meteors of most magnitude levels, but somewhat discriminatory
against fast, faint meteors and slow, bright meteors.  

* An interesting relationship is that found for the meteor trail
orientation with respect to the plane of radio wave propagation, Beta.  The
rather anti-intuitive effect is that a higher peak reflected power will
occur from a trail which is parallel to the plane of propagation, with a
somewhat lower power being reflected from a trail which is perpendicular to
the plane of propagation (all else held constant).

** The Secant Squared Phi Effect

The key ingredient which attracted early researchers to the possibilities
of radiometeor forward scatter -- both in the realm of meteor science and
meteor burst communication -- was the sec^2(phi) terms which appear in the
duration equations for both the underdense and overdense expressions.
Additionally, helpful sin^2(phi) terms also appear in the expressions for
echo peak power.  What this implies is that the further transmitter and
receiver are from each other, The more power the meteor trail will reflect,
and the *much* longer will the duration of the echo be.  At some point, the
attenuation due to distance (the (r1*r2)*(r1+r2) terms) will override the
advantage of continuing to increase distance and phi, but for a time
(depending upon transmitter power) the advantage over the back-scatter
condition is significant.  

This can be illustrated (and is in Chapter 9) by looking at the best
regions of atmosphere to point a transmitting and receiving antenna for a
particular forward-scatter link, that is, where the highest number of
echoes, highest powers, and longest durations will be obtained.    if the
sky is uniformly filled with meteor radiants, the highest concentration of
potential reflection-causing meteor trails (those which have the proper
geometry) will be located in an elliptical ring at the 95 km altitude
level, having transmitter and receiver as foci.  This ring corresponds to
radiants having angular altitudes of about 30-60 deg, peaking near 45 deg.
If the forward-scatter link is short, the elliptical ring will be fairly
uniform in meteor density, but if the link is long, the ring will show
higher concentrations of likely echo candidates closer to the ends of the
ellipse major axis -- nearer to the vicinities of the transmitter and
receiver on the ground.  This would tend to support the common desire among
radiometeor  amateurs to point their receiving antennas at some very high
elevation angle in order to catch these end-point reflections.  The effect
of angle Beta, discussed above, would also tend to support this notion,
since a higher proportion of end-point meteors will have lower Beta's.

HOWEVER, when the effect of angle phi is taken into account, this picture
shifts abruptly.  Meteor trails located near the midpoint between the two
stations will have the highest phi's, and thus give back the best power
levels and significantly longer echo durations.  Meteors located near the
path endpoints will have lower reflected powers and much shorter durations.
 The effect is that the regions of best echo characteristics will be the
so-called hot spot regions, located about 50-100 km to either side of the
transmitter-receiver great circle path midpoint.  McKinley shows some very
nice theoretical echo density maps for this type of situation, and meteor
burst communication firms make almost exclusive use of hot spot
reflections.  This is not to say that end-point reflections do not occur, I
do know of one military sponsored forward scatter experiment using a
hardened below-ground antenna for meteor burst communication employing
endpoint reflections, but this was a rather singular effort.  For most
medium and long distance forward-scatter links, relatively low antenna
elevation angles, with transmitting and receiving antennas aimed at one or
both hot spot regions, yield the best and most consistent results.   The
one exception that I know of is for a very short-range link (less than
about 150 km), in which better performance in the northern hemisphere is
gained by pointing the transmitting and receiving antennas to the north in
order to take advantage of the higher concentration of ecliptical radians
to the south.  This special case is more akin to the back-scatter
situation, in which phi will always be rather small, and the highest
concentration of echo candidates should be sought.  

I'll close this epistle (sorry about the length) with a repeat post of a
table I placed on MeteorObs a couple of years ago, showing hot spot
relative azimuth and elevation angles for a variety of link great circle
distances.  This model was created in a Maple worksheet, and gives the
reflection location (altitude and azimuth) for a meteor trail occurring
midway between transmitter and receiver, having a radiant at 45 deg
elevation, and a flight path perpendicular to the plane of propagation.
Such a meteor is indicative the center of one of the two hot spot regions
for the given link.  The two angles are shown in degrees.  Note the rapid
drop in antenna beam elevation angle.


     RANGE (km)     TEST ALTITUDE  TEST AZIMUTH OFFSETS

     50   44   75
100  41   62 
     150  38   51
200  34   43
     250  30   37
     300  27   32
     350  24   29 
     400  22   26 
450  20   23 
     500  18   21 
     550  17   20
     600  15   18 
     650  14   17 
     700  13   16 
     750  12   15 
     800  11    15
     850  10   14
     900  9    14
     950  9    13 
     1000 8    13 
     1050 8    12
1100 7    12
1150 6    12
1200 6    11
     1250 6    11
     1300 5    11
     1350 5    11
     1400 4    11
     1450 4    10
     1500 4    10
     2000 1    10

I hope that all of the above has been elucidating and helpful.  

Best regards,

     Jim



James Richardson
Tallahassee, Florida
richardson@digitalexp.com

Operations Manager / Radiometeor Project Coordinator
American Meteor Society (AMS)
http://www.amsmeteors.org

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