Some Notes and Equations for Forward Scatter
compiled by James Richardson
Here are some basic notes on the canonical equations for meteor
forward-scatter which I originally put together for another
email list, but which I thought might be of interest here as
well. There is a little math involved, but the information
which can be gathered from the equations is quite informative as
to how a forward scatter system will behave under different
system and link configurations (on the ground), and different
meteor velocities and flight directions (in the atmosphere).
The basic geometry requirement for forward-scatter is as
follows:
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 are 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 it
is necesary to dig in a little deeper to gain a working
understanding of how to optimize a particular link setup. For
this presentation, I draw 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, which have been derived from theory and validated
empirically during the heyday of radiometeor astronomy (1945-
1970) , 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 the r1 and r2 lines,
beta = angle between meteor 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 (polarization
coupling factor).
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 = time 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
are:
* 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 when the trail altitudes drop to 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 the reflection angle, phi, is taken
into account, this picture shifts very 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. As an example, echoes from the midpoint region of a
600 km link will have durations about 15 times longer than
echoes from the endpoint regions, while echoes from the midpoint
region of a 1200 km link will have echo durations which are
about 92 times longer than those echoes from the endpoint
regions. 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 quite small, and the highest
concentration of echo candidates should be sought.
The below table lists the elevation angles (measured from the
horizon), and relative azimuths (measured from the bearing of
the great circle path between receiver and transmitter) needed
to point the beam of a transmitting/receiving antenna at the cen
er of the hot spot region for a particular forward-scatter link.
These are given 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 trail is indicative of a
reflection from the center of one of the two hot spot regions
for the given link. The two ngles are shown in degrees. Note
the rapid drop in antenna beam elevation angle.
RANGE (km) ALTITUDE AZIMUTH OFFSET
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 Richardson
AMS Radiometeor Project Coordinator