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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.

 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

Creation date : 2009/04/19 @ 19:55
Last update : 2011/01/01 @ 23:58
Category : Formula
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