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  

Creation date : 2009/04/19 @ 19:55
Last update : 2009/08/19 @ 19:06
Category : Formula
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