Epsilon echoes

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This page is dedicated to the Epsilon shaped reflection seen by radio forward scatter meteor observers. 

Ah, Epsilon reflections....

The traditional 'glint and blob' and wind-shear mechanisms don't, in my opinion, quite match the results of our observations.

I think it was Gaspard who came up with an explanation involving a 'tube' of ionisation, rather than a filled cone.

Here's my pet theory, which is based on standard theory, but allows for multiple, or extended reflection points from the same meteor trail, simply as a result of the trail increasing faster at greater altitudes. I'm not sure this idea is the correct explanaition and would welcome comments and corrections, so that we can fully understand the phenomena.

In my opinion, epsilon reflections are common under the right circumstance. So common, as to almost the norm, if conditions are right.

I suggest that it is all to do with the trail expanding more rapidly at greater heights. We know that the trail expansion is governed by the ambipolar diffusion rate and that this rate decreases with decreasing altitude. So, if a meteor travels down through the atmosphere the trail that it generates will expand at a greater rate at greater altitudes. As a consequence, the reflecting surface of the trail will develop a concave characteristic along the length of the trail. This may allow multiple, or extended reflection points to develop. 

You can easily demonstrate the same effect. The image below shows the effect of viewing a candle, representing the transmitter, reflected by a concave reflecting surface, in this case represented by a shaving mirror. The mirror represents the ionisation trail left by the meteor.

The first part of the above image shows the reflection from a flat surface, representing the initial trail. The second, third and forth images were taken using the concave surface of the mirror and, in this case, changing the distance between to the mirror slightly. The same effect could have been achieved by flexing the reflective surface, had that have been possible. You can see that under the right circumstances the reflection point extends across the width of the mirror.

Having spent some time working through some examples, it does appear that there are circumstances under which the meteor trail adopts a expanding concave curved shape which allows for two separate reflection points for a forward scatter system. The second reflection point may extend over a considerable length of the trail. For this to be possible the following need to be true...

1. The total path Tx - meteor - Rx needs to be long (2000 km plus) (I've now modified that view that it may be less than 2000km, but can't demonstrate that at the moment)

2. The meteor's path needs to be inclined to the horizontal by a significant amount, so that the path passes through a wide range of altitudes and the trail expands at different rates along its length.

3. The electron line-density needs to at the top end of the accepted range.

Under the above conditions, and with the right geometry, the trail can have two reflection points. The first is the traditional reflection point about a minimum in the total propagation distance. However, I note that, given the path length requirement, this traditional reflection point will likely be below the radio horizon and not seen.

The animation below shows a representation of the meteor trail and the development of the extended reflection point in the upper portion of the trail. The lower portion of the trail shows the traditional reflection point. 

The first attached chart shows the total propagation path distance (x) against the height for a typical trail. the reflection points occur where the total propagation path stays constant along the path. These areas correspond to vertical lines. The traditional reflection points are in the lower part of the chart. You can see the development of the secondary reflection point in the top left lines after 18 and 20 seconds.

The second reflection point is higher along the trail and represents the points either side of a maximum in the total propagation distance.

Again, if the circumstances are correct, the concave curvature of the trail at the second reflection point can match the convex curvature of the system ellipsoid. Under these conditions, the second reflection point can extend for many kilometres along the trail. In the example I calculated, there is less than 1 metre difference in total propagation distance over 6 km of ionisation trail 40 seconds after the meteor passed! In other words the curved trail exactly lies along the system ellipsoid for 6km of its length.

I believe that it is the way in which the signals, reflected from these extended reflection points, constructively and destructively interfere that produces the epsilon patterns seen. I've got a bit more work to do here, but I believe the above mechanism may explain both the single epsilon and the multiple repeating epsilons.

The above calculations are based on a 1000 km system-axis and a 1200 km off-axis distance for the meteor. In this case the meteor is almost vertical - just tilted enough to provide the correct reflection geometry. The trail radius assumes the trail is over-dense with an electron line density of 1E21 and system operate at a wavelength of 6m. r0, ambipolar diffusion rate and over-dense trail radius are based equations from McKinley chapter 8. The geometrical calculations are a bit tricky, so I may have made some errors. I'll be happy to pass on the method if anyone wishes to check them. At the moment I have to manually move the meteor end points to find the correct reflection geometry...

So why an Epsilon?

I believe that we can get extended reflection points under certain circumstances and that is when the expanding trail touches the system ellipsoid over an extended distance. Under these circumstances, we'd expect to see an interference pattern when the reflecting surface is illuminated by a single frequency (monochromatic) source. In the case of a signal reflected from a meteor trail, the circumstances are complicated by the movement of the reflecting surface.

If the reflection point is extensive, then the reflected signal will be given a different frequency shift by virtue of the differing speeds with which the trail is expanding, at different points. As a consequence of this, the reflection splits into strands of differing frequency.

I believe it's the development of the interference pattern, across the extended reflection point, which results in the 'epsilon'. I also believe that it is the expansion of the trail at different rates, along the extended reflection point, the 'splits' the reflection into 'strands' of different frequency. In the above diagram, the frequency shift f2 will be greater than frequency shift f1, because the trail is expanding more rapidly at greater altitude.

In the diagram below, strand 1 would correspond with the top of the extended reflection within the trail and strand 3 the lower end of the extended reflection point.

Epsilons and Radiant Altitude

I went on to try and verify that radiant altitude was important by looking at the distribution of 'epsilon' reflections in the Quadrantids. I thought I saw a relationship, but that's open to interpretation. 

The first is for 20:00 UT 03/01/07. The Quadrantid radiant is active, but less than 20 degrees above the horizon. There are plenty of reflections, but no epsilon characteristic. Given the time of day, sporadic rates would be low and the vast majority of those reflections would be Quadrantids.

The second image is for 08:00 UT with the Quadrantid radiant overhead - many reflections show epsilon characteristics.


Having examined all spectrum charts across two days of the 2007 Quadrantids, my 'Epsilon' counts are along the lines of the following chart. They are only approximate as they are based on a subjective judgement of what is and what is not an 'Epsilon'. But, Epsilon happy-hour seemed to be around hour 33, or between 09:00 and 10:00 UT on the 4th January 2007. The Quadrantids peak somewhere around hour 24 or 00:00 UT on 4th January 2007. The x-axis is messed up and should count from 0 to 48, rather than 1 to 49.

Most (all?) 'epsilons' I observed were from the shorter (665 km) west / east baseline to Liege Bol d'Air. Very few, if any, 'Epsilons' were observed off the longer (1545 km) north / south baseline to Lousa. This possibly explains the lack of a similar pattern in Andy's spectra. I suspect this is related to the 'Epsilon' generating area being significantly off-axis, but I've more work to do to convince myself of that.

My radiant-height calculation is based on my receiver location. If I recalculate the radiant-height based on the mid-point of the Liege Bol d'Air - Rx path, I expect the fit, between the curves, may be even better. However, we need to be careful in interpreting the above chart, as Quadrantid rates will obviously increase with increasing radiant altitude.

Trail Radius

I need to check the following figures for trail radius and half andgle, as I know I made a mistake early on. I can't remember if I corrected these numbers or not...

When a meteor enters the earth's atmosphere, it creates a trail of ionisation with an initial radius. The initial radius varies with height. The trail also expands at different rates for different heights. The chart below illustrates the trail radius, for height from 75km to 120 km and for times up to 10 seconds after the initial meteor event.

ht (km)  1 sec  2 sec  3 sec  4 sec  5 sec  6 sec  7 sec  8 sec  9 sec  10 sec
120 288.01 563.44 838.86 1114.28 1389.70 1665.13 1940.55 2215.97 2491.40 2766.82
115 132.66 260.01 387.36 514.71 642.06 769.41 896.76 1024.11 1151.46 1278.81
110 61.12 120.01 178.89 237.78 296.66 355.54 414.43 473.31 532.20 591.08
105 28.17 55.40 82.63 109.85 137.08 164.31 191.53 218.76 245.99 273.21
100 12.99 25.58 38.17 50.76 63.34 75.93 88.52 101.11 113.70 126.29
95 5.99 11.81 17.63 23.45 29.27 35.09 40.92 46.74 52.56 58.38
90 2.76 5.45 8.15 10.84 13.53 16.22 18.91 21.60 24.29 26.99
85 1.27 2.52 3.76 5.01 6.25 7.50 8.74 9.99 11.23 12.47
80 0.59 1.16 1.74 2.31 2.89 3.47 4.04 4.62 5.19 5.77
75 0.27 0.54 0.80 1.07 1.34 1.60 1.87 2.13 2.40 2.67


Trail Half Angle

The trail is initially a very narrow cone, but rapidly expands to be a horn shape. The angle by which the surface of the trail departs from the path of the meteor, the half angle, is detailed below. This can be over 20 degrees at great heights, but is typically less than 10 degrees. Below 95 km, the half angle will seldom exceed 1 degree.

ht (km)  0 sec  1 sec  2 sec  3 sec  4 sec  5 sec  6 sec  7 sec  8 sec  9 sec  10 sec
120 0.14 2.82 5.49 8.14 10.75 13.31 15.82 18.28 20.66 22.97 25.21
115 0.06 1.30 2.54 3.78 5.01 6.24 7.46 8.68 9.89 11.08 12.27
110 0.03 0.60 1.17 1.75 2.32 2.89 3.46 4.03 4.60 5.17 5.74
105 0.01 0.28 0.54 0.81 1.07 1.34 1.60 1.87 2.13 2.40 2.66
100 0.00 0.13 0.25 0.37 0.50 0.62 0.74 0.86 0.99 1.11 1.23
95 0.00 0.06 0.12 0.17 0.23 0.29 0.34 0.40 0.46 0.51 0.57
90 0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.18 0.21 0.24 0.26
85 0.00 0.01 0.02 0.04 0.05 0.06 0.07 0.09 0.10 0.11 0.12
80 0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.06
75 0.00 0.00 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.03