Originally Posted By: PeteM
Don,

If there was no wad, I would agree with your math. But the pellets exit together. The Rayleigh distribution is not dependent on separate exit. It has been proven to describe single exit (rifle, artillery, arrows, etc.) and multiple exit (sticks of bombs, shotguns, etc.) equally well.

If you look at the photography, They do not begin to separate until nearly 6" from the muzzle. At 12" the column is deformed. The centers of impact relative to the aiming point of multiple patterns will be described by a Rayleigh distribution. The distribution of shot, assuming the pellets separate, relative to the center of impact of each pattern will also be described by a Rayleigh distribution in the radial direction.

Add in the fact that some pellets are compressed, thus deforming them and changing their co-efficient. The wad, hardness of the shot and any buffering material all play a role. Given that, I believe that a mathematically pristine Rayleigh distribution is impacted. The theory at least assumes all components of the distribution to be equal. In practice an allowance is always made for reality. Disagree. The Rayleigh distribution assumes that there are differences and tells you how, on average, those differences will influence the point of impact relative to center of impact. This is the same principle as describing the tallness of humans via a normal distribution.
The pellets do not impact the plane of the target simultaneously. Not just because of the length of the column, but also because of minor variations in velocity and ballistic co-efficient. This becomes obvious to the shotgunner at the patterning board. smile True, and, along with the exacting pressures on the pellet at time of separation which determine the pellets initial departure velocity and direction, determine where, relative to the other pellets the individual pellet will impact.

On average, the radial direction distribution of pellets will approximate a Rayleigh distribution and the circumferential distribution should be even (same number of pellets per segment of a "washer").Pete