Forensic Bk InMti, 55 (1992) 105  130 Elsevier Scientific Publishers Ireland Ltd.
105
AN INVESTIGATION INTO PELLET DISPERSION BALLISTICS
N.K. NAG and P. SINHA State Forensic Science Laboratory, Calcutta 700 087 (In&u) (Received December 13th, 1990) (Revision nxeived August ‘7th 1991) (Accepted April 6th, 1992)
Summary Existing works on pellet dispersion ballistics are confined to some databased models derived from statistical analysis of observed patterns on targets but the underlying process causing the dispersion lacks due attention. The present article delves into the relatively unexplored areas of dispersion phenomena, and attempts to develop a theoretical model for general application. The radial velocity distribution of pellets has been worked out by probing into the physical process of dispersion based on transfer of momentum from undispersed shot mass to dispersed pellets. The ratio Bu/v,, (U = root mean square (r.m.s.) radial velocity and v, = muzzle velocity of the pellets) is found to be fairly constant for a fixed gunammunition combination and has been suitably designated as ‘Dispersion Index’ @I) characterising its dispersion capability. The present model adequately accounts for pellet distribution on targets and it appears that ‘Effective Shot Dispersion’ (ESD) as introduced by Mattoo and Nabar [ESD = [(4/IV&Ri2] 1’2,where Ne is the total number of pellets and Ri is the radial distance of the ith pellet from centre of pattern], gives a faithful numerical measure of overall dispersion at a given distance. A relationship between ESD and firing distance, incorporating the effects of air resistance and gravity has been worked out, which reveals that DI controls the dispersion at a given distance. For small distances (< 20 m) the relation reduces to a linear one, as already observed empirically and looks like ESD = Ee + DI x firing distance, E,, being a parameter dependent on gun and ammunition. The present model, unlike earlier ones, is versatile enough to explain the natures of the dependence of dispersion on firing distance as well as on gunannmmition parameters, which are essential for a faithful reconstruction of a crime scene. The model has been tested with such experimental data as are available and reasonable agreement is observed. Key words: Ballistics; Pellet dispersion; Pellet statistics; Dispersion index; Effective shot dispersion; Firing distance; Choke; Barrel length; Shotgun
Introduction
The observation of pattern reproducibility of pellets discharged from a shotgun at a given distance has induced several forensic examiners to look for a fundamental basis governing the dispersion process and also to explore the ability of pattern characteristics to estimate the firing distance [l  161. The dynamical Corresponrleace to: N.K. Nag, State Forensic Science Laboratory, 03790738/92/$05.00
0 1992 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
Calcutta 700 03’7, India.
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behaviour, however, of a projected charge of pellets from a shotgun does not appear to be as simple as that of a single projectile ejected from a firearm. A complete analytical solution to the exact pellet dynamics is a formidable task, since it is a very complicated manybody problem with dynamical forces, constraints and boundary conditions not fully known. As such, even though experimental data on pellet dispersion already exist, our present knowledge of pellet dynamics is confined to some empirical/semiempirical relations obtained by statistical analysis of the data. The position is so perhaps because little attempt has so far been made to probe into physical features of the pellet dispersion and thereby work out the dynamical behaviour of dispersing pellets. The problem, therefore, demands a detailed investigation in this direction for a better understanding so as to develop a theoretical methodology for estimating distance of firing. The present article attempts to provide a fresh look at the statistical evaluation of pellet dispersion in conformity with the underlying physical process. The following aspects of shotgun ballistics have been considered: (i) the pellet dispersion mechanism, (ii) the radial pelletvelocity distribution, (iii) the distribution of pellets on target, (iv) the effects of air resistance and gravity on pellet trajectories, (v) the estimation of firing distance and (vi) the effects of gunammunitiondependent parameters (e.g. barrel length, choke, etc.) on pellet dispersion. The proposed model pays attention to processes involved in interior as well as exterior shotgun ballistics. Pellet statistics, representing the radial velocity distribution of pellets, emerged from such considerations and were conveniently used for range estimates. The model has been quantitatively examined with a wide variety of data observed by different workers and its validity has been tested. As the present paper attempts to offer theoretical explanations to the observations made by earlier workers it would be useful to review briefly previous works in general and applications of Gaussian/Maxwellian statistics to shotgun pellets in particular, because of its pertinence in developing the present statistical model. Review In his comprehensive early work, Burrard [l3] covered many aspects of shotgun ballistics including excellent illustrations of test patterns of pellets due to various gunammunition combinations at different distances. Although these patterns bore some noticeable consistency the characteristics were not explored by forensic scientists in practical problems until Mattoo and Nabar [4] put forward a deterministic approach to extract numerical information from the patterns. They rightly noted the approximate circular symmetry of the pellet pattern and introduced the concept of ‘Effective Shot Dispersion’ (ESD) as a characteristic parameter of the pattern. The ESD was defined [4] as
where NOis the total number of pellets and Ri is the distance of the ith pellet from the centre of pattern. Thus, ESD incorporates the integrated information
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of the entire pellet pattern, but with the merit of being a single numerical parameter. These authors also noticed an approximate linear relationship between ESD and firing distance and hence suggested a simple method to estimate the latter from pellet distribution for the cases where the entire pattern was available for ESD estimation. Jauhari et al. [5,6] also made a statistical analysis of the behaviour of six parameters derived from combinations of horizontal and vertical pellet dispersions, as observed by enclosing the total pattern by the smallest rectangular area. These authors and also Heany and Rowe [7] later observed that the square root of the area of the smallest circumscribing rectangle (JA), could be used as a characteristic parameter of the pellet pattern and termed as ‘spread’ [7] (Jauhari et al. [6] called it ‘Pattern Size’), which varied approximately linearly with firing distance. However, Rios and Thornton [8] raised an objection to the inadequacy of the statistical analysis of Heany and Rowe [7] in arriving at such a conclue8ion. After an elaborate study Wray et al. [9] made a comparative analysis of the suitability of using three different parameters, viz., (i) ESD, (ii) spread and (iii) radius of smallest circle enclosing the total pattern for choosing an effective parameter in order to estimate firing distance. While all these parameters could be expressed as linear functions of distance of firing, ESD was found to give the best fit. Later, the George Washington University group [lO,ll] tried to validate the method of range estimation via regression analysis after conducting blind test firings. Moreau et al. [12] studied the effect of barrel length on pellet patterns fired by shotguns with barrels sawn off stepwise to 6 in. However, their findings on increased dispersion due to decrease in barrel length lacked any physical explanation for such behaviour. Rios et al. [13] conducted a multivariate statistical analysis to investigate the effects of barrel length, discharge distance, choke constriction and shot size on dispersion and found a large correlation of dispersion with the discharge distance. They were critical about using the area (A) of the smallest circumscribed rectangle as a suitable parameter of pellet pattern, because of its approximate quadratic dependence on firing distance, but remained silent on other parameters, viz., ESD or spread (dA) having approximate linear relationships. Guided b!y the precedence of relating degree of choke with percentage of shots within a ;30u circle at 40 yd, they decided to characterise a pattern using the general term ‘dispersion’ to define a new parameter as the percentage of shots being dispersed outside a circle of 75 mm diameter about the patterncentre. The authors attempted to fit their data on ‘dispersion’ and discharge distance with a logistic response curve and thereby tried to verify the behaviour of socalled ‘logits’ as linear function of firing distance, leading to a negative outcome. Thus they finally left the problem of ascertaining the functional relationship between their ‘dispersion parameter and target distance unsolved and at the same time did not offer any physical justification for using such a logistic model, which is classically used to explain doseresponse relationship of drugs and apparently applied to biological systems. All the works on range estimation problem discussed so far were dependent
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on the availability of a complete pattern and confined to pellets small in number. But in general, the total pattern is not always available due to outfliers, which are specially significant for ammunition containing a large number of pellets. So, to deal with such common cases the necessity of providing a statistical account for pellet pattern became evident. Investigation in this direction required a knowledge of distribution of velocities of the dispersing pellets and it was assumed that the pellet velocities obey some random distribution law like Gaussian [14] or Maxwellian [15], which were empirically parameterised and tested with some experimental data. These statistical approaches appeared interesting especially as the results were encouraging, but the assumptions of the velocity distribution laws were somewhat arbitrary and no plausible attempt was made to examine whether the physical process involved in the dispersion phenomenon was consistent with the proposition of Gaussian or Maxwellian velocity distribution laws for the dispersing pellets. Again, the assumptions of Gaussian or Maxwellian laws make no effective difference here, since both are expressed by the same exponential form of probability function that a pellet will possess a particular velocity component. Recently, Boyer et al. [16] have presented some data on distribution of No. 2 birdshot and 00 buckshot. They failed to match their data with the Gaussian model and remained sceptical about the general applicability of Gaussian distribution. The GaussianMaxwellian
model and dispersion
phenomenon
It is worthwhile to briefly outline the GaussianMaxwellian (GM) model applied to a shotgun pellet dispersion process in order to prepare necessary background for a study of the statistical law governing the pellet dynamics. For the sake of simplicity let us, for the time being, deal with an ideal situation by neglecting effects of gravity and air resistance, which will ultimately be accounted for. According to GM distribution law the number of pellets possessing a radial velocity component lying between vu,and v, + dv, is given by u2vr2vr dv, MGM
=
(2)
2ar2No e
where, No is the total number of pellets and (IIis the characteristic parameter of the model having dimension (v) ‘. This corresponds to the number of pellets likely to be observed on a normal target plane and enclosed within circles of radii r and r + dr about the centre of mass and is given by r dr where, t is the time of flight. In deriving Eqn. (2a) from Eqn. (2), vu,has been replaced by r/t assuming that initially all the pellets had their r coordinates zero, i.e., r = 0 at target distance (x) = 0. Under the present ideal condition, t = z/v,,; where, v. is the muzzle velocity. So, the fraction of pellets enclosed within a circle of radius r is given by the distribution function
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#JGM(r) =:
r2/x2 1  e a2v02
(3)
The number of pellets per unit area at a distance T from the patterncentre is Nopo&r) with the density function,
PGMtr)
=:
2 2 r2iz2
&
em” vO
The Eqns. (3) and (4) characterise the pattern according to the GM model. Now, for ascertaining the validity of this statistical law the physical process leading to pellet dispersion needs to be examined critically. Had there been a single projectile, it would have followed a straight trajectory along azilmuthal axis under the ideal conditions so assumed and hit the target at the origin of T  8 plane. But this does not happen in the case of multiple pellets which delmonstrate a certain distribution on the target plane about the centre of pattern at the origin. Thus, for a single shot the probability for possessing any velocity component other than v, is zero, while for multiple pellets it is not so. Therefore, for multiple pellets the possession of a nonvanishing radial velocity component (vJ, by any pellet is due to the presence of others, i.e., v, must be acquired by a pellet from momentum transfer by others through interpellet interactions. Initially, the pellets are in a closely packed condition in the form of a cylindrical column inside the barrel where they move en masse with the same velocity along the Zaxis (barrel axis) and possess no other velocity component that may give rise to any kind of dispersion. Thus, the pellets are at rest with respect to each other inside the barrel and any kinetic interaction hardly takes place there. However, the requisite process leading to interpellet interaction is initiated within the barrel. The pressure developed due to propellant burning acts on the shot mass as a tremendous compressive stress in the direction of the barrel axis (azimuthal axis) and as a consequence, a radial tensile stress is also produced depending on the former and also on Poisson’s ratio of pellet material. As the shot mass is free to move along the bore of the gun in the direction of the azimuthal axis the strain energy due to the azimuthal compressive stress is readily converted into kinetic energy and the shot mass is set in motion. By virtue of the cylindrical symmetry of the shot mass system the resultant tensile stress is directed radially outwards and the associated tensile stress wave is reflected back from the barrel wall as a compressive stress. As a result, the peripheral pellets experience a high compressive istress from the barrel wall and this is subsequently transmitted radially inwards .and develops a significant amount of strain energy in the shot mass. During tlne passage of shot mass through the barrel the gas pressure falls off rapidly after reaching a peak value within a few inches of travel. The strain energy at any moment depends on the instantaneous stress and hence on the gas pressure at that moment. So, during the motion of the shot mass through the barrel, its kinetic energy being the integrated effect of gas pressure, goes on increasing lcontinuously. However, the strain energy decreases as the gas pressure
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falls off gradually after the peak value in the barrel. Thus, the shot mass acquires kinetic energy due to azimuthal stress and potential strain energy due to the radial stress inside the barrel. As soon as the pellets emerge from the barrel, the instantaneous radial compressive stress is released and the associated strain energy is converted into kinetic energy. This new kinetic energy depends on the discharge gas pressure at the moment of emergence of the shot from the muzzle and leads to interpellet kinetic interactions which could not be effected by the initial kinetic energy inside the barrel. As a result, the pellets after their emergence from the barrel can no longer move together with same velocity along the firing axis and start dispersing. The outermost pellets gather radial momentum from the rest through interpellet interactions and are first to disperse followed by the next pellets and so on. Therefore, an arbitrary pellet during its dispersion process can gather radial momentum by interaction only with those pellets yet to disperse from the mass, while those already dispersed fail to influence its dispersion through any interaction. As soon as the interpellet interactions are over the pellets get separated from each other more and more and traverse free paths. So, due to nonoccurrence of any further collision, the distribution of velocities amongst the pellets, as produced by the above mechanism, remains unaltered throughout the motion of the dispersed pellets. The pellets possessing higher radial velocities have higher radial coordinates, manifesting an axial symmetry and the separation among the pellets goes on increasing with time leading to a continuous decrease in pellet density (in space), which does not remain steady. It is well known that a system of gas molecules, obeying the classical Maxwellian law of distribution of velocities, is in a steady state in which the density and law of distribution of velocities remain the same at every point of the’gas throughout all time and unaffected by chaotic molecular collision. Thus, the molecules having velocity components lying within a specified limit are, at every instant, distributed at random independent of the positions and velocities of other molecules, i.e., each point in space has the same properties. Further, Maxwellian law is valid when there is no external force, that is, the molecules do not possess any potential energy. It is now evident that the dynamical system of dispersing pellets does not have such properties as those of a Maxwellian system. The pellets possess potential strain energy which causes kinetic interactions. The interactions undergone by a pellet and hence the radial velocity acquired by it depends on its position in the undispersed shot mass and the radial velocity distribution produced in the pellets is a result of a series of sequential interactions and thus, is axially symmetric. Pellets possessing radial velocities within a specified limit have also their radial coordinates lying within a specified limit and thus not randomly distributed. So, even though the distribution of the radial component of pellet velocities produced by the above mechanism, may so turn out that Eqn. (2) is approximately satisfied, it cannot be apriori assumed that the Maxwellian velocity distribution law is valid for the dispersing pellets. Statistical
evaluation
of pellet dynamics
Let us suppose that at any moment during the dispersion process, N pellets out
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of a total number of No are yet to disperse from the shot mass. Let the kinetic interactions among these N pellets cause a momentum (p) transferred to the pellet which is ready to disperse at this moment. Then the pellets which will disperse subsequently will interact with a gradually decreasing number of undispersed pellets and will receive momenta less than p in the course of interpellet kinetic interactions. So, the probability that any momentum up to p will be transferred to a pellet to disperse it from the shot mass, should be P(p) = N/No
(5)
In the case of a large number of pellets, the direction of momentum transfer will be rando:m and as such the individual probabilities for the respective orthogonal components of p, P(pJ, P(pr) and (P(pJ will be independent and equal. So, we should have
P(Px)= IFPJ = P(P3,and P(p) = P(pJ . P(pJ . P&), so that
(6)
P(pJ = (:N/N,Ju3 Now, let P*@.J be the probability that any xcomponent of momentum up to p, will not be transferred to a pellet through interpellet interactions. The probability tlhat the same pellet will not receive any further xcomponent of momentum up to p, ’ is p*(p, + px’) = p*(px)  P*(pX’)
(7)
under the assumption that P*(pJ and P*(pX’) are independent. Differentiation with respect to px gives 
d
dp,
P* (p, + p,‘) = P*(pX’)  d P*@X) dp,
Dividing (8) by (7) yields d d In F’*& + pX ‘) = ~ In p*(pJ = constant dpx dp, since p’ is arbitrary. Therefore, we should have P*@.J = Ki e_Qpx
(10)
where Ki and K2 are constants. Since P*(O)must be 1, we have K1 = 1 and thus p*@3
=
[email protected] (104
Now, the sum of the complementary probabilities P&J and P*(px) should be unity, Therefore, we should have from Eqns. (6) and (1Oa)
112 e 4? P, =
1 
(N/No)113
(11)
For a particular value of N, the upper limit of the ycomponent of momentum transfer will also satisfy Eqn. (ll), i.e., p, = p, and hence the corresponding upper limit of radial velocity, v, = p,/nz = d2 *P&L, m being the mass of each pellet. Thus, we have finally for the number of pellets acquiring a radial velocity up to v, as N = No (1  e BVr)3
(12)
with /3 = m K2/d2. So, the number of pellets having their radial velocities lying between v, and v, + dv, is given by w
= 30
. jjT,, (1

eB+)2
. eh
dv r
(13)
The root mean square (r.m.s.) and most probably radial velocities, u and v, respectively, may be easily evaluated as, u = 2.173&I, and v, = l.O9361/3= 0.5 u
(14)
The observed pellet pattern on target at a distance x will satisfy a radial distribution of the type H(r) = (1  ey3
(15)
with h = &/z and $Jbearing the same physical meaning as &M. The associated density function may be written as
I&) = g
(1
_
em”>” . eh’
Eqns. (15) and (16) describe the distribution of pellets on a target according to what we call ‘Pellet Statistics’ and are significantly different from the corresponding Eqns. (3) and (4) according to GM statistics. As such, before formulating a model for range estimation, it will be worthwhile to digress upon the properties of the distribution and density functions given by Eqns. (15) and (16) which are actually governed by the behaviour of radial pellet velocities obeying Eqns. (12) and (13). The interpellet interactions cause finite momentum transfers throughout the interaction process until the last two pellets interact between themselves and get dispersed. Thus, dN in Eqn. (13) will be vanishing for both v, = 0 and v, = 00. Consequently, for a charge of pellets, p(O) = P(Q)) = 0. While, for the case of a single shot, i.e., in the absence of any interpellet interaction p(O) = 1 and p(r) = 0 for r > 0. The r.m.s. and the most probable radial displacements (
[email protected] and rm) will be given by
113
@=
2.1.731Xand r,,, = 10.5 m
(17)
in analogy with Eqn. (14). The most densely populated area corresponding to prnaxis determined by the value of r(r,J for a maximum value of dNd(xr2) and may be evaluated as rd
=
0.5227/X = 0.24 @
(13)
But according to the GM model, this peak value should occur at the centre (r = 0). So, the occurrence of such a density peak at some distance away from the centre demands a critical analysis by examining the corresponding radial velocity acquired by the pellets leading to this maximum density. It is obvious that the number of pellets acquiring a radial kinetic energy 5, (= Yzmv,2) is zero for both 4, = 0 and [, = 00 as in the case of v,. The most probable kinetic energy occurs for the maximum value of dNld&, which corresponds to V, = 0.5227/o, i.e., the pellets possessing the most probable kinetic energy have radial displacement r = rd. Since the kinetic energy acquired by a pellet through interpellet interaction is the work done on it (W = [,), the above maximum corresponds to the minimum of dW/dN, which is the work done per pellet. Thus, the pellets tend to get most thickly assembled in a region as determined by the least work done per pellet and such an occurrence is always favoured by the fundamental principles of mechanics. For the GM model, the most probable 5, is zero and prnaxoccurs at r = 0. Thus, pellet statistics and GM statistics are vividly distinguished on this point and the analysis of real patterns in this respect will decide between the two. Range e#stimation model
Since, a definite gunammunition combination produces a reproducible pellet pattern at a given distance, the observed pellet distribution on a target should be related to target distance through a gunammunition dependent parameter which has to be properly propounded for developing a faithful range estimation model. For this purpose we should also select a characteristic parameter for pellet pattern and express it as a function of firing distance and the gunammunition dependent parameter. It is desired that this pelletpattern parameter ought to be a simple one, at the time a sensitive function of firing distance as well as a good representation of the entire pattern. The gunammunition parameter, governing its dispersion properties, should have a simple phydcal significance so that it is possible to analyse how factors related to gun and ammunition affect the dispersion process. Apart from constructing the above model parameters, we should also account for the initial distribution of pellets in the undispersed shot mass as well as for the effects of air resistance and gravity which have not been considered so far. We would first construct the parameters of the model under an ideal situation neglecting the above corrections. In ord.er to characterise the observed pellet pattern, we note that the
114
parameter ‘spread’ used by earlier workers was expressed by the size of the enclosure of the total pattern, while the other parameter ‘dispersion’ defined by Rios et al. [13] represented the percentage of total pellets lying within an enclosure of a definite size. Thus, both these parameters characterise the pattern on the basis of solitary information concerning the number of pellets and the size of their enclosure. So, in their attempt to choose a better candidate for characterising pellet patterns Rios et al. [13] have merely reversed the way of defining the parameter ‘spread’ and in doing so they have made the parametric dependence of pellet pattern on firing distance much more complicated and unsuitable for an easy estimation of firing distance. But, the distribution characteristic of certain objects in a pattern should be most adequately determined by statistically evaluating the total pellet pattern, extracting as much information as possible. ESD [4] satisfies this requirement and is, therefore, the best choice for characterising the pellet pattern. The comparative study of Wray et al. [9] also indicated the superiority of ESD over other parameters. In that case the pellets had their initial radial coordinates r = 0 at x = 0, ESD as defined by Mattoo and Nabar [4] turns out to be E’
=
[email protected]=
4.346/X = 2ut
(19)
which under the ideal situation may be written as E’=(Bu/w,,)x=Dx
(194
This new dimensionless parameter D = 2u/v0, is exclusively determined by the gunammunition combination and relates ESD with firing distance. The parameter D controls the dispersion capability of a gunammunition combination and may be termed as ‘Dispersion Index’ (DI) of this combination which will exhibit greater pellet dispersion at a given target distance for a higher value of DI. In order to account for the initial pellet distribution in the undipersed shot mass, we note that initially the centers of all the pellets remain within a cylindrical column of diameter 4 = CJ&,  4, where d,,, and di, are, respectively the muzzle and pellet diameters. Thus, ESD as observed on a target at z = 0, should be E,, = d&2
(20)
E’ as defined in Eqn. (19) involves the radial displacements during the flight
alone and is zero at x = 0. Thus, actual observed ESD on a target pattern should be E=Eo+E’=E,,+Dz
(21)
When there are N undispersed pellets in a shot mass, the radial coordinate of the outermost pellet, which is ready to disperse, will be r’ = (d3/2) * (AvNfJ)1’2
(22)
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so that after undergoing a radial displacement r during its flight up to target, it attains a radial coordinate R=r+r’
(23)
on the target. It is easy to find from Eqns. (15), (19) and (23) that E’ = 4.346 (R  r’)/ln [l  (AVN0)“3]
(24)
Eqns. (2lt 24) enable one to estimate ESD of a pattern by counting the number of pellets N enclosed within a circle of radius R on the target. Next, we shall account for the effects due to gravity and air resistance which will alter the pellet trajectories and their time of flight. Thus, the actual pellet trajectories are not straight lines and the horizontally fired shot does not normally hit a vertical target as has been assumed in the above analysis. The deviation of actual trajectory from the ideal one would be larger for longer target distances. However, the above derived formulae will hold true in a frame of reference at rest with the centre of mass of the pellets, the azimuthal axis being directed along the direction of motion of the centre of mass and the target plane being normal to the axis. Thus, for horizontal firing, a usual vertical target would be inclined to a normal plane in the centre of mass frame by the same angle, 8’, that the actual trajectory makes with the horizontal direction at target. McShane’s equations solved by Jauhari e;kL [17], give for a projectile velocity and the angle, 8’ = tan  1 v at a distance x as, v = v0e [
[email protected]((lkto) + (At/2c))], with A = 1117696ft‘, g = acceleration due to gravity, ballistic coefficient of projectile (dimensionless), = 0.061861~YZ, for C = spherical pellets, n being the number of pellets per oz. The time of flight is t =
(c/Avo) . (eAdc  1)
(25)
Since the dispersed pellets traverse free paths the ballistic coefficient of each pellet (same for all) should be used in Eqn. (25). A circular area of radius R on the actual vertical target would appear elliptical on a plane normal to the direction of trajectory of the centre of mass and will have its semimajor and semiminor axes, R/COST’ and R, respectively. For small values of T ’, such an elliptical area may be approximated to a circle of radius RIdcosT ’, having an equal area only with slightly modified boundary. So, to account for the altered trajectory, we should replace R by RldcosT ’ in Eqn. (24). However, it is found that this correction is indeed negligible for target distances of practical interest (usually < 50 m). But, the time of flight for target distances larger than 20 m, substantially differs from x/v0 and as such, we should modify the time of flight in Eqn. (19) by the expression in Eqn. (25). Thus, instead of Eqn. (19a), we have finally from Eqn. (21), E = EO + D  B(edB 
1)
(26)
where B = c/A. The Eqn. (26) is the desired functional relationship between
116
pellet pattern parameter and firing distance. For negligible effects due to gravity and air resistance, i.e., for small target distances, Eqn. (26) reduces to a simple linear form, E = Es + D *z, as was originally suggested by Mattoo and Nabar 141.Thus, for a particular gunammunition combination and a questioned pellet pattern the practical procedure for range estimation is summarised as follows.
The first step is to estimate ESD of the questioned pattern for which the direct method of Mattoo and Nabar [4] is the best if the complete pattern is available. In the case of a large number of pellets, however, small parts of the pattern are often missing from the boundary and such patterns may be analysed by resorting to pellet statistics. The centre of the pattern is again better located by the method of Ref. (4) rather than by visual estimation. The pellets (Ni) contained in concentric circles (radii Ri) should be counted in as many circles as possible for 0.1 zoo),appears as, Pd a e ’ Pd = A2 e“,
(30)
AZ being a constant, since 3comay be taken to be a constant for a gunammunition. So the velocity, V of shot mass at z should satisfy dV/dt = A3e (12 for x > x0 (A3 being a constant). If V’ be the velocity at z = x0, then we may
write,
since, dz = Vdt. Thus, after some simplification we obtain V = V, (1  b eaz)1’2
(31)
where, V,,, and b are constants. We have verified the validity of Eqns. (30) and (31) from the data in Table LII of Ref. (2) from 2 = 6  30 in and noted that both the data on pressure and velocity fit Eqns. (30) and (31) with almost the same value of the decay constant (a). The results are quite consistent particularly in the range from x = 8” onwards. So, we may assume that z. is not greater than 8” as is normally expected. Now, from Eqn. (19a), the DI for a gunammunition combination with barrel length z may be written from Eqns. (29) and (30) as D = (Q eT/(1
 b ey”’
(32)
where, Q is a constant. The Eqn. (32) gives us scope to explain the observations of Ref. (12) on pellet patterns due to sawnoff shotguns. The data presented in their Table 1 are suitable for this purpose, since the ‘dispersal’ values (which are just half of the ESD values) for different patterns have been estimated in this table and these data will allow us to calculate the dispersion indices for the gunammunition combinations. It may be noted that when their original weapons with fulllength barrels were first shortened by 4 in, all those behaved as true cylinders, since the chokes were located near the muzzle (within 3 in from muzzle). Thus, for barrel lengths 20, 12 and 6 in for a particular gunammunition combination but initially with different chokes, the respective dispersal data agreed within experimental errors. So, excluding the data for unsawn weapons, we had the dispersal values of three different gunammunition combinations, viz., (i) Remington, 12gauge, 00 buckshot (series l), (ii) Winchester Western Super X, 12gauge, 00 buckshot (series 2) and (iii) Winchester Western Super X Magnum, lkgauge, 00 buckshot (series 2). The DI for each pattern was calculated from Eqn. (19a) and the DI values for a particular gunammunition combination determined at three distances (10, 30 and 50 ft) were found to be more or less constant, as expected. We used the mean DI for each barrel length
127
for the data on the above three different combinations in order to examine the validity of Eqn. (32). Thus, for the first combination we had mean DI values for barrel lengths 6, 12, 20, 24, 26 and 32 in all corresponding to true cylinders. These data have been plotted in Fig. 10. The curve according to Eqn. (32) has also been shown in the same figure and it accounts for the variation of dispersion index with change in barrel length. The data for the third combination also showed similar agreement between theory and experiment. However, the experimental results for the second combination indicated a small positive slope in the DI versus barrel length curve for barrel lengths in the region 2028 in, while the theoretical curve predicted a small negative slope. However, the agreement below 20 in was as good as in Fig. 10. It may be noted that the effect of change in barrel length is very small above 20 ” as has also been observed by Rios et al. [13]. In fact the variations in pellet dispersion in this region are themselves comparable with the observational errors as is evident from Table 1 of Ref. (12). Thus, the changes of significance in dispersion due to variation of barrel length are produced below 20 ” and Eqn. (32) explains such changes in all cases reported in Table 1 of Moreau et al. [12]. Phase V
The shooter’s Year Book [18] provides a generalised Table for the percentage of pellets enclosed within a 30in circle at different target distances and for varying chokes. It was mentioned in this publication that the table might be used for estimating number of pellets within such a target area and, as examples, for using it, iseparate tables for 11/16oz and 1% oz of Nos. 4, 5, 6, ‘7shots have been provided without any reference to the guns employed. We have seen that according to pellet statistics the percentage values are not the same for different gunammunition combinations even for the same DI. Anyway, it may be of some interest to compare the pellet statistics results with this table. As such, we have computed the pellet statistics predictions for No. 6 shot fired by a 16bore gun and presented the results in Table 2 along with the corresponding values from TABLE
2
PERCENT.AGE OF PELLETS IN 30” CIRCLE AT DIFFERENT RANGES AS PREDICTED BY PELLET STATISTICS (GUN: 16BORE; LOAD: NO. 6 SHOT; DI ASSUMED TO BE 0.02 FOR TRUE CYLINDER)
so?i?lg of
Raw W)
Qun
True cylinder Imp. cylinder l/4 choke l/2 choke 314 choke Full choke
SO
65
40
45
50
55
60
72(60) 73(72) 75(77) 79(83) 83(91) 86(100)
60(49) 62(60) 64(65) 68(71) 72(77) 77(84)
48(40) 50(50) 52(55) 57(60) 61(65)
39(33)
30(27) 32(33)
2ry22) 25(27) 27(30) 30(33) 34(37)
18( 18)
66(70)
56(59)
40(41) 42(46) 46(50) 51(55)
34(38) 37(41) 42(46) 47(49)
38(40)
W22) 21(25) 24(27) 27(30) 31(32)
128
Ref. (18) within parentheses. In our computation the dispersion index has been assumed to be 0.02, a typical value for a true cylinder, while those for choked cylinders were determined by Eqn. (27) and the predictions agree reasonably well with those in Ref. (18), presumably based on test firings. Discussion The pioneering work of Mattoo and Nabar [4] was followed by many investigations on pellet dispersion leading to much experimental data, but the works really lacked the causation of dispersion and ballistics involved, which could enable a forensic investigator to understand the dispersion process and apply to practical problems. The present paper pays attention to the physical process and mechanism involved in pellet dispersion ballistics leading to the emergence of pellet statistics, representing the dynamical behaviour of the dispersing pellets. There may be some small contributions to the dispersion process from effects other than momentum transfer in interpellet interactions, such as those due to air resistance or gases emerging from muzzle. The contribution of air resistance to dispersion process is negligibly small compared to momentum transfers in interpellet interactions. The persisting effect of air resistance during the flight of the dispersed pellets has been accounted for and found to have some significance only for sufficiently large target distances. The shot mass after its emergence from the muzzle escapes the influence of the muzzle gas effluents within a few feet and only a few peripherial pellets start dispersing within this short span of time. Thus, the dispersion process of most of the pellets is free from any effect of gases emerging from the muzzle, while some peripherial pellets being affected by this factor may show small departures from pellet statistics. Such effects will, therefore, be manifested only in the boundary region of the pattern and the pellet distribution therein in the case of actual patterns show slight deviations from what was predicted by pellet statistics. Anyway, the distribution of pellets, except for about 10% from the boundary, was remarkably consistent with pellet statistics and consideration of circular enclosures excluding the outer zone of patterns for ESD evaluation in practical cases should not involve any errors due the above perturbing effect. The involvement of different physical parameters in the dispersion process being interpretable through the present model, a wide variety of data from different sources could be explained. Thus, the basic concepts on which the present model rests appear compatible and justified. But, the agreement of the GM model with pellet patterns at regions away from pattern centre is also noteworthy and this poses a question  why is it so when the dispersing pellets are not expected to be governed by the GM velocity distribution law? An insight into the derivation of pellet statistics might answer this question. Had we considered energy transfer instead of momentum transfer in interpellet interactions and assumed that the entire energy transferred in this way contributed to radial kinetic energy, the distribution of radial kinetic energy according to the pellet dispersion mechanism so turned out that we would have ended up with Eqn. (2) rather than Eqn. (13). Now, as long as the undispersed column of pellets has a
129
significant radial spread, the major part of energy transferred in interpellet interactions is likely to contribute to radial kinetic energy. Thus, except for the central zone of the pattern, the distribution of pellets does not deviate much from GM model expectations as is apparent from Fig. 2B. The proposed dispersion mechanism gives a physical picture of a dispersion process for a fairly large number of pellets, presumably of small size (c&,< < c&,).The radial momentum transferred to an arbitrary pellet was found to depend on its position (as defined by its radial coordinate) in the undispersed shot mass just at the moment of exit from the muzzle. Thus, when choke is introduced at the muzzle end, the rearrangement of pellets in the shot mass alters the dispersion properties and the dispersion mechanism could explain the resultant change in pellet dispersion due to altered pellet distribution in the undispersed shot mass. The Eqn. (27) derived in this context, suggests that other factors remaining unaltered the smaller the shot size the greater would be the dispersion. However, due to nonavailability of relevant experimental data the dependence of pellet dispersion on pellet size, as suggested by the present model, could not be verified. It was emphasised earlier that the r.m.s. radial velocity was determined by the gas pressure at the moment of emergence of shot mass from the muzzle, while muzzle velocity is already known to be the integrated effect of gas pressure on the shot mass althrough its motion inside the barrel. Thus, the varying propelling gas pressure determines the DI. Such effects of gas pressure on pellet dispersion could be verified from the data on pellet dispersion produced by sawnoff shotguns.. The shot mass emerges from the muzzle after being energised by the gas pressure in the forms of both kinetic energy from potential strain energy. Since the pellets acquire their radial kinetic energy from potential strain energy, DI, as defined, appears proportional to the square root of the ratio of potential strain energy to kinetic energy possessed by the shot mass at the muzzle end. Hence, this ratio plays the key role in determining the pellet dispersion exhibited by a particular gunammunition combination. DI, basically being a property of a gunammunition, should be in principle deducible from gunammunition parameters. But, to do so is difficult, since the gas pressure has a number of significant effects apart from energising the shot mass and our present knowledge of interior ballistics does not allow us to assess the exact amount of energy transferred to the shot mass and its fractions possessed by it as kinetic and potential strain energies. As such, determination of DI and subsequent verification of the model was based on the study of observable effects, i.e., on analysis of pellet patterns and all the relevant data available were found to support the pellet dispersion mechanism and pellet statistics formulated on that basis. Acknowledgements
The authors thank Dr. D. Roy, Physics Department, Jadavpur University, Calcutta, for critical comments and Dr. B.N. Mattoo, Director, Forensic Science Laboratory, Maharashtra State, Bombay, for an enlightening letter discussing some general aspects of the problem including the work of Ref. (15).
130
References
7 8 9 10
11 12 13 14 15 16 17 18
G. Burrard, The Modern Shotgun, Vol. I, Herbert Jenkins, London, 1960. G. Burrard, The Modern Shotgun, Vol. II, Herbert Jenkins, London, 1960. G. Burrard, The Modern Shotgun, Vol. III, Herbert Jenkins, London, 1960. B.N. Mattoo and B.S. Nabar, Evaluation of effective shot dispersion in buckshot patterns. J. Forensic Sci., 14 (1969) 263269. M. Jauhari, S.M. Chatterjee and P.K. Ghose, Statistical treatment of pellet dispersion data for estimating range of firing. J. Formaic Sk, 17 (1972) 141 149. M. Jauhari, S.M. Chatterjee and P.K. Ghose, A comparative study of six parameters for estimating range of firing from pellet dispersion. J. Indian Acad Fomwic SC%., 13 (1974) 1724. K.D. Heaney and W.F. Rowe, The application of linear regression to rangeoffire estimates based on the spread of shotgun pellet patterns. J. Forensic Sci., 28 (1983) 433436. F.G. Rios and J.I. Thornton, Discussion of ‘The application of linear regression to rangeoffire estimates baaed on the spread of shotgun pellet patterns’. J. Fowmic Sci., 29 (1984) 695  696. J.L. Wray, J.E. McNeil and W.F. Rowe, Comparison of methods for estimating range of fire Sci., 28 (1983) 846857. based on the spread of buckshot patterns. J. For& W.F. Rowe and S.R. Hanson, Rangeoffire estimates from regression analysis applied to the spreads of shotgun pellet patterns: results of a blind study. Fomnsic SC%.Znt., 28 (1985) 239250. C.H. Farm, W.A. Ritter, R.H. Watts and W.F. Rowe, Regression analysis applied to shotgun rangeoffire estimations: results of a blind study. Forensic Sci. Znt., 31(1986) 840854. T.S. Moreau, M.L. Nickels, J.L. Wray, K.W. Bottemiller and W.F. Rowe, Pellet patterns fired Sci., 30 (1985) 137 149. by sawedoff shotguns. J. For& F.G. Rios, J.I. Thornton and K.S. Guarino, Muhivriate statistical analysis of shotgun pellet dispersion. Forensic 5%. Znt., 32 (1986) 2128. N.K. Nag and A. Lahiri, An evaluation of distribution of pellets due to shotgun discharge. Fomwic Sci. Znt., 32 (1986) 151 159. C. Bhattacharyya and P.K. Sengupta, Shotgun pellet dispersion in a Maxwellian model. Foresnic Sci. Znt., 41 (1989) 205217. D.A. Boyer, L.D. Marshall, L.M. Trzicak and W.F. Rowe, Experimental evaluation of the distribution of the pellets in shotgun pellet patterns. Fommic Sci. Znt., 42 (1989) 5159. M. Jauhari, S.M. Chatter+ and A. Sen, Trajectory of shotgun wadding. Fomwic Sci. Znt., 22 (1983) 123 130; see also Refs. (2) and (3) cited in this paper. The Shooter’s Year Book, Imperial Chemical Industries Ltd., London, (1956) 26.