**Variability Management**

Litz (Ref. 4) quite correctly points out that a shooter must understand the whole distribution of the variation of shots and seek to minimize the spread. Since you cannot correct what you cannot identify, the intent of this section is to help shooters better understand where variability might be coming from. There are many important (perhaps even most important) variables for which the point mass model here is useless, but we work with what we have.

With the tools so far, we can examine the effect of variation in muzzle velocity, shot propellant variation, and bullet mass variation effects. Later in this section we note some effects of errors in headspace and base to ogive distance. Among the many variables we cannot deal with are recoil, rifle inhomogeneities (including cleanliness), bullet inhomogeneities, variability in ignition due to packing, orientation, and temperature variation and effect of tolerances in the chamber such as extent of "jump" and intrabarrel yaw.

From the reloading handbooks, one can get an estimate of the change in velocity for change in propellant load. Similarly one can get the change in velocity for change in bullet weight. The question still remains as to the variability of the loads in manufacture. One hand loader said that a 1/2 grain error is hard to avoid.

Some manufacturers publish the extent of bullet mass variation, and while raw mass fluctuation does not seem important, imbalances in the mass location could be.

**Variation of Muzzle Velocity**

Let's say you have some parameters for a particular bullet, and want to estimate the effect of a different muzzle velocity. There are different ways to handle the problem depending on the velocity profile -- for now as a parabolic profile or as the leading terms of an exponential profile. The exponential profile is probably the more accurate and is the easiest to deal with.

**Vz for Exponential Profile**

The exponential profile is the easiest to adjust since V = Vz exp(-Q X), and Q needs no adjustment.

**Vz for Parabolic Profile**

The parabolic profile (V(S) = A S^2 + B S + Vz) is a bit more difficult. One way is to rewrite it as V(S) = Vz( A/Vi S^2 + B/Vi +1) where Vi is the starting maximum velocity and Vz is some subsequent maximum velocity.

A slightly different way is to move in range along the existing curve until we get to the desired muzzle velocity, then incorporate this change in distance into the A and B parameters, and reset the distance measure to start at the new velocity point.

Starting with A ,B and Vz, we first find the offset distance d for the new velocity Vy = Vz + y by solving the following.

A S^2 + B S + Vz = A(S+d)^2 + B(S+d) + Vz+y (18)

Having found d we expand the S+d terms, and regroup the constants into the parameters. The results are that the new parameters (A` , B` , Vy) work with the new S` measured from our fictitious gun with higher muzzle velocity.

The solution to (18) d= (B + sqrt(B^2 + 4 A y) )/ (2A). And d should be negative if y, the increase in muzzle velocity, is positive. Then:

A` = A (19A)

B` = B - 2 A d = sqrt(B^2 + 4 A y) (19B)

Vy = Vz + y (19C)

S` = S + d (19D)

**Adjusting A and B for External Variables**

The drag coefficient CMcC = Q = r a Cd/(2m) where r is air density, a is area, m mass, and Cd the drag function. For rifles, CMcC = r pi Cd/ 8Cb where Cb is the usual ballistic coefficient. For the exponential profile Q is readily adjustable using the definitions.

Using the approximation Vz exp(- Q S), if we assume that the A, B, C parameters are the leading terms of a Taylor series for the exponential, then

A = (1/2) Vz Q^2 (though Vz Q^2 is better) (15E)

B' = - Vz Q (15F)

C = Vz (15G)

Reference the section on "Estimates of A,B from a Ballistic Coefficient".

**Wind**

The Didion model for wind drift works as follows. In a vacuum there would be no wind drift (duh!). In the atmosphere the air that does the retardation is moving and imparts that motion to the bullet. The amount of drift depends on the amount of time the wind had available to retard the bullet relative to the vacuum state. Using the transit time T(S) from Eq. 3, the wind drift W is given by the wind speed times the amount of time added by the drag.

W = (T(S) - S/Vz) (transverse wind velocity relative to bullet) (20)

Even for idealized uniform wind the drift is not uniform throughout the trajectory since d( T(S) - S/Vz)/ dS is not constant. For a shooter who cannot have a detailed wind velocity profile the averages over the range would seem to be as close to exact as is possible in the field.

The interaction of the bullet with tornadic wind speed or with horizontal motion (like a shot from helicopter) would be where issues on the isotropy of the ballistic drag could be important. For low wind speeds the bullet apparently turns into the wind in such a way as to keep the drag unchanged.

Rifleist would like to speculate that highly turbulent but otherwise undirected winds should have the effect of increasing drag through increased yaw resulting from conversion of gyroscopic torque to precession as the wind forces buffet the bullet.

McCoy's derivation of eq. 20, (his eq. 7.27) is interesting. He artfully imposes an air velocity reference shift so to include the wind speed, and completely finesses the dynamics of getting a bullet to move sideways. He does a much better job in his 6 axis analysis.

The Sierra article by W. T. McDonald on wind drift is well worth reading for those who want to dig more deeply.

**A Note on How to Think about Variability**

People are taught to think about variability in terms of standard deviations and "the bell curve", also known as the "Normal" or Gaussian. This curve shape is a pure abstraction and is never observed -- no, not ever. The word "normal" in this context does not mean "usual", it means orthogonal in the sense that the curve is constructed from the addition of a large number of independent, or orthogonal, variables. The "dead" targets left at my range seem often to have modes of variation, not uniformly dispersed around a center. Modal behavior is to be expected in chaotic situations. In chaotic situations there are different effects coming and going, like wind, barrel heating, accumulated barrel crud, systematic bad bullets, cold fingers, sore shoulders and such. So start off by looking at the pattern to see if there is something obvious, and think about how the pattern changes as you proceed. As an example, suboptimal loading of a bipod was causing a diagonal streak on the target as the rifle jumped and rotated on one leg. To get high accuracy, the first step is to isolate the variability modes, then eliminate or control them. Statistics probably will not be as useful as actual shooting and keeping accurate track of what happens as you change your inputs.

**Effect of Temperature and Humidity on Trajectory**

Given a perfect bullet fired from a perfect gun, once the bullet leaves the gun, the largest impact on trajectory will be air density, wind and gravity. Many devices can measure pressure which is the largest variable in air density, but temperature and humidity also play a role. Humidity is important only at high temperatures and high relative (and absolute) humidity. At 90F and 100%RH the water vapor concentration is enough to lower the density of air by 1.4% so drag will be reduced.

Temperature certainly will have an effect, perhaps the biggest, on the way gun powder reacts to propel the bullet, but it is not something that can be analyzed here.

**VARIABILITY IN A 7 MM REM MAG--SOME INTERIOR BALLISTICS**

At the start, I did not get the accuracy I expected from my new 7 MM Rem Mag. One by one I have eliminated issues. These include a KRG stock to help the recoil upkick, and a muzzle brake to help reduce recoil. But there was still a strange and more or less systematic variation between bullet manufacturers. The pattern of errors was that 3 or 4 out of 5 shots would have MOA or better, then one would be 3-5 away. I bought both a head space reference and an ogive gauge ring and began measuring. It was soon evident that cartridge bolt to ogive measurements (CBTO) vary substantially from brand to brand, as listed here with dimensions in inches.

Brand Wt. Ave. CBTO delta Approx Accuracy

B 140 2.567& 0.159 Sub MOA

F 160 N 2.611& 0.115 MOA

F 165 S 2.619& 0.107 MOA

W 150 2.657& 0.069 mult MOA

W 175 2.660& 0.066 mult MOA

H 180 2.686& 0.040 high MOA, tumbling

SAAMI 2.726

Hand Load 150 2.752& -0.026 Bolt won’t close

(The & indicates an unknown reference shift.)

On reviewing my target data, it was clear the commercial loads that had been giving me largest variation had CBTO 2.65 or longer. A hand load of uncertain origin was fully .02 in longer than spec, and it was so long the bolt would not close. When forced into the chamber then removed, these long bullets would score noticeably. Brand B gives consistent sub MOA accuracy. F is a bit higher, but still roughly MOA. W and the rest are upwards of 2 MOA and higher. I think if I had kept my raw data, I might have a decent plot of variation versus CBTO. Whatever, it certainly shows why hand loading may be necessary for highest accuracy. The Berger Bullets Blog has a good discussion of CBTO and cartridge overall length, COAL.

Another variable to be considered is "headspace", the distance from the base to the shoulder of the casing. At the end of the firing, the cartridge will have expanded to whatever headspace the rifle has. The case displacement adds to the CBTO. Rifleist does not know if this occurs before, during, or after engagement of the bullet with the lands, but expansion of the case probably happens early in the firing. The hand loads noted above have essentially zero headspace, while the commercial bullets have 10/1000" and higher.

In summary, if the bullet is too long for the rifle, and/or does not engage coaxially there can be in-barrel yaw, and throw off when the bullet exits the barrel. Such interactions can induce random jumps that can vary within a manufacturing lot.

**NRA Shooting Accuracy Test**

The NRA magazine publishes test results on rifle accuracy. It states these are the "average of 5 consecutive 5 shot groups". I wondered what this test was telling us in terms of single shot numbers, so I ran a simulation with Gaussian distributions. I found that the NRA test produce a quite reliable method of setting boundaries for single shots. The test outcome is about 2.25 times the single shot standard deviation. A summary is that over 90% of the time the NRA test result will include, meaning to fall outside of, single shots.