Limitations of Extreme Distance Shooting

I’ve always been interested in shooting at long range since I was a kid. Back in the day, I thought one hundred yards was far. As time went on, my view of what “far” really is has increased significantly. Being able to put a bullet in the air at an extreme distance is easy, but connecting with your intended target at a distance is not. There are a few companies out there who are building large caliber extreme range rifles; they are pissing money into the wind.

Physics being physics, if you can get an aerodynamic object going fast enough and send it in a direction at a forty-five-degree angle, it will go as far as it can. Just because a bullet will go 4k, doesn’t mean that the rifle it came out of will be accurate. You can watch you tube videos of “world records” being set, mainly some old fat guys laying on their food blisters ALL DAY, pissing hundreds of rounds down range. Then the wind blows the round into the target (a massive one), and they all cheer and call it a record. ANYONE can do that. If you put enough projectiles into the air, you will hit something at some point in time.

The companies building these super rifles seem to forget science, or perhaps they just don’t bother with it for it would smash their hopes of DOD buying their shiny new 20mm sniper rifles. Maybe they think that the folks they are trying to sell them to are dim whits, and will purchase the concept hook line and sinker. Once you push past a certain distance (atmospheric dependent), being able to accurately engage a target with a 1st or 2nd round hit drop off the charts. It’s not because the rifle isn’t accurate, the optics not of high quality or the projectiles not perfect. It’s a combination of light, math and a lack of technology.

I’ll go into each one and try not to lose you in the process.

First Issue: Light, more specifically the effects of atmospheric refraction on line-of-sight accuracy.

Time for some science and math. Air has an index of refraction very close to vacuum (n=1.00), and for many applications and circumstances, approximating this index of air to be n=1.00 is sufficient. However, for applications that require looking through long distances/columns of air, the approximation breaks down. At room temperature (T=20°C, sea-level), the index of air is n=1.000272, while at a temperature of 50°C, the index of refraction drops to 1.000243. This small drop in the index, when combined with long viewing distances, can cause some effects that distort or displace objects. Since air of different temperatures and pressures varies in the index, light does not necessarily travel in a straight line.

By Fermat’s principle, light follows a path that is stationary (usually either the shortest or longest optical distance). Optical path or distance is simply the product of the index and the physical distance. For inhomogeneous media like air, this requires integration along the road, such that:

where n is the refractive index, So and S are the start and end points, ds is an increment along the curve, and L is the resulting optical path length. Since n is a three-dimensional function that depends on atmospheric conditions, calculating L or knowing the endpoint S is challenging over long distances.

In the case of atmospheric/thermal turbulence, an object is blurred/distorted by that turbulence. In such a case, identification of an object can be difficult, but there is clear feedback (due to poor image quality) that the actual vector to the object is not consistent with the visual line-of-sight. In many calm conditions, however, the effects of atmospheric refraction can be more confusing and misleading. The two categories of “calm air” distortion are mirage and displacement.

Mirages result in an apparent second object, either above or below the actual object.

  • Inferior mirage – hot, less dense air near the surface causes light at glancing angles to be deflected upward. This creates an unstable inverted image of the actual object.

  • Superior mirage – cold, dense air near the surface causes light at glancing angles to be deflected downward. This creates the illusion of an object floating above the surface in addition to the direct viewing of the line-of-sight object.

Displacement effects are perhaps the most challenging for which to compensate. Unlike either turbulence or mirage, displacement does not necessarily provide an indication that the object “isn’t where it’s supposed to be.” These two effects are;

  • Looming and sinking – unlike a mirage, looming and sinking do not create secondary images. Instead, the object being viewed appears to be floating above the horizon (in the case of looming) or depressed relative to the horizon (sinking). Both effects displace the apparent location of the object relative to its true position.
  • Towering and stooping – similar to looming and sinking, towering and stooping do not create secondary images but instead stretch or squash the object. The object will appear either taller or shorter than it really is, depending on the type of hot/cold layering in the atmosphere.

In short, the future you go out, the worse this problem becomes. You may think you are aiming at a target, yet the target you are aiming at is not at the location you are aiming at. To make matters worse, when you are using optics that have internally adjustable windage and elevation corrections, you have prismatic aberration issues. That’s a topic for another article.

The second issue is wind.

There is no current unclassified and field portable systems to instantly and accurately measure the winds aloft in the bullets flight path. The shooter must guess at what the winds are doing all the way to the target. The longer the distance, the worse the problem gets.

The third issue is lack of ballistic programs that use real math vs. heuristics

One of the first problems solved by the new (in the 60’s) electronic computers (at that time, using vacuum tubes) was the calculation of ballistic trajectories for artillery cannons during World War II. These computations were done on the very first general purpose computer, the ENIAC at the University of Pennsylvania.

The first complete publication of ballistic modeling and solution techniques was “Modern Exterior Ballistics,” by Robert McCoy of the US Army’s Ballistic Research Laboratory. This book has become the “Bible” of technical ballistic trajectory theory and practice.

There are significant sources of error in the presenting the modeling and standard mechanisms used for solving the ballistic trajectory equations. The most important of these is the simplification of the problem by leaving out all modeling of the rotational dynamics of the projectile.

The full ballistic trajectory equation suite consists of six (6) coupled second order differential equations. Each of these equations represents the dynamics of one “degree of freedom” of the bullet. The enumerated degrees of freedom for a bullet are:

  • Down-range position
  • Horizontal cross-range (windage) position
  • Vertical cross-range (height above like of sight) position
  • Projectile roll (rifling twist) angle
  • Projectile pitch angle
  • Projectile yaw angle

The first three of these are the positional equations, while the second three are the rotational dynamical equations (also known as the “attitude” equations). Each of the equations has a number of forcing function terms that cause some part of the dynamics for that particular degree of freedom. Gravity, for example, is a forcing function term, as are the drag forces (including wind).

The coupling of the equations means that there are forcing function terms in each of the equations that depend on the current solutions of one or more of the other equations in the set. For example, the instantaneous drag on the projectile along the down-range direction depends on the instantaneous projectile pitch and yaw angles while the time rate of change of the pitch and yaw angles depend on the down-range velocity, which, in turn, depends on the summed instantaneous drag. The solution of the entire set of fully coupled equations is a daunting task, indeed.

From the standpoint of the shooter, projectile rotational dynamics is not particularly interesting. The shooter wants to get the projectile to the target, which has everything to do with its position. It is true that there are attitude terms in the positional equations, but their effects are rather small compared to the other forcing terms like gravity, drag and wind. Leaving out the computations of the rotational dynamics does not have a huge effect (but it is not insignificant, either) on the position solution at closer ranges and considerably reduces the complexity of actually getting a solution. The most obvious effect, but not the only effect, of this simplification, is that the force term that results in spin drift is no longer being computed. Modeling spin drift without the attitude terms requires an after-solution add-on using some form of heuristic approximation. Different products use different, usually proprietary heuristics for this purpose or simply ignore the spin drift altogether. For a standard 168 grain .308 bullet with a muzzle velocity of about 2650 feet per second and a right-hand twist of one turn in 11.75 inches, the spin drift results in about a 6.7 inch position offset to the right[1] at one thousand yards range. Dropping the angle equations reduces the mathematical model to what is known as a point mass model.

Another source of error in the present solution methods is due to the “flat fire” approximation. The item of highest interest to the shooter is the vertical angle that the bullet must have as it exits the muzzle (i.e., the vertical scope setting). The method used to obtain this angle is to assume that the shot is taken flat, that is, with the barrel completely horizontal. The method assumes that there is no initial angle on the gun and computes the trajectory out to the target range as if the Earth would not stop the bullet if it hit the ground. The projectile will be pulled down by gravity during its trip to the target and the solution to the ballistic equations will result in a number representing how far down it got pulled. Simply computing the arc-tangent of this number divided by the range will result in the required angle… but not quite. The flat fire solution presumes, by its very nature, that the initial down-range velocity is the muzzle velocity. However, when we raise the gun by the angle specified by the flat fire solution, it results in both an initial down-range velocity and an initial upward velocity. These two velocities have a Pythagorean relationship to the muzzle velocity. The result is a small reduction in the down-range initial velocity, so the bullet actually takes a little longer to reach the target than the flat fire approximation predicts, so it falls a little farther and the shot arrives a little low.

Other sources of inaccuracy are encountered when shooting at long ranges on inclines (up or down). The first is that the gravity vector is no longer aligned with the vertical axis. This puts some gravitational acceleration into the down-range axis. The result is that there is either more or less effective drag on the projectile, depending on whether the shot is going up- or downhill. The second is much more subtle. For a long-range shot on a significant up or downhill angle, there is an associated change in actual, Earth oriented altitude. This altitude change results in changes in density, pressure, and temperature along the trajectory, and associated changes in Mach number and drag.

All of the sources of imprecision discussed above are very small to non-existent at short ranges. However, the physical processes and time/distance evolution of the effects are nonlinear and grow to considerable significance at extreme long range.

There are other causes of error in addition to those mentioned above. They are much more mathematically and/or physically obscure and go beyond the scope of the current document, but they are no less significant than those discussed here.

The common point mass approximations that, per force, ignore attitude force terms and subtle geometric and atmospheric effects simply cannot accurately compute precise solutions beyond about eight hundred meters range. This is the reason ballistic calculators resort to “truing”, or correcting for errors occurring in the first (or second…) round shot. Getting past these limitations requires an entirely new approach to solving the ballistic problem, and there is only one NASA guy I know that has this 90% solved.

I hope this clarifies my initial statement of why some companies are literally pissing money into the winds.