Ballistic coefficient

In ballistics the ballistic coefficient (BC) of a body is a measure of its ability to overcome air resistance in flight. It is inversely proportional to the deceleration—a high number indicates a low deceleration. BC is a function of mass, diameter, and drag coefficient. It is given by the mass of the object divided by the diameter squared that it presents to the airflow divided by a dimensionless constant i  that relates to the aerodynamics of its shape. Ballistic coefficient has units of lb/in² or kg/m². Normally BC's are stated in lb/in² by gun projectiles producers without referring to this unit.

Formula

The formula for calculating the ballistic coefficient for a body is as follows :

where:

• BC_Physics = ballistic coefficient as used in physics and engineering
• M = mass
• A = cross-sectional area
• C_d = drag coefficient
• ρ (rho) = average density
• l = body length

• This is not the same thing as the BC used by bullet manufacturers. This is the BC as defined by and used in Physics and Engineering. Although it would not be incorrect to use this equation on bullets, the BC obtained from this equation would not give the same value as the BC from a bullet manufacturer because their value is a comparison to the G1 bullet model.^{[citation needed]}

Bullet performance

The formula for calculating the ballistic coefficient for bullets ONLY is as follows:^[1][2]

where:

• BC_Bullets = ballistic coefficient
• SD = sectional density, SD = mass of bullet in pounds or kilograms divided by its caliber squared in inches or meters; units are lb/in² or kg/m².
• i = form factor, i = ; (C_G ~ 0.5191)
• C_B = Drag coefficient of the bullet
• C_G = Drag coefficient of the G1 model bullet
• M = Mass of object, lb or kg
• d = diameter of the object, in or m

This BC formula gives the ratio of ballistic efficiency compared to the standard G1 model projectile. The standard projectile originates from the "C" standard reference projectile defined by the German steel, ammunition and armaments manufacturer Krupp in 1881.^[3] The G1 model standard projectile has a BC of 1.^[4] The French Gavre Commission decided to use this projectile as their first reference projectile, giving the G1 name.^[5][6]

A bullet with a high BC will travel farther than one with a low BC since it will retain its velocity better as it flies downrange from the muzzle, will resist the wind better, and will “shoot flatter” (see external ballistics).^[7]

When hunting with a rifle, a higher BC is desirable for several reasons. A higher BC results in a flatter trajectory which in turn reduces the effect of errors in estimating the distance to the target. This is particularly important when attempting a clean hit on the vitals of a game animal. If the target animal is closer than estimated, then the bullet will hit higher than expected. Conversely, if the animal is further than estimated the bullet will hit lower than expected. Such a difference in bullet drop can often make the difference between a clean kill and a wounded animal.

This difference in trajectories becomes more critical at longer ranges. For some cartridges, the difference in two bullet designs fired from the same rifle can result in a difference between the two of over 30 cm (1 foot) at 500 meters (550 yards). The difference in impact energy can also be great because kinetic energy depends on the square of the velocity. A bullet with a high BC arrives at the target faster and with more energy than one with a low BC.

Since the higher BC bullet gets to the target faster, it is also less affected by the crosswinds.

General trends

Sporting bullets, with a calibre d ranging from 0.177 to 0.50 inches (4.50 to 12.7 mm), have BC’s in the range 0.12 to slightly over 1.00, with high being the most aerodynamic, and low being the least. Very-low-drag bullets with BC's ≥ 1.10 can be designed and produced on CNC precision lathes out of mono-metal rods, but they often have to be fired from custom made full bore rifles with special barrels^[8].

Ammunition makers often offer several bullet weights and types for a given cartridge. Heavy-for-caliber pointed (spitzer) bullets with a boattail design have the high BC's, whereas lighter bullets with square tails and blunt noses have lower BCs. The 6 mm and 6.5 mm cartridges are probably the most well known for having high BC bullets and are often used in long range target matches of 300-1000 meters. The 6 and 6.5 have relatively light recoil compared to larger calibers with high BC bullets and tend to take matches where accuracy is key. Examples include the 6mm PPC, 6mm Norma BR, 6x47mm SM, 6.5x47mm Lapua, 6.5 Grendel and the 6.5-284. The 6.5 mm is also a very popular hunting caliber in Europe.

In the United States, hunting cartridges such as the .25-06 Remington (a 6.35 mm caliber), the .270 Winchester (a 6.8 mm caliber), and the .284 Winchester (a 7 mm caliber) are used when high BCs and moderate recoil are desired. The .30-06 Springfield and .308 Winchester cartridges also offer several high-BC loads, although the bullet weights are on the heavy side. The .308 is also a favorite long-range target cartridge.

In the larger caliber category, the .338 Lapua Magnum and the .50 BMG are popular with very high BC bullets for shooting beyond 1000 meters. New competitors in the larger caliber category are the .375 and .408 Cheyenne Tactical and the .416 Barrett.

The transient nature of bullet ballistic coefficients

Variations in BC claims for exactly the same projectiles can be explained by differences in the ambient air density used for these BC statements or differing range-speed measurements on which the stated G1 BC averages are based. The BC changes during a projectile's flight and stated BC's are always averages for particular range-speed regimes. Some more explanation about the transient nature of a projectile's G1 BC (it rises above or gets under a stated average value for a certain speed-range regime) during flight can be found at the external ballistics article. This article implies that knowing how a BC was established is almost as important as knowing the stated BC value itself.

For the precise establishment of BCs (or perhaps the scientifically better expressed drag coefficients), Doppler radar-measurements are required. The normal shooting or aerodynamics enthusiast, however, has no access to such expensive professional measurement devices. Weibel 1000e Doppler radars are used by governments, professional ballisticians, defense forces, and a few ammunition manufacturers to obtain exact real world data on the flight behavior of projectiles of interest.

Doppler radar measurement results for a lathe turned monolithic solid .50 BMG very-low-drag bullet (Lost River J40 .510-773 grain monolithic solid bullet / twist rate 1:15 in) look like this:

The initial rise in the BC value is attributed to a projectile's always present yaw and precession out of the bore. The test results were obtained from many shots not just a single shot. The bullet was assigned 1.062 for its BC number by the bullet's manufacturer Lost River Ballistic Technologies.

Measurements on other bullets can give totally different results. How different speed regimes affect several 8.6 mm (.338 in calibre) rifle bullets made by the Finnish ammunition manaufacturer Lapua can be seen in the .338 Lapua Magnum product brochure which states Doppler radar established BC data.^[9]

Differing mathematical models and bullet ballistic coefficients

Most ballistic mathematical models and hence tables or software takes for granted that one specific drag function correctly describes the drag and hence the flight characteristics of a bullet related to its ballistics coefficient. Those models do not differentiate between wadcutter, flat-based, spitzer, boat-tail, very-low-drag, etc. bullet types or shapes. They assume one invariable drag function as indicated by the published BC. Several different drag curve models optimized for several standard projectile shapes are however available.
The resulting drag curve models for several standard projectile shapes or types are referred to as the:

• G1 or Ingalls (by far the most popular)
• G2 (Aberdeen J projectile)
• G5 (short 7.5° boat-tail, 6.19 calibers long tangent ogive)
• G6 (flatbase, 6 calibers long secant ogive)
• G7 (long 7.5° boat-tail, 10 calibers tangent ogive, preferred by some manufacturers for very-low-drag bullets^[10])
• G8 (flatbase, 10 calibers long secant ogive)
• GL (blunt lead nose)

Since these standard projectile shapes differ significantly the Gx BC will also differ significantly from the Gy BC for an identical bullet. To illustrate this a bullet manufacturer has published a G1 BC of 0.659 and a G7 BC of 0.337 for their 7 mm Match Target VLD bullet and has since published the G1 and G7 BCs for most of their target bullets.^[11] In general the G1 model yields compariatively high BC values and is often used by the sporting ammunition industry.

Satellites and reentry vehicles

Satellites in Low Earth Orbit (LEO) with high ballistic coefficients experience smaller perturbations to their orbits due to atmospheric drag.

The ballistic coefficient of an atmospheric reentry vehicle has a significant effect on its behavior. A very high ballistic coefficient vehicle would lose velocity very slowly and would impact the Earth's surface at higher speeds. In contrast a low ballistic coefficient would reach subsonic speeds before hitting the ground.

In general, reentry vehicles that carry human beings back to Earth from space have high drag and a correspondingly low ballistic coefficient. Vehicles that carry nuclear weapons launched by an Intercontinental Ballistic Missile (ICBM), by contrast, have a high ballistic coefficient, which enables them to travel rapidly from space to a target on land. That makes the weapon less affected by crosswinds or other weather phenomena, and harder to track, intercept, or otherwise defend against.

See also

• External ballistics - The behavior of a projectile in flight.

• Trajectory of a projectile Ballistic coefficient can be use to calculated the trajectory of a projectile. Although the above page currently lacks any practical application of BC. For information on calculating the trajectory of a projectile excluding air resistance see Trajectory of a projectile. BC is an easy way to account for air resistance.

Freeware small arms ballistic coefficient calculators

• JBM Ballistic Coefficient (Velocity)
• JBM Ballistic Coefficient (Time)

References

• Aerospace Corporation Definition
• Chuck Hawks Article on Ballistic Coefficient
• Ballistic Coefficient Tables
• Exterior Ballistics.com
• How do bullets fly? The ballistic coefficient (bc) by Ruprecht Nennstiel, Wiesbaden, Germany