Miller twist rule

Miller twist rule is a mathematical formula created by Don Miller to calculate the optimum rate of twist for a given bullet traveling through a rifled barrel. Miller suggests that while Greenhill's formula works well there are better and more precise ways to calculate what the proper twist rate for a bullet should be that are not much more difficult to calculate.

Formula
The following formula is one recommended by Miller.

$${t}^2 = \frac{30m}{sd^3l(1+l^2)}$$

where:
 * m = bullet mass in grains
 * s = gyroscopic stability factor (dimensionless)
 * d = bullet diameter in inches
 * l = bullet length in calibers
 * t = twist in calibers per turn

Given those definitions we can expand:

$${t} = \frac{T}{d}$$

where $$T$$ = twist in inches per turn, and

$${l} = \frac{L}{d}$$

where $$L$$ = bullet length in inches.

Stability factor
Using Miller's formula we can also calculate the stability factor assuming we already know the twist. Simply solve for $$s$$.

$${s} = \frac{30m}{t^2d^3l(1+l^2)}$$

Twist in inches
It is, of course, possible to solve for twist in inches directly by simply solving for $$T$$

$${T} = \frac{30m}{sdl(1+l^2)}$$

Safe values
When computing this formula Miller states several safe values that can be used in place of some of the more difficult to determine variables. For example he states that assuming a mach number of $$M$$ = 2.5 (or roughly 2800 ft/sec assuming standard conditions at sea level where 1 Mach is roughly 1116 ft/sec) is a safe value to use in regards to velocity. He also states that when making rough estimates that involve temperature that using $$s$$ = 2.0.

Example
If we take the Nosler Spitzer .30-06 Springfield round which is similar to the one pictured above we can easily fill in the variables and calculate the estimated twist rate. Starting with the formula

$$t = \sqrt{\frac{30m}{sd^3l(1+l^2)}}$$

We then fill in:
 * m = 180 grains
 * s = 2.0 (the safe value noted above)
 * d = .308 inches
 * l = 1.180" /.308" = 3.83 calibers

$$t = \sqrt{\frac{30 * 180}{2.0 * .308^3 * 3.83(1+3.83^2)}} = 39.2511937$$

Which tells us we have 39.2511937 calibers per turn. We calculate $$T$$ using $$t$$ and see that

$$T = 39.2511937 * .308 = 12.0893677$$

So our rifling should be roughly 12" per turn. The twist, as listed on Wikipedia's .30-06 article is 10" inches per turn as an average of .30-06 caliber rifles; thus 12" inches per turn is fairly accurate. The discrepancy seen here also helps explain why certain bullets seem to work better in certain rifles when fired under similar conditions.

Comparison to Greenhill's formula
Greenhill's formula is actually much more complicated in full form. The rule of thumb that Greenhill devised based upon his formula is actually what is seen in most places, including Wikipedia. That rule was:

$$Twist = \frac{C D^2}{L} \times \sqrt{\frac{SG}{10.9}}$$

The actual formula was:

$$S = \frac{s^2 * m^2}{C_M\alpha \div \sin(a) * t * d * v^2}$$

where:
 * S = gyroscopic stability
 * s = spin rate in radians per second squared
 * m = polar moment of inertia
 * CM α = pitching moment coefficient
 * a = angle of attack
 * t = transverse moment of inertia
 * d = air density
 * v = velocity

Thus, Miller, in essence, took Greenhill's rule of thumb and expanded it slightly while, at the same time, keeping the formula simple enough to calculate without advanced training in mathematics. In order to improve on Greenhill, Miller used mostly empirical data and some basic geometry.

Corrective equations
Miller, in his work, notes several corrective equations that can be used:

The velocity ($$v$$) correction for twist ($$T$$): $$f_v{^{1/2}} = [\frac{v}{2800}]^{1/6}$$

The velocity ($$v$$) correction for stability factor ($$s$$): $$f_v = [\frac{v}{2800}]^{1/3}$$

The altitude ($$a$$) correction under standard conditions: $$f_a = e^{3.158x10^{-5} * h}$$ where $$h$$ is altitude in feet.

Calculators for stability and twist

 * Bowman-Howell Twist Rate Calculator
 * Miller Formula Calculator
 * Drag/Twist Calculator based on Bob McCoy's "McGyro" algorithm