[probably more than you wanted to know....]
I have a copy of Greenhill's original 1879 paper, and I can tell you that it is actually a good bit more complicated than what you generally see given as "Greenhill's Formula." The original formula was actually derived for an "oblate spheroid" shape....somewhat of a "football" shape, as a mathematical convenience. It is derived neither for a cylinder nor a ogive-conical although it has always worked fairly well for either.
That being said, don't worry about it, you can basically use it for either shape and get approximate minimum twists rates that are ok and that is why the simplified formula works as well as it does:
twist/d = constant x d/L x sqrt(D/10.9)
where
twist/d = twist in calibers
constant = 150
d = caliber
L = length of bullet
D = density of bullet (lead = 11.34gm/cc)
sqrt() = square root
(You often see Specific Gravity instead of Density.....if Density is given in metric units of grams per cubic centimeter (gm/cc) then that is numerically equal to the Specific Gravity because the Density of Water is 1gm/cc.)
The fact is that the original complicated form of Greenhill Formula has an additional weak dependence on length that is not represented in the usually given simple version....This one of the reasons that the Constant given above has different values for differnt lengths of bullet. For a very short bullet, ie, a ball, some will use 120. I think that is too small, but somethign smaller than 150 is fairly obvious from studying Greenhill's 1879 paper.
The reason that the constant is sometimes taken as 180 for modern spitzer suppository guns is because the drag curve is lower at velocities way above 1500fps and a constant =180 is usually given for these higher velocities, ie 2800fps or so. This does not apply, obviously to our muzzleloaders, which bullets fly in the 1500fps +/- 500fps region, including the trans-sonic region of highest drag.
Using constant = 150 for a ball
twist/d = 150 x .490/.490 x sqrt(11.34/10.9) ~ 153
and actual twist rate = .490 x 153 = 75" or one turn in 75"
That is a minimum twist rate. You usually see anywhere from 1in66" to 1in72" for a PRB .50cal barrel.
You will often see someone opine that a ball does not need much spin because no matter which way it turns, it is just the same round ball....This is not true for at least two reasons:
1) the ball gets flattend some on its nose, and around its periphery upon loading with a tight patch.....
2) even if the ball remained mathematically round, because it has a spin (whether random or given by too-slow rifling) if it turns sideways it becomes a curve-ball (caused by air-flow effect called the Magnuson Effect). This is the same effect that causes drift at extended range due to spin, as the trajectory curves away from the direction of the ball's spin axis.
Better too give your ball enough spin to stay oriented than to have Magnuson in your pants at close range....unless of course you intend it to be a
smoothbore...in which case it goes like a knuckleball at some point...
YHS,
rogerw
