Quote of the Day

Eggshells smashing each other with hammers. — Winston Churchill, describing his feelings on battleship combat.

Introduction

I must admit that I am a bit of a battleship junkie. I have been reading some old US Navy manuals on battleship fire control, which discuss the various effects that must be corrected for to ensure accurate fire (Figure 1). In this post, I want to examine how the curvature of the Earth affected the gunnery direction. Curvature corrections are only needed for very long-range artillery.

Gunnery direction calculations usually begin with a range table (Figure 2), which tells the gunner the angle that projectile must be fired at to hit a target at a given range on the same horizontal plane as the gun (i.e. no difference in height between the gun and target). The target height relative to the gun can be either positive or negative, which affects the range that is used to index into the range table . For example, battleships in WW2 doing shore bombardment sometimes needed to attack fortifications on mountains (e.g. Mount Suribachi on Iwo Jima). For sea-level sea battles, the targets are below the horizontal plane of the ship firing the projectile.

Figure 3 shows that firing at a target that is at sea level also involves a difference in heights. The rangefinders on a battleship determined a Line-Of-Sight (LOS) distance, but that distance is not the same as the horizontal distance listed in the table of Figure 2. The LOS distance must be corrected to an effective horizontal distance that can be looked up in the range table. My goal in this post is to show how we can correct the LOS distance to provide the required horizontal distance, which can then be used to read the gun elevation from the table in Figure 2.

All calculations are performed in Excel – my workbook is here.

Background

Earth Curvature Calculation

I have written about how to compute the curvature of the Earth over a given distance in another post using Equation 1, which relates the deviation from horizontal to the distance from the measurement origin.

Eq. 1

where

δ deviation from horizontal, which is called curvature in gunnery.

R is the radius of the Earth (3963.2 miles)

R LOS is the LOS distance.

These parameters are illustrated in Figure 5.

We can use Equation 1 to compute a curvature versus range table (Figure 4). This table duplicates the results shown in this reference.

To illustrate how to read this table, consider the range of 19,800 yards. We go to the row that corresponds to 19,000 yards and find the column that corresponds to 800 yards. At the intersection of the row and column, we find a curvature of 84 ft.

Rate of Height Change

The US Navy manuals refer to "Column 19" and the "Change in height of impact for variation of 100 yards in sight bar." While this sounds like a complex parameter, it is simply the tangent of the projectiles impact angle with respect to horizontal, which is called the angle of fall and is listed in the range table shown in Figure 2. The tangent of the angle of fall tells you how many feet the projectile loses in height for every foot of horizontal distance. We will use this parameter to relate the height difference to the range correction.

Analysis

Earth Curvature Correction Calculation

Figure 5 defines some variables using the illustration of Figure 3. You can see in Figure 5 hitting target on a requires reducing the range setting of the gun (R H ) from the distance measured along the line of the sight (R LOS ) by Δ, i.e. .

For modeling purposes in Figure 6, we can treat the trajectory of the shell near the target as a straight line. This allows us to use a simple trigonometric function to compute Δ, i.e. .

Example

I copied a section of the range table from the US Navy manual and used it to compute: (1) curvature; (2) change in height of impact for variation of 100 yards in sight bar (i.e. LOS range); (3) danger space (discussed in this blog post). I can verify that (1) and (2) agree with the manual. Item (3) is discussed but not listed in the manual tables.

Conclusion

I am interested in understanding the gunnery corrections for the Earth's curvature and the Coriolis effect. I believe this post thoroughly covers the curvature correction. I will put out a post shortly on the correction for the Coriolis effect.