ROTATION OF THE EARTH

GENERAL

1.         Although the earth rotates at a constant rate, the correction for rotation varies with a number of factors and, therefore, rotation is more readily considered a non-standard condition. Factors influencing the effect of rotation of the earth on the travel of a projectile are:

a. direction of fire,

b. A/D,

c. velocity of projectile,

d. range to target, and

e. latitude of the gun.

2.         The correction tables provide all the data needed to compensate for rotation in the

gunnery problem, however, some background theory of rotational effects may assist in an

understanding of their application.

Rotational Effects on Range x

3.         Because of rotation of the earth, a point on the equator has an eastward linear velocity of approximately 457 m/s. This linear velocity decreases to zero at either pole. Consider a gun on the equator firing due east at a target (Example 1, Figure x). During TOF of the projectile, the gun and target will travel from G to G' and T to T', respectively, along the circumference of the earth. The projectile, however, travels in a vertical plane, the base of which is parallel to the original plane of departure established at the time of firing; that is, it is pivotal to the circumference of the earth at the gun but not at the target. At the end of a given TOF, the projectile will be at P' when the target is at T'. Hence, the projectile will continue along an extended trajectory and land farther east or, in this instance, beyond its target. The normal trajectory of the projectile is interrupted.

4.         Consider the same gun firing westward (Example 2, Figure x). Again, the projectile falls to the east of the target, but in this instance east is short. The effect in each example is as if the QE fired has been in error by the amount of angle "a", which is the angle formed by the base line G' P' and a tangent to the earth at G'. When the gun is firing eastward, angle "a" is plus (range long); when the gun is firing westward, angle "a" is minus (range short).

5.         A second effect on range is known as projectile lag. This is best explained by use of a diagram (see Figure xx). Assume that a projectile is fired straight up into the air, ie at an A/D of 1 600 mils. When the projectile is fired, it will have a horizontal velocity equal to the rotational velocity of the earth. During the time that the projectile is in flight the earth rotates moving the gun from G to G' and the projectile moves through an arc P to P'. As this occurs in the same time, and the horizontal velocity of the gun and projectile are the same, distance G to G' equals distance P to P'. However, P to P' is further away from the centre of the earth and the angle subtended is less, therefore, the round lands at "X". Furthermore, the effect of gravity on the projectile is acting through the centre of the earth causing the projectile to lag.

6.         Tabular Firing Tables list a single range correction for rotation of the earth that combines the rotation effect and the lag effect. These two effects are opposing and they reach their maximum values at different angles of departure as follows:

a. At an A/D of approximately 530 mils the rotation effect reaches its maximum.

b. At an A/D of approximately 1 070 mils the two effects are equal and cancel each other.

c. At an A/D of 1 600 mils, the effect of projectile lag reaches its maximum.

Projectile Lag

Figure xx

7.         A third consideration is the curvature effect. Curvature effect exists because of the use of a map range for which the surface of the earth is assumed to be flat, but the actual range is measured on a sphere. The gun-target (GT) range is computed for a plane tangent to the surface of the earth at the gun. When the projectile reaches this range, it is still above the curved surface of the earth and will continue to drop, resulting in a slightly longer true range than desired. This effect is less than 1 meter in 1 000 meters [1] and is of little significance except at very long ranges. It is disregarded when FTs are used, since FT ranges include curvature effect.

ROTATION EFFECTS ON BEARING

8.         A final rotational effect is described as the latitudinal effect. When the gun and target are at different latitudes, the eastward rotational velocity imparted to the projectile and target is different. For example, if the gun is nearer the equator, the projectile will travel faster and, therefore, further to the east than the target (see Example 1, Figure xxx). The reverse is true if the target is nearer the equator.

9.          When the gun and target are at the same latitude the projectile will also be deflected away from the target. This is because the projectile tends to travel in the plane of the great circle containing the gun and the target at the time of firing. Because of the rotation of the earth, this great circle plane is continually changing with respect to its original position. As viewed from above, it would appear that the great circle containing the gun and target is turning with respect to the great circle followed by the projectile (see Example 2, Figure xxx). In the northern hemisphere the latitudinal effect is to the right; in the southern hemisphere it is to the left.

REFERENCE

[1]        B-GL-306-004/FP-001, Field Artillery, Volume 6, Duties at Regimental

Headquarters and the Gun Position;

Glossary

The following symbols and abbreviations are used in this document:

A/D     - angle of departure

MV      - muzzle velocity

m/s       - metres per second

f          - angle of elevation

g          - gravitational force (9.8 m/s2)

Hv          - horizontal component of velocity

Vv          - vertical component of velocity

R         - range to the level point

T          - time of flight to the level point

T          - any given time

h          - projectile height at time t.