# Field Artillery Manual accounts for Earth's Rotation

Manual that shows that in Field Artillery the rotation of the Earth is taken into account. They have to correct for the vertical (range) and horizontal (azimuth deviation) components of the Coriolis acceleration.

Note: MILS = Milliradian is an angular unit. 6400 mil = 360°.

## Coriolis Dependence on Latitude and Elevation Angle

#### Coriolis with Zero Elevation Angle

The following simplified version of the Coriolis formula is only valid for objects that move parallel to the surface of the earth:

(1)
with
see
where'
 $c$ ' =' 'Horizontal component of the Coriolis acceleration, acting perpendicular to the direction of motion. Positive values are acting to the right. $v$ ' =' 'Speed of the moving object with respect to the surface of the sphere $\Omega$ ' =' 'Angular velocity of the sphere $T$ ' =' 'Rotation period $\lambda$ ' =' 'Latitude of the moving object

According to this formula the Coriolis acceleration acts always to the right in the northern hemisphere and always to the left in the souther hemisphere with respect to the direction of motion.

#### Coriolis with Non-Zero Elevation Angle

This is not true for objects on a ballistic trajectory. If an object is not traveling parallel to the surface, but instead climbing and descending on its path, the Coriolis acceleration may change direction, depending on the current elevation angle at the current position.

For example: Lets assume we are on latitude 30° north and fire a bullet with an elevation angle of 45° due north. Then the angle between the trajectory and the rotation axis of the earth is 45° − 30° = 15°. This corresponds to an elevation angle of 0° at 15° south latitude. So it should be expected that this bullet, fired in the northern hemisphere, should deviate to the left, not to the right, despite beeing fired in the northern hemisphere.

Is this really the case and is this taken into account in Artillery Firing Tables?

The answer is yes! In the calculations of ballistic trajectories we have to use the general vector form for the Coriolis acceleration:

(2)
where'
 $\vec c$ ' =' 'Coriolis acceleration in 3D space $\vec v$ ' =' 'Velocity vector in 3D space $\vec \Omega$ ' =' 'Angular rotation vector. The vector is pointing in the direction of the rotation axis and its magnitude is the angular rotation Ω = 2π/T $T$ ' =' 'Rotation period

You can use my Coriolis, Centrifugal and Gravitational Forces on the WGS84 Globe Model calculator, which uses the formula above and gives you the Coriolis acceleration vector in different reference frames.

The fomula above gives the whole Coriolis effect as a vector in 3D space. This vector can be decomposed into a vertical and a horizontal component. The vertical component is also called the Eötvös effect and changes the distance the bullet travels. The horizontal component causes a deviation to the left or right, depending on the elevation angle, direction of flight and latitude. Because the elevation angle constantly changes along the trajectory of ballistic object, the direction and magnitude of the Coriolis acceleration vector also does change along the flight path with respect to the direction of motion. To calculate the net effect, the accleration has to be integrated over the whole path. This can only be done numerically.

#### Azimuth Correction Table I

If we look into the Azimuth Correction Table I e.g. for 30° north latitude, firing azimuth 0 (north) up with a certain elevation angle we have to correct the aiming to the left (yellow L range), because the Coriolis acceleration in the northern hemisphere is to the right for shallow elevation angles. If the elevation angle exceeds a certain angle, more of the high speed flight path is upwards, causing a deflection to the left, and the overall deviation can tend to the left, so you have to correct to the right (red R range).

So the overall correction direction and magnitude depends on the summation of the Coriolis effect along the whole flight path. It can change direction depending on the elevation angle and firing range, even the whole trajectory is in the same hemisphere.

Note: In the table you provide the target distance, not the elevation angle. But the elevation angle depends on the target distance. Above a certain canon elevation angle the reachable target distance decreases, see values below the line with the stars.

Azimuth Correction Field Artillery Table I for 30° latitude north and south

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