Apart from the present well-known transient acceleration errors of the bubble sextant in flight, there exists a persistent and predictable acceleration error caused by the rotation of the earth. This error results from the fact that the earth’s rotation combined with the speed of the plane causes a pendulum or a bubble to indicate a false zenith (or horizon). This error is based on the same principle as that which controls the law of storms, the flow of ocean currents, and the drift of shells in their trajectory.
The rotation of the earth gives rise to an acceleration at right angles to the direction of motion relative to the earth, known as the acceleration of Coriolis. The acceleration of a particle moving in a rotating path is equal to the sum of the acceleration of the particle in the path, the acceleration of that point along the path at which the particle is situated, and a supplementary, or Coriolis’ acceleration at right angles to the path. The vector of the latter is so directed that if applied to the end of the vector of the first acceleration (that in the path) it would tend to rotate it in the direction in which the path is being rotated.
The acceleration of the particle in the path influences the magnitude but not the direction of gravity at the position. The acceleration of that point along the path at which the particle is situated is that which causes the pendulum, although perpendicular to the surface, to point away from the geometric center of the earth. The effect of this acceleration is allowed for in the construction of navigational maps. The Coriolis’ acceleration at right angles to the path is one which deviates the pendulum being transported but does not affect a non-transported pendulum at the same position.
The Coriolis’ acceleration is one which is:
- Directly proportional to the ground speed.
- Directly proportional to the sine of the latitude.
- Directly proportional to the angular rate of the earth’s rotation (a constant).
- Wholly independent, in magnitude, of the direction of flight.
- Always directed perpendicular to the track.
- Directed to the left of the track in the Northern Hemisphere and to the right in the Southern.
The formula for the horizontal component of this acceleration, a, is: a = 2wV sin L
where w is the angular velocity of the earth 0.0000729 radians per second; V is the aircraft velocity; L, the latitude.
If the velocity is in miles per hour, then, a/g = 0.023V sin L
where V is, again, in miles per hour.
The effect on an airplane itself is negligible, merely causing the apparent level to differ from the true horizon by the angle d'. The effect on the bubble sextant is precisely the same, but may not be negligible.
For example, the angular amount of this deviation of the vertical in Lat. 40° (Wright Field) at a ground speed of 200 statute miles an hour is 3 minutes of arc; the deviation being such that altitudes measured to the right are too small and those measured on the left are too great by the above amount. If uncorrected, an error of 3.0 nautical or 3.4 statute miles may occur in position determination in the above case. Since the error increases directly as the ground speed and the sine of the latitude it is evident that intolerable errors would occur in Alaskan operation at modern plane speeds. Altitude measures in the direction of motion are not affected; those at right angles are most in error. To show more clearly the theory involved, we will illustrate and derive the formulas given above.
In Fig. 1, V is the velocity vector of the airplane along a path which makes an angle A with the meridian. The meridian and parallel components V cosine and sine A, respectively, are denoted by Vm and Vm, respectively. The component of Vm in the plane of the parallel is denoted Vm sin L.
The surface components of the Coriolis’ accelerations resulting from velocities Vp and Vm sin L are shown in Fig. 2, the two being separated for clarity.
The total and the surface components of the Coriolis’ acceleration are illustrated in Fig. 3. The subject acceleration being of the type 2Vw, the surface components are
Acm = 2Vw cos A sin L
Acp = 2Vw sin A sin L
The square of the total surface component:
Act2 = 4V2w2 cos2 A sin2 L+4V2w2 sin2 A sin2 L
= 4V2w2 sin2 L (cos2 A +sin2A)
Act2 = 2Vw sin L
From Fig. 4, it is shown that the total Coriolis’ acceleration is directed at right angles to the track.
Figure 5 shows that the error is corrected by translating all celestial position lines an amount equal to the tabulated error. The original and corrected position lines in the figure are dotted and solid, respectively. Observations directly to the rear (or forward) are not affected by the error. The positions of such lines should therefore be unchanged by the translation (note that the line is slid along itself). Lines from observations taken abeam suffer the greatest change in position. The positions of other lines are altered an intermediate amount by the translation.
To facilitate the correction to compensate for the subject error the accompanying table has been computed. If the ground speeds in the table are taken as statute miles, the tabulated corrections are like-wise in statute miles. If the ground speeds are regarded as knots, the tabulated values are distance corrections in nautical miles and also minutes of arc deviation of the vertical. To correct for the acceleration, all celestial position lines are translated according to instructions forming a part of the table.
Lat | 100 | 150 | 200 | 250 | 300 | 350 | 400 |
|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
|
10 | 0.4 | 0.7 | 0.9 | 1.1 | 1.4 | 1.6 | 1.8 |
|
20 | 0.9 | 1.3 | 1.8 | 2.2 | 2.7 | 3.2 | 3.6 |
|
30 | 1.3 | 2.0 | 2.6 | 3.3 | 3.9 | 4.6 | 5.3 |
|
40 | 1.7 | 2.5 | 3.4 | 4.2 | 5.1 | 5.9 | 6.8 |
|
50 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.1 | 8.1 |
|
60 | 2.3 | 3.4 | 4.6 | 5.7 | 6.8 | 8.0 | 9.1 |
|
70 | 2.5 | 3.7 | 4.9 | 6.2 | 7.4 | 8.7 | 9.9 |
|
75 | 2.5 | 3.8 | 5.1 | 6.4 | 7.6 | 8.9 | 10.17 |
|
Translate all lines R/L in N/S Hemisphere
Perpendicular to Track.
Correction Is Statute/Nautical as G.S.
These data were sent to the British Royal Air Force and to others in England and Canada, as well as to the Naval Aircraft Factory, Navigation Section, and other places and have created considerable interest.
Obviously, this material should be brought to the attention of naval aviators in this country, as well as air corps officers, and we know of no better place for disseminating this information than through the Naval Institute Proceedings.
In a previous article in the Naval Institute Proceedings about Pan American Airways in navigation, I noted that Pan American pilots found the error on the beam to be greater than the error on the head, but they attributed this to the motion of the plane. It now seems clear, however, that this is due to the conditions described in this article rather than to the motion of the plane.