COORDINATE SYSTEMS FOR NAVIGATION

Fig. 1 Shape of earth

            Earth was born in a molten state. While cooling, it bulged out due to its rotation and its surface settled in a way that it approximately represented points of equal gravitation. The earth has finally taken the shape of an ellipsoid with an uneven surface. Surface of seawaters, however does contains points of equal gravitation more precisely, ignoring tidal disturbances and water flows. This surface can be theoretically extended under landmasses to form an enclosed surface, which is known as Geoid. The Geoid is again not a precise ellipsoid in mathematical terms. For navigational computations a perfect ellipsoid is assumed with following qualifications:

1. Its center coincides with mass center of the earth (EC) including atmosphere.
2. Mean of squares of deviations between direction of gravity and normal to ellipsoid is minimum. The above said deviations mostly remain below 10 seconds of arc.

            See Fig. 2. Due to rotation of the earth (at 15.041067 degrees per hour), a body at a point P experiences a centrifugal force away from the axis of rotation. Thus, the apparent gravity is not only due to the mass of the earth, but a resultant (g_l) of the actual gravitation (g) and the acceleration a_cf due to centrifugal force given by:

            a_cf = w_iex(w_iexR)  = |w²_ie Rcosf|U_cf                        (1)

where U_cf is unit vector in direction of centrifugal force.

Fig. 2 Different longitudes of earth fA Astronomical     latitude fT Geodetic latitude fC Geocentric latitude

              g_l represents local gravitation, which is a vertical line as seen with a plumb line. We can see that a_cf is maximum in equatorial plane and approaches zero at the poles. It is always directed along radial from axis of rotation and parallel to equatorial plane. This direction is shown by unit vector U_cf as shown in the above figure. Its magnitude as you can find from Equation (1) above, is proportional to cosf, wheref is latitude of the point P under consideration. We define three types of latitude angles.

·         Geocentric latitude: f_c made by line PO

·         Geodetic latitude: f_T made by line PCBA; this line is normal to the ellipsoid.

·         Astronomical latitude: f_A made by line PD

Note: The centrifugal force plays important role in function of accelerometers in inertial navigation.............