486 Table 166 



RADIUS OF CURVATURE ON A POLAR STEREOGRAPHIC PROJECTION 



In computing gradient wind speeds (Table 40) and in other problems it is necessary to 

 determine a factor r which depends on curvature of the trajectory. This factor arises in 

 taking account of the horizontal component of the centrifugal force acting on a particle. 

 The problem is twofold: (1) to determine the trajectory of the particle on a map, and 

 (2) to determine the required value of r if the trajectory on the map is known. The first 

 problem is of such nature that it cannot be treated adequately here. (Note. — In many 

 cases an approximation is made from the curvature of the isobars or streamlines.) The 

 second problem has been solved for the case of a polar stereographic projection, since on 

 this projection a "small circle" on the earth projects as a circle on the map. Table 40 

 provides a means for computing the desired r for trajectories on a polar stereographic 

 projection. 



Let R be the radius of the earth, r' the true radius of the "small circle" on which the 

 particle is assumed to be traveling at a given instant, and a its angular radius (as seen 

 from the center of the earth). Then r' =R sin a. Since we are concerned with the hori- 

 zontal component of the centrifugal force, the effective horizontal radius of the curvature 

 required in the gradient wind equation is given by r =r' sec a = R tan a. If an arc on a 

 map representing the instantaneous trajectory of a particle of air is determined, this arc 

 may be regarded as a portion of a "small circle." 



To determine r for a given arc of a trajectory on the map: 



1. Complete the circle by extending the arc (a set of circular templates will prove 



very useful). 



2. Find the meridian which passes through the center of this circle. 



3. Determine the latitudes fa and fa of the points where this meridian intersects the 



circle (extend the meridian across the pole if necessary). 

 4A. If the circle found in step 1 does not contain the pole, find the difference between 



fa and fa and enter part A of the table with this difference as the argument. 



The corresponding tabular value is the required radius r in statute miles, from 



the formula r = R tan i(fa — fa). 

 4B. If the circle found in step 1 contains the pole, find the sum (fa + fa) and enter 



part B of the table with this sum as the argument. The corresponding tabular 



value is the required radius r in statute miles, from the formula r = R tan [90° 



— i(#i + *»)]. 



(continued) 



SMITHSONIAN METEOROLOGICAL TABLES 



