How is the radius of curvature of the prime vertical (N) defined?

The radius of curvature of the prime vertical, denoted as N, is defined by the formula that incorporates the semi-major axis (a) and the square of the first eccentricity (e²) of the ellipsoid, adjusted for the latitude (Φ). The correct formula is expressed as a/(1 - e²sin²Φ)^(1/2).

This definition arises from the need to describe the geometry of an ellipsoid, as used in geodesy. The prime vertical refers to a vertical plane that contains the zenith and is perpendicular to the celestial meridian. The radius of curvature in the prime vertical provides a measure of how the surface of the ellipsoid curves at a particular latitude.

In the correct formulation, the denominator (1 - e²sin²Φ)^(1/2) particularly accounts for the effect of the Earth's ellipsoidal shape on the curvature. The terms e² and sin²Φ represent the influence of the Earth’s flattening and the latitude on this curvature, respectively.

The other options incorporate variations of these components; however, they either do not align with the correct dimensionality or geometric interpretation according to the definitions derived from the ellipsoidal geometry. Thus, the choice reflecting a/(1