WGS84RPT.tif:Corel PHOTO-PAINT - Henry A. Rowland Department ...

The equations listed here for the angles ( α , ε,
θ ) are applicable to both the
northern and southern hemispheres. For positive h, each of these angles is zero when point P is
directly above one of the poles or lies in the equatorial plane. Elsewhere for h > 0, they have
the same sign as the geodetic latitude for point P. For h = 0, the angles α and θ are equal and
ε = 0. Numerical results have indicated that the angular separation (ε) between the component
r
γ
h
and the total normal gravity vector γtotal
satisfies the inequality ε < 4 arcseconds for
geodetic heights up to 20,000 meters. For completeness the component ( γ φ ) of the total
r
normal gravity vector γtotal
at point P in Figure 4.2 that is orthogonal to γ
h
and lies in the
meridian plane for point P is given by the expression:
γ
φ
( α) + γ cos( α)
= −γ
r
sin
ψ
(4-23)
The component γ
φ
has a positive sense northward. For geodetic height h = 0, the γ φ
component is zero. Numerical testing with whole degree latitudes showed that the magnitude of
γ remains less than 0.002% of the value of γ for geodetic heights up to 20,000 meters.
φ
Equations (4-16) and (4-23) provide an alternative way to compute the magnitude
total normal gravity vector through the equation:
h
r
γtotal
of the
r
γ
total
=
γ
2
h
+ γ
2
φ
(4-24)
In summary then, for near-surface geodetic heights when sub-microgal precision
is not necessary, the Taylor series expansion Equation (4-3) for γ
h
should suffice. But, when
the intended application for γ
h
requires high accuracy, Equation (4-4) will be a close
approximation to the exact Equation (4-16) for geodetic heights up to 20,000 meters. Of
course, γ
h can be computed using the exact Equation (4-16) but this requires that the
computational procedure include the two transformations, R 1 and R 2 , that are shown in
Equation (4-17). Because the difference in results between Equations (4-4) and (4-16) is less
than one µgal (10 -6 gal) for geodetic heights to 20,000 meters, the transformation approach
would probably be unnecessary in most situations. For applications requiring pure attraction
(attraction without centrifugal force) due to the normal gravitational potential V, the u- and β-
vector components of normal gravitation can be computed easily in the ellipsoidal coordinate
system by omitting the last term in Equations (4-5) and (4-6) respectively. These last attraction
terms account for the centrifugal force due to the angular velocity ω of the reference ellipsoid.
4-6