Natural coordinates [\(\varPhi \), \(\varLambda \), H] are related to geodetic latitude, \(\varphi \), longitude, \(\lambda \), and ellipsoidal height h, by (Heiskanen and Moritz 1967):
$$\begin{aligned} \left. \begin{array}{ll} \xi = \varPhi - \varphi &{} (a) \\ \eta = (\varLambda - \lambda )\cos \varphi &{} (b) \\ N = h - H &{} (c) \\ \end{array} \right\} , \end{aligned}$$
(2)
where, \(\xi \) is the meridional component of the DoV, \(\eta \) the prime vertical component of the DoV and N is the geoid undulation. These equations relate the geodetic quantities [\(\varphi , \lambda , h\)] to the physical quantities [\(\varPhi \), \(\varLambda \), H] associated with the gravity field. If the geodetic coordinates [\(\varphi , \lambda ,~h\)] of a point have been determined with respect to a given datum, and if the natural coordinates [\(\varPhi \), \(\varLambda \), H] have been determined also (e.g., from astronomical observations and from spirit leveling), then one can obtain the deflection components (\(\xi _\mathrm{astro}\), \(\eta _\mathrm{astro}\)) and the geoid undulation (N) at that point from Eq. (2). Today, such a geometric determination can be realized by making astronomical observations of \(\varPhi \) and \(\varLambda \) at a point whose geodetic coordinates [\(\varphi , \lambda , h\)] have been determined from GNSS positioning, and whose orthometric height, H, has been determined from spirit leveling. Such a geometric determination yields astrogeodetic DoV components that are specific to the particular datum to which the geodetic coordinates, \(\varphi \) and \(\lambda \), refer.
The DoV components (\(\xi \), \(\eta \)) also represent the slopes of the geoid surface with respect to the surface of an equipotential reference ellipsoid along the meridian and the prime vertical, respectively, (Heiskanen and Moritz 1967, section 2–22):
$$\begin{aligned} \left. {\begin{array}{ll} {\xi _{{\text {grav}}}} = - \frac{{\partial N}}{{R\,\partial \varphi }} &{} (a) \\ {\eta _{{\text {grav}}}} = - \frac{{\partial N}}{{R\cos \varphi \,\partial \lambda }} &{} (b) \\ \end{array}} \right\} , \end{aligned}$$
(3)
where R is a mean Earth radius. The sign convention in Eqs. (2) and (3) is such that \(\xi \) is positive when the astronomical zenith, \(Z_\mathrm{a}\), is north of the geodetic zenith, \(Z_\mathrm{g}\); and \(\eta \) is positive when \(Z_\mathrm{a}\) is east of \(Z_\mathrm{g}\) (Torge 2001, p. 219). These slopes can be determined gravimetrically, either using the integral formulas of Vening Meinesz, or using a spherical harmonic representation of the Earth’s gravitational potential. The gravimetric determination of DoV requires dense measurements of gravity and detailed mapping of the topography near the computation point for the effective use of the integral formulas, or, equivalently, a very high degree and order gravitational field model. EGM2008 is a gravitational model that is complete to spherical harmonic degree 2190 and order 2159 (Pavlis et al. 2012) and carries enough resolving power to support the determination of gravimetric deflections with adequate accuracy, to estimate the astronomical coordinates for the Airy Transit Circle by:
$$\begin{aligned} \left. {\begin{array}{ll} {\varPhi _{{\text {grav}}}} = \varphi + {\xi _{{\text {grav}}}}&{} (a) \\ {\varLambda _{{\text {grav}}}} = \lambda + {\eta _{{\text {grav}}}} \cdot \sec \varphi &{} (b) \\ \end{array}} \right\} . \end{aligned}$$
(4)
These “synthetic” astronomical coordinates can then be compared to independent astronomical observations. If the orientation of the zero meridian plane is the same for both astronomical and geodetic longitudes, Eqs. (2b) and (3b) will yield consistent values for \(\eta \) (Bomford 1980, p. 100).
The authors used the EGM2008 spherical harmonic coefficients to degree 2190, and calculated the deflections at Greenwich via harmonic synthesis, to obtain \(\xi _\mathrm{grav}\) = 2.156\(''\) and \(\eta _\mathrm{grav} \cdot \) sec \(\varphi \) = 5.502\(''\) assuming zero elevation, a result which agrees with the estimates of 2.15\(''\) and 5.51\(''\) made by Ekman and Agren (2010) via numerical differentiation of the gridded version of the EGM2008 geoid. The results predict an astronomical longitude of \(\varLambda _\mathrm{grav}\) = 00\(^{\circ }\) 00\('\)00.19\(''\)E for the Airy Transit Circle. Within the estimated EGM2008 commission error in \(\eta _\mathrm{grav}\)sec \(\varphi \), \(\pm \) 0.47\(''\) 1\(\sigma \) (see also Pavlis et al. 2012, section 5), the predicted astronomical longitude is in good agreement with its originally adopted value of zero.
Figure 2 illustrates the reference ellipsoid as seen from a vantage point along its minor axis above the north pole (point O). The line-segment QC identifies the plane of the astronomical reference meridian of Greenwich, \(\varLambda \) = 0. It is parallel to \(OC''\), the plane of the geodetic reference meridian of \(G''\), \(\lambda \) = 0. The angle \(C'\)
\(OC''\) is exactly \(\eta \cdot \) sec \(\varphi \), the east–west component of the DoV at Greenwich, projected onto the equator. As illustrated, the shift eastward at Greenwich represents only a lateral transfer of the trace of the geodetic longitude \(\lambda \) = 0 meridian across the surface of the Earth. Thus, the geodetic prime meridian at Greenwich has the same orientation as a function of time as the astronomical meridian. In order for the plane of the geodetic prime meridian to pass through the geocenter, its trace on the ground in the vicinity of Greenwich must move to the east by about 102 m, the sign and magnitude of which is correctly predicted by EGM2008.