2.2 Astronomic Coordinates
2.2.3 Differences Between Geodetic and Astronomic Quantities
As we will see in Section 2.3, the astronomic latitude, longitude, and azimuth are observable quantities based on a naturally defined and realized coordinate system, such as the astronomic system or the terrestrial reference system alluded to in Section 2.2. These quantities also depend on the direction of gravity at a point (another naturally defined and realizable direction). However, the quantities we use for mapping purposes are the geodetic quantities, based on a mathematically defined coordinate system, the ellipsoid. Therefore, we need to develop equations for the difference between the geodetic and astronomic quantities, in order to relate observed quantities to mathematically and geographically useful quantities. These equations will also be extremely important in realizing the proper orientation of one system relative to the other.
Already in Problem 2.2.2.1-4, the student was asked to derive the difference between astronomic and geodetic azimuth. We now do this using spherical trigonometry which also shows more clearly the differences between astronomic and geodetic latitude and longitude. In fact, however, the latter differences are not derived, per se, and essentially are given just names, i.e., (essentially) the components of the astro-geodetic deflection of the vertical, under the following fundamental assumption. Namely, we assume that the two systems, the astronomic (or terrestrial) and geodetic systems, are parallel, meaning that the minor axis of the ellipsoid is parallel to the z- axis of the astronomic system and the corresponding x-axes are parallel. Under this assumption we derive the difference between the azimuths. Alternatively, we could derive the relationships under more general conditions of non-parallelism and subsequently set the orientation angles between axes to zero. The result would obviously be the same, but the procedure is outside the present scope.
Figure 2.22 depicts the plan view of a sphere of unspecified radius as seen from the astronomic zenith, that is, the intersection of the local coordinate axis, w, with this sphere. The origin of this sphere could be the center of mass of the Earth or the center of mass of the solar system, or even the observer’s location. Insofar as the radius is unspecified, it may be taken as sufficiently large so that the origin, for present purposes, is immaterial. We call this the celestial sphere; see also Section 2.3. All points on this sphere are projections of radial directions and since we are only concerned with directions, the value of the radius is not important and may be assigned a value of 1 (unit radius), so that angles between radial directions are equivalent to great circle arcs on the sphere in terms of radian measure.
Clearly, the circle shown in Figure 2.22 is the (astronomic) horizon. Za denotes the astronomic zenith, and Zg is the geodetic zenith, being the projection of the ellipsoidal normal through the observer, P (see Figure 2.21). As noted earlier, the angular arc between the two zeniths is the astro-geodetic deflection of the vertical, Θ (the deflection of the tangent to the plumb line from a mathematically defined vertical, the ellipsoid normal). It may be decomposed into two angles, one in the south-to-north direction, ξ, and one in the west-to-east direction, η (Figure 2.23). The projections of the astronomic meridian and the geodetic meridian intersect on the
celestial sphere because the polar axes of the two systems are parallel by assumption (even though the astronomic meridian plane does not contain the z-axis, the fact that both meridian planes are parallel to the z-axis implies that on the celestial sphere, their projections intersect in the projection of the north pole). On the horizon, however, there is a difference, ∆1, between astronomic and geodetic north. celestial sphere local horizon astronomic meridian geodetic meridian
astronomic north geodetic north
O Za Zg F 90° − Φ 90° − φ α A ua u g ∆1 ∆2 ∆λ zg za Q Qa Qg ∆p • • • • • • • • north pole • • H
Za Zg F ∼ α • • • ξ η H ε • Θ
Figure 2.23: Deflection of the vertical components.
Now, the angle at the north pole between the meridians is ∆λ=Λ–λ, again, because the two systems presumably have parallel x-axes (common origin on the celestial sphere). From the indicated astronomic and geodetic latitudes, we find by applying the law of cosines to the triangle
ZgOF:
cos 90°–φ = cosηcos 90°–Φ+ξ + sinηsin 90°–Φ+ξ cos 90° . (2.163)
Since η is a small angle (usually of the order of 10 arcsec, or less), we have
sinφ≈sin Φ–ξ , (2.164)
and hence
ξ=Φ–φ . (2.165)
Applying the law of sines to the same triangle, ZgOF, one finds sinη
sin∆λ =
sin 90°–φ
sin 90° ; (2.166)
and, with the same approximation,
η= Λ–λ cosφ . (2.167)
Thus, the north and east components, ξ and η, of the deflection of the vertical are essentially the differences between the astronomic and the geodetic latitudes and longitudes, respectively.
point, Q, while the great circle arc (approximately, since the two zeniths are close), ugQg, is the same as the geodetic (normal section) azimuth, α, of the target point. Thus, from Figure 2.22, we obtain:
A –α= uaQa– ugQg=∆1+∆2 . (2.168)
It remains to find expressions for ∆1 and ∆2.
From the law of sines applied to triangle ugOua, we find sin∆1
sin∆λ = sinφ
sin 90° ⇒ ∆1=∆λsinφ , (2.169)
with the usual small-angle approximation. Similarly, in triangle QgQQa, the law of sines yields
sin∆2 sin∆p =
sin 90°– zg
sin 90° ⇒ ∆2=∆p cos zg . (2.170)
Also, triangle ZaQH (see also Figure 2.23) yields sin∆p sin ξ+ε ≈ sinα sin za ⇒ ∆p≈ ξ+ε sinα sin za . (2.171)
Finally, from the approximately planar triangle ZgFH we obtain
ε≈ η
tan 180°–α , (2.172)
which could also be obtained by rigorously applying the laws of cosines and sines on the spherical triangle and making the usual small-angle approximations.
Substituting (2.171) and (2.172) into (2.170), we find
∆2= ξ+ε sinαcot z
= ξsinα–ηcosα cot z ,
(2.173)
where the approximation z = z g≈za is legitimate because of the small magnitude of ∆2. We come to the final result by combining (2.169) and (2.173) with (2.168):
which, of course, in view of (2.165) and (2.167) is the same as (2.162). Equation (2.174) is known as the (extended) Laplace condition. Again, it is noted that α is the normal section azimuth. The second term on the right side of (2.174) is the extended part that vanishes (or nearly so) for target point on (or close to) the horizon, where the zenith angle is 90°. Even though this relationship between astronomic and geodetic azimuths at a point is a consequence of the assumed parallelism of the corresponding system axes, its application to observed azimuths, in fact, also ensures this parallelism, i.e., it is a sufficient condition. This can be proved by deriving the equation under a general rotation between the systems and specializing to parallel systems. The geodetic (normal section) azimuth, α, determined according to (2.174) from observed astronomic quantities is known as the Laplace azimuth.
The simple Laplace condition (for z = 90 °),
A –α= Λ–λ sinφ , (2.175)
describes the difference in azimuths that is common to all target points and is due to the non- parallelism of the astronomic and geodetic meridian planes (Figure 2.22). Interestingly, the simple Laplace condition is also the Bessel equation derived for geodesics (2.97) which, however, is unrelated to the present context. The second term in the extended Laplace condition (2.174) (for target points with non-zero vertical angle) depends on the azimuth of the target. It is analogous to the error in angles measured by a theodolite whose vertical is out of alignment (leveling error).
2.2.3.1 Problems
1. Suppose the geodetic system is rotated with respect to the astronomic system by the small angle, ωz, about the polar axis. Repeat all derivations and thus show that the components of the deflection of the vertical and the Laplace condition are now given by
ξ=Φ–φ,
η= Λ–λ–ωz cosφ,
A –α= Λ–λ–ωz sinφ+ Φ–φ sinα– Λ–λ–ωz cosφcosα cot z .
(2.176)
2. Suppose that an observer measures the astronomic azimuth of a target. Describe in review fashion all the systematic corrections that must be applied to obtain the corresponding geodesic azimuth of the target that has been projected (mapped) along the normal onto an ellipsoid whose axes are parallel to the astronomic system.