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Re: Horizons, was Summary of Bowditch Table 15
From: Fred Hebard
Date: 2005 Jan 30, 20:20 -0500
From: Fred Hebard
Date: 2005 Jan 30, 20:20 -0500
I can only confess that I considered the non spherical shape of the earth while writing, but thought to keep to one component at a time. George is correct in what he says. On Jan 30, 2005, at 5:52 PM, George Huxtable wrote: > Jim Thompson and Fed Hebard are making somewhat heavy weather about > this > question of the difference between local gravity (the direction of a > plumb-bob) and the direction of a line to the centre of the Earth. > These > directions are not the same (except at the poles and the equator) > because > the Earth is not a uniform sphere, but a non-uniform spheroid, with > some > local bulges. > > Fred wrote- > >> This is called deflection from the vertical, where the vertical points >> to the center of the earth. It is not a significant factor until you >> get next to the Andes or some other huge mountain close to the sea. >> As >> I recall, the errors are on the order of 1' of arc or so, which would >> make it more a problem for surveyors than navigators. > > But in that passage Fred is referring only to local gravitational > distortions that are due to the local non-uniformities in the Earth's > crust. They exist, but there are bigger factors at work, because of the > non-spherical Earth. > > You can see easily, by simply sketching an exaggerated ellipse as a > slice > through the Earth from pole to pole, that a line at right-angles to the > sea-surface through a point P is not in the same direction as a line > between P and the Earth's centre. Except, that is, when P is at a pole > or > on the equator. The biggest divergence is at a latitude of 45 degrees, > North or South, when the difference amounts to 11' 33". > > The direction of bodies in the sky (the declination part, anyway) is > defined by its direction, up or down from the Earth's equator. > Similarly, > the geographic latitude is defined by the angle that the local vertical > makes with the plane of the Earth's equator. Devices that measure > altitudes, such as sextants, bubble-sextants, or surveyors' > theodolites, > all measure with respect to the local horizontal or sea-surface or > artificial horizon surface, always 90 degrees from that local vertical. > Astronomer's telescopes are set up with respect to that local vertical. > > So, when we observe a body on the meridian, we can relate directly its > altitude to its declination and to our latitude. If our latitude had > been > defined in a different way, such as the direction of a line between P > and > the centre, that simple relation would not apply, and all sorts of > complications would result. > > It's all been made rather easy for us navigators by defining latitude > in > that way. Mercator charts, beside their stretching toward the poles > that > we're all familiar with, also have a bit of extra distortion in them, > to > allow for the length of a sea-mile to vary somewhat with latitude. For > that's part of the price that has to be paid for the simplicity of > defining > latitude in the way we do. Because the Earth's surface is (a bit) more > tightly curved, in the N-S direction, near the equator than it is at > the > poles, the length of a sea-mile varies , being less near the equator, > greater near the poles. And the length of a sea-mile, measured in the > E-W > direction, differs from a sea-mile measured in the N-S direction. Just > slightly; not so much as you would notice. Doesn't affect navigation > much, > does it? It adds up to that 11' 33" divergence, around 45deg latitude. > > In navigation, when we set a course and distance to our destination, > we are > not usually bothered by discrepancies of a few miles in distance, or a > fraction of a degree in the course, so for those purposes a sphere is a > close-enough approximation. Formulae exist that take the spheroidal > shape > into account, for both rhumb-line and great-circle navigation. Tables > for > "meridional parts" allow for the true shape. But who bothers? > > So the result of it all, in celestial navigation, is that everything > that > we measure in the sky has to be with respect to the direction of a > plumb > line, or the plane of the horizontal (which are so closely related that > either one exactly defines the other). > > George. > > ================================================================ > contact George Huxtable by email at george@huxtable.u-net.com, by > phone at > 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy > Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. > ================================================================ >