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Re: What is the radius of the Prime Vertical?
From: Frank Reed
Date: 2019 Mar 15, 08:52 -0700

David, you wrote:
"I wish you hadn't told me that.  You've caused me to question my faith in bubble and pedulous reference sextants.  The above being true, how acurate and how joined up were the pioneer astronomer's attempts to map the Universe, and how do modern astronomers get their declination measurements correct?  Do they have to apply a correction to what they measure?"

It's weird, right? And it's not just bubble sextants and similar, with their more obvious dependence on local gravity. Remember, too, that the marine sextant is also dependent on a great "carpenter's level" or "spirit level"... It's just that the fluid is outside the sextant: the sea itself. When we use the sea horizon for a local level, we are getting the local vertical from gravity just as we do with a bubble sextant.

There are various different definitions and sub-definitions of latitude and longitude, but let's consider just two: one we'll call "astronomical latitude and longitude" and the other we'll call "geometric latitude and longitude". There are other names for both of these ...I'm trying to use names here that are descriptive and relatively transparent in meaning.

Astronomical latitude and longitude is the original, observable form of lat/lon. We build up a map of the globe by first measuring the angular coordinates of the stars and other celestial bodies on the celestial sphere and then pulling those coordinates down to the ground using the apparent local vertical which is determined strictly by local observations of "up" and "down" based on local gravity (bubble levels, hanging plumb bobs, the mean surface of the ocean which we see as the ocean horizon, and so on).

First, we need a map of the stars with exact coordinates. So we set up a national observatory (with an annex in the other hemisphere to get the "southern" stars), and we hire some folks to make all those observations and publish them. That's Royal Astronomer James Bradley at Greenwich, e.g. We thus know the declinations and right ascensions of all the bright stars. We then define the latitude of a place to be equal to the declination of a star that passes through the zenith at that place. If we set up a properly calibrated sidereal clock, the sidereal time is equal to the right ascension at the zenith (and the process of calibrating the sidereal clock is equivalent to determining longitude).

The latitudes and longitudes that we get from this astronomical process are not quite identical to the coordinates that we have drawn on the celestial sphere. The coordinates on the sphere are pure geometric coordinates. We know that one star is some 26.531° south of another, for example, because we rotate our telescopes by that angle to get to the second star from the first --after we have corrected for variable factors like refraction in the atmosphere and aberration of starlight (which, of course, Bradley discovered while working on this project). The coordinates on the celestial sphere are pure angles.

Meanwhile, as our astronomers are compiling that great star catalog, we realize that the Earth is an oblate spheroid, an ellipsoid, rather than a perfect sphere. It's flattened by about one part in 300 due to its rotation. What do we do about that? There are two options. We can either draw up perfect geometric latitude and longitude coordinates on the globe, where a degree measured at the center of the Earth corresponds exactly to a degree on the surface of the ellipsoidal globe, or we can "cheat" and change the definition of latitude to correspond as closely as possible with astronomical latitude and longitude. As it turned out historically, it was simpler to cheat. This choice spares us from having to go back and correct all of our earlier latitudes (longitude, as it turns out, isn't affected), and it also spares us from having to correct every astronomical latitude that we observe in the future with a correction that adds no genuinely useful information. Both of these are "geometric latitude and longitude", but in the former case, the latitude has a simple definition taken directly from standard spherical coordinate systems --an angle at the center of the globe, while in the latter case --the system we actually use-- latitude is not quite a simple coordinate, and we adjust it to match as nearly as possible the astronomically-observed latitude.

Read more on Wikipedia: geodetic latitude.

There's an interesting present/future cartographic issue to consider. What should we do on other planets? Should we define mean ellipsoids and set up latitude as it is defined on Earth? Or instead, maybe just say "the hell with it" and use pure spherical coordinates? This is an open issue. There is already inertia to use earth-like latitude on planets like Mars, where many people believe in a future of human civilization, but on other small worlds, it all seems rather pointless. Coordinate systems on asteroids, for example, are generally drawn up as pure spherical coordinates. Cartographers determine a center, an axis (usually the axis of rotation, but sometimes a body-symmetry axis) and some "prime meridian" and then latitudes and longitudes are pure geometric coordinates. In most respects, this is a simpler, more general approach.

Frank Reed

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