NavList:

Gary had asked a question about the celestial-derived coordinates of Howland Island and how they compare with GPS (WGS84) coordinates. As I described generally previously and Paul Hirose described in greater detail, this is primarily an issue of the "deflection of the vertical".

GPS coordinates refer to a nice smooth ellipsoid (a sphere flattened by about 1 part in 300 to match the polar flattening of the Earth). There have been many different choices for the parameters of this ellipsoid over the decades but the overwhelmingly dominant choice today is the WGS84 ellipsoid. Modern latitudes, longitudes, and heights are referenced to this perfect mathematical surface which globally approximates the shape of the "mean sea level" surface of the Earth with deviations in some areas amounting to about a hundred meters.

The Earth's mean sea level is a hypothetical surface determined by the Earth's gravitational field. It's a surface defined by physics rather than mathematics. If the Earth were a spinning spheroid composed of some uniform density substance with no topography, then this "mean sea level" surface or "geoid" could be made identical to the mathematical ellipsoid of the WGS84 coordinates. Instead, the Earth has a slightly lumpy distribution of mass, both at the surface and in the mantle and concentrations of mass create slight bulges in the geoid. If we subtract away tides and currents, these bumps and undulations in the geoid are nearly identical to bumps and undulations in the surface of the ocean. Ocean water is drawn to large concentrations of mass and piled up in small "bulges" in the ocean surface. Similarly when there are voids in the distribution of dense matter, like deep sea canyons and trenches, there is less gravity, and the ocean water is drawn away from those low gravity regions creating slight depressions in the ocean's surface. Note that these bulges and valleys are very slight. The valleys are never deep enough to make actual concave features in the surface of the geoid, though in principle they could be.

Celestial navigation is necessarily tied to the local gravitational vertical, whether it's determined by a hanging plumb line, or by the bubble in an aircraft sextant, or the sea horizon in a nautical sextant (with the ocean serving as a great "spirit level") or from the output of a sophisticated inertial system that finds the vertical with accelerometers. The whole system of celestial navigation works because, at any instant of time, there is a one-to-one mapping from points on the surface of a sphere (or spheroid) to directions on the celestial sphere. Or in simpler terms, at any instant, each point of latitude and longitude on the Earth is marked by a specific star, possibly very faint, which is almost exactly at the zenith for that location. And the zenith is the astronomical zenith as determined by the gravitational vertical by "looking up" a plumb line.

The gravitational vertical is perpendicular to the geoid at every spot on the Earth. So as we're sailing along on the oceans riding up and down those bulges and valleys caused by the "lumpy" distribution of the Earth's mass, the zenith from which we reference our celestial observations is shifting about by some small angle. It tilts astern as we climb uphill, and it shifts towards the bow as we sail downhill. This is that "deflection of the vertical" that Paul described in detail.

Many islands, like Howland Island out in the Pacific, have relatively small "hills" of water around them. The geoid bulge there is only about two meters high over a region that's about fifty miles wide. Nearer to home for many of us, Bermuda in the eastern Atlantic has a rather sizable geoid bulge which could, in fact, be detected with very accurate celestial navigation.

Imagine a voyage from Newport, Rhode Island (which I can see out my window despite some fog today) to Bermuda. As we sail south, for every nautical mile that we travel, the location of our zenith in the heavens shifts by one minute of arc. To keep things simple, let's leave out the rotation of the Earth for now. This simple one-to-one relationship would hold for most of the voyage. Now suppose 60 nautical miles from Bermuda, we begin comparing celestial fixes and GPS fixes. The one-to-one relationship between nautical mile traveled and minutes of arc shift in the zenith guarantees that these positions are essentially identical ...until we start to climb the hill. The bulge in the geoid around Bermuda is rather substantial. The island sits atop a mountain, and the gravitational attraction of that mountain draws water towards it. When we're about 20 miles from Bermuda, our boat is tilted back towards the north by about half a minute of arc. From whatever direction we approach the island, we have to climb this hill so we find our zenith is shifted back, away from the island by somewhere around half a minute of arc when we are couple of dozen miles away (the geoid hill appears to be steepest towards the southwest). Our celestial fixes at this time will lag behind the GPS fixes showing our position about a half mile further from the island than it should be. It is as if we have slowed down slightly. Then, as we crest this hill in the geoid, the zenith will begin to rotate back to where it should be, and the celestial position would catch up with the GPS position. This would apply to any and all celestial fixes. The net effect is to create the illusion of slowing down by a few percent starting a few dozen miles out and then speeding up again as we reach the island itself. Note that Bermuda sits nearly at the top of the local hill in the geoid so celestial fixes taken at anchor in the harbor will match the GPS positions, at least as far as the geoid is concerned.

Back to Howland Island out in the Pacific, it also sits atop of a bulge in the geoid, but it's a considerably smaller hill to climb with deflections of the vertical on approach of only one or two tenths of a minutes of arc. But back to Gary's original question, since the island itself is very near the peak of that bulge in the geoid, like Bermuda and most islands in mid-ocean, a celestial fix taken on the island itself would agree very closely with a GPS position. The gravity field is flat at the top of the hill.

Also, just covering all the bases, we can't forget that simple offset of 5 seconds of arc due to the difference between the WGS84 Prime Meridian and the original Prime Meridian through the transit instrument at Greenwich. If we can measure celestial altitudes (somehow) with second of arc accuracy and get UT with equivalent accuracy, then that little extra bit matters.

-FER

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