# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Summary of Bowditch Table 15**

**From:**Trevor Kenchington

**Date:**2005 Jan 28, 23:08 -0400

Couple of days away on a business trip and I get back to a score or more of messages on this thread! Hopefully, this one won't repeat anything posted by others in the interim. Bill wrote: >> 3: However, I cannot confirm numerically that Table 9/15 is >> correct. >> >> >> Consider a 100ft mast seen from 5 miles away by an observer with >> 10ft height of eye. The 90ft of the mast extending above his >> sensible horizon should subtend an angle of about 10' at his eye, >> yet entering the table with (H-h)=90ft and angle 10' gives a >> range of 4.3 miles. >> > > You forgot to subtract dip at 10 ft. of 3.1' from your calculated 10' before entering the tables. That would make the entering argument > 6.9' tabular value approx. 6 nm. I did not forget dip. I calculated the 10' for the 90ft part of the mast projecting above the observer's sensible horizon. Dip is the angle below the sensible horizon, down to the visible horizon. >> Let the observer climb a further 60ft up his own mast and the remaining >> 30ft of the observed mast showing above the sensible horizon >> should subtend about 3.4'. However, Table 9/15 would then reduce >> the range to 3.3 miles. >> > > If the observer was at 10 ft, then climbed an additional 60 ft, he/she > would be at 70 ft above water level, correct? Yes > Using the simplified formula for distance to horizon, 1.169(sqrt height > of eye) the observer's horizon would be 9.8 nm away. If the observed > vessel is 5 nm away, as stated above, the observer should be able > to see the hull and waterline, in which case Table 16 would apply. > (Note that the geographical range would be 21.5 nm from 70 ft > aloft.) As George has noted, despite its title and explanatory note, Table 15 works perfectly well with objects closer than the observer's horizon distance. If the distant vessel's waterline is visible, Tables 16 or 17 would be alternative ways of finding its distance but Table 15 is still an option. Geographic range is something quite different: Not the actual distance to the vessel but the maximum distance at which its truck could just be seen, if it was that far away. > I also do not understand how you arrived at the 3.4' angle. Roughly speaking > we have 2 right triangles with opposite sides of 30 and 70 feet, with approx. 5 nm (30380 ft) adjacent sides. Using the tangent, find > and add the angles of both. I get approx. 11.3'. Do a rough sanity > check with the law of cosines, (c^2 = a^2 + b^2 -2ab cos c) where > c=100, a and b = 30380, and again we get approx. 11.3'. Once again, I was working with only the part of the mast above the observer's sensible horizon, hence only the right triangle 30ft in the perpendicular and 5 miles in the base. My calculator gives 3.39468... arc minutes. The almanac gives a dip of -8.1' for a height of eye of 70ft, so the entire angle from visible horizon to mast truck would be 3.4+8.1=11.5'. I'd guess that the extra 0.2' is the result of refraction, which is included in the almanac's value for dip but ignored in the above tangent calculations. > Getting down to short strokes, I am assuming 100 ft of mast visible > above the deck (as opposed to a 100 ft mast keel stepped, showing > maybe 92 ft above deck). My example assumed that the 100ft mast was measured above the waterline of the vessel, since that is the reference level for the observer's height of eye. A 100ft mast would not extend 90ft above the sensible horizon of an observer with height of eye of 10ft unless both heights were taken from a surface with constant geopotential. It was a bit off the topic of the thread but Jim included in one of his posts, as an explanation for Bill: > 1. The Visible Horizon is the one you see with your naked eye when you look > at the apparent bounday between the sky and sea (or earth). A line from > your eye to that horizon is called the Visible Horizon. > 2. The Geometric Horizon is the real line between your eye and the actual > boundary between sky and sea/earth. Since light bends between the horizon > and your eye, then the Visibile Horizon is rarely coincident with the > Geometric Horizon. The difference is accounted for mainly by terrestrial > refraction. > 3. The Sensible Horizon is part of the Horizon Coordinate System. The > Sensible Horizon is a line from your eye that runs out into space parallel > to the Celestial Horizon. The angle between the Sensible Horizon and the > Visible Horizon is called the Dip. The Sensible Horizon is a very > non-intuitive concept for learners. The sensible horizon might be better understood as a plane, perpendicular to the direction of gravity acting on the observer and drawn through the observer's eye. It is parallel to the celestial horizon because that too is a plane perpendicular to the direction of gravity acting on the observer but drawn through the centre of the Earth. The visible horizon isn't a line drawn from the observer's eye to the apparent sea/sky boundary. For piloting purposes, it is a circle drawn on the Earth which coincides with that apparent boundary but for celestial navigation it is another plane perpendicular to the direction of gravity (as are all the horizons). Specifically, the visible horizon is a plane drawn through the points where rays of light from the observer's eye which just graze the sea/sky boundary cut the celestial sphere. But that is a really weird concept since the celestial sphere doesn't actually exist. The geometric horizon is closely similar, in concept, to the visible horizon. But I don't think it is right to say that the difference is a matter of the geometric horizon being defined by the "actual boundary between sky and sea". The actual boundary between sky and sea is the curved (and often rough) surface of the water. The geometric horizon is defined in terms of a cone, centred on the observer's eye and tangential to the water surface, in contrast to the visible horizon's definition in terms of curved (by refraction) light rays. Bill responded to Bill Noyce's explanation of part of this with: > Is the sensible horizon is a plane perpendicular to the vertical (line > through center of the Earth and viewer), AT THE VIEWER'S HEIGHT OF EYE, as I > understand Jim's drawing? At the observer's height of eye, yes. But not perpendicular to a line drawn from the observer to the centre of the Earth. It is perpendicular to the direction of gravity, which means very nearly tangential to the surface of the geoid. But living here at close to 45 degrees North, a line perpendicular to the geoid is appreciably different to one directed towards the centre of the Earth. > General instructions for a pan of oil etc. instruct the user to place it on > the ground. No dip correction. If it relates to the sensible horizon and > my understanding of the definition is true, then dip correction would be > required if the pan were placed on a stool? That doesn't seem right to me. If your oil pan were big enough to reach to your horizon, then the oil surface would be curved as the ocean surface is and you might have to worry about that curvature if you set the reflected image of the Sun near the far side of the pan. However, realistic artificial horizons are so small that their surfaces can be assumed to be flat -- hence no reason to worry about dip. Trevor Kenchington -- Trevor J. Kenchington PhD Gadus@iStar.ca Gadus Associates, Office(902) 889-9250 R.R.#1, Musquodoboit Harbour, Fax (902) 889-9251 Nova Scotia B0J 2L0, CANADA Home (902) 889-3555 Science Serving the Fisheries http://home.istar.ca/~gadus