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

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Re: Distance by Vertical Angle
From: Murray Buckman
Date: 2020 Oct 7, 11:18 -0700

OK, I’ll play along. I may be totally on the wrong track.

Without testing by some form of measurement on my screen, it appears that the sextant angle and the observed angle are 2:1.  In other words the angle of the true horizon and the masthead bisects the angle of the visible horizon and the mast head.  This is a very shallow angle indeed, which brings with it significant risk of error even before we consider the refraction visible in the photo.  The effect of a measurement error at shallow angles in greater than the same error at greater angles.

Just to “ball park” the sextant angle, assume a height of eye of about 10 feet.  My height of eye is 5’6” and, knowing that refraction is worsened by a low HoE at the observation point, I have walked up the beach a little to get a clearer view of the horizon.  I’d do better if I had some higher land around – but the photo presented appears to have been taken from close to sea level.

I use a constant in the distance-off formula (nautical miles) of 1.17, which is arguable and I believe factors in a “standard” refraction (yeah – right).  As a side note, I know that the dipping range of the mast at my HoE is about 15.7 miles.  I can’t see the hull or the booms, and I reckon a topmast schooner has about a third of the height of the mast above the gaff.  So I reckon I can see about 70 feet of the main mast above the visible horizon.  Again, at my height of eye, the visible horizon is 3.7 miles away.

I could just guess at this point based on these heights and the dipping distance that the ship is somewhere plus or minus 5 miles away and no further than 6 miles away.

So – simple trig using the tan formula, I get the length of the adjacent side as 3.7 and the length of the opposite side as 70/6076 (being feet per NM) or 0.011521.  The inverse tan of opposite over adjacent would give me a sextant angle of about 10’42”, half of which is contributed by the height of my eye.  So after correcting for dip I get a really small angle.  This is an estimate and does not account for any other sextant error but – hey – the angle is so small is a risk.  The distance off as computed increases or decreases relative to the angle as that angle gets smaller.  Any angle this shallow will be basically worthless.  My guess is likely just as good.

I am relying on a crude constant of 1.17 (v 1.06) to cover refraction, but in the climate indicated and observing close to sea level this should not be considered very accurate.  I could ask about the temperature and the barometer reading, but I think I just assume that whatever I compute will have an unmeasured error.  I probably do not have those tools with me on account of being shipwrecked.

Of course if I take this sextant altitude to a table for distance of by vertical altitude do I get any validation?  Bowditch has a table (2017 Table 15) which allows me to apply a difference between my HoE and the observed object (so dip correction is built in).  So I can run with my cruse estimate of 10.7' and a heigh difference of 95 feet.  That would give me give me about plus or minus 4.0 miles off (with mental interpolation only).  I should refine that last step, but as I am using a crude estimate of the visible height of the mast to derive an estimated sextant angle, I will not do better than "rubbish in rubbish out".  For example, at this low sextant angle, 10 feet either way on my estimate of visible mast height has a mile or more impact on the derived distance off (if the mast is shorter) and a little less if the mast is taller.

In this case we are not determining a danger angle - rather we are seeking to be rescured from cockroaches.  However if a danger angle was the intention, I'd not be using such a shallow vertical sextant angle.

OK, have fun putting me onto the right track ;-)

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