# NavList:

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

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Re: Longitude by calculator -theodolite
From: Paul Hirose
Date: 2013 Jun 29, 16:28 -0700

```Bruce J. Pennino wrote:
> Actually, for my
> next set of sights I'm going to use some early rising stars at twilight.
> Should work ok.  Someone asked  about my using the moon. I used the moon
> because I can't put any shades on the theodolite, and there would be
> optical distortion (maybe) as suggested.

Wouldn't it be more convenient to work in full darkness? Then you
wouldn't have to observe within a specific time window. The selection of
stars would be better too.

> I am measuring time to nearest whole second. So I'm probably accurate to
> plus or minus 1/2 second for a single measurement, at very best .

It helps to use a stopwatch with a split-action feature (to stop the
display without stopping the watch). Record the start time. Take a split
at each observation. After all the observations, take a final split
against your time standard to verify the start time.

I used to own a dedicated stopwatch with 10 memories to record the
splits. Sadly, it quit working several years ago. My wristwatch simply
displays the split for 10 seconds then resumes running. That's not as
convenient, but still usable.

> The theodolite is a "6 second gun", which
> means I can directly read to 3 seconds, and maybe estimate to the
> nearest second or so.

To realize the potential of that instrument, altitudes should be
computed to 1 second or better. By careful choice of stars you
can minimize refraction error. A stable temperature helps. That's
another reason to observe when the sky is fully dark. You avoid the
rapid temperature decrease around sunset.

I wonder if you have considered deflection of the vertical. At your
location, xi = -4.26 and eta = 1.37 seconds, according to the National
Geodetic Survey calculator:
http://www.ngs.noaa.gov/cgi-bin/GEOID_STUFF/deflec12A_prompt.prl

That means an instrument, exactly level with respect to gravity,
actually has its vertical axis inclined 4.26 seconds south and 1.37
seconds east with respect to the ellipsoid. Its observations yield
astronomic latitude and longitude, different from the geodetic
coordinates on a map or GPS receiver.

At the precision to which you are working, I believe deflection of the
vertical will be a significant part of your error budget unless corrected.

For example, at 2013 June 30 0200 UTC, Antares is observed from N42
W072, 0 height above ellipsoid. UT1-UTC = +0.05773 s. Computed
unrefracted altitude from the Tinyac program:

20°45'10.0" no deflection of vertical
20°45'14.4" with deflection of vertical

One solution is to compute altitude at the geodetic position where a
perpendicular to the ellipsoid is parallel with the deflected plumb line
at the true position. The adjusted position is north latitude plus xi,
and east longitude plus (eta divided by cosine latitude). In this case,
the adjusted position is N41 59 55.74 W071 59 58.16.

The Tinyac program uses that simple method. I later realized it is
effective for altitude only. Azimuths are inaccurate, the error
increasing with latitude. Lunar3 does it right, though. Compare azimuth
and unrefracted altitude of Antares, including deflection of the vertical:

169°13'36.2"  20°45'14.4"  Tinyac
169°13′35.0″  20°45′14.4″  Lunar3

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