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40 South asked-

| Supposing one was in a small boat with an accurate timepiece and the
| necessary tables, how accurate could you determine your longitude by
| observing the rising or setting of the sun or any other celestial
| body?

There have been several responses, but I suggest that they have missed an
important point.

Yes, in theory, if you happen to know your latitude well, then an
observation of the moment of Sunrise will give your local time, which can be
compared with Greenwich time to give your longitude. You could use the
predictions in the Nautical Almanac, but these are to the nearest minute of
time only, and for Greenwich noon at the middle day of a block of three
days, and for broad bands of latitude. You could interpolate, to some
extent, but a better way would be to do the job by trig or tables, for that
known latitude, date, and time, allowing for Sun semidiameter, refraction,
and dip.

The best you can expect is to observe a sunrise from the Equator, near the
equinox, when it comes vertically out of the sea at 15 degrees in each hour.
At other latitudes, and other times, the ascent of the Sun is at a shallower
angle, much shallower at high latitudes, so its rate of rise is
correspondingly less, and any error in measuring Sun altitude reflects in an
enhanced error in time, and therefore in longitude.

Now, here's the rub. Although sunrise/sunset predictions are based on a
presumed value of refraction, at an observed altitude of 0 degrees, which
will be something like 33 or 34 arc-minutes, that is no more than a rough
approximation. At sunrise/sunset, the light is having to pass through an
immense thickness of atmosphere, at a glancing angle, and is badly affected
by unknown and unpredictable refraction as it passes through the various air
layers, at unknown temperatures. That is why the Sun's image can get so
distorted as it nears the horizon, and even if there's no apparent
distortion, the whole Sun can get displaced, vertically, by these refraction
effects. It's not unusual for the actual refraction at the horizon to be
half a degree out from that predicted value, sometimes even more, so
depending on its sign, it could double, or nullify, the predicted
refraction.. In polar latitude, the effects can be greater still. That's why
navigators keep clear of celestial altitudes when they are less than 10
degrees or so. At higher altitudes, refraction is much less, and, more
important, more precisely predictable.

Measured at the equator, every minute of refraction error will result in a
4-second error in timing sunrise, and thus a 1 minute error in longitude. At
higher latitudes, the errors will be enhanced. So timing sunrise/sunset will
never provide a precise value for longitude. But if a rough ball-park figure
will suffice, it could do the job.

40South continues-

| Also, assuming you had a compass and were north or south of the
| tropics, would it be possible to estimate your latitude by taking a
| bearing of the setting or rising of a celestial body?

I'm not sure why he applies the restriction of being "north or south of the
tropics". Again, in theory, yes. Again, it's affected by the vagaries of
refraction near the horizon. But a bigger source of error is the difficulty
of measuring that bearing. On land, with a theodolite and a known
North-South line, it can be measured precisely. At sea, it's affected by the
uncertanties of the magnetic compass. It's hard to read azimuth to better
than a degree or so at the best of times, and then there are additional
errors in allowing for deviation and variation. Historically, a
sunrise/sunset azimuth was used the other way round. With latitude known
(from a noon Sun sight, followed by a bit of DR) it was used, on a wooden
vessel, to check magnetiic variation.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.




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