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Nigel Gardner asked-


>Anyone out there with a good algorithm and process for longitude by lunars?
>
>NG

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I am aware that what follows is not a full answer to Nigel's question. It
proposes a successive-approximation procedure for correcting the
chronometer error: not finding the longitude directly, but a first step
toward it. It's a procedure which could be implemented on a computer, but
mould be quite inappropriate to other situations such as hand-calculation
using tables.

The produre described below has not been written or tested by me, so it is
rather a tentative suggestion. Some details have been omitted, such as
automatic decision of which limb of the moon is the appropriate one to
measure to. It's quite possible that there errors or ambiguities, and I
would be grateful if any list member would bring them to my attention. The
proposal has been stood up to be shot down, if necessary.

==============

I would treat the problem in its simplest form to start with, not
attempting to establish longitude immediately, but simply to find the
chronometer error. Once this error is known, longitude can then be deduced
by measurement of an altitude of a body that's well away from the meridian,
as a separate problem.

To find the chronometer error, the observer only needs to take the lunar
distance and needs to measure no altitudes at all. The necessary altitudes
can be computed rather than measured, so no horizon is needed. Star-moon
distances can then be measured in pitch-darkness, when stars are bright.

I would presume a rough initial position (lat and long) and an assumed time
of the measurement: probably the chronometer reading with its unknown
error.

I know of no way of expressing time as an explicit funtion of lunar
distance; only how to calculate lunar distance in terms of time. So the
method will be one of successive approximation.

The accurate position in the sky of the moon and the other body must be
calculated, to a precision of 10 sec of arc or so, for the assumed time.
For this I would use the methods given by Meeus, in his Astronomical
Algorithms, using all the many coefficients that he provides. These sky
positions should be converted to altitude and azimuth, as seen from the
assumed lat and long. These are the positions the bodies would be seen at
if viewed from the centre of the earth, and if our atmosphere did not
refract.

Correct these two altitudes for the effect of refraction and also parallax
of moon and sun (augmented parallax in the case of the moon). Note that
these corrections are to be made in the opposite sense to what we are used
to, because we are aiming to calculate the direction in the sky the
observer would see the body if he were to measure its altitude, by
correcting the geocentic position. Not the other way round, as we usually
do it. It may be worthwhile to adapt those corrections slightly because we
are using a slightly different angle as the argument, but this is a minute
adjustment.

Now we calculate the angle-in-the-sky between these two apparent bodies by
one of the standard spherical-geometry formulae. Exactly the same as
calculating the great-circle distance between A and B on a spherical earth.
This is the calculated lunar distance. Note that in this process we have
not needed to "clear" the lunar distance of parallax and refraction in the
complex way that is usually called for; it was simple addition and
subtraction.

Now we take our measured lunar distance, corrected for index error. We
allow for the semidiameter of the moon (remembering which was the
appropriate limb) and if called for, the Sun. These semidiameters and the
parallaxes used previously are taken from the calculated earth distance,
which was a by-product of the Meeus calculations.

After all this, we have a measured lunar distance and a calculated lunar
distance to compare. The difference between these quantities, if we divide
by the angular velocity of the moon in its orbit, is roughly the error in
the chronometer. Adjust the assumed time by this quantity, and if necessary
repeat the whole calculation again, using the new assumed time. At each
iteration the time-error will shrink and you choose when to stop the
process.

George Huxtable

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