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"George Huxtable"  wrote:
>   I can see, then, how Frank's proposal could give one position line,
>   placed at right angles to the azimuth of the Moon, on the chart,
>   because the Moon's altitude has been deduced from its parallax. But
>   it's only the Moon's altitude that can be deduced by this method,
>   because it has such a large parallax. No information is provided on
>   the altitudes of stars.
>
>   I haven't yet understood Frank's explanation of how a second position
>   line can be deduced by measuring the distance between the Moon and a
>   second star, and, I too, ask for a more detailed explanation with a
>   numerical example.

I think I see the principle involved.  Suppose we measure the angle
between a star and the nearest limb of the moon to be 10 degrees.  (We
neglect refraction.)  Where are we?

We have to be on the surface of a particular cone, with a half-angle
of 10 degrees.  The axis of the cone goes through the center of the
moon and the star.  A ray from us to that nearest limb of the moon
lies on the surface of the cone, and points to its apex.  Anywhere on
that cone an observer would measure the same angle from star to moon
limb.

We are also on the surface of the earth.  Therefore, our line of
position is the intersection of a cone with a sphere.  Recall that the
axis of the cone went through the center of the moon, not the center
of the earth.  Therefore, our line of position will in general not be
a circle.

A measurement from the moon to a different star gives us another cone,
and a second line of position.  If the first star were directly above
the moon, we could use a second star to the right or left.  However, I
would want to use at least three stars, preferably at the "10
o'clock", "12 o'clock", and "2 o'clock" positions with respect to the
moon (i.e. above it, so as to minimize the effect of refraction, and
at 60 degree intervals to optimize the geometry).

If the moon is low in the sky, then we get a very poor estimate of our
distance to the moon's GP.  However, I think the estimate of our
azimuth from the moon's GP gets better when the moon is low.

The mathematics will be challenging, though.

We might be better off picking stars at "9 o'clock" or "3 o'clock"
because the geometry gets simpler.  If the angle from the star to the
nearest limb is 90 degrees the geometry is even simpler, because the
cone becomes a flat plane, and its intersection with the earth is a
circle.  If the angle from the star to the further limb were 90
degrees we could use that the same way.

I'm also looking forward to that worked example :-)

           - Jim Van Zandt


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