Celestial navigation explained  page 4
Celestial mechanics  a blueprint
The
hatched triangle on the top of the celestial sphere is the one we will use to
solve the celestial navigation problem.
The three sides of the triangle are:
col, the
colatitude (90° minus latitude); z,
the zenith distance (90° minus altitude h); Delta,
the polar distance (90° minus declination delta).
The angle of the triangle opposite to the side z
is called the polar angle P (180°W to 180°E).
This angle is also the angular distance QD at the celestial equator:
Here: PE = gw  GHA*
This is a spherical triangle, not a plane triangle.


While you may remember the formulas to solve a plane triangle in geometry, the formula for spherical geometry may not be quite so memorable?
As a recap, the triangle in plane geometry and the cosinus rule for the spherical geometry in the general case:


The
application of the general case to our problem:


We found a mathematical relation between what we know (delta,
GHA, h) and what we
are looking for (latitude, longitude).
With
two observations, we get a system of 2 equations with 2 unknowns that we
are able to solve.
The celestial navigation problem is now resolved.
Of course, this resolution is not really simple by hand but for a computer the process is quite straightforward: it solves the system of equations by an iterative method using the estimated latitude and longitude as starting values.
With more than 2 observations, it’s even possible to improve the traditional method and to perform a statistical analysis:
 to give a certain weight to each observation according to its reliability
in the normal law model;
 to compute and eliminate the possible systematic error of the observer;
 to correct the assumed course and speed if enough observations are provided
(exactly the same way the GPS is able to give the course and speed of the
vessel if enough satellites are visible).
To do this, a program like ASNAv is using the leastsquare method
with iterative weighting adjustment by the Biweight function on a system of
equations given by the differential correction method. Each equation i looks
like:

For a normal human being, solving such a celestial navigation equations system will take hours ! 
To check manually the results of celestial navigation programs, we can use
the traditional method of the Lines of Position (LOPs).

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