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Re: Moving Navigator LAN: Culmination to Meridian Passage fully solved
From: Bill Ritchie
Date: 2024 Jan 25, 19:10 -0800

Re: Moving Navigator LAN: Culmination to Meridian Passage fully solved

A second ‘innovative’ solution – the ‘Ritchie’ method.

Antoine, in your post of 231219, you said that to your best knowledge ‘Step 3 has not been specifically documented yet as having been exactly solved mathematically.’ Also, earlier, you said that you looked forward to bringing ‘either rediscovered or even 100% new innovative solutions’.

You then folow with your new innovative solution. Excellent work, very detailed and obviously a labour of love.

Your post stimulated me to seek an different "100% new innovative solution" to improve upon the “Wilson 2” method with higher observer speeds and when the Moon (frequently) has a high declination rate. I have come up with the “Ritchie” method. Although my independent work, I do not claim to be the first. If any reader knows of any prior similar method, I’m sure they will post details. It uses binary search techniques, so it has no hard copy equivalent. It is a two-stage method, first calculating the moving observer’s position at the time of culmination, followed by deducing the time and position where the observer would have observed meridian passage. I start with assuming that the body’s maximum sextant altitude has been noted and that the exact time of culmination has been deduced by statistical or graphical analysis of a series of sights bracketing culmination.

‘Ritchie A’ method to find the observer’s position at time of culmination.

(Nothing new in stages 1 – 3.)

1. From the body's sextant altitude at the known culmination time, deduce the observed altitude, (Ho), using the usual altitude corrections.

2. Determine the latitude and longitude of the position on the Earth’s surface where the body is overhead at the instant of culmination. (GP, Geographical Position.)

3. The observer, at culmination, must be on an arc of a circle of position, (CoP), at the angular Zenith Distance (90° – Ho) from GP.

4A. Binary search between a range of points on the relevant arc of this CoP to find the point where the calculated rate of change of altitude (ΔHc) is zero. The found point is the observer’s position at culmination.

4B. ΔHc for each search point above is found by using the observer’s velocity (COG/SOG) to create hypothetical positions either side of that point at (say) 1 minute before and after culmination time and noting the difference in the determined Hc at those positions and times.

5. For the Moon, repeat steps 1 to 4 at least twice, using the found latitude each time. This is because the parallax correction needed to find Ho is dependent on the observer’s latitude which was initially only assumed.

6. The found position at culmination time is a valid fix and is all the navigator needs to glean from the culmination observation.

7. The process is common for upper and lower culminations once the ‘relevant’ arc has been specified.

‘Ritchie B’ method to find the position and time when the body was on observer’s meridian.

This further process to find the time and position at meridian passage is, from a navigator’s viewpoint, academic. However, it is a traditional reference point and is useful for comparing results with other methods that calculate the meridian passage event directly.

1. The track of the moving observer, specified by COG/SOG, must pass through the above found culmination position.

2. At the time of meridian passage, the observer must be on such track, though how far before or beyond the culmination position is yet to be determined.

3A. Binary search the observer’s positions over a range of times to find the point and time on the observer’s track where the LHA at that point and time was 000°. This yields the time and position of the observer’s meridian passage.

3B. Finding the LHA at each above search point/time is done by first calculating the observer’s position along that track at that search time and then calculating the LHA of the body at that position/time.)

4. Lower culminations must search for an LHA of 180°, otherwise the method is unchanged.

I have compared results with examples B1 to B4 in the latter part of Antoine’s submission, with maximum differences in culmination positions of 1.1^s UT, 0.06’ Lat and 0.18’ Long. The exaggerated examples used an observer speed of 50 knots and the Moon declination rate was over 16’/hour. I suspect that closer agreement would result if common refraction corrections had been used, and if my parallax and figure of the Earth corrections were as detailed as Antoine’s. However, such differences are trivial compared with the likely errors in measuring the altitude and time of culmination.

The meridian passage page of Astron now uses the Ritchie method. I can post links to my code should any reader ask. My former “Wilson 2” code, for the worst of the cases compared above, shows differences in meridian passage of 13^s UT, 0.2’ Lat and 3.1’ Long.

Bill Ritchie.

Usually Brixham, UK. Presently Tauranga, NZ

PS: Is there a better term for lower culmination? Lower nadir isn’t quite right.

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