NavList:

Dear Robert,

(1) - As Bela Kovacs reminds us, Noon sights / Local Apparent Noon (LAN) sights have been extensively addressed on NavList lately.

(2) - To the best of my knowledge, and not until less than a couple of years back, no mathematical solutions for exactly solving for LAN sights had ever been published for a Moving Navigator.

Assuming for all second and subsequent derivatives of motion (both Observer and Body) to be equal to zero - 2 such solutions independently derived have been published (Dec 2023 and Jan 2024).

Both methods yield identical results.

(2.1) - By Bill Ritchie : https://navlist.net/Moving-Navigator-LAN-Culmination-Meridian-Passage-fully-Ritchie-jan-2024-g55312   . And,

(2.2) - By Antoine M. Couëtte : https://navlist.net/Moving-Navigator-LAN-Culmination-Meridian-Passage-fully-Couëtte-dec-2023-g55131 , under the form of 2 comprehensive studies on LAN sights :

(2.2.1) - LAN sights Part 1 of 2

(2.2.2) - LAN sights Part 2 of 2

(3) - Methods (2.1) and (2.2) / (2.2.2) are not suited for quick computations since they extensively rely on successive approximations.

3.1 - So we are left with the more "classical" methods, not forgetting the very clever "folder sheet of paper" method by F.E. Reed and the "Wilson 1" and "[amazing] Wilson 2" hand drawn methods.

3.2 - By reference to documents (2.2.1) and (2.2.2) the Classical numerical methods essentially boil down to :

3.2.1 - ΔUTmp-c (in seconds of time) = (UT Meridian Passage - UT Culmination) s = 48/π * (tgφ - tgD) (μφ-μD) ’/h .

HINT ! Here the "48/π" coefficient applies only for the Sun seen from ... steady observers.

Hence it can be expected that such "48/π" coefficient may and will significantly depart from reality for moving Navigators observing different bodies (e.g. the Moon).

And :

3.2.2 - (h₀ Meridian Passage - h₀ Culmination) ’ =  0 ’ 

3.3 - Instead of Formulae 3.2.1 and 3.2.2 here-above and at the expense of some minor increase in "numerical complexity" , i.e. introducing the exact ratio "k=(15/μT) " of the actual Rate of Body Local Hour Angle seen from a moving observer, and as an important  practical conclusion Document (2.2.2) strongly advocates using rather :

3.3.1 - kΔUTmp-c (in seconds of time) = (UT Meridian Passage - UT Culmination) in seconds of time = (15/μT)² * 48/π * (tgφ - tgD) (μφ - μ’/hD) .

And:

3.3.2 - (h₀ Meridian Passage - h₀ Culmination) ’ = - 1/2 * |(kΔUTmp-c) h * (μφ-μD) ’/h|

As is evidenced by the various numerical examples in the last part of Document (2.2.2), even under extreme conditions (very fast moving observers at even 50 kts) Formulae 3.3.1 and 3.3.2 here-above enable one shot estimates to fall extremely close from the "true" results yielded by either (1.1) or 1.2) .

Hope it helps,

Antoine M. "Kermit" Couëtte