NavList:

Paul,

       Thanks for looking into this. For my purpose I was numerically solving for the time of conjunction and iterating until the error was comfortably under 0.1 seconds. Since the differences between all values you give and mine are well inside that things appear to be as behaving as expected. Just for interest I cranked up the precision a bit and find apparent geocentric conjunction in R.A. at

TT = 2019-01-18 19:33:52.732.

84.6970144 zeta Tauri
84.6970144 moon

I am using de430.bsp as de431.bsp seems a bit more trouble to acquire and at https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/aareadme_de430-de431.txt

it states

“The difference between DE430 and DE431 is mainly in the dynamical model for the Earth's Moon.

DE430 uses a more complete dynamical model which produces a slightly more accurate ephemeris for the Moon.”

As noted in my earlier post I will probably make inclusion of the quadratic terms in the Besselian elements an option. Your information indicated that the offset of the Moon’s geometric centre and centre of mass does not have a significant impact at least for the events we were looking at. Including that would require handling lunar librations which consider “out of scope”. Limiting Besselian Elements to be linear has the advantage that you only had to solve quadratic equations in the reductions which could be handled in manual calculations. Chauvenet shows how to do this using trigonometric substitutions https://archive.org/details/manualofspherica031436mbp/page/n539

Incidentally this is not the method that Captain Frank Worsely was using in 1915 (https://www.canterburymuseum.com/assets/DownloadFiles/Navigation-of-the-Shackleton-Expedition-on-the-Weddell-Sea-pack-ice.pdf). He was using Raper’s Method which when performed manually requires many interpolations between tabulated values. Although I haven’t looked into it in detail I assume the errors from the multiple interpolations mount up and degrade the final result. Besselian Elements have always seemed to me to be a much more straightforward,

Robin Stuart