NavList:

Dear Lars,

Thanks for your reply.

(1) - Chauvenet

In order to carry out computations mathematically better than say 3", the closeness of Lady Moon and its relatively important semi-diameter require that we use an ellipsoidal Earth rather than a spherical one. Chauvenet gives interesting formulae here but I have not found a comprehensive set from him to carry out all computations. Hence I have resorted to 3D computations to accurately compute both Moon topocentric Parallax and augmented SD. Just performing such computations on a sphere can give mathematical end results off by almost 0.2' under extreme cases.

(2) - Coordinates and SD computing method (Full computation method)

As regards the "computing method" you are again describing Lars, yes I think that all the few "number crunchers" among us are using the very same one.

Like yourself and others, for any UT during the time period considered I am able to accurately compute all parameters of interest :

2.1 - Moon Center height minus Sun LL height, subsequently referred to as "Δh"

2.2 - (Moon Center Azimuth - Sun Center Azimuth) * cos (Sun Center height) , subsequently referred to as " ΔAzcos⊙ "

2.3 - Moon augmented semi-diameter refracted, subsequently referred to as  " ☾SD " .

Such computation method is subsequently referred to as " Full Computation Method ".

(3) - Equation to solve,

Computing such 3 variables of interest is a required first step towards solving our problem.

We then need identifying the UTi values - if any - solving the following equation :

f (UTi) =  0.00' , with

f(UT) = SQRT (Δh **2 +  ΔAzcos⊙ **2) -☾SD

(4) - "Search" method vs. "Computing" method

Equation f(UT) is a tool to solve for our problem. Again as number crunchers we are all using this same equation.

How are we going to use this tool ? We need some kind of "research method" here, no ?

Hence my earlier question to you Lars should have been worded as follows :" Between 17:00:00 UT and 18:30:00 UT, where do you start searching and narrowing your search to obtain the 2 solutions ? "

As a search method, Frank indicates : QUOTE I did this once every thirty seconds for the period of interest. Then I ran a line through the points and calculated the time when the azimuths matched. I get 17h07m44s and 18h20m36s. UNQUOTE

In other words, my question to you Lars could have been : "How did you determine the period of interest and how did you narrow your research into finding your corrected end-results" ?

(5) - One method using successive approximations

Since computing f(UT) is quite a lenghty task to me, I have attempted to determine then use approximation functions for Δh, ΔAzcos⊙ and☾SD in order to program f(UT) on my hand held and expedite my "search" for solutions. I was expecting such approximations to be quite simple.

Well, it unexpectedly turned out to be quite a lengthy task ...

Since it might be instructive to our NavList Community, let me elaborate further down on the solving saga I went into.

*******

For the record and own subsequent calculations (non significant digits retained on purpose here in order to wipe out any cause for round-off errors) :

For position N28° / W 070° on Oct 14th, 2023 and from Full Computation Method, I am getting the following results :

At UT = 17:00:00.0 : Δh = + 16.91820', ΔAzcos⊙ = + 05.77953' and ☾SD = 15.23200'

At UT = 18:30:00.0 : Δh =  - 00.57960', ΔAzcos⊙ = - 18.48716' and ☾SD = 15.23850'

*******

5.1 - Linear interpolation (1st order approximation) 

For each of the 3 variables - i.e. Δh, ΔAzcos⊙ and☾SD - I first used a linear interpolation through simply drawing a straight line to join the end values. Hence at both ends the linear values exactly match the real ones. 

5.1.1 - 1 st order approximation results : running 1st order f(UT) indicates that Sun LL is covered by the Moon between 17:10:13.5 and 18:17:32.0 s

5.1.2 - 1st order results check from Full Computation Method

The Full computation method indicates that at 17:10:13.5 and 18:17:32.0 s, the 1st order approximation f(UT) values are in excess of 3', while they were expected to show values very close to 0 .

Hence the 1st order approximation of Δh, ΔAzcos⊙ and☾SD is insufficient.

5.2 - Second order approximation

I am therefore looking for a 2nd order correction to the 1st order approximation, such second order correction to be also exactly zero at both ends, in order to not "spoil" the 1st order approximation at interval ends.

With T = UT - 17:00:00.0, when UT varies from 17:00:00.0 to 18:30:00.0 then T varies from 0:00:00.0 to 1:30:00.0

With T' = 4/3T - 1 - which makes T = 4/3 (T' + 1) - then T' varies from -1 to +1.

We then need to consider 2nd order Corrections of the form " k * (T'-1)(T'+1) " since this 2nd degree polynom in T' is to vanish at T'=-1 and T'=+1.

At mid interval, i.e. T' = 0, we compute 1st order approximations for Δh, ΔAzcos⊙ and☾SD and compare them to their true values derived from the Full Computation Method.

The required corrections at mid-interval - i.e. at T' = 0 - are as follows : 

for Δh : -3.21390'  , for ΔAzcos⊙ : +2.66967' and for☾SD : +.00425'

To the 1st order approximation, we choose to apply the following 2nd Order corrections :

correct Δh with -3.23190' * (T'**2 - 1) , correct ΔAzcos⊙ with +2.66967' * (T'**2 - 1) and correct ☾SD with +.000425' .

As aimed at, the 2 nd order approximations end values are exact, as well as the mid-interval one now (i.e. the one for T' = 0 ).

5.2.1 - 2nd order approximation results : running 2nd order f(UT) indicates that Sun LL is covered by the Moon between 17:07:21.5 and 18:21:23.0 s

5.2.2 - 2nd order results check from Full Computation Method

The Full computation method indicates that at 17:07:21.5 and 18:21:23.0 s, the 2nd order approximation f(UT) values are is slightly in excess of 0.3'.

That's about a tenfold improvement brought by the 2nd order approximation over the 1st order one. Nonetheless the 2nd order approximation of Δh, ΔAzcos⊙ and☾SD is still insufficient.

5.3 - Third order approximation

Let's then look for a 3rd order additionnal correction to the 1st and 2nd orders approximations.

Again, such 3rd order correction is to be also exactly zero at both ends, in order to not "spoil" the 1st order approximation at interval ends. And also it is to be exactly 0 at mid interval - i.e. T' = 0 - in order to not "spoil" the 2nd order interpolation there.

We then need to consider 3rd order Corrections of the form " k'' * (T'-1)*T*(T'+1) " since this 3rd degree polynom in T' is to vanish at T'=-1, T' = 0 and T'=+1.

To make a long story not too long, I have retained the following k'' values for the 3rd order corrections :

forΔh : k''= +0.3517' , for ΔAzcos⊙: k'' = +0.47191' and for ☾SD : k'' = 0.00000'.

5.3.1 - 3rd order approximation results : running 3rd order f(UT) indicates that Sun LL is covered by the Moon between 17:07:44.5 and 18:21:00.5 s

5.3.2 - 3rd order results check from Full Computation Method

The Full computation method indicates that at 17:07:44.5 the 3rd order approximation f(UT) value is 0.006' and at 18:21:00.5s such 3rd order approximation  f(UT) value is 0.002 ', with the following trends :

at 17:07:44.5s Δf(UT)/ΔUT = -0.34' /minute of UT= -20.3 "/minute of UT, and at 18:21:00.5s Δf(UT)/ΔUT = +0.35' / minute of UT= -21.3 "/minute of UT

HERE WE ARE !

These results [17:07:44.5 - 18:21:00.5] are quite close from :

Lars (last results) :  [17:07:43 - 18:20:47] 

Geoff :  [17:07:48 - 18:20:40] 

Frank :  [17:07:44s - 18:21:36s] 

Last note : 

ENTRANCE time : 17:07:46 +/- 2 seconds of time . We all fully agree here :

EXIT time shows more dispersion : 18:21:08 +/- 26 seconds of time.

In your last post, Lars you give us what seems a good explanation for the bigger "EXIT time" dispersion : QUOTE " Thus any very small difference in say moon's semidiameter will have an observable impact on the exact time of contact amounting to several seconds of time, due to the slow passing of the moon relative to the sun. "  UNQUOTE.

That's the opportunity for me to indicate that for the EXIT time, for the MOON AUGMENTED + REFRACTED SD I am using a value of 15.24' which I trust to be accurate to +/- 0.01' . Can anyone prove or disprove such value ? Paul Hirose, maybe ? Thanks in advance.

To all of you brave readers, if you are still here ... thanks for your Kind Attention,

Kermit