בס"ד

by Dr. Irv Bromberg, University of Toronto, Canada

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If I compute that a New Moon occurred a few thousand years ago, who is going to know if my result was correct or not?

In *Kalendis* I implemented calendrical calculations and astronomical algorithms as published by Dershowitz & Reingold, based on their printed math notation, often following their provided LISP, Java or Mathematica program code. I scarcely understand LISP. My *Kalendis* implementation is currently in Windows Visual Basic 6. In many cases their Java code was the easiest for me to convert to Visual Basic. Their errata, however, have only been published as math notation and LISP. I have validated my implementation against the examples that they published in their book and errata.

Likewise for astronomical algorithms published by Meeus, as printed math notation, I have validated my results against the few examples that he provided.

However reproducing the cited authors' examples merely confirm that my implementation can reproduce their calculations. If their algorithms or formulae are erroneous then mine will "inherit" the same errors. Therefore I have sought independent ways to validate my astronomical calculations.

Professor Aldo Vitagliano, of the Chemistry Department of Universitá di degli studi Napoli Federico II, offers *Solex*, an astronomical calculation engine that is based on numerical integration algorithms. *SOLEX* can be obtained from: <http://main.chemistry.unina.it/~alvitagl/solex/>.

The astronomical calculations in *Kalendis* are based on analytical astronomical algorithms, which execute much more quickly than numerical integration methods. Numerical integration, however, is considered the most accurate available means of carrying out celestial mechanics calculations today, if one has a fast computer and is willing to wait!

I compared *Kalendis* to *SOLEX* 9.0 (rev 04) using its DE409 with major asteroids setting and random dates ranging from 5000 BC to 15000 AD (dynamical time). The parameters compared were the geocentric ecliptic longitude of Sun and Moon, as well as the lunar latitude and distance. For all compared dates these parameters always agreed to within 4 significant figures, often agreed to 5 significant figures, and sometimes agreed to 6 significant figures.

*SOLEX* version 9.1 was released at the beginning of January 2007, and at the end of January 2007 the NASA Eclipses web site published a newly updated and extended Delta T approximation. Since then, I am using these together as the "gold standard" for all astronomical evaluations.

Book 3 Treatise 8 of the Code of Maimonides, translated to English by Solomon Gandz and published by the Yale University Press, New Haven, Gregorian 1956 AD.

It is not clear when this treatise was originally published but in the text he mentioned the Hebrew year 4938 = Julian **1178** AD and apparently his calculation examples referred to that year.

Chapter 17, Section 14: "First elongation of Moon" (=Lunar Phase) at sunset in Jerusalem on the evening of Friday, **2nd of Iyyar, 4938** is given by Maimonides as 11°27' = **11.45°**.

The coordinates that *Kalendis* version 6.6T(392) uses for Jerusalem are:

Latitude 31.7778°N, Longitude 35.2344°E, Elevation 800 m, Time Zone +2 hours.

Hebrew dates displayed by *Kalendis* correspond to the date as of midnight, Universal Time. Thus the date to use for a Hebrew date that is shortly after local sunset is the __prior__ day, with the addition of the fraction of day that defines the correct moment.

The corresponding day number in *Kalendis* is 430007, and *Kalendis* gives sunset at 18:20h Israel standard time = 16:20 Universal Time = 0.68 as a fraction of a day. **Thus the Kalendis moment of sunset was 430007.68, for which Kalendis displays a "Moon Phase" of 11.2°, which is in excellent agreement with Maimonides.**

Although sunset marked the beginning of Iyar 2, *Kalendis* displays the Hebrew date as Iyar 1 until the day number is incremented to 430008.

Total Solar Eclipse in Babylon on 15 April **136 BC** (Julian, no year zero), which seems to be the most ancient eclipse about which we can be reasonably certain of its date and time.

Stephenson gives 4 minutes as the duration of a "time-degree", which was apparently based on the time it takes for the celestial sphere of stars to appear to rotate by 1°. Today the Earth rotates 360° with respect to the stars (=one sidereal day, which is slightly shorter than the solar day) in 86141.1 seconds = 3 minutes 59.3 seconds.

The solar eclipse began 24 time-degrees after sunrise.

After 18 time-degrees the eclipse was total — planets and stars were visible.

Totality ended after 35 time-degrees.

In *Kalendis* version 6.6T(392) I entered a user-defined locale for Babylon:

Latitude 32°34'N, Longitude 44°22'E, Elevation 100 m, Time Zone +3 hours.

The corresponding day number in *Kalendis* is -49571, and *Kalendis* gives sunrise at 05:34h Babylon standard time.

(Note that Stephenson quoted the sunrise time as 05:53h. Perhaps he ignored elevation and atmospheric refraction, or used a longitude for Babylon, not specified in his lecture, that is further west than the above?)

The eclipse started 24 x 3h59.3s = 95.7 minutes = 1h35m after sunrise = 07:09h.

Totality started 18 x 3h59.3s = 71.8 minutes later at 08:21h, and lasted until 35 x 3h59.3s = 139.6 minutes after onset = 09:29h.

The mid-point of the total phase was therefore halfway between 08:21h and 09:29h = 08:55h.

Thus the lunar longitude ought to have been zero degrees at 08:55h - 3h = **05:55h** Universal Time on that day.

Searching from midnight that day, *Kalendis* found a lunar longitude of zero degrees at the Universal Time moment -49570.7438.

"Kalendis" displays 12279 seconds as the DeltaT for that moment.

(To see that moment, users of *Kalendis* can simply input this moment value into the day number text box, then press the "Tab" key.)

Convert this to a time of day fraction by taking it modulo 1 = 0.256183, which corresponds to **06:09h** Universal Time.

**Thus the Kalendis estimate of the moment of lunar conjunction is within 14 minutes of the deduced historical record — not bad, for an eclipse that occurred almost 2140 years ago!**

Can we account for the residual 14 minutes?

- Perhaps the coordinates that I used in
*Kalendis*for Babylon were were further to the west of the site where the historical observations were made. - Perhaps the Babylonian definition of sunrise was different from
*Kalendis*, which uses the moment of the first appearance of the upper limb of the solar disk at the horizon. The Babylonians may have used a moment that was later by a few moments, such as the moment when half of the solar disk was above the horizon, or the moment when a time reading could first be taken on a horizontally mounted sundial. Stephenson gave the sunrise time as 02:53h UT which was 05:53h Babylon time, or 19 minutes later than*Kalendis*, but he did not state how that was calculated or the criteria used for the definition of sunrise. - We don't know how accurate the Babylon time-of-day measurements were.
- Perhaps I have misunderstood the translation or Stephenson's comments, deriving the wrong time-of-day moments. Certainly the times given above differ from Stephenson, but that is mainly due to the difference in the moment estimated for sunrise. Also he did not explicitly consider the moment of lunar conjunction, being more interested in the beginning and ending eclipse contact points.
- The value used for DeltaT could easily be the largest source of error, in the future this calculation could be revisited after we have more accurate approximations of historical DeltaT values.
- Delta T cannot be directly estimated from historical records of solar eclipse first / last contact moments because the exact timing depends not only on the Earth rotation rate but also on the speed of movement of the lunar umbral shadow on Earth's surface, which must be calculated for each eclipse and locale. For further information see Jean Meeus, "The effect of Delta T on astronomical calculations",
*Journal of the British Astronomical Association*1998;**108**(3): 154-156.

The *Kalendis* estimate of the Mean Synodic Month is in time units of Mean Solar Days. This is quite different from either of the two formulae widely available on the internet and attributed to the ELP2000 Lunar Theory (**E**phemerides **L**unaires **P**arisiennes) of Chapront *et al*, which calculate the Mean Synodic Month in uniform time units of Ephemeris Days:

- the older and less accurate non-linear quadratic formula = 29.5305888531 + 0.00000021621
*T*- 3.64 x 10^{-10 }*T*^{2} - the newer and more accurate linear formula = 29.530588853 + 0.0000002162
*T*

where *T* is the number of Julian Centuries (intervals of 36525 ephemeris days) relative to J2000.0 (January 1, 2000 AD at Noon).

Mean Solar Days are getting longer with time due to tidal deceleration of Earth's rotation, whereas Ephemeris Days are by definition of constant duration over the centuries. In terms of Ephemeris Days the durations of lunations are getting __longer__, because Moon is moving further away, due to tidal acceleration, at an average rate of about 3.8 cm per year (according to on-going Laser Lunar Ranging measurements). However, in terms of Mean Solar Days (that is, Calendar Days), durations of lunations are getting __shorter__, because solar days are getting longer at a greater rate than the lunations.

**After conversion of Ephemeris Days to Mean Solar Days, Chapront's linear formula corresponds to a Synodic Month length that is essentially identical to and is an important validation of the astronomical algorithms of Kalendis**, shown in the following charts:

**Length of Solar Day in Atomic Seconds relative to Year 1900, by Gregorian Year**109 KB**Chapront**144 KB*vs.*Kalendis: Synodic Month Comparison, by Gregorian Year

Updated 20 *Shevat* 5767 (Traditional) = 20 *Shevat* 5767 (Rectified) = February 11, 2006 (Symmetry454) = February 8, 2006 (Gregorian)