The crescent criteria and the Babylonian reproduction

This appendix closes the two points that appendix A left open: the criterion for the visibility of the first lunar crescent is here built up from the ground and applied, and the Babylonian first-of-Nisan rule (Parker & Dubberstein) is recomputed against its own anchor. It makes the narrowest link of the book — which evening carries the first crescent, and hence which day is 1 Nisan — as hard as astronomy can make it. All figures can be reproduced with a small Python script (Python 3 with the ephem library).

C.0 · The visibility criterion, and why it is needed

The whole calendar hangs on one observation: on which evening is the young crescent first seen with the naked eye? That evening begins 1 Nisan; thirteen days later falls 14 Nisan, the day of death. The book claims (ch. 7) that the crescent of 11 April AD 31 lies "well above every published threshold". This appendix tests that, not with a rule of thumb but with the most-cited modern model.

The model. In 1997 B. D. Yallop (A method for predicting the first sighting of the new crescent moon, HM Nautical Almanac Office, NAO Technical Note 69) brought the older criteria of Fotheringham, Maunder, Schoch and Bruin together into one testable quantity, q, computed at the "best time" — the moment on the road between sunset and moonset at which the chance of observation is greatest (sunset plus four ninths of the time between sunset and moonset). The quantity combines two angles and the width of the crescent:

- ARCV — the height of the moon above the sun (the "arc of vision"); - ARCL — the moon–sun angle in the sky (the "arc of light", which fixes the phase); - W — the width of the illuminated crescent in arcminutes, W = s·(1 − cos ARCL) with s the apparent semidiameter of the moon;

summarised as q = [ARCV − (11.8371 − 6.3226 W + 0.7319 W² − 0.1018 W³)] / 10.

Yallop's zones. q > +0.216: crescent easily visible with the naked eye (A). +0.216 to −0.014: visible under favourable conditions (B). −0.014 to −0.160: borderline zone, optical aid possibly needed (C). Lower: improbable to impossible without a telescope (D–F). A 1 Nisan that begins on a zone-A evening is as hard as an ancient observation can be.

Tools and place. The lunar and solar positions were computed topocentrically for Jerusalem (31.78° N, 35.24° E, 754 m) with PyEphem, the same library as appendix A. The dates are proleptic Julian. [CALCULATED]

C.1 · The decisive crescent: the two evenings of April AD 31

The March moon of AD 31 (conjunction 11 March) falls eleven days before the equinox (measured to the conjunction; its first crescent some nine days) and is too early as a Nisan moon (ch. 8); Nisan shifts to the next lunation, with conjunction on 10 April, around the middle of the day. The question is then narrow: on which evening does its crescent first stand visible? [CALCULATED]

The evening of 10 April is excluded — the crescent is then not yet five hours old, stands on the horizon and sets four minutes after the sun: unobservable. The evening of 11 April carries a crescent of nearly thirty hours, a good eleven degrees high, setting a full hour after the sun, with q = +0.32 — deep in zone A. The leap between the two evenings is enormous; there is no doubt which evening is the first. Hence: 1 Nisan begins on the evening of 11 April; 14 Nisan falls on Wednesday 25 April. This is the astronomical core of the book, here recomputed independently — no rule of thumb, but a model that places the crescent well within the most easily visible class.

C.2 · The three race years at the threshold

Set the first-crescent evenings of the three candidates side by side. The weakness of a calendar computation always lies in the borderline cases — years in which the crescent stands at the edge of visibility, so that one day either way shifts the whole month. [CALCULATED]

And this appendix need not stand alone, for there is an independent second computation and it comes from the other side of the debate. Bradley Schaefer, then at the Goddard Space Flight Center, computed in QJRAS 31 (1990) the visibility of every young crescent above Jerusalem for AD 26–36 with his own brightness model — different ephemerides (Tuckerman interpolation), a different moment of assessment (forty minutes after sunset rather than Bruin's 4/9 × lag), and a different measure. His measure is R: the logarithm of the ratio of the moon's brightness to the minimum brightness required to distinguish it against the twilight sky. R = 0 means a fifty per cent chance of detection; negative means a prediction of invisibility; R < 1 he calls "difficult to spot" and R > 2 "easy visibility" (p. 58). And he supplies a measure of confidence: DR, roughly one sigma, with the rule that at |R/DR| > 2.5 his criterion gave "correct predictions for all 201 observations".

Thirteen evenings have been laid side by side in this way, and there is not one on which the two scales contradict each other. More: they rank the doubt in the same places. The two evenings Schaefer expressly calls "not certain" — 23 March 30 and 30 March 32 — both fall here in zone C, the borderline; his one "difficult and uncertain" (18 April 33) falls in zone B; his invisible evenings in zone F; his "easy" in zone A. [COMPUTED and DOCUMENTED — Schaefer (1990), Table I on p. 57 and the definitions on p. 58, read at first hand from page images]

⛔ And one sentence of Schaefer's belongs in this appendix because it repeats its own limit. He prints the first four solar months of each year, and he says why: "because it is not clear which lunation was identified as Nisan, because of uncertainties in the equinox and intercalations" (p. 58). That is precisely the boundary C.2 below sets for itself: the crescent says when, the calendar rule says which. The independent second computer arrives at the same limitation.

Two things emerge. First: AD 31 stands furthest from every edge. Its crescent is not borderline-visible but amply visible (zone A), and the evening before is utterly invisible (zone F) — the ±1-day wobble that astronomers acknowledge for borderline cases does not touch AD 31. Second — and this belongs here in honesty: the visibility model does not decide which lunation is Nisan, only when each crescent first appears. That AD 33 has an amply visible March crescent (20 March, zone A) is exactly what makes the case sharp: under a rule that admits the March moon, 14 Nisan AD 33 falls on ~3 April (the consensus "Good Friday"); under a rule that rejects it because 1 Nisan falls before the equinox, Nisan shifts to the April moon and 14 Nisan falls on ~2 May (ch. 8, ch. 15). The crescent says when; the calendar rule says which — and that rule, not this appendix, is the narrowest link.

C.3 · The Babylonian rule, recomputed against its own anchor

The Jewish month names are Babylonian, and the Babylonian calendar kept 1 Nisanu after the spring equinox — a rule that Parker & Dubberstein (Babylonian Chronology 626 B.C.–A.D. 45) have reconstructed in full for these centuries, and which Sacha Stern cites from their tables (Calendar and Community, pp. 61–62). That rule is testable against one dated anchor: the spring festival at which Vitellius is in Jerusalem in AD 37 (Antiquitates 18.89–90, 120–124). Precision is obligatory here, for it is easy to say too much. Josephus dates that festival to no calendar day; what he supplies is that it fell in April and not in March — the travel time of the news of Tiberius's death does not allow March. The 19 April comes from Parker & Dubberstein, not from Josephus. And that the festival was the Passover is a construal: Josephus writes ἑορτῆς πατρίου, "the festival of the fathers". [CONSTRUED] [DOCUMENTED, Parker & Dubberstein; CALCULATED, the reproduction] The Babylonian rule is one of two documented practices: in Egypt, from Aristobulus to Philo, the Jewish Passover fell at or around the equinox, on average a month earlier (Stern, ZPE 130, 2000, p. 171 n. 69). This reproduction tests the Babylonian rule; which practice Jerusalem kept, it does not decide (ch. 8).

Parker & Dubberstein give Nisanu 1 of AD 37 as 6 April (their table, p. 46). The independent recomputation confirms this to the day: the April conjunction of AD 37 falls on 4 April, its first visible crescent on the evening of 5 April, so that 1 Nisanu begins on 6 April — precisely Parker & Dubberstein's date. Thirteen days on, 14 Nisan falls on 19 April, and that is exactly the day on which Josephus's dated festival comes out. The rule thus passes the check against both the Babylonian table and the ancient anchor. But that is a necessary check and not a distinguishing argument, and the book must make no more of it than it is: the more lenient equinox rule too, and Lanser's shifted series too, arrive at that same 19 April for AD 37 (ch. 8). The anchor confirms that a late-April Passover was possible; it does not choose between the reconstructions.

Applied to the three race years, the same rule gives (summarised from appendix A.3):

Under this rule AD 30 keeps its Friday, AD 31 comes out on Wednesday 25 April, and AD 33 loses its "Good Friday" — which survives only under the more lenient equinox rule. Two honesties belong with it, and both cut toward this side. First, the rule derives its authority not from the AD 37 anchor, for that anchor does not distinguish (see above); it derives it from the first-century Judean data that ch. 8 measures out — few, and against a documented Egyptian counter-practice, so that the rule stays an assumption for Jerusalem (above; appendix D). Second, under the other ancient rule — that of the ancients, which requires only the festival after the equinox — AD 31 has a second outcome: then the March moon stays and 14 Nisan falls on Tuesday 27 March. That too is not a Friday, and there lies the asymmetry with AD 33: for AD 33 the choice of rule decides whether there is a Friday, for AD 31 only which midweek day it becomes (ch. 8, ch. 15).

C.4 · What this appendix does not deliver

- The model is statistical, not a guarantee. Yallop's zones are calibrated on hundreds of observations, but the atmosphere on one evening in the year 31 is unknowable. What the computation delivers is not "the crescent was seen" but "the crescent lay far inside the class that is almost always seen" — and the evening before far outside it. For AD 31 that margin is so large that no reasonable atmosphere overturns it; for the true borderline years (AD 30 at the edge) that reservation is real. And for AD 31 there is a second line behind the first, which owes nothing to any visibility model: the thirty-day cap on a month (ch. 7). Cloud on 11 April, and on 12 April as well, moves 14 Nisan at most to Thursday 26 April — a Friday would require an Adar II of thirty-one days. [CALCULATED] - The crescent does not decide the month. As C.2 says: which lunation carries Nisan hangs on the equinox and barley rules (ch. 8), not on visibility alone. This appendix hardens the one half (when the crescent appears); the other half remains a documented calendar rule, not a measurement. - Parker & Dubberstein's table is not reproduced here. It has been worked through in full for 200 BC–AD 45 in the machine-readable edition (appendix A.6); the anchor (Nisanu 1 AD 37 = 6 April) has in addition been recomputed and holds — a successful necessary check, not an independent ground for the rule. - The topocentric corrections are standard, not refined. Elevation, refraction at the horizon and the moon's semidiameter were computed with the library's standard values; a more refined horizon model shifts q by hundredths, not tenths — too little to lift AD 31 out of zone A.

Reproduction. The script is `yallop_q.py` (Python 3 with the ephem library, version 4.2.1). It computes, for any evening, the conjunction, sunset, moonset, lag, best time, ARCL, ARCV, W, q and the zone, and it carries a `--verify` flag that sets its output against fourteen figures printed in this appendix: the q, moon altitude, ARCL, lag and zone of 11 April AD 31; the q, lag and zone of 10 April; the zone and lag of 13 March; the q of 23 March AD 30 and 20 March AD 33; and the anchor Nisanu 1 AD 37 = 6 April, in the form 4 April zone F, 5 April zone A. All fourteen are reproduced.

⚠ One reservation on the convention, and it belongs here rather than in a footnote. Yallop's formula can be evaluated with ARCV and ARCL each taken geocentrically or topocentrically, and the four combinations give, for 11 April AD 31, q = +0.317, +0.287, +0.408 and +0.377. This appendix's figures follow the first: ARCV topocentric, ARCL geocentric. That combination is the only one that closes all three printed quantities at once — q = 0.32, ARCL = 13.7 and a moon altitude of 11.4°. It was established from that agreement, not from Yallop's own text, which has not been consulted at first hand. The reproduction is therefore an internal consistency check: it shows that the script does what this appendix did, not that either is what Yallop intended. [CALCULATED; the convention CONSTRUED from agreement with the printed figures]

The same reservation carries a consequence worth naming. On the evening of 12 March AD 31 the convention of this appendix gives q = −0.313, zone F; the two geocentric variants give −0.219 (zone D) and −0.240 (zone E). Earlier workings of this material quoted the geocentric figures. Under the convention the appendix actually uses, 12 March falls in zone F. The withdrawn figures are recorded in the script so that they remain visible.

The calibration, and what it costs this appendix

Reproducing printed figures shows only that a script does what the appendix did. So the computing core was run against observations it had never seen: the three world-record crescent sightings by naked eye held by the International Astronomical Center, which publishes for each one the location, the conjunction, the lag and its own ARCV, ARCL and relative azimuth.

The core passes. For the Tennessee record of 25 February 1990 the script gives ARCV 7.53° against the published 7.60°, relative azimuth 0.55° against 0.60°, lag 38.3 minutes against 39. For the Ashdod record of 20 September 1990: ARCV 5.93° against 5.90°, relative azimuth 18.41° against 18.40°, lag 28.6 minutes against 29. Angles to within seven hundredths of a degree, lag to within forty seconds, at latitudes and dates the script was never tuned on. [CALCULATED against ICOP's published figures, which are taken from that source and not from the observing reports themselves]

⚠⚠ Yallop's criterion does not pass, and the consequence falls on this appendix. The Tennessee sighting — the youngest crescent on record seen with the unaided eye — computes to q = −0.32 on the script's angles and −0.335 on ICOP's own. That is zone F: below the Danjon limit, not visible. The criterion rejects an observation that was made. The Ashdod record, by contrast, lands in zone C, where it belongs.

A zone-F verdict is therefore not a reliable negative, and this appendix must not use it as one. The distinction that matters is distance from the boundary. The evening of 10 April AD 31 computes to q = −1.08 — more than three times as far below the threshold as the Tennessee record — and that judgement stands. The evening of 12 March AD 31, at −0.313, sits essentially on the Tennessee value, and it does not. On that evening the zone letter settles nothing: a crescent at that value has been seen.

This is where the calibration confirms a judgement made earlier on other grounds. Chapter 7 does not rest on 12 March: it calls that evening a borderline case and adds that the borderline does not matter, since both outcomes lead thirty days on to the same 1 Nisan of 12 April, and since the thirty-day cap runs from an Adar II beginning on 13 or 14 March. The chain was built so as not to need the F, and the calibration shows why that was right. What the calibration removes is only the stronger claim — that 12 March can be excluded — which the argument never used. [CALCULATED]

Three records are extremes, not a validation: they calibrate the boundary, not the middle, and ICOP notes that the observers are experts who know where and when to look. The reliability of the Tennessee sighting itself has not been assessed here. Whoever would set that observation aside removes this objection with it — but that requires a reason, not a convenience.