Astronomical and calendrical tables

*This appendix makes the astronomical and calendrical statements of the book recomputable. Every table can be reproduced with freely available tools; the script that produces it is a small Python programme (Python 3 with the ephem library). Wherever a table rests on a documented source rather than on my own computation, that is stated explicitly.*

A.0 · Method, tools and calibration

Tools. All astronomical positions were computed with PyEphem 4.2.1 (the same library the book names), running under Python 3. The lunar phases (conjunction/new moon) and the spring equinox are Ephem's own `next_new_moon`, `previous_new_moon` and `next_equinox`.

Calendar. All civil dates are given in the proleptic Julian calendar — the calendar actually in force in the first century. (Whoever converts these dates to the proleptic Gregorian subtracts two days for this century (25 April 31 Julian is 23 April Gregorian); that does not change the weekday.)

Weekday. The weekday is computed from the Julian day number (JDN mod 7) — a quantity that does not depend on which calendar one uses, and is therefore exact and verifiable.

Calibration. The weekday machine was tested on five dates whose weekday is fixed; all five come out exactly:

AD 33 appears here twice, and deliberately: under the two calendar rules its 14 Nisan falls on two different dates (3 April or 2 May). Both stand here only as weekday checks on the machine — which of the two is the correct calendar rule is decided by Chapter 8, not by this table.

As long as these five are right, every other weekday in this appendix is right. [CALCULATED]

A.1 · The Passover lunation against the equinox — the backbone of the two-horse race (ch. 7)

For each year AD 26–37, the spring equinox and the new moon (conjunction) of the lunation falling closest to the equinox — the "March moon" from which Nisan would begin. The last column is the distance of that conjunction from the equinox (negative = before the equinox). [CALCULATED, ephem 4.2.1]

The three race years stand here in black and white. The distances are measured at the conjunction; the first visible crescent (1 Nisan) follows roughly a day and a half later, so that the crescent distances the book names are about two days smaller:

- AD 30 — conjunction virtually on the equinox; the March moon stands just before it, the crescent just after. Every observant spring rule leaves Nisan in March → 14 Nisan = Friday 7 April, at most Thursday 6 April under the generous crescent threshold — in no case a Wednesday. - AD 31 — conjunction eleven days before the equinox (crescent ~nine days before, deep in winter). Under a rule that requires the month after the equinox, Nisan shifts to April → 14 Nisan = Wednesday 25 April; under the strict rule that requires only the festival after the equinox, the March moon stays → Tuesday 27 March (ch. 8). Both are midweek; neither is a Friday. - AD 33 — conjunction a good three days before the equinox (crescent ~a day and a half before). This is the borderline case: here, and only here, the choice of rule decides whether Nisan stays in March (the consensus Friday 3 April) or shifts to April/May.

That is the measurement behind ch. 7 and ch. 15. AD 31 lies far from the boundary (eleven days): the choice of rule does move its fourteenth — from 25 April to 27 March — but it does not move it to a Friday. AD 30 lies instead right beside the equinox (−0.2 d), but its robustness is of another kind — under every observant rule Nisan stays in March and 14 Nisan is not a Wednesday but a Friday; the threshold evening itself is a borderline Thursday/Friday (Appendix C.2, q = −0.15). AD 33 lies on the boundary, where the choice of rule decides the weekday.

A.1b · The two descriptive data of Josephus — and 25 April on the edge of both

Josephus gives two places in which Passover is fixed against something outside the Jewish calendar. Neither is a prescription — there is no "one must" with either, and ch. 8 weighs them as descriptions. But both put 25 April AD 31 at the outer limit, and this appendix counts that itself rather than leaving it to a reviewer.

The first is the sun in Aries (Ant. 3.248). On the tropical reckoning in which the vernal equinox is Aries 0° — "the beginning of Aries was (and still is) the traditional astronomical designation of the vernal equinox" (Stern, Calendar and Community, p. 55 n. 215) — the apparent solar longitude on 25 April AD 31 comes out at ≈ 32.1° at noon and ≈ 32.4° by the evening of the slaughter, that is 2.1–2.4° of Taurus: the sun has left Aries, by roughly two days. The boundary itself is datable — the sun crosses 30° of longitude on 23 April AD 31 — so the claim can be checked rather than taken. (The span between the two figures is the hour of day plus the correction for ΔT, not a disagreement between computations.) [CALCULATED]

The second is Pharmuthi. Ant. 2.311 equates the Jewish Nisan with the Egyptian month Pharmuthi. In the Alexandrian calendar of Josephus' day Pharmuthi 1 = 27 March Julian and, the month having thirty days, Pharmuthi 30 = 25 April. 25 April AD 31 is therefore Pharmuthi 30 — the last day of the month with which Josephus identifies Nisan. Stern uses the same equation in the opposite direction, as an indication that Passover generally fell after the equinox (p. 55, with n. 216 for the 27 March), and he adds the limitation that belongs with it: Nisan is lunar and Pharmuthi solar, "it would be unreasonable to expect an Egyptian month to correspond exactly to Jewish calendrical parameters" (n. 218). [DOCUMENTED — Ant. 2.311 and Stern p. 55; CALCULATED — the Alexandrian month arithmetic]

Both together: under both of Josephus' descriptive data 25 April lies at the extreme edge — just past the last degrees of Aries, and on the last day of Pharmuthi. That is not a falsifier, for a description prescribes nothing, and Stern himself states of the Aries passage that it "does not necessarily imply a formal or well-defined calendrical rule" (p. 61). It is a cost, and it is entered here on this book's own side of the ledger: this book demands a base rate of the consensus, and it may not write off a double edge-position of its own.

A.2 · 14 Nisan and its weekday, AD 26–36 — the weekday base rate (ch. 7, ch. 16)

Two ways of reckoning — and where they part. This book computes 14 Nisan in two ways, and it pays not to confuse them. One is the strict equinox rule (the Babylonian one, table A.3 below): the March moon counts only when 1 Nisanu falls after the spring equinox, otherwise Nisan shifts to April. The other is the observational rule with the barley threshold (this table A.2): Nisan follows the visible crescent, and a month is intercalated as long as the barley is not yet aviv. For almost every year the two give the same weekday. They part on exactly one race year: AD 33 depends on the choice of rule — Friday 3 April under the barley threshold (A.2), Saturday 2 May under the strict equinox rule (A.3) — and AD 31 does not: under both rules it falls on Wednesday 25 April, for its March moon lies nine days from the boundary. *(To avoid confusion with A.1: this concerns the two ways of reckoning of this book. There exists besides these a third, ancient rule — the one Anatolius reports from the ancients, which requires only the festival after the equinox — and under it the March moon of AD 31 does stay and 14 Nisan falls on Tuesday 27 March. That too is not a Friday; ch. 8 measures it out. Two things about that third rule belong here, because this table would otherwise pass them over: it is an attribution by Anatolius and not a citation from Philo or Josephus, and in Eusebius its fourteenth-day form stands in the derivational layer, where the day of Passover is introduced as a given premise, while every prescriptive formulation is equinox-relative (HE 7.32.17–19). It is computed here because later readers made a rule of it, not because the ancients are known to have laid it down.)* Where the ch. 7 table puts AD 33 on 2 May and this table on 3 April, that is therefore no contradiction but the difference between the two rules; which rule is the correct one is weighed by ch. 8 / ch. 15.

The date of 14 Nisan under the observational (crescent) calendar with the barley threshold of t. Sanhedrin 2:2 (Machon Mamre edition; Galilee binding, ≈ 5 April; intercalation whenever the March route falls too early). The weekday is again computed with the calibrated machine. This table reproduces the calibrated crescent/barley computation for AD 26–36. [CALCULATED]

Weekday distribution across the eleven years: Sun 1 · Mon 3 · Tue 1 · Wed 2 · Thu 2 · Fri 2 · Sat 0 — nearly flat, as expected.

The third column — Lanser's shifted series

For completeness, alongside the two ways of reckoning of this book a third outcome belongs in view, for it is published and it is internally coherent. Richard Lanser's "Exodus Hebrew Calendar" runs the Jewish intercalation series one cycle-year before the Babylonian; his Nisan 1 is each time the thirteenth lunation of the preceding Babylonian leap year. In the window under study he therefore diverges in precisely the years after a leap year:

| year | P&D / Parker (Jerusalem crescent) | Lanser, Exodus Hebrew Calendar |

|---|---|---|

| AD 30 | Nisan 1 = 25 Mar → 14 Nisan = Friday 7 April | same |

| AD 31 | Nisan 1 = 12 April → 14 Nisan = Wednesday 25 April | Nisan 1 = 14 Mar → 14 Nisan = Tuesday 27 March |

| AD 32 | Nisan 1 = 1 April → 14 Nisan = Monday 14 April | same |

| AD 33 | Nisan 1 = 19 April → 14 Nisan = Saturday 2 May | Nisan 1 = 21 Mar → 14 Nisan = Friday 3 April |

| AD 34 | Nisan 1 = 9 April → 14 Nisan = Thursday 22 April | same |

His window as a whole lies about eleven days earlier than that of P&D (Nisan 1 from 13 March to 11 April, against 25 March to 22 April). The dispute lies not in the astronomy — both are genuine first crescents, and his figures were recomputed and found correct — but in the barley: on his placement the omer of 16 Nisan in AD 31 falls on 29 March. That makes his placement highly improbable; it does not refute it, and this appendix says no more than ch. 8 and ch. 15 carry. See ch. 8 and the excursus in ch. 15. [CALCULATED — from his own chart and spreadsheet, laid alongside the machine-readable P&D tables]

The base rate. In this window 14 Nisan falls twice on a Wednesday (AD 28 and AD 31): 2/11 ≈ 0.18, against a chance level of 1/7 ≈ 0.14. *That 2/11 holds within the rule of this table, which places AD 34 in April (→ Thursday). Under the strict "festival-after-equinox" rule AD 34's March option of 24 March lives and also gives a Wednesday; the base rate then becomes 3/11. Chapter 7 explicitly counts that edge case and accounts for why AD 34 still drops out — not on the equinox but on the ministry window (ch. 9). See ch. 7, box "The base rate". The measure should lie equally on both years, and the conservative value is 3/11.* A Wednesday "by itself" is thus chance-level in any case; the strength of the case lies not in the Wednesday but in the crossing of that rare weekday with the measured day-axis (ch. 3). And note AD 28: also a Wednesday, but excluded by the wider chronology (a baptism around AD 24 falls before the fifteenth year of Tiberius; ch. 9). So the barley neither brings the Wednesday nor takes it away — it leaves \{AD 28, AD 31\}, and from that pair Tiberius chooses AD 31.

A.3 · The Babylonian (Parker & Dubberstein) check — the core years (ch. 8, ch. 15)

The Babylonian calendar kept 1 Nisanu after the spring equinox (Stern, Calendar and Community, pp. 61–62, from Parker & Dubberstein). Under that rule — calibrated on Josephus's dated spring festival of AD 37 (called by Josephus "the festival of the fathers", ἑορτῆς πατρίου; almost certainly Passover, but the identification is CONSTRUED) = 19 April — the core years fall as follows: [CALCULATED on the documented rule; month names in P&D "completely secure"]

Two things stand out. First: the rule passes the check at the only dated anchor (AD 37 = April, 14 Nisan 19 April). (That is a necessary check and not a distinguishing argument: the lenient rule too, and Lanser's shifted series too, arrive at that same 19 April for AD 37. See ch. 8 for the limits.) Second: under this rule AD 33 yields not 3 April but 2 May (Saturday) — the famous "Good Friday" of the consensus survives only under the lenient rule, not under the Babylonian. That is the measurement behind Chapter 15.

A.4 · The two Passover lunar eclipses (ch. 8, ch. 11, ch. 15)

From the NASA/Espenak–Meeus five-millennium canon (LE), crossed with my own computation of the moon's height for Jerusalem. [DOCUMENTED, canon; CALCULATED, height]

Two caveats, both made in the book. (1) A partial eclipse does not turn the moon blood-red; only the eclipsed part changes colour. This row stands here as image, not as proof — it counts on neither side (ch. 7, ch. 15). (2) That AD 31 had a well-visible Passover eclipse and AD 30 none takes from the consensus its sole claim to "the eclipse", but yields no independent weight in favour of AD 31: a Passover full moon coincides with an umbral lunar eclipse in roughly 12% of years [CALCULATED].

A.4b · The candidate eclipses for Herod's death (ch. 10)

From the NASA/Espenak–Meeus five-millennium canon (LE), crossed with my own computation of the moon's height for Jerusalem. [DOCUMENTED, canon; CALCULATED, height]

Three caveats. (1) The canon dates the maximum in dynamical time. "10 January 1 BC" is the local date in Jerusalem; in universal time the same eclipse still stands on 9 January. Whoever finds the two dates side by side sees no contradiction but two time reckonings. (2) The Passover dates. They are computed with this book's crescent rule: 1 Nisan is the evening on which the crescent first becomes visible after conjunction, in the month whose first crescent falls on or after the spring equinox. That gives 14 Nisan on 20 April 5 BC, 10 April 4 BC, 7 April 1 BC and 26 April AD 1. The literature commonly reckons to 11 April 4 BC and therefore speaks of 29 days where this rule gives 28; that single day lies within the margin of any visibility rule. The distances are full days; counted inclusively, as Josephus counts, each figure is one higher. (3) What this table does not do. It points to no year. It shows which eclipses were visible, and how conspicuous: 5 BC carries two total eclipses above Jerusalem, both more deeply eclipsed than the partial one of 13 March 4 BC, and the one of 23 March stands at the same distance from the next Passover as that of 13 March 4 BC — twenty-eight days — but total instead of partial. The year rests in Chapter 10 on the two clamps and the three sons; the eclipse is a filter, not a pointer.

A.5 · The passion week of AD 31, day by day (ch. 11)

Given 1 Nisan = 12 April (Julian), hence 14 Nisan = Wednesday 25 April. [CALCULATED]

The rows are indexed on the Jewish day; the Julian date runs from midnight. The evening of 14 Nisan thus falls on Tuesday evening, 24 April Julian.

The three nights of the sign (Matthew 12:40) thus fall between Wednesday evening and Saturday evening: Wednesday-to-Thursday, Thursday-to-Friday, Friday-to-Saturday — three full days, exactly as the sign requires.

A.6 · What this appendix does not deliver

- The crescent visibility itself is not modelled to the minute. The first-visibility dates (1 Nisan) rest on the calibrated crescent/barley computation and on Parker & Dubberstein, not on a Schoch/Yallop criterion built up here from scratch. The ±1-day wobble touches the borderline years (AD 30 Thu/Fri) but not AD 31, which lies nine days from the boundary. - What remains undelivered is the printed page for every line. Parker & Dubberstein's table has been worked through in full (28 September 2026), but in the machine-readable edition corrected against the printed SAOC 24, for 200 BC to AD 45, the end of the table in SAOC 24 (the 1956 edition runs on to AD 75). For AD 26–37 it gives Nisanu 1 on 8 April 26, 28 March 27, 15 April 28, 5 April 29, 25 March 30, 12 April 31, 1 April 32, 19 April 33, 9 April 34, 30 March 35, 17 April 36 and 6 April 37; every Babylonian month in table A.3 agrees with it. The earliest Nisanu 1 in those 245 years is 24 March (85 BC), the latest 23 April (150 BC). [CALCULATED from the table] Until then the Babylonian rule and its anchors (among them Nisanu 1 AD 37 = 6 April, P&D p. 46; 130 BC = 11 April, p. 42) had been applied via Stern's primary citations and calibrated on the AD 37 anchor. The literal P&D line for AD 31 has also been read off the printed table itself: the scan of Parker & Dubberstein p. 46 was seen with my own eyes, and the line "342 | 31 | 4/12" gives Nisanu 1 AD 31 = 12 April — exactly the 1 Nisan on which this book reckons. [DOCUMENTED, P&D p. 46] That is the Babylonian calendar, and the link from Babylon to Judea does not hold of itself: just as AD 33 does not founder on this table, the 25 April of 31 does not rest on it either (ch. 8, ch. 15). - The solar longitude of A.1b carries a margin of a few tenths of a degree. Between the noon value (≈ 32.1°) and the evening value (≈ 32.4°) lie the hour of the day and the correction for ΔT (some three hours in AD 31). What carries is the direction, not the second decimal: the sun has left Aries. And the competing ancient conventions do not rescue the point — a zodiac that puts the equinox at Aries 8° or 10° places 25 April further outside Aries, not inside it. - The barley threshold (≈ 5 April) is a reading, not a measurement. With an earlier threshold (Jericho/omer alone, ≈ 31 March) AD 31 is even unique; with a later one AD 28 joins it. The discriminating power of the barley hangs on that threshold and so cannot be pinned down — which is why the book carries it as a necessary condition, not as a filter.

Reproduction: a small Python script (Python 3 with the ephem library, version 4.2.1) prints the anchor check and all the tables above.