Doc #10 — Nautical Almanac: Precomputed Astronomical Data for Navigation

COMPUTED DATA: SOLAR ALMANAC

View the computed Solar Almanac Data for 2026–2028 → — Daily sun GHA, declination, and equation of time, generated from the Meeus solar position algorithm. This sample demonstrates that almanac computation is a routine task for any computer with Python and standard astronomical libraries — the dataset can be extended to 100 years with hourly values, moon, planets, and all 57 navigational stars. Ideally precomputed and printed before any disruption.

View the Mathematical Reference Tables → — Logarithms, trigonometric functions, square roots, and conversion factors needed for manual navigation calculations.

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RECOMMENDED ACTIONS

Immediate (Days 1–3) — Phase 1

1. Download and archive the DE440 ephemeris file from the JPL servers. Store on at least three independent physical media (USB drives, hard drives). Also download DE421 as backup. Also download Skyfield source code, PyEphem source code, and all Python dependencies.
2. Locate and secure all copies of the USNO/UKHO Nautical Almanac currently held in NZ — at ports, on vessels, in naval facilities, in maritime training institutions, in libraries. These are validation references and must be preserved.
3. Locate and secure copies of Meeus Astronomical Algorithms and the USNO Explanatory Supplement. These are the fallback references for manual computation. Check university libraries, private astronomical society collections, and maritime training libraries.
4. Identify personnel with Python programming skills and basic astronomical knowledge. This is a common combination — any competent programmer can implement the computation with Meeus or Skyfield documentation. Astronomers at NZ universities (particularly the University of Canterbury’s Department of Physics and Astronomy, and the University of Auckland) are obvious candidates.¹

Short-term (Days 3–14) — Phase 1

5. Write and validate the computation script. Cross-check against published almanac data (Section 6). A competent programmer familiar with astronomical algorithms can complete this in an estimated 3–7 days of focused work, though debugging edge cases (e.g., polar twilight, lunar interpolation near perigee) may extend this.
6. Compute the complete 100-year dataset. Store raw data in multiple formats on multiple media.
7. Format for printing. Generate print-ready output for all 100 years.

Medium-term (Days 14–60) — Phase 1

8. Print the almanac. Begin with archive copies and major port copies. Coordinate with overall printing schedule (Doc #5). Printing 100–150 sets of 5,000 pages each will require sustained printer operation — schedule accordingly.
9. Distribute to ports and maritime training institutions. Include instructions for use (referencing Doc #138).
10. Begin celestial navigation training for all offshore navigators, using the new almanac with GPS as a cross-check to build confidence and skill (Doc #138).

Long-term (Months 3–12) — Phase 1–2

11. Distribute to vessels as they are fitted out or assigned for offshore voyages.
12. Conduct validation exercises — navigators take celestial fixes using the almanac and compare with GPS (while GPS remains usable) to verify accuracy and build skill.
13. Establish manual computation training as a fallback capability, using the reference texts (Section 9.4).
14. Plan for almanac extension beyond 100 years — either through manual computation, through computers rebuilt under Doc #14, or through international collaboration if trade relationships are restored.

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1. PURPOSE AND CONTEXT

1.1 Why a nautical almanac matters

Celestial navigation — determining a vessel’s position by observing the sun, moon, stars, and planets with a sextant — requires three things: a sextant (the instrument), sight reduction tables (the mathematical tables for converting observations to position — Doc #11), and a nautical almanac (the astronomical data telling the navigator where each celestial body is at any given moment). Without the almanac, the sextant is useless and the sight reduction tables have nothing to operate on.²

New Zealand currently relies on the annual Nautical Almanac published jointly by the United States Naval Observatory (USNO) and the United Kingdom Hydrographic Office (UKHO).³ This publication has been produced continuously since 1767 (initially by the British Astronomer Royal Nevil Maskelyne) and is the global standard for marine celestial navigation.⁴ NZ imports copies each year.

If global trade is permanently severed, no further copies arrive. Existing copies cover only the current year and perhaps a year or two ahead. Celestial navigation — which NZ’s maritime trade will depend on as GPS satellites degrade (Doc #29) — becomes impossible within 1–2 years unless NZ computes and prints its own almanac.

1.2 Why 100 years

The computation cost of extending the almanac is essentially zero once the software is written and validated. Computing one year of data takes seconds on a modern computer; computing 100 years takes minutes. The marginal cost of each additional year is only the paper and toner to print it — approximately 50 pages per year.

One hundred years of almanac data (2026–2125) costs roughly 5,000 printed pages. This is a small investment against the possibility that NZ will not have the computational infrastructure to regenerate this data later. It provides navigational independence for a century — long enough for NZ to rebuild the industrial base needed to produce computers capable of recomputation (Doc #135), or to establish manual computation methods (Doc #138).

Assumption: This 100-year horizon assumes that NZ will either rebuild computational capability or develop alternative navigational methods within a century. If neither occurs, the almanac’s coverage would need to be extended by manual computation — a laborious but documented process (Section 9).

1.3 Urgency: why this prints first

The almanac must be among the very first documents printed (alongside Docs #5, #11, #29, and other Phase 1 materials). The reasoning is simple:

• Printing infrastructure is a wasting asset. Laser printers require toner cartridges, fusers, and drums that NZ cannot manufacture. Every page printed from a finite supply of consumables must be high-priority. The almanac qualifies because it provides a century of navigational capability from roughly 5,000 pages — an exceptional return on printing investment.
• Computers are needed for computation. While NZ’s existing computers work today, they will not work indefinitely. Hardware failures, power supply degradation, and the inability to replace components mean that computational capability is time-limited. The almanac must be computed and validated while computers are functional.
• The data is extremely labour-intensive to recreate without computers. Manual computation of a single year’s almanac data requires hundreds of hours of skilled human labour (Section 9). Computing 100 years manually would require tens of thousands of hours. Doing it by computer takes minutes.

The urgency calculus is clear: compute the data now, print it now, before either capability is lost.

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2. CONTENTS OF THE NAUTICAL ALMANAC

The standard nautical almanac, as published by USNO/UKHO, contains the following data, all of which the NZ almanac must replicate.⁵

2.1 Daily pages (the core data)

Each day of the year occupies one page (or a shared page for three days in the standard format). For each day, the tables provide:

Sun:

• GHA at each whole hour (00:00–23:00 UT), tabulated to 0.1 arcminute
• Declination at each whole hour, tabulated to 0.1 arcminute
• Semi-diameter (approximately 15.8–16.3 arcminutes, varies slowly over the year)
• Equation of time (difference between apparent and mean solar time)
• Meridian passage time (time the sun crosses the Greenwich meridian)

Moon:

• GHA at each whole hour, tabulated to 0.1 arcminute
• Declination at each whole hour, tabulated to 0.1 arcminute
• Horizontal parallax (HP) — typically 54–61 arcminutes, needed because the moon is close enough that parallax is significant for navigation⁶
• Semi-diameter (approximately 14.7–16.8 arcminutes, varies with distance)
• Meridian passage time
• v and d correction factors for interpolation between hourly values

Navigating planets (Venus, Mars, Jupiter, Saturn):

• GHA at each whole hour, tabulated to 0.1 arcminute
• Declination at each whole hour, tabulated to 0.1 arcminute
• v and d correction factors
• Magnitude (brightness, needed for identification)
• Meridian passage time

Aries (the vernal equinox reference point):

• GHA of Aries at each whole hour — this is the reference from which star positions are measured⁷

2.2 Star data

The 57 navigational stars used in celestial navigation are tabulated with:

• Sidereal Hour Angle (SHA) — the angular distance west of Aries, essentially fixed for practical purposes (changes slowly due to precession)
• Declination — also essentially fixed on the timescale of a year
• Magnitude — for identification

The 57 stars are a subset chosen for brightness (mostly magnitude 2.0 or brighter), broad distribution across the sky, and unambiguous identification. They include well-known stars such as Sirius, Canopus, Rigel, Arcturus, Vega, and the Southern Cross stars Acrux and Gacrux.⁸

Fact: The SHA and declination of stars change slowly due to precession of the equinoxes (approximately 50.3 arcseconds per year) and proper motion (unique to each star, generally small). For a 100-year almanac, these changes are significant and must be computed for each epoch. A star’s SHA may shift by approximately 1.4 degrees over 100 years due to precession alone.⁹

2.3 Rise, set, and twilight tables

For selected latitudes (typically every 2 or 5 degrees of latitude):

• Sunrise and sunset times — tabulated for every third day or so, interpolated for intermediate dates and latitudes
• Civil twilight — sun 6 degrees below horizon (enough light for normal outdoor activity without artificial light)
• Nautical twilight — sun 12 degrees below horizon (horizon still visible for sextant observations; the optimal observation window)
• Astronomical twilight — sun 18 degrees below horizon (sky fully dark)
• Moonrise and moonset times — tabulated similarly

These tables are essential for planning observation times. Celestial navigation observations are most accurate during nautical twilight, when both the horizon and stars/planets are visible simultaneously.¹⁰

2.4 Interpolation tables (“Increments and Corrections”)

The hourly GHA values must be interpolated to the exact minute and second of an observation. The almanac includes:

• Increments tables: GHA correction for minutes and seconds past the whole hour, for sun/planets and for moon (which moves faster and requires separate interpolation)
• v and d correction tables: Small adjustments for the fact that GHA and declination change at slightly varying rates through the day

Key point: The interpolation tables are the same for every year. They need to be printed only once and can be bound into every annual volume, or printed as a separate permanent reference sheet.

2.5 Additional data

• Polaris (Pole Star) tables: For determining latitude in the Northern Hemisphere (less relevant for NZ but included for completeness and for vessels sailing north)
• Dip correction table: Correction for observer’s height above sea level
• Refraction correction table: Correction for atmospheric bending of light
• Planet identification diagram: Shows which planets are visible and where to look at different times of year
• Star charts: For identifying navigational stars (Section 7)

2.6 Page count estimate

Based on the format of the standard USNO/UKHO Nautical Almanac:¹¹

Section	Pages per year	Notes
Daily pages (sun, moon, planets, Aries)	30–33	Three days per double page, ~122 double pages, printed both sides
Star tables	2	Changes slowly; could be updated every 5–10 years
Rise/set/twilight tables	6–8	For relevant latitudes
Interpolation tables	4–6	Same every year (print once)
Additional tables and instructions	4–6	Mostly constant
Total per year	~46–55

Estimate: Approximately 50 pages per year, yielding roughly 5,000 pages for 100 years. This is consistent with the catalog estimate. The interpolation tables and instructions (approximately 10 pages) need only be printed once, reducing the total slightly.

Fact: The published USNO/UKHO Nautical Almanac runs to approximately 320 pages, but this includes extensive explanatory text, examples, commercial editions’ supplementary material, and data for all latitudes. The NZ edition can be substantially shorter by including only the latitudes relevant to NZ’s likely sailing range (roughly 60°S to 30°N, covering routes from Antarctica to the tropics) and by minimizing repeated explanatory text.

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3. THE MATHEMATICS: ORBITAL MECHANICS FOR ALMANAC COMPUTATION

3.1 What must be computed

The fundamental computation is: given a date and time (in Universal Time), calculate the geocentric position (right ascension and declination, or equivalently GHA and declination) of the sun, moon, each planet, and the vernal equinox (Aries). From these, all other almanac quantities (rise/set times, twilight, semi-diameters, parallax) can be derived.

3.2 Solar position

The sun’s apparent position is computed from the Earth’s orbital elements. The dominant terms are:

• Mean longitude and mean anomaly: Linear functions of time, defining the Earth’s average position in its orbit
• Equation of center: Correction for the eccentricity of Earth’s orbit (maximum approximately 1.9 degrees), computed from the mean anomaly using Kepler’s equation
• Nutation and aberration: Small corrections (up to ~18 arcseconds for nutation, ~20 arcseconds for aberration) arising from the wobble of Earth’s axis and the finite speed of light

The resulting accuracy using algorithms from Meeus (1991) or the USNO/UKHO computational methods is better than 1 arcsecond for the sun’s position — far more accurate than needed for marine navigation.¹²

Fact: Marine celestial navigation typically achieves position accuracy of 1–2 nautical miles under good conditions. This requires the almanac data to be accurate to about 1 arcminute (1 arcminute of GHA or declination corresponds to 1 nautical mile). Solar position algorithms accurate to 1 arcsecond are therefore roughly 60 times more accurate than needed.¹³

3.3 Lunar position

The moon’s motion is the most complex computation in the almanac. The moon is perturbed by the sun, by the Earth’s oblateness, and by the other planets. The principal lunar theory (currently the ELP/MPP02 series, derived from decades of work by Chapront and others) includes hundreds of periodic terms.¹⁴

The key perturbation terms include:

• Evection: ~1.27 degrees amplitude, period ~31.8 days
• Variation: ~0.66 degrees amplitude, period ~14.8 days (half a synodic month)
• Annual equation: ~0.19 degrees amplitude, period ~365.3 days
• Parallactic inequality: ~0.11 degrees amplitude

Using the full Chapront ELP series or the JPL Development Ephemeris (DE series), lunar positions can be computed to better than 1 arcsecond. However, simplified algorithms (as in Meeus Chapter 47, using the principal perturbation terms) achieve accuracy of approximately 10 arcseconds — still adequate for navigation.¹⁵

The moon is the critical accuracy test for the almanac. If the lunar computation is correct, everything else is trivially correct. Validation should focus on lunar positions (Section 6).

3.4 Planetary positions

The positions of Venus, Mars, Jupiter, and Saturn are computed from their orbital elements and perturbation theories. The main computational approaches:

• VSOP87 (Variations Seculaires des Orbites Planetaires): A semi-analytical theory developed by Bretagnon and Francou at the Bureau des Longitudes, providing planetary positions with sub-arcsecond accuracy over several millennia.¹⁶
• Keplerian elements with perturbation corrections: Simpler but less accurate; adequate for navigation
• JPL Development Ephemeris (DE series): Numerically integrated ephemeris, the most accurate available; used by Skyfield and other modern software¹⁷

For navigational purposes, planetary positions accurate to 1 arcminute are sufficient. All three approaches achieve this level of accuracy.

3.5 Stellar positions

The 57 navigational stars are effectively at infinite distance and do not orbit. Their apparent positions change due to:

• Precession: The slow wobble of Earth’s rotational axis, causing the celestial coordinate grid to shift. The dominant term is a westward motion of the vernal equinox of approximately 50.3 arcseconds per year (the “precession of the equinoxes”).¹⁸
• Nutation: A small periodic wobble superimposed on precession (maximum ~18 arcseconds, period ~18.6 years)
• Proper motion: The actual motion of the star through space, unique to each star. For most navigational stars, proper motion is very small (a few arcseconds per century or less). A notable exception is Arcturus, with proper motion of approximately 2.3 arcseconds per year.¹⁹
• Aberration: A small displacement (up to ~20 arcseconds) due to the finite speed of light and Earth’s orbital velocity. Annual aberration is periodic and well-modeled.²⁰

For a 100-year almanac, precession is the dominant effect on star coordinates and must be computed accurately. Proper motion is significant only for a few stars over this timeframe but should be included for all 57 for correctness.

3.6 Rise, set, and twilight computation

Sunrise, sunset, and twilight times are computed from:

• The sun’s declination on that date
• The observer’s latitude
• The geometric condition (sun’s center at the horizon for rise/set, adjusted for refraction of approximately 34 arcminutes and semi-diameter of approximately 16 arcminutes; sun at 6°, 12°, or 18° below horizon for twilight)

The computation is standard spherical trigonometry, well-documented in Meeus and Bowditch. For a given latitude and solar declination, the hour angle at which the sun reaches the specified altitude is:

cos(H) = (sin(a) - sin(lat) * sin(dec)) / (cos(lat) * cos(dec))

where H is the hour angle, a is the altitude criterion, lat is the observer’s latitude, and dec is the solar declination.²¹

Moonrise and moonset are computed similarly but are more complex because the moon moves significantly in right ascension and declination during the day (up to ~13 degrees in right ascension per day), requiring iterative computation.

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4. SOFTWARE FOR COMPUTATION

4.1 Recommended primary tool: Skyfield (Python)

Skyfield is an open-source Python library developed by Brandon Rhodes specifically for high-accuracy positional astronomy. It is the recommended primary tool for computing the NZ nautical almanac.²²

Key characteristics:

• Uses JPL Development Ephemeris data files (DE421 or DE440) for planetary and lunar positions — the same data source used by NASA for spacecraft navigation
• Computes positions of sun, moon, planets, and stars with sub-arcsecond accuracy
• Handles precession, nutation, aberration, and proper motion of stars correctly
• Computes rise, set, and twilight times
• Well-documented, actively maintained (as of 2025), pure Python (no compiled dependencies beyond NumPy)
• Licensed under MIT license (freely usable)

Practical note: Skyfield requires downloading ephemeris data files (DE421 is approximately 17 MB; DE440 is approximately 100 MB). These files should be downloaded and stored locally before any disruption of internet access. DE421 covers the years 1899–2053; DE440 covers approximately 1549–2650. DE440 is recommended for the 100-year almanac.²³

Critical action: Download and locally store the DE440 ephemeris file, Skyfield source code, and all dependencies immediately. Multiple copies on multiple storage media (USB drives, hard drives, optical discs).

4.2 Alternative: PyEphem (Python)

PyEphem is an older Python library for astronomical computation, wrapping the XEphem C library by Elwood Downey.²⁴ It computes planetary and stellar positions using VSOP87 and other analytical theories rather than JPL numerical ephemerides.

Key characteristics:

• Self-contained (does not require external ephemeris files)
• Slightly less accurate than Skyfield for lunar positions (typically 1–5 arcseconds vs. sub-arcsecond) — still adequate for navigation
• Mature and stable codebase
• Licensed under LGPL

PyEphem is a suitable backup if Skyfield or its ephemeris files are unavailable. The two libraries should be cross-validated against each other and against published almanac data.

4.3 Alternative: IAU SOFA Library (C/Fortran)

The Standards of Fundamental Astronomy (SOFA) library is maintained by the International Astronomical Union. It provides fundamental positional astronomy routines in C and Fortran.²⁵

Key characteristics:

• Maintained by the authoritative international body for astronomical standards
• Provides all necessary coordinate transformation routines (precession, nutation, Earth rotation, sidereal time)
• Does not include a complete ephemeris — must be paired with an ephemeris source (JPL DE series or analytical theory)
• Fortran version useful if NZ has Fortran-capable systems

4.4 Reference texts for algorithms

If software is unavailable or needs to be rewritten from scratch, the following texts provide complete algorithms:

• Jean Meeus, Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998: The standard reference for practical positional astronomy computation. Provides complete algorithms for solar, lunar, and planetary positions, rise/set times, precession, nutation, and all related quantities. Clear enough for a competent programmer to implement from scratch.²⁶
• USNO Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books, 2012: The definitive reference for the methods used in the official astronomical and nautical almanacs. More detailed and rigorous than Meeus.²⁷
• Peter Duffett-Smith, Practical Astronomy with your Calculator, Cambridge University Press: A simpler introduction, suitable for manual or basic calculator computation.²⁸

These texts should be in the Recovery Library’s reference collection. If they are not already physically present in NZ, they should be acquired or their key algorithms transcribed before supply chains are severed.

4.5 Computation time estimates

Task	Hardware	Estimated time
Compute 1 year of hourly sun/planet GHA and dec	Modern laptop (2020s)	< 1 second
Compute 1 year of hourly moon GHA, dec, HP	Modern laptop	< 2 seconds
Compute 100 years of all daily page data	Modern laptop	1–5 minutes
Compute 100 years of rise/set/twilight for relevant latitudes	Modern laptop	5–15 minutes
Format output for printing	Modern laptop	Minutes
Total computation for 100-year almanac	Modern laptop	< 30 minutes

Estimate: These times assume use of Skyfield or PyEphem with competent scripting. The actual limiting factor is not computation but formatting, proofreading, and printing.

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5. DATA FORMAT AND PRINT LAYOUT

5.1 Design principles

The printed almanac must be:

• Unambiguous: No entry should be open to misinterpretation. Column headers, units, and time references must be explicit.
• Readable in poor conditions: Mariners use almanacs on pitching vessels, in dim light, with wet hands. Font size should be no smaller than 8 point. High contrast (black on white) is essential.
• Compact: Every page printed costs irreplaceable toner and paper. Minimize wasted space without sacrificing readability.
• Consistent: Use the same layout conventions as the standard USNO/UKHO almanac where possible, because NZ navigators are trained on that format. Unnecessary innovation in layout will cause errors.

5.2 Recommended layout

Follow the standard three-day spread format used in the USNO/UKHO Nautical Almanac:²⁹

Left page (for a double-page spread):

• Three columns, one per day
• Each column contains: Aries GHA at each hour (00–23 UT), then Venus GHA and Dec, Mars GHA and Dec, Jupiter GHA and Dec, Saturn GHA and Dec
• Star SHA and Dec table at bottom (the 57 navigational stars, updated as needed for precession)

Right page:

• Three columns, one per day
• Each column contains: Sun GHA and Dec at each hour, Moon GHA and Dec at each hour, with v, d, and HP values
• Sunrise/sunset and twilight times for selected latitudes at bottom

Back of book (same every year):

• Increments and corrections tables (approximately 4–6 pages)
• Altitude correction tables (dip, refraction — approximately 2 pages)
• Star charts (Section 7)
• Instructions for use (1–2 pages, kept brief; detailed instruction in Doc #138)

5.3 Latitude selection for rise/set tables

The standard almanac tabulates for latitudes from N72° to S60° at intervals that vary from 2° to 10°. For the NZ edition, the following latitudes should be included at minimum:

Latitude	Relevance
60°S	Southern Ocean / sub-Antarctic islands (Campbell Island, Auckland Islands)
55°S	Southern Ocean routes
50°S	Southern route to South America
47°S	Southernmost NZ (Invercargill, Stewart Island/Rakiura)
45°S	Dunedin, southern South Island
42°S	Christchurch, central South Island
40°S	Cook Strait, Wellington, Nelson
38°S	Tasman Sea routes, central North Island coast
36°S	Auckland
34°S	Northern NZ (Whangarei, Bay of Islands)
30°S	Tasman route to Australia, Kermadec Islands
25°S	Tropical Pacific routes (Tonga)
20°S	Fiji, tropical Pacific
15°S	Samoa, Cook Islands
10°S	Equatorial Pacific
5°S, 0°, 5°N, 10°N	Equatorial and near-equatorial routes
15°N, 20°N, 25°N, 30°N	Northern Pacific / Hawaii route (less likely but possible)

This covers NZ’s likely maritime operating area while remaining compact. Additional latitudes can be interpolated.

5.4 Paper and binding

Paper: Standard 80 gsm office paper is adequate for printing but has limited durability. For the master archive copies, heavier paper (100–120 gsm, acid-free if available) is preferred. Acid-free paper resists yellowing and brittleness; conventional wood-pulp paper becomes fragile after 20–50 years depending on storage conditions.³⁰

Binding: Each annual volume (approximately 50 pages) should be saddle-stitched (stapled) or perfect-bound. For durability, consider:

• Spiral or comb binding (allows the book to lie flat — useful on a chart table)
• Laminated or heavy card covers
• Multiple copies: even if individual copies deteriorate, redundancy ensures the data survives

Distribution format options:

• Individual annual volumes: 100 separate booklets, one per year. Simple to produce, easy to store and distribute. A vessel carries only the current year’s volume (plus the interpolation tables).
• Decade volumes: 10 volumes covering 10 years each, approximately 500 pages per volume. More durable binding possible.
• Single 100-year volume: Approximately 5,000 pages — impractically large for a single book but possible as a reference archive.

Recommendation: Print individual annual volumes for distribution to vessels and ports, plus a complete set bound in decade volumes for the national archive. This maximizes both usability and durability.

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6. VALIDATION

6.1 Why validation matters

A navigational error caused by an almanac error can result in a vessel running aground, missing its destination by tens of miles, or being unable to fix its position at all. The almanac must be correct. Validation is essential.

6.2 Validation methods

Cross-check against published almanacs: The most reliable validation. Compare the computed NZ almanac against the published USNO/UKHO Nautical Almanac for years where published data is available (typically 2024–2027, depending on how many years’ editions are currently held in NZ). Every entry must match to within 0.1 arcminute for the sun and planets, and 0.2 arcminute for the moon.

Cross-check between software implementations: Compute the same data using both Skyfield and PyEphem (and SOFA if available). Discrepancies larger than 1 arcsecond for the sun, or 5 arcseconds for the moon, indicate a problem.

Cross-check against direct observation: Using a sextant and accurate timekeeping, observe the sun’s altitude at known times from a known position. Compare the observed altitude (corrected for refraction, dip, and semi-diameter) against the altitude predicted from the computed almanac data and the known position. Agreement to within 1–2 arcminutes confirms the almanac is correct. This is the ultimate validation — it tests the entire chain from computation through formatting.

Internal consistency checks:

• GHA of sun should increase by approximately 15° per hour (exactly 15.0° for mean sun; apparent sun varies slightly)
• GHA of moon should increase by approximately 14.5° per hour (varies from about 14.2° to 14.9°)³¹
• GHA of Aries should increase by approximately 15.041° per hour (the sidereal rate)³²
• Declination of sun should vary between approximately +23.44° and -23.44° over the year (the obliquity of the ecliptic)
• Moonrise/moonset times should shift later each day by approximately 30–70 minutes (average approximately 50 minutes; varies with lunar orbital geometry and observer latitude)
• Sunrise/sunset times should be consistent with NZ’s seasonal expectations (e.g., Auckland earliest summer sunrise approximately 05:55–06:00 NZDT, latest winter sunrise approximately 07:30–07:35 NZST)³³

6.3 Error handling

If errors are found after printing, an errata sheet must be prepared and distributed to all almanac holders. This is the historical practice — published nautical almanacs have always included errata, and navigators are trained to expect and apply them.³⁴

For the NZ edition, the errata risk is reduced by the fact that the computation is automated (eliminating transcription errors that plagued historical almanacs) and that multiple software implementations can be cross-validated. The main error risk is in the formatting and printing pipeline — a misaligned column, a truncated digit, or a page printed out of sequence.

Mitigation: Automate the formatting pipeline entirely. The script that computes the data should also generate the formatted print-ready output. Human review should focus on spot-checking the printed output against the raw computed data at random dates throughout the 100-year span.

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7. SOUTHERN HEMISPHERE STAR IDENTIFICATION

7.1 Navigational stars visible from NZ latitudes

NZ spans approximately 34°S to 47°S latitude.³⁵ From these latitudes, the navigator has access to most of the 57 navigational stars, but with important differences from Northern Hemisphere practice:

Always visible (circumpolar from NZ): Stars with declination south of approximately -43° to -56° (depending on observer latitude) never set and are always available:

• Acrux (Alpha Crucis) — Dec -63°, magnitude 0.8
• Gacrux (Gamma Crucis) — Dec -57°, magnitude 1.6
• Hadar (Beta Centauri) — Dec -60°, magnitude 0.6
• Rigil Kentaurus (Alpha Centauri) — Dec -61°, magnitude -0.0
• Miaplacidus (Beta Carinae) — Dec -70°, magnitude 1.7
• Canopus (Alpha Carinae) — Dec -53°, magnitude -0.7
• Achernar (Alpha Eridani) — Dec -57°, magnitude 0.5
• Peacock (Alpha Pavonis) — Dec -57°, magnitude 1.9

These are the Southern Hemisphere navigator’s primary stars — reliably available on any clear night from NZ waters.³⁶

Never visible from NZ: Stars with declination north of approximately +43° to +56° never rise above the horizon:

• Polaris (the North Star) — Dec +89°, never visible from NZ
• Alioth — Dec +56°, barely rises or never from southern NZ
• Dubhe — Dec +62°, never visible from NZ
• Kochab — Dec +74°, never visible

Fact: Polaris is the primary latitude-finding star in the Northern Hemisphere. From NZ, it is below the horizon. Southern Hemisphere navigators determine latitude from the Southern Cross and pointer stars (Acrux, Gacrux, Hadar, Rigil Kentaurus), from Sigma Octantis (the south celestial pole star, magnitude 5.4 — very faint, difficult to use), or from meridian transits of any star of known declination.³⁷

Seasonally available: The majority of the 57 stars are visible from NZ for part of the year. The navigator must know which stars are available in which season.

7.2 Star charts for NZ

The almanac should include star charts showing the sky as seen from NZ latitudes. At minimum:

• Four seasonal charts: Sky views for January/February (NZ summer), April/May (autumn), July/August (winter), October/November (spring), each showing the sky at approximately 21:00 local time from latitude 40°S
• Southern circumpolar chart: Showing the region around the south celestial pole, with the Southern Cross, Centaurus, Carina, and other permanently visible constellations
• Star identification key: For each of the 57 navigational stars visible from NZ, a brief description of how to find it (e.g., “Canopus — second brightest star in the sky, high in the southern sky in NZ summer; follow the keel of the ‘false cross’ pattern in Carina”)

These charts are the same for all years (precession shifts star positions by only about 1.4° over 100 years, which is negligible for visual identification). They should be printed once and included in every annual volume or distributed as a separate, durable reference card.

7.3 The Southern Cross method for finding south

The Southern Cross (Crux) is the most important constellation for Southern Hemisphere navigators. The method for finding the south celestial pole:

1. Identify Acrux and Gacrux (the long axis of the cross)
2. Extend the long axis of the cross approximately 4.5 times its length toward the south
3. Alternatively, draw a perpendicular bisector of the line joining Hadar and Rigil Kentaurus (the “pointer stars”); the south celestial pole is approximately where this bisector meets the extended cross axis

This method gives an approximate bearing to south, useful for orientation even without a sextant. It should be documented in the almanac’s introductory pages and in Doc #139 (Celestial Navigation).³⁸

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8. PRODUCTION WORKFLOW

8.1 Phase 1: Software preparation (Days 1–7)

1. Install and verify software. Ensure Skyfield (or PyEphem) is installed and functional on at least two independent computer systems. Download and locally cache all required ephemeris data files (DE440, star catalogs). Verify installation by computing known positions and checking against published data.

2. Write the computation script. A single Python script (estimated 500–1,500 lines) that:

   • Computes all daily page data (GHA, Dec, v, d, HP, semi-diameter) for sun, moon, four planets, and Aries for every hour of every day for 100 years
   • Computes SHA and Dec for all 57 navigational stars for each year (accounting for precession and proper motion)
   • Computes sunrise/sunset, twilight, and moonrise/moonset for all selected latitudes for each day
   • Outputs the data in a print-ready format (PDF or formatted text suitable for direct printing)

3. Validate the script against published almanac data for at least two known years (Section 6).

8.2 Phase 2: Computation (Day 7–8)

Run the validated script to generate the complete 100-year dataset. This takes minutes of computer time but should be followed by:

• Automated consistency checks (Section 6.2)
• Spot-check of randomly selected dates across the full range
• Cross-validation against a second software implementation if available

Store the raw computed data in multiple formats (CSV, JSON, formatted PDF) on multiple storage media.

8.3 Phase 3: Formatting and review (Days 8–14)

Format the computed data into print-ready layouts. Review a sample of formatted pages (at minimum: the first page of each decade, plus 10 randomly selected pages) against the raw data to verify that the formatting pipeline has not introduced errors.

8.4 Phase 4: Printing (Days 14–30, or as printing capacity allows)

Print the required number of copies. Minimum print run:

Copy purpose	Quantity	Notes
National Archive	3 sets	Stored in geographically separated locations (e.g., Wellington, Christchurch, Auckland)
Major ports	12–15 sets	One per significant NZ port (Auckland, Tauranga, Wellington, Lyttelton, Dunedin, etc.)
Maritime training institutions	3–5 sets	For navigator training
Offshore-capable vessels	50–100 sets	Estimate of NZ vessels expected to navigate offshore over coming decades
Reserve stock	20–30 sets	For replacement and future vessels
Total	~90–150 sets

At 5,000 pages per set, the total printing requirement is approximately 450,000–750,000 pages. This is significant but feasible — it represents roughly the output of one laser printer running full-time for 2–5 weeks, depending on printer speed and consumable availability.³⁹

This printing load must be scheduled alongside other Phase 1 printing priorities (Doc #5). The almanac is high priority but not the only critical document. Coordination with the overall printing schedule is essential.

8.5 Phase 5: Distribution and training (Ongoing)

Distribute completed volumes to ports, vessels, and training institutions. Ensure that navigators know how to use the NZ edition (which should be identical in format to the USNO/UKHO edition they are trained on). Conduct spot-check validation exercises where navigators take sextant observations and verify their position fixes against the new almanac.

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9. MANUAL COMPUTATION FALLBACK

9.1 Why this section exists

If computers become unavailable before the 100-year almanac is fully computed and printed, or if the almanac must be extended beyond its 100-year coverage, NZ will need to compute almanac data manually. This is laborious but well-documented — it is exactly how nautical almanacs were produced for two centuries before electronic computers.⁴⁰

9.2 Historical precedent

The British Nautical Almanac Office employed human computers from 1767 to the 1950s to compute the Nautical Almanac by hand. The work was organized as follows:⁴¹

• Computers (the job title) performed the arithmetic: evaluating trigonometric series, solving Kepler’s equation iteratively, computing corrections term by term
• Comparers checked the computers’ work by independent recalculation
• Work was distributed among multiple computers, each responsible for a portion of the tables
• Standard computation forms were used to reduce errors

The labour requirement was substantial. A single skilled human computer, working full-time, could compute approximately one month of solar ephemeris in 2–3 days, but one month of lunar ephemeris in 1–2 weeks (due to the much larger number of perturbation terms). A complete year’s almanac required roughly 6–12 person-months of labour, including checking — the wide range reflecting the precision required and the number of celestial bodies tabulated.⁴²

9.3 Simplified manual methods

If full-precision manual computation is not feasible, simplified methods can produce adequate navigational accuracy. However, they involve a significant performance gap relative to computer-generated data: position fix accuracy degrades from 1–3 nautical miles (with full-precision almanac data) to approximately 3–8 nautical miles (with simplified manual data), primarily due to larger lunar position errors and coarser interpolation:

• Solar position: The sun’s GHA and declination can be approximated to within 1–2 arcminutes using only the equation of center (the dominant correction for orbital eccentricity) plus mean longitude. This requires computing only a handful of trigonometric terms. Meeus Chapter 25 provides the method.⁴³
• Stellar positions: Star positions change slowly due to precession. Given the 57 stars’ positions at a known epoch, positions at a future epoch can be computed using the precession formulas (which involve only basic trigonometry) and the known proper motions. This is manageable manual work — one skilled person could update all 57 stars for a new year in a few hours.
• Lunar position: The most difficult to compute manually. Simplified series with 20–30 terms (rather than the hundreds used in full theories) can achieve accuracy of approximately 2–5 arcminutes — adequate for navigation but not ideal. The navigator accepts somewhat larger position uncertainties when using simplified lunar data.
• Planetary positions: Simplified perturbation tables (as in Meeus or Duffett-Smith) allow planetary positions to be computed to within a few arcminutes with moderate effort.

9.4 Training for manual computation

If manual computation becomes necessary, the following skills must be preserved:

• Trigonometry (sine, cosine, tangent and their inverses) — both conceptual understanding and the ability to evaluate them from tables (Doc #29 includes trigonometric tables)
• Iterative solution methods (particularly for Kepler’s equation)
• Series evaluation (summing periodic terms)
• Awareness of significant figures and error propagation
• The use of pre-computed auxiliary tables (tables of common sub-expressions) to reduce repetitive work

Doc #139 (Celestial Navigation) should include training material for manual almanac computation as a fallback capability.

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10. COMPLEMENTARY DOCUMENTS AND CROSS-REFERENCES

The nautical almanac does not function in isolation. The complete celestial navigation system requires:

Document	Content	Relationship to almanac
Doc #10 (this document)	Nautical Almanac — astronomical data	Provides the “where are the celestial bodies” data
Doc #11	Sight Reduction Tables	Provides the mathematical tables for converting sextant observations + almanac data into position lines. Without Doc #11, the almanac is usable only with manual spherical trigonometry.
Doc #135	Computer Construction	Long-term path to rebuilding computational capability for regenerating almanac data beyond the 100-year coverage
Doc #29	Mathematical Tables	Trigonometric, logarithmic, and other tables needed for sight reduction if Doc #11 is unavailable, and for manual almanac computation (Section 9)
Doc #138	Sailing Vessel Design	The vessels that will use this almanac for navigation
Doc #138	Celestial Navigation	The complete guide to using the sextant, almanac, and sight reduction tables to determine position. The operational manual for the data this document specifies.
Doc #5	Printing Strategy	The printing schedule and resource allocation that determines how many copies of the almanac can be produced

All of these documents should be cross-referenced in each other’s text. A navigator receiving a copy of the almanac should know where to find the sight reduction tables and the navigation guide.

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11. GPS DEGRADATION AND THE TRANSITION TO CELESTIAL NAVIGATION

11.1 GPS satellite constellation status

The Global Positioning System (GPS) constellation consists of approximately 31 operational satellites as of 2025, with a designed operational life of approximately 10–15 years per satellite.⁴⁴ Without ground station uploads (which require a functioning US Air Force Space Command), the satellites’ onboard clocks and orbital parameters drift:

• Clock drift: Without corrections, individual satellite clocks drift by approximately 1–3 nanoseconds per day. After one month without uploads, position errors grow to approximately 5–15 metres. After 6 months, errors grow to 100+ metres. After 1–2 years, individual satellites become unreliable for navigation.⁴⁵
• Orbital parameter drift: Without updated ephemeris data, predicted satellite positions diverge from actual positions, contributing further errors.
• Satellite failures: Without replacement launches, the constellation degrades as individual satellites fail. The constellation may provide degraded but usable coverage for 3–8 years after the last upload, but will progressively lose accuracy and availability.⁴⁶

Estimate: GPS will provide useful (if degraded) navigation for approximately 2–5 years after a global disruption, with progressively increasing position errors. By years 5–10, it will be largely unusable. Celestial navigation must be the primary offshore navigation method from approximately year 3 onward — and navigators should transition earlier to build proficiency while GPS is still available as a cross-check.

11.2 GLONASS, Galileo, BeiDou

Other satellite navigation systems (Russian GLONASS, European Galileo, Chinese BeiDou) face the same degradation timeline. Their availability to NZ navigators post-disruption depends on whether their respective ground control segments survive, which is uncertain. NZ should not plan on any satellite navigation system being available beyond approximately 5 years.

11.3 The almanac as insurance

The 100-year almanac is insurance against a foreseeable transition: from satellite navigation (accurate, effortless, but dependent on foreign infrastructure) to celestial navigation (less precise, skill-intensive, but fully self-contained). NZ’s maritime trade — which will rely on sail (Doc #138) — requires this transition to succeed. The almanac is a critical component.

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12. ACCURACY BUDGET

How accurate is the complete celestial navigation system, and what does the almanac contribute?

Error source	Typical magnitude	Notes
Almanac error (sun, stars)	< 0.1 arcminute	Effectively zero contribution with computed almanac
Almanac error (moon)	0.1–0.5 arcminute	Small; dominated by other errors
Sextant instrument error	0.3–1.0 arcminute	Depends on instrument quality and maintenance
Sextant observation error	0.5–2.0 arcminute	Depends on observer skill, sea state, visibility
Time error	1 arcminute per 4 seconds of clock error	4 seconds of time error = 1 nm of longitude error; accurate timekeeping is critical (Doc #29)
Refraction anomalies	0.5–2.0 arcminute	Atmospheric conditions deviate from standard refraction model
Sight reduction rounding	0.1–0.5 arcminute	Depends on table precision (Doc #11)
Plotting/charting error	0.5–1.0 nm	Chart accuracy and plotting technique
Combined system accuracy	1–3 nautical miles	Under good conditions with a skilled navigator

Key insight: The almanac’s contribution to total system error is negligible compared to observation and timing errors. An almanac accurate to 0.5 arcminutes is more than adequate. This means that even simplified computational methods (Section 9.3) produce almanac data good enough for navigation. The emphasis on high accuracy in the primary computation is about insurance and professionalism, not about practical necessity.

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13. CRITICAL UNCERTAINTIES

Uncertainty	Impact if unfavorable	Mitigation
Printing capacity may be insufficient for 100+ complete sets	Fewer copies mean fewer vessels with almanacs; loss of any copy is more consequential	Prioritize archive copies and port copies first; vessel copies second. Store digital data for later reprinting if printing capability is restored.
Software dependencies may be unavailable (corrupted files, failed hardware)	Cannot compute the almanac by the primary method	Maintain software on multiple independent systems; pre-compute and store raw data immediately; have manual computation fallback (Section 9)
DE440 ephemeris file may not be downloaded in time	Skyfield cannot compute high-accuracy positions without it	Download immediately (this is a critical action); PyEphem as fallback (self-contained, no external files needed)
Toner and paper stocks may be smaller than assumed	Fewer pages available for almanac printing	Reduce page count by compressing format, printing double-sided, or covering only 50 years instead of 100
NZ may lack navigators trained in celestial navigation	Almanac exists but nobody can use it	Doc #139 (Celestial Navigation training); begin training programs immediately using GPS as cross-check
Timekeeping may degrade (quartz clock failures, no replacements)	Time errors > 4 seconds cause position errors > 1 nm	Doc #29 addresses timekeeping; maintain quartz chronometers; develop mechanical chronometer repair capability
Nuclear winter atmospheric effects (dust, aerosol loading) on astronomical observation	Reduced visibility; abnormal refraction; difficulty observing stars	Nuclear winter dust settles primarily in stratosphere; surface-level visibility for bright stars/sun should be adequate in most conditions, but extended overcast periods may limit celestial observations47
Accuracy of long-range (50–100 year) planetary predictions	Accumulated orbital perturbation errors for distant dates	For sun and stars, errors remain negligible over 100 years. For moon, errors using DE440 remain sub-arcminute for centuries. For planets, VSOP87/DE440 accuracy is sub-arcsecond over this timeframe.48

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14. THE 57 NAVIGATIONAL STARS — REFERENCE LIST

The following table lists all 57 navigational stars used in the standard nautical almanac, with their approximate SHA, declination (epoch J2000.0), visual magnitude, and visibility from NZ latitudes (40°S reference). Stars are listed by number as assigned in the Nautical Almanac.^{49 50}

No.	Name	Constellation	Mag.	SHA (°)	Dec (°)	Visibility from 40°S
1	Alpheratz	Andromeda	2.1	358	+29	Seasonal (low)
2	Ankaa	Phoenix	2.4	353	-42	Year-round
3	Schedar	Cassiopeia	2.2	350	+57	Never rises
4	Diphda	Cetus	2.0	349	-18	Seasonal
5	Achernar	Eridanus	0.5	335	-57	Circumpolar
6	Hamal	Aries	2.0	328	+24	Seasonal (low)
7	Polaris	Ursa Minor	2.0	319	+89	Never rises
8	Acamar	Eridanus	3.2	315	-40	Year-round
9	Menkar	Cetus	2.5	314	+04	Seasonal
10	Mirfak	Perseus	1.8	309	+50	Never/barely rises
11	Aldebaran	Taurus	0.9	291	+17	Seasonal
12	Rigel	Orion	0.1	281	-08	Seasonal
13	Capella	Auriga	0.1	281	+46	Never/barely rises
14	Bellatrix	Orion	1.6	278	+06	Seasonal
15	Elnath	Taurus	1.7	278	+29	Seasonal (low)
16	Alnilam	Orion	1.7	276	-01	Seasonal
17	Betelgeuse	Orion	0.4	271	+07	Seasonal
18	Canopus	Carina	-0.7	264	-53	Circumpolar
19	Sirius	Canis Major	-1.5	258	-17	Seasonal
20	Adhara	Canis Major	1.5	255	-29	Seasonal
21	Procyon	Canis Minor	0.4	245	+05	Seasonal
22	Pollux	Gemini	1.1	243	+28	Seasonal (low)
23	Avior	Carina	1.9	234	-59	Circumpolar
24	Suhail	Vela	2.2	223	-43	Year-round
25	Miaplacidus	Carina	1.7	221	-70	Circumpolar
26	Alphard	Hydra	2.0	218	-09	Seasonal
27	Regulus	Leo	1.4	208	+12	Seasonal
28	Dubhe	Ursa Major	1.8	194	+62	Never rises
29	Denebola	Leo	2.1	183	+15	Seasonal
30	Gienah	Corvus	2.6	176	-17	Seasonal
31	Acrux	Crux	0.8	173	-63	Circumpolar
32	Gacrux	Crux	1.6	172	-57	Circumpolar
33	Alioth	Ursa Major	1.8	166	+56	Never rises
34	Spica	Virgo	1.0	158	-11	Seasonal
35	Alkaid	Ursa Major	1.9	153	+49	Never/barely rises
36	Hadar	Centaurus	0.6	149	-60	Circumpolar
37	Menkent	Centaurus	2.1	148	-36	Year-round
38	Arcturus	Bootes	-0.0	146	+19	Seasonal
39	Rigil Kentaurus	Centaurus	-0.0	140	-61	Circumpolar
40	Zubenelgenubi	Libra	2.8	137	-16	Seasonal
41	Kochab	Ursa Minor	2.1	137	+74	Never rises
42	Alphecca	Corona Borealis	2.2	126	+27	Seasonal (low)
43	Antares	Scorpius	1.0	112	-26	Seasonal
44	Atria	Triangulum Australe	1.9	108	-69	Circumpolar
45	Sabik	Ophiuchus	2.4	102	-16	Seasonal
46	Shaula	Scorpius	1.6	96	-37	Year-round
47	Rasalhague	Ophiuchus	2.1	96	+13	Seasonal
48	Eltanin	Draco	2.2	91	+51	Never/barely rises
49	Kaus Australis	Sagittarius	1.8	84	-34	Year-round
50	Vega	Lyra	0.0	81	+39	Barely rises (low)
51	Nunki	Sagittarius	2.0	76	-26	Seasonal
52	Altair	Aquila	0.8	62	+09	Seasonal
53	Peacock	Pavo	1.9	54	-57	Circumpolar
54	Deneb	Cygnus	1.3	50	+45	Never/barely rises
55	Enif	Pegasus	2.4	34	+10	Seasonal
56	Al Na’ir	Grus	1.7	28	-47	Year-round
57	Fomalhaut	Piscis Austrinus	1.2	15	-30	Year-round

Notes on the table:

• “Circumpolar” = never sets from 40°S; always available on clear nights
• “Year-round” = rises and sets but is above the horizon for a substantial portion of most nights
• “Seasonal” = visible for part of the year
• “Seasonal (low)” = visible but always near the northern horizon; observations less accurate due to greater atmospheric refraction
• “Never rises” = below the horizon from 40°S; not available to NZ navigators
• “Never/barely rises” = may technically rise a few degrees above the horizon at the northern tip of NZ but is practically unusable

Fact: Of the 57 navigational stars, approximately 40–45 are usable from NZ’s latitude range. This is more than sufficient — a navigator needs only 3–4 well-placed stars for a fix, and typically has 10–20 available on any clear night.⁵¹

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15. SAMPLE DATA FORMAT

The following illustrates the format for one day’s sun and moon data, to be used as a template for the production script’s output formatting. This example shows the layout only — the numerical values are illustrative.

2027 MARCH 15 (TUESDAY)

  UT    SUN              MOON
  hr  GHA      Dec      GHA      Dec      v     d     HP
       °  '    °  '      °  '    °  '
  00  176 38.2  S 2 12.4  148 22.1  N14 37.6  11.3  8.2  57.4
  01  191 38.2  S 2 11.4  162 51.8  N14 45.8  11.3  8.2  57.4
  02  206 38.2  S 2 10.4  177 21.6  N14 53.8  11.4  7.9  57.4
  03  221 38.2  S 2 09.4  191 51.5  N15 01.7  11.4  7.9  57.3
  ...
  23  356 38.1  S 1 49.5   90 56.2  N16 42.1  11.7  5.4  57.1

  SD = 16.1'    Eqn of Time: -09m 12s    Mer. Pass: 12h09m
                Moon SD = 15.6'

Notes:

• GHA in degrees and arcminutes to 0.1’
• Declination in degrees and arcminutes to 0.1’, with N/S hemisphere indicator
• v and d are interpolation correction factors (arcminutes per hour deviation from standard rate)
• HP is horizontal parallax in arcminutes
• SD is semi-diameter in arcminutes
• All times in Universal Time (UT/GMT); NZ navigators convert to local time as needed

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16. DIGITAL PRESERVATION

16.1 The raw data

In addition to the printed almanac, the complete computed dataset should be preserved digitally:

• Format: Plain text CSV files (one file per year, or one master file) containing all computed values. CSV is human-readable and requires no special software to interpret.
• Storage media: USB flash drives (multiple copies), external hard drives, optical discs (DVD-R or Blu-ray — more durable than hard drives for long-term archival). If available, M-DISC optical media is rated for 1,000+ years of data retention.⁵²
• Quantity: At minimum 5 copies of the complete digital dataset, stored in geographically separated locations.

16.2 The computation software

The Python scripts used to generate the almanac should also be preserved digitally, along with:

• Complete source code for Skyfield and/or PyEphem
• The DE440 ephemeris file
• Python interpreter source code (in case the language itself needs to be reinstalled)
• Meeus algorithms implemented in pseudocode (language-independent)

If NZ rebuilds computing capability in the future (Doc #135), this software archive allows the almanac to be recomputed, extended, and validated.

16.3 Printed backup of algorithms

The key algorithms from Meeus and the Explanatory Supplement should be printed as part of the Recovery Library, independent of the digital archive. Algorithms printed on paper survive indefinitely under reasonable storage conditions and require no technology to read. This is the ultimate fallback — even if all computers, all digital media, and all printed almanacs are lost, the algorithms on paper allow reconstruction from scratch.

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FOOTNOTES

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1. NZ astronomical institutions: University of Canterbury (Physics and Astronomy); University of Auckland (Physics); Victoria University of Wellington (School of Chemical and Physical Sciences); Carter Observatory (Wellington); Stardome Observatory (Auckland). The Royal Astronomical Society of New Zealand (RASNZ) maintains an active amateur community with computational skills. https://www.rasnz.org.nz/↩︎

2. Bowditch, N., The American Practical Navigator (Pub. No. 9), National Geospatial-Intelligence Agency. The standard reference for marine navigation, continuously updated since 1802. Chapter 16 covers celestial navigation and the use of the nautical almanac. Available at: https://msi.nga.mil/Publications/APN↩︎

3. The Nautical Almanac, published jointly by HM Nautical Almanac Office (UK Hydrographic Office) and the US Naval Observatory. Published annually since 1767 (UK) and 1855 (US, jointly from 1958). The current edition is approximately 320 pages and covers one calendar year. Available commercially and in maritime reference libraries worldwide. https://aa.usno.navy.mil/publications/nao↩︎

4. Forbes, E.G., The Birth of Scientific Navigation: The Solving in the 18th Century of the Problem of Finding Longitude at Sea, National Maritime Museum, 1974. Also: Croarken, M., “Astronomical Labourers: Maskelyne’s Assistants at the Royal Observatory, Greenwich, 1765–1811,” Notes and Records of the Royal Society, 2003. The Nautical Almanac has been published continuously since 1767. Historical editions contained errors that were gradually reduced as computation methods improved.↩︎

5. The Nautical Almanac, published jointly by HM Nautical Almanac Office (UK Hydrographic Office) and the US Naval Observatory. Published annually since 1767 (UK) and 1855 (US, jointly from 1958). The current edition is approximately 320 pages and covers one calendar year. Available commercially and in maritime reference libraries worldwide. https://aa.usno.navy.mil/publications/nao↩︎

6. The moon’s horizontal parallax is the angle subtended by the Earth’s radius as seen from the moon. At the moon’s average distance of approximately 384,400 km, this is approximately 57 arcminutes (nearly 1 degree). This is large enough to affect navigational observations significantly — a navigator at the Earth’s surface sees the moon displaced from its geocentric position by up to 1 degree, depending on the moon’s altitude. The parallax correction must be applied to all lunar observations. Source: Explanatory Supplement to the Astronomical Almanac, 3rd ed., 2012.↩︎

7. The vernal equinox (Aries) is the reference point from which celestial positions are measured. Its GHA increases at the sidereal rate of approximately 15.0411° per hour (360.9856° per day), reflecting Earth’s rotation. Star positions are given as SHA (Sidereal Hour Angle), measured westward from Aries. The GHA of a star equals GHA Aries + SHA of star. Source: Meeus, J., Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998.↩︎

8. The 57 navigational stars were standardized in the 20th century for use in the Nautical Almanac and the sight reduction tables (HO 249 / AP 3270). Selection criteria: brightness (mostly magnitude 2.5 or brighter), broad distribution across the celestial sphere, and ease of identification. The list has remained stable for decades. Source: The Nautical Almanac; also Blewitt, M., Celestial Navigation for Yachtsmen, Adlard Coles, various editions.↩︎

9. The precession of the equinoxes is caused by the gravitational torque of the sun and moon on Earth’s equatorial bulge. The rate is approximately 50.3 arcseconds per year (1 degree in approximately 71.6 years). Over 100 years, this produces a shift of approximately 1.4 degrees in the coordinates of all celestial objects. Source: Lieske, J.H., et al., “Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants,” Astronomy and Astrophysics, 1977. Updated by the IAU 2006 precession model.↩︎

10. Celestial navigation observations are most accurate during nautical twilight (sun 6°–12° below the horizon), when the horizon is still visible as a reference line for the sextant while stars and planets are already visible. This window is typically 20–30 minutes long, occurring twice daily (morning and evening). Source: Bowditch (note 1); Blewitt (note 6).↩︎

11. The Nautical Almanac, published jointly by HM Nautical Almanac Office (UK Hydrographic Office) and the US Naval Observatory. Published annually since 1767 (UK) and 1855 (US, jointly from 1958). The current edition is approximately 320 pages and covers one calendar year. Available commercially and in maritime reference libraries worldwide. https://aa.usno.navy.mil/publications/nao↩︎

12. Meeus, J., Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998. This is the standard reference for practical astronomical computation. Chapter 25 covers solar coordinates (accuracy better than 1 arcsecond using the full method, or approximately 1 arcminute using the simplified method). Chapter 47 covers lunar coordinates (accuracy approximately 10 arcseconds using the principal terms). Chapter 33 covers planetary positions using orbital elements.↩︎

13. One arcminute of arc on the Earth’s surface corresponds to one nautical mile (1,852 metres), by definition. A position error of 1 arcminute in the almanac therefore corresponds to 1 nautical mile of position error — negligible compared to the 1–3 nautical mile accuracy of the overall celestial navigation system. Source: International Hydrographic Organization definition; Bowditch (note 1).↩︎

14. Chapront-Touze, M. and Chapront, J., “The Lunar Ephemeris ELP 2000,” Astronomy and Astrophysics, 1983. Updated as ELP/MPP02. This semi-analytical lunar theory includes over 35,000 terms in its most complete form. Truncated versions with 100–200 terms achieve accuracy of approximately 1–10 arcseconds. The JPL Development Ephemeris (numerically integrated) achieves sub-arcsecond accuracy.↩︎

15. Meeus, J., Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998. This is the standard reference for practical astronomical computation. Chapter 25 covers solar coordinates (accuracy better than 1 arcsecond using the full method, or approximately 1 arcminute using the simplified method). Chapter 47 covers lunar coordinates (accuracy approximately 10 arcseconds using the principal terms). Chapter 33 covers planetary positions using orbital elements.↩︎

16. Bretagnon, P. and Francou, G., “Planetary Theories in Rectangular and Spherical Variables: VSOP87 Solutions,” Astronomy and Astrophysics, 1988. VSOP87 provides planetary positions accurate to better than 1 arcsecond for the inner planets and a few arcseconds for the outer planets, over a timespan of several thousand years centered on J2000.0.↩︎

17. Folkner, W.M., et al., “The Planetary and Lunar Ephemerides DE430 and DE431,” Interplanetary Network Progress Report, 2014. The DE series are numerically integrated ephemerides produced by JPL, fitted to centuries of astronomical observations and spacecraft tracking data. DE440 (2021) is the current standard, covering approximately 1549–2650 CE. https://ssd.jpl.nasa.gov/planets/eph_export.html↩︎

18. The precession of the equinoxes is caused by the gravitational torque of the sun and moon on Earth’s equatorial bulge. The rate is approximately 50.3 arcseconds per year (1 degree in approximately 71.6 years). Over 100 years, this produces a shift of approximately 1.4 degrees in the coordinates of all celestial objects. Source: Lieske, J.H., et al., “Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants,” Astronomy and Astrophysics, 1977. Updated by the IAU 2006 precession model.↩︎

19. Proper motion data for navigational stars from the Hipparcos and Gaia catalogs. Most navigational stars have proper motions below 1 arcsecond per year. Notable exceptions: Arcturus (~2.3”/yr), Rigil Kentaurus (~3.7”/yr), Sirius (~1.3”/yr). Over 100 years, these stars shift by several arcminutes — significant and must be included in the computation. Source: ESA Hipparcos and Gaia catalogs. https://www.cosmos.esa.int/web/hipparcos; https://www.cosmos.esa.int/web/gaia↩︎

20. The precession of the equinoxes is caused by the gravitational torque of the sun and moon on Earth’s equatorial bulge. The rate is approximately 50.3 arcseconds per year (1 degree in approximately 71.6 years). Over 100 years, this produces a shift of approximately 1.4 degrees in the coordinates of all celestial objects. Source: Lieske, J.H., et al., “Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants,” Astronomy and Astrophysics, 1977. Updated by the IAU 2006 precession model.↩︎

21. Meeus, J., Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998. This is the standard reference for practical astronomical computation. Chapter 25 covers solar coordinates (accuracy better than 1 arcsecond using the full method, or approximately 1 arcminute using the simplified method). Chapter 47 covers lunar coordinates (accuracy approximately 10 arcseconds using the principal terms). Chapter 33 covers planetary positions using orbital elements.↩︎

22. Skyfield: a Python library for astronomical computation by Brandon Rhodes. Uses JPL DE ephemeris files for high-accuracy computation. MIT license. Documentation at https://rhodesmill.org/skyfield/ — Source code at https://github.com/skyfielders/python-skyfield↩︎

23. JPL ephemeris files: DE421 covers 1899–2053 (17 MB); DE440 covers approximately 1549–2650 (100 MB). DE440 is recommended for the 100-year almanac because it covers the full required date range. Files available from https://ssd.jpl.nasa.gov/planets/eph_export.html and via Skyfield’s built-in download functionality.↩︎

24. PyEphem: a Python wrapper for the XEphem astronomical computation library by Elwood Downey. Uses VSOP87 and other analytical theories. Does not require external ephemeris files. LGPL license. Documentation at https://rhodesmill.org/pyephem/ — Note: the same developer (Brandon Rhodes) maintains both Skyfield and PyEphem. PyEphem is considered legacy; Skyfield is the recommended successor.↩︎

25. IAU Standards of Fundamental Astronomy (SOFA) library. Maintained by the International Astronomical Union. Provides canonical implementations of fundamental positional astronomy routines. Available in C and Fortran. http://www.iausofa.org/↩︎

26. Meeus, J., Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998. This is the standard reference for practical astronomical computation. Chapter 25 covers solar coordinates (accuracy better than 1 arcsecond using the full method, or approximately 1 arcminute using the simplified method). Chapter 47 covers lunar coordinates (accuracy approximately 10 arcseconds using the principal terms). Chapter 33 covers planetary positions using orbital elements.↩︎

27. Urban, S.E. and Seidelmann, P.K. (eds), Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books, 2012. The definitive reference for all computational methods used in the Astronomical Almanac and Nautical Almanac. Approximately 700 pages.↩︎

28. Duffett-Smith, P., Practical Astronomy with your Calculator, 4th ed., Cambridge University Press, 1988. Provides simplified algorithms suitable for pocket calculator computation. Less accurate than Meeus but more accessible and sufficient for navigational accuracy.↩︎

29. The Nautical Almanac, published jointly by HM Nautical Almanac Office (UK Hydrographic Office) and the US Naval Observatory. Published annually since 1767 (UK) and 1855 (US, jointly from 1958). The current edition is approximately 320 pages and covers one calendar year. Available commercially and in maritime reference libraries worldwide. https://aa.usno.navy.mil/publications/nao↩︎

30. Paper durability depends on acidity (pH) and storage conditions. Acid-free paper (pH 7.0 or higher) can last several hundred years under favorable storage. Standard wood-pulp paper (pH 4.5–5.5) typically becomes brittle within 50–100 years, faster in humid or warm conditions. Source: Library of Congress preservation guidelines. https://www.loc.gov/preservation/↩︎

31. The moon’s horizontal parallax is the angle subtended by the Earth’s radius as seen from the moon. At the moon’s average distance of approximately 384,400 km, this is approximately 57 arcminutes (nearly 1 degree). This is large enough to affect navigational observations significantly — a navigator at the Earth’s surface sees the moon displaced from its geocentric position by up to 1 degree, depending on the moon’s altitude. The parallax correction must be applied to all lunar observations. Source: Explanatory Supplement to the Astronomical Almanac, 3rd ed., 2012.↩︎

32. The vernal equinox (Aries) is the reference point from which celestial positions are measured. Its GHA increases at the sidereal rate of approximately 15.0411° per hour (360.9856° per day), reflecting Earth’s rotation. Star positions are given as SHA (Sidereal Hour Angle), measured westward from Aries. The GHA of a star equals GHA Aries + SHA of star. Source: Meeus, J., Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998.↩︎

33. Auckland sunrise times: earliest summer sunrise approximately 05:57 NZDT (early December); latest winter sunrise approximately 07:33 NZST (late June). Note that NZ observes daylight saving time (NZDT, UTC+13) from late September to early April; NZST (UTC+12) applies the rest of the year. Navigators using the almanac work in UT and convert to local time. Source: https://www.timeanddate.com/sun/new-zealand/auckland↩︎

34. Forbes, E.G., The Birth of Scientific Navigation: The Solving in the 18th Century of the Problem of Finding Longitude at Sea, National Maritime Museum, 1974. Also: Croarken, M., “Astronomical Labourers: Maskelyne’s Assistants at the Royal Observatory, Greenwich, 1765–1811,” Notes and Records of the Royal Society, 2003. The Nautical Almanac has been published continuously since 1767. Historical editions contained errors that were gradually reduced as computation methods improved.↩︎

35. NZ’s latitude range: Cape Reinga (approximately 34°25’S) to Slope Point, Southland (approximately 46°40’S). Stewart Island/Rakiura extends to approximately 47°17’S. NZ’s sub-Antarctic islands extend further south (Campbell Island at 52°33’S, but these are unlikely to require routine navigational service). Source: Land Information New Zealand (LINZ). https://www.linz.govt.nz/↩︎

36. The 57 navigational stars were standardized in the 20th century for use in the Nautical Almanac and the sight reduction tables (HO 249 / AP 3270). Selection criteria: brightness (mostly magnitude 2.5 or brighter), broad distribution across the celestial sphere, and ease of identification. The list has remained stable for decades. Source: The Nautical Almanac; also Blewitt, M., Celestial Navigation for Yachtsmen, Adlard Coles, various editions.↩︎

37. The Southern Cross method for finding south is documented in all Southern Hemisphere navigation references. The accuracy is approximately 2–3 degrees for rough compass checks, improving with practice. For precise direction, a compass or sextant observation of a celestial body at known azimuth is preferred. Source: Blewitt (note 6); also Cunliffe, T., Celestial Navigation, Adlard Coles, various editions.↩︎

38. The Southern Cross method for finding south is documented in all Southern Hemisphere navigation references. The accuracy is approximately 2–3 degrees for rough compass checks, improving with practice. For precise direction, a compass or sextant observation of a celestial body at known azimuth is preferred. Source: Blewitt (note 6); also Cunliffe, T., Celestial Navigation, Adlard Coles, various editions.↩︎

39. A typical office laser printer produces approximately 20–40 pages per minute. At 30 pages per minute, printing one complete 5,000-page almanac set takes approximately 2.8 hours. Printing 100 sets takes approximately 280 hours (12 days of continuous operation). Toner consumption is approximately 1 cartridge per 2,000–5,000 pages depending on coverage and cartridge size. Source: Manufacturer specifications (HP, Canon, Brother).↩︎

40. Forbes, E.G., The Birth of Scientific Navigation: The Solving in the 18th Century of the Problem of Finding Longitude at Sea, National Maritime Museum, 1974. Also: Croarken, M., “Astronomical Labourers: Maskelyne’s Assistants at the Royal Observatory, Greenwich, 1765–1811,” Notes and Records of the Royal Society, 2003. The Nautical Almanac has been published continuously since 1767. Historical editions contained errors that were gradually reduced as computation methods improved.↩︎

41. Croarken, M., Early Scientific Computing in Britain, Oxford University Press, 1990. Documents the organization and methods of the British Nautical Almanac Office and other computation offices. Human computers typically worked in pairs (computer + comparer) to reduce errors. The same organizational model was used at the US Naval Observatory, Greenwich Observatory, and other institutions.↩︎

42. Forbes, E.G., The Birth of Scientific Navigation: The Solving in the 18th Century of the Problem of Finding Longitude at Sea, National Maritime Museum, 1974. Also: Croarken, M., “Astronomical Labourers: Maskelyne’s Assistants at the Royal Observatory, Greenwich, 1765–1811,” Notes and Records of the Royal Society, 2003. The Nautical Almanac has been published continuously since 1767. Historical editions contained errors that were gradually reduced as computation methods improved.↩︎

43. Meeus, J., Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998. This is the standard reference for practical astronomical computation. Chapter 25 covers solar coordinates (accuracy better than 1 arcsecond using the full method, or approximately 1 arcminute using the simplified method). Chapter 47 covers lunar coordinates (accuracy approximately 10 arcseconds using the principal terms). Chapter 33 covers planetary positions using orbital elements.↩︎

44. GPS satellite constellation status: the US Space Force operates approximately 31 GPS satellites (as of 2025) in 6 orbital planes at approximately 20,200 km altitude. Design life is 10–15 years per satellite (varies by generation: Block IIF ~12 years, GPS III ~15 years). Source: GPS.gov (https://www.gps.gov/systems/gps/space/); US Space Force fact sheets.↩︎

45. GPS clock drift and degradation without ground control uploads: Estimated from satellite clock stability specifications (rubidium and cesium atomic clocks with stability of approximately 1–3 parts in 10^13 per day). Without uploads, the navigation message (broadcast ephemeris) becomes stale; position errors grow as predicted satellite positions diverge from actual. The specific degradation rate is classified but publicly available analyses suggest useful accuracy (~100 m) for weeks to months, degrading to kilometers over 1–2 years. Source: Betz, J.W., “Engineering Satellite-Based Navigation and Timing,” Wiley, 2015.↩︎

46. GPS clock drift and degradation without ground control uploads: Estimated from satellite clock stability specifications (rubidium and cesium atomic clocks with stability of approximately 1–3 parts in 10^13 per day). Without uploads, the navigation message (broadcast ephemeris) becomes stale; position errors grow as predicted satellite positions diverge from actual. The specific degradation rate is classified but publicly available analyses suggest useful accuracy (~100 m) for weeks to months, degrading to kilometers over 1–2 years. Source: Betz, J.W., “Engineering Satellite-Based Navigation and Timing,” Wiley, 2015.↩︎

47. Nuclear winter atmospheric effects on astronomical observation: The primary effect is stratospheric aerosol loading from soot and dust, which reduces surface insolation by approximately 30–70% depending on severity. Stellar and solar observations may be affected by increased atmospheric extinction and scattering, but the sun and first-magnitude stars should remain observable through all but the most severe conditions. The main impact on navigation would be extended periods of overcast weather rather than aerosol opacity per se. Source: Robock, A. et al., “Nuclear Winter Revisited with a Modern Climate Model and Current Nuclear Arsenals,” Journal of Geophysical Research, 2007.↩︎

48. Bretagnon, P. and Francou, G., “Planetary Theories in Rectangular and Spherical Variables: VSOP87 Solutions,” Astronomy and Astrophysics, 1988. VSOP87 provides planetary positions accurate to better than 1 arcsecond for the inner planets and a few arcseconds for the outer planets, over a timespan of several thousand years centered on J2000.0.↩︎

49. The 57 navigational stars were standardized in the 20th century for use in the Nautical Almanac and the sight reduction tables (HO 249 / AP 3270). Selection criteria: brightness (mostly magnitude 2.5 or brighter), broad distribution across the celestial sphere, and ease of identification. The list has remained stable for decades. Source: The Nautical Almanac; also Blewitt, M., Celestial Navigation for Yachtsmen, Adlard Coles, various editions.↩︎

50. Proper motion data for navigational stars from the Hipparcos and Gaia catalogs. Most navigational stars have proper motions below 1 arcsecond per year. Notable exceptions: Arcturus (~2.3”/yr), Rigil Kentaurus (~3.7”/yr), Sirius (~1.3”/yr). Over 100 years, these stars shift by several arcminutes — significant and must be included in the computation. Source: ESA Hipparcos and Gaia catalogs. https://www.cosmos.esa.int/web/hipparcos; https://www.cosmos.esa.int/web/gaia↩︎

51. The 57 navigational stars were standardized in the 20th century for use in the Nautical Almanac and the sight reduction tables (HO 249 / AP 3270). Selection criteria: brightness (mostly magnitude 2.5 or brighter), broad distribution across the celestial sphere, and ease of identification. The list has remained stable for decades. Source: The Nautical Almanac; also Blewitt, M., Celestial Navigation for Yachtsmen, Adlard Coles, various editions.↩︎

52. M-DISC is an optical disc format using an inorganic recording layer (stone-like material) rather than organic dye. Manufacturer testing and independent testing (US Department of Defense) suggest data retention exceeding 1,000 years under standard storage conditions. Available in DVD and Blu-ray formats, writable on standard optical drives with M-DISC capability. Source: Millenniata/Verbatim product specifications; US Naval Air Warfare Center testing report, 2009.↩︎