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Consider as an example, AstroPixel's ``Six Millennium Catalog of Phases of the Moon''---

http://astropixels.com/ephemeris/phasescat/phasescat.html

My question is two-fold:

  1. How are these tables which give (for example) dates and times of new moons and full moons to the nearest minute, generated?

  2. How precise is this information? From http://astropixels.com/ephemeris/phasescat/phases0001.html we see, for example, that on Jan 13 0001, a New Moon occurred at 10:58 (UT). How much of an error bound is there likely to be on these data?

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The first question (how are the tables generated?) is easy to answer. The acknowledgement on the Astropixels page says:

It is based on procedures described in Astronomical Algorithms by Jean Meeus (Willmann-Bell, Inc., Richmond, 1998).

For the second question (how accurate are the calculations?), there are 3 sources of error that I can think of:

  1. The procedure may be for a limited time span. Meeus' book does not mention what time span the formulas are good for, but the source from which the procedure was based on (Chapront's ELP-2000/82) used the years 1900 to 2000 to determine the variables in the equations. (See NASA Astrophysics Data System - The lunar ephemeris ELP 2000.) It is hard to know what the accuracy is when going backward or forward several hundred years. Mathematically, extrapolating beyond the chosen time span could be reasonably accurate, or it could be "completely wrong". (I did not read all of the Astropixels' pages, but I know that a lot of historical research was done when calculating the solar eclipse tables. So I would anticipate some cross-referencing between the Moon phases and eclipses. After all, you cannot have a solar eclipse unless you have a New Moon!)
  2. Time is based on the rotation of the Earth, such as GMT (Greenwich Mean Time), UT (Universal Time), or the time in a local time zone. This type of time is not consistent over long time intervals. The Earth's rotation slows down and speeds up which requires adjustments to our clocks. The motion of the planets is (more) consistent and runs at a steady rate. Astronomers call this type of time Dynamical Time (formerly known as TDT Terrestrial Dynamic Time or ET Ephemeris Time). The difference between UT and Dynamical Time is explained in the Astropixels' page Uncertainty in Delta T. The uncertainty in Delta T is over an hour for the year 2000 BCE, so the time of the phase can be off by this amount just due to the uncertainty of clock time (the Earth's "clock") versus Dynamical Time (the solar system's "clock").
  3. Computer languages can introduce inaccuracies. For example, it may not be possible to store the number of days, hours, minutes between now and the year 2000 BCE to full precision due to the limited accuracy of storing really large number. There are approximately 1.05 billion (1.05E9) minutes in 2000 years. In order to calculate a result 2000 years ago to a precision of 1 minute requires the ability to store a number such as 1,050,000,001 and 1,050,000,002. (I am not a computer expert, but these numbers can probably be stored easily and to a higher accuracy. The potential problem is that the procedure uses values of time, time squared ($T^2$), time cubed ($T^3$), and time to the fourth power ($T^4$)! Thus, numbers can quickly become larger than what the computer can store to full precision.)

When put all together, I would expect that the times 2000 years ago (or 2000 years in the future) may only be accurate to a couple of hours.

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    $\begingroup$ The IEEE double precision format remains accurate to the 1 second level for 158 million years, so that's not a big problem. The ELP 2000 uses Barycentric Dynamic Time (TDB) as its independent variable, so that's not nearly as big a problem as using Universal Time. Extrapolation is always a big problem, as is using Universal Time. $\endgroup$ – David Hammen Oct 15 '20 at 18:23
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    $\begingroup$ Thanks @DavidHammen, that is good to know. I was thinking of the discussion in this thread, which now that I look at it again, I see that it is referring to single precision numbers. softwareengineering.stackexchange.com/questions/392723/…. The gist is that although single precision can handle numbers between -3E+38 to 3E+38, it cannot handle all numbers in that range. $\endgroup$ – JohnHoltz Oct 15 '20 at 21:16
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    $\begingroup$ Single precision? That's good to about seven decimal places. This is merely terrible for position and velocity. It is beyond terrible for time. The Standards of Fundamental Astronomy use a pair of doubles whose sum conceptually add to form time. By making one of the numbers a value that can be represented exactly (e.g., an integral number of days) and the other a small offset (e.g., fraction of days), it is possible to obtain much better precision than can afforded by just one double precision number, ten picoseconds for any time since the Big Bang. $\endgroup$ – David Hammen Oct 15 '20 at 21:32

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