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Accuracy of the
Astronomical Algorithms

General rule

As a general rule, you can assume that all of the astronomical displays within Emerald Chronometer are at least as accurate as the precision with which you can read the display, from about 4000 BCE to about 2800 CE. For example, the azimuth displays on Haleakala can be read approximately to the nearest degree, and there is far more accuracy in the data than that.

Similarly, the For example, the rise and set dials, which can be read to the nearest minute, are at least that accurate. But note of course that they are only valid if the rise or set point on the horizon is at exactly the same level as you are; if the Sun is setting behind a hill above you, for example, it will set much earlier than the time indicated.

The displays depend heavily on having the correct time and location. If you have NTP synchronization and Location Services turned on, the correct time zone set in the iPhone's Settings app, and have green lights at the top of the display, you can presume that everything you are looking at is as accurate as you can read it. In general, those green lights mean you have about a kilometer (half a mile) of accuracy and are within a half second of the correct time.


The algorithms employed in Emerald Chronometer are very high-precision series calculations originally developed by astronomers at the Bureau des Longitudes in Paris in the 1980s and 1990s. They are particularly well-suited for the iPhone because the data tables they are based on can fit in about 500 kilobytes of memory (this includes data for most planets for the same period), and yet still produce accuracy of less than 1/100 of a degree over the entire period mentioned. No Internet connection is required for any astronomical calculation.

Specifically, the tables employed are from Lunar Tables and Programs from 4000 B.C. to A.D. 8000, by Michelle Chapront-Touzé & Jean Chapront, copyright 1991, and Planetary Programs and Tables from -4000 to +2800, by Pierre Bretagnon & Jean-Louis Simon, copyright 1986, both published by Willmann-Bell, Inc. (the latter includes the Sun motion tables).

The algorithms presented in those books were ported to C for use in the iPhone development environment, and local caches were developed to avoid recalculation of common quantities. The time conversions in those books have been superceded by more accurate ones, described below.

As a side note, the iPhone is a potent floating-point calculating machine; every time you move the Sun or Moon hand on Mauna Kea a little bit, over a thousand double-precision sines and cosines are calculated, whose arguments are themselves each polynomial expressions with several terms, and it all gets done in much less than a tenth of a second. Early in their careers the authors of Emerald Chronometer used machines that filled whole rooms with less horsepower than the iPhone you're holding in your hand.

The RA display on Geneva displays the RA "of date", meaning the RA applied from the equniox current on the given date. This means that the Local Sidereal Time (per its definition) also displays the rotation from the equinox, and not from the equinox of J2000. It also means that the Sun, Moon, and lunar nodal points' RA "of date" may be read from that dial. The constellations are displayed in the exact orientation found in J2000, but rotated according to the precession to match their locations at the displayed time. The P03 formulae for general precession are used.

In contrast, on Mauna Kea, the RA display is fixed in the zodiac dial to be the J2000 RA numbers of the constellations, and so on that dial may be read the J2000 RA of the Sun and Moon. The local sidereal time may thus be read from the front of MK only in centuries close to J2000; at other times, the more accurate LST display on the back of Mauna Kea may be used.

Historical data

As mentioned, you can use Emerald Chronometer to display astronomical information going back to 4000 B.C. However, it is important to understand that the time scale going back that far is subject to considerable uncertainty, due to uncertainties in the exact rotational speed of the Earth in the past.

What that means in practice is that the astronomical events happened as shown, but the time of day may not be the exact time that they happened. You can find a discussion of the various time scales involved, notably Ephemeris Time, or Terrestrial Dynamic Time, and Universal Time, in many places on the web.

And, of course, the dates of events depend on which calendar is used. Emerald Chronometer uses the Gregorian calendar for future dates and for past dates back to 1582, and the Julian calendar from 1 BCE to 1582 CE. Prior to 1 BCE Emerald Chronometer uses a proleptic Julian calendar, with leap years on 1BCE, 5BCE, etc, back every four years. Until very recently calendar representations around the world have varied widely, so using Emerald Chronometer for historical dates requires an advanced knowledge of Chronology.

Emerald Chronometer always displays UTC, and uses that as the starting point for its calculations, converting to Terrestrial Dynamic Time (TDT) using the polynomial expressions suggested by Fred Espenak at this NASA site based on the data in Morrison & Stephenson, 2004.

The Watches

Copyright © 2009, Emerald Sequoia LLC; last updated: 2009 November 25

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