Celestial coordinates are typically given in one of several different ways, depending on who is using them. First, we have to decide the "origin" (math talk for where we imagine we are standing when we measure the positions). The choices are heliocentric (measured from the center of the Sun), geocentric (from the center of the earth), or topocentric (from a point on the earth's surface) form. The difference between geocentric and topocentric measurements is likely to be noticeable only for the Moon.
Secondly, we need to choose a coordinate system: the plane from which to measure the angular distance up-or-down, and the direction from which to measure the angular distance left-or-right. There are three principal coordinate systems: equatorial, ecliptic and "alt-az."
Equatorial Coordinates. At a planetarium, you could project a grid onto the screen, representing the "lines" of celestial latitude and longitude. In the equatorial system, the plane of reference is the celestial equator: the projection, into the sky, of the terrestrial equator. Likewise, the North Celestial Pole (NCP) is directly above the Earth's North Pole, and the South Celestial Pole below the Earth's South Pole. Celestial latitude, measured from the equator, is called "declination," and is measured in degrees, arc-minutes ('), and arc-seconds (").
Just as we needed a terrestrial prime meridian from which to measure terrestrial longitude, we need to arbitrarily fix a location for the celestial prime meridian in order to determine right ascension. In 1950, it was defined as passing through the "first point" in the constellation Aries.
There's no east or west celestial longitude; it is measured either eastward from the first point as 0-24 hours ("right ascension", RA), or westward as 0-360° ("sidereal hour angle"). RA is stated either like declination, or, because of the relationship of longitude to local time, in hours, minutes (m) and seconds (s) (one minute RA is 15' ; one second, 15").
For any celestial object, there will be a point on the earth's surface such that the object would be directly overhead. That's called the "sub-point" (or "geographical point," GP). As the Earth rotates, etc., the GP will move.
Ecliptic Coordinates. Geocentric equatorial coordinates work well for the Sun and the stars, at least in the short term (years as opposed to centuries), but for the planets, it helps to carry out computations in ecliptic coordinates. The earth's orbital plane is called the ecliptic, and a line drawn through the center of the Earth, and perpendicular to the ecliptic, defines the North and South Ecliptic Poles. Depending on what you are trying to compute, you can use geocentric or heliocentric coordinates.
Because of precession (the wobbling of the Earth's axis relative to the plane of the Earth's orbit), the equatorial coordinates of even the Sun and stars changes slowly with time. (One full precession cycle takes about 26,000 years). Celestial North has to be defined on the basis of the orientation of the Earth's axis, relative to its orbit, as of a particular time ("epoch").
Precession causes the NCP to revolve around the North Ecliptic Pole. Thanks to precession, the celestial prime meridian passes through the constellation Pisces.
The Earth's orbit itself is perturbed by the rest of the Solar System, resulting in changes in the orientation of the major axis and the orbital plane relative to the rest of what I will loosely call "Distant Outer Space." These changes are just too small and too slow to worry about here.
Horizontal (Alt-Az) Coordinates. A navigator's observation of a celestial object isn't likely to be recorded, initially, in equatorial coordinates, but rather in terms of the object's altitude and azimuth. The altitude is the vertical angle between it and the "celestial horizon," which in turn is a distant imaginary circle, centered on the observer and level with the observer's eye, and in a plane perpendicular to the zenith line (from the observer to the point directly overhead, opposite the direction of gravity). The azimuth is its horizontal direction, an angle measured from the direction which points to the North Geographic Pole. The imaginary semicircle running across the sky from north to south is the observer's meridian.
Alt-Az coordinates are relative to an observer on the earth's surface, and thus are topocentric. After correcting for observational errors, they can be converted into other coordinates.
Sources of Error in Celestial Navigation
There are several kinds of error which can occur. The first are observational errors, wherein the "read" position, in alt-az coordinates, doesn't correspond to the actual position of the object at that time. Or there is an error in determining the time at which the observation was made.
Secondly, there can be an error in the prediction of the celestial coordinates. If the navigator is using a published star atlas or catalogue, then this could be an error on the part of whoever computed the published coordinates, or on the part of the navigator, in taking the value from the table, and perhaps in updating it as needed.
Finally, there can be a sight reduction error, that is, an error in the use of the observation and the reference data to compute the latitude and longitude of the ship.
It does no good to worry about computing star positions to the correct milli-arc second if your observational instrument is only accurate to the nearest degree. Hence, in improving the art of navigation, you need to tackle sources of error in their order of importance. Nunez' Defense of the Sea Chart (1537) said that there was no point in correcting for the meridian of observation, in using solar declination tables, unless the longitude difference was at least six hours, because of the grosser errors resulting from the imprecision with which the astrolabe measured altitude. (Taylor 181).
I am going to ignore sources of error which are always smaller than 1'. Usually those mean an error of about a mile on the ground, but if you are using the "lunar distance" method to measure longitude, a 1' error in lunar distance corresponds to a 0.5° error (up to 35 miles) in longitude.
Up-time Computer Software
It isn't likely that there is any navigation software in landlubberly Grantville. However, the high school science department is reported to have at least one astronomy program, and those have data useful in celestial navigation.
There were many amateur astronomy programs available in late 1999/early 2000, but the ones I think most likely to have been acquired for educational purposes are:
Distant Suns 5.1 (3/2000)
Deep Space 5.56 (by 1998)
Dance of the Planets QED edition (1994)
Red Shift 3 (1998)
Starry Nights Deluxe 2.0 (by 1999) or Pro (1/2000)
TheSky v.4 (by early 2000)
Voyager 2 (by 1999)
DeepSky 2000 (1/2000)
Canon says that Johnnie Farrell has a telescope with a "goto." A certain amount of astronomical data could be extracted, somewhat laboriously, from the "goto."
Star Data
If we know the locations of the stars in ecliptic or equatorial coordinates, we can use them for navigation. There are three possible sources of this information:
—down-time star catalogs (including atlases and globes), corrected for the passage of time
—up-time star catalogs (books and software), ditto
—post-RoF observations
Down-time Star Data. The most useful compilation is the star catalogue of Tycho Brahe. His "cat D" (1598) provides ecliptic coordinates (nearest 0.5') for 1004 stars. Tycho was well aware of precession (see below) and, since the catalog was the fruit of years of observation, all star positions were corrected to what they should be for epoch "1601.03".
Tycho's accuracy is excellent. Rawlins compared Tycho's positions to those predicted by combining the Yale Catalog (1982) with "Newcomb's traditional precession constants" (see "Precession" below). For his 100 "select stars" (the bright stars likely to serve as navigational beacons), the error in either equatorial coordinate was never as much as 6'. The mean error was 1.62' in RA and 1.48' in declination.
The greatest weakness of Tycho's data is that his observatory was in Denmark, and hence his coverage of southern hemisphere stars is poor.
Up
-time Star Data. Books for amateur astronomers will explain how to locate stars (and other celestial objects). Ideally, they will specify the location of the star in celestial equatorial coordinates (right ascension and declination) and also state the standard date for which those coordinates were determined.
Pasachoff, Stars and Planets was in the Mannington Middle School Library. The 2000 edition came out in 1999. Appendix 2 gives the calculated mid-1999 equatorial coordinates (to nearest 0.1') for the 314 brightest stars (down to apparent magnitude 3.55), each identified by its name as given in Bayer's 1603 atlas. There is also a copy of Burnham's Celestial Handbook in the high school science department.
The mysterious computer program should have star data specified at least to the nearest arc-minute (as with TellStar, 1985) and more likely to the arc-second (as in Dance of the Planets, 1994).
Precession. The modern star positions, and even those of Tycho's, are not quite accurate in the 1630s. The discrepancies are primarily the result of precession. Encyclopedia Americana "Equinox" says that the cycle is about 26,000 years, and that precession is at a rate of about 50 arc-seconds/year (which implies a cycle length of 25,920 years). EB11 ("Precession of the Equinoxes", "Earth") gives two values for "general precession", 50.2453 (1850) and 50.2564 (1900) arc-seconds/year.
Both the down-time and up-time star data can be roughly corrected for precession, by any competent down-time astronomer, by assuming that precession occurs at the constant rate suggested by EB11, and then carrying out the appropriate spherical trigonometry calculation. Tycho and Kepler both corrected older data; the value used by Tycho was 51" (Rawlins 17.)
That astronomy program should be able to precess the modern star positions back to the 1630s and the underlying algorithm is probably more complex (and accurate) than the simple constant precession contemplated by Tycho .
Proper Motion. For some navigational stars (Rigil Kent, Arcturus, Polaris, Zuben-ubi) proper motion (the real motion of the star relative to the solar system) can create a noticeable error (Reis). Obviously, the stars will experience more than ten times as much proper motion in the nearly four centuries separating Pasachoff from the 1630s, as in the three decades elapsed since Tycho's catalog. The astronomy programs may take proper motion into account.
Recommendations. Assuming, as is likely, that the available software gives star positions with better than the Tychonian arc-minute accuracy, and supports precession to the 1630s, we will probably use the software to read off the correct 163x equatorial coordinates for all the navigational stars. Otherwise, we will probably use Pasachoff's data for southern hemisphere stars and Tycho's for the rest, with both adjusted for precession.
In the long-term, astronomers will use telescopes to obtain star positions which are both current and accurate. One of the first catalogs compiled with telescopic assistance, Flamsteed's, was accurate to 10" arc (Wakefield 51).
Astronomical and Nautical Almanacs (Ephemerides)
An ephemeris is an almanac which tabulates the positions of an astronomical object as of different times. The difference between the nautical and astronomical almanac is one of emphasis. A modern nautical almanac will list predictions only for the Sun, Moon, and the "navigational" planets, and a stellar reference point, the constellation Ares. It will also identify the locations of the navigational stars (57 nowadays) relative to Ares. An astronomical almanac will cover the other planets and moons, and will provide coordinates for many additional stars. In either case, the solar, lunar and planetary predictions are usually good only for a few years, unless you have a computerized version.
Some internet sources would have you believe that the first nautical almanac was published in 1767. That was merely the first one with "lunar tables" for calculating longitude. The 1545 almanac of Martin Cortes was a long term (1545-1580) almanac, in which the solar declination was calculated by combining values for the month/day, and the year, to obtain the zodiacal position of the sun, and that then used to find the actual declination. In contrast, the almanac of William Bourne (1576) featured a simple look up, but was useable for a much shorter period. A more recent almanac was Davis's Seaman's Secrets (1594). It provided a table of the sun's declination for noon each day for the years 1593-97. (Graham; EB11/Navigation). The down-timers actually could do better than that; Digges' Prognostications (1553) had a table of the sun's altitude for every hour of the day at latitude 51.5° N (Taylor 187).
Some of the down-time almanacs also had star data; Mariner's Mirror (1588) offered the declination and right ascension coordinates for 100 "fixed stars" (Taylor 209).
A modern nautical almanac provides the declination and the Greenwich Hour Angle (GHA), to nearest 0.1', for each celestial object useful in navigation. The GHA is essentially the angle between the celestial meridian of the object, and the celestial meridian over Greenwich. Values are given for every hour (Greenwich Mean Time, GMT) of every day for the Sun, Moon, and planets, and for every day for the first point of Aries. To get the GHA for a particular star, you add the star's SHA. (GHA changes 1°/day for the stars, because, thanks to the earth's orbital movement, the earth doesn't have to quite complete a full rotation to face the same star a second night.) There is a correction value given, for each day, for the Sun and planets, to allow for interpolation between whole hours. And the Moon's movement is so irregular that a separate correction value is given for each hour.
The major concern with regard to the down-time manuals is accuracy. For example, for July 23, 1579, when Drake left the California coast, Bourne's declination tables (1574) were in error by six arc-minutes (Graham). The problem was that Bourne, not knowing Kepler's laws of planetary motion, had miscalculated the apparent solar movements.
While that was a "model error," computational errors were common. According to Bowditch, Tables 1 and 2 of Moore's Practical Navigator (1800) had 3,500 errors. And Astronomer-Royal Maskelyne's "Requisite Tables" were equally faulty, with 1,024 mistakes in Table 21. (Callaghan 215). The safest course of action is clearly to generate the numbers, and print the manuals, by computer.
Errors can also occur in using tables. Unlike a computer program, a table can't give celestial positions for every location at any instant of the day. If the observation isn't for the location and time of day assumed by the table, then for greatest accuracy, you must interpolate between table values. Errors could be made in interpolation, or the seamen could decide not to bother interpolating at all.
Solar declination tables, for example, were calculated as of the local time at a particular location. If the ship were at a different longitude, then its local time was different, and the navigator should make a longitude correction before using the declination, as taught by Hariot. Wright (1599) said that by ignoring longitude, the mariner might be "deceived sometimes 10 or 12 [arc] minutes in taking the sun's declination. Drake, in circumnavigating the world, ignored the problem. (Graham)
It is worth noting that seamen made calculations using the abacus (Swanick 42) and Gunter's line, sector and scale (basically devices for graphical solution of trig and log problems).
* * *
Solar Positions. The apparent motion of the Sun is a direct consequence of the real orbital motion of the Earth. There is a systematic error in many seventeenth century predictions of the Sun (and hence of the planets) because of Tycho's erroneous value (0.018) for the eccentricity of the Earth's orbit. (Gingerich xix). Cassini (1667) recalculated it as 0.017. Dutton-Smith says that it was 0.01675104 in 1900, and, using his formulae (86), when Grantville popped into the seventeenth century (May 25, 1631, Gregorian), it was 0.016862.
Planetary Positions. The apparent motion of a planet results from the combination of the real motions of that planet and the Earth.The only planets used for navigation are Venus, Mars, Jupiter and Saturn. Their advantage is that they are bright; their disadvantage, it is more difficult to predict their position than that of the "fixed" stars.
Before the up-timers arrived, down-timers predicted planetary appe
arances using the solar system models of Ptolemy, Copernicus, Tycho, or Kepler.
Kepler, for example, predicted planetary positions through 1637 in the Rudolphine Tables (1627). These predicted planetary positions through 1637. Lorenz Eichstadt (1596-1660) produced sequels in 1634, 1637, and 1639.
Kepler's Rudolphine Tables competed with the 1632 ephemeris of the Copernican Philip van Lansberge (1561-1632) and the 1622 Astronomia Danica of the "Tychonian" Christen Longomontanus (1562-1647).
Andrea Argoli (1570-1657) based his 1621 ephemeris on pure Copernican theory (adjusted circular heliocentric orbits). In 1634 he published new tables which followed the "Tychonian" model (all planets except the Earth circularly orbit the Sun). Argoli's predictions for Mars (1650s) were within 10' arc. His accuracy was less for other planets: Saturn (~40'), Jupiter (30'), Venus (2°), and Mercury (9°). For the "Sun", it was 8'. (Gingerich xi-xx).
It is perhaps worth mentioning that several later ephemerides authors are alive as of the RoF, including Ismael Boulliau (1605-1694), Noel Durret (1590-1650), Jeremiah Horrocks (1618-1641), and Thomas Streete (1622-1689). They may play a role in post-RoF astronomy.
Down-time mathematician-astronomers are going to learn some very important lessons from the up-timers and their books:
(1) Kepler was on the right track; the planets are, to a first approximation, in elliptical orbits with the Sun at one focus (his first law), they don't move at a constant velocity (his second law), and the periods are related to the size of the orbits (his third law);
(2) the Keplerian laws aren't really laws, they are a corollary to a special case of the real law governing planetary motion—Newton's law of universal gravitation. (Kepler's laws can be derived if one assumes that there are just two bodies in the universe and one is much more massive than the other.)