(3) the other bodies "perturb" the orbit of interest, changing (usually slowly) its orbital elements (size, ellipticity, orientation).
(4) there is no exact solution to the N-body problem, but a pragmatic solution is obtainable by approximation methods.
With this information, they can make reasonably accurate short-term predictions of the movements of the planets even without up-time data.
In fact, with just the Keplerian laws, Kepler successfully predicted (in 1627) the transit of Mercury in 1631. The theory of gravitation, in its turn, made it possible for Edmund Halley to recognize that the comet seen in 1682 had previously been observed in 1531 and 1607, and that it would return in 1758.
A planetary orbit is defined by five orbital elements (and a sixth indicates where the planet is in that orbit). Two describe the orbital geometry and the other three the orientation in three dimensions relative to Earth's orbit. EB11/Planets provides all of the orbital elements for the planets as of 1900/10. Pasachoff (Appendix 9) provides the masses of the Sun and planets, and three of the orbital elements (semimajor axis, eccentricity, and inclination), as of 2000. There are differences; e.g., Mars' eccentricity is 0.093309 (EB11) or 0.09341 (Pasachoff). Kepler's value was 0.09265 (Murray 20).
The six elements necessary to specify an elliptical orbit can be determined from three observations. However, there are many different routes of getting from the observational data to the orbital elements, and they vary in terms of accuracy and computation time. Historically, the major post-Newtonian contributions to orbit determination were those of Euler (1744), Lambert (1761-71), Lagrange (1778), Laplace (1780), Olbers (1797), and Gauss (1809)(Dubiago 7-14). The point to note is that some very heavy hitters studied the problem and that it still took over a century to get from Newton's Principia (1687) to the Gaussian method.
The observations need to be far enough apart so as to "see" the orbit from significantly different perspectives, but close enough together so that the elements haven't had time to be significantly perturbed. Once the elements are known, you can use additional observations to try to figure out how that orbit is being perturbed. Then you can calculate a more definitive present orbit, and that in turn allows you to later detect smaller or less frequent perturbations.
The basic concept of perturbation is one familiar to seventeenth century mathematicians; the epicycles engrafted on the Ptolemaic (and Copernican) models of the solar system perturbed the basic "circular" orbits of the planet in such a way as to account for discordant observations.
However, perturbation theory is an advanced topic in mathematics and it is by no means clear that the mathematicians of Grantville can teach it. It will need to be rediscovered. Once that happens, what you end up with is that each orbital element, instead of being constant as in the Keplerian model, is a complicated function of time. This is called an "analytical model."
In theory, if you know the masses, positions and velocities of all significant bodies in the solar system at the same point in time, you can instead use "numerical integration" to determine their positions and velocities in the past and in the future. In essence, you calculate the gravitational forces on each object, and determine how their positions and velocities change over a small time interval. Then you calculate the forces acting over the next time interval, and so on.
This wasn't feasible until high-speed computers were developed.
In practice, you are combining observational data, of varying reliability, from different dates. So the initial state for the simulation is determined using an analytical model. The simulation is run, generating an ephemeris for a period for which observations exist. The initial state is tweaked until the predicted ephemeris is a "best fit" to the observations. The simulation can be extended to make predictions concerning the past and future perambulations of the bodies.
The accuracy of the predictions will depend on the accuracy of the starting data, and on use of a sufficiently small time step. The smaller the interval, of course, the more computation is necessary.
There were some pre-RoF amateur astronomy programs, such as Dance of the Planets, which had some numerical integration capability.
Among professionals, the trend has been to use numerical integration to generate a "background ephemeris," and then find the analytical expressions which best fit the data. As of the RoF, the "gold standard" for the planets was the VSOP87 "semi-analytical" model, in which analytical (polynomial and trigonometric) expressions were fitted to the DE200 ephemeris (covering 1600-2169) generated by numerical integration. VSOP87 is believed to be accurate to 0.05 arc second for the modern period and to one arc second over a period of several thousand years. Unfortunately, it also contains thousands of correction terms for each planet (Bretagnon, Table 6). Amateur astronomy programs typically use, at best, a simplified version of VSOP87.
Lunar Positions. The Moon's orbit about Earth is only approximately elliptical, because of the effect of the Sun, the Earth's equatorial bulge, the planets, etc. It is thus incredibly difficult to predict.
The average lunar motion is about 30'/hour, but three anomalies (eccentricity, evection and variation) were known to the down-timers. In theory, those are enough to predict the lunar position with an accuracy of about 10' (Fitzpatrick).
Flamsteed thought that the lunar theory of 1683 was capable of predicting lunar position with an accuracy of at most 12', and Kollerstrom believes that Newton's 1702 theory was accurate to 7-8'. The theoretical accuracy would be degraded by computation errors. For example, in 1695-1701, a French almanac had lunar longitude errors in 1695-1701 which sometimes exceeded 30'. (Kollerstrom; cp. Williams 79).
The "gold standard" for the Moon is the "semi-analytical" ELP2000. It is accurate to 1.5" for 1900-2100 and 20" for 1500-2500 (Giesen). Amateur astronomy software would probably use a truncated version with arc-minute accuracy.
* * *
The computation of astronomical tables by hand calculations is laborious and error-prone. The process of constructing almanacs would be simplified by amateur astronomy software.
The marketing appeal of these programs often resided in their graphical representations of the sky. But what we want is a program which can print an ephemeris, showing either, for a particular day, the positions of the sun, moon, and planets, or, for a series of days, the positions of a single celestial object.
Secondly, the program must allow the user to specify a date in the 1630s. Starry Night Deluxe, for example, allowed an earliest date of 4713 BC.
Finally, the program must provide accurate results for seventeenth century dates. But we don't need VSOP87. If the software correctly implements the algorithms published by Meeus' Astronomical Algorithms (a popular source), it will be accurate for planetary positions in the 1630s to within an arc-second (ProjectPluto.com), which for us is overkill. An NOAA reviewer (Code) said that Voyager II had "planet calculations good to a couple of seconds of arc over 500 years" and implied that it still was not as accurate as Red Shift or Starry Night.
The moon is more of a problem; Meeus only provides about ten arc-second accuracy even for modern times, which, for calculating longitude by lunar distance (see below), will result in longitude errors of about 5'.
If the software becomes unusable (e.g., the disk is copy-protected and becomes unreadable), then we have to compute the tables ourselves. We can use Kepler's orbital elements, the "up-time" elements, or elements determined by new post-RoF observations. New computer programs (or spreadsheets) can be written, both to derive elements from observations, and to produce ephemerides from the elements.
* * *
For some years after RoF, only Grantville and a few other towns in Europe will have computers and thus those places will produce most ephemerides. The output will probably be to dot matrix printers, which is considered to be a sustainable technology, in order to avoid the transcription errors associated with typesetting.
Navigational Use of the Pole Star
In the northern hemisphere, o
bservation of the Pole Star, Polaris, allows you to determine your latitude, as well as the direction of True North (as distinguished from Compass North). Of course, it is important that you know your constellations. "Columbus's celestial navigation was almost invariably unfortunate, a litany of wildly wrong latitudes caused by his mistaking other stars for Polaris." (Phillip-Birt 178)
If Polaris were in fact located at the NCP, then altitude of Polaris would be your latitude, and its bearing would be the direction of True North. Petrus Peregrinus' Epistola de Magnete (1269) recognized that Polaris moved in a small circle about the north celestial pole. (Polaris is now less than 1° from the NCP. In 1601, Polaris was at 87°08.7' declination, 5°56.6' RA—about 3° from NCP (Rawlins 99-S2). Wright (1599) said that the distance was 2d52' (Graham).)
The Regimento do astrolabio e do quadrante (Lisbon, 1510) evidences that down-timers know how to use the "rule of the north"—based on the orientation of the "guard stars" alpha and beta Ursa Minor—to find the latitude from the altitude of Polaris. If they formed a vertical line, then Polaris was at the same altitude as the celestial north, and no correction was necessary. If they were horizontal, then you had to add or subtract several degrees, depending on whether they pointed west or east. (Taylor 146).
Determining Latitude
Polestar Altitude. I have already alluded to use of the Pole Star to find latitude. That doesn't work in the daytime, or in the southern hemisphere.
Meridian Sight. The second method requires observing the altitude of a celestial object when it crosses the observer's meridian. For the sun, this will occur at local noon. (A circumpolar star will cross the meridian twice a day, and the crossings are upper and lower culmination.) Knowing the declination of the sun for the date in question, simple arithmetic yields the latitude.
Of course, that requires both an almanac with a declination table, and the ability to recognize when local noon has arrived. The sun ascends during the morning, and descends during the afternoon. Local noon is the moment at which the solar disk seems to hang motionless in the sky.
Obviously, if you don't measure the altitude at, precisely, local noon, the computed latitude will be in error. However, at moderate latitudes, around local noon the trajectory of the Sun is fairly flat, i.e., its altitude doesn't change rapidly. For 40°N, the maximum change of altitude is 0.1' at 2 min before or after local apparent noon, 0.6' at 4 min. Sail up to 80°N, and the altitude changes are 0.4' and 6', respectively. (Mixter 317)
Double Altitude. Sometimes the weather doesn't permit a meridian sighting of the sun. If you make two successive observations, and you know the time interval between them, and the sun's declination (almanac), you can calculate the latitude without bothering to observe the sun at local noon. For this purpose, the watch doesn't need to keep accurate time over the long term; it just needs to be able to measure a time interval of an hour or two. The measurements should be close to when the object reaches the meridian, and the procedure is more prone to error when the meridian crossing is high in the sky. (Bowditch/1826, 128) A complication is that the first altitude must be corrected for the estimated movement of the ship.
Ex-Meridian Altitude. The weather may be so bad that you can only make one sighting, but close to noon. You can still compute the latitude from the altitude, albeit less reliably. The first item you need is a well-regulated watch; one which keeps good time and which was recently set (perhaps the preceding morning), based on celestial observations, to the local time. The second is an estimated latitude (Bowditch) or longitude. And you need almanac information.
Equal Altitude. This is a special case of the double altitude method. You take a timed morning sighting, and then, in the afternoon, time when the Sun drops back to the same altitude. (You keep the sextant set at the original altitude and let the Sun swim into view.) The time midway between, suitably adjusted, is considered the time of local noon. (Williams 111). Polter (1605) objected to the use of equal altitudes because the declination changes between readings, even though the change is only 1'/hour, at most (Taylor 218).
If the Sun is hidden all day, or poorly located for use of the double altitude method, one may instead observe a star (if there is a visible horizon), a planet, or the Moon, but the latter changes its celestial position rapidly, and this poses computational complications.
Accuracy. Drake's accuracy (1579) was about 9' for measurements on shore and 21' for those at sea (DNG).
Determining Longitude
All methods of determining longitude require comparing local time with the simultaneous time at a reference meridian (e.g., Greenwich). The difference in time, multiplied by 15°, yields the difference in longitude.
No celestial object hovers over a single point of the Earth's surface (although Polaris comes close). During the course of a day, as a result of the Earth's rotation, the GP of a star traces a circle on the Earth's surface. That circle is at a fixed latitude (determined by the declination of the star) but the longitude of the GP can only be determined if you know the local time. And you need to know the longitude of the GP if you want to calculate the longitude of the ship.
Local time. The simplest way, in theory, to know the local time of an observation is to carry it out when the sun has hung in the sky (reached meridian altitude), which is, approximately, local noon. A watch time can be corrected, after the fact, to local time by using the equal altitude method (see "determining latitude" above) to determine the watch time at which local noon occurred. (Preston 172) At sea, it was more common to shoot the Sun when it was bearing east or west and use its altitude, together with computed latitude and estimated longitude, and the Sun's declination, to calculate the time of observation (Bowditch 155). Star positions can also be used to estimate a local time.
If local noon is determined by a sighting, the time since noon can be tracked by means of an hourglass or, better yet, a simple timepiece. (Even a timepiece that was not suitable for keeping accurate time over the length of a voyage might be reasonably accurate over the hours between a noon-sight and a twilight-sight.) And at night, local time could be determined to perhaps the nearest quarter-hour using a "nocturnal" (see part 1).
Reference time. The reference time may be determined either by observing some celestial event (which happens essentially simultaneously for both the reference observatory and the ship's location), or by inspection of a chronometer set previously to the reference and which has kept "consistent" time (it loses or gains time in a predictable manner) since then.
The celestial events which have been used for longitude determinations include jovian moon eclipses, lunar eclipses, lunar occultations, and particular angular separations of the moon from the Sun or stars.
Jovian Moon Method. In theory, a reference time could be determined by noting when the moons of Jupiter passed into or out of its shadow, and comparing it to the times stated in an ephemeris computed for a location of known longitude. When a predicted immersion or emersion was observed, a clock was set to the ephemeris time. The next day, the observer noted the clock time at which the sun peaked (local noon). You then calculated the longitude, hoping that in the course of a day, the clock hopefully wouldn't lose or gain too much time from the true reference time.
The ephemeris for Paris was calculated by Cassini in 1668 and by 1696 Cassini published a map of the world which used longitudes determined by this method. Unfortunately, the method was impractical on shipboard. The necessary telescope (15-20 feet long) had a narrow field of view, so it would be difficult to keep the moons under observation while the ship pitched and rolled. If you were using a pendulum clock, then there was the further problem that the clock wouldn't work properly, even over the relatively short time interval between the two necessary observations The experienced astronomer Halley tried, but concluded that the Jovian eclipses were "absolutely unfit at sea" (Mentzer; Wakefield 86-7).
Just as well. Cassini's tables were in substantial error because he failed to consider the effect of the finite speed of light
(discovered by Roemer, 1676) on the time of observation of Jovian eclipses (Wakefield 164).
Lunar eclipse. A lunar eclipse occurs when the Moon passes through the Earth's shadow. It is observable from anywhere on the night hemisphere, and begins and ends at the same time for all observers. If you have an almanac giving the time the eclipse begins or ends for a reference site of known longitude, you can compare that reference time to the local time. Unfortunately, lunar eclipses occur only a few times a year, are difficult to time, and in practice yield an accuracy of only perhaps 0.5-1.5° (Espenak; Oliver).
Lunar occultation. The moon takes about 29.5 days (its synodic period) to travel 360° in celestial longitude, so its change in celestial longitude is about half a degree per hour. In contrast, the stars have essentially fixed celestial longitudes. Hence, the movement of the moon, relative to the stars, could be used to judge the reference time.
Initially, it was proposed that astronomers predict the times that the moon would "occult" (pass in front of) various stars. Unfortunately, while we think of the moon as large, its angular size is half a degree—that of a penny held 2.29 meters away. On a ship at sea, you are normally going to be able to identify the brighter stars, and the odds are not great that, on a particular night, one of these will be occulted by the moon.
Lunar Distance ("Lunars"). Hence, astronomers instead predicted the "lunar distance," the angular distance between the moon and a celestial reference point (a star or the Sun), for different hours of the day, day after day. (Although the Sun moves against the sky, its celestial longitude can be predicted with accuracy.)