Sources of Computations and Cautions concerning Accuracy


Visibility phenomena have been computed since antiquity, but are still among the most uncertain astronomical phenomena to calculate accurately. First, there are the obstacles of weather, atmospheric transparency, and ambient light, which are difficult or impossible to control by any calculation. But even assuming optimal observing conditions, there remains the problem of the absence of an adequate body of reliable observations of the phenomena for comparison with calculation. In the case of the first visibility of the moon, the SAAO Lunar Crescent Visibility Database contains about 120 unaided observations of first visibility, not all completely reliable, which are reproduced reasonably well by the formula for first visibility we have used. For the planets, reliable observations are almost nonexistent. The most extensive body is in the Babylonian Astronomical Diaries, fragments of which, in varying state of preservation, survive from -651 to -60, but these do not appear to be reliable, as shown by the many erroneous lengths of true synodic periods between observations of phenomena bounding single synodic periods; errors of two or three days are very common and far larger errors also occur. It is next to impossible to evaluate the reliability of a single observation within a range of ±2 or ±3 days. Karl Schoch based his values of the arcus visionis on his own observations near Berlin and upon Babylonian observations, and one knows the reliability of neither his observations nor the values of the arcus visionis he derived. What is required to establish an adequate empirical foundation for planetary visibility is many years of observations, since one would need to observe all phenomena through the zodiac and the period of Saturn is thirty years, and made from different stations to test the effects of geographical latitude. Even this would leave uncertainties for Venus since its phenomena occur in only five points in eight years which regress slowly through the zodiac. Observations of this kind could be carried out by amateur astronomers in a cooperative project, but no one after the Babylonians has observed these phenomena consistently. Hence, there is no adequate empirical evidence for testing calculations of planetary visibility. And this is even more true of stellar visibility since, with the exception of Sirius and a very few other stars, there are virtually no observational records of visibility phenomena.

This brings us to the theory and parameters underlying our calculations. The ephemeris calculations for the sun, moon, and planets are based upon Steve Moshier's ephemeris (www.moshier.net), which agrees closely with Jet Propulsion Laboratory's DE404, modified by corrections in the Alcyone Ephemeris to agree more closely with JPL's long ephemeris DE406. (For comparisons of Alcyone Ephemeris with DE406 as well as JPL's Horizons, and an evaluation of sources, see the documentation of Alcyone Ephemeris at www.alcyone-ephemeris.info, from which the ephemeris and its documentation may be downloaded.) The computation of the coordinates of stars is specific to this program, using equatorial coordinates for J2000 and proper motions from the Yale Bright Star Catalogue. There is no problem with the accuracy of ephemeris calculations of longitude, latitude, and times of rising and setting. Although accuracy does decrease with increasing intervals of time, greater than, say, ±2,500 years from the present epoch, the uncertainties are too small to affect the dates of visibility phenomena. The coordinates of the ephemeris used in PLSV differ from DE406 by only a few seconds for the planets and less than 30" for the longitude and 20" for the latitude of the moon in -2999. There are, however, additional uncertainties for the moon because of uncertainties in delta T, the correction for secular acceleration in the earth's rate of rotation. Delta T = ET - UT, the difference between (uniform) Ephemeris Time and (nonuniform) Universal Time, used in PLSV, is computed from the formulas of Chapront-Touzé & Chapront (1991, see www.phys.uu.nl/~vgent/astro/deltatime.htm.). The differences between these and other formulas amount to about 0' to 30' in the longitude of the moon in AD 0 and 4' to 2° 30' in -2999, which, in the worst case can affect the date of first or last visibility by one day although for the most part should have no effect. The uncertainties to be described below are far greater than anything in the underlying ephemeris.
The visibility diagrams themselves, which depend upon nothing more than the ephemeris computations, can therefore be considered accurate, particularly within their intervals of 4 minutes of time and 0.1° of altitude and azimuth. The problem comes in the dates of visibility phenomena computed from the fixed and variable arcus visionis. The method of computation from the arcus visionis has been explained in the preceding treatment of the superior and inferior planets. Here we consider the sources and uncertainties of the parameters. We have used Schoch's values of the arcus visionis for first and last visibility as our default fixed values simply because they have been in use for many years and no other complete sets are known aside from Ptolemy's, which are given below. If one computes many phenomena from Schoch's values, looks at the range of magnitudes at the phenomena, and then graphs magnitude against arcus visionis, one finds that they lie close to a line of the sort expressed by our variable arcus visionis, which suggests that a linear relation between magnitude and arcus visionis is at least reasonable. Strictly, the linear relation derived from Schoch's values is AV = 10.24 + 1.45 x magnitude, but with some outliers, which suggest the following corrections, three of which are limits:
Phenomena Schoch Corrected
Mercury EF 10.5° 9 -10.5°
Mercury EL 11 11-13
Mercury MF 13 11-13
Mars F 14.5 13.5
When these values have been substituted, the result is AV = 10.5 + 1.44 x magnitude, from which our default AV = 10.5 + 1.4 x magnitude is derived. A comparison of Schoch's fixed AV and the range of our variable AV for the limits of magnitudes found at first and last visibilities, computed for Babylon, since that is where the greatest number of observations have been made, with a critical altitude of 1°, is as follows:
Phenomenon Magnitude Fixed Variable
Saturn F 0.2 to 1.4 13° 10.8° to 12.5°
L 0.3 to 1.4 10 10.9 to 12.5
Jupiter F -1.6 to -1.2 9.3 8.3 to 8.8
L -1.6 to -1.2 7.4 8.3 to 8.8
Mars F 1.4 to 2.0 14.5 12.5 to 13.3
L 1.4 to 2.0 13.2 12.5 to 13.3
Venus EF -3.4 to -3.3 6 5.7 to 5.9
EL -4.0 to -3.0 5.2 4.9 to 6.3
MF -3.4 to -3.1 5.7 5.7 to 6.2
ML -3.4 to -3.3 6 5.7 to 5.9
Mercury EF -1.4 to 0.0 10.5 8.5 to 10.5
EL 1.0 to 2.2 11 11.9 to 13.6
MF 0.5 to 1.5 13 11.2 to 12.6
ML -1.3 to -0.2 9.5 8.7 to 10.2
The fixed AV's shown here, although not without question, are the default parameters. The range of the variable AV results from the default AV = 10.5 + 1.4 x magnitude. In any case, the fixed AV for each phenomenon for each planet may be altered separately, although probably no fixed AV is adequate, and the constant and coefficient of the variable AV may be altered for each planet, including to produce something closer to Schoch's fixed values modified by variation of magnitude; the alterations will be saved until changed or the default values replaced. The alterations may be made to 0.01, but in truth these parameters are so imperfectly known that anything beyond 0.1 is probably false precision. We believe the variable AV must be preferable to the fixed AV; the problem is determining the best constant and coefficient, either for all the planets together or for each planet separately if they do in fact require or benefit from individual values.
Since they are of historical interest and may prove useful, we also give Ptolemy's values of the arcus visionis. These differ in the Almagest and in the later Handy Tables and Planetary Hypotheses, which are probably based upon later observations and appear to be somewhat better.
Phenomenon Almagest HT & PH
Saturn F & L 11° 13°
Jupiter F & L 10 9
Mars F & L 11.5 14.5
Venus EF & ML 5 7
Venus MF & EL 5 5
Mercury All 10 12
For acronychal rising and cosmical setting of the superior planets, no arcus visionis has been determined, although both phenomena probably use the same value as the planet is in both cases close to opposition at the horizon opposite the sun. Ptolemy says in the Planetary Hypotheses that the values for acronychal rising should be about half those for heliacal rising and setting. Our default fixed values are based upon an extrapolation from the values for first and last visibility based upon the greater brightness of the planet near opposition and the darker horizon opposite the sun. It is easy to show, however, that fixed values are not sufficient and a variable arcus visionis is preferable, as is clear from the table:
Planet Magnitude Fixed Corrected Variable
Saturn -0.8 to -0.3 8 8 to 8.5 8.0 to 8.6
Jupiter -2.5 to -2.0 6 6 to 6.5 6.2 to 6.7
Mars -2.6 to -0.9 6 6 to 8 6.0 to 7.9
The range of magnitudes suggests that the fixed AV should be replaced with a range, small for Saturn and Jupiter, but quite large for Mars. Using this range, gives the variable AV = 8.9 + 1.15 x magnitude, from which our default AV = 8.9 + 1.1 x magnitude is derived; its results are shown in the last column. It is possible that the values for Saturn and Jupiter, both fixed and variable, are somewhat larger than required; because of its large variation in magnitude, no fixed value appears to be satisfactory for Mars. Here again, the parameters for both fixed and variable AV may be altered separately for each planet to 0.01, although 0.1 is more realistic; the alterations will be saved until replaced or the default values restored.
Little is known of the arcus visionis of stars, for which we have used the formulas for variable magnitude for the superior planets, not because the magnitude of a single star varies significantly, but because the formulas can be used for the range of magnitudes of stars. The following table gives the arcus visionis computed by the formulas with default parameters for magnitudes at intervals of 0.5 to magnitude 3 and then 1.0 to magnitude 6:
Magnitude Heliacal Acronychal
-1.5 8.4 7.25
-1.0 9.1 7.8
-0.5 9.8 8.35
0.0 10.5 8.9
0.5 11.2 9.45
1.0 11.9 10.0
1.5 12.6 10.55
2.0 13.3 11.1
2.5 14.0 11.65
3.0 14.7 12.2
4.0 16.1 13.3
5.0 17.5 14.4
6.0 18.8 15.5
We do not know whether these values of the arcus visionis are correct; it is possible that the values for heliacal phenomena are slightly high and for acronychal phenomena slightly low. They can be changed in the stellar visibility page, either individually in the in use and default columns or by changing the parameters of the formulas applied to all stars. The critical altitude, which can vary with magnitude as well as local observing conditions, may also be set separately for individual stars.
Aside from the distinction between rising and setting phenomena close to the sun and acronychal phenomena at the horizon opposite the sun, we have made no attempt to take into account variation of the difference of azimuth between the planet and the sun. Variation of difference of azimuth probably has only a small effect opposite the sun, but can have a large effect close to the sun, particularly at higher geographical latitudes where the variation can be large. For stars, which can rise and set at any difference of azimuth from the sun, although the difference for each star is constant except for the long-term effects of precession, it is surely a significant factor in the dates of visibility phenomena. It is possible that by computing a large number of phenomena of planets and stars and examining the differences of azimuth, which are tabulated in the more information window of heliacal dates, some relation may be derived which would allow a refinement of the arcus visionis, but we have not attempted to do this.

Summary remarks. The calculation of visibility phenomena is plagued by uncertainties that will only be resolved by a body of reliable observations that does not yet exist. We have done our best to provide flexible methods of computation based upon the arcus visionis, either fixed or variable, with all parameters adjustable individually for each planet and star. In addition, the critical altitude for visibility is adjustable for each planet separately and for each star separately or for all stars together. It appears from trial calculations that changes in the critical altitude produce greater differences in the dates of phenomena than reasonable changes in the arcus visionis, so the critical altitude must be set with great care. The user is encouraged to experiment with different parameters to find which appear to produce the most accurate, or most reasonable, results, although in the absence of reliable observations for comparison, it is difficult to say what most accurate or most reasonable is.