Some remarks concerning solar eclipse data predictions

Size: px

Start display at page:

Download "Some remarks concerning solar eclipse data predictions"

Mark Sutton
5 years ago
Views:

1 Some remarks concerning solar eclipse data predictions Robert Nufer, Switzerland Computing solar eclipse predictions consists of two main "steps": 1.) Accurate computing of the positions of Sun, Earth and Moon and 2.) computing the geometry of the Moon's shadow on Earth. The accurate positions of the solar system bodies can be computed with (two) different mathematical methods. Numerical integration and analytical approach (calculation of the perturbations caused by other planets to the orbits of Earth and Moon). To compute the planetary positions for a desired time point with numerical integration the positions and velocities of the involved bodies must be known for a certain time point and from this time point Newton's gravitational law (and relativistic effects...) is applied in small time steps until the desired time point is reached. The advantage of numerical integration is the "intrinsic" accuracy which is limited only by the computer's numerical representation of numbers (rounding errors). The chosen step size is a compromise between cumulative rounding errors and precision. To compute the planetary positions for any desired time point with the analytical approach (ten)thousands of terms (perturbations) are summed. For practical reasons very small terms are ignored. For the calculation of solar eclipses over long time periods both methods may be used. The "mathematical errors" of both methods are much smaller than our knowledge about the physical parameters (mass, velocity, figure...) of the bodies. The Moon's position is influenced not only by the well known gravitational law but also by a "secular acceleration" caused by tidal effects. This additional acceleration yields to a difference in the Moons position of -26"/century 2. Several eclipse prediction works use(d) different values between "/century 2 and -26"/century 2 ("Six Millennium Catalog of Solar Eclipses" by Fred Espenak). The most modern value is "/century 2 ("" by Fred Espenak and Jean Meeus). To compute a "true" planetary position to an accuracy of 100 meters over a time period of five thousand years the planets position must be known to an accuracy of 100 meters on a certain date (near the present), the EXACT masses of all involved bodies must be known, and NO ROUNDING ERRORS are allowed! The calculation of the eclipse phenomena is also not possible in a mathematical exact manner: We do not know the flattening of the Earth to more than five to six significant digits! We do not know the exact offset of the Moon's center of figure from its center of mass. Furthermore, the surface of the Moon is not polished, but there are mountains and depressions, as on Earth. How should we define the size of the Moon? If an average lunar radius is used the changeover between annularity an totality will not be "true". This case happened on October 3, Some authors stated this eclipse as annular-total, even with a maximum totality duration of zero seconds, where it was in fact never total. There was always sunlight shining through some lunar depressions.

2 There are several assumptions, adoptions and definitions that can change the type of an eclipse or even an eclipse to become inexistent. In any carefully performed eclipse calculation these effects are always observed at the moment when the shadow axis is very tangent to the Earth's surface. Example: The annular-total eclipse of December 17, begins total, remains total until approx. five seconds! before sunset. This can be visualized with my solar eclipse program FInspektor ( To get a "feeling" for the "impact" of different eclipse predictions I compared the "Five Millennium Catalog of Solar Eclipses" (5MCSE) from Fred Espenak and Jean Meeus and the "" (6MCSE) from Fred Espenak with my own calculations, a numerical integration (using the program Solex91 from Aldo Vitagliano) over ten millennia from to To make it not too complicated, I compare here the first century common to all catalogs, the 20th century BC from to From my numerical integration I compiled a seven millennia catalog with Saros counting information, and complete sets of Besselian elements in polynomial form. These data are now used in my program FInspektor. The only different eclipse type between my compilation and the 5MCSE in this century is the eclipse of December 17, (FInspektor "labels" this eclipse as total, but when inspecting the eclipse near sunset one can observe the annularity approx. five seconds before sunset! The differences between the two catalogs 5MCSE and 6MCSE are significant greater than the differences between my calculations and the 5MCSE. The root mean squares (rms) are given at the end of the table.

3 The following table contains eclipse prediction data from three data sources: ( ( Robert Nufer The columns contain from left to right: Nr Number of eclipse in century Date and time (TD) at eclipse maximum For all three data sources: Tp γ mg W D Eclipse type according to "Key to Catalog of Solar Eclipses" (Fred Espenak and Jean Meeus). The letters "b", "m" and "e" for begin, middle and end of Saros are omitted. Gamma. Minimal distance of the shadow axis from the Earth's center (Equatorial Earth radii) Eclipse magnitude Path width at eclipse maximum (kilometers) Center line duration of total or annular phase at eclipse maximum (seconds) For the "Six Millennium..." and "my own..." data four columns are given containing the differences to the "Five Millennium..." data: Unit is the last digit of the corresponding data column. γ mg W D Difference between γs. (1 = Equatorial Earth radii (~640 meters)) Difference between eclipse magnitudes (1= magnitude) Difference between path widths (kilometers) Difference between Center line durations at eclipse maximum (seconds)

4 Jun 12 03:14:51 T T T Dec 5 23:45:23 A A A Jun 1 18:09:16 T T T Nov 25 05:57:03 A A A Apr 22 13:19:56 P P P May 22 02:45:35 P P P Oct 16 08:01:52 P P P Nov 14 18:48:49 P P P Apr 10 13:54:52 A A A Oct 4 23:23:37 T T T Mar 30 17:24:52 A A A Sep 24 10:31:54 A A A Mar 20 03:59:50 T T T Sep 13 14:32:00 A A A Feb 8 11:41:40 P P P Mar 9 19:48:09 P P P Aug 3 21:35:06 P P P Jan 29 02:34:14 T T T Jul 23 04:03:17 H H H Jan 17 11:29:42 A A A Jul 12 17:34:12 T T T Jan 6 13:10:01 P P P Jul 2 10:30:06 T T T Nov 26 19:21:37 P P P May 23 16:56:38 H H H Nov 16 03:39:33 H H H May 11 23:28:58 A A A Nov 4 17:46:47 T T T May 1 00:29:51 A A A Oct 25 09:21:08 T T T

5 Mar 21 15:06:51 P P P Apr 20 02:55:09 P P P Sep 15 07:15:14 P P P Mar 11 03:26:36 T T T Sep 4 09:29:03 A A A Feb 28 19:51:38 T T T Aug 23 09:09:06 A A A Feb 17 11:43:36 T T T Aug 12 13:43:07 A A A Jan 8 07:26:45 P P P Jul 3 17:34:50 P P P Aug 2 01:40:26 P P P Dec 28 07:38:06 A A A Jun 23 10:42:06 T T T Dec 17 07:52:15 A A A Jun 12 01:14:01 T T T Dec 5 14:32:25 A A A May 2 19:42:02 P Missing P Jun 1 09:17:39 P P P Oct 26 16:52:42 P P P Nov 25 03:41:17 P P P Apr 21 20:18:29 A A A Oct 16 08:02:30 T T T Apr 11 00:17:50 A A A Oct 5 18:43:53 A A A Mar 30 11:26:06 T T T Sep 23 22:12:57 A A A Feb 18 19:47:20 P P P Mar 20 03:32:45 P P P Aug 14 04:48:27 P P P

6 Feb 8 10:34:06 T T T Aug 3 11:35:27 H H H Jan 28 19:13:23 A A A Jul 24 01:18:22 T T T Jan 17 20:43:43 P P P Jul 12 18:09:32 T T T Dec 7 03:38:57 P P P Jun 2 23:50:31 A A A Nov 26 12:22:23 H H H May 23 05:53:04 A A A Nov 16 02:41:09 T T T May 12 06:45:38 A A A Nov 5 18:10:59 T T T Mar 31 22:13:17 P P P Apr 30 09:34:02 P P P Sep 25 15:13:22 P P P Mar 21 11:03:12 T T T Sep 14 16:59:19 A A A Mar 11 03:42:14 T T T Sep 3 16:35:18 A A A Feb 28 19:33:01 T T T Aug 23 21:25:13 A A A Jan 19 15:11:49 P P P Jul 14 01:14:53 P P P Aug 12 09:35:44 P P P Jan 7 15:19:18 A A A Jul 3 18:12:59 T T T Dec 27 15:54:04 A A A Jun 23 08:20:59 T T T Dec 16 23:03:15 A A A

7 Jun 12 15:51:46 P P P Nov 7 01:46:11 P P P Dec 6 12:31:33 P P P May 2 02:39:40 A A A Oct 26 16:45:10 T T T Apr 21 07:07:29 A A A Oct 16 03:01:44 A A A Apr 10 18:45:52 T T T Oct 5 06:02:20 A A A Mar 2 03:43:57 P P P Mar 31 11:10:33 P P P Aug 25 12:12:43 P P P Sep 24 05:12:58 P P P Feb 19 18:25:24 T T T Aug 13 19:17:10 H H H Feb 8 02:47:50 A A A Aug 3 09:08:54 T T T Jan 28 04:09:29 A A A Jul 24 01:54:42 T T T Dec 18 11:52:50 P P P Jun 14 06:46:26 P P P Dec 7 21:03:10 H H H Jun 2 12:18:03 A A A Nov 26 11:34:24 T T T May 22 13:02:16 A A A Nov 16 03:00:41 T T T Apr 12 05:14:40 P P P May 11 16:12:21 P P P Oct 6 23:18:01 P P P Apr 1 18:33:43 T T T

8 Sep 26 00:37:49 A A A Mar 21 11:25:32 T T T Sep 14 00:12:27 A A A Mar 11 03:11:51 T T T Sep 3 05:17:59 A A A Jan 29 22:46:25 P P P Jul 25 09:01:49 P P P Aug 23 17:40:35 P P P Jan 18 22:51:51 A A A Jul 15 01:49:25 T T T Jan 7 23:51:29 A A A Jul 3 15:30:05 H H H Dec 27 07:31:39 A A A Jun 22 22:27:06 P P P Nov 17 10:40:46 P P P Dec 16 21:18:48 P P P May 13 08:59:51 A As A Nov 7 01:29:55 H H H May 2 13:57:20 H H H Oct 27 11:21:18 A A A Apr 21 02:04:36 T T T Oct 15 13:56:25 A A A Mar 12 11:31:58 P P P Apr 10 18:42:36 P P P Sep 4 19:47:20 P P P Oct 4 13:00:58 P P P Mar 2 02:05:12 T T T Aug 25 03:09:40 H H H Feb 19 10:09:18 A A A Aug 14 17:09:41 T T T

9 Feb 8 11:24:38 An A An Aug 3 09:46:19 T T T Dec 28 20:02:32 P P P Jun 24 13:43:01 P P P Dec 18 05:41:01 H H T Jun 13 18:42:52 A A A Dec 7 20:26:30 T T T Jun 2 19:20:26 A A A Nov 27 11:49:40 T T T Apr 22 12:14:51 P P P May 21 22:52:55 P P P Oct 17 07:27:29 P P P Apr 12 01:59:44 T T T Oct 6 08:23:27 A A A Apr 1 19:02:33 T T T Sep 25 07:58:25 A A A Mar 22 10:43:01 T T T Sep 14 13:20:52 A A A Feb 10 06:11:20 P P P Aug 4 16:56:04 P P P Sep 3 01:53:33 P P P Jan 29 06:15:48 A A A Jul 25 09:31:57 T T T Jan 18 07:40:04 A A A Jul 14 22:44:43 H H H Jan 7 15:52:48 H H H Jul 4 05:07:17 P A P Nov 28 19:33:50 P P P Dec 28 06:00:28 P P P May 23 15:21:27 P P P

10 Nov 17 10:15:36 H H H May 12 20:45:04 H H H Nov 6 19:44:18 A A A May 2 09:19:50 T T T Oct 26 21:55:31 A A A Mar 23 19:12:17 P P P Apr 22 02:10:29 P P P Sep 16 03:32:55 P P P Oct 15 20:56:53 P P P Mar 12 09:36:57 T T T Sep 4 11:10:54 H H H Mar 1 17:22:05 A A A Aug 25 01:18:07 T T T Feb 18 18:29:52 A A A Aug 14 17:45:39 T T T Jan 9 04:06:28 P P P Jul 5 20:44:02 P P P Aug 4 07:51:37 P P P Dec 29 14:13:35 T T T Jun 24 01:12:09 A A A Dec 18 05:14:10 T T T Jun 13 01:41:44 A A A Dec 7 20:35:33 T T T Jun 2 05:36:42 P P P Oct 28 15:39:52 P P P Apr 23 09:22:31 T T T Oct 17 16:15:51 A A A Apr 12 02:33:41 T T T Oct 5 15:53:35 A A A Apr 1 18:05:15 T T T

11 Sep 24 21:33:45 A A A Feb 20 13:23:40 P P P Mar 22 03:40:50 P P P Aug 16 00:59:14 P P P Sep 14 10:16:33 P P P Feb 9 13:30:39 A A A Aug 5 17:20:50 T T T Jan 29 15:22:41 A A A Jul 25 06:03:48 H H H Jan 18 00:08:49 H H H Jul 14 11:50:45 A A A Dec 9 04:25:19 P P P Jan 7 14:37:00 P P P Jun 3 21:45:37 P P P Nov 28 18:58:47 H H H May 24 03:36:37 H H H Nov 18 04:06:24 A A A May 12 16:35:44 T T T Nov 6 05:56:20 A A A Apr 3 02:45:51 P P P May 2 09:35:23 P P P Sep 26 11:27:50 P P P Oct 26 04:57:57 P P P Mar 23 16:57:33 T T T Sep 15 19:22:56 H H H Mar 13 00:23:10 A A A Sep 5 09:35:37 T T T Mar 1 01:25:12 A A A Aug 25 01:52:52 T T T rms rms

GAMINGRE 8/1/ of 7

GAMINGRE 8/1/ of 7 FYE 09/30/92 JULY 92 0.00 254,550.00 0.00 0 0 0 0 0 0 0 0 0 254,550.00 0.00 0.00 0.00 0.00 254,550.00 AUG 10,616,710.31 5,299.95 845,656.83 84,565.68 61,084.86 23,480.82 339,734.73 135,893.89 67,946.95