On Accuracy
Mark Pottenger
I just sent the final master disks for the 1988 release of the CCRS Horoscope Program to Astrolabe. This article is an adaptation of one of the appendices in the manual which I thought might be of interest.
This release extends the ephemerides to the years 4,000 B.C. through 2,500 A.D., adds timed transits and dated progressions, and puts back planetary returns after a 5-year absence. The accuracy of all planetary calculations has been increased, and none of these changes would have made sense without that improvement. The rest of this article is devoted to qualifying and clarifying that claim.
Delta t
One of the reasons any claims for accuracy in ancient dates are subject to question is a fundamental problem with time. All theories of planetary motion work with some form of time variable based on a consistent, uniform and nonvarying system of time measurement. Our ordinary day of solar (or sidereal) time based on the rotation of the earth is not such a system. The earth’s rotation is gradually slowing on a long time-scale and also shows small and irregular changes on a short time-scale. Delta t, meaning change or difference in time, is a correction (fudge factor) to get from clock times tied to the rotation of the earth to the uniform ephemeris time or dynamical time used in planetary theories. Delta t is determined empirically by comparing planetary and lunar theory to observations, so values for delta t are always a year or two behind the present. The standard Astronomical Almanac gives estimated values only a year or two into the future.
In the CCRS Horoscope Program, values for delta t from 1620 to the present are in a disk file (DELTAT) which the program reads. For future dates, the program just uses the last value of delta t in the file. For dates before 1620, the program uses a formula to estimate delta t. It is in estimating ancient values for delta t that we run into a major uncertainty. In versions up through 1987, I used a formula in Meeus’ calculator book. In the 1988 version I have switched to a formula in Bretagnon-Simon. The table below shows the effect of this change.
DELTA T ESTIMATES (IN SECONDS & HOURS)
|
YEAR |
PREVIOUS S |
PREVIOUS H |
CURRENT S |
CURRENT H |
|
-4000 |
100021 |
27.78 |
109692 |
30.47 |
|
-3900 |
96589 |
26.83 |
105948 |
29.43 |
|
-3800 |
93217 |
25.89 |
102269 |
28.41 |
|
-3700 |
89904 |
24.97 |
98655 |
27.40 |
|
-3600 |
86652 |
24.07 |
95106 |
26.42 |
|
-3500 |
83459 |
23.18 |
91622 |
25.45 |
|
-3400 |
80327 |
22.31 |
88203 |
24.50 |
|
-3300 |
77254 |
21.46 |
84849 |
23.57 |
|
-3200 |
74242 |
20.62 |
81560 |
22.66 |
|
-3100 |
71289 |
19.80 |
78336 |
21.76 |
|
-3000 |
68396 |
19.00 |
75177 |
20.88 |
|
-2900 |
65563 |
18.21 |
72083 |
20.02 |
|
-2800 |
62790 |
17.44 |
69054 |
19.18 |
|
-2700 |
60076 |
16.69 |
66090 |
18.36 |
|
-2600 |
57423 |
15.95 |
63191 |
17.55 |
|
-2500 |
54830 |
15.23 |
60357 |
16.77 |
|
-2400 |
52296 |
14.53 |
57588 |
16.00 |
|
-2300 |
49823 |
13.84 |
54884 |
15.25 |
|
-2200 |
47409 |
13.17 |
52245 |
14.51 |
|
-2100 |
45055 |
12.52 |
49671 |
13.80 |
|
-2000 |
42761 |
11.88 |
47162 |
13.10 |
|
-1900 |
40527 |
11.26 |
44718 |
12.42 |
|
-1800 |
38353 |
10.65 |
42339 |
11.76 |
|
-1700 |
36239 |
10.07 |
40025 |
11.12 |
|
-1600 |
34185 |
9.50 |
37776 |
10.49 |
|
-1500 |
32190 |
8.94 |
35592 |
9.89 |
|
-1400 |
30256 |
8.40 |
33473 |
9.30 |
|
-1300 |
28381 |
7.88 |
31419 |
8.73 |
|
-1200 |
26567 |
7.38 |
29430 |
8.18 |
|
-1100 |
24812 |
6.89 |
27506 |
7.64 |
|
-1000 |
23117 |
6.42 |
25647 |
7.12 |
|
-900 |
21482 |
5.97 |
23853 |
6.63 |
|
-800 |
19907 |
5.53 |
22124 |
6.15 |
|
-700 |
18392 |
5.11 |
20460 |
5.68 |
|
-600 |
16937 |
4.70 |
18861 |
5.24 |
|
-500 |
15541 |
4.32 |
17327 |
4.81 |
|
-400 |
14206 |
3.95 |
15858 |
4.41 |
|
-300 |
12930 |
3.59 |
14454 |
4.02 |
|
-200 |
11715 |
3.25 |
13115 |
3.64 |
|
-100 |
10559 |
2.93 |
11841 |
3.29 |
|
0 |
9463 |
2.63 |
10632 |
2.95 |
|
100 |
8427 |
2.34 |
9488 |
2.64 |
|
200 |
7451 |
2.07 |
8409 |
2.34 |
|
300 |
6535 |
1.82 |
7395 |
2.05 |
|
400 |
5679 |
1.58 |
6446 |
1.79 |
|
500 |
4883 |
1.36 |
5562 |
1.55 |
|
600 |
4146 |
1.15 |
4743 |
1.32 |
|
700 |
3470 |
0.96 |
3989 |
1.11 |
|
800 |
2853 |
0.79 |
3300 |
0.92 |
|
900 |
2297 |
0.64 |
2676 |
0.74 |
|
1000 |
1800 |
0.50 |
2117 |
0.59 |
|
1100 |
1363 |
0.38 |
1623 |
0.45 |
|
1200 |
986 |
0.27 |
1194 |
0.33 |
|
1300 |
669 |
0.19 |
830 |
0.23 |
|
1400 |
412 |
0.11 |
531 |
0.15 |
|
1500 |
215 |
0.06 |
297 |
0.08 |
|
1600 |
77 |
0.02 |
128 |
0.04 |
As you can see, the two formulae for estimating delta t differ by hours for ancient dates (almost 3 hours at the new earliest date, and over 1/2 hour at the previous earliest date). I switched because I hope the newer formula is a more accurate estimate, but I don’t know how close either is to “reality”. There have also been questions over the years about the validity of delta t, though they mostly seem to have been settled.
Also, you can see from the table that delta t is literally changing the time for which you are calculating planets positions by hours for ancient dates (more than one full day before -3300).
Ephemerides
All ephemerides and theories of planetary motion involve some attempt to fit the predictions of gravitational theories to a real world of observations. This is essentially a statistical process—compare your numbers to observed planetary positions, tweak your numbers or your theory, compare again, and keep tweaking until you get the best match you can between your theory and the observations you are using to test it. This process has given us planetary theories which fit modern observations very well—well enough to guide space probes to the planets and their moons. For ancient observations, we have more uncertainty, and the delta t question. But it is important to remember in any discussion of long-term planetary calculations, especially for Neptune and Pluto, that the theories we are using are primarily elaborate curve-fitting to observations in this century and some in the last century.
For several years up through 1983, the standard ephemerides used internationally by the astronomical community were based on Sun, Mercury, Venus and Mars positions from formulae in Volume 6 of the Astronomical Papers, outer planets positions from Volume 12 of the Astronomical Papers, and Moon positions from formulae in the Improved Lunar Ephemeris.
The formulae I have used in versions of the CCRS Horoscope Program through 1987 were only intended to get positions good to the nearest minute of arc, with Mars occasionally off by as much as two minutes. For the 1988 version, I have upgraded to formulae which should be good to at worst a few seconds of arc. I am using full Volume 6 formulae for the Sun, Mercury, Venus and Mars, Volume 22 formulae for the outer planets, and all longitude terms of one tenth of a second of arc or larger from the Improved Lunar Ephemeris formulae for the Moon. Volume 22 is the same kind of formula as Volume 12, but it was published about 15 years later and has better values for Pluto. (I was using Volume 22 in the earlier versions, but had an accuracy problem for ancient dates from having used an approximate precession formula instead of an exact one.)
The standard ephemerides used internationally by the astronomical community were changed in 1984 to use positions calculated at JPL (Jet Propulsion Laboratory) with a numerical integration including the Sun, Moon, planets, five asteroids, and effects of lunar libration and relativity.
Positions from the CCRS Horoscope Program match pre-1984 ephemerides within less than one second of arc. Most positions are within two-tenths of a second, with Sun positions mostly within a few hundredths of a second. Many Moon longitudes are within one second of arc, with almost all the rest within two seconds. The Moon’s latitude and distance can be off quite a few seconds since I’m not using as many terms. (I plan to do more work with the Moon formula for next year’s release.) Since the astronomical standards changed in 1984, the match to ephemerides from 1984 on is not as close. Many positions are still within a second of arc, most are within two seconds, but I’ve noticed differences up to eight seconds with Neptune. (Note: this is a reduction in rated accuracy that does not reduce the internal consistency of the positions used—see the section on returns below.)
The new astronomical standard ephemerides were developed primarily by fitting to observations in this century, with some observations from last century. Many old observations couldn’t be fitted to the theory and weren’t used. For longer term ephemerides, I compared against two other sources: charts calculated at Astro Computing Services and positions from the program in the book Planetary Programs and Tables from -4000 to +2800 by Bretagnon and Simon. This program gives positions for the Sun, Mercury, Venus, Mars, Jupiter and Saturn for the years -4000 to +2800, and Uranus and Neptune for the years +1600 to +2800. All positions from Bretagnon-Simon are stated to be accurate to within 0.01 degrees (36 seconds).
The following tables show results of a spot-check against ACS and against Bretagnon-Simon:
CCRS vs. ACS differences in seconds of arc
|
YEAR |
Sun |
Moon |
Mer |
Ven |
Mar |
Jup |
Sat |
Ura |
Nep |
Plu |
|
-4000 |
0 |
15 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
|
-3000 |
0 |
12 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
-2000 |
0 |
4 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
|
-1000 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
|
0 |
0 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
|
1000 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
1600 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
|
1700 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
|
1800 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
1900 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
2000 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
2100 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
|
2500 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
CCRS vs. Bretagnon-Simon differences in minutes and seconds of arc
|
YEAR |
Sun |
Mer |
Ven |
Mar |
Jup |
Sat |
Ura |
Nep |
|
-4000 |
6’47” |
7’ 9” |
7’ 4” |
3’54” |
1’19” |
1’14” |
|
|
|
-3000 |
3’52 |
4’ 5 |
3’35 |
4’13 |
2’ 8 |
1’46 |
|
|
|
-2000 |
2’ 8 |
2’20 |
2’14 |
1’51 |
1’33 |
1’34 |
|
|
|
-1000 |
58” |
1’ 8” |
1’ 1” |
54” |
52” |
1’ 6” |
|
|
|
0 |
27 |
36 |
34 |
37 |
33 |
35 |
|
|
|
1000 |
10 |
12 |
14 |
28 |
12 |
15 |
|
|
|
1600 |
1 |
9 |
2 |
4 “ |
5 |
3 |
6” |
3 |
|
1700 |
2 |
1 |
1 |
2 |
3 |
1 |
1 |
8 |
|
1800 |
2 |
0 |
0 |
3 |
2 |
1 |
1 |
3 |
|
1900 |
1 |
2 |
1 |
1 |
1 |
1 |
1 |
2 |
|
2000 |
4 |
3 |
3 |
1 |
1 |
3 |
5 |
11 |
|
2100 |
10 |
14 |
14 |
0 |
2 |
5 |
3 |
31 |
|
2500 |
1’12 |
1’44 |
20 |
52 |
2 |
11 |
17 |
2’ 9 |
These tables show that CCRS positions are quite good in comparing to the astrological standard for excellence (ACS). The comparison to Bretagnon-Simon shows that there is room for work in comparison to recent astronomical work. The Neptune column is especially suspicious, and I will be looking into it for possible improvement in a future release. The entries for 2000, 2100 and 2500 clearly show the effect of delta t. My method and that used at ACS is to use the last confirmed value for delta t into the future, but Bretagnon-Simon uses the estimate formula for the future as well as the past. This gets a delta t of 1,532 seconds by 2500, compared to 56 seconds from the latest table entry—a difference of 25 minutes.
Considering the accuracy of available birth data, I believe the planetary positions calculated by CCRS ‘88 are good enough for astrological use.
Return times
The time of a solar, lunar, or planetary return is very dependent on the accuracy of the basic longitudes calculated.
The Sun moves an average of 59 minutes of arc per day (with only small variations between slowest and fastest), or 2 1/2 minutes of arc per hour, or 2 1/2 seconds of arc per minute of time. A Sun calculated to plus or minus one minute of arc accuracy would give you a solar return time with a worst possible error of nearly an hour. A one minute of arc error in the natal Sun is an error of 24 minutes of time. An additional one minute of arc error in the return Sun is another error of 24 minutes of time. Errors will often be in opposite directions and of different magnitudes, partially canceling each other, but if both errors add together you get a total error of 48 minutes of time in your return. A Sun good to plus or minus one second of arc gives you return times with a worst possible error of 48 seconds of time. The new Sun routine used in the CCRS Horoscope Program is accurate to a couple seconds of arc when compared to the current astronomical standard, giving worst possible time errors of 1 1/2 minutes. However, it is internally consistent to a few hundredths of a second of arc, which I believe gives a better picture of return time accuracy with a worst possible time error of 2 seconds. I mention consistency here for a reason: if your calculations are internally consistent to a higher precision than the accuracy of their match to observations, I think they give return times accurate at the level of their consistency rather than at the level of your match to observations. As an example, a watch that keeps time correctly to a tenth of a second but is set ten minutes off gives times consistent at the tenth of a second level, not the ten minute level. (Even if this line of thought is full of holes, the return times are still adequate.)
A slow Moon moves under 12 degrees per day, or 30 minutes of arc per hour, or 30 seconds of arc per minute of time. With the same logic as described for the Sun above, we get worst possible return time errors of 4 minutes of time for a Moon good to the nearest minute of arc and 4 seconds of time for a Moon good to the nearest second of arc. The Moon in the CCRS Horoscope Program is good to a couple seconds of arc, giving lunar return times within at worst 8 seconds of time.
Planetary return times are slightly more complicated because retrogrades and velocity changes make average figures much less meaningful. The uncertainty figure printed on the tabular pages of planetary returns is derived from the natal and return velocities. Minutes in a day (1440) is divided by seconds in a degree (3600), and the result is divided by the planetary velocity in degrees per day. The answer is possible error in minutes of time per second of arc. This figure is calculated for the natal velocity and for the return velocity and the results are added together and multiplied by 2 for the two second of arc accuracy level of the planetary calculations. The answer, a worst possible error in return time, is printed as the uncertainty figure. From this, you can see that the slower a planet is moving, the larger your uncertainty in time. Outer planet return times are inherently less certain than inner planet or solar and lunar returns. Return times of planets at stations or retrograde are less certain than for direct planets.
The following table illustrates the dependency of possible error on velocity:
ERROR IN TRANSITING TIME FOR 1 SECOND OF ARC ERROR IN LONGITUDE
|
VELOCTY |
ERROR |
VELOCTY |
ERROR |
|
10.000 |
0:02 |
0.100 |
4:00 |
|
9.000 |
0:03 |
0.090 |
4:27 |
|
8.000 |
0:03 |
0.080 |
5:00 |
|
7.000 |
0:03 |
0.070 |
5:43 |
|
6.000 |
0:04 |
0.060 |
6:40 |
|
5.000 |
0:05 |
0.050 |
8:00 |
|
4.000 |
0:06 |
0.040 |
10:00 |
|
3.000 |
0:08 |
0.030 |
13:20 |
|
2.000 |
0:12 |
0.020 |
20:00 |
|
1.000 |
0:24 |
0.010 |
40:00 |
|
0.900 |
0:27 |
0.009 |
44:27 |
|
0.800 |
0:30 |
0.008 |
50:00 |
|
0.700 |
0:34 |
0.007 |
57:09 |
|
0.600 |
0:40 |
0.006 |
66:40 |
|
0.500 |
0:48 |
0.005 |
80:00 |
|
0.400 |
1:00 |
0.004 |
100:00 |
|
0.300 |
1:20 |
0.003 |
133:20 |
|
0.200 |
2:00 |
0.002 |
200:00 |
|
0.100 |
4:00 |
0.001 |
400:00 |
This shows time error in minutes and seconds for a range of velocities in degrees per day. The table gives figures for a one second of arc error. To use it to judge the accuracy of times from any program, multiply the times by the rated accuracy (or real accuracy determined from experience if the rated accuracy is unrealistic) of the planetary positions involved (2 for CCRS ’88). For returns, possible error is always the sum of natal error and transiting error, based on the velocities at each time. Always remember that this figure is the maximum possible error and would only occur in the worst-case situation of both natal and transiting longitudes having the maximum error and both errors adding together instead of canceling.
An experiment for the curious: request the same planetary return several times, giving dates from a couple days before to a couple days after the date of the return. Look at the return times that come out. They will be close to each other, but not the same. This effect comes from the way returns are calculated. The program calculates the planet for the date you enter, then corrects the date and time to get closer to the natal position. It keeps correcting the return date and time until it gets a return position within 0.0001 degrees (0.36 seconds of arc) of the natal position. With slow moving planets, 0.36 seconds of arc can represent considerable time, and the exact moment the program finds depends on where it started from.
All of this is my way of saying that planetary return times and cusps (from anybody’s program) need to be viewed with a little healthy skepticism.
Transiting times and progressed dates
The same kind of uncertainty discussed above for returns also applies to any transiting times or progressed dates from any program. In this, internal consistency of calculations doesn’t make any difference—accuracy is the only measure to use.
For transiting times, the above formula for calculating uncertainty of return times only needs to be changed to say that you use the velocities of two different planets instead of two velocities for the same planet. You still get two possible error figures from velocity and rated accuracy and add them together for the total uncertainty.
For secondary (day for a year) progressions, apply the same logic, then convert total uncertainty from time to date by multiplying by the day-for-a-year ratio:
1 day : 1 year
24 hours : 365 days (approximately)
2 hours : 1 month (approximately)
1 hour : 15 days (approximately)
4 minutes : 1 day (approximately)
For every four minutes of time uncertainty in the planetary calculations involved, you have about one day of uncertainty in the dates of secondary progressions. (All of these ratios except the first are approximate. The actual ratio used in the CCURRENT module of the CCRS Horoscope Program is one day to one year of 365.24219879 days for tropical charts or 365.25636042 days for sidereal or precessed charts.)
Details
The geocentric positions used in the CCRS Horoscope Program are apparent positions for the true equinox of date. This means they are positions as you would have seen things if you had gone out and looked at the sky at that date and time. In contrast, the heliocentric positions and the positions shown from the perspectives of other planets are all geometric positions for the mean equinox of date. Geometric positions are where the planets are at the moment, rather than where we see them. The difference between geometric and apparent positions is the motion of the planet in the time it takes light to get from the planet to the viewer. It is called aberration or the light-time correction. The difference between the mean equinox and the true equinox is called nutation. At the moment, the CCRS program corrects for nutation in longitude, but does not correct for nutation in obliquity when deriving right ascension and declination from longitude and latitude.
Another kind of correction to positions that some people use is parallax. A parallax correction allows for the difference in perspective between the center of the earth and the actual observer’s position on the surface of the earth. The CCRS Horoscope Program doesn’t currently offer a parallax correction option, though it is under consideration for a future release.
A concept related to parallax is whether we give positions for the centers or the surfaces of the Sun, Moon and planets. All positions normally given are for centers, but the Sun and Moon are so large that their surfaces extend about 15 minutes of arc in each direction. People dealing with conjunctions, rise & set times, and ingresses of these two bodies need to be conscious of this extended size.
New and old
The new disk ephemerides are used everywhere in the program. The new Sun, Moon, Mercury, Venus, and Mars routines are used in calculating all permanently stored charts, timed progressions and transits, and creating or printing ephemerides. The same inner planet and Moon routines as last year are used in quick on-screen charts, untimed charts, and dynamic astrology graphs. The reasons for keeping the older routines in these places is to keep run times reasonable (the new routines are considerably slower than the old routines) and because these sections don’t need second of arc accuracy.
Standards and thanks
I view Neil Michelsen’s Astro Computing Services as a standard for accuracy in the astrological community. I want to say a special thank you to Neil for a listing letting me match his dynamic astrology routines, planetary masses letting me get a barycenter position consistent with his, and a listing that let me find why my Moon positions were differing from his more in the past.
I have used values from other astrological sources for the origin and orientation of the invariable plane and the longitude and latitude of the galactic center, but I will revise them if I find more recent values in an astronomical source.