Period of unique horoscopes? What is the period of unique horoscopes? In other words how long does it take before any configuration of the classical planets repeats?

The seven classical "wandering stars" are: Sun, Moon, Saturn, Jupiter, Mars, Venus, and Mercury.

Assume a horoscope is created by dividing the ecliptic into 12 equal regions and then for a particular day noting which region each of the 7 "wandering stars" is occupying relative to the earth. For example, a horoscope could be: Sun in Aries, Moon in Capricorn, Saturn in Taurus, Jupiter in Libra, Mars in Pisces, Venus in Aries, Mercury in Aries.

Since Saturn revolves around the sun once every 29 earth years, Jupiter every 12 years, and Mars 1.88 or so, multiplying together we have 654. So, relative to each, they would seem to repeat their configuration every 654 years.

This is potentially a complicated question which might require a simulation to determine the answer. For example, today (5 December 2016), the following is true for the tropical zodiac:

Sun in Sagittarius
Moon in Aquarius
Mercury in Capricorn
Venus in Capricorn
Mars in Aquarius
Jupiter in Libra
Saturn in Sagittarius

So, one instance of this problem would be to determine the next date at which this same configuration will next recur.
I wrote https://github.com/barrycarter/bcapps/tree/a944a8d8315fd6169907f73e47698f030befe3bb/ASTRO/bc-zodiac.c to solve this. I may've tweaked bc-zodiac.c since then, so you'll need the exact link above (not the most recent version of bc-zodiac.c) to confirm/replicate my results. In the directory https://github.com/barrycarter/bcapps/blob/master/ASTRO/


The file "houses.txt.bz2" contains a list of all house changes for now +- ~15000 years in a mostly-English-readable format.
The file "house-states.txt.bz2" is a list of all "horoscopes"(house states for all planets) in machine readable format for the same period. Please see https://github.com/barrycarter/bcapps/blob/master/ASTRO/bc-zodiac.txt (which also contains the text of this answer) for details on how to read this file.


Important caveats below, but first, some notes:


You'll notice that lines like:


BCE 13200-JUL-12 09:48 SUN ENTERS ARIES PROGRADE -479626193452 S0+

suggest the vernal equinox was on July 12th about 15000 years ago (of course, no one actually used our calendar back then), which seems a bit off. Further, you'll notice:

CE 17191-FEB-23 00:27 SUN ENTERS ARIES PROGRADE 479386211293 S0+

suggesting the vernal equinox will occur on February 23rd (about a month early) 15000 years from now (assuming they continue using our calendar).

See https://astronomy.stackexchange.com/a/13009/21 for why this happens (my answer there has slightly different dates/times, because I used a less accurate model).


In the 11,100,017 days (about 30,391 years) covered, there are 6,359,874 horoscope changes, for an average horoscope change roughly every 1.75 days. Broken down by planet:


The moon changes house 4,875,276 times, about once every 2.3 days.
Mercury changes house 450,523 times (including retrograde house changes), about once every 25 days.
Venus changes house 384,971 times (including retrograde), about once every 29 days.
The Sun comes in third with 364,691 changes, about once every 30 days. A more exact number is once every 30.4367697585 days. When you multiply this by 12, you get 365.2412371020 days, which, as expected, is a good estimate of the length of the tropical year (the exact value given by http://hpiers.obspm.fr/eop-pc/models/constants.html is 365.242190402, which is 82 seconds different).
Mars changes house (including retrograde) 209,501 times, about once every 53 days.
Jupiter (w/ retrograde) 49,215 times, once every ~225.5 days.
Saturn (w/ retrograde) 25,697 times, ~432 days.

As expected, most of these house changes (6,283,141) are in the forward or "prograde" direction and very few (76,733) are in the retrograde direction. Of course, the Sun and Moon always move prograde, another reason the prograde number is so large.
Since there are 12 houses and 7 planets, it would seem there are 12^7 = 35,831,808 possible horoscopes. However, there are actually only 3,732,480:


Since Mercury's greatest elongation is 28 degrees, it must remain within one house of the Sun. Given the Sun's house, there are only 3 possibilities for Mercury's house (either the Sun's house or one of the two adjacent houses).
Since Venus' greatest elongation is 47 degrees, it must remain within two houses of the Sun, so there are only 5 possibilities given the Sun's house.
Combining these, we see there are 12*12*3*5*12*12*12 or 3,732,480 possible horoscopes.

Of these possibilities there are 2,565,570 (about 69% of the total possible) actual horoscopes. Of these,


969,067 horoscopes occur only once
The horoscope "4045424" (translated below) occurs 18 times (most recently in 1799 and next in 5391), but two others occur 17 times, 10 appear 16 times, and many appear 15 times, so it's only the most frequent by coincidence (probably just because of where DE431 happens to start and end).
"4045424" translated is Sun in Leo, Moon in Aries, Mercury in Leo, Venus in Virgo, Mars in Leo, Jupiter in Gemini, Saturn in Leo.



Note that this horoscope is probably only the most frequent because of how I chose the start and end dates (based on the DE431 ephemeris).

Important caveats:


My standard disclaimer: please do not rely on anything I do/post anywhere. I can and do make mistakes. Feel free to check my work and alert me to mistakes.
If you actually want to check my results, you will need to download kernels from NASA. See http://naif.jpl.nasa.gov/naif/tutorials.html for more details on how this all works.
Note that NASA can't predict planet positions (or even the Earth's own orientation) into the far future: they solve differential equations to predict planetary positions and then publish an approximation of those solutions. The further away we get from today, the less accurate these predictions are.
We have a pretty good approximation of how the Earth precesses, but we can't even predict THAT super accurately more than a few months in advance. Note that https://naif.jpl.nasa.gov/pub/naif/generic_kernels/pck/ shows the high precision precession kernel is only good through 6 Mar 2017, and even THAT is updated twice a week (see https://naif.jpl.nasa.gov/pub/naif/generic_kernels/pck/aareadme.txt for details). Since precession affects the location of the first point of Aries, this adds another uncertainty.
Precession doesn't change the ecliptic itself, but nutation does. We don't have a super-accurate model of nutation, and I am using the undocumented SPICE functions zzmobliq_ and zzwahr_ as per a private reply to my https://naif.jpl.nasa.gov/pipermail/spice_discussion/2016-January/000423.html
The calendar on https://cafeastrology.com/astrologyof2017horoscopes.html (and the 2016 equivalent) should agree with my results within +- 1 minutes. Remember to convert between ET on their site and UTC in my file.
http://aa.usno.navy.mil/data/docs/EarthSeasons.php can also be used to check some of my calculations (within +- 1 minute), recalling that, in the northern hemisphere:


"SUN ENTERS ARIES" (as in "first point of Aries") marks the vernal equinox.
"SUN ENTERS CANCER" (as in "Tropic of Cancer") marks the summer solstice.
"SUN ENTERS LIBRA" marks the autumnal equinox. As nearly as I can tell, there is no well-known associated phrase with this point in the zodiac.
"SUN ENTERS CAPRICORN" (as in "Tropic of Capricorn") marks the winter solstice.

If you use HORIZONS (http://ssd.jpl.nasa.gov/?horizons) to check my work, you'll notice my results are accurate to the minute for about +- 1000 years from now, but drift in accuracy after that. I'm not exactly sure why this is.
I used a "stepsize" of 86400 seconds (1 day), so if a planet changed house and then changed back in under a day, my program may miss it. Someone w/ more computing power may wish to re-run my program with a smaller stepsize.
The moon's position varies with your location on Earth. I used a geocentric model, giving the moon's position for a theoretical observer located at the center of the Earth.
I actually computed planetary barycenters (the center of mass of the planet and its satellites) because NASA doesn't provide planet position information for the entire 30,000 year period I computed (see http://ssd.jpl.nasa.gov/eph_spans.cgi?id=A for details).


For Mercury and Venus, this no effect (no moons), and the Mars' system barycenter is within a few inches of Mars' own center (Phobos and Deimos have very little mass). Jupiter is fairly massive, so its system's barycenter is fairly close to its own center, and even the effect Titan has on Saturn isn't that great.


The human-readable time in houses.txt.bz2 is the output of the C function http://naif.jpl.nasa.gov/pub/naif/toolkit_docs/C/cspice/timout_c.html (format ""ERAYYYY##-MON-DD HR:MN ::MCAL ::RND" modified slightly by Perl). They are reasonable for current dates, but not necessarily for past dates because the Gregorian Reformation of the calendar occurred at different times for different countries, and the BCE/CE time system didn't exist prior to 1 CE (and probably not even then). For precision work, see bc-zodiac.txt for details.
As above, I use "ARIES" to mean the "1st house", ie, the constellation that starts at the vernal equinox ("the first point of Aries", where the ascending ecliptic meets the celestial equator) and ends 30 ecliptic degrees of ecliptic longitude later.


This was reasonably accurate in 56 BCE, but is certainly not true today as the vernal equinox is in Pisces and drifts towards Aquarius, heralding the "Age of Aquarius".

In the 30,000+ year period covered by my program, the first point of Aries makes a complete circuit (and a bit more) of the Zodiac, even ending up in Ophiuchus for a while.

Of course, the constellation boundaries I mention above are those defined by the IAU. I don't know how (or if) the ancients drew their constellation boundaries.


I am leaving my old answer below, because NASA is a trusted site for astronomers, and WebGeoCalc and HORIZONS are important resources. The step-by-step method I show below can help people here make their own calculations or help others. I am in no way affiliated with NASA or the government of the United States, except as a citizen.


EDIT: As I should have guessed, people have already done this: https://cafeastrology.com/astrologyof2017horoscopes.html (and I'm guessing there are many many other sites too, just google "Mercury enters Aries" "Venus enters Taurus" type stuff).

I hope to provide a better answer later, but for now:


Visit http://wgc.jpl.nasa.gov:8080/webgeocalc/#CoordinateFinder and do this:






and click "Calculate". This will tell you every time Mercury has an ecliptic longitude of 30 degrees, the cusp between astrological Aries and Taurus.

If you do this for all 7 planets and every multiple of 30 degrees (12*7 = 84 times), you can use the results to see when the combination of constellations change. However, this is has several problems:


It's extremely inefficient
If you hover over "Solar System Kernels kernel set" above, it says the kernels are valid from 1950 to 2050. However, with the options I chose, you can only get results from 1966 to 2034, possibly because the ECLIPDATE frame is only defined in that interval (when I choose years outside this range, I get an unhelpful and inaccurate error-- you may get different results for other planets).
The results are in calendar times; however, this may be my fault. You may be able to get the results in a more numerical and useful format.


The "correct" way to do this is to use CSPICE (http://naif.jpl.nasa.gov/naif/tutorials.html) which lets you make calculations +- 10000 years from now.

Leaving that aside, let's check our results. If you scroll down a bit in the results window, you'll see:



Let's check the result for 2017-04-20 by visiting http://ssd.jpl.nasa.gov/horizons.cgi and entering these parameters:



to get these results:



As you can see, Mercury's ecliptic longitude crosses 30 degrees between 2017-04-20 and 2017-04-21, so our results are reasonable. Interestingly, Mercury is in retrograde at this time, so it actually crosses from Taurus into Aries, not vice versa.