Tropical Astrology

Jamie Michelle

June 16, 2022

To Urania, our heavenly muse.

Originally published at the Internet Archive on June 11, 2020, ark:/13960/t0wq8t60r. Herein revised on June 16, 2022. This document is released under Version 3.0 of the "Attribution (By)" Creative Commons license and/or Version 1.3 of the GNU Free Documentation License.


Order Sign Meaning Referent Symbol Constellation Sign Ruler (Domicile) Starts Classical Element Alchemical Process
1 Aries Ram the golden-fleeced ram that rescued Phrixus and Helle ♈︎ Aries Mars on the northward equinox fire (🜂) calcination
2 Taurus Bull the form that Zeus took in order to seduce Europa ♉︎ Taurus Venus 1/3 between the northward equinox and the northern solstice earth (🜃) congelation
3 Gemini Twins Castor and Pollux ♊︎ Gemini Mercury 2/3 between the northward equinox and the northern solstice air (🜁) fixation
4 Cancer Crab the giant crab that Heracles killed ♋︎ Cancer Moon on the northern solstice water (🜄) dissolution (🝡, 🝢)
5 Leo Lion the Nemean lion that Heracles killed ♌︎ Leo Sun 1/3 between the northern solstice and the southward equinox fire (🜂) digestion
6 Virgo Virgin Astraea ♍︎ Virgo Mercury 2/3 between the northern solstice and the southward equinox earth (🜃) distillation (🝠)
7 Libra Balance the scales of justice held by Astraea, Dike, Themis and Justitia ♎︎ Libra Venus on the southward equinox air (🜁) sublimation (🝞, ☊)
8 Scorpio Scorpion the giant scorpion that killed Orion ♏︎ Scorpius Mars 1/3 between the southward equinox and the southern solstice water (🜄) separation
9 Sagittarius Archer the satyr Krotos ♐︎ Sagittarius Jupiter 2/3 between the southward equinox and the southern solstice fire (🜂) ceration
10 Capricorn Horned Goat the sea-goat form that Pan took in order to escape Typhon ♑︎ Capricornus Saturn on the southern solstice earth (🜃) fermentation (🝤)
11 Aquarius Water-Carrier Ganymede ♒︎ Aquarius Saturn 1/3 between the southern solstice and the northward equinox air (🜁) multiplication
12 Pisces Fishes the ichthyocentaurs Aphros and Bythos who carried Aphrodite from the sea ♓︎ Pisces Jupiter 2/3 between the southern solstice and the northward equinox water (🜄) projection

Note that astrology's influence upon individuals is real, although its effects upon humans is not based upon the distant stars, but rather the seasonal effects of the Sun. This of course means that the effects of the Sun's seasonal variance upon humans (particularly during gestation, of which has lasting lifelong consequences upon one's personality and upon one's susceptibility to various diseases) are diminished (though not eliminated) the closer one is to the equator; while these yearly effects are reversed for the Southern Hemisphere as compared with the Northern Hemisphere (since the seasons are reversed for said hemispheres). For some details on this, see the following papers:

Classical Planets

Order Name Symbol Roman Deity Greek Deity Norse Deity Mesopotamian Deity Hindu Deity Day of the Week Metal
1 Sun Sol Helios; Apollo Sól Utu/Shamash Surya Sunday (1) gold (aurum, Au; atomic number: 79; group: 11; period: 6)
2 Mercury Mercury Hermes; Apollo Odin Nisaba; Nabu Budha Wednesday (4) mercury (hydrargyrum, Hg; atomic number: 80; group: 12; period: 6)
3 Venus Venus Aphrodite Frigg Inanna/Ishtar Shukra Friday (6) copper (cuprum, Cu; atomic number: 29; group: 11; period: 4)
4 Moon Luna; Diana Selene; Artemis Máni Nanna/Sīn Chandra Monday (2) silver (argentum, Ag; atomic number: 47; group: 11; period: 5)
5 Mars Mars Ares Týr Nergal Mangala Tuesday (3) iron (ferrum, Fe; atomic number: 26; group: 8; period: 4)
6 Jupiter Jupiter Zeus Thor Marduk Bṛhaspati; Indra Thursday (5) tin (stannum, Sn; atomic number: 50; group: 14; period: 5)
7 Saturn Saturn Cronus Njord Ninurta/Ninĝirsu Shani Saturday (7) lead (plumbum, Pb; atomic number: 82; group: 14; period: 6)

The ancient world's concept of planet was as a wondering star (ἀστήρ πλανήτης, astēr planētēs), i.e., a regularly-occurring light in the sky (a "star") which unlike the many fixed stars of the celestial sphere, moved across said fixed stars in regular patterns (as opposed to, say, meteors, which were thought of as shooting stars, or falling stars). According to the Oxford English Dictionary's entry for "planet", referring to the ancients, "The seven planets, in the order of their accepted distance from the Earth, were the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn." (See John A. Simpson and Edmund S. C. Weiner [Eds.], The Oxford English Dictionary [Oxford, UK: Clarendon Press, 2nd ed., 1989].) The classical planets were variously also called the Seven Stars, or the Seven Luminaries.

The ancient Greeks initially thought that Mercury was two different planets: they named it Apollo when visible in the morning; and Hermes when visible in the evening. Later the Greeks realized that these seemingly two different planets were actually the same planet, and they kept the name Hermes for it. Apollo later came to be identified with the Sun.

Additionally, the ancient Greeks initially thought that Venus was two different planets: they named it Phosphorus when visible in the morning; and Hesperus when visible in the evening. Again, eventually the Greeks realized that these seemingly two different planets were actually the same planet, and they then associated it with the goddess Aphrodite. Coming later, the ancient Romans knew that Venus in its Morning Star and Evening Star appearances was actually a single planet, but when wishing to specify which appearance aspect they were referring to, called the morning appearance Lucifer, and the evening appearance Vesper (the Roman equivalents of their Greek counterparts); while their general name for the planet was Venus, the Roman version of Aphrodite.

Due to the ancient conception of a planet as being a wondering star, often when wishing to specify that they were referring to the planet rather than the actual god/goddess, the ancients would refer to it as, e.g., the Star of Aphrodite, etc.

Modern Planets

Order Name Symbol Sidereal Orbit Period Orbital Eccentricity Sidereal Rotation Period Axial Rotation in Relation to the Sun Number of Moons
1 Mercury 87.969257 SI day 0.2056302929816634 58.6463 SI day prograde 0
2 Venus 224.70079922 SI day 0.006755786250503024 243.018484 SI day retrograde 0
3 Earth 🜨, ♁ 365.256363004 SI day 0.01670236221760735 0.9972695663290843 SI day prograde 1
4 Mars 686.98 SI day 0.09331510156759697 1.02595675 SI day prograde 2
5 Jupiter 4332.589 SI day 0.04877487712602974 0.41353831 SI day prograde 79
6 Saturn 10755.698 SI day 0.05572339502033634 0.4440093 SI day prograde 82
7 Uranus ♅, ⛢ 30685.4 SI day 0.04440556667821134 0.718333 SI day retrograde 27
8 Neptune 60189 SI day 0.01121522948737634 0.67125 SI day prograde 14

The foregoing table's orbital parameters are taken from the below National Aeronautics and Space Administration's (NASA) Horizons On-Line Ephemeris System (except for the Earth's orbit and rotation periods, which are taken from the below section's IERS citation). To use the website-interface to obtain ephemeris data for Mercury, select Ephemeris Type: Orbital Elements; Target Body: Mercury; Center: @sun; and Time Span: 2000-01-01 12:00 to 2000-01-02, with Step Size: 1 day (the Step Size simply needs to be longer than the two Time Span parameters, otherwise one gets multiple ephemeris datasets, each at the interval of the Step Size). To obtain data for other planets, change the Target Body parameter to the desired planet. The values for the orbital eccentricities were obtained by setting the Target Body to the respective planet's Barycenter. The given Time Span parameters set the time to January 1, 2000, noon Barycentric Dynamical Time (acronymized as TDB, from the French: Temps Dynamique Barycentrique), which corresponds to the international astronomical epoch standard of J2000.0. When the unit of a year is given in the Horizons system's data output, it is often defined as the astronomical standard Julian year of 365.25 SI day.

See also the following resource for additional physical data on the major celestial objects within the Solar System:

Metrological Units of Time

From the foregoing reference:

mean solar day = 86400.002 second


mean solar week = 604800.014 second

From the foregoing reference:

day (in the International System of Units; Système International d'Unités; SI) = 86400 second

tropical year (or solar year; its period determines the seasons) = 31556925.2507328 second ≈ 365.2421819473199 mean solar day ≈ 52.17745456390284 mean solar week

sidereal year (its period is in reference to the fixed stars) = 31558149.7635456 second ≈ 365.2563545489918 mean solar day

sidereal month (a lunar month; its period is in reference to the fixed stars) = 2360591.55792 second ≈ 27.32166091755415 mean solar day


month ([tropical year]/12) = 2629743.7708944 second ≈ 30.43684849560999 mean solar day ≈ 4.348121213658570 mean solar week

Using the equation from the foregoing reference with the above values from IERS:

synodic month (a lunar month; its period determines the phases of the Moon) = 1/(1/[sidereal month] - 1/[sidereal year]) = 1/(1/[2360591.55792 second] - 1/[31558149.7635456 second]) ≈ 2551442.877200854 second ≈ 29.53058817291294 mean solar day ≈ 4.218655453273277 mean solar week

Astronomical Software

For a useful command-line program that can accurately compute the positions of various celestial bodies for given past and future times (while conversely capable of computing times for various events such as equinoxes and solstices; phases of the Moon; sunrises and sunsets; etc.), see the following websites for Skyfield, which is cross-platform, free and open-source software, and which uses the Python programming language:

For a planetarium program useful for visualizing the arrangement of celestial objects in the sky for given past and future times, see the below website for KStars, which is cross-platform, free and open-source software:

See also the below website for XEphem, which is an ephemeris and planetarium program that runs on Unix-like operating systems (and it will run under the Microsoft Windows operating system using virtual machine software such as VirtualBox with Linux installed as the operating system on the virtual machine). It is free and open-source software. (Just to note, the above PyEphem program uses XEphem's 'libastro' C library.)

For Unix-like operating systems, see also the following websites for Sunclock, which displays different maps of the Earth with the overhead positions of the Sun and Moon for desired times, additionally showing which parts of the Earth are illuminated by the Sun at the set times. Sunclock is free and open-source software.

The following command-line program for Unix-like operating systems is able to output the lunar phases for given times, and the times for sunrises and sunsets for given locations and days. It is free and open-source software.

Mathematical Software

Below are some free and open-source Computer Algebra Systems (CAS). Such systems can perform symbolic computations, arithmetic, series operations (e.g., summations and products), calculus operations, and more. Most such systems can also create graphs of functions (i.e., plots). These systems can perform arbitrary-precision calculations with integers and floating-point numbers (e.g., the significand of floating-point numbers can be precise to millions of digits on 32-bit computers, and billions of digits on 64-bit machines). All the below CAS run natively on Unix-like operating systems, and some have native ports to the Windows operating system.

Below is a very advanced virtual-desktop calculator which is able to perform many of the functions that Computer Algebra Systems are able to perform. It comes with both graphical user-interface and command-line versions. It features arbitrary-precision arithmetic with integers and floating-point numbers. It is cross-platform, free and open-source.

The following resources feature two emulators of the HP 48GX scientific graphing calculator by Hewlett-Packard, which was produced from 1993-2003. The first emulator is Emu48, which is Windows software but runs well under the WINE (Wine Is Not an Emulator) Windows-compatibility layer on Unix-like operating systems. The second is x48, which runs on Unix-like platforms. Both are free and open-source software. Hyperlinks to the necessary ROM file are included below. Lastly, the documentation for the HP 48GX is also included.

The following cross-platform, free and open-source program is able to convert between many different units of measurement:

See also the below mathematics reference work which describes several special functions:

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