Ancient Planetary Model Animations
A very brief introduction to the models and an equally brief User’s Guide.
For more detail and background, try Six Easy Lectures on Ancient Mathematical Astronomy.
I. The following links point to animations which run in web browsers, and thus need the latest Flash plug-in from http://www.macromedia.com. Note that right-clicking a link will generally offer the option of opening the animation in a new window, thus allowing you to easily open several animations simultaneously if you choose.
- The Sun (or try this version if you are using Netscape).
- See how the Season Lengths change as you vary the eccentricity and apsidal line direction of the Sun (or try this version if you are using Netscape). You might want to zoom in to get a clear view of the eccentricity.
- The concentric equant (or try this version if you are using Netscape), wherein the motion is on a concentric deferent but is uniform with respect to an offset point (the equant). The animation shows that the concentric equant is equivalent to either an eccentre with varying eccentricity or an epicycle with varying radius.
- The simple Moon (or try this version if you are using Netscape). These first two are not to scale. The eccentricities are made larger to show more clearly.
- The final Almagest Moon (or try this version if you are using Netscape) and with korny soundtrack (or try this version if you are using Netscape).
- The Ibn ash-Shatir model for the Moon (or try this version if you are using Netscape), also adopted by Copernicus. This exaggerated model is somewhat clearer (or try this version if you are using Netscape).
- A variety of models relevant to my paper “The Second Lunar Anomaly in Ancient Indian Astronomy”, Archive for History of Exact Sciences, (in press, 2007). The article is also available at www.springerlink.com.
- Mercury (or try this version if you are using Netscape), with a greatly exaggerated eccentricity for the sake of clarity
- The Ibn ash-Shatir model for Mercury (or try this version if you are using Netscape), also adopted and modified by Copernicus.
- Venus (or try this version if you are using Netscape).
- Regiomontanus’ moving eccentric models for Mercury and Venus (Netscape versions: Mercury Venus). The slider takes you from the usual epicycle model, to the moving eccentric model (which Ptolemy claims in Almagest XII.1 does not apply to inferior planets), and finally to the positions (first geocentric, then heliocentric) that Swerdlow proposes as a factor in Copernicus’ considerations of heliocentric models (see Proc. Amer. Phil. Soc. 177 (1973) p 476).
- Mars (or try this version if you are using Netscape).
- Jupiter (or try this version if you are using Netscape).
- Saturn (or try this version if you are using Netscape).
- Arabic models for replacing the equant for the outer planets and Venus (or try this version if you are using Netscape) compared to the Almagest equant model. These are the models of Nasir al-Din al-Tusi, Mu’ayyad al-Din al-Urdi, and Ibn ash-Shatir. The models of al-Urdi were also used at a later date by Qutb al-Din al-Shirazi, and it is not known if al-Shirazi was aware of the al-Urdi models. For the outer planets Copernicus adopted the version of al-Urdi or al-Shirazi, while for Venus and Mercury he adapted other ash-Shatir models. In no case is it known how Copernicus became aware of any of these models. In all cases the epicycle of the planet is optionally included for clarity, and of course is not needed in any event for the heliocentric Copernican models. The eccentricity is also greatly exaggerated for clarity.
- An interactive Tusi couple (or try this version if you are using Netscape). The Tusi couple is a way to produce linear simple harmonic motion using only combinations of uniform circular motions (i.e. just the inverse of the usual method of producing uniform circular motion by combining two orthogonal simple harmonic motions), and as far as is known, using the couple to produce linear motion was the only use by Arabic astronomers. In this modern, and hence ahistorical, version you may vary the relative radii of the two circles, which will change the path to an ellipse, or you may vary the relative frequencies of the rotations to get other patterns (try the values 25 and 75, and do not even think about asking if there is any connection to the Da Vinci Code). See http://mathworld.wolfram.com/Hypocycloid.html for more information.
- The transformation between a geocentric model and a heliocentric model for an outer planet (Jupiter/Mars) and an inner planet (Venus) (or try this version for Jupiter/Mars and this version for Venus if you are using Netscape). The primary eccentricities are neglected.
- Motion in latitude of an outer planet (or try this version if you are using Netscape) and an inner planet ((or try this version if you are using Netscape). The inclinations are greatly exaggerated, and some of the minor details in the Almagest models are omitted, but the animations are qualitatively correct.
- Ptolemy’s Cosmology (or try this version if you are using Netscape). Make scale smaller to bring in Jupiter and Saturn, and make it larger to see Mercury and the Moon. Zoom far in and see Ptolemy’s geography of the Earth.
- Tycho Brahe’s Cosmology (or try this version if you are using Netscape). Note how Mars’ orbit intersects the orbit of the Sun. However, after observing the comet of 1577, Tycho became convinced that there are no solid celestial spheres, and so hence no reason not to prefer his geocentric version of Copernicus’ heliocentric model (for which, ignoring the difference between the mean and real Sun, see Kepler’s 3rd Law below).
- A comparison of Kepler motion and equant motion (or use this version if you are using Netscape). You will have to use a rather large eccentricity to see much difference at this scale.
- Kepler’s Cosmology (or use this version if you are using Netscape). Make the scale smaller to bring in Jupiter and Saturn.
- Isaac Newton’s approach to Kepler’s area law (or try this version if you are using Netscape), with a short explanation.
II. The following links point to stand-alone versions of the animations, for both Windows and Macintosh computers, which can be run in full-screen mode (ctrl-f in Windows, something similar for Macs):
- Regiomontanus’ Moving Eccentric Models for the Inner Planets
- Geocentric-Heliocentric Transformation
Outer planet (Jupiter) Windows Macintosh
Inner planet (Venus) Windows Macintosh
The above files have the advantage that you may save the executable files locally on your computer and thus avoid any dependence on a network connection when you want to use them. They have the disadvantage that if you do that, you might not have the latest version of the animations.
III. Some technical details which might be useful for anyone who wants to understand how the models work, or to create similar models.
IV. Links to Related Sites
- George Saliba’s essay on Arabic/Islamic Science includes useful references to more extensive discussion of the Arabic planetary models.
- Robert van Gent’s Almagest Ephemeris Calculator gives quantitative values for all the parameters of the various Almagest models at a given value of t.
- Glen Van Brummelen’s site gives a set of computer animations for the Almagest geometric models of planetary motion. These animations use the Geometer’s Sketchpad.
- Craig Sean McConnell’s site gives a number of animations and visualizations of ancient planetary models from Euduxos to Copernicus.
- Giampiero Barbieri’s site gives animations of the models of Ptolemy, Copernicus, Kepler and Newton (still under construction).
- Three essays by Hugh Thurston: the Planets, Babylonian Planetary Models, and Formulas Useful in Astronomy
Please consider all the animations as works in progress. Anyone is welcome to use them freely for any non-commercial purpose. They are particularly intended to be useful for teaching, independent study by students, and perhaps contemplation of just how clever the ancient astronomers were. Please with any suggestions for improvements, and especially if you notice any errors.
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