Monday, 26 November 2012

Mercury's meandering deferent

This week I'm back with my equatorium, trying to pin down precisely how well it models Ptolemy's planetary theories (and, in turn, finding out the strengths and weaknesses of those theories in predicting the locations of the planets in the night sky).  Today it's the turn of the Winged Messenger, Mercury.

In my last "medieval craftsmanship" post I described the procedure for using the Equatorie of the Planetis to find the longitude of most planets.  I mentioned that the procedures for the Sun, Moon and Mercury vary to a greater or lesser extent from this basic procedure.

So here's how it works for Mercury.

Let's first discuss why the Ptolemaic model doesn't work as well for Mercury as the other planets.  Note: the simplest way for me to explain this is by referring to modern astronomical theories.  Before I do, I should stress that the success or failure of medieval theories was obviously not judged by how well they approximated modern astronomy; instead, their success was measured, in good empirical fashion, by how closely they matched observed data.  Ptolemy's genius - building on the work of earlier, less celebrated men such as Apollonius and Hipparchus - was to devise theories for each planet that produced predicted locations that would match observations to an astounding degree of accuracy.

To recap: deferent=big dashed circle
centred on X; equant=big black dot;
epicycle=little dashed circle.
So, that said, what's up with Mercury?  Well, as you may have read in one of my previous posts, the Ptolemaic system of planetary motion is based on two circles for each planet, the deferent and epicycle.  Together these model the movement of the planets as observed from the earth.  In other words, [ANACHRONISM ALERT] they account for both the earth's and the planets' motion around the sun.  But if you're familiar with the basics of modern astronomy you may be aware that, according to Kepler's first law of planetary motion, the planets' (and earth's) motion around the Sun is elliptical, not circular.  How can we make circles into ellipses?

We get a pretty good approximation of one ellipse from the fact that (a) the deferent is not centred on the earth, and (b) the centre of the epicycle moves around the deferent at a constant speed relative to another point, the equant.  But the epicycle is still a circle, and the planet is assumed to move around it at a constant speed.  Problem, no?

Well, no, not really.  Why not?  Precisely because Ptolemy was not trying to model modern systems: the deferent and epicycle are not pretending to be two ellipses; they're just accounting for what we see in the sky.  As it happens, in the case of Venus, Mars, Jupiter and Saturn, the location of the deferent and equant can effectively factor in a bit of extra eccentricity, so the epicycle doesn't need it.  The resulting level of (in)accuracy will vary depending on the orbits of the various heavenly bodies, but this system gives us a maximum inaccuracy of about 9 minutes of arc (three-twentieths of a degree) for Mars, and is even better for the other planets.

But not for Mercury, though - with this system the locations for Mercury might be almost a degree out.  You might think that's still pretty accurate (remember that we're talking about naked-eye observation here), but it wasn't good enough for Ptolemy.  Here's his solution:

You may remember I mentioned before that we have to mark an aux, or apogee, for each planet; I drew a line (sometimes called the line of apsides) through earth, the deferent centre and equant, out to the rim of the equatorium.  Because the deferent centre is on this line, the aux is the point on the deferent circle that is furthest from earth.  With all the other planets, as the diagram above shows, the centre of the deferent is halfway between earth and the equant point.  But Mercury's deferent centre (D) moves: it can be any point on a circle which straddles the line of apsides, goes through the equant (E), and has a radius equal to the distance from earth to E.  You can see this little circle in the photo below: the pin is at the centre of earth, and E is on the right of the little circle, nearest the pin.


OK then.  We start, as we did before, by finding the mean motus (let's call it λ) of Mercury in our handy astronomical tables.  As usual, we find the point on the rim of the equatorium that is λ° round anticlockwise from the vernal equinox.  Now we subtract the value of the aux from the mean motus; if we get a negative answer, we add 360°.  Let's call the result ψ (technically, it's known as the mean centre).  Now, to find your deferent centre, count clockwise ψ° from the point on the little circle opposite E (on the left on the picture above, furthest from the pin, where I've unhelpfully marked a tiny D).  You might be able to see a little dashed line on the picture above, showing a value of 79° for ψ (as it was on 31 December 1392).

That's it - once you have the location of your deferent centre, you can go ahead with the method as I described before, laying out your parallel threads, placing the common centre deferent of the Epicycle on D, and so on.

I hope all that makes sense!  Next time I'll explain the (much simpler) procedure for the Sun.  Then it's the Moon, which will raise the question of latitude for the first time.  Please visit again soon!

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