Monday, 17 December 2012

How to predict an eclipse

Who's read Prisoners of the Sun?  In this enthralling Tintin story our hero is captured by a group of Inca who have survived in secret.  He and Captain Haddock are sentenced to be burned alive; their pyre is to be set alight by the rays of the sun.  Tintin is allowed to choose his time of death and cleverly picks the execution to coincide with a solar eclipse.  The amazed tribesmen believe he can command the sun god and immediately set him free.

This trick is not entirely fictional: Christopher Columbus pulled off something similar.  His son Fernando (considered a fairly reliable biographer) writes that in 1504 Columbus, stranded on Jamaica and undermined by mutiny, needed more supplies.  So he invited the local chiefs to a feast and, according to Fernando,
told the gathering [that] God was very angry with them for neglecting to bring us food ... and had determined to punish them with famine and pestilence...  They should therefore attend that night the rising of the moon: She would rise inflamed with wrath, signifying the chastisement God would visit upon them... At the rising of the moon the eclipse began, and the higher it rose the more complete the eclipse became, at which the Indians grew so frightened that with great howling and lamentation they came running from all directions to the ships, laden with provisions, and praying the Admiral [Columbus] to intercede with God. (p. 261)
Useful party piece, eh?  Well, you'll be pleased to know that you can pull this one off using your own planetary equatorium, which you've built using the instructions provided on this very blog.  In the last post I explained how to find the Moon's longitude.  If you're looking for the planets, longitude is usually enough, because they stick fairly close to the Sun in its annual journey (the ecliptic) across the background of fixed stars.  But for an eclipse, "fairly close" is not enough.  The Moon's latitude can be ±5° (celestial latitude is measured from the ecliptic), but an eclipse will only take place when it is 0° - that's why the ecliptic is called the ecliptic!  So in this case latitude is rather important to us, and to medieval astronomers.

Fortunately, it's quite easy to predict.  Here's what you do.  You need two pieces of information: the Moon's true motus, and the true motus of Caput Draconis.  Caput Draconis (which of course means the head of the dragon) is one of the two lunar nodes (the other, directly opposite it, is called Cauda [tail] Draconis) - the two places where the Moon's inclined orbit intersects with the ecliptic.  The nodes move steadily through the sky at a rate of about 19° a year, and their true motus is listed in any good set of astronomical tables like the one in our manuscript.

The Moon's true motus crops up naturally in the process described in my last post.  In the picture on the right, the green thread has been used to mark the Moon's mean motus (which we took from our tables), and the white thread has been stretched across the pole (middle) of the epicycle from the centrum oppositum.  The angle between the green and white thread is called the equation of centre, and that angle plus the mean motus is the true motus.

Now all we have to do is find the difference between the two true motuses.  (If it's more than 90° we may have to fiddle the figures a little, but that's not too tricky.)  Then find that number on the face of your equatorium, and stretch a thread horizontally across to the other side (as I've done in the photo below).  In the middle it will cross a line which our author has called line alhudda.  When we marked up the face, we divided that line into five (corresponding to the Moon's full range of latitude), so we can simply read off the point where our thread crosses the line.  In the case pictured below, it's about 2.6°.  Whether it's positive or negative will depend on whether we subtracted the true motus of the Moon from that of Caput/Cauda Draconis, or vice versa.

That's it!  As I said, if the Moon's latitude is 0° (and its longitude is either the same as the Sun's or 180° away), we'll have a lovely eclipse.  Remember that, it may save you from execution...


Wednesday, 12 December 2012

The Moon on a stick*

*well, on a disc of MDF, anyway.  

Want to know how to find the moon?  Of course you do!

So far on this blog I have constructed and calibrated a medieval planetary equatorium, and shown you how to use it to find the longitudes of the planets (most recently Mercury).  Now it's the turn of the Moon.

This will be a two-part post, because - unlike for the planets - our equatorium will not only tell us the longitude of the Moon, but also its latitude.

Celestial longitude is measured on the ecliptic, which is a circle in the sky angled at about 23½° to the celestial equator (the projection of the earth's equator outwards onto the sky).  Throughout the year, the Sun, Moon and planets move steadily around the ecliptic.  The Sun, by definition, sits on it; the Moon and planets may stray a little from it, but won't go far.  That's why medieval astronomers didn't worry about the planets' latitude (which is a measure of angle above or below the ecliptic).

The Moon, though, is different.  Classical and medieval astronomers were certainly interested in knowing its latitude.  Can anyone guess why?  (Feel free to comment; you may even win a prize.)  So our equatorium will tell us the Moon's latitude, and that will be the subject of my next post.

But let's talk about longitude now.  The Moon's longitude was modelled by Ptolemy in broadly the same way as for the planets, with a deferent and epicycle (for a refresher, read my explanation here).  There's one crucial difference, though: there's no equant.  Remember that the equant point was the centre of the motion of the epicycle around the deferent, which helped to model the irregular motion of the planets.  But since, for obvious reasons, the Moon's motion as seen from Earth is much more regular, there's no need for an equant point.

But we do still have a rather complicated setup with a moving deferent centre.  This accounts for the fact that, for medieval astronomers, the Moon's movement still seemed somehow to be linked to the Sun's.  (Hardly surprising to us, since the Earth orbits the Sun and the Moon orbits the Earth, but not so obvious to medieval scholars.)  In order to find the Moon's deferent, we need to look up the Moon's mean motus (the location of the centre of the epicycle, as measured from the constellation Aries) in our tables.  From this, we subtract the Sun's mean motus to find the deferent centre (if you get a negative answer, add 360°).

The deferent centre thus moves in a circle around the Earth.  Opposite it on that circle is another point known as the centrum oppositum.  We'll make use of that in a minute.

So, once we've worked out the location of those two points and marked them with our black and white string (pictured right), we get out our big Epicycle.  We put it with its common centre deferent reference point on the Moon's deferent centre.  Then:

1. As with the planets, we move the "black" string to mark the Moon's mean motus on the edge of the face of the equatorium.
2. We move the centre ("pole") of the big Epicycle so it's under the black string, while keeping the common centre deferent in place.  Note: we haven't done the trick with the parallel white string - remember there's no equant this time.  (This change seems to have flummoxed the author of the Equatorie of the Planetis manuscript I'm studying - he was a bit confused on this point.)
Step 3
3.  Run your white string from the centrum oppositum, over the pole of the Epicycle, to the far side of the Epicycle (see picture).
4.  Starting from where the white string crosses the far side of the Epicycle, turn the movable pointer ("label") clockwise [unlike with the planets where we turn it anticlockwise] to mark out the number of degrees corresponding to the Moon's mean argument, which we find in our tables.
5.  As with the planets, stretch the black thread to the mark for the Moon on the label (see below).
6.  Where the black thread crosses the edge of the face, is the Moon's true place (longitude).

And that's it!  Now you can find the Moon, even on a cloudy night.  In the next post, we'll learn how to use the equatorium to find its latitude.  Don't pretend you're not excited...