Dante’s Heaven

In previous posts (Inferno, Purgatorio, Paradiso), I have mentioned the scientific content of Dante’s incredible theological poem, the Divine Comedy. Above, just for fun, is a chart of Heaven (the Solar System) in his Paradiso. Notice the sphere of fire which was believed to surround the Earth.


The Tropical Year: 31.6888 nHz

One of the most important cycles we live by is the tropical year, measured from equinox to corresponding equinox (or solstice to corresponding solstice). The tropical year lasts, on average, 365.2422 days (365 days, 5 hours, 48 minutes, 45 seconds), which means that it is an oscillation with a frequency of 31.6888 nanohertz (nHz). This is the cycle of the seasons.

Spring, summer, autumn, and winter are the conventional seasons, but the tropical year may be split up into more than or less than four seasons, and these need not be of equal length. In northern Australia, a frequent division is “the dry” (May to September), “the build up” (September to December), and “the wet” (December to April). Local Aboriginal people, however, may recognise as many as six seasons.

The Sidereal Year: 31.6875 nHz

A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the stars. This is the time that it takes for the sun to move through the “signs of the zodiac.” Because of the precession of the equinoxes, the sidereal year is 365.2564 days, which is about 20.4 minutes longer than the tropical year. As a result, ancient rules assigning dates to the signs of the zodiac are now completely wrong. The sidereal year corresponds to an oscillation of 31.6875 nanohertz.

The Synodic Month: 391.935 nHz

A synodic month is a cycle from new moon to new moon or full moon to full moon. This period actually varies by several hours, but it averages out to 29.530588 days (29 days, 12 hours, 44 minutes, 3 seconds).

In 432 BC, Meton of Athens noted that 235 synodic months (6939.7 days) is almost exactly equal to 19 years (6939.6 days). This period is called the Metonic cycle, and is used for predicting solilunar events such as the date of Easter.

The synodic month is also strangely similar to the average menstrual cycle (28 days), and this is reflected in the word (“menstrual” derives from the Latin mēnsis = month).

The Week: 1.65344 µHz

The week has an origin among the ancient Hebrews. It also has a Babylonian origin (the relationship between the two origins is unclear). The Babylonians related the 7 days of the week to the sun, moon, and 5 visible planets. They also related them to various gods. Our days of the week derive from the Babylonian week, via Greece and Rome: Sunday (Sun), Monday (Moon), Tuesday (Tiw, god of war = Mars), Wednesday (Woden = Mercury), Thursday (Thor = Jupiter), Friday (Frigg = Venus), and Saturday (Saturn).

Early Christians related the two week concepts together, pointing out that the day of the Resurrection (the day after the Jewish Sabbath) corresponded to the day of the Sun in the Roman system. The week corresponds to an oscillation of 1.65344 microherz.

The Sidereal Day: 11.6058 µHz

A sidereal day is the time that it takes the earth to rotate once around its axis. It often surprises people to discover that this time is 23 hours, 56 minutes, 4.1 seconds. It can be measured by the time to go from a star being overhead to the same star being overhead again.

The Solar Day: 11.5741 µHz

A solar day (24 hours, give or take some seconds) is the time from noon to noon. It is longer than a sidereal day because, while the earth is rotating around its axis, it is also moving around the sun. To put it another way, the sun is not a fixed reference point for the earth’s rotation. The difference between the sidereal and solar days mean that the stars seem to rise about 3 minutes and 56 seconds earlier every night.

Some posters

For people with children, these high-resolution posters are intended for printing on A3 paper, and can be freely downloaded:

Mathematics posters: Flat shapes, Solid Shapes, Prime Factors (colour-coded)

Biology posters: Ladybirds of Australia, Plants and Fungi, Some Flowering Plants (monocots marked with a dot)

Astronomy/geography posters: Southern Cross (showing colours and magnitudes of stars), Orion (ditto), Geographical Features

Technology posters: Vehicles, Milestones in Materials

A planetary conjunction and a Christmas greeting

It’s time for a Christmas blog post, and there’s really only one thing to write about. On 21 December, there will be a great conjunction, in which Jupiter and Saturn will appear very close together in the early evening sky. Look for them near the western horizon, as they get closer and closer over the next two weeks. The diagram below (from fourmilab.ch) shows what the solar system will look like at conjunction. A line from Earth to Jupiter continues on to Saturn, but skims by the Sun (which is why the “kissing planets” are only visible in the early evening). These two giant planets have not appeared so close for several centuries.

The Christmas connection relates to the Magi mentioned in the Bible: “After Jesus was born in Bethlehem in Judea, during the time of King Herod, Magi from the east came to Jerusalem and asked, ‘Where is the one who has been born king of the Jews? We saw his star when it rose and have come to worship him.’” (Matthew 2:1–2, NIV)

One theory as to what the Magi might have seen was a set of similar conjunctions of Jupiter and Saturn in 7 BC (the great astronomer Johannes Kepler was the first to suggest this – and yes, thanks to a calendrical error 500 years later, Jesus was born around 7–4 BC). There were three such conjunctions in 7 BC: in May, in late September, and again in December. Here is a view of the September one, as seen from Ctesiphon in Parthia (from fourmilab.ch again). The planets Jupiter (♃) and Saturn (♄) would have risen just before sunset, and been visible in the evening twilight (and then throughout the night):

The Babylonians and Persians had an elaborate system of seeing omens in the sky. For example: “If Jupiter becomes steady in the morning: enemy kings will be reconciled. … If Jupiter passes Regulus and gets ahead of it, and afterwards Regulus, which it passed and got ahead of, stays with it in its setting, someone will rise, kill the king, and seize the throne.” So a conjunction is the sort of thing that would have gotten the attention of star-gazers in the Parthian Empire (the planet Jupiter was associated with the god Marduk and with kingship). Others have suggested a nova recorded in China in March of 5 BC. Yet others have suggested that a sequence of astronomical events led the star-gazers to search for a newborn king specifically in the frontier province of Judaea:

Countless paintings show the journey of the Magi across the desert (the one above is from James Tissot). If they were sensible, they would likely have travelled along the trade routes via Palmyra and Damascus. “A cold coming we had of it” wrote the poet T.S. Eliot (adapting lines from a 1622 homily by Lancelot Andrewes), “Just the worst time of the year / For a journey, and such a long journey.”

The real danger was Herod, of course. He had murdered his own sons Alexander and Aristobulus in 7 BC (and was to murder a third son, Antipater, in 4 BC). According to Macrobius (Saturnalia Book 2, IV, 11), Caesar Augustus had quipped “Better to be Herod’s swine [Greek hus] than his son [huios],” making a Greek pun referencing Herod’s Jewish religion and its prohibition on pork. Naturally, a man like that would be less than thrilled at the suggestion that another heir to the throne might exist:

When King Herod heard this he was disturbed, and all Jerusalem with him. When he had called together all the people’s chief priests and teachers of the law, he asked them where the Messiah was to be born. ‘In Bethlehem in Judea,’ they replied, ‘for this is what the prophet has written:

“But you, Bethlehem, in the land of Judah,
are by no means least among the rulers of Judah;
for out of you will come a ruler
who will shepherd my people Israel.”’
[a summary of Micah 5:2–5]

Then Herod called the Magi secretly and found out from them the exact time the star had appeared. He sent them to Bethlehem and said, ‘Go and search carefully for the child. As soon as you find him, report to me, so that I too may go and worship him.’” (Matthew 2:3–8, NIV)

This is the kind of trouble you get when you mix star-gazing boffins and international diplomacy (I have been to enough international scientific events to know how that works). The gospel account makes a further confusing reference to the “star” and mentions the famous gifts of “gold, as to a king; myrrh, as to one who was mortal; and incense, as to a God”:

After they had heard the king, they went on their way, and the star they had seen when it rose went ahead of them until it stopped over the place where the child was. When they saw the star, they were overjoyed. On coming to the house, they saw the child with his mother Mary, and they bowed down and worshiped him. Then they opened their treasures and presented him with gifts of gold, frankincense and myrrh. And having been warned in a dream not to go back to Herod, they returned to their country by another route.

When they had gone, an angel of the Lord appeared to Joseph in a dream. ‘Get up,’ he said, ‘take the child and his mother and escape to Egypt. Stay there until I tell you, for Herod is going to search for the child to kill him.’” (Matthew 2:9–13, NIV)

This “Flight into Egypt” has been a common subject of Christian art. The painting above, by Adam Elsheimer (1609), includes a beautifully painted Milky Way.

Egypt, of course, was a logical destination. We know first-century Alexandria primarily as the scientific and mathematical centre of the world of that time, but it also had a thriving Jewish community, with hundreds of thousands of Jews living in the city, and hundreds of thousands more in the rest of Egypt. The Hellenistic Jewish philosopher Philo was still a boy at this time, as was the Greek scientist Heron, but the Musaeum was fully active. Eventually, Alexandria was also to become one of the most important Christian cities, and the statement “gold, as to a king; myrrh, as to one who was mortal; and incense, as to a God” was from Alexandria. An active Coptic Church is still there (though suffering hardship).

But troubled as the world of 2020 may be, let me wish all my readers a Merry Christmas, and a better 2021!

The Sydney Observatory

Observatory exterior (photo by Greg O’Beirne, 2006)

An unusual free science museum in Sydney, Australia is the Sydney Observatory. This opened in 1858 as a working observatory. The time ball, which dropped each day to mark the exact time, is still operating at 1:00 PM each afternoon. The observatory now operates as a small museum, having been refurbished during 1997–2008. The telescopes can also be used on paid night tours.

The observatory is a stiff climb up Observatory Hill. The exhibits are limited in number, but include some excellent orreries. Unless you have some astronomical expertise, the paid guided tours will be helpful. My brief visit was an enjoyable one.

An orrery at Sydney Observatory (photo by Anthony Dekker)


Above is an analemmatic sundial. The idea is to orient the sundial facing south, and then place a vertical pointer on the central figure-8 track, in a position corresponding to the date. The sundial above shows a simulated shadow for 2:15 PM yesterday. It can be seen that the sundial tells the time reasonably well, thanks to the inbuilt adjustment for variation in solar position.

For large-scale analemmatic sundials, like the one below, people can stand on the central figure-8 track and act as a human pointer. A sundial like this is fun to have in the garden.

Here are blank sundials for some Southern Hemisphere cities:

The ShadowsPro software will also generate sundials like these, if anyone is particularly enthusiastic.

The constellation Scorpius

Winter is here (in the Southern Hemisphere, at least), and the constellation Scorpius always heralds the southern winter’s icy sting. The image below is based on a vintage astronomical illustration, but I have corrected the star positions of the major stars and indicated their apparent magnitude (brightness) and approximate colour (based on spectral class). It is interesting to compare the image with this quality photograph.

Generations of astronomers have memorised the O–B–A–F–G–K–M stellar classification system developed by Annie Jump Cannon with the mnemonic “Oh, Be A Fine Girl/Guy/Gal/Gentleman, Kiss Me.” Scorpius does not contain any bright O-class stars, but it is easy to see stars ranging from the hot blue-white B class to the cooler orange-red M class (stars which are only “red hot”).

The most obvious star in Scorpius is the enormous red supergiant Antares, which has that name because it is easily confused with the planet Mars (Ares). It is also known as “Cor Scorpii” (the heart of the scorpion). It is easy to recognise the curved tail as well, with the stingers Shaula and Lesath at its tip. It is less obvious which stars are the scorpion’s claws – the artist here has drawn the left claw extended so as to reach the dim white star Psi Scorpii. Other artists draw the scorpion facing more to the right, with the line of blue-white stars being the claws.

Infographic constructed using R (with lm to map true sky coordinates to image coordinates, rasterImage for the background, and the showtext package for fonts).

Her pale fire she snatches from the sun

Shakespeare writes “the moon’s an arrant thief, and her pale fire she snatches from the sun” (Timon of Athens, Act 4, Scene 3). He is, of course correct. The moon merely reflects sunlight, and produces no light of its own. One way of telling this is that moonlight actually displays the same telltale absorption spectrum as sunlight:

Our eyes tend to perceive moonlight as “blueish” or “silvery,” but that is because of the way our eyes work at low light levels. Long-exposure photographs under moonlight, like this one, look much like daytime shots:

Anaxagoras (499–428 BC) seems to have been the first to discover that the moon shines only by reflected light:

Anaxagoras also explained that solar eclipses occur when the moon moves between the earth and the sun. Total solar eclipses are dark precisely because the moon produces no light of its own: