It’s all in a day

Here’s a question. How long does it take for the earth to spin exactly once on its axis?

If you answered “24 hours”, then I’m sorry to tell you that you’re wrong. Not wrong by very much, it’s true, compared with people who live in the crazier parts of the internet (YouTube, where else?).

Probably not as wrong as Mr Potters Clay, who doesn’t understand that the earth can spin on its axis and orbit the sun at the same time. To be fair, the only book he refers to offers no useful astronomical information.

Whereas Mr Thrive and Survive claims to have read the wikipedia but he still believes the earth is flat.

And you’re almost certainly not as wrong as Mr Robert Sungenis, who thinks that the earth is the center of the universe, says that evolution is false, and that Stanley Kubrick filmed and faked the NASA moon landings.

These guys are, in physicist Wolfgang Pauli’s well-known phrase “not even wrong”. (Pauli’s full comment was, “Das ist nicht nur nicht richtig, es ist nicht einmal falsch!” — “That is not only not right, it is not even wrong!”.)

A few minutes more, or less

The correct answer (which I hope you knew) is currently 23 hours, 56 minutes, and 4.1 seconds. During this time, the Earth has, in addition to spinning on its axis, travelled through space on its orbital path around the sun at about 67,000 miles per hour, so it’s already about 1.6 million miles away from where it started at the beginning of the day. So it needs to rotate for a few more minutes if it’s to turn so that the sun is more or less at the same point in the sky as the day before. And, at that speed, this takes about four minutes.

Image from wikimedia

In 24 hours the Earth rotates about 361 degrees, or a bit more than one revolution. In fact, the familiar 24 hour time period was defined precisely to include that extra 1 degree of rotation.

In each single rotation of 23 hours and 56 minutes, the earth returns back to face the same distant stars. A day measured relative to the stars like this is called a sidereal (pronounced “Sigh-Dear-Eee-Al”) day, sidus being Latin for star. The earth takes one sidereal day to complete a single rotation.

And, just as a sidereal day is shorter than a solar day, a sidereal hour is shorter than a solar hour, at 59″ 50′.22 solar minutes. And a sidereal minute has 59.84 solar seconds. There are 365.2422 solar days and 366.2422 sidereal days in a (solar) year. Sidereal time runs a bit faster than the sun-centered time we keep, and is locked to the rotation rate of the earth.

These two ways of measuring time on the earth – the time by the sun and the time by the stars – have been used over the centuries and they weave together through history.

Apparently

The most visible result of the difference between star and solar time is that the sun appears to travel across the starry background a small amount every day during the course of a year.

To visualize this it’s now easy to use astronomy software which lets you “switch off” the sun and see the stars behind it during the day. The following video shows the sun’s position once every day early afternoon in March, as seen at the same sidereal time — so about 4 minutes earlier each day according to a traditional clock. You can see the stars even though it’s early afternoon. The earth has made exactly one revolution between each frame. As a result, the sun – and the displayed solar time – appear to move backwards by about 4 minutes a day. It looks like the sun is moving anticlockwise against the background stars, instead of clockwise across the sky during the day.

Obviously both these persuasive illusions are wrong – we should try to imagine the sun more or less fixed in space, while we’re watching the starry background slowly change behind it as we hurtle around our star. (When you’ve mastered this, try to imagine as well the Earth and the solar system orbiting around the center of the galaxy at about 420,000 miles per hour.)

E-quinox, E-quator, E-cliptic

As the month of March progresses, the sun appears to move slowly eastwards against the stellar background scenery. By the 20th or so, it crosses the celestial equator. (Imagine the celestial equator as a piece of rope running all the way around the earth’s surface at the equator, then lifted off the ground high into the air, hoping it stretches as it does so, and way up into the sky.)

The apparent path followed by the sun in the sky is the ecliptic. With our flattish solar system, most of the planets also appear in this band of the sky. The zodiac constellations are here too, because the ancient astronomers observed the way the sun moved through them. At the beginning of March, the sun’s backdrop was provided by the constellation Aquarius, the water carrier. By the end of March, the sun has waded through Pisces and is leaping into Aries.

The point where the equator and ecliptic cross is called the First Point of Aries or the Vernal Point, named as such by greek astronomer Hipparchus in 130 BC, because at the time that point was in the Constellation of Aries. It’s now moved to the right and is in the constellation of Pisces, but everyone still calls it by its antique name.

The video suggests that the vernal equinox, the moment when the sun crosses the celestial equator, is about midday on March 20. NASA give the precise time as 10:29 UTC.

You might have expected the spring equinox to occur at 12 noon, or midnight. But in fact the exact moment of the equinox occurs when the Sun passes through the vernal point. The sidereal day starts when the vernal point is at the meridian. The two aren’t necessarily the same, and probably rarely are.

Greenwich, solar, and local sidereal time

As with timekeeping on the earth, there are two versions of star time. The first is a single universal sidereal time for the earth that’s defined for a global reference point, in the same way that universal solar time was once relative to Greenwich in London. The other is your local sidereal time, which may be hours ahead or behind, depending on how far away you and your longitude are from Greenwich. As with solar time, your location on the earth means that you’re in a different position relative to the stars at any moment compared to others.

Each Greenwich sidereal day begins (0h 0m 0s) when the Vernal Point is on the Greenwich Meridian, ie directly overhead – or at least as high as possible in the sky, as seen from the northerly latitudes of London.

Greenwich sidereal time will be 12h 0m 0s at the moment of the autumn equinox (when the sun passes through the vernal point again, in the opposite direction). But again it’s not equal to the solar time, because the autumn equinox in 2017 is estimated to be 20:02 UTC, and that’s below the horizon on a London September evening. But a few hours later the two times, solar and sidereal, are momentarily identical. Then the sidereal time pulls ahead again, and will stay in front, increasing its lead, for another year.

A sidereal clock has to gain about 4 minutes a day — 2 hours a month — over an ordinary clock. Two clocks side by side, one mean solar, one sidereal, will both show 24 hours of 60 minutes, but the sidereal time will run faster by nearly 4 minutes every day, and after a month it will be two hours ahead. They’ll only agree once every year. That moment is shortly after the Autumn Equinox, when the first point of Aries is directly overhead when and the mean solar time is midnight (not forgetting the inconvenience of Daylight Savings Time).

Cassio clock

If you live in the Northern Hemisphere, there’s a convenient virtual sidereal clock in the night sky. Find the right-most star of the easy-to-spot constellation of Cassiopeiea, the big “W” that rotates around the Pole Star. (It’s known as Caph, or Beta Cassiopeia, named for the Arabic word for palm of a hand). Imagine an hour hand extending from Polaris to this star. Because the star has an RA close to “sidereal midnight”, at 0h 09m, it’s almost perfect to act as a big hour hand.

Now imagine a 24 hour clock face centred on the Pole Star as you look due north. Cassiopeia rotates around Polaris, the Pole Star, once a sidereal day, and indicates your local sidereal time. The clock runs counter-clockwise. The sidereal time in this simulated sky picture is about 18h 0m for Greenwich.

It’s all Greek

The small but significant difference between the length of the sidereal and solar day was very important to the ancient astronomers in Sumeria, Babylon, and Egypt. Without the benefit of computers or astronomy software, they had to observe the constellations and stars that preceded sunrise, and constructed their annual calendar accordingly. Here’s the rising of Sirius in 1872 BCE, as visualized by Stellarium (without the hazy atmosphere):


You can’t see the stars during the day, but you can observe the stars that appear just before dawn, and see them slowly shift over the course of a few days.

I expect they would have seen it more like this:

The first “just before the sun” rising of Sirius (its heliacal rising) in the summer was noteworthy because the star had disappeared from the evening sky about two to three months previously. While the sun is moving in front of a given star, the star can’t be seen because it is only above the horizon during the day. The heliacal rising occurs when the sun has moved far enough past the star so that the star rises and becomes visible just before sunrise. When the Egyptians saw Sirius rising before sunrise they knew it would soon be time for the flooding, or inundation of the Nile River, around which all ancient Egyptian life revolved.

The 70 or 80 days day period during which Sirius disappeared from the night sky before its heliacal rising again was also the period of time between mummification and entombment, when the dead person was believed to reawaken to eternal life in the underworld. Perhaps the common mythological idea of a visit to some underworld and a welcome return to life has some links with these early beliefs and with a star’s temporary oblivion before its subsequent resurrection.

The date of the heliacal rising of Sirius was taken as the beginning of the Egyptian calendar — the first day of Thoth. But because of the precession of the equinoxes, the star-based calendar and the civil calendar moved slowly apart, to reunite after 1461 years. The Egyptians were aware of the cyclical nature of this drift, called the Sothic cycle.

Like clockwork

The ingenuity of the early astronomers is evident in the devices that survived them, such as the astrolabe, the “star taker”, and the Antikythera mechanism, the only surviving clockwork device from the ancient Greek civilization, currently on display in the National Archaeological Museum, Athens, and needing a bit of oil:

Image from Wikimedia

The astrolabe, possibly invented by Greek astronomer Hipparchus, is a clever combination of two coordinate systems (sky and earth) and two time systems (the solar calendar and sidereal time). The rotating disk on the top represents a squashed celestial sphere, and holds the stars and the ecliptic, and the bottom disk represents the earth, and shows the horizon and latitude lines. You can rotate the star-studded top to move to any point in the year, and read off the solar and sidereal times, and star positions, for any date.

Here’s an online astrolabe:

and here’s an interactive one:

The great Indian mathematican and astronomer Aryabhata, 470-540 CE, calculated the sidereal rotation (the rotation of the earth referenced the fixed stars) as 23 hours 56 minutes and 4.1 seconds. This value is very close to the modern value of 23:56:4.091.

It’s possible that the early medieval clocks were direct descendants of clockwork-powered astrolabes, probably developed by Islamic scholars based on Greek ideas. And on many of the earlier European astronomical clocks, including the famous Prague Orloj, and similar designs, you can see the same astrolabe-style stereographic projection that combines the two systems.

On clocks like the Orloj, look for the line between Aries (♈︎) and Pisces (♓︎), the Vernal Point, for a line that stretches out to the dial with a little gold star on the end. This effectively shows 0h 0m sidereal time.

Image from wikimedia

Clock mechanisms, particularly the more elaborate astronomical creations, are often described as ‘computers’. There are some obvious parallels. The clock designer, after much head-scratching, specifies in advance all the clock’s operations by calculating sequences of integers that, when multiplied and divided, produce the required output. These sequences are specified in terms of toothed interlocking gear wheels. The completed design is then ‘compiled’ into iron and brass, assembled, and starts running: the continuous rhythmic input signals from the escapement are transformed by the instructions in the clockmaker’s program to produce useful information on the output display.

The additional clockwork mechanisms required to show a starry disk or sidereal time in addition to the standard solar timekeeping mechanisms don’t have to be too complex. With 86400 seconds for a solar day, if one sidereal day is 3 minutes and 55.9 seconds shorter, it’s 86164.1 solar seconds long, and 86400/86164 is 1.002737909297. That looks like it might be a difficult number to achieve by combining a few gear teeth. But, consider a wheel with 96 teeth rotating once in 24 hours, which drives another wheel of 79 teeth joined to an arbor which carries a wheel of 157. This then drives a wheel with 133 teeth joined to an arbor which carries a wheel of 72, and drives one of 103. A hand on this last arbor will show sidereal hours. For a skilled clockmaker, this is probably one of the easier calculations they had to make:

79/96 * 133/157 * 103/72 => 0.9972695874616654 * 86400 => 86164.09235668789

The moon’s complex motions would have presented a much sterner challenge.

Probably the most accurate mechanical clock in the world is in Copenhagen. It’s Jens Olsen’s World Clock, shown here in this picture from Flickr user Garret Ziegler:

The eight gear wheels making up the sidereal train have the following teeth counts:

80/118, 143/127, 173/163, and 125/101

which makes a drive conversion ratio equivalent to

80/118 * 143/127 * 173/163 * 125/101 => 123695000/123357259 => 86164.0905258903

which is accurate to 100th of a second in a year.

Although the celestial sphere appeared on early clocks, sidereal dials didn’t appear on clocks until the 17th Century. An obvious reason is that clocks weren’t accurate enough to justify having two separate hands that were only 4 minutes apart after 24 hours.

Dial time

The Dutch scientist Christian Huygens developed Galileo’s pendulum idea further from 1657 onwards, and eventually, after solving the mathematical problems, won some patent rights for manufacturing pendulum clocks. Robert Hooke’s introduction of the anchor escapement in 1660s, and George Graham’s introduction of the dead-beat escapement in 1715, meant that the accuracy available to clockmakers was sufficiently good to build clocks accurate enough to distinguish between solar and sidereal time. This dual dial example, with the sidereal dial on the right, is from 1715 and is in the British Museum, attributed to Daniel Quare and Stephen Horseman.

Thomas Tompion and others built clocks with sidereal dials and astrolabe dials, reflecting the increasing number of professional and amateur scientists.

At Greenwich and other observatories, there was a demand for pure sidereal clocks, which allowed the astronomers and time-maintainers to measure star and planet transits directly.

Howver, some of the more ingenious amateurs would have asked their local clockmaker to shorten the pendulum of their clock, from 39.14 inches down to 38.87 inches, perhaps, to convert it from a solar clock to a slightly faster sidereal clock.

John Harrison, the famous clockmaker, often used sidereal time to check how accurate his clocks were running.

A great advantage of John Harrison’s home was that it faced south. From his bedroom window, he was able to line up sights with a neighbour’s chimney across the road, so that when a star emerged from behind the chimney, it crossed the north-south meridian, and this enabled him to know the precise sidereal time, and check his solar time clocks.

(From The Cosmic Elk’s history: John Harrison and the Problem of Longitude )

This, by the way, was a technique originally recommended by Huygens.

Right Ascension, hour angles, and grid references

The main users of sidereal time and sidereal clocks and watches are astronomers. Sidereal time corresponds with the coordinate system that is used for locating objects in the sky. On Earth we use a pair of numbers, latitude and longitude, to specify the locations of places, with the 0/0 origin point being Greenwich, in London, (chosen for convenience rather than for any real scientific reason). For coordinates of things in the sky, the equivalent of earthly longitude is right ascension, and the equivalent of earthly latitude is declination.

Right ascension is (slightly confusingly) measured in hours, and there are 24 RA hours in a complete circle, each hour consisting of 15 degrees (24 * 15 = 360). This isn’t a coincidence, of course, but it can be a bit odd, because the sky (appears to) rotate once a sidereal day, and so the RA grid system with its 24 hour divisions also rotates once every sidereal day. It’s like a giant 24 hour dial rotating (anticlockwise) in the sky once every 23 hours and 56 minutes. The location of 0h 0m (the top of this imaginary dial) is the vernal point, the first point of Aries.

The position of any star (or galaxy) in the sky as seen from the earth can be specified by its right ascension and declination. For example, Betelgeuse, the famous red giant star in the constellation of Orion has an RA of 05 hours 55 minutes and 10.3 seconds, and a declination of 7° 24′. At the time of writing it hasn’t yet exploded but it will do one day. The Andromeda Galaxy, the nearest galaxy to ours, has an RA of 0 hours 42 minutes and a declination of 41° 16′.

When a sidereal clock says 5h 55m 10s, Betelgeuse will reach the highest point in the sky, on the observer’s meridian … . And the Andromeda Galaxy will cross the meridian (“culminate”) some 19 hours later. (We’re sticking to the Greenwich meridian and Greenwich sidereal time here.)

So the object will be at its highest in the sky, on the meridian, when the current (Greenwich) sidereal time is the same as the object’s right ascension. This is why you’ll find sidereal clocks in observatories and on astronomer’s desks.

Astronomers also use the hour angle, which is the time in hours and minutes since an object in the sky last crossed your local meridian (probably the nearest it can get to being overhead). The hour angle of the vernal point is your local sidereal time. The object’s hour angle indicates how much sidereal time has passed since the object was on the local meridian. The hour angle is also the angular distance between the object and the meridian, measured in hours, where 1 hour is 15 degrees. For example, if an object currently has an hour angle of 2.5 hours, it crossed the local meridian 2h30m ago. 15 * 2.5 is 37.5, so the object is 37.5 degrees west of the meridian.

This diagram tries to relate the Greenwich and Local Hour Angles, the Local Mean Sidereal Time, and the Greenwich Mean Sidereal Time to the vernal point and right ascension.

Image from Wikimedia

This clock in London’s Science Museum, designed by Joseph Vines and made by Walsh of Newbury, Berkshire in about 1936, has twin dials and is well equipped with hands:

There’s apparently a moon hand showing the phases of the moon and its position relative to the sun—perhaps that’s the big hand pointing to II, so the small hand with a star on it pointing down to XII is the sidereal hour. The timekeeping is controlled by Harrison’s temperature compensating gravity escapement and gridiron pendulum.

You’ll often see sidereal clocks in observatories. This picture from Flick user Cheryl Colan shows the sidereal clock that sits next to the Clark 24 inch refracting telescope at the Lowell Observatory in Flagstaff, Arizona:

In your pocket or on your wrist

During the 18th and 19th centuries it was possible to buy clocks and pocket watches that show sidereal time, sometimes in addition to solar time. Here’s a rare example by John Arnold, the great Cornish watchmaker who continued the work that John Harrison started, building high-quality chronometers for navigation and astronomy:


This is the John Roger Arnold, No°2, George III mean and sidereal pocket chronometer with bimetallic Z balance.

The movements of these two watches, made between 1796 and 1799, featured some of the most famous inventions of father and son, including their fabulous thermo-compensated Z balance, expansion escapement and gold helical spring.

And here’s the modern Arnold & Sons DBS Equation Sidereal watch, a modern-day Swiss tribute to the original:

Images and text from blog.perpetuelle.com

I like to think of astronomers and astrophysics professors striding around with a sidereal watch in one waistcoat pocket and a solar watch in the other. They could then check these against the sidereal and solar clocks in the observatories.

This Waltham 24 hour sidereal pocket watch from the late 1890s was on sale on Ebay recently for a few thousand dollars:

Judging from this page of the Admiralty Manual of Navigation, sidereal watches were until quite recently issued as standard to seafaring navigators. Although, this model appears to be the sidereal equivalent of a 12 hour dial:

Of course, if money is really no object, there are many ways of showing sidereal time and your fortune on your wrist. Here, for example, is the Patek Philippe Calibre 89, the most complicated watch in the world.

Only four were made, and each would sell for more than $5 million today. This side (the back) of the watch shows the sidereal time, including the sidereal seconds in the small dial at the bottom.

A bit more affordable at about $250,000 is the Jaeger-LeCoultre Master Grande Tradition Grande Complication watch, which has a prominent sidereal display. Here’s a video:

For a similar price, how about this newly made double pendulum double dial clock by David Walter, showing solar time on the left, sidereal time on the right?

Computer time

The ubiquitous computer provides sidereal time on demand, to everyone.

On the internet, there are converters, such as Jürgen Giesen‘s sidereal converter, the one at Washington University, or the US Navy‘s.

This modern control panel is attached to the world’s largest optical telescope, the Gran Telescopio Canarias (GranTeCan or GTC), also known as the Great Canary Telescope. At the top its showing UTC and Local Sidereal Time (LST), along with much more, including the RA and Declination of presumably the point at which the telescope is currently looking.

(Image taken from a BBC documentary about astronomy presented by Jim Al-Khalili.)

We still use sidereal time to correct and control solar time, like we did in the 18th century. But today, instead of observations of stars crossing the meridian, we now use observations of quasi-stellar objects — quasars — outside our own galaxy. UTC is now calibrated according to a super-large and incredibly accurate 3D grid system called the International Celestial Reference Frame (ICRF), a reference frame based on the radio positions of 212 extragalactic sources. The positional accuracy of these sources is better than about 1 milliarcsecond (something like the apparent size of an iPhone screen on the moon, as seen from the earth?).

There’s an app for that

The pocket watch and wristwatch have mostly been replaced by the smartphone, and there are a few apps for showing sidereal time. On iOS, the simplest is SkyTime (below), which has an Apple Watch app (which shows sidereal, universal, and Julian time). Something like PolarScope Align also tells the sidereal time.

Or you can use something with a more traditional approach, such as Emerald Chronometer’s watch simulators — there are sidereal dials on the Geneva and Mauna Kea watches. (Shown below in the iPad version:)

On Android phones, you can try apps such as MySiderealTime (below) or Astro Sidereal Time.

Do it yourself?

Making a quartz sidereal clock

With the advent of cheap and accurate quartz clocks, you no longer have to spend a lot of money to have a sidereal clock. Of course, most quartz movements are conventional 12 hour versions for conventional people, but you can buy 24 hour single-revolution-in-a-day versions as well, and these can be modified to produce a clock that runs slightly faster or slower, ideal for making more specialized clocks.

Buying ready made

Brian Mumford of Mumford Micro Systems makes sidereal clocks (and others) for sale:

I make the clocks myself here in Santa Barbara. I use my own microprocessors to change the rate of a quartz movement to be correct for the different types of timekeeping. I hand-adjust each movement to run within one or two seconds per week of true sidereal
time. Greater precision is not warranted since the temperature fluctuations of a quartz oscillator are in that scale.

These clocks are an outgrowth of my interest in clocks in general and the manufacture of the MicroSet Precision Clock and Watch Timer. Since I specialize in the measurement of precision pendulum clocks, I have developed the tools and methods needed to fabricate and calibrate precision quartz clock motors.

To buy one of Brian’s sidereal clocks, visit http://www.bmumford.com/clocks/sidereal/.

Making your own sidereal clock

If you’re good at making things, perhaps you’d like to try assembling your own sidereal clock. A good place to visit is Geppetto Electronics, suppliers of the Crazy Clock, a quartz movement that has been modified with firmware that can run custom source code.

You can install different software routines that can be used to alter the timing of the standard clock to make, for example, a Martian Clock (on Mars one day is 24 earth hours, 39 minutes, 36 seconds), a Tide clock showing a day of 24 hours, 50 minutes, 28 seconds, or of course a Sidereal Clock. (There are more way-out clocks that you can install: with names such as Crazy, Wacky, Loopy, Early, and Warpy, you can probably guess what kind of time-keeping they offer…

Meanwhile, elsewhere on planet Earth

Sidereal time gives us links with the stars. So it probably isn’t surprising that there are a few creative thinkers who have found unusual ways of using sidereal time to promote their own beliefs.

For one, there are a number of astrologers who call their version of the ancient (pseudo)science sidereal astrology. This takes into account the current locations of the planets and stars, rather than the old Greek/Egyptian methods, which are out of date due to the slowly changing orientation of the solar system – the First Point of Aries has been in the constellation of Pisces for ages. The wikipedia will tell you all you need to know.

More exotic are the meditation and remote viewing practitioners, who believe that it’s possible to use extra-sensory perception to see distant objects. They claim that there are ‘good’ and ‘bad’ times for remote viewing, and that these coincide with sidereal time:

There is a 400% increase in spiritual/psychic/energy healing at 13:30 Sidereal time; 200-300% increase in abilities between 13:15 and 13:45 Sidereal time, and 200% increase in abilities between 12:45 and 14:15 sidereal time.

source

These are presumably Greenwich Sidereal Times, to be converted to Local Sidereal Time according to their current terrestrial location.

One possible cause of the variable ‘reception’ that remote viewers and meditators claim to experience is thought to be the super massive black hole at the center of our galaxy (which is in the direction of the constellation of Saggitarius). With an RA of 17h 45m it rises at about 17:45 Greenwich Sidereal Time every day, so the 13:30 time suggested might indicate that when it’s above or below you it has more effect on the human brain.

Considerably further out are the Kennedy Death Conspiracy Theorists. The claim is that those members of the Kennedy family who suddenly died did so at 16:00 sidereal time (presumably Greenwich sidereal time, adjusted for local time?). But, JFK was assassinated at 18:30 UTC on November 22, 1963, and Robert Kennedy was assassinated at 07:15 UTC, June 5 1968. These convert to sidereal times of 22h 34m and 0h 10m. I don’t think that’s close enough for a convincing conspiracy theory.

Closing music

Finally, switching away from the paranormal back to the normal: here’s a link to a video from a band called Sidereal:

a Beach Rock band based out of Jacksonville Beach, Florida that features rich harmonies along with musical versatility influenced by various artists and a collective of different musical backgrounds. While multiple genres shine through the group’s music, Sidereal is heavily rooted in a bouncy, dance-friendly Reggae style that is infused with a Rock edge. 

They have nothing at all to do with the subject of sidereal time, and anyway they pronounce their name “Side Reel”. But it ticks along nicely, and so makes a suitable close for this overlong post…