Science Clock Series: Part XI

Today's number comes from astronomy and is given by:

\[\approx\ \text{diameter of ♃(in \(\beta\); \(\oplus=1\beta\)}\] This is a slightly roundabout way of saying "approximately the diameter of Jupiter in Earth-diameters." Let's look at it a little more closely:

First of all, what in the world is ♃ supposed to be? Or \(\oplus\)? To answer those questions we need to go back in time. About 2,000 years in fact, give or take. You see, one thing that I've learned from idly inspecting ancient writing, whether written, inscribed, or etched, is that ancient people liked to abbreviate.

Although it surprised me at first, this is entirely reasonable when you think about it; we do it all the time in everyday life, especially with the proliferation of instant messaging. Ancient peoples had to write everything by hand, which in my opinion is very dull and tiresome. You start looking for ways to reduce the amount you have to write, and before you know it you've got abbreviations all over the place.

Anyway, writing goes back a long time, but for much of history it was limited to a thin slice of the most educated in society. The study of astronomy also goes back a long time, and was one of the most common subjects for that educated elite to study, given its importance to pre-Industrial societies in helping to determine things like the proper time to plant and harvest crops in order to ensure everyone didn't starve over the winter.

Put those fact together, and people have been writing about astronomy for a very long time. Some of the oldest writings we find have been discovered to be about astronomy. Since it was so important, and given that most people like to save time and effort when writing, ancient astronomers in the Hellenistic period around the time of Christ came up with a set of symbols to refer to the "planets."

Note that the word "planets" in this context refers to the seven "planets" of the Ptolemaic (and originally Aristotelian) heliocentric system: the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn.These are the objects which, if you're familiar with the night sky, appear to move across it against the background of the fixed stars. Anyway, ancient astronomers came up with symbols for them that were used up through the Renaissance period. In fact, their use was so common that when astronomers such as William Herschel started discovering new planets astronomers rapidly came up with new symbols for them too. Anyway, here's a table with the symbols for the Sun, and the eight planets discovered before 1900:
\begin{align*}
\text{Sun}&\dots☉\\
\text{Mercury}&\dots☿\\
\text{Venus}&\dots♀\\
\text{Earth}&\dots\oplus\\
\text{Mars}&\dots♂\\
\text{Jupiter}&\dots♃\\
\text{Saturn}&\dots♄\\
\text{Uranus}&\dots♅\\
\text{Neptune}&\dots♆
\end{align*}You may be familiar with the symbols for Mars and Venus, as they have come to stand for “male” and “female” respectively in modern usage. Other than that, the only symbols commonly used in astronomy any more are the ones for the Sun and Earth. It's standard practice in astronomical journals for the symbols \(\text{R}_☉\), \(\text{M}_☉\), and \(\text{L}_☉\) to stand for the mass, radius, and luminosity of the Sun, respectively (and similarly for the Earth using the symbol for Earth).

It might give you some indication just how little known these symbols are today if I told you that right up until I looked them up to write this post I thought the symbol for Jupiter on my clock stood for Neptune!

Now that I know it stands for Jupiter, we can look at what the clock actually says: approximately the diameter of Jupiter in terms of “beta”, where “Earth” = 1 “beta.” I actually looked up beta to make sure there wasn't some special use for it that I wasn't aware of and couldn't find anything, so I'm not entirely sure what the point of introducing it only to immediately define it as one Earth was. Anyway, if we then check with the diameters of both Earth and Jupiter, we find that Jupiter does indeed have a diameter about 10.9377 times greater than Earth's.

So there you have it. And I realize this post isn't actually as short as I promised last time, though hopefully it was still interesting. There's a lot related to the astronomical symbols that I didn't cover, such as the fact that several were created for the first nineteen asteroids discovered before people realized that creating unique symbols for every asteroid would be effectively impossible and gave up (given that we now know of over a hundred thousand asteroids and suspect there may be ten times that number in the solar system, we can see that this was a good decision!).

Anyway, check back for the final post in this series, with a number from meteorology! Click here to jump directly to it.

Science Clock Series: Part II

In part two of this series, we look at a subject from nuclear physics. Or chemistry. It's kind of at the point where the two overlap.

Today's number is two, and it is approximately equal to:

\[2\approx\text{T}_{1/2}\,^{237}\text{Np}\,(\times10^6\,\text{y})\] T\(_{1/2}\) refers to the half-life of a substance, which means the amount of time, on average, that it takes for half of a sample of a radioactive substance to decay into something else. \(^{237}\)Np is the chemical symbol for the element neptunium (specifically, the isotope neptunium-237), and “\(\times10^6\)” is scientific notation for “multiply this number by 1,000,000”. So the whole expression means “approximately equal to the half-life of neptunium-237 when multiplied by two million years,” referring to two.

Neptunium is the element with atomic number 93 and the first transuranic element. This means it is the first element after uranium (atomic number 92), and is thus only found in nature in extremely tiny amounts (after uranium no element is found in nature in anything other than trace amounts). Neptunium has at least nineteen known isotopes, of which the most stable is neptunium-237 (also written \(^{237}\)Np) with 93 protons, 144 neutrons, and a half-life of 2.144 million years.

So the full expression can be read as “two (million years) is approximately equal to the half-life of neptunium-237.” And now you know where it comes from. Check back next time for something from cosmology! Click here to jump directly to it.

As an aside, the name neptunium comes from the planet Neptune which follows the planet Uranus out from the Sun, just as neptunium follows uranium in the periodic table. (Plutonium also follows neptunium just as Pluto follows Neptune [most of the time, anyway].) Uranium was named after the seventh planet from the Sun, which is now known as Uranus, but which was not always the case. When it (the planet) was originally discovered there was some controversy over what it should called: Herschel, the discoverer, wanted “Georgium Sidus” (“George's Star” in Latin), after his patron King George III of England. Astronomers from other countries were (understandably) a bit miffed at a celestial object bearing the name of a foreign monarch, and several alternate names were proposed, including "Uranus" by the German astronomer Johann Bode, who first determined Uranus' orbit. A few years later when Bode's colleague Martin Klaproth discovered a new metal (in 1789) he named it uranium in support of Bode's proposed name (which eventually beat the competition to become the standard today). By the time neptunium was discovered (officially in 1940) the name Uranus was long the standard, so neptunium and later plutonium were simply nice additions. 

Moons and Months

It's probably not a big surprise to most of you to learn that the words for "moon" and "month" are related in English (and some other languages as well). Our Moon's orbital period of 27 days, 7 hours, and 41.1 minutes comes very close to the number of days you get when you divide the Earth's orbital period by twelve, and makes a nice natural division of time.

But have you ever thought about the moons of other planets? For example Mars' two moons, Phobos and Deimos, orbit their parent planet in just 7 hours 40 minutes and 30.3 hours respectively. Many of Jupiter and Saturn's close-in moons likewise orbit in less than an Earth day. In fact, there are dozens of moons with a shorter orbital period than our Moon.

On the flip side of the scale, there are also dozens of moons with longer orbital periods than our Moon. Jupiter and Saturn both also have lots of small, irregular moons that orbit far from their parent body, which can take months or even years to complete one orbit. Saturn's moon Phoebe, for instance, takes 550.3 days to make a complete circuit, nearly two Earth years. Prior to last week, I knew of a few Jovian moons with orbital periods measured in days in the 600's and 700's. Given Jupiter's humongous mass, you'd expect that it would be able to hold onto satellites further out than other planets, which would have correspondingly long orbital periods.

So you can imagine my surprise when I, on a whim, looked up the satellite with the longest orbital period and discovered it belonged to...Neptune?? And not just by a few days or even a few months – we're talking years here.

In fact, it turns out the four longest orbital-period moons all belong to Neptune. The two inner ones, Sao and Laomedia, have orbital periods of 7.97 and 8.68 years respectively. The two outer ones, Psamathe and Neso, take 24.84 and 26.67 years to orbit Neptune once, respectively.

I found this revelation absolutely mind-boggling. Neither of these moons has completed an orbit since I've been born. They have longer orbital periods than the first five inner planets. They orbit Neptune at a mean distance of around 48-49 billion kilometers (about 30 million miles), which is nearly a third of the distance from the Earth to Sun. At its furthest point, Neso can be further from Neptune than Mercury ever gets from the Sun!

If you wondered, like me, how Neptune and not Jupiter can have the furthest-out and longest-orbiting satellites, it has to do with something called the Hill sphere (named after 19th-century American astronomer and mathematician George William Hill). The Hill sphere is basically the region of space in which an object's gravitational pull dominates the attraction from other objects in the region. For a moon to remain in orbit about a planet, it must remain entirely inside the planet's Hill sphere, or it will eventually be pulled loose by the gravitational perturbations of other planets. This limits how long of an orbital period a moon (or other satellite) can have before it is no longer stably bound to its parent planet. For instance, the mathematics suggests that it is impossible for the Earth to have a satellite with an orbital period of longer than about seven months.

To get to the point, a planet's Hill sphere depends both on its mass, and its distance from the Sun (and other massive sources of gravitational perturbation). Jupiter, of course, is many times more massive than Neptune (and all the other planets combined), but Neptune is several times further from the Sun. Add in the inverse-square nature of gravity, and Neptune manages to eke out a victory in the "largest planetary Hill sphere" competition. (Interestingly, of the four outer planets, Jupiter has the smallest Hill sphere; it increases slightly but steadily in size from Jupiter through Saturn and Uranus on to Neptune. Turns out increased distance from the Sun is more important than decreasing mass.) Neso and Psamathe are orbiting nearly at the outer limit of Neptune's Hill sphere, so they are likely to remain the moons with the longest orbital periods for the foreseeable future.

Of course, they were only discovered in 2002 and 2003, respectively, so who knows what else could be out there! It's an exciting time for us lovers of planetary science and Solar System dynamics.

Anyway, I hope you found that as interesting as I did. If you're interested in other comparisons between the moons of the Solar System, this page on Wikipedia has a nice table that you can sort by various categories.

Globular Cluster Series (Part 25): NGC 6293

Wow, number twenty-five already. It seems like just yesterday that I decided to make a photographic catalog of the Milky Way's globular clusters, but it's already been a year. Today's picture is the moderately-sized cluster NGC 6293 in the constellation Ophiuchus, the Serpent Bearer. NGC 6293 is pretty similar to the cluster I showed off last time, NGC 6541. NGC 6293 is a bit further away at 31,000 light-years (compared to 22,800), and also a bit physically smaller at 71 light-years in diameter (compared to 100), which combine to give it much smaller size on the sky, only 7.9 arc-minutes compared to 15.0.

Globular cluster NGC 6293 in Ophiuchus.

Both NGC 6293 and 6541 are at about the same distance from the galactic center: 6,200 light-years for NGC 6293, and 6,800 for NGC 6541. Interestingly, despite the fact that NGC 6293 is both fainter and smaller than 6541, it was discovered first, in 1785 by William Herschel (discoverer of the planet Uranus), while NGC 6541 wasn't discovered until 1826. This may have something to do with the fact that NGC 6541 is much more southerly than 6293, which would make it appear much fainter and harder to see for the northern European astronomers who discovered both of them.

Our Star

Have you stopped to ponder just how mind-blowingly huge the Sun is lately?

Last week while volunteering up at the Vis I took a picture of the Sun through the solar telescope on a whim. I noticed a large sunspot group on it, but didn't think anything else of it until this week when I learned that said sunspot group (called Active Region 1339) is one of the larger ones on record. I'd also heard somewhere along the line that it was larger than Earth, so I decided to do some visual comparing of my own. After seeing how Earth and Jupiter looked against the Sun, I decided to go all the way and add the rest of the planets. This image is the result. It shows the 8 planets of our Solar System against the Sun with AR 1339, all of them correctly sized relative to each other. (The distances between the planets are not to scale, due to the way I set up the picture.)

Our Solar System.

Look at this image, and let it sink in for bit. The Sun accounts for a whopping 99.86% of all matter in the Solar System. It's big. For fun, see how many other sunspots you can spot in this picture that are larger than Earth.

Edit (11/25/11): One other thing I like about this picture that I forgot to mention the first time is the sense of security it gives, when you really think about it. Stable orbits, despite their ubiquity in nature, are still nothing to take for granted, and it's sort of comforting seeing just how huge the Sun is compared to the Earth, and just how firmly we're caught in its gravitational embrace.

“Tremble before Him, all the Earth; indeed, the world is firmly established, it will not be moved. Let the heavens be glad, and let the Earth rejoice” -- 1 Chronicles 16:30-31a