Kepler’s third law is so simple to state that (as shown last time) it is something that any grade school kid, armed with Copernicus’s data and a calculator, can verify. Yet it was 75 years from Copernicus’s publication til Kepler discovered this formula! Why did it take Kepler until 1618, nearly 50 years of age, to recognize such a simple relationship? Were people just dumber than high-school students back then?
Here’s a clue. We take all sorts of math for granted that didn’t exist four hundred years ago, and calculations which take an instant now could easily take an hour or even all day. (Imagine computing the cube root of 4972.64 to part-per-million accuracy by hand.) In particular, one thing that did not exist in Copernicus’ time, and not even through much of Kepler’s, was the modern notion of a logarithm.
Much of this work was done by Nicolai Copernicus himself, the most famous of those philosophers who argued for a Sun-centered universe rather than an Earth-centered universe during the millennia before modern science. He had all the ingredients we have, minus knowledge of Uranus and Neptune, and minus the clues we obtain from telescopes, which would have confirmed he was correct.
Copernicus knew, therefore, that although the planetary distances from the Sun and their cycles in the sky (which astrologers [not astronomers] have focused on for centuries) don’t seem to be related, the distances and their orbital times around the Sun are much more closely related. That’s what we saw in the last post.
Let me put these distances and times, relative to the Earth-Sun distance and the Earth year, onto a two-dimensional plot. [Here the labels are for Mercury (Me), Venus (V), Mars (Ma), Jupiter (J), Saturn (S), Uranus (U) and Neptune (N).] The first figure shows the planets out to Saturn (the ones known to Copernicus).
Figure 1: The planets’ distances from the Sun (in units of the Earth-Sun distance) versus the time it takes them to complete an orbit (in Earth-years). Shown are the planets known since antiquity: Mercury (Me), Venus (V), Mars (M), Jupiter (J) and Saturn (S).
The second shows them out to Neptune, though it bunches up the inner planets to the point that you can’t really see them well.
Figure 2: Same as Figure 1, but now showing Uranus (U) and Neptune (N) as well.
You can see the planets all lie along a curve that steadily bends down and to the right.
Copernicus knew all of the numbers that go into Figure 1, with pretty moderate precision. But there’s something he didn’t recognize, which becomes obvious if we use the right trick. In the last post, we sometimes used a logarithmic axis to look at the distances and the times. Now let’s replot Figure 2 using a logarithmic axis for both the distances and the times.
Figure 3: Same as Figure 2, but now with both axes in logarithmic form.
Oh wow. (I’m sure that’s the equivalent of what Kepler said in 1618, when he first painstakingly calculated the equivalent of this plot.)
It looks like a straight line. Is it as straight as it looks?
Figure 4: Same as Figure 3, but with a blue line added to show how well a straight line describes the distance/time relationship of the planets, and with a grid added that passes through 1, 10, 100, 1000 Earth-years and distances equal to 1, 10, 100 Earth-Sun distances.
And now we see three truly remarkable things about this graph:
First, the planet’s distances to the Sun and orbital times lie on a very straight line on a logarithmic plot.
Second, the slope of the line is 2/3 (2 grid steps up for every 3 steps right) rather than, say, 7.248193 .
Third, the line goes right through the point (1,1), where the first horizontal and first vertical lines cross.
Whether you’re a die-hard who insists on measuring the distances between the planets and the Sun yourself (which I’ve shown you how to do here and here), or you are willing to accept what other people tell you about them, it’s interesting to look at the pattern among these distances. They are shown at right, starting with the smallest — Mercury (Me) — and proceeding to Venus (V), Mars (Ma), Jupiter (J), Saturn (S), Uranus (U) and Neptune (N), nearly 100 times further out than Mercury. The inner planets up to Mars are very close together, all bunched within 1.5 times the Earth-Sun distance, whereas the outer planets are much further apart.
Figure 1: (Left) the planet-Sun distances relative to the Earth-Sun distance; (Right) the time between solar conjunctions, when the planet disappears (roughly) behind the Sun from Earth’s perspective; “>” indicates the planet moves into the morning sky (east) after conjunction, whereas “<” means it moves into the evening sky (west).
Also shown in the figure are the lengths of the planet’s cycles. Remember, a cycle starts when a planet reappears from behind the Sun and ends when a planet again disappears behind the Sun… the moment of “solar conjunction,” or just “conjunction” for short in this post. Some planets have short cycles, others have long ones. Interestingly, now it is the outer planets that all bunch up together, with their cycles just a bit longer than an Earth year, whereas Mercury, Venus and Mars have wildly different cycles ranging from a third of an Earth-year to two Earth-years. In the figure I’m also keeping track of something that I didn’t mention before. As their cycles begin, Mercury and Venus initially move into the evening sky, in the west, setting just after sunset. I’ve indicated that with a “<” Meanwhile Mars, Jupiter and Saturn move into the eastern morning sky, rising just before sunrise, as indicated with a “>”. (Mars, Jupiter and Saturn just reappeared from behind the Sun this winter; that’s why they’re all in the morning sky right now.) This difference is going to prove important in a moment.
Before going on, let me make another version of the same figure, easier to read. This involves making a “logarithmic plot”. Instead of showing the step from 1 to 2 as the same as the step from 0 to 1, as we usually do, we replot the information so that the step along the axis from 1 to 10 is the same as the step from 0.1 to 1. It’s gives exactly the same information as the Figure 1, but now the planet-Sun distances don’t bunch up as much.
Orbits Vs. Cycles
Figure 2: exactly the same as Figure 1, but with the vertical axis in “log-plot” form.
Now, the cycles from one solar conjunction to the next, long beloved of astrologers, are not beloved of astronomers, because they involve a combination of two physically unrelated motions. A solar conjunction happens when a planet disappears behind the Sun from Earth’s perspective, so the time between one conjunction and the next combines:
the orbital motion of the planet around the Sun;
the yearly rotation of the line between the Sun and the Earth. (So far, we haven’t found evidence as to whether the Sun moves around the Earth or the Earth moves around the Sun — and we’ll remain agnostic about that today.)
There was a big solar flare on Monday, which as often happens created a flash of X-rays (travel time to Earth 8.5 minutes), a blast of high speed protons and electrons (travel time minutes to hours), and a “Coronal Mass Ejection” (CME) of slower subatomic particles (travel time 1 to 3 days.) It’s the CME that is one of the main triggers for auroras borealis (north) and australis (south).
Update 23:00 Eastern Time (0300 UT): the CME has arrived in the past hour. There are northern lights going on, but so far they are still quite far north compared to predictions. This could change, but it will soon be too late for me to report on it, and it’s pretty cloudy here so I’m unlikely to stay up late hoping. Checking “northern lights” or “aurora” on Twitter is a good way to get real-time information, and there are some webcams on https://aurora.live/camera/ .
Aurora forecasts are not very reliable yet, because of a lack of data about the CMEs themselves and the ever-changing environment between the Sun and the Earth through which they are moving. So neither timing of arrival nor strength of impact can be predicted with great confidence. With that caveat, it seems likely that the flare will arrive late tonight or early in the morning in Europe, late afternoon to evening in the United States. Its effects could be as strong as any we’ve seen in a number of years; auroras may well be visible well into central Europe and the northern third of the United States, or even further south than that, on and off over a period of many hours. Fingers crossed.
We will have mostly cloudy skies here, so I probably won’t see this one. But I hope many of you are lucky enough to see a spectacle tonight!
Update: Also today, an even stronger flare occurred. If this one also created a CME that’s aimed toward Earth, we may have another round of strong auroras within a couple of days. See the data below showing X-rays detected from today’s flare, and from Monday’s earlier flare.
Measurements by two of our geostationary weather satellites of X-rays coming from the Sun; large peaks are caused by solar flares which produce X-rays in great abundance. The more X-rays, the stronger the flare.
Oh, and by the way, sometime soon I’ll show you how to use solar flares and CMEs to solve one of the hardest problems in do-it-yourself astronomy. If only the Greeks had known!
Now, the last step in mapping out the other planets, before heading for more intriguing territory.
In a previous post I showed you how you can measure the distance between Venus and the Sun, RVS, relative to the distance between Earth and the Sun, RES. Under the assumption that Venus’s orbit around the Sun is circular (or nearly so), you can use the fact that when the angle between Venus and the Sun reaches its maximum (the moment of greatest elongation, and also approximately the moment when Venus appears half lit by the Sun), there’s a simple right-angle triangle in play. High school trigonometry then gives you the answer: RVS/RES ≈ 0.72 ≈ 1/√2. The same trick works for Mercury, which, like Venus, is a near Sun-orbiting planet, closer to the Sun than Earth.
But there’s no maximum angle for Mars, Jupiter, or the other far planets. These planets are further out than Earth and can even appear overhead at midnight, when they are 180 degrees away from the Sun. Fortunately there’s another right triangle we can use, again under the assumption of a (almost-)circular orbit, and that can give us a decent estimate.
The Triangle for the Far Sun-Orbiting Planets
Let’s focus on Mars first, although the same technique will work on the outer planets. Mars has a cycle in which it disappears behind the Sun, from Earth’s perspective, on average every 780 days. (That start of the cycle is called “solar conjunction,” or just “conjunction” when the context is clear.) About half a cycle later, after on average 390 days, it is at “opposition”: closest to Earth, largest in a telescope, appearing overhead at midnight, and at its brightest. But if we wait only a quarter cycle, on average 195 days after conjunction, then the Mars-Sun line is at a 90 degree angle to the Earth-Sun line. That means that Mars, Earth and the Sun form a right-angle triangle with the right angle at the location of the Sun.
So on the day of first quarter we should measure the angle on the sky between Mars and the Sun. That’s the angle A on the figure below. Then the Mars-Sun distance RMS and the Earth-Sun distance RES are the two sides of a right-angle triangle. That means they are related by the tangent function:
RMS/RES = tan A.
For a planet whose distance RPS from the Sun is greater than Earth’s, an estimate of its distance compared to the Earth-Sun distance REScan be made using the fact that (were its orbit perfectly circular) it would make a right-angle triangle after the first or third quarter of its cycle, with the right angle located at the Sun. Sizes of Earth, Sun and planet not shown to scale.(more…)
This Sunday, Venus reaches a special position from which it is easy to estimate roughly how large the average Venus-Sun distance (RVS) is relative to the average Earth-Sun distance (RES). (I say “on average” because the Venus-Sun distance isn’t quite constant. Venus’s orbit, like that of all the planets, isn’t quite circular. But this is a small effect that we can ignore for the purpose of rough estimates.)
If you are a true diehard astronomy-geek, by all means get up at 5 or 5:30 in the morning on Sunday (or really any of the next few days) to check this directly. I can assure you (since I have been up at that time recently, due to chronic insomnia more than astronomy-geekhood) that Venus looks absolutely gorgeous against the deep blue of the pre-dawn sky. But if you have no intention on getting up that early, or clouds intervene, there’s a shortcut — on your phone.
Greatest Elongation, Near-Circular Orbits, Half-Lighting and Right Angles
On Sunday, Venus moves to a position where, from Earth’s perspective, the angle between Venus and the Sun on the sky reaches its maximum. This position is called “greatest elongation“, and it is reached twice per cycle, once in the evening sky and once in the morning. If Venus’s orbit were perfectly circular, this would also be the moment when Venus appears half-lit; as we’ve been seeing in two recent posts (1,2), that’s an effect of simple geometry:
if Venus’s orbit were circular, then at greatest elongation, the triangle formed by Earth, Venus and the Sun would be a right angle where Venus is located, and Venus would be half-lit.
This holds for Mercury too, as it would for any near Sun-orbiting planet.
Since Venus’s orbit isn’t quite circular, this isn’t precisely true; half lit and the right angle come together, but greatest elongation is off just a few days. This is a minor detail unless you’re an astronomy-geek, and won’t keep us from getting a good estimate of the Venus-Sun distance.
What else can we learn just with simple observations? Since the stars’ daily motion is an illusion from the Earth’s spin, and since the stars do not visibly move relative to one another, our attention is drawn next to the motion of the objects that move dramatically relative to the stars: the Sun and the planets. Exactly once each year, the Sun appears to go around the Earth, such that the stars that are overhead at midnight, and thus opposite the Sun, change slightly each day. The question of whether the Earth goes round the Sun or vice versa is one we’ll return to.
Let’s focus today on the planets (other than Earth) — the wanderers, as the classical Greeks called them. Do some of them go round the Earth? Others around the Sun? Which ones have small orbits, and which ones have big orbits? In answering these questions, we’ll start to build up a clearer picture of the “Solar System” (in which we include the Sun, the planets and their moons, as well as asteroids and comets, but not the stars of the night sky.)
The Basic Patterns
If we make the assumption (whose validity we will check later) that the planets are moving in near-circles around whatever they orbit, then it’s not hard to figure out who orbits who. For each possible type of orbit, a planet will exhibit a different pattern of sizes and phases across its “cycle“ when seen through binoculars or a small telescope. Even with the naked eye, a planet’s locations in the sky and changes in brightness during its cycle give us strong clues. Simply by looking at these patterns, we can figure out who orbits who.
Once you’ve convinced yourself the Earth’s a spinningsphere of diameter about 8000 miles (13000 km), and you’ve estimated the Moon’s size and distance(diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter), it’s easy to convince yourself the Sun’s bigger than the Earth, and much further than the Moon. It just takes a couple of triangles, and a bit of Moon-gazing.
Since that’s all there is to it, you can guess that the ancient Greek astronomers, masters of geometry, already knew the Sun’s the larger of the two. That said, they never did quite figure out how big and far the Sun actually is; we need modern methods for that.
It’s Just a Phase
The Moon goes through a monthly cycle of phases, lasting about 291/2 Earth days, in which the part that glows brightly with reflected sunlight grows and shrinks, from crescent to full and back again. The phases arise because there are two simple ways of dividing the Moon in half:
At any moment, the half of the Moon that faces Earth — let’s call it the near half of the Moon — is the only half that we can potentially see. (We’d only be able to see the far half, facing away from Earth, if the Moon were transparent, or a big mirror was sitting beyond the Moon.)
At any moment, the half of the Moon that faces the Sun is brightly lit — let’s call it the lit half. The other half is dark, and its presence can only be detected by the fact that it can block stars that it moves in front of, and through a very dim glow in which it reflects sunlight that first reflected from the Earth (called “Earthshine.”)
The phases arise because the lit half and the near half aren’t the same, and the relationship between them changes from night to night. See the diagram below. When the Moon is more or less between the Sun and the Earth (it rarely passes exactly between, because its orbit is tilted by a few degrees out of the plane of the drawing below) then the Moon’s lit half is its far half, and the near half is unlit. We call this dark view of the Moon the “New Moon” because it is traditionally viewed as the start of the Moon’s monthly cycle.
Figure 1: The Moon’s phases, assuming the Sun’s much further than the Moon. When the Moon is roughly between the Earth and Sun, its near half coincides with the unlit half, making it invisible (New Moon). As the cycle proceeds, more of the near half intersects with the lit half; after 1/4 or the cycle, the Moon’s near half is half lit and half unlit, giving us a “half Moon.” At the cycle’s midpoint, the near side coincides with the lit half and the Moon appears full. The cycle then reverses, with the other half Moon occurring after 3/4 of the cycle.
When the Moon is on the opposite side of the Earth from the Sun (but again, rarely eclipsed by Earth’s shadow because of its tilted orbit), then its near side is its lit side, and that creates the “Full Moon”, a complete white disk in the sky.
At any other time, the near side of the Moon is partly lit and partly unlit. When the line between the Moon and Earth is perpendicular to the Earth-Sun line, then the lit side and unlit side slice the near side in half, and the Moon appears as a half-disk cut down the middle.
When I was a child, I wondered why half this half-lit phase of the Moon, midway between New Moon (invisible) and Full Moon (the bright full disk), was called “First Quarter”, when in fact the Moon at that time is half lit. Why not “First Half?” Two weeks later, the other half of the near-side of the Moon is lit, and why is that called “Third Quarter” and not, say, “Other Half”?
This turns out to have been an excellent question. The fact that a Half Moon is also a First Quarter Moon tells us that the Sun is large and far away!
Last time I described an easy way for you to determine the size of the Moon — easier than the famous techniques used by the classical Greeks. (We don’t need to know the Earth’s circumference, as they did, if we’re ok with a moderately precise estimate.) Once you’ve done that, there’s an simple method, well known since classical times, for figuring out how far away the Earth’s companion is. That’s what I’ll describe in this short post.
(What’s not so easy is to determine the distance and size of the Sun. The classical Greeks failed in their efforts. We’ll need a more modern approach… but that’s for next week.)
Size Versus Distance
Even the early classical Greeks knew something about the Sun, just from the fact that the Moon and Sun appear roughly the same size to our eyes — that is, they occupy about the same amount of sky. If the Sun is twice as far away as the Moon, its diameter must be twice as big, in order that it appear the same size. That’s illustrated in the figure below. If it is ten times as far away, its diameter must be ten times as big. If it’s four hundred times as far away, its diameter is four hundred times as big. (Spoiler: that last one’s the truth; but we’ll get to it later.)
If the Moon is a distance L away from you, and another object twice as far away appears to be the same size in the sky, then that object’s diameter must be twice the Moon’s diameter D. This logic applies more generally to objects further and nearer than the Moon.
You can run this logic in the other direction; if something perfectly blocks the Moon, then if it’s ten times closer than the Moon its diameter must be ten times smaller. If it’s a billion times closer than the Moon, it must be a billion times smaller.
Having confirmed we live on a spherical, spinning Earth whose circumference, diameter and radius are roughly 25000, 8000, and 4000 miles (40000, 13000, and 6500 km) respectively, it’s time to ask about the properties of the objects that are most obvious in the sky: the Sun and Moon. How big are they, and how far away?
If the Moon were close to Earth, then at any one time it would only be visible over a small part of the Earth, as indicated in light blue. But in fact (except at new moon) about half the Earth can see it at a time.
Historically, many peoples thought they were quite close. With our global society, it’s clear that neither can be, because they can be seen everywhere around the world. Even the highest clouds, up to 10 miles high, can only be seen by those within a couple of hundred miles or so. If the Moon were close, only a small fraction of us could see it at any one time, as shown in the figure at right. But in fact, almost everyone in the nighttime half of the Earth can see the full Moon at the same time, so it must be much further away than a couple of Earth diameters. And since the Moon eclipses the Sun periodically by blocking its light, the Sun must be further than the Moon.
The classical Greeks were expert geometers, and used eclipses, both lunar and solar, to figure out how big the Moon is and how far away. (To do this they needed to know the size of the Earth too, which Eratosthenes figured out to within a few percent.) They achieved this and much more by working carefully with the geometry of right-angle triangles and circles, and using trigonometry (or its precursors.)
The method we’ll use here is similar, but much easier, requiring no trigonometry and barely any geometry. We’ll use eclipses in which the Moon goes in front of a distant star or planet, which are also called “occultations”. I’m not aware of evidence that the Greeks used this method, though I don’t know why they wouldn’t have done so. Perhaps a reader has some insight? It may be that the empires they were a part of weren’t quite extensive enough for a good measurement.
