MATT STRASSLER
Of Particular Significance

Breakthrough! Do-it-Yourself Astronomy Pays Off: A Law of Nature (and a Bonus)

In recent posts (here and here), I showed you methods that anyone can use for estimating the planets’ distances from the Sun; it just takes high-school trigonometry. And even more recently I showed how, using just algebra, you can easily obtain the planets’ orbital periods from their cycles as we see them from Earth, starting with one solar conjunction and ending at the next.

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.

What do they mean?

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POSTED BY Matt Strassler

ON April 5, 2022

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How You Can Determine Each Planet’s Year from its Distances and Cycles

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.)

So what astronomers want to know is the orbital period of each planet — it’s own year. That is, how long does it takes each planet to orbit the Sun, from the planet’s perspective, or from the Sun’s perspective. This is the time that an observer on the Sun would see for the planet to complete a circle relative to the fixed stars, and vice versa. (Remember we gathered evidence that the stars are fixed, or extremely slowly drifting from the perspective of the Earth, using a gyroscope, whereas either or both the Sun or the Earth are rotating relative to one another by about one degree per day. We also know the stars are much further than the Sun from our two measurements of the Moon’s radius.)

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POSTED BY Matt Strassler

ON March 31, 2022

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Alert — Aurora (Northern Lights) Possible Tonight

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!

Picture of POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON March 30, 2022

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How to Estimate the Distance to the Outer Planets Yourself

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 RES can 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.
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POSTED BY Matt Strassler

ON March 28, 2022

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A Prediction from String Theory

(An advanced particle physics topic today…)

There have been various intellectual wars over string theory since before I was a graduate student. (Many people in my generation got caught in the crossfire.) But I’ve always taken the point of view that string theory is first and foremost a tool for understanding the universe, and it should be applied just like any other tool: as best as one can, to the widest variety of situations in which it is applicable. 

And it is a powerful tool, one that most certainly makes experimental predictions… even ones for the Large Hadron Collider (LHC).

These predictions have nothing to do with whether string theory will someday turn out to be the “theory of everything.” (That’s a grandiose term that means something far less grand, namely a “complete set of equations that captures the behavior of spacetime and all its types of particles and fields,” or something like that; it’s certainly not a theory of biology or economics, or even of semiconductors or proteins.)  Such a theory would, presumably, resolve the conceptual divide between quantum physics and general relativity, Einstein’s theory of gravity, and explain a number of other features of the world. But to focus only on this possible application of string theory is to take an unjustifiably narrow view of its value and role.

The issue for today involves the behavior of particles in an unfamiliar context, one which might someday show up (or may already have shown up and been missed) at the LHC or elsewhere. It’s a context that, until 1998 or so, no one had ever thought to ask about, and even if someone had, they’d have been stymied because traditional methods are useless. But then string theory drew our attention to this regime, and showed us that it has unusual features. There are entirely unexpected phenomena that occur there, ones that we can look for in experiments.

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POSTED BY Matt Strassler

ON March 20, 2022

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This Weekend, Measure the Distance to Venus Yourself

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.

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Picture of POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON March 18, 2022

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