(This is the fourth post in a series, though it can be read independently; here are post #1 , post #2 , and post #3.)
For many years, I thought that measuring the distance to the Sun was quite difficult for a non-astronomer. I had the impression that it requires precision measurements, often involving telescopes or information from satellites, and that it was only easy to obtain a minimum distance and a maximum distance that were still quite far apart, as I explained in my last post.
But it’s not true. As I’ll explain today, it turns out that anyone can estimate the distance to the Sun, at night, with nothing more than the naked eye, basic reasoning, and… meteors.
Just from the fact that a long meteor crosses the sky in a few seconds, you can infer that the Earth-Sun distance is something like 100 million miles (km). If the Sun were only 10 million miles (km) away, the meteors would lazily drift among the stars, only a bit faster than the motions of the space station and other satellites, which take minutes to cross the sky. Meanwhile, if the Sun were a billion miles (km) away, then meteors would flash across the sky in a fraction of a second.
With a little more work and knowledge, you can use meteors to make an estimate of the Sun’s distance that’s well within a factor of 2 of the truth. It’s not even that hard to get a precise measurement that’s good to 10% or so.
It may seem odd that one can use such little specks of dust in the Earth’s atmosphere to determine, without a telescope, how far it is to the Sun. But in fact the method is simple. It’s so simple that it must have been pointed out two centuries ago. Curiously, though, I’ve never seen it written down anywhere. It seems to be little-known, even to scientists.
The Method
A visible meteor (the moving dot in the sky that often leaves a trail behind it) occurs when a stone (called a meteoroid), usually no larger than a small pebble, enters the Earth’s atmosphere roughly 100 km (60 miles) above the ground. As the meteoroid proceeds, friction from air molecules heats it to high temperature, causing the air around it to glow. This glow is what creates the bright dot seen from the Earth’s surface. The dot remains visible until the meteoroid has largely vaporized, usually while still at a high altitude of at least 50 km (30 miles). [Occasionally, larger objects survive their fall and hit the Earth; but these meteorites are rare and not our focus here.] I’ve written a bit more about meteors, as relevant to this discussion, here.
A typical meteor resembles a rapidly moving star, hence its common names “shooting star” and “falling star”. But compared to real stars, which are giant and distant, meteors are tiny and close.
How can observations of meteors teach us the distance to the closest star of all, the Sun? Here’s the basic conceptual idea.
First Step
First, you need to estimate the meteors’ speeds. You can do this yourself, with the naked eye. I’ve written a webpage to explain the details of how this is done, but here’s the method:
- Working with a friend a few miles away and using the geometry of parallax, you can confirm that meteors begin glowing at a height of roughly 50-100 miles (100-150 km) .
- Knowing a meteor’s height, you can then use the rate at which it crosses the sky to estimate its speed.
If you do this, you will find that all meteors move at tens of km (tens of miles) per second, and never slower or faster than that.
Second Step, the Rough Idea
Next, you need to understand how and why the speeds and angles of meteors indicate that the Earth is traveling at roughly the same speed as the meteors. The details of why this is so are explained on this webpage, but here are a couple of ways to understand the logic.
First, consider a rough analogy to meteors striking the Earth — an oversimplification, but it does capture the rough idea. Imagine raindrops falling on your car in a storm with breezy, swirling winds, and think about how things change depending on the speed of your car (see Fig. 1).
- If your car is stationary or moving slowly, then the raindrops’ speed and direction as they strike your car are set mainly by the wind speed and direction; over time, all your car’s windows will be struck equally with raindrops.
- If your car is moving at highway speeds, much faster than the storm’s winds, then the raindrops might as well be stationary; they will strike nearly head-on to your front windshield, at a speed that is roughly the speed of the car. In this case, the details of the winds don’t matter much.
- If your car is moving at a speed comparable to the winds, then the situation is in between: on average the raindrops will be somewhat more likely to hit your front windshield, and they will, on average, do so at a speed somewhat higher than your car’s speed, and considerably higher than the raindrops that hit your rear windshield.
Mere naked-eye observation of various meteor showers will show you that the meteors in most (but not all) meteor showers arrive from a direction that is overhead in the post-midnight sky, the Earth’s “front windshield”, and their meteors are faster than those whose that arrive from a direction that is overhead in the pre-midnight sky. In other words, the patterns exhibited by meteors in meteor showers resemble the last case, and not the first two cases. This pattern then reveals that the speed of the Earth around the Sun is indeed comparable to, and neither much slower than or much faster than, the speeds of the meteors as they move around the Sun. And so the Earth’s speed must be a few tens of km (miles) per second.
This similarity between the speed of the giant Earth and the speed of these tiny meteoroids might seem like a strange accident. Why should these two speeds be related? Well, it’s no accident.
Second Step, More Precisely
Knowing some facts about how objects move under gravity — specifically Newton’s laws of motion, his law of gravity, and the conservation of energy — we can prove that the Earth’s speed and that of meteors should indeed be similar. (The detailed reasons are explained on this webpage.) Specifically,
- For any object that is bound gravitationally to the Sun, its speed (relative to the Sun) as it passes the Earth must be less than √2 = 1.4 times the speed of the Earth around the Sun.
- From the Earth’s perspective, that means that the speed (relative to the Earth) of any meteor that strikes the Earth must be less than (√2 + 1) = 2.4 times the speed of the Earth around the Sun; and the fastest meteors will be those that emerge from a direction that is overhead just before sunrise.
Thus, if we assume that all meteoroids in meteor showers are gravitationally bound to the Sun (an assumption which can be checked), then this logic implies the Earth’s speed is 1/2.4 = 0.4 times the speed, relative to Earth, of the fastest observed meteors.
Now, depending on how precisely you’ve measured meteors’ speeds in the first step, you can determine the Earth’s speed with the same precision. Even just knowing that meteors travel at tens of km per second relative to Earth will already tell you that the Earth moves at a few tens of km per second. If you do precise measurements, you’ll find that the fastest meteors are seen to move, relative to the Earth, at about 70 km (45 miles) per second. Dividing by 2.4, you can conclude that the Earth’s speed in its orbit is about 30 km (20 miles) per second.
Last Step
The final step is the easiest: if you know the Earth’s speed, then you know how far it travels in a year, which is the circumference of its near-circular orbit. Dividing the orbit’s circumference by 2𝝅 gives you the average radius of that near-circular orbit — namely RES, the distance of the Earth from the Sun. (See Fig. 2.)
Specifically,
- RES = ( 30 km per second x 31.5 million seconds per year x 1 year ) / (2𝝅) = 950 million km / (2𝝅) = 150 million km = 93 million miles
It works!
Final Comment
All of this is done at night, and nothing in it requires a telescope, a satellite measurement, precise geometry, or any inherently challenging observation. In contrast to Aristarchus’ geometrical method for measuring the Earth-Sun distance, which requires measurement of a very small angle, and later parallax measurements which required excellent telescopes, this method is relatively easy. We can all see that meteors take seconds to cross the sky, not minutes or fractions of a second. We just need to pay more attention to the implications of that fact.
Again, if you want more details on why this works and how one would carry this measurement out for real, check out my new webpages on
13 Responses
Was this method done historically? If so, when?
It does depend on understanding that shooting stars are orbiting bodies burning up in the atmosphere, which I seem to recall took quite a long time to become consensus.
Indeed, it wasn’t done historically because, as you noted, meteors weren’t understood at the time when Cassini and Richter used precision telescopes to measure the distance to Mars (and thus to everywhere else) using parallax. Instead, historically, it was the other way round; knowing the distance to the Sun, one could confirm the properties of meteors and confirm that they are in orbits around the Sun. It is often the case in science that you can run the logic in multiple directions, and what makes sense at the time of discovery is opposite to what makes sense once one knows more.
The height of a meteor was first measured by Halley using parallax and the observations of two different observers. With some patience and a friend, one can certainly learn that essentially all meteors are 50-100 miles up, which confirms (using other measurements) that they are atmospheric.
However, confirming the assumption that they are orbiting objects hitting the Earth requires some thought and measurement, and one should make consistency checks on the assumption. Study of meteor showers is needed. Many of the ingredients necessary are discussed in my three webpages linked on this page: https://profmattstrassler.com/articles-and-posts/relativity-space-astronomy-and-cosmology/meteors/ . On the last of the three pages, Figure 4 shows a powerful consistency check that confirms the understanding of meteors is correct.
You done a great job, and got out throughout the solar system without instruments. You’ve said that instruments are needed to get out to our galaxy and beyond. I agree. But isn’t it true that to get the order of magnitude, as good as you got in Part 3 of this series, it can be done with affordable (if not exactly simple) methods. It would seem to me that a good equitorial mount with even a cheesy guide scope, a plain ordinary current generation “mirrorless” camera, and either a standard Schmidt-Cassegrain scope or a (more expensive) 800mm fluorite camera lens would do.
The idea is to use the Sun as a standard candle, and some (not obviously true, but true) assumptions. The method would give some assurance that the Sun is not exceptional. That’s
verifiable with a good telescope and a bit of construction.
What one would do is take photos of clusters in our galaxy, both globular and open, and measure the brightness of the stars in the three channels of the sensor. This would create a color-luminosity diagram, and verify that in fact it is, more or less, a temperature luminosity diagram. One would notice the main sequence, and place the Sun on it, assuming that its a typical main sequence star. Then cluster distances would follow.
One could also build a simple spectrograph and get, for the brightest stars, spectra, which is the verification that the main sequence is “typical” and the sun is on it. This is easily possible … I actually did it when in high school in the early 1960s, using a 1″ grating, long focus achromatic lenses, and a home-made tubing system … and film. The exposure time was 5 minutes for Sirius. Today with silicon sensors it would be much less (even with Bayer filters).
Suitable gratings are available for good to amazingly cheap (for what they are) prices on that auction site. But spectra are really not necessary.
One notes that the “trouble ratio” of this to your methods is orders of magnitude less than the size ratio of our galaxy to our solar system.
Is there some systematic reason my suggestion would not work?
I definitely think it’s worth pursuing these ideas, and seeing how simple you can make the process and the inferences.
I myself have some order-of-magnitude ideas to get us out of the solar system. But they require assumptions similar to the ones you will need, and are not so easy to verify. I hadn’t been thinking about them very hard, since knowing more about the Sun’s distance and size is so much more important. Now that I’ve solved that problem, I’ll turn my thoughts to it when time permits.
There are, of course, lots of closer-to-home problems. Is there a way to measure G, Newton’s gravitational constant? It’s easy to measure c, the cosmic speed limit, if you assume that radio waves are a form of light, but how do you prove that they are related to visible light? Or measure c directly in visible light? And so on. As the above blog post shows, some problems which seem very hard may actually not be… if we can just be clever enough.
Noted re the above, Matt. When it comes to measuring c, we use the motion of light to define the second and the metre, which we then use… to measure the motion of light! See https://arxiv.org/abs/0705.4507 where Magueijo and Moffat described it as a tautology. I think it would be great if a guy like you could get into this kind of thing. IMHO it leads to all sorts of interesting stuff. Really interesting stuff.
Its easy to show that light and radio waves are the same thing!
All you need do is show that light (wavelength 500 nm) and radio waves (wavelength 50m), a factor of 100,000,000, are connected by a continuum of intermediates showing the same properties. At radio frequencies from 6MHz to 60 GHz one can do lots of tests … easiest is making a two-element Yagi antenna and varying the element spacing by a factor of 20 or so,
noting that the amount of signal from a station has a periodic variation proportion to the inverse frequency, thus measuring c. From 6 GHz to 60GHz one can do it with Lecher wires (google that). From 6 GHz (5 cm) to 400nm one can use a single method using planar
interferometers made from very thin Mylar coated with aluminum stretched over ultra-flat frames. Mylar and aluminum transmit and reflect well enough over the whole range – you need several ones with different diameters. All these methods overlap in wavelength quite a bit, and you’d need only three methods, all overlapping, to “detect” the waves: your eyes, a thermocouple bolometer, and radios (though one single bolometer would in fact work over the whole range!). You could even use a tiny ordinary thermometer painted black with a few concave mirrors.
For the whole range you could measure frequency c would remain constant. The key is the use of a bolometer over the whole range. I’ve actually done it with bolometers, down to 3 Ghz.
Very nice. I used to shoot stars myself to measure distances 🙂
That’s a remarkably accurate method requiring very little knowledge.This really *should* be more well known; even a rough Fermi estimation could be worked out using a cheap camera and the logic laid out here. Of all the work you’ve put up on this blog perhaps this is the most impressive to me, not on a technical level, but in how simple it is to get such a groundbreaking answer.
Thank you.
Very good Matt! When I first read you claim I thought “oh come on, no way”. However after reading the article, I stand corrected. Very well explained!
Glad you like it. I think you and other experienced readers will appreciate some of the plots that I laid out in this article which gives more details: https://profmattstrassler.com/articles-and-posts/relativity-space-astronomy-and-cosmology/meteors/how-meteor-speeds-measure-the-earths-speed/ . See in particular Figure 4.