*(This is the second post in a series; here’s post #1.)*

It’s not too hard to measure the distance to the Moon; the Greeks did it over two thousand years ago. First you measure the size of the Moon, which can be done in various ways; for instance, you can use the occultation (i.e. blocking) by the Moon of a star or planet, or the outer edges of a solar eclipse, as viewed in different locations on the Earth. These measurements do require multiple people at different locations to accurately report what they have witnessed, but they don’t require any fancy equipment or highly precise observations. Then you can quickly determine the distance to the Moon using the angular diameter of the Moon on the sky.

However, measuring the distance to the Sun, or to any other planet, is much more difficult. Today I’ll explain why… and this will help us envision a way around the problem.

## Aristarchus’ Method and its Limitations

If the Sun were closer, perhaps just twice as far as the Moon, there would be an easy way to measure its distance from Earth, using geometry, as pointed out and attempted by the Greek thinker Aristarchus about 2300 years ago. When the Moon is approximately half full (known as “First Quarter” or “Last Quarter” Moon, since this occurs at 1/4 and 3/4 of the cycle between New Moons), it can be easily seen in daylight, about 90 degrees away from the Sun. At precisely the moment when the Moon and Sun appear 90 degrees apart in the sky, so that the Earth, Moon and Sun form a right-angle triangle, the fraction of the Moon that is lit tells you another angle of the triangle. Then trigonometry and our knowledge of the distance to the Moon tells us the distance to the Sun.

Following Aristarchus, it’s not hard to check yourself (using the principles of Figures 7 and 8 of this post) that the Sun must be at least ten times further from the Moon, and must therefore be larger than the Earth. Unfortunately, for a large and very distant Sun, the fraction of the Moon that is lit at 90 degree separation becomes imperceptibly close to one-half, making an extraction of the Sun’s distant all but impossible.

In our own solar system, this is our situation; the Sun is so far away that the lit fraction of the Moon differs from one-half by less than one percent. Consequently this method isn’t very good for estimating the Sun’s distance, because it requires an extremely precise, difficult, and potentially ambiguous measurement. *(For this reason, I’m skeptical about pre-Renaissance astronomers’ measurements. Astronomers after Aristarchus claimed the Sun is about 50 million miles away, which is only off by a factor of two, but I suspect that they at best only knew that it was at least 50 million miles away. I doubt they could have measured the difference between 50 million, 500 million or 500 billion miles. If someone knows of evidence that contradicts this view, please let me know.) *

Once this simple geometric method fails for the Sun, so do all the other possible simple measurements of other objects such as planets and their moons.

Later scientists invented more elaborate ways to measure the distance to the Sun. But these require powerful telescopes and precise observations, as were used in the 17th century to actually make the first accurate measurement. Here we are interested in measurements that can be made by the average person. Why are such measurements so difficult to find?

The difficulty is that the easiest things to measure — apparent locations and sizes of objects in the sky, and the time it takes for objects to orbit the Sun and one another — are unchanged if one rescales, by a fixed factor, all distances and sizes (except the Earth and Moon’s sizes and the distance between them, which are known and mustn’t be rescaled). This is called “rescaling the solar system.”

## Rescaling the Solar System

Again, to be very clear, **rescaling the solar system** in this context means:

- We multiply all the distances between planets and the Sun, including that between the Earth and the Sun, by a fixed factor
*X* - We multiply all the distances between planets and their moons, excepting the distance between the Earth and our Moon, by the same factor
*X* - We multiply all the sizes of the Sun, its planets and their moons, excepting the Earth and our Moon, by the same factor
*X*

The claim is that nothing that is easy to determine about the rescaled solar system — nothing easily measured by eye or in an small amateur telescope, or that can be inferred from such measurements — changes at all.

That’s not exactly true. We can’t choose any ** X** we want; if

**is too small, then the rescaling will bring the Sun relatively close to Earth, and Aristarchus’ method will work. But we could make**

*X***as large as we want, pushing the Sun further and further away, without changing what’s easy to measure.**

*X*The two types of properties of the solar system that are easy to measure are

- Time periods — for example,
- The time between a planet’s opposition (i.e. when it is overhead at midnight) and its previous opposition
- The time it takes for a moon to orbit its planet
- Etc.

- Ratios of lengths — for example,
- The distance of a moon from its planet
**divided by**the diameter of its planet - The distance of a planet from the Sun
**divided**by the distance of the Earth from the Sun - Etc.

- The distance of a moon from its planet

## Why Appearances Don’t Change When We Rescale the Solar System

The crucial point is that what we see on the sky by eye or in a telescope is an **angle**, not a length. We can directly see how large the Sun appears to be on the sky. Its angular diameter * θ_{S} *is about half a degree (out of the 360 degrees in a circle), which is about .009 in radians.

For small angles, it is approximately true that

=**θ**_{S}**D**/_{S}**R**_{ES}

where * D_{S}* the Sun’s actual diameter and

*is its distance from Earth. From this we learn that the Sun’s diameter is 0.009 (just under one percent) of its distance from Earth. But we learn neither its size nor its distance.*

**R**_{ES}Now suppose everything in the solar system (except the sizes of the Earth and Moon and the distance between them) were doubled — i.e., we take ** X = 2.** Then the size of the Sun would double, but so would its distance, so it would appear the same to our eyes. The same would be true for all planets, moons, and distances between them. From eyes and telescopes on Earth, nothing would look different. The same would be true for any

**that’s not too small.**

*X*## Why Timings Don’t Change When We Rescale the Solar System

In addition to the angular sizes of things, we also measure timing. For instance, we can measure the time that it takes each of Jupiter’s moons to orbit Jupiter, and the time it takes Jupiter to orbit the Sun. Why don’t these timings change when we double the size of the solar system?

Kepler’s third law of planetary motion (which we were able to derive ourselves) states that the square of the time ** T** that it takes for a planet to orbit the Sun (its “orbital period”) is proportional to the cube of the distance

**between the planet and the Sun:**

*R**T*∝ R^{2}^{3}

where “∝” means “is proportional to.” (Here I am assuming the planet is in a near-circular orbit and not worrying about fine points.)

At first glance, it would seem that rescaling the solar system, which changes all the distances * R* , would change all the planets’ orbital periods

**, too. But not so! When we rescale the sizes of objects,**

*T***we also make their masses larger**, and this exactly compensates for the increase in distances.

Kepler’s law took its final form following discovery of Newton’s law of gravity. What it actually say, for a planet orbiting the Sun in a near-circular orbit, is

*T*= (2^{2}**𝝅***)*/ (R^{2}^{3}*G M*)_{S}

where ** M_{S} **is the Sun’s mass (and

**is Newton’s constant, of no importance here.) When we rescale the solar system, we can imagine that all objects remain made of the same material, so that their average densities remain constant.**

*G**[For experts: If we don’t imagine this, then anything is allowed to happen, and we know even less.]*Then the mass of the Sun is equal to its average density

**times its volume**

*⍴*_{S}**:**

*(***𝝅***/6) D*_{S}^{3}*M*_{S}= (**𝝅***/6) ⍴*_{S}D_{S}^{3}

Combining this with Kepler’s law, we learn

*T*= (24^{2}**𝝅***/G*) ( R/*⍴*_{S})*D*_{S}^{3}

and thus **the orbital time of the planet is actually proportional to the cube of a ratio of lengths** — the ratio of the planet’s distance from the Sun to the Sun’s diameter.

And so, since rescaling the solar system leaves this ratio of lengths unchanged, it leaves the planet’s orbital period the same as before. This is true for all planets, and also for their moons; in the latter case, the squares of the moons’ orbital periods are again proportional to the cube of a **ratio** between the moon-planet distance and the planet’s diameter.

It’s no use; when we rescale the distances and sizes of objects in the solar system, neither angles nor overall timings change.

## Temperature?

You might think, “Wait, if we move the Sun much closer or further away, that should make it much hotter or colder here on Earth. Shouldn’t we be able to tell how far away the Sun is by how warm it is here on Earth?”

It’s a good idea. But it doesn’t work either.

The temperature of the Sun at its visible edge, where it becomes transparent, can be inferred from sunlight; the energy it emits in particular frequencies of light reveals that it is approximately 5500 degrees Centigrade (roughly 10000 degrees Fahrenheit). So its temperature is known, and has to be held constant while rescaling the solar system.

But if you double Earth-Sun distance ** R_{ES}** and the Sun’s diameter

**without changing the Sun’s temperature or the size of the Earth, then**

*D*_{S}- because the Sun is further away, the fraction of the Sun’s energy absorbed by the Earth does indeed go down — by a factor of 2
^{2}=4, - but a larger Sun has a larger surface area by a factor of 4, and therefore it emits 4 times as much energy at the same temperature.

The two effects cancel; the Earth absorbs the same amount of energy as before, and the Earth’s temperature remains unchanged.

## What Options Remain?

If we are to figure out the overall scale of the solar system, and with it the distance to the Sun, the key is to find something that we can measure that actually **does** change when we rescale the solar system. What options do we have?

Parallax measurements do work, because they involve a ratio of the distance to a faraway object (which rescales) to a distance on the Earth (which does not.) In other words, rescaling the solar system but not the Earth would leave the baseline of a parallax measurement unchanged while doubling the true distance to the distant object. Unfortunately, the distances of all other large objects in the solar system are so large that parallax angles are very small, and thus parallax measurements are very challenging.

What about velocities? The speed of a planet involves dividing the circumference of its orbit by its time period, ** 2𝝅R/ T** . That ratio

**does**change under rescaling, because

**, the distance of the planet from the Sun, rescales, while**

*R***does not. The same is true of the speeds of many other objects moving around the solar system. If we can measure the speed of something in the solar system that does not belong to the Earth-Moon system, then we will potentially be able to measure the size of the solar system.**

*T*Doppler effects, in which frequencies of light emitted by an object shift upward if the object moves toward you, and downward if it moves away, can be used to determine relative speeds. (This is the principle behind a radar gun, which uses radio waves and measures shifts in frequency caused when the waves reflect off a vehicle.) But these effects are not visible to the naked eye or simple telescope, and are very hard for most of us to measure.

Nevertheless, velocities will resolve our conundrum. We’ll see in my next post how to get a weak but definite upper bound to the Sun’s distance, complementing our lower bound of a few million miles from Aristachus’ method. Then we’ll make our way toward more powerful approaches, and put the Sun in its place.

## 11 Responses

How about this: a substantial asteroid comes whizzing between the earth and the moon. Its close enough

to measure its path and velocity by the same methods used for Earth and Moon. It keeps on going around the sun.

One can easily measure the orbital period and size in the “solar system as a whole” scale. This takes a respectable but ordinary telescope. That and the velocity at the known (near earth) point gives the scale.

Such near passes are seen fairly frequently these days, even ones discovered by amateurs.

Its easiest to vizualize if the asteroid had a circular orbit the same size as the Earth’s, but inclines 90 degrees to the earth’s orbit. And it would reappear the next year (modulo perturbations of course).

This is indeed the right kind of idea. We’ll get to it after one more post on another method.

This was done professionally by optical astronomers in the 1950s to determine the AU (or “solar parallax”), by using the asteroid Icarus. which was chosen because it regularly passes close to the Earth and thus has a large parallax. This effort was overtaken by events in the 1960s when Irwin Shapiro and others started measuring planetary distances with radar. Icarus passes close to the Sun, and so has a large general relativistic precession of perihelion, and has been observed since then by radar as a separate test of general relativity.

Every orbit of the moon gives a first-quarter right angle and a third-quarter right angle. Measuring the lunar orbit times between the first-to-third-quarters and the third-to-first-quarters and vice versa gives another way to get the non-right angle at the moon. This seems entirely do-able by Huygens (I am not an archaeohorologist; I’m just skimming https://en.wikipedia.org/wiki/History_of_timekeeping_devices#Pendulum_clocks ). With daily sidereal corrections (so that only error accumulated on the day of the measurements affects the measurement), and knowledge of the 15-ish degrees of revolution of the Earth in its orbit per measurement, it seems that this time should be measurable around 1650 to +/- 20 s per 1.18 Ms, or about 20 ppm. (Warning: computation is off-the-cuff and I haven’t checked it, but that’s (I think) 100x precision compared to the angle being measured.) Of course, the moon’s orbit is not circular, so several orbits should be averaged together. Viewing is not guaranteed for every first- and third- quarter moon. Your mileage may vary, et c.

*Maybe* this could have been done using earlier clocks by a bit more heroic experimentalists. One could degrade the above precision and still get in the right OOM of the answer.

It would be helpful if you went through the logic a little more slowly here; your terminology is in some places ambiguous and your method therefore not entirely clear. Are you defining “first quarter right angle” through the fraction of the Moon that is sunlit, the Moon’s angle relative to the Sun, or literally 1/4 of its orbit time?

Your estimate of systematic errors seems rather aggressive, and you do have to be very careful not to degrade the precision very far, because the distance goes as the inverse of the angle. A lower bound on the angle translates into an upper bound on the distance. So if you can’t sharply distinguish the angle from 0 degrees, you get no upper bound. I’d like you to clarify what you are actually measuring and how that translates into the relevant angle.

In any case, this currently sounds more difficult than a parallax measurement of Mars, whereas I’m looking for something simpler.

Place coordinates so that the Earth and Sun are (approximately) stationary. In these coordinates, the Moon orbits the Earth. Every orbit of the moon produces both a first-quarter moon and a third-quarter moon. So twice per lunar orbit there is an opportunity to measure the angle in the Sun-Moon-Earth system when there is a right angle at the Moon. (This is not the right triangle you described in your post, and I was *NOT* clear about this. Apologies.) Pick any two right triangles (one at one-quarter moon and one at three-quarter moon) that are successively produced as the Moon orbits. The diagram we have in mind has the Earth and the Sun at the ends of a line segment which is the hypotenuse of each of two right triangles, a circle around the Earth — the orbit of the Moon, and the two right triangles having right angles on that orbit. (This choice of coordinate systems hides a 15 degree rotation of the later triangle, since it is actually formed a half month after the earlier right triangle. Describing the geometry with the 15 degree rotation included is wordier, but do-able. These 15 degrees make the inner circular segment time shorter and the outer circular segment time longer (both times described below), by an entirely predictable amount, so these consistent delta-times are removable.)

Note that one does not have to measure exactly when the illuminated portion of the Moon is exactly 50%. One can make several measurements over a few days to interpolate the time at which this happens. (Imagine a slightly fuzzy discretely sampled sinusoid, with segments comprising half the data points missing — because the moon is not visible half the time. The maximum amplitude corresponds to full moon and the minimum amplitude corresponds to new moon. A handful of data points around the zero-crossing are used to estimate the time of one- (or three-, depending whether the function is increasing or decreasing at the zero-crossing) quarter moons, i.e. times when the view of the Moon from the Earth is at right angles to the view of the Moon from the Sun. I would be unsurprised to learn that this data actually exists and was circulated among Arabic astronomers, and early Renaissance astronomers had very precise estimates of the times between half-moons and some idea what the periodicity of the deviations in these times is. But again, this isn’t my area of study, so I don’t have ready references.)

There are two segments of the orbit of the moon in this diagram, the inner circular segment between the Moon-Sun triangle sides that meets the Earth-Sun line segment, and the outer circular segment between the Moon-Sun triangle sides that does not meet the Earth-Sun segment (because that circular segment is further from the Sun that the Earth is). The inner circular segment is subtended by an angle that is pi radians reduced by the Earth-Sun-Moon angle of both right triangles. The outer circular segment is pi radians increased by those two angles.

Since these angles are hard to measure, instead measure the time it takes the Moon to orbit along one circular segment versus the other circular segment. That is, the data we wish to collect is the inter-half-moon times. These are about 1 Ms durations. Assuming the marked time of half-moons is accurate to +/- 10 s relative to a sidereal reference on the day they occur (the stated accuracy of Huygen’s clocks and method of clock discipline he used). This gives precision around 20 ppm (20 microradians) if a Huygens-class experimenter uses a single clock. (More clocks, especially clocks that have characterized relative dispersions, say by measuring their deviations at every sidereal reference, can give much better time precision and accuracy.)

The angle one wishes to measure is the Earth-Sun-Moon angle which is approximately distance(Earth to Moon)/distance(Sun to Moon) ~= 1/400, so a couple of milliradians.

The Earth and Moon orbits are *not* circles, so one should observe periodic deviations in the imputed angle of a few percent, but one gets a new measurement in two weeks, so this can be averaged down by repeated measurement. A couple of milliradians >> 20*10^(-6). In fact, a couple of milliradians * ( a few percent) >> 20*10^(-6), so the periodic deviations should also be measurable by a Huygens-class astronomer with Huygens’s equipment.

Thanks, I’ll go through this. I don’t know if I’ll have a clear sense as to whether you’ve estimated the uncertainties fairly or forgotten something, but maybe another reader will chime in.

Huygens, maybe yes. Not sure about before.

If astronomers before the telescope had been capable of this, then some of them should have gotten the right answer. Since they were off by a factor of 2 on the distance, that means they got the angle twice as large as it should be, i.e. a 50% error on their central value. With that large a systematic uncertainty, they could not have firmly known that the angle was non-zero. Thus their answer was, at best, a lower bound on the distance, and the systematic uncertainty implies that their lower bound should be reduced by another factor of 2, making them off from the real answer by a factor of 4.

Again, this is not my field, so I cannot vet this source, but https://www.sebfalk.com/post/how-to-read-a-medieval-astronomical-calendar writes “Let that sink in for a moment. John Somer has given us every new moon for 76 years, calculated to the nearest minute.” in a calendar produced in 1380. This suggests that there was enough data available at that time to estimate times of quarter-moons to about a minute. Shortly after that quote is a description of timings of eclipses “to the nearest second”.

It’s not clear to me how this precision is real unless *much* data went into computing the periods of various component motions. Astrolabes were used for timekeeping by the 10th century and the precision of “hand sized” astrolabes is about 0.25 degree. https://www.tandfonline.com/doi/full/10.1080/00253359.2013.766999

I also know their dataset was large enough to produce a table of the daily precession of the stars and planetary apogees with precision of 1 part in 10^16 (although I have no idea what the accuracy is). http://astrolabesandstuff.blogspot.com/2015/01/precision-accuracy-medieval-astronomy.html

Thanks for this. (It would be important to check that the accuracy is as good as the precision, but let’s assume so.) As you said, “This suggests that there was enough data available at that time to estimate times of quarter-moons to about a minute.” Maybe. But does that allow them to actually measure the effect of the Sun not being infinitely far away, which is a 1/400 effect? We don’t know their systematic errors.

Here’s what I continue to struggle with: if what you suggest were really true, and the measurement should have been within their grasp, why were they off by a factor of 2? I.e., if you could reliably measure a 1/400 effect (meaning you have a precision of about 1/1000 or so, or at worst, 1/800), why would your answer come out to be 1/200, somewhere between 2 and 3 standard deviations too large?

I continue to suspect that the right interpretation is that they really only placed an upper bound of 1/100, convinced themselves they’d seen 1/200, and had no true lower bound.