*(This is the third post in a series, though it can be read independently; here are post #1 and post #2; and post #4 blows this one out of the water, so don’t miss it!)*

Measuring the distance to the Sun is challenging, for reasons explained in my last post. Long ago, the Greek thinker Aristarchus proposed a geometric method, which involves estimating the Moon’s sunlit fraction on a certain date. Unfortunately, because the Sun is so far away, his approach isn’t powerful enough; Aristarchus himself underestimated the distance. *[This last remained true for later astronomers before the 17th century, though they got closer to the truth, presumably by using more precise methods than you or I could easily apply. I doubt anyone truly found a maximum possible distance to the Sun just using geometry.]* The best we can do, using Aristarchus’ method and our naked eyes, is determine a

**minimum**possible distance to the Sun: a few million miles.

Today we’ll see how to obtain a **maximum** distance to the Sun, using an approach suggested in the previous post: by measuring * speeds*. Specifically, we’ll make use of a speed that the ancient astronomers weren’t aware of:

**the speed of light**, also known as

**the cosmic speed limit**. That’s 186,000 miles (300,000 km) per second, or 5.9 trillion miles (9.5 trillion km) per year. We’ll find the Sun’s distance is less than 12 billion miles… still much larger than its true distance, but a significant improvement on our starting point!

*c*Here’s what we’ll consider:

- The Earth’s speed around the Sun should be less than
**c** - The Sun should not be a black hole (i.e. light should be able to escape from its visible edge)
- Clouds of particles blasted from the Sun should not be able to travel faster than
**c**

## What We Can’t Learn From Light

Before we make progress, let’s quickly dispense with an idea that is tempting but won’t work.

If we could just measure the time that it takes for light to travel from the Sun to Earth, that would directly tell us the distance. An obvious idea is to try to use solar flares, giant explosions that occur on the Sun and release powerful blasts of X-rays (an invisible form of light.) If we could just compare the time when the X-rays arrive at Earth to the time they left the Sun, we could multiply that time by the speed of light and get the distance to the Sun. Super easy!

The only problem is that we don’t know when they left the Sun. We see the X-rays when they arrive at Earth. We don’t know when they started their journey. And so, we don’t have enough information, and the idea fails.

More generally, in order to use light directly to measure a distance, we have to know both the start time and the end time. This is what is used by professionals when they bounce a powerful pulse of radio waves (another invisible form of light) off a distant planet and listen carefully with enormous antennas to the response: the time to go out and back, divided by two and multiplied by the speed of light, provides the distance. But you and I can’t do that ourselves. And there’s no natural process where we know both the departure time and the arrival time.

## Putting the Speed Limit to Use

But even without light, **c**** sets a limit on the relative speeds between any two nearby objects **— that’s the sense in which it is a cosmic speed limit. That means the Earth can’t move faster than * c* relative to the Sun.

We know that the Earth goes round the Sun once a year on a nearly-circular orbit whose radius is the Earth-Sun distance ** R_{ES}** , and whose circumference is

**. Its average speed**

*2***𝝅***R*_{ES}**relative to the Sun**,

**, is its orbital distance divided by its orbital time, and that has to be less than**

*v*_{E}**, so:**

*c*=*v*_{E}/ (1 year) <*2***𝝅***R*_{ES}= 9.5 trillion km / year*c*

from which we learn a maximum possible distance to the Sun:

< (1 year)*R*_{ES}/*c*= 9.5 trillion km /*2***𝝅**= 1.5 trillion km = 0.94 trillion miles*2***𝝅**

Not great, but at least we know the Sun can’t be a light-year away!

## Black Hole Sun?

But we can do much better than that. Rescaling the solar system to make it larger and larger, putting the Sun far away while keeping the Earth’s orbital period unchanged, requires making the Sun’s mass enormous. The pull of its gravity at its surface becomes greater, and if it is strong enough, even sunlight won’t be able to escape, and the Sun will form a black hole. *(We might not want to assume Einstein’s view of gravity is correct, since we haven’t checked it ourselves. Still, we can be sure something rather drastic will happen to sunlight once it can’t escape in the usual manner.)*

The **escape velocity** of an object is the minimum speed required to escape its gravitational pull from a particular location outside it. But if we don’t know either the Sun’s mass or its radius, it is impossible to calculate the escape velocity from its visible surface. Fortunately, the escape velocity can also be computed from the Sun’s radius **and its density** — and we do know the density of the Sun from ocean tide patterns, as I explained last week. It’s about 40% of the Moon’s density, and thus 25% of Earth’s.

Requiring the escape velocity be less than the cosmic speed limit gives us a maximum radius *R _{S}* for the Sun in terms of

*, Newton’s gravitational constant*

**c****, and the sun’s density**

*G***. The formula for this turns out to be**

*⍴*_{S}*R*_{S}[*< c*]*(***8𝝅***/3) G ⍴*_{S}^{-1/2}

*If you’re curious, click here to see how this can be derived using simple math and physics.*

From Newton’s law of gravity, one can show that just outside the Sun’s visible surface, the escape velocity, in terms of the Sun’s mass and radius, is

*(v*_{escape})^{2}= 2GM_{S}/R_{S}

(which one could guess, except for the 2, just using the physicist’s trick of dimensional analysis.) We can write the Sun’s mass in terms of the Sun’s volume, **(4𝝅/3) R_{S}^{3}** , multiplied by the Sun’s density

**. This gives us**

*⍴*_{S}**(v**_{escape})^{2}*=*R*(***8𝝅***/3) G ⍴*_{S}_{S}^{2}

Finally, we require ** v_{escape}** <

*for sunlight to escape the Sun, and solve for*

**c****to get the above result.**

*R*_{S}Plugging in numbers we find

*R*340 million km = 210 million miles_{S}<

Meanwhile, the Sun’s angular size in the sky tells us that ** R_{ES}** is about 215 times larger than

**(and for the same reason, the same ratio relates the Moon’s distance and its radius... or about a ratio of 100 between its distance and its diameter.) So we learn that the very fact that the Sun looks like an ordinary hot glowing object requires that**

*R*_{S}= 215**R**_{ES}< 73 billion km = 45 billion miles.*R*_{S}

Now we’re making real progress!

## Lesson From a Solar Flare

Earlier on I pointed out that we can’t just use timing of a solar flare’s X-rays to measure our distance from the Sun. But we can use the “coronal mass emission” (CME), the eruption of a great swarm of subatomic particles, that often accompanies the solar flare. The cloud glows, so we can see it on satellites as it travels away from the Sun.

Particularly powerful flares often generate the fastest CMEs. Here’s one blasting sideways off the limb of the Sun, shown in three stills taken from this NASA video *(see time 1:35-1:40 for the CME in question.)* The blue and white image is the STEREO-B satellite’s data; the black central region is physically shielded, blocked so that full sunlight doesn’t blind the satellite’s camera; and the central red sphere indicates the size and location of the Sun behind the shield.

We can see from the images that the CME travels 4 times the diameter of the Sun, or 8 times its radius, in 45 minutes. Since light travels 810 million km (500 million miles) in 45 minutes, the fact that the CME’s speed can’t exceed * c* tells us that the Sun’s radius can’t be more than 1/8th of that 810 million miles. Specifically,

*R*810 million km / 8 = 100 million km = 60 million miles_{S}<

which implies, roughly, that

= 215**R**_{ES}< 20 billion km = 12 billion miles.*R*_{S}

That’s a lot better than when we started! Our range of possible distances is now below 10,000.

Incidentally, each of the three limits in Fig. 6 on the maximum distance to the Sun is probably an overestimate by a factor of 2 or 3. We’ve required that the various speeds can’t be greater than * c*, but actually they have to be somewhat smaller than that, because if any of them were near

**, unusual phenomena would be observable by telescope or the naked eye. We should therefore restrict the speeds in question to be perhaps 1/3 of**

*c***, reducing the maximum distance to the Sun by a corresponding factor in each case.**

*c*But these are minor details. As we’ll see in the next post, we can do much better.

## 15 Responses

From high-school mechanics v_E^2/r_E = a_E = centripetal acceleration of the Earth around the sun.

Hence: r_E = v_E^2/a_E. Since I can’t feel this acceleration, it must be tiny as confirmed by plugging in some values to calculate a_E = (30Km/s)^2/(150 billion meters) = 0.006 m/s^2. But after thinking about this more carefully, the correct explanation is that we’re in free fall around the sun so that this acceleration can’t be measured here on Earth even if we were clever enough. Ah well.

On the other hand: as the velocity of the Earth around the sun approaches c, so will the internal stresses needed to rotate the Lorentz contracted Earth, as seen from the frame of the Sun, as its velocity changes direction; which should set a lower bound for for the maximum orbital velocity of the Earth. But this is then getting too far away from the spirit of Matt’s post IMO, which is to present an easy way of measuring the distance to the Sun via check-it-yourself science that the majority of us can do.

Your last paragraph can’t be right either; from the point of view of the Earth, there are no such stresses. Indeed, the whole point of Einstein’s relativity is that length contraction **does not create** any stresses; it is not a real compression of objects but a reorganizing of space and time. [This is in contrast to what Lorentz and those around him would have expected.] To work through how a Lorentz-contracted spinning object is interpreted by a passing observer is very tricky and I can’t do it off the top of my head. But the fact that there is no problem from the Earth’s point of view is enough to prove that there is no problem from anyone’s point of view, and thus, no limit on the Earth’s speed other than the speed of light.

I think I should have said _additional internal forces_ to keep the Earth together as its orbital speed and hence acceleration increases around the Sun.

If I accelerate two physical points with identical forces in flat space in my lab frame, they’ll move apart from one another as measured from one another’s stationary frame. For interested readers, this is related to the ‘Bell’s spaceship paradox’ which can be found on Wikipedia. To keep them together and ‘rigid’, an additional force has to be given to either which I physically interpret as causing them to move towards one another in the lab frame and hence ‘Lorentz contract’. This is what I had in mind for the Earth orbiting the sun, with these additional internal forces changing as the direction of the velocity and acceleration of the Earth changes as it orbits the Sun, leading to a change in how the Earth is Lorentz contracted in the Sun’s frame.

But since the Earth is in free fall, I now can see problems with the above physical interpretation since the above forces disappear and likewise for the internal forces; if I keep things simple by assuming space-time remains ‘flattish’ over the dimensions of the Earth.

This is great, but I wanted to note that the green and red colors in your diagrams are quite difficult for colorblind people (like myself) to distinguish, especially when they are in thin lines.

Thanks for pointing this out. I’ll try to avoid this in future, and if time permits I’ll come back and fix this.

However, how could we “check it ourselves” the speed of lite and gravitational constant?

The first successful measurement of the speed of lite used irregularities in orbits of Jupiter moons and required knowledge of the Earth-Jupiter distance.

Excellent questions.

There are a number of ways to approach the speed of light with modern technology. The easiest indirect measurement is to use the time delay in past communications with astronauts on or near the Moon. The Moon recordings, which are widely available to the public, immediately show that the delay can’t be longer than a couple of seconds — otherwise the conversations wouldn’t have been possible — so that sets a lower bound on c. (Recall that measuring the distance to the Moon is something that one can do easily, so that’s not a bottleneck.) But the recordings are better than that, because often there is feedback. A statement by one party is heard in the speakers of the other party, recorded by the latter’s microphone and sent back to the original party; and the time between the original statement and the echo gives a good estimate of c. (I had a clip that demonstrates this clearly, but I’ll have to find it; it seems buried in my computer and I can’t think where.) Measurements of c on Earth aren’t that difficult now either, and there are plenty of videos and websites explaining how to do it, but so far I think there’s some effort and some experience required. In any case, the point is that computers now operate in the nanosecond realm, and light travels 1 meter every three nanoseconds, so it’s not really that challenging anymore. Experts on high-speed trading can tell you all the details; if the speed of light were much faster than is claimed, then the economics of high-speed trading locations would be quite different; see for example Donald MacKenzie’s book “Trading at the Speed of Light”. Nothing like real people making real decisions to make money to prove how the world really works. So one could use a variety of methods to infer the speed of light, and of course, they all give the same answer.

As for G, that is much, much harder to measure directly. However, everything in this post (and in all of the do-it-yourself astronomy posts that I’ve written) rely only on G times a mass, and never on either G or the mass M separately — even if the way I wrote things might have given an impression otherwise. That’s because in gravitational accelerations, the only thing that appears is G times M. Therefore one needs either to go beyond acceleration or beyond gravity — measure an actual force, or take advantage of a non-gravitational fact.

One can estimate G if one knows the mass of any one astronomical object around which things orbit, or if one can measure the force between two known objects. Cavendish did the latter, but it’s a very difficult measurement to repeat (despite the existence of videos in which an amateur claims to have done it, but which obviously have electrostatic issues that haven’t been addressed.) The fact that it is so difficult puts an upper bound on G; and there are probably other effects whose absence restricts G in interesting ways. I don’t think a Cavendish-like experiment will ever be do-it-yourself, though I’m happy to be proven wrong.

Another approach to G, which can get you the right order of magnitude, is to estimate a planet’s or moon’s density and thus its mass separately from G; this was Newton’s controversial approach. Satellites and gravity provide an estimate of the Earth’s relative density profile, and then direct measurements of the crust provide the actual density of its outer regions, so this constrains the Earth’s mass and thus G — but I’m not sure how tightly. You could also rely on some knowledge of how rocks behave under pressure (i.e. how dense can the Earth’s core possibly be?) which is probably hard to obtain oneself. So I do think this is difficult, and it is hard to do from direct observation.

Some things are inevitably possible but hard — confirming the world is made of quanta, for instance, is probably very hard. But I used to think measuring the distance to the Sun is hard, and now (as you’ll see this week) I know it isn’t. Sometimes one just needs the right insight.

I’d wager, given Doppler shift, that Earth’s speed couldn’t be 0.1c, or we’d see it in the sky. I’d even wager less than 0.01c if we made careful comparisons, side-by-side photos of the sky in either direction. Given the orbits of other planets, and pinning them as <c would also tighten the bounds a bit.

It saddens me that most of the points in this article I have seen directly refuted by people such as Electric Universe believers as conspiracy. It's tainted do-it-yourself science for me forever.

I’m sure 0.3 c would be visible by eye, but whether 0.1 c is visible by eye isn’t something I attempted to think through carefully. If you can come up with a foolproof method, that would be interesting and good to know. The question of 0.01 c using photography is interesting, but I’m not sure it is as easy as you suggest.

Mercury’s speed definitely tightens the bound, but by less than a factor of 2, so I decided not to add that in to the post, which is really more about orders of magnitude.

Everything about conspiracy theorists is a sad commentary on the human species. But the point of “do-it-yourself” science or “check-it-yourself” science, whatever you want to call it, is not to convince conspiracy theorists. Such people have decided that everything around them is conspiracy, and no amount of evidence could possibly change their minds. The fact that the engineers who actually send satellites to other planets, and who build computers, happen to use the conventional scientific wisdom … well, that’s irrelevant to such folks. You can’t fight willful illogic.

The targets of “check-it-yourself” science lie elsewhere. Children, who are still open-minded. Teachers, who need to be able to understand and explain *why* scientists believe the results of science. People who are subjected to seductive conspiracy theory arguments, and want to know what foundation there is for the mainstream scientific worldview. And people who already have some confidence in science but harbor doubts about what’s really known and what’s really not.

It seems to me that there are a lot of people in our society who want and/or need to understand that science isn’t simply received wisdom; it is something you can check and use, over and over and over again, anytime you want. And for them, it’s important to show how it can be done.

First, Tycho Brahe did naked eye positioning to roughly the arc minute level, or ~0.3 milli radians. His failure to detect parallax type effects means that he could have limited the speed of the Earth to < about 100 km / sec, if he had known the speed of light, which he didn't.

Note that with modern VLBI we can determine the aberrational annual motion of the Earth to order 10^-10 radians, or 3 cm/s and, yes, it all checks out. (I believe I was the first person to do that, with Demetrios Matsakis at the USNO. It was, in our time, more a check of the accuracy of the code than any profound physics.)

Just to be clear, which parallax effect exactly are you referring to in order to limit the speed of the Earth? What’s your target and what’s your baseline?

I said “parallax type.” I can see now that vagueness was likely to confuse.

The Earth’s orbital aberration causes an ~ 20 arc second, ~annual, ~ elliptical, motion in the position of distant objects that depends on the location of the object in the sky, but is independent of distance (as long as the object is indeed distant). Parallax causes an ~annual ~elliptical motion in the position of distant objects that depends on location, but is also inversely proportional to distance*. The biggest stellar parallax is that of Proxima Centauri, 0.768067 arc seconds. People have been looking for stellar parallaxes for a long time, and the existence of the much larger annual aberration mightily confused matters until James Bradley explained it in 1727 (and used it to estimate c).

* We can’t (yet) determine parallax well enough for cosmological effects to be detectable. Matthew McQuinn, University of Washington, wants to change that and just won a NIAC Phase 1 for that.

I’ve no method for checking .01c, and I’ve not said it’s easy. Only that, if someone asked me to bet money that it was impossible, I’d decline the wager. I’ve just seen a man light a match, with a match,shot out of a pistol in his garage. The ingenuity of many people, many better engineers than me, puts a lot within the range of the backyard hobbyist.

Conspiracy,I fear has co-opted a lot of backyard science.With a lot of ‘Look at this simple demonstration\picture’ rhetoric. Phone camera videos of the moon can apparently prove it’s a hologram. I don’t knot know what this will mean going forward, but I don’t like it.