https://profmattstrassler.com/2024/01/19/how-far-could-the-sun-possibly-be/ Skip to content [ ] Menu * Home * About + About Me + About This Site and How to Use It + New? Start Here * Reference Articles + The Higgs Particle + Large Hadron Collider FAQ + Large Hadron Collider News + Particle Physics Basics + Relativity, Space, Astronomy and Cosmology + Some Speculative Theoretical Ideas for the LHC + Thoughts on the Scientific Process + Technical Zone * Waves in an Impossible Sea + Waves in an Impossible Sea: Home Page + Waves in an Impossible Sea: Commentary and Discussion * Exotica * Video Of Particular Significance Conversations About Science with Theoretical Physicist Matt Strassler How Far Could the Sun Possibly Be? January 19, 2024 by Matt Strassler (This is the third post in a series, though it can be read independently; here are post #1 and post #2.) 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. [image-63]Figure 1: A simple application of Aristarchus' method tells us that the minimum distance to the Sun is a few million miles (km), ruling out the red region. But the entire green region is still allowed. 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 c. 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! [image-64]Figure 2: By the end of this post, we'll know a maximum possible distance for the Sun, too. 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 2R[ES] . Its average speed relative to the Sun, v[E] , is its orbital distance divided by its orbital time, and that has to be less than c, so: * v[E] = 2R[ES] / (1 year) < c = 9.5 trillion km / year from which we learn a maximum possible distance to the Sun: * R[ES] < (1 year) c / 2 = 9.5 trillion km / 2 = 1.5 trillion km = 0.94 trillion miles Not great, but at least we know the Sun can't be a light-year away! [image-68]Figure 3: Requiring Earth's speed relative to the Sun be below the cosmic speed limit gives us a maximum (and thus a finite allowed range) for the Earth-Sun distance. 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 c, Newton's gravitational constant G , and the sun's density [?][S] . The formula for this turns out to be * R[S] < c [(8/3) 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, (4/3) R[S]^3 , multiplied by the Sun's density [?][S]. This gives us * (v[escape])^2 = (8/3) G [?][S] R[S]^2 Finally, we require v[escape] < c for sunlight to escape the Sun, and solve for R[S] to get the above result. Plugging in numbers we find * R[S] < 340 million km = 210 million miles Meanwhile, the Sun's angular size in the sky tells us that R[ES] is about 215 times larger than R[S] (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[ES] = 215 R[S] < 73 billion km = 45 billion miles. Now we're making real progress! [image-73]Figure 4: Requiring the Sun have an escape velocity below c limits its size, and thus its distance from Earth. 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. [image-66]Figure 5: A cloud of particles known as a CME blasts off from the edge of the Sun following a solar flare. (The Sun itself is blocked by the black disk, though an image has been placed within the disk to show the Sun's location and size.) The green box shows the CME's general location at a time 2:40; by 3:25, the CME's front edge has traveled a distance equal to four times the Sun's diameter away from the green box. 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[S] < 810 million km / 8 = 100 million km = 60 million miles which implies, roughly, that * R[ES] = 215 R[S] < 20 billion km = 12 billion miles. That's a lot better than when we started! Our range of possible distances is now below 10,000. [image-74]Figure 6: The fastest CME's must travel slower than c, and so this further reduces the maximum possible distance to the Sun. 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 c, 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. But these are minor details. As we'll see in the next post, we can do much better. Share this: * Twitter * Facebook * LinkedIn * Reddit * Like this: Like Loading... Related Categories Astronomy, general relativity, History of Science Tags astronomy, earth, Newton, SolarFlare, SolarSystem, sun Post navigation Why is it So Hard to Measure the Distance to the Sun? 8 thoughts on "How Far Could the Sun Possibly Be?" 1. [b483d3] Kudzu January 22, 2024 at 2:31 AM 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