How big is a rainbow?

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1:04I should say, I'm in a dark room. Hello and welcome to CrowdScience from the BBC World Service. I can't see my guest. I'm Marnie Chesterton and I'm in an almost pitch black studio with rainbow expert Philip Lavin. and, I think, an object in front of me. What we've got here is a cylinder full of water. It's a big vertical glass cylinder. And perhaps we could switch the spotlight on. As Philip looks for the switch in the dark, my eyes are drawn to a projection on the wall behind him. I'm now shining white light at our cylinder full of water and down here we can see... A rainbow! A rainbow!

2:00Yes, the world needs more rainbows, but why did we set up a whole experiment to make one? Well, any crowd science adventure always starts with a curious listener. My name is Sakura. I live in Cambridge, UK. Anne, what is your question for crowd science? How big is a rainbow? Can I ask how you came up with it? My husband goes for a walk every morning, and one morning when he came home, he said he'd seen a rainbow. So I said, was it a big rainbow? And he said, I don't think so. It was from that road and those houses. I thought that sounds like a big rainbow but when he showed me the photo it didn't look particularly big so I wondered are all the rainbows the same size or some rainbows bigger than others?

3:01Thank you Sakura. We have all seen a rainbow. At some point in our lives we've been in that spot with a bit of rain, a bit of sunshine, and somehow this magical colour show turns up too. And it doesn't have to be your first one to fill you with awe, especially when it dominates the landscape, stretching right across the horizon. But how large is that? I have this gut feeling that chasing rainbows to measure them might be difficult. So how do I measure a rainbow? and if I can measure one, what's the biggest one I can find? To get to that point, I first need to understand how a rainbow actually works and to do that, I'm back in the dark studio with our rainbow expert, Philip.

3:54A rainbow is the pattern that you get when sunshine, white light, is reflected and refracted from millions of raindrops in the sky. OK, so using your laser pointer, if you point a beam of green light through that cylinder of water and you can tell me which bits are reflection and which are refraction. You're going to look down onto the cylinder. Yep. Imagine a green wave of light entering the water. The light goes in, is refracted. It changes angle. The light travels more slowly in water and it has to bend to make it work. And then heads towards the back of the glass cylinder. Reflected from the opposite side.

4:47Where part of it bounces back like a pinball. And then back out the other side where it's refracted out. and hits the white wall behind Philip to create a bright green dot. But what happens if Philip moves the laser pointer further towards the side of the cylinder? It goes further that way and then comes back again. We reach this crucial bit called the rainbow angle where a huge amount of concentration of light occurs. OK, so it's geometry, and if you calculated all of the different possible angles, you get this point where light concentrates, and that's the rainbow angle. Yeah, exactly. Now, this rainbow angle is slightly different for different colours.

5:41So if you shine white light into the glass cylinder... White light from the spotlight. And remember, white light is made up of all the different colours of the spectrum. They all refract differently. Then each colour changes to a slightly different angle. The rainbow angle, we call it, it's 42 degrees for red light, it might be 40 degrees for blue light. The angles split the rainbow into its constituent colours. and ta-da on the wall there's not just a single dot of green but all the colors of the rainbow i can see the blue i can see the the the green the yellow the red there are all the colors are there the interesting thing is that it's a straight rainbow okay why is that not what we see in the sky with natural rainbows.

6:39Although it's a very nice rainbow, it's caused by a cylinder of water. Raindrops are not cylinders. They are close to spherical. Okay, so I'm going to replace the cylinder of water by an electric light bulb, which I have taken the insides out of, and I have filled it with water. So the beam of white light is shining onto the light bulb filled with water. Do not try this at home, kids. Straight away, here we go. You can see this arc which goes actually up all around there. On the wall, we can now see one section of the beautiful curve of a rainbow, not too different from something you might have come across in the wild.

7:32When you see a rainbow, you're seeing the process that we were just showing here, not from just one raindrop, but from millions of raindrops. So, in the wild, it's millions of raindrops refracting light, reflecting light off the back of the raindrop, refracting it a bit more, and then you see the portion of that refracted light coordinated into this structure called rainbow. Yes, exactly. When you see a rainbow, if you were to draw a line from your head to a raindrop and then from that raindrop to the sun, that angle is the rainbow angle Philip mentioned. And the exact colour you see depends on whether it's at around 40 degrees for violet, 41 degrees for green, or around 42 degrees for red.

8:28That's the nuts and bolts of how to make a rainbow. Now it's time to move on to Sakura's actual question. How big does that make each rainbow? My name is Raymond Lee and I am a retired professor, formerly teaching and doing research at the US Naval Academy in Annapolis, Maryland. I'm just going to get straight to it. Raymond, how big is a rainbow? It has a fixed size in angle, but it is not an object, so it doesn't have a linear size. We think we see a distant rainbow near the horizon as an enormous thing, but that's just because we're so used to inferring linear size from angular size in context.

9:20So tell me about the angular size. I feel like I know that you're right on some level and it ties into no one managing to find the end of rainbows. By angular size, what we mean is the angle in degrees that goes from one side, vertical side of the rainbow to the other, as seen from your position. It turns out that that angle doesn't depend on your position or what kind of rain is causing the rainbow. It varies only a little. the radius is about 42 degrees so you double that to get to the diameter which makes it 84 degrees. 84 degrees. Let's put that into practice. Imagine stretching your arms out in front of you and pretending you're going in for a hug so spreading your arms until they're almost at a right angle.

10:18If you see the full half circle arc of a rainbow this is approximately how wide it will be. No whether you move. I knew all this when I was a graduate student learning about rainbows and yet my compelling mistake when I saw one and I didn't have a wide angle lens on my camera was to back up to try to get more of it in the frame. Of course what's going to happen? Nothing. Nothing. The landscape will change you'll see more of distant objects but since the rainbow isn't an object, I was doomed to frustration or embarrassment. Fortunately, I was there by myself, but I remember it is well enough to embarrass myself in the future or right now.

11:08With the World Service audience. That's right. It's fine. Our millions of listeners are very forgiving. I feel that I understand on one level what you're saying, but then I just can't get my head around the fact that I've seen rainbows in my back garden a metre and a half away from me. So it's in front of the wall, therefore it's closer than that, therefore it's smaller. And then I've seen them further away going from Big Bend to the other side of the River Thames, and so it's got to be kilometres across. Right. Well, we are very used to inferring linear size from context. If something looks like it is off in the distance, we automatically assume, and it's a reasonable assumption, it works most of the time, that what we're seeing is at that distance.

12:06and therefore if it's at that distance and yet it is in close competition with distant objects that we know are large, we'll transfer that linear size assumption to what we're seeing in this case, the rainbow. We make the reasonable but incorrect assumption based on our daily experience that there's a large thing out there in the distance. But does that mean, are you saying that all rainbows are the same size? Yes. All rainbows to within a fraction of a degree have the same angular size. They occupy the same angle in space in front of you. And that doesn't change whether you get closer or farther away from the rain shower.

12:59So rainbows, according to Raymond, don't do big or small. He says they are an optical phenomenon with a particular angular size. The big or small, says Raymond, just comes from judging it against other objects in your view. But maybe there is still a size difference in how much of the rainbow you actually see. The rainbow is always centred on the shadow of your head. That's something known as the anti-solar point, which really helps you to work out whether you're going to see a rainbow at all. As the sun gets higher in the sky, the shadow of your head, of course, gets closer to your feet. And that means you're looking at a lot of ground in the directions where the rainbow could occur, but because you don't have a long, unimpeded path length of sunlit rain in that direction, that part of the rainbow doesn't exist.

13:58So that means that the rainbow gets bigger and more impressive, all things being equal, as the sun gets lower in the sky. And the best rainbows in terms of how much of the semicircle you can see are going to occur near sunset or sunrise. Sometimes you just see more rainbow, more of that semicircle, which I think counts as larger. If we take that as our definition of size, how big can a rainbow get?

14:35My name's Colin Leinhardt. I'm an aerial photographer based in Perth, Western Australia. You put your name into Google and the first thing that comes up is Rainbow, which is quite nice. It's like the one-hit wonder. Colin's claim to fame happened accidentally. He was in a helicopter coming back from a rainy and largely unsuccessful photo shoot. It was raining and it was a miserable day. And at the end of the day, we were flying home and it was sunset. set. When Colin and Pilot flew past the edge of a rain cloud. As we went into the mist there was this huge round rainbow surrounding us. A massive full circle with all the colours of the rainbow.

15:28Me and the pilot just looked at each other we just like we didn't know what it was. I started to take photos with a normal lens that I had on the camera and it only took in about 15 percent of the rainbow so I always carry a 180 degree fisheye lens put that on straight away and then I can get the full rainbow in the shot and I had no idea at all what it was. Colin gets home posts his circular rainbow pic on social media and his website and forgets about it until almost a year later when an astrophysicist who curates the NASA astronomy picture of the day page gets in touch with him. And he goes, oh, you know what you've got there, don't you?

16:17And I just, I said, look, I have no idea at all, you know. And he was the one that told me all about it and he was just so excited. He's always wanted to see one, he's never seen one. And he said, look, you don't mind if we do a little story about it, do you? The next day, my emails were just going off. I had never had so many emails from media organisations, just hundreds. We're just the latest in a long line of people that are like, tell me about the rainbow. Yes, 10 years later. It turns out rainbows are, in fact, circles, which I would argue is the biggest kind of rainbow. But we usually see one as an arc because land or sea get in the way of seeing the whole ring.

17:05Next up, I'm hunting for other ways to find a larger one.

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18:13You're listening to CrowdScience from the BBC World Service. I'm Marnie Chesterton, and in this episode, we're answering a question from listener Sakura, who wants to know how big a rainbow is. So far, we've discovered that rainbows have an angular size, but not a linear size. Despite me really wanting to measure one in metres, scientists say no. But we've also learnt that rainbows are actually circles, which I would argue is one way to get yourself a bigger rainbow. And we're about to find another.

18:51Hello, my name is Ping Wa Li. Right now I'm working as an assistant professor at the Chinese University of Hong Kong. How big is a rainbow? The size of the rainbow is a very interesting question, but there will not be any simple answer to that. Penghua agrees that a rainbow just has an angular size, but sometimes there isn't just one. If the sky between you and the rainbow is kind of clear, you may be able to see so-called secondary rainbow. Is this a double rainbow? It's a double rainbow, yeah. When light enters a raindrop, it can get reflected more than once. Not just single internal reflection, but maybe double reflection.

19:42So in that case, you will be able to see two different arcs of rainbow, one on top of the other one. Because the double rainbow is sort of on top of the outside of the rainbow, I'd say that the double rainbow is bigger than the primary rainbow, right? Yes, yes. So it turns out, because of some complex geometry, if the light bounces around inside the raindrop twice, it leads to a bigger angular size, around 10 degrees larger than the primary rainbow. And it's not just that. If we go back to white light entering a raindrop... Different colours. Each colour refracts. They have different reflection angle.

20:29Bending at a different angle. When it bounces off like an internal reflection. At every reflection, those colours will spread out more. The angular difference between the violet or blue colour and the red colour one will become larger and larger. So after a second reflection, when that light exits the raindrops, it's more spread out or bigger. That's why the secondary rainbow will look wider than the primary rainbow. If you can have double rainbows, can you have triple rainbows? Yeah, of course. Brilliant. So light enters the raindrop, bounces around three times and comes out the other end of the raindrop, confusingly creating a rainbow in the direction of the sun.

21:23Actually the third one emerged in parallel with the sunbeam. If the light bounces four times, the resulting rainbow is also towards the sun. But if it bounces five times... The fifth rainbow actually emerges to your direction. It's in the same direction as our lovely double rainbow. This is sounding more and more like you're in a massive pinball machine. Yes. You can create a map of where to look with each higher order rainbow. The problem is, with each reflection, the light also gets fainter and fainter. So what's the highest order we've actually seen in nature? I'm Harald Aedens. I'm originally from the Netherlands.

22:12I did a PhD at New Mexico Tech and worked on a mountaintop observatory during the summers. And this is where I documented many of the rainbows in my life. Harald was working away on his studies when some important rainbow discoveries were made. Someone had photographed the third order rainbow. So the third order was discovered and the fourth order was discovered the same summer. And then the hunt was on for the fifth. So I thought, well, maybe I'll go look in my photo collection. If you don't mind, I can share my screen for a picture. Yeah, sure. Harold shows me a photograph. There's some trees in the foreground, a large mountain in the back, and in between, a beautiful half-circle double rainbow.

22:58And this program I was using has a simple slider for contrast, so you can browse through pictures quickly and you can slide the contrast to adjust it. So I did. And this happened. The screen switches to another picture, zoomed in on just one section of the rainbow and with much higher contrast between the colours. And in between the primary and secondary rainbow, there's a ghostly presence, a faint bit of green. And I was just looking at it like, huh, that's it, that's it. You know, I'm like stuttering in my head. And that green band is about as wide as the secondary rainbow, if you agree. Yes, yeah.

23:40Just on the inside of that green band, you see a hint of violet. Yes, yes, yes, yes, yes. Okay. And that greenish tint. Yes. That's basically your middle of your fifth order rainbow. Yes, that's exactly right. The upper edge of the fifth order rainbow ends at a similar point to the secondary rainbow but it is technically wider. The question is can we count the fifth order rainbow as bigger? Harold told me he hasn't seen it with his own eyes although it should technically be possible under perfect conditions. So maybe not. I feel like the ghostly waft of a higher order rainbow that you can barely see shouldn't count.

24:31I want my rainbow impressively coloured and, well, big, which I was told earlier they can't be. But maybe, just maybe, there's another way to look at size. I wanted to check with Harold. Do rainbows really just have an angular size?

24:53Harold, I spoke to Raymond Lee earlier and he's pretty insistent that rainbows just have an angular size, that there's no kind of linear way of measuring them. OK. You're nodding. I feel like he's... I can't get my head... I know he's right, but there's also a part of me that thinks, Well, hang on. If you see a rainbow behind some mountains, but then in front of the next mountains, surely that gives them a linear distance. Raymond Lee is correct. It's all about angles. But. Excellent. I like a but. The scientist answer is yes, it's only an angular size. There is no distance involved here. However, I agree with you saying there surely is some kind of distance involved there.

25:43And that's if you talk about where the rainfall was that the rainbow occurs in. And for example, your example of a rainbow being, let's say, in front of a distant mountain range or behind some object in the foreground. What you're seeing there is the rainbow appearing in rainfall that's between those two structures. and that could be a kilometre away and then you can say, well, this rainbow is in rain, that's a kilometre away. There is nothing between you and this rainfall that contributes to the rainbow light. Maybe I was right. There is a distance to the rainbow and that would mean that if we see the rainbow two kilometres away, the rainbow must be bigger, huge in fact, right?

26:26Well, unfortunately, when it comes to rainbows, nothing is as it seems. The rainbow is not a two-dimensional image or object. It's a three-dimensional cone-shaped light effect that starts at your eye and it extends out into, well, not infinity, but as far as the rainfall is falling. So if you're standing in the rain and this shower is, let's say, between you and a kilometer away from you, that rainbow cone is a kilometer deep. Okay, so I'm imagining something like an ice cream cone, like a waffle cone going out from my eye. That cone has a volume and therefore has size. But the viewing geometry is such that you don't see it bigger or smaller, because you're looking at this cone of light going to your eye, with the apex at your eye.

27:25I know this fact is a bit mind-boggling. If you're standing at the edge of a massive rain shower with the sun right behind you, if you pick, say, one spot, a red patch on the upper left side of the rainbow, the red you see is the cumulative colour reflected out of all of the raindrops with just the right angle to bounce red light into your eye. Some of those drops may be a few centimetres away from your eye, some may be right at the back of the shower. They all contribute to the final illusion, which means that a rainbow is 3D. So maybe we should measure rainbows in volume? I love this idea of a rainbow actually having a volume.

28:10Like you can measure, I don't know, litres of rainbow. You can. It's a rainbow producing volume of droplets. Yes, that has a volume. And so you can speak about, yeah, litres of, you know, rainbow producing, well, droplets or even, yeah. Cool.

28:36Listener Sakura asked, how big is a rainbow? One of those deceptively simple science questions that opens a can of optical worms. On one level, a rainbow is 42 degrees big, because that's the sweet spot that allows you to see sunshine refracted and reflected off the back of raindrops and into your eyes. Move closer and you still get exactly the same sized rainbow. But there are different ways to see a bigger one. If you go to higher ground, ideally up in an aeroplane and look out with the sun behind you, then you have a chance to see the full circle, which is more rainbow. You might even get a secondary one on the outer margin of the first, fainter but outside and therefore bigger.

29:24And if the rain shower is very big, the volume of rainbow-producing droplets might be very high, which arguably might mean a bigger volume of rainbow. The thing is, I can't stop my brain from judging actual distances. Which means that, despite lots of conversations with my producer and our experts, I will still think of a rainbow that spreads across my city's skyline as bigger than the one that I can make in the back garden. Pleasingly, I'm not alone. Harold also struggles to explain this to his four-year-old daughter. I have a daughter who's going outside, and whenever there's a rainbow, I will make sure she's outside.

30:08My daughter's like, wow, this rainbow is huge. And then now I have to laugh. Because I'm not going to talk about light paths and diffractions. Angular size. Actually, darling, that's an optical illusion. Thank you so much to listener Sakura for her colourful question and back to her for the credits. That's it for this episode of CrowdScience from the BBC World Service. It was presented by Marnie Chesterton and produced by Florian Bohr. Today's question came from me, Sakura, in Kenrichhire, in the UK. If, like me, you have a question you're curious about and would like the CrowdScience team to look into, Do what I did and email crowdscience at bbc.co.uk.

31:04Thanks for listening. Bye.