## Rainbows

###### When the Sir Isaac Newton explained the colours of the rainbow with refraction the poet John Keats was horrified. Keats complained (through poetry of course) that a mathematical explanation robbed these marvels of nature of their magic.

Whether you, dear reader, agree with Keats’ view or not, it is time to deep dive into the mathematical explanation, requiring just basic geometry of lines and circles. As we will see, the explanation is just as elegant as the rainbows themselves.
When sunlight enters a droplet from any angle some light is reflected and some is refracted into the raindrop. We most commonly encounter refraction when we look at a straw in a glass, it seems distorted and cut off. How much light is refracted is determined by the refraction index $Latex formula$ which is simply $Latex formula$; the speed of light in vacuum divided by the speed of light in the new medium which makes n a number between one and (usually) two.
Due to this underlying mechanism, the angle of the light beam changes according to Snell’s law $Latex formula$ , where $Latex formula$ is the refractive index of the first medium (air, $Latex formula$ = 1) and $Latex formula$ is the incidental angle. The secondary angle is a bit smaller because the refractive index of water is around 1,34.  From here some of the sunlight reflects off the back side of the raindrop and then leaves the raindrop through the “bottom” where the light is refracted again, same as when it entered the raindrop.

Now, depending on where the light enters the raindrop it will exit the raindrop at different angles. We can calculate the total deflection D by adding up all grey angles $Latex formula$. We then calculate the derivative for the incidental angle to find the minimum. This minimum angle has the most intense light and creates what we can see as a rainbow. Remember that we are plotting the total deflection, the angle between the sunlight and you looking up will be 180°-D. This ends up to be around 42° and does not depend on the size of the water droplet.

Supposedly Descartes (mostly known for his philosophy) figured all this out graphically, but he did not understand why the rainbow showed different colors. He didn’t know that every medium has a different refractive index for each color which is a wavelength in the electro-magnetic spectrum, Blue’s refractive index is around 1,342 while red’s is around 1,331 resulting in different deflection angles.

There are so many further interesting facts to point out but I will try to keep the list short:

In theory there are many more orders of the rainbow, each one reflects light once more inside of the rain trop. The second order rainbow which has its colors reversed is the only one you can frequently see with bare eyes and is located about 10° above the first order rainbow.

Rainbows seen from the ground can only occur in the morning or the evening due to the 42°-degree angle between you and the sun. If the sun is higher than 42° -degrees, the rainbow will be below the horizon (unless you are up high and looking down). Seen from an airplane, rainbows are full circles directly opposite of the sun!

Last but not least, seawater has a higher refractive index than rain water, so the radius of the seabow is a bit smaller, making for some crazy photos!

So next time you see one of these colorful arcs appear in the sky, try to remember the elegant math behind what you’re seeing!

And as always, stay curious!

## Hawai’i

Now, other than Kiruna, I do not have to explain where Hawai’i is or why you would want to there, do I? So this will be more like a travel blog. Before we start with “travelstuffs” its time for some SCIENCE, so bear with me.

Hawai’i as a region subsists of several islands emerging out of the water in the middle of the pacific. The leading and widely accepted theory on stationary hotspots proposed by geophysicist Tuzo Wilson as late as 1963 explains the existence of the Hawai’in islands. Another well-known example of a hotspot is Yellowstone. Imagine hot magma moving upward from the lower part of the earth’s core up to the lithosphere which is the region just below the earth’s crust.

These hotspots are stationary; the crust above though moves ever so slowly (about as fast as your fingernails grow), creating a line of volcanos behind the current location of the hotspot.

The whole Hawai’i seamount chain is almost 6000 kms long! Of course the old volcano islands are worn down by erosion and eventually disappear below the waster surface.

Although this explanation should be enough for now, I do need to point out that there are many open questions and different theories concerning this topic.

The two most popular and populated islands are Big Island and O’ahu. I will be concentrating on O’ahu.

Before we begin here, I would like to mention your chance of seeing at least one rainbow per day is pretty high (not the license plate of Hawai’i, the real ones). Rainbows are fascinating! So after you have red how and why they occur you can boast about your knowledge whenever you see one on Hawai’i.

So, for a beginning we can start with some classical stuff: Surfing. Chances are, you haven’t tried this “exotic” sport. If you are ever going to try, you should when you’re on O’ahu. Depending on the time of year, you can learn surfing on “small” waves on the north beach or on Waikiki beach which is the most famous beach in Honululu. Surf lesson will cost you around 50-80\$ per hour so best do group lessons and find some options to get the lowest price. It’s fairly easy to get some first results (cinda standing on the board) after an hour or two. (By the way, did you know that water waves are really complicated to explain and need some extensive theories to roughly model them? Maybe I’ll do an article about that sometime)

Another classic thing to do is snorkeling at Hanauma Bay. You need to arrive at least before 8 a.m. because the bay has a maximum capacity of tourists in order to preserve the corals. The earlier you come the less other tourists and the calmer the water but more importantly less sun. Snorkeling with your back exposed to the midday sun is not a good idea, believe me, I tried it :/. You may also want to keep the tide in mind. If it is low tide you will have fairly limited depth of water which can cause you to hit the corals while you are swimming. Also, if you buy some snorkeling gear beforehand it may be cheaper than to rent equipment at the bay itself.

You should definitely do some hiking around the island. From Diamond head you have a very nice view onto Waikiki beach. Inland there are several hikes to nice waterfalls through the rain forest. Or you can climb the 1048 steps up Koko head which allows for a great view on non-cloudy days.

Lastly I would recommend the National pacific cemetery and or the pearl harbor museum. I know you probably don’t like museums but these two are a fast way to learn about the WWII pacific war. You don’t need to go on the pearl harbour tour that brings you to the sunken USS Arizona.
It’s just a sunken ship and you can’t really see anything. But the museum part of pearl harbor and the cemetery which is free show the whole marine war against the Japanese. As a European we only concentrate on how bad Hitler was and how the Allies kicked ass in Europe. I knew nothing about the pacific war before I was on O’ahu and it was really fascinating.

To do all the things mentioned above plus some other obvious and easy to find tourist destinations you will probably need around a full week.

So as you see, there is more to do on Hawaii than chilling on the beach and boasting on Facebook about you being in Hawaii. You can still do that, just don’t miss out on the other good stuff.

And as always, stay curious!