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## The length of sunrise and sunset

In a fit of boredom, I decided to work out how long it takes the Sun to set, i.e. the length of time between the instant that the bottom of the Sun appears to touch the horizon, and the instant that the top of the Sun disappears below the horizon. Of course, the reverse applies to sunrises as well.

I’d already worked out in my head, in an effort to get to sleep one night, that the quickest sunset will take very close to 2 minutes, but this assumes that the Sun is setting vertically downwards, i.e. it is only correct if you are located at the equator. At other latitudes, the Sun will be coming down at an angle to the horizon, and so will take longer to set.

A bit of simple trigonometry gives a better, latitude-dependent answer for the time taken:

$\displaystyle t = \frac{d}{v\cos\theta},$

where d is the angular diameter of the Sun as seen from Earth, v is the angular velocity with which the Sun appears to move through the sky, and θ is the observer’s latitude on Earth.

Now, the angular diameter of the Sun is “about half a degree”, and the best value that I could get for it is actually d = 0.533°. Next, you can get the angular velocity of the Sun from the fact that it appears to travel a whole 360° in 24 hours, giving v = 360° / (86,400 seconds). Putting these values into the formula gives:

$\displaystyle t \approx \frac{127.92}{\cos\theta}$

for the time taken in seconds for the Sun to rise or set.

However, this simple approach assumes that the Sun is setting directly due West of the observer (or alternatively, rising directly due East), i.e. it is only valid at the equinoxes. At other times of the year, the Sun will set at a point somewhat North or South of due West, and will take longer to do so, with this effect being maximised at the solstices.

This nice webpage, which I subsequently found, suggests that the longest (solstice) sunsets have durations better described by:

$\displaystyle t \approx \frac{142}{\cos(1.14\,\theta)}.$

So, the above two formulae can be used to set upper and lower limits for the duration of a sunrise or sunset at a given latitude.

I’m writing this at a latitude of +51.41°, meaning that the sunset should take somewhere between 3 minutes, 25 seconds (in March and September) and 4 minutes, 33 seconds (in June and December). Maybe, if I’m still bored, I’ll time it. 😐

## The Y-Δ transformation

In the spirit of generosity, I offer here the equations for converting between two equivalent, three-terminal resistor networks: the “Y” (sometimes called “star”) arrangement, and the “Δ” (often called “triangle”) arrangement. Here’s a diagram of the two networks:

For this to make sense, the three outer nodes must be the same in each of the two networks (for this reason, I think that a better name for this would be the “Y-∇ Transformation”).

For obtaining the equivalent triangle resistances from known Y resistances:

$\displaystyle \\ R_{\Delta1} = \frac{R_{\rm Y1}R_{\rm Y2} + R_{\rm Y1}R_{\rm Y3} + R_{\rm Y2}R_{\rm Y3}}{R_{\rm Y1}},\\[1em] R_{\Delta2} = \frac{R_{\rm Y1}R_{\rm Y2} + R_{\rm Y1}R_{\rm Y3} + R_{\rm Y2}R_{\rm Y3}}{R_{\rm Y2}},\\[1em] R_{\Delta3} = \frac{R_{\rm Y1}R_{\rm Y2} + R_{\rm Y1}R_{\rm Y3} + R_{\rm Y2}R_{\rm Y3}}{R_{\rm Y3}}.$

For obtaining the equivalent Y resistances from known triangle resistances:

$\displaystyle \\ R_{\rm Y1} = \frac{R_{\Delta2}R_{\Delta3}}{R_{\Delta1} + R_{\Delta2} + R_{\Delta3}},\\[1em] R_{\rm Y2} = \frac{R_{\Delta1}R_{\Delta3}}{R_{\Delta1} + R_{\Delta2} + R_{\Delta3}},\\[1em] R_{\rm Y3} = \frac{R_{\Delta1}R_{\Delta2}}{R_{\Delta1} + R_{\Delta2} + R_{\Delta3}}.$

For more on this, check out this detailed webpage.

## Algebra

One of the reasons that I choose WordPress for this blog is that I was told that it can render equations nicely. And why use easily understood words when you can use equations? To test out the beautiful rendering, I thought I’d try the most beautiful equation ever, Euler’s identity:

$\displaystyle {\rm e}^{i\pi} + 1 = 0.$

You’ll want to save a picture of that to your secret stash!

## The Thrill of InterBlagging

So, I figured, there aren’t enough blogs on the Internet, and there are too many fascinating things happening in my life that are going undocumented.

Furthermore, all these Internet millionaires keep telling me to start a blog, then monetize it. Alas, I instantly messed up by choosing WordPress (no ads allowed).