# The equation of time strikes again

Some of us are rather looking forward to getting to 22 June. That's when the days get longer again. Yes, the reality is that no-one's really going to notice much difference for a while, but it's encouraging to think that the days will be getting lighter again, if only by a little bit. Don't confuse that with temperatures getting warmer – the coldest day (on average, of course) lags the darkest day quite considerably. Here it's around the end of July.

But there's an interesting effect going on with sunrise and sunset. We've already had the darkest evening (hooray!) yet the darkest morning is still to come. Look at the sunrise and sunset times (for Hamilton) on the MetService website: today we've had sunrise at 7.32am and with cloudless skies the sun may stay out all the way to sunset at 5.07pm. But tomorrow sunset is recorded at 5.08pm (later!) and on Saturday sunrise has shifted to 7.33am (also later!). How can that be?

The point here is that the length of a day, meaning now the time between when the sun is at its highest to the next time the sun is at its highest, is only 24 hours **on average **(for some periods of the year its greater, for some periods is less), and isn't equal to the time it takes for the earth to spin once on its axis.

Let's take this last point first. It's solar midday, meaning that the sun is at its highest. Now, let the earth rotate exactly once on its axis. Do we get back to solar midday the next day? No. That's because, in the time taken for the earth to rotate once, it has also moved along its orbit (about 1/365th of the way around). That means it's got to spin a little bit more before the sun reaches its higherst point. The time to spin once (the siderial day) is about 23 hours 56 minutes – four minutes less than the mean solar day. Note that 4 minutes x 365 = 24 hours – which means one more revolution than you might expect – the earth actually does 366 and a quarter revolutions each year.

However, the movement along the earth's orbit in a day is only on average 1/365th of the orbit. When the earth is closest to the sun (called perihelion – 3 January at present) it moves faster. That's Kepler's second law. When it's further away (at this time of year) it moves slower. That would mean that in January, we should perceive that solar midday gets later every day by our watch (since the earth needs extra time to spin that extra bit more), but that in July, the solar midday should be getting earlier. However, that's not what is observed. Our prediction for January is true, but for July it's the other way around – solar midday actually gets later as measured by our watches.

There's another effect going on. This is because the earth is tilted on its axis. However, it's quite tricky to explain why that makes a difference. Consider the transition from winter to summer, in the southern hemisphere. If we look at the position of the sun at sunrise and sunset, we see it move southward from one day to the next. What is significant is that at sunset the sun is further southward than for the previous sunrise. That gives us a shift in the measured time between solar midday and the next solar midday. A better explanation is given here. This effect is 'zero' at the solstices and equinoxes, and does two cycles a year. Add this to the effect of Kepler's second law, and we get the odd-looking curve that is called 'the equation of time', and means that, at present, each solar day is slightly longer than 24 hours, giving both ligher evenings and darker mornings.

You can see a net result displayed on the ground under the sundial in Hamilton Gardens. The elongated figure-of-eight is called an 'analemma'. It will show you the position of the tip of shadow of the pole at different times at different times of year.