I did an amusing back-of-the-envelope calculation on the drive home tonight. I didn’t actually have an envelope….
The days are getting longer! But how much longer does the day get each day? Let’s suppose that the periodicity of the day length is described by a sine wave. And lets suppose that where we live the shortest day in winter has eight hours of sunlight and the longest day in summer has 16 hours. Also, to keep things simple, lets say day 0 is an equinox, for example the upcoming Spring Equinox on March 21 (or is it March 20? it varies from year to year). Hmm, we’ll need some variables sprinkled in there — how about y for the length of the day and x for the day of the year. Therefore our assumption can be written this way:
Let’s invoke some good old calculus. The rate of change of day length is the first derivative of y with respect to x. Trust me, this works out to
Exactly at the equinox, the cosine expression goes away (becomes 1), thus the rate of change of day length is
So at the equinox, I’d estimate that the day gets about 4 minutes longer each day. That comes out to about 28.9 minutes per week.
So how close is my back-of-the-envelope appoximation? I used a military sunrise/sunset calculator to find out what the actual sunrise and sunset times are for Minneapolis. First I checked my assumptions : The shortest day of the year here (Dec 21) is 8:46, and the longest day is 15:37. My guess of 8:00 and 16:00 is reasonably close, but if you redo the calculation above with those numbers you arrive at 3.53 minutes per day instead of 4. Next, let’s look at how long the days are near the equinox:
| date | sunrise | sunset | length of day (minutes) |
| March 19 | 0618 | 1825 | 727 |
| March 20 | 0616 | 1826 | 730 |
| March 21 | 0614 | 1827 | 733 |
| March 22 | 0612 | 1828 | 736 |
| March 23 | 0610 | 1830 | 740 |
The data from the navy indicates that the days are getting longer by just over three minutes each day. That is still a little bit lower than expected. I suppose maybe the length of the day is not exactly a sine wave, what with the earth being in an eliptical, not circular orbit and all…
3 replies on “The days are getting longer!”
car math huh………..
Equinox occurs with the advent of spring, right? Equinox means that night and day are of equal length, i.e., 12 hours each. So spring starts on March 20th this year, but it is daylight for 10 minutes longer than 12 hours on that day, so shouldn’t spring start earlier?
My best guess is that there is some bias in terms of how they measure sunrise and sunset. Probably the sunrise and sunset are not measured from when the center of the sun is at horizon — rather it is when the top of the sun’s disc touches the horizon. It probably takes five minutes for the sun to move from that position to where the center of the sun’s disc is at horizon.