Autumn Equinox

September 21, 2012

We already have most of what we need: a previous exercise provides functions to calculate sunrise and sunset times, and the Standard Prelude provides functions that translate dates to integers. All we need to do is extend the functions from the Standard Prelude to handle times, which we do by adding fractions of the 24 × 60 × 60 = 86400 seconds in a day:

(define (julian year month day hour minute second)
  (let* ((a (quotient (- 14 month) 12))
         (y (+ year 4800 (- a)))
         (m (+ month (* 12 a) -3)))
    (+ day (/ hour 24) (/ minute 24 60) (/ second 24 60 60)
       (quotient (+ (* 153 m) 2) 5)
       (* 365 y)
       (quotient y 4)
       (- (quotient y 100))
       (quotient y 400)
       (- 32045))))

(define (gregorian julian)
  (let* ((j (floor julian))
         (hms (* (- julian j) 24 60 60))
         (hour (floor (/ hms 60 60)))
         (hms (- hms (* hour 60 60)))
         (minute (floor (/ hms 60)))
         (second (- hms (* minute 60)))
         (j (+ j 32044))
         (g (quotient j 146097))
         (dg (modulo j 146097))
         (c (quotient (* (+ (quotient dg 36524) 1) 3) 4))
         (dc (- dg (* c 36524)))
         (b (quotient dc 1461))
         (db (modulo dc 1461))
         (a (quotient (* (+ (quotient db 365) 1) 3) 4))
         (da (- db (* a 365)))
         (y (+ (* g 400) (* c 100) (* b 4) a))
         (m (- (quotient (+ (* da 5) 308) 153) 2))
         (d (+ da (- (quotient (* (+ m 4) 153) 5)) 122))
         (year (+ y (- 4800) (quotient (+ m 2) 12)))
         (month (+ (modulo (+ m 2) 12) 1))
         (day (+ d 1)))
    (values year month day hour minute second)))

(define (duration j-start j-finish)
  (if (< j-finish j-start) (duration j-finish j-start)
    (let* ((diff (- j-finish j-start))
           (days (floor diff))
           (hms (* (- diff days) 24 60 60))
           (hours (floor (/ hms 60 60)))
           (hms (- hms (* hours 60 60)))
           (minutes (floor (/ hms 60)))
           (seconds (- hms (* minutes 60))))
      (values days hours minutes seconds))))

To be complete, we wrote the gregorian function of the Standard Prelude, even though it is not necessary for today’s exercise. Then it is easy to calculate the length of the day:

> (call-with-values
    (lambda () (solar 0 0 2012 9 21 0))
    (lambda (set rise)
      (let* ((start-hour (string->number (car (string-split #\: rise))))
             (finish-hour (string->number (car (string-split #\: set))))
             (start-minute (string->number (cadr (string-split #\: rise))))
             (finish-minute (string->number (cadr (string-split #\: set)))))
        (duration (julian 2012 9 21 start-hour start-minute 0)
                  (julian 2012 9 21 finish-hour finish-minute 0)))))
0
11
53
0

That’s 0 days, 11 hours, 53 minutes, and 0 seconds. Oops. Our sunrise/sunset calculation is imprecise. But at least we can now handle times as well as dates in our standard library functions.

You can run the program at http://programmingpraxis.codepad.org/lb9mRWuW.

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4 Responses to “Autumn Equinox”

  1. Mike said

    Day and night are (more or less) equal length at the equator all year round. I think you mean day and night are also equal everywhere else!

  2. programmingpraxis said

    Hmmm. You’re right.

    I just looked at the morning newspaper. Where I am, in St Louis, Missouri, USA, today will be 12:10 of sunlight.

    Am I getting my astronomy wrong?

    I just sent an email to the local weatherman. We’ll see what he has to say.

  3. programmingpraxis said

    I had an answer already; that was fast! Dave Murray says:

    the earth is not a perfect circle..it is a little more pear shaped…thats the difference…the time difference works its was out over about 5 to 6 days

  4. treeowl said

    Even a hypothetical spherical Earth doesn’t give equal day and night times at the equinox, in general. Wikipedia defines the equinox thus: “An equinox occurs twice a year (around 20 March and 22 September), when the tilt of the Earth’s axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth’s equator. The term equinox can also be used in a broader sense, meaning the date when such a passage happens.”

    It says also that “The date at which sunset and sunrise become exactly 12 hours apart is known as the equilux. Because times of sunset and sunrise vary with an observer’s geographic location (longitude and latitude), the equilux likewise depends on location and does not exist for locations sufficiently close to the Equator. The equinox, however, is a precise moment in time which is common to all observers on Earth.”

    Why doesn’t the equilux exist too close to the equator? Because sunrise occurs when the sun just barely begins to appear on the horizon (because of refraction, this occurs a little before the upper limb of the sun is actually tangent to the horizon). When the Earth has turned by an angle of π, the sun will be just about to start setting (around the equinox at the equator). It won’t actually set until the disk of the Sun just disappears behind the horizon (which occurs a little late because of refraction). Even ignoring refraction, the angular diameter of the Sun is about 31.6′–32.7′ (according to Wikipedia) (about 0.00919-0.009512 radians), so the day gets about half a degree more rotation than the night gets by simple geometry, and refraction makes it even longer. The lesson I just learned from this: the equilux isn’t directly tied to the equinox, and despite common usage, the equinox is best understood as an instant of alignment, rather than a particularly well-balanced day.

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