The course began with the spinning Earth: a tilted axis turning once a day. This last lesson cashes that picture in. The Sun's crossing of a meridian defines the day; the meridians of the first lesson each carry their own version of it; and the tilt decides how much of each day is daylight.
The mean solar day and LMT
The real Sun is an irregular timekeeper: the Earth's orbit is slightly elliptical and its axis tilted, so the interval between the Sun's successive crossings of a meridian, the apparent solar day, varies through the year. Timekeeping therefore invents a better Sun: the mean sun, a fictitious sun moving at the average rate of the real one, whose successive transits of a meridian are exactly 24 hours apart: the mean solar day. LMT (local mean time) is time kept by the mean sun against the observer's own meridian: the mean sun crosses the observer's meridian at 1200 LMT and the anti-meridian at 0000. Every meridian keeps its own LMT, which is precisely why the world needed something shared: UTC (coordinated universal time), in this arithmetic the LMT of the Greenwich meridian.
Arc to time
The Earth turns 360° in 24 hours, and the whole conversion follows:
15° of longitude = 1 hour;
1° of longitude = 4 minutes;
15' of longitude = 1 minute of time.
The direction rule rides on the sunrise: the Sun crosses eastern meridians first, so east of Greenwich the LMT is ahead of UTC, and west of Greenwich behind: LMT = UTC + east longitude in time, or − west longitude in time. At 045°E when it is 1200 UTC, the LMT is 1200 plus 3 hours, 1500. At 073°30'W the correction is 73° × 4 minutes plus 30'/15 minutes, 4 hours 54 minutes behind: at 0900 UTC the LMT is 0406. Run the same sums backwards to turn an almanac's LMT into UTC.
Standard time and the date line
No country runs its clocks on each town's meridian; two conventions do the rounding:
Standard time — the legal time a state adopts, normally a whole (sometimes half or quarter) number of hours from UTC, listed for every state in the almanac's tables;
Daylight saving time — where used, sets the clocks one hour ahead of standard time in summer.
The calculations are bookkeeping: a clock reading 1730 in a zone of UTC+3 means 1430 UTC; if daylight saving is in force the same clock face means 1330 UTC, the extra hour peeled off first.
The zones accumulate to a seam in the mid-Pacific. Crossing the International Date Line westbound, the date advances by one day; crossing it eastbound, the date is set back by one day, the local clock time carrying on regardless: a westbound crew's Sunday becomes Monday mid-ocean, and an eastbound crew lives one calendar day twice. The line itself zigzags around territories, but the rule at the seam does not.
Sunrise, sunset and twilight
The day's edges have exact definitions, all built on how far the Sun sits below the horizon:
sunrise and sunset are the moments the Sun's upper edge sits on the visible horizon, rising and setting;
civil twilight is the interval while the Sun's centre lies between the horizon and 6° below it, ending in the evening and beginning in the morning at that 6° depression, the working limit of outdoor light;
the times are extracted from an almanac, tabulated against latitude and date, and given in LMT, converted to UTC with the arc-to-time of this lesson.