Longitude, time zones and universal time

Builds on: Sections 1.3, 1.4 and 2.2
Reading time: 35 min
Exercises: 4
Olympiad problems: 1

All of China keeps Beijing time, eight hours ahead of universal time, although the country spans more than 60° of longitude. In Kashgar, in the far west, the Sun crosses the meridian on 16 April at about 14:56 by the clock, and in Beijing at about 12:14. The same Sun reaches the meridians of the two cities nearly three hours apart, because Kashgar lies 40° of longitude further west. In this section we see how the time of day depends on longitude, how countries choose their clocks, and how to turn the time on a watch into the position of the Sun.

2.3.1Local mean time and universal time

Before you readMake a guess first

Madrid is at longitude 3.7° W and Ankara at 32.9° E. By how much do their local mean times differ?

The mean solar time of Section 2.2 depends on where we are, because it is set by the hour angle of the mean Sun, and hour angles are measured from the observer's own meridian. Mean solar time at a given place is called its local mean time. As Earth turns, the mean Sun crosses one meridian after another, moving westward by 15° of longitude every hour, or 1° every 4 minutes. A place further east meets it sooner, so its local mean time is later.

The local mean time at Greenwich, at longitude 0°, is called universal time, written UT. Astronomers give the times of events in UT so that everyone can use them, whatever their own longitude.

ResultLocal mean time(2.5)

LMT = UT + \frac{λ}{15 °} h

$LMT$	local mean time at longitude $λ$
$UT$	universal time, the local mean time at Greenwich
$λ$	longitude, positive east of Greenwich and negative west of it

Each degree of longitude is 4 minutes of time, and each 15° is one hour.

NoteThe universal time sent out by radio signals and the internet is UTC, which is kept by atomic clocks. Earth's turning is very slightly irregular, so UTC is held within 0.9 s of the time set by Earth's rotation by adding an occasional leap second. Olympiad problems treat UT and UTC as the same.

Figure 2.4Local mean time and the time on the clocksinteractive

Universal time

Madrid, clocks UT+1
Ankara, clocks UT+3
Kashgar, clocks UT+8
Beijing, clocks UT+8
New York, clocks UT−5

The figure shows Earth seen from above the north pole N, with meridians every 15°, as at an equinox. The Sun is to the left, so along the heavy radius it is 12:00 local mean time, and the hatched half is in night. Earth turns anticlockwise, carrying each place towards the Sun and past it. The slider sets universal time, and the table gives the local mean time and the time on the clocks of each city. At it is noon by mean time in Kashgar, while the clocks there show 14:56. At it is noon by mean time in Ankara, where the clocks show 12:49, and at it is noon at Greenwich.

2.3.2Time zones

If every town kept its own local mean time, clocks a few kilometres apart would disagree, which became a real problem once railways and telegraphs linked distant towns. Most of the world therefore uses zone time. A country, or part of one, sets its clocks to the local mean time of a chosen meridian, usually a whole number of hours, that is a multiple of 15°, from Greenwich. Türkiye keeps UT+3, the mean time of the meridian at 45° E, all year round. India keeps UT+5:30, the meridian at 82.5° E, Nepal keeps UT+5:45, and China keeps UT+8 across the whole country. In summer many countries move their clocks one hour ahead, a practice called summer time or daylight saving time, so that Spain, for example, changes from UT+1 to UT+2.

ResultZone time(2.6)

ZT = UT + Z

$ZT$	zone time, the time on the clocks
$Z$	the zone's offset from universal time, in hours, positive east of Greenwich, including any hour of summer time

The offset

Z

is fixed by law and is often not

λ / 15 °

, so the clocks of a place can be well ahead of its local mean time or behind it.

Ankara is at 32.86° E, so its local mean time is UT plus 2 h 11.4 min, while its clocks show UT+3. Ankara's clocks therefore run 48.6 minutes ahead of its local mean time, and the Sun crosses the meridian there at about 12:49 by the clock, give or take the equation of time.

Quick checkUse the result above

Madrid is at longitude 3.70° W and keeps UT+1 in winter. Its clocks are then ahead of its local mean time by one hour and how many minutes?

minutes

2.3.3From the clock to the Sun

Combining (2.6), (2.5) and (2.3) gives the chain that every timing problem needs. The time on a watch gives universal time, the longitude gives local mean time, and the equation of time gives apparent solar time, which is the Sun's hour angle plus 12 hours.

\begin{aligned} UT & = ZT - Z \\ LMT & = UT + λ / 15 ° \\ T_{app} & = LMT + E \end{aligned}

Going the other way, from the Sun to the clock, we run the same steps backwards.

Example 2.6Noon in Ankara on 3 November

At what time by Turkish clocks does the Sun cross the meridian of Ankara (32.86° E) on 3 November, when the equation of time is +16.4 min?

Solution

The Sun crosses the meridian at 12:00 apparent solar time. By (2.3) the local mean time is then $12 : 00 - 16.4 min = 11 : 43.6$ . Ankara is $32.86 / 15 = 2.191$ h, or 2 h 11.4 min, east of Greenwich, so by (2.5) the universal time is $11 : 43.6 - 2 : 11.4 = 9 : 32.2$ . Turkish clocks are 3 hours ahead of UT.

AnswerThe Sun crosses the meridian of Ankara at about 12:32 by the clock on 3 November.

Example 2.7Noon in Madrid in summerfinish it

On 21 June the equation of time is −1.8 min. At what time by the clocks of Madrid (3.70° W, UT+2 in summer) does the Sun cross the meridian? Give the answer in hours.

The start

At apparent noon the local mean time is $12 : 00 - (- 1.8 min) = 12 : 01.8$ . Madrid is $3.70 \times 4 = 14.8$ min west of Greenwich, so the universal time is $12 : 01.8 + 14.8 min = 12 : 16.6$ .

Your stepNow find the time on the clocks of Madrid.

hours

2.3.4The date line

Going round the world eastward, a traveller sets the clock forward by an hour in each new time zone, and after a full circle has gained a whole day. To keep the date consistent, the zones meet on the far side of the world from Greenwich at the international date line, which runs mostly along the meridian at 180° but bends round island groups. Crossing it eastward, the date goes back one day, and crossing it westward, it goes forward one day. The neighbouring islands of Samoa and American Samoa lie on opposite sides of it, so their clocks show the same time but their calendars are a day apart.

Common errorsSection 2.3

A place east of Greenwich has a later local mean time, so its longitude is added to UT. A sign error here shifts the answer by twice the longitude correction.
The time on a watch is zone time, which is rarely equal to local mean time. Before finding the Sun's hour angle from a watch, both the zone offset and the longitude are needed.
Summer time adds an hour to the zone offset. A problem set in summer in a country that uses it needs $Z$ one hour larger.
A longitude in degrees is converted to time at 4 minutes per degree. Using 15 minutes per degree, by confusing it with 15° per hour, gives nonsense.

ExercisesSection 2.34 questions

2.3.1
Kashgar is at longitude 75.99° E. How many hours is its local mean time ahead of UT?
hours
2.3.2
A city at longitude 100° W keeps UT−6. How does its local mean time compare with its clocks?
2.3.3
An observer at longitude 30° E, in a country that keeps UT+2, sees the Sun cross the meridian on 11 February, when the equation of time is −14.2 min. What time is it by the observer's clock? Give the answer in hours.
hours
2.3.4
A ship crosses the international date line sailing west on 5 March. What date is it just after the crossing?

Olympiad problemsPast IOAA problems, restated1 problem

2007Sunrise on the shortest day2 points

Adapted from IOAA 2007 (Chiang Mai), theory problem 1.1. We have restated it in our own words and kept the original data. The solution was last checked on 17 October 2026.

An observer is at latitude 42.5° N and longitude 71° W, where the clocks show UT−5. Estimate the time of sunrise by the clock on 21 December. Ignore refraction and the size of the Sun's disc.

I solved it without the solution

AnswerThe Sun rises at about 7:16 by the clock, or 7:18 if the equation of time is ignored. The first official solution treats the small triangle near the horizon as if it were flat and gets 7:10, which is 8 minutes out at this latitude. The alternative solution printed with it uses the exact formula and gets 7:18.