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Time across the planets

Measurement · level 6, lesson 5 of 6

A question to start

Imagine you are living on Mars, and you keep an ordinary Earth alarm clock set for 7:00 a.m. On your first morning, the alarm rings just as the Sun comes up. A month later, the alarm rings at 7:00 a.m. again, and you look outside: it is the middle of the night.

Nothing is broken. A day on Mars, called a sol, is about 39 and a half minutes longer than a day on Earth. Each morning the Sun rises about 40 minutes "later" by your Earth clock, and after 30 days that adds up to nearly 20 hours.

This lesson is about how we measure time with the sky: why an Earth day is 24 hours, why the world is cut into time zones, why the Earth really turns once in less than 24 hours, and how long days and years are on Mars.

The idea

A day is a turn of the planet

A day is the time from one noon to the next, when the Sun is highest in the sky. It happens because the Earth rotates (spins) on its axis, an imaginary line through the North and South Poles.

Why 24 hours? Not because of anything in nature: the hour was made to fit the day. Historians trace the 24 hours back to ancient Egypt, where the daylight and the night were each divided into 12 parts. Splitting each hour into 60 minutes and each minute into 60 seconds goes back to the Babylonians, who counted in sixties. So "one day = 24 hours" is a definition. Today the second is defined much more precisely, by counting the vibrations of caesium atoms, and the day is very close to \(24 \times 60 \times 60 = 86\,400\) seconds.

Time zones: 15° per hour

The Earth turns 360° in a day, so in one hour it turns

\[360° \div 24 = 15°\]

When it is noon in one place, the Sun is highest there, but for a place 15° of longitude further west the Sun still has an hour to go. If every town used its own Sun time, clocks in towns a few hundred kilometres apart would disagree by many minutes, which was a real headache when railways started running fast trains. The answer was time zones: wide strips of the Earth where everyone agrees to use the same clock time, usually changing by one whole hour from one zone to the next. The idea was strongly pushed by the Canadian railway engineer Sandford Fleming in the 1870s and 1880s.

Canada spans six main time zones. In standard time (winter):

Zone Example Hours behind UTC
Pacific Vancouver 8
Mountain Calgary 7
Central Winnipeg 6
Eastern Toronto 5
Atlantic Halifax 4
Newfoundland St. John's 3½

UTC (Coordinated Universal Time) is the world's reference clock, roughly the time in London in winter. Newfoundland's half-hour zone is a reminder that zones are decided by people, not by the 15° rule alone; their borders also follow provinces and countries. In summer most of Canada moves its clocks one hour ahead (daylight saving time), though most of Saskatchewan does not.

To convert a time between zones, find the difference in hours and add (going east) or subtract (going west). It is like converting units: the clock time changes, the moment does not.

The Earth turns once in less than 24 hours

Here is a surprise. If you time the Earth's spin against the distant stars instead of the Sun, it turns once in about 23 hours 56 minutes 4 seconds. This is called the sidereal day ("sidereal" means "of the stars"). Why the difference?

While the Earth spins once, it also moves along its orbit around the Sun, about \(\tfrac1{365}\) of the way round. So after one full spin, the Sun is no longer in quite the same direction, and the Earth has to turn a little extra to bring the Sun back overhead.

Sun start: noon one spin later (23 h 56 min): not yet noon (the move along the orbit is drawn much bigger than 1/365)

How much extra? In a year of about 365.25 days the Earth goes 360° round the Sun, so in one day it moves about \(360° \div 365.25 \approx 0.986°\). Turning 0.986° takes about \(\tfrac{0.986}{360}\) of a day, which is about \(0.986 \div 360 \times 24 \times 60 \approx 3.9\) minutes. So the solar day (Sun to Sun) is about 4 minutes longer than the sidereal day (star to star). Our 24-hour clocks follow the Sun, because the Sun is what we live by.

Days and years on Mars

Mars also spins and orbits, at its own pace:

Earth Mars
spin, against the stars 23 h 56 min 4 s 24 h 37 min 23 s
solar day 24 h 24 h 39 min 35 s (one sol)
year about 365.25 days about 687 Earth days (about 669 sols)
tilt of the axis about 23.4° about 25.2°

Mars's tilt is close to Earth's, so Mars has seasons too, but because its year is nearly twice as long, each season lasts nearly twice as long. The gap between Mars's star day and its sol is only about 2 minutes, not 4, because Mars moves round the Sun more slowly, about 0.54° per sol.

Mission teams who drive rovers sometimes live on "Mars time" for the first months of a mission, so that they can work while their rover sleeps and plan its next sol. Their meetings start about 40 minutes later every Earth day, and after about 36 or 37 Earth days their schedule has slipped all the way round the clock. For a single time across the whole planet, scientists use Coordinated Mars Time, the mean solar time at Mars's prime meridian, which runs through a small crater called Airy-0.

A sol is often split into 24 "Mars hours", each \(\tfrac{88\,775}{24} \approx 3699\) seconds, or about 61.65 Earth minutes.

Worked examples

1. How much longer is a sol?

  1. A sol is 24 h 39 min 35 s \(= 24 \times 3600 + 39 \times 60 + 35 = 86\,400 + 2340 + 35 = 88\,775\) s (more exactly 88 775.244 s).
  2. An Earth day is 86 400 s. Difference \(= 2375\) s \(=\) 39 min 35 s.
  3. As a percentage: \(2375 \div 86\,400 \approx 0.0275\), so a sol is about 2.75% longer.

2. A Mars year in sols. A Mars year is about 686.98 Earth days.

  1. In seconds: \(686.98 \times 86\,400 \approx 59\,355\,000\) s.
  2. Divide by the length of a sol: \(59\,355\,000 \div 88\,775 \approx 668.6\) sols.

3. Across Canada. A plane leaves Toronto at 8:00 a.m. and flies for 5 hours to Vancouver. What time is it in Vancouver when it lands?

  1. In Toronto time it lands at 1:00 p.m.
  2. Vancouver is 3 hours behind Toronto, so the local time is 10:00 a.m.

4. Your age on Mars. You are 13 Earth years old. How many Mars years is that?

\(13 \times 365.25 = 4748.25\) days, and \(4748.25 \div 686.98 \approx 6.9\) Mars years. You would be about 6.

5. The drifting schedule. A rover team starts on Mars time, with their first meeting at 9:00 a.m. Earth time. After 10 sols, how far has their meeting moved on the Earth clock?

\(10 \times 2375.244 = 23\,752.44\) s \(= 6\) h 35 min 52 s. The meeting is now at about 3:36 p.m. Earth time.

Try it

  1. It is noon (standard time) in Toronto. What time is it in Vancouver, Winnipeg, Halifax and St. John's?
  2. How many degrees does the Earth turn in 1 hour? In 1 minute? In 4 minutes?
  3. How much longer, in hours, minutes and seconds, are 10 sols than 10 Earth days?
  4. How many sols pass in 30 Earth days?
  5. You are 14 Earth years old. How old are you in Mars years?
  6. How long, in Earth minutes, is a "Mars hour" (one twenty-fourth of a sol)?
  7. Mars moves about 0.538° along its orbit each sol. Use this to explain why its sol is about 2 minutes 13 seconds longer than its star day.
  8. In the question at the start, after how many Earth days does an Earth alarm clock come back into step with Mars sunrise?
Answers
  1. Vancouver 9:00 a.m. (3 h behind); Winnipeg 11:00 a.m.; Halifax 1:00 p.m.; St. John's 1:30 p.m.
  2. \(360 \div 24 = 15°\) per hour; \(15 \div 60 = 0.25°\) per minute; \(4 \times 0.25 = 1°\) in 4 minutes.
  3. \(10 \times 2375.244 = 23\,752.44\) s \(\approx\) 6 h 35 min 52 s.
  4. \(30 \times 86\,400 \div 88\,775.244 \approx 29.2\) sols.
  5. \(14 \times 365.25 \div 686.98 \approx 7.4\) Mars years.
  6. \(88\,775.244 \div 24 \approx 3699\) s, and \(3699 \div 60 \approx 61.65\) minutes.
  7. Mars has to turn an extra 0.538° to face the Sun again, which is \(\tfrac{0.538}{360}\) of a turn. \(\tfrac{0.538}{360} \times 88\,775 \approx 133\) s, about 2 min 13 s.
  8. The clock slips 2375.244 s per day, and needs to slip a full day, 86 400 s: \(86\,400 \div 2375.244 \approx 36.4\) Earth days.

Common mistake

Treating time like a decimal. 1 h 30 min is 1.5 h, not 1.3 h; 39 min 35 s is about 0.66 h, not 0.3935 h. Convert everything to seconds (or to one unit) before you subtract, divide or compare.

Adding instead of subtracting when going west. Going west, clocks go back. Check with a fact you know: when people in Toronto are eating lunch, people in Vancouver are just starting their morning.

Where it leads