The Mathematics of Time

1. The Mathematics of Time A/P Helmer Aslaksen Dept. of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/

2. What is the goal of this talk? • Understanding the Heavens was among the major problems in all civilizations • Appreciate some of the problems involved and see how different cultures attempted to solve them • Appreciate the centrality of mathematics to human civilization

3. What is mathematics? • In ancient times, there was no divisions between mathematics, physics and astronomy • Until around 1850, astronomy was part of applied mathematics before it became astrophysics

4. What is time? • Year • Month • Day • Hour • Second • What is not in the above list? • Week • Minute

5. Year • Use heliacal rising (or setting) of stars to trace the backward motion of the Sun relative to the stars along the ecliptic • Revolution of the Earth around the Sun • Precession of the equinox • Sidereal year (365.256 days) vs. tropical year (365.242 days) • Hipparchos ca 150 BCE and Yú Xǐ (虞喜) ca 320 CE • Why do Indian astronomers use sidereal year?

6. Month • Sidereal month (27.32 days) vs. synodic month (29.53 days) • Synodic month varies between about 29.27 and 29.84 days. • Lunar months can start with the new Moon (China and South India), the first visibility of the lunar crescent (Islam) or the full Moon (North India) • Lunar months are 29 or 30 days

7. Lunar visibility • First visibility of the lunar crescent is an “unsolvable” scientific problem • Different Islamic countries tries to solve it in different ways • Islamic criteria tend to be “optimistic”

8. Types of calendars • Solar calendar • Lunar calendar • Lunisolar calendar

9. Solar calendars • Gregorian calendar • Basic unit is the day • Approximates the tropical year by adding leap days • Ignores the Moon • The year is 365 or 366 days

10. Leap days • Since the tropical year is a bit less than 365.25 days, we should add a leap day a bit less than ever fourth year • The Julian calendar had leap years every four years • Year n is a leap year if n is divisible by 4, but not by 100, or n is divisible by 400

11. Lunar calendar • Islamic calendar • Basic unit is the month • Ignores the Sun • The year is 12 months or 354 (sometimes 353 or 355) days • Muslim holidays are not tied to the seasons • No Muslims use the arithmetical Islamic calendar!

12. Lunisolar calendar • Chinese and Jewish calendars • Basic unit is lunar month • Approximates the tropical year by adding leap months • The year is 12 or 13 months. A 12-month year is 354 (sometimes 353 or 355) days. A 13-month year is 384 (sometimes 383 or 385) days • Holidays are tied to the seasons within a month

13. Leap months • A lunar consisting of 12 lunar months is about 12 x 29.5 = 354 days • A lunar years is about 365-354=11 days shorter than a tropical year • Lunisolar calendars have leap years about every third year • The Metonic or zhāng (章) cycle: 19 solar years is almost exactly 235 lunar months • Seven leap months in 19 years

14. Chinese New Year • Beginning of spring, lì chūn (立春), on February 4 is halfway between winter solstice and spring equinox • Chinese New Year is meant to approximate the beginning of spring • New Moon closest to Feb 4 • January 21 to February 21 • Subtract about 11 days, but if that would take us before January 21, add 19 days

15. The calendar in Chinese culture • Why were there more than 100 calendar reforms? • Foreign talent • The current Chinese calendar was designed by a German Jesuit • No linear year count in the Chinese calendar • Needham dismissed calendrical studies

16. Day • Nychthemeron is the 24 hour day • NOT the revolution period of the Earth! • Sidereal day (23h 56m 4s) vs. solar day • True solar day can be from 22s shorter to 30s longer than 24hr • Mean solar day • Equation of time • Analemma

17. Analemma picture

18. Rising and setting analemma

19. Sunrise and sunset in Singapore • Singapore lies almost on the equator, so we would expect the Sun to rise at the same time each day of the year • Sunrise varies between 6.46am and 7.17am, with the earliest on November 1 and the latest on February 9 • Sunset varies between 6.50pm and 7.21pm, with the earliest on November 5 and the latest on February 13 • The difference between the earliest and latest sunrise in 30 minutes, but the difference between the longest and shortest day is only 8 minutes • Ptolemy predicted, but could not measure this

20. Eccentricity

21. Why does obliquity matter?

22. Obliquity

23. Combined effect

24. Analemma

25. Globe

26. Globe with analemma

27. Hours • The Egyptians used temporal (unequal) hours that divided the day and night into 12 hours, whose lengths depends on the season • Hipparchos introduced equinoctial hours of equal length • Equal hours did not catch on until the fourteenth century • O’clock refers to equinoctial hours

28. Seconds • Tidal friction slows down the Earth’s rotation speed • The changes in the Earth’s rotations speed are not fully understood and cannot yet be predicted properly • Leap seconds are occasionally added at the end of the year to compensate