# Which Day Does the Sun Rise Earliest in Singapore?

## Introduction

Singapore lies almost on the equator, so most people would expect the Sun to rise at more or less the same time each day of the year. In fact, the sunrise time varies between 6:46 a.m. and 7:17 a.m., with the earliest sunrise on November 1 and the latest on February 9.

In the same way, the sunset time varies between 6:50 p.m. and 7:21 p.m., with the earliest sunset on November 5 and the latest on February 13.

If you know about the equation of time, it takes on its maximum value (16m 25s) around November 3 and its minimum value (-14m 15s) around February 11. (There is a nice table of the equation of time and declination on freepages.pavilion.net/users/aghelyar/sundat.htm.)

If you want to understand this, you can read the paper The Analemma from a Tropical Point of View that I have written together with my honours student Shin Yeow TEO. This is still an early draft.

You may also want to check out the excellent Analemma site. (I have downloaded two great programs from this site, SunGraph and Analemma.)

## The rising analemma in the tropics

We can use the rising analemma to help us deduce the dates when the earliest and the latest sunrise occur. Figure 1 shows us the “analemma-rise” for the northern hemisphere, which is the time when the lowest part of the figure eight leaves the southeastern horizon. This happens at the same clock time every morning. The Sun’s position at this particular clock time will change throughout the year, along this analemma. When the Sun reaches the particular spot, which is the lowest part of this tilted analemma, the latest sunrise of the year occurs. Similarly, the earliest sunrise occurs when the Sun reaches the spot on the analemma that is the first point to leave the horizon every morning. It is clear from the diagram that the earliest sunrise does not occur at the summer solstice and the latest sunrise does not occur at the winter solstice. Instead the earliest sunrise occurs before the summer solstice, at around early June; and the latest sunrise occurs after the winter solstice, at around early January.

 Figure 1: Analemma-rise in the northern hemisphere

Now visualize the analemma-set in the western sky at a place in the northern hemisphere. At the same time each afternoon, the lowest part of the analemma dips below the horizon. On the calendar date when the Sun reaches this spot on the curve, the earliest sunset of the year occurs. Similarly, when the Sun reaches the spot on the curve that is the last to dip below the horizon, the latest sunset occurs. Figure 2 shows the analemma-set. Again, we see that the earliest sunset does not fall on the winter solstice, instead it occurs before the winter solstice, at around early December; the latest sunset falls after the summer solstice at around late June.

 Figure 2: Analemma-set in the northern hemisphere

The shortest day lies about midway between the dates of latest sunrise and earliest sunset. Similarly, the longest day lies about midway between the dates of the earliest sunrise and the latest sunset. All of these events rely on the fact that the analemma has a curve at either end. If it did not (if there were no equation of time), then the analemma would be a north-south line instead of a figure-eight. The earliest sunrise, longest day, and latest sunset would coincide on the same day, the summer solstice. The latest sunrise, the shortest day and the earliest sunset would coincide on the winter solstice. The angle between the axis of the rising analemma and the horizon is equal to the latitude of observer. Similarly the angle between the axis of the setting analemma and the horizon is equal to the latitude of the observer. Thus in Figure 3, the angle between the eastern horizon and the axis should be equal to the angle between the western horizon and the axis. Note that the points A, B, C and D represent the earliest sunrise, latest sunset, earliest sunset and latest sunrise, respectively. If the analemma were a perfect figure-of-eight, we would expect the number of days between A and the summer solstice to be equal to the number of days between B and the summer solstice. Similarly, the number of days between C and the winter solstice should be equal to the number of days between D and the winter solstice. However since the analemma is slightly distorted, there is no exact symmetry. Nevertheless, the deviation is very slight.

 Figure 3: Relationship between the axis and the horizon

The curve on the top of the analemma is smaller than the curve at the bottom of the analemma. This is expected as we can see that the 2 maxima of the equation of time graph are not of the same magnitude. As a result, the number of days in which the earliest sunset and the latest sunrise are away from the winter solstice may not be equal to the number of days in which the latest sunset and the earliest sunrise are away from the summer solstice. Since the angle between the analemma and the horizon changes with latitude, we would expect that the day on which the earliest sunrise occurs varies with latitude too. In the next section we will discuss more on the relationship of the latitude with the date of the earliest and latest sunrise.

It is important to note that the rising analemma for the southern hemisphere is orientated differently from that of the northern hemisphere. The rising analemma in the northern hemisphere has its bottom tipped down and the top tipped up. However, the rising analemma of the southern hemisphere has the bottom tipped up and the top tipped down. The dates of the summer solstice and winter solstice in the southern hemisphere are reversed from those of the northern hemisphere. Thus we would expect the earliest sunrise and the latest sunset to occur somewhere in December; the latest sunrise and the earliest sunset to occur somewhere in June. Figures 4 and 5 illustrate the case for the southern hemisphere.

 Figure 4: Analemma-rise in the southern hemisphere Figure 5: Analemma-set in the southern hemisphere

Now, let us compare the case for the equator with those of higher latitudes. For higher latitudes, the extrema of the sunrise and sunset are found near the tips of the analemmas. But at the equator, due to the rising and setting analemmas being horizontal, these extrema are located further away from the tips. As a result the earliest and the latest sunrise are located more than one month away from the winter solstice. The earliest sunrise occurs on November 3 and the latest sunrise occurs on February 10. Figure 39 illustrates why this is so. The earliest sunset and the latest sunset also fall on November 3 and February 10 respectively. This is because the setting analemma is actually a 180 degrees rotation of Figure 6. Therefore the point that rises first will also set first.

 Figure 6: Rising analemma at the equator

Notice that the two points that represent November 3 and February 10 on the analemma are actually the points farthest away from the line representing zero equation of time. Recall that the maximum magnitude of the equation of time is around 15 minutes. Therefore these two points are around 30 minutes away from each other in terms of equation of time. Thus this explains why the clock time of the earliest sunrise and the clock time of the latest sunrise are around 30 minutes apart, even though the equator experiences equal amount of daylight every day.

It may seem that the earliest sunrise only occurs once in the course of the year. But this is not true for all latitudes. In particular for the latitude of 5°N, we notice that the analemma has two humps as it rises. As a result there are two earliest points that leave the horizon together. When the Sun is at these two points, May 23 and October 24, a place at latitude 5°N experiences its two earliest sunrises of the year.

 Figure 7: Morning analemma at latitude 5°N

Next, observe that although the analemma curves sharply at the top and bottom, it is almost straight in the middle. Thus, if there exists a particular latitude that will result in this straight part of the analemma to be parallel to the horizon, we would expect a period of dates that will have the same clock time for the sunrise. In particular, a location at latitude 14°N satisfies this condition.

 Figure 8: Morning analemma at latitude 14°N

In fact, from about September 9 to September 30, while the Sun travels along this portion of the analemma, sunrise occurs at virtually the same time every morning. Each day the Sun appears on the horizon at a point progressively farther south in azimuth.

## Tug of War

From the section above we see that the tilt of the rising analemma and the width of the analemma determine which points leave the horizon first or last. The width of the analemma tends to move the earliest sunrise and the latest sunset away from the summer solstice towards the points on the analemma furthest away from the analemma axis, November 3 and February 10, respectively. However as the tilt of the analemma increases, the width effect is undermined by the tilt effect and the earliest sunrise and the latest sunset shift towards the summer solstice.

The tilt of the rising analemma is dependent on the latitude of the observer and the width of the analemma is dependent on the equation of time. Thus this means that there is a constant tug-of-war between the latitude of the observer and the equation of time in determining the dates of the earliest sunrise and latest sunset.

According to the equation of time, the Sun is fastest on November 3 and slowest on February 10. Thus if we only consider the equation of time, the earliest sunrise should be on November 3 and the latest sunset should be on February 10. However as latitude increases, the earliest sunrise and the latest sunset shift towards the summer solstice. As a result, if the equation of time were the dominating factor, the earliest sunrise would be near to November 3 and the latest sunset would be near to February 10. If the latitude of the observer were the dominating factor, the earliest sunrise and the latest sunset would be near to June 21. Specifically due to the orientation of the months on the analemma, the earliest sunrise would lie in late May to June 21 and the latest sunset would lie in mid July to June 21.

So when does the effect of one dominate that of the other? Let us look at the following table.

 Latitude Earliest Sunrise Latest Sunset Latest Sunrise Earliest Sunset 0°N Nov 3 Feb 10 Feb10 Nov 3 1°N Nov 1 Feb 12 Feb 9 Nov 4 2°N Oct 30 Feb 14 Feb 7 Nov 5 3°N Oct 28 Feb 16, Jul 20 Feb 5 Nov 7 4°N Oct 26 Jul 19 Feb 4 Nov 8 5°N Oct 24, May 23 Jul 17 Feb 2 Nov 10 6°N May 24 Jul 16 Jan 31 Nov 11 7°N May 25 Jul 15 Jan 30 Nov 12 15°N June 2 Jul 8 Jan 21 Nov 21

From the table, we see that with respect to the earliest sunrise, from latitude 0°N to 5°N, the equation of time component dominates. At latitude 5°N, both effects are on par, thus two earliest sunrises occur, one near to November 3 and one near to June 21. After latitude 5°N, the latitude component dominates and from then on, the earliest sunrise jumps to late May and progresses towards June 21.

With respect to the latest sunset, from latitude 0°N to 3°N, the equation of time component dominates. At latitude 3°N, both effects are on par, thus two latest sunsets occur, one near to February 10 and one near to June 21. After latitude 3°N, the latitude component dominates and from then on, the latest sunset jumps to mid July and progresses towards June 21.

Similarly we would expect the earliest sunset to be located near November 3 at low latitudes and progresses towards December 21 as latitude increases; and the latest sunrise to be located near February 10 at low latitudes and progresses towards December 21 as latitude increases. However the earliest sunset and the latest sunrise progress towards December 21 smoothly, unlike the earliest sunrise and the latest sunset, which progress towards June 21 with a jump in between. To explain this, note that for the earliest sunset, to progress from November 3 to December 21 only take approximately 1½ month. This is similar for the case of the latest sunrise. However for the earliest sunrise to progress from November 3 to June 21 requires approximately 4½ months. Thus a smooth progression is not possible and a jump resulted. The same applies to the latest sunset.

## Daylight Saving Time in Kuching

In Europe and the United States, the clocks are switched an hour forward in the summer. This converted time is known as Daylight Saving Time. The main purpose of Daylight Saving Time is to make better use of daylight and save energy. Energy use and the demand for electricity for lighting in our homes are directly connected to when we go to bed and when we get up. During the summer months, sunrise is very early in the morning. Without Daylight Saving Time, most people will still be asleep many hours after sunrise. By moving the clock ahead one hour, we can make use of this one-hour daylight to do work. In addition, Daylight Saving Time “makes” the sun “set” one hour later and therefore reduces the period between sunset and bedtime by one hour. This means that less electricity would be used for lighting and appliances late in the day.

In the United States, Daylight Saving Time begins on the first Sunday of April and ends on the last Sunday of October. In Europe, Daylight Saving Time begins on the last Sunday in March and ends on the last Sunday in October. Although there is a small irregularity in the actual date when Daylight Saving Time starts, it occurs during the summer. Equatorial and tropical countries (lower latitudes) do not observe Daylight Saving Time since the daylight hours are similar during every season, so there is no advantage to moving clocks forward during the summer.

In Kuching (latitude 1.55°N) from 1935 to 1941, the clocks were switched forward by twenty minutes from September 14 to December 14. This Daylight Saving Time in Kuching appears unusual. The existence itself is unexpected since we mentioned earlier that places in the tropics usually do not observer Daylight Saving Time. In addition the time period September 14 to December 14 do not correspond to any season markers. The time adjustment of twenty minutes is different from the usual one hour. However, this Daylight Saving Time is not chosen without reason.

Figure 9is obtained from the Mathematica package Calendrica. The solid curve in figure 9represents the time of sunrise for Kuching if there is no Daylight Saving Time. The local extrema A and C fall on February 8 and July 31 respectively. The local extrema B and D fall on May 16 and October 31 respectively. In fact February 8 and October 31 mark the latest and earliest sunrise in Kuching respectively. The time difference between the earliest sunrise and the latest sunrise is around 30 minutes.

Suppose we want to reduce the time between the earliest and latest sunrise, by forcing the earliest sunrise to fall on May 16 instead of October 31. Let us draw a horizontal line tangential to point B (May 16). This tangent line will cut the curve at two points, which we will name as E and F. If we shift the section of the curve from point E to F upward so that point E and F are in line with point A, we would be able to achieve our aim. In fact this is the idea behind the Daylight Saving Time in Kuching.

Figure 10 shows how the time of sunrise will change after incorporating the Daylight Saving Time. Point E and F correspond to September 14 and December 14 respectively. It is necessary to note that the time of sunrise at these two points, after the upward shift, are not exactly the same as that of point A (February 8). I suppose it is more convenient to incorporate Daylight Saving Time for a full 3-month period, rather than choosing two dates that exactly correspond to the time of sunrise of point A but result in an awkward duration. To explain why the time adjustment is twenty minutes, we need to look at figure 43 again. This time interval is simply the amount by which point E and F need to be shifted upward to be in line with A. In other words, this is the time difference between the time of sunrise on February 8 and the time of sunrise on May 16 (figure 10). As a result after incorporating Daylight Saving Time, the time difference between the earliest sunrise and the latest sunrise is reduced to around 20 minutes.

 Figure 9: Time of sunrise in Kuching
 Figure 10: Time of sunrise in Kuching with daylight saving time

## Resources

In order to use the software from the Analemma site, you need to know your longitude and latitude. To find the coordinates of your home town, use some of the links at the geography section at about.com.

You may also enjoy my course called Heavenly Mathematics & Cultural Astronomy.

Together with Tey Meng Khoon and Frederick H. Willeboordse of CITA, I have developed several interactive Java applets that I hope will help you understand the motion of the Earth and the Sun.

If you are the do-it-yourself type, you must check out the cool indoor analemma site. And don't forget to admire the classic analemma picture from Sky & Telescope.

The popular astronomy magazine Sky & Telescope used to have a great comic strip called SkyWise. One month SkyWise discussed Latest Sunrise, Earliest Sunset. Sky & Telescope no longer has a free archive, but you can buy the article or look it up in a library.

If you want to make a nice table of sunrise and sunset times for your location, you can go to the Sun or Moon Rise/Set Table for One Year page at the Astronomical Applications Department of the U.S. Naval Observatory. Their site has lots of interesting stuff!

I also have a Mathematica notebook about this. It uses the code of Nachum Dershowitz and Edward M. Reingold from their book Calendrical Calculations. Their Lisp functions were translated into the Mathematica package Calendrica by Robert C. McNally. Please note that this is the code from the first edition, which is freely available. The code from the new edition will only be available with the book.

Back to Helmer Aslaksen's page on Calendars in Singapore.

Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com

Web Server Statistics for Helmer Aslaksen, produced by Analog.

I use the W3C MarkUp Validation Service and the W3C Link Checker.