Solar motion from Australia.

At noon throughout the year the Sun has a north-south and east-west motion around the meridian. Earliest/latest sunrises and sunsets do not occur at the solstices and the effect is more pronounced with decreasing latitude. This phenomenon is calculated for 25 Australian cities and the following observations are recorded:

1. The latest sunrise occurs before the June solstice and the earliest sunset after.

2. The earliest sunrise occurs before the December solstice and the latest sunset after.

3. The effect at the December solstice is more pronounced than for the June one.

4. The effect increases with decreasing latitude.

INTRODUCTION

An observer plotting the location of the Sun at noon over a year would notice it altering its position in two directions. It changes its elevation (angle above the horizon) and shifts east and west of the meridian (line passing from north to south over the viewer's head). These two displacements of the Sun in the sky create a figure eight shape.

This paper investigates the causes of these observations. Specific values are calculated with the use of simple algorithms. The perspective is an Earth-centred view.

The time from sunrise to the Sun's crossing of the meridian, and from the meridian to sunset depends on both the declination of the Sun (angular distance from the celestial equator) and the latitude of the observer. Even though the longest and shortest days are the same date for all places in Australia, the earliest and latest sunrise and sunset are latitudinally dependent. Darwin, as the most northerly Australian city, is selected as an example. Finally, the procedure is repeated mathematically to obtain a table of results for 25 Australian cities.

PLACE IN THE CURRICULUM

The astronomy material presented here encompasses the first two of Kepler's Laws of Planetary Motion, namely, that the planets move in elliptical orbits with the Sun at one focus, and an imaginary line joining the Sun to a planet sweeps out equal areas in equal intervals of time. In addition, an effect of the tilt of the Earth's spin axis to its orbital plane, other than the traditional one of its being the cause of seasons, is examined. The topic presents a varied treatment of reference frames, as the shift here is from laws and concepts couched in a view from outside the Earth to those of an observer on Earth.

One of the aims of the National Science Curriculum is, 'an interest in and understanding of the natural world' (National Science Curriculum, 1988:18) and the Sun has a great influence on this planet. Just some of the environmental applications of this work involve the placement of rooms in a home for natural lighting and warmth, the pitch of the roof over an outdoor living area (as well as its positioning) and the location of gardens and selection of vegetation for them.

The variation of the Sun over a year at noon could be observed in a lesson with free downloadable astronomical software. However, an extended activity of a plot with a shadow stick to a horizontal or vertical surface could be added in the junior or primary science curriculum. The details given should allow teachers to translate the concepts into simpler terms for their younger students but the analysis also encourages physics students to derive results for their own locality as a research or investigative topic. The theme provides a ideal context for the use of Information and Communication Technology within Science or ICT itself. This real situation is an opportunity for advanced mathematics students to incorporate trigonometric functions, parameterised curves, graphs of single effects and the compounding of sinusoidal ones.

NORTH-SOUTH MOTION

The elevation or north-south variation of the position of the Sun at noon over a year is a result of the tilt of the spin axis of the Earth to the vertical of its orbit. Since the spin axis of the Earth is inclined at 23.44[degrees] to the vertical of its orbital plane, the inclination of the ecliptic (path of Sun) is 23.44[degrees] to the celestial equator (circle through the stars directly over the equator of the Earth). This parameter is known as the obliquity of the ecliptic. Declination is the angle from the celestial equator moving away from it at right angles with + to the north and--to the south. The Sun crosses the celestial equator (0[degrees] declination) moving northwards at the March equinox, reaches its maximum northerly position over the Tropic of Cancer (23.44[degrees]N) at declination +23.44[degrees] at the June solstice, returns across the equator at the September equinox and reaches its most southerly point at the December solstice over the Tropic of Capricorn (23.44[degrees]S) of declination -23.44[degrees]. The dates vary a little depending on the closeness to a leap year, but in 2006 they were March 20, June 21, September 23 and December 22 (Dawes et al, 2005: 82) and these are the dates used here.

The tropical year is the length of time between successive March equinoxes. It is the basic unit for our calendar and is of 365.242 19 days duration. (Ridpath, 1989: 58) A simple calculation (Fletcher, 2002) to determine the declination of the Sun for each day is given by

declination = 23.44 sin {(360/365.242 19) (286 + D)} = 23.44 sin {0.986(286 + D)}. (1)

This algorithm is based on 0[degrees] declination at the March equinox. 286 is the number of days from the March equinox until December 31 of the previous year (2005 in this example). D is the day number, starting with 01 January (2006 in this example). (Figure 1)

[FIGURE 1 OMITTED]

Simple geometry can be used to show that the angle of elevation of the Sun on the meridian near noon is 90[degrees] minus the angular difference between the Sun's declination and an observer's latitude (Kaler, 2002: 43) and the result could be graphed for this specific latitude. This exercise is suitable for lower secondary students.

EAST-WEST MOTION

The east-west variation of the Sun from the meridian at noon depends on the two factors of an elliptical rather than a circular orbit for Earth (Kepler's First Law) and the 23.44[degrees] tilt. The two factors are independent but their effects are combined. The following derivation is from Yeow (2002:9-21).

Firstly, the tilt angle is taken as 0[degrees] and the elliptical effect treated. From Kepler's Second Law the speed of the Earth in its orbit is greatest at perihelion (closest point to the Sun) and least at aphelion (furthest point from the Sun), which in 2006 were January 4th and July 3rd respectively (Dawes et al, 2005: 82). From the point of view of the Earth the average angular movement of the Sun is:

360[degrees]/365.242 19 days = 0.986[degrees] per day.

If N is the number of days after perihelion, m = 0.986N gives the angular change from perihelion of a uniformly-moving Sun on a circle centred on the Earth.

The position of the real Sun uses the Earth at one focus of an ellipse. The following equation gives the angular change v of the actual Sun (Duffett-Smith, 1992: 87).

v = m + 360/[PI] e sin m = m + 1.926 sin m (2)

where e is the eccentricity of Earth's orbit and is 0.016 81 (Dawes et al, 2005: 82). The position is calculated each five days and then averaged to give the angular speed over time, compared with a uniformly moving Sun. (Figure 2)

[FIGURE 2 OMITTED]

A1 = m - v = -1.926 sin m (3)

is the angular difference between a circular and elliptical orbit for the Sun. A positive value gives a faster -moving Sun so that at noon it is west of the meridian. Conversely, a negative value gives a slower-moving Sun which at noon is east of the meridian. The signs have been changed for a southern hemisphere observer looking north. Figure 3 gives the position of the Sun relative to the meridian, based on the elliptical orbit only.

[FIGURE 3 OMITTED]

The second factor is now treated independently. A circular orbit is selected but the tilt is now considered. Movement on the celestial equator is the basis for clock time but the real Sun moves at an angle to this. At the intersection points of the ecliptic and the celestial equator, a 1[degrees] motion on the ecliptic approximates a 1[degrees] cos 23.44[degrees] = 0.917[degrees] on the equator. Further from these times, spherical geometry (Kaler, 2002: appendix 3) is required. For a circular orbit the real Sun and mean (uniformly-moving) Sun are in step at the equinoxes and solstices. If 0[degrees] is taken as the March equinox, this corresponds to angles of 0[degrees], 90[degrees], 180[degrees] and 270[degrees]. The maximum angular difference would occur between these angles. Select 45[degrees]. The angular movement on the ecliptic is given by arctan (0.917 tan 45[degrees]) = 42.52[degrees] (Mills, 1978: 214-216).

45[degrees] - 42.52[degrees] = 2.48[degrees] This is a sinusoidal variation of amplitude 2.48. While the period for the elliptical factor is one year, it is 6 months (182.6 days) for the obliquity. In 2006 there were 75 days between perihelion and the March equinox. One quarter of a year averages 91.3 days. If N is taken as previously, the effect here is (similar to Yeow, 2002: 20):

[A.sub.2] = 2.48 sin ((N - 75) 182.6/91.3). (4)

This factor is graphed in Figure 4.

[FIGURE 4 OMITTED]

The two effects are now combined.

A = [A.sub.1] + [A.sub.2]

A = -1.926 sin 0.986N + 2.48 sin ((N - 75) 182.6/91.3). (5)

This is graphed in Figure 5.

[FIGURE 5 OMITTED]

The curve has deep turning points near February 11 and November 03 and secondary changes about May 14 and July 26. (Meeus, 1991: 173)

ANALEMMA

The union of the north-south and east-west changes results in the construction of an analemma, which is the figure-eight shape created by plotting the path of the noon Sun over a year. The word is derived from the Greek for the pedestal of a sundial. The vertical axis is the declination of the Sun (Equation 1 and Figure 1), but in the southern hemisphere the values need to be reversed to show the shape of the locus of the Sun. The actual position in the sky is obtained by raising or lowering the entire curve to match the elevation at one's latitude. The horizontal axis gives an angle, with positive values being east and negative west of the meridian (Equation 5 and Figure 5).

[FIGURE 6 OMITTED]

The Sun is not on the meridian at noon at the solstices, so the analemma is skewed and does not line up with the vertical axis. (Kaler, 2002:169)

RISING AND SETTING TIMES

The sidereal rotation of the Earth is 0.997 269 68 days (Dawes et al, 2005:82) = 23 hours 56 minutes.

((23 x 60) + 56) minutes / 360[degrees] = 3.989 minutes per degree.

Thus, from the position of the Sun one can determine the time at which it crosses the meridian (Figure 7).

[FIGURE 7 OMITTED]

The algorithm for the time interval between rising and setting of a star and its transit over the observer's meridian for particular declinations is derived from spherical geometry (Kaler appendix 3) and is given by Ridpath (1989: 40).

cos (semi-diurnal arc) = - tan (declination) x tan (latitude) (6)

where the semi-diurnal arc is the angle that a star makes from either horizon to the meridian and the latitude is taken as negative for the southern hemisphere. The star can be the Sun and the angle from sunrise to the meridian or the meridian to sunset can be converted to a time in minutes by using the rate of movement of 3.989 minutes per degree. The following procedure is from Wagon, 1990, as mentioned by Yeow (2002: 21-24). Figure 8 is based on the semidiurnal time for the latitude of Darwin (12[degrees] 28' south).

[FIGURE 8 OMITTED]

Subtraction of the semi-diurnal time in Figure 8 from the minutes before or after noon in Figure 7 gives the actual time of sunrise (Figure 9), while addition gives the time of sunset (Figure 10).

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Around the June solstice the peak of the second graph precedes the trough of the first. In other words, the earliest sunset precedes the June solstice and the latest sunrise follows it. Also, for the December solstice, the earliest sunrise precedes it and the latest sunset follows. As the Sun is closer to the meridian at noon in June than in December, the separation of earliest sunrise-latest sunset is greater than for earliest sunset-latest sunrise, (Kaler, 2002:169) adjusted for the southern hemisphere.

The situation is next displayed in a close-up view for Darwin around the June solstice (June 20 in 2006, day 172 of the year in Figures 11 and 12).

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

The earliest sunset occurs on day 151,21 days before the June solstice, as seen on the second graph. From the first graph, the latest sunrise occurs on day 190, 18 days after the June solstice. The graphs may be taken as one as the horizontal axis values are in line. The vertical distance between the two curves is the length of daylight. The thick dotted line gives the shortest vertical distance (and therefore time) between the curves. This is the June solstice.

The deviation of earliest/latest sunrise/sunset increases from high to low latitude; that is, the effect is more pronounced near the equator.

THE AUSTRALIAN SITUATION

The above derivation was used for Darwin as it gives a good separation around the solstice. Twenty five other cities in Australia are now analysed in the same way (see Table below). The Australian Bureau of Statistics census data (2005) were used to determine areas with populations above 60,000. Those in close proximity to larger cities have been excluded. Alice Springs has been added for its centrality. The Sunshine Coast is given for Caloundra and the Gold Coast at Southport.

CONCLUSION

The elevation of the Sun at noon is affected by the obliquity of the ecliptic. This factor and the elliptical orbit of the Earth determine the Sun's position east or west of the meridian at noon. This paper treats these three contributions independently and then combines them to produce an analemma. The position of the Sun can be converted into a time before or after the meridian at noon. The length of time from sunrise to meridian cross or meridian cross to sunset is dependent on latitude. A combination of this time and that before or after the meridian at noon leads to a calculation of sunrise and sunset throughout the year for any latitude. This was performed for 25 Australian cities. Analysis of the data leads to the date for earliest and latest sunrise and earliest and latest sunset at each locality. The longest and shortest days are the same throughout Australia but these do not correspond to earliest/latest sunrise/sunset. This resulted in the following observations:

1. The latest sunrise occurs after the June solstice and the earliest sunset before.

2. The earliest sunrise occurs before the December solstice and the latest sunset after.

3. The effect at the December solstice is more pronounced than at the June one.

4. The effect increases with decreasing latitude.

REFERENCES

Australian Bureau of Census (2005). htta://www.abs.gov.au

Dawes G, Northfield P and Wallace K. (2005). Astronomy 2006 Australia, Quasar Publishing, Georges Hall.

Duffett-Smith, (1992). Practical Astronomy with your Calculator, 3rd edition, Cambridge University Press.

Fletcher D. (2002). Solar Declination, http://holodeck.st.usm.edu/vrcomputing/vic_t/tutorials/solar/ declination.shtml

Kaler B. (2002). The Ever-Changing Sky--A Guide to the Celestial Sphere, Cambridge University Press, Cambridge, 2002.

Meeus J. ( 1991 ). Astronomical Algorithms, Willmann-Bell, Inc., Richmond, Virginia.

Mills, H.R. (1978). Positional Astronomy and Astro-Navigation Made Easy, Stanley Thornes Ltd.

National Science Curriculum. (2008). Framing paper, National Curriculum Board, Carlton South.

Ridpath, Ian (ed) (1989). Norton's 2000.0 Star Atlas and Reference Handbook, 18th edition, Longman Group UK Ltd, Essex.

Wagon, S. (1990). Why December 21 is the Longest Day of the Year, Mathematics Magazine, 63, 307-311.

Yeow, T. S. (2002). The Analemma for Latitudinally-Challenged People, http://www.math.nus.edu.sg.aslaksen/projects/tsy.pdf, Singapore.

Keith Treschman is a Teacher at Brisbane Girls' Grammar School.
```Table) Earliest sunset before and latest sunrise after June
solstice and earliest sunrise before and latest sunset after
December solstice for 25 Australian cities.

EARLIEST      LATEST
SUNSET       SUNRISE
Days before   Days after
LATITUDE        June         June
CITY             [degrees].'S    Solstice      Solstice

Darwin              12.28           21            18
Cairns              16.55           18            14
Townsville          19.16           17            13
Mackay              21.09           16            11
Rockhampton         23.23           15            10
Alice Springs       23.42           15            10
Sunshine Coast      26.48           13             9
Brisbane            27.24           13             8
Toowoomba           27.33           13             8
Gold Coast          27.58           13             8
Perth               31.56           11             7
Newcastle           32.56           11             6
Bathurst            33.25           11             b
Sydney              33.52           11             6
Wollongong          34.25           11             6
Canberra            35.17           11             5
Albury              36.05           10             5
Bendigo             36.46           10             5
Ballarat            37.34           10             5
Melbourne           37.49           10             5
Geelong             38.08           10             5
Burnie              41.04            9             4
Launceston          41.26            9             4
Hobart              42.53            9             3

EARLIEST        LATEST
SUNRISE        SUNSET
Days before    Days after
December       December
CITY               Solstice      Solstice

Darwin                36            35
Cairns                32            29
Townsville            30            27
Mackay                28            25
Rockhampton           27            23
Alice Springs         27            23
Sunshine Coast        25            20
Brisbane              24            19
Toowoomba             24            19
Gold Coast            24            19
Perth                 22            16
Newcastle             21            16
Bathurst              21            15
Sydney                21            15
Wollongong            20            14
Canberra              20            14
Albury                20            13
Bendigo               19            13
Ballarat              19            12
Melbourne             19            12
Geelong               19            12
Burnie                17             9
Launceston            17             9
Hobart                16             9
```
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