# Misseri-calendar - Simulation of the Sun Track

Author(s):

The axial tilt of the Earth is the angle between the Earth’s rotational axis and the normal vector to the orbital plane around the sun. The axial tilt is currently about 23.44°. The axis remains tilted in the same direction throughout the year; however, as the Earth orbits the Sun, the hemisphere tilted away from the Sun will gradually become tilted towards the Sun, and vice versa. Whichever hemisphere is currently tilted toward the Sun experiences more hours of sunlight each day. In the northern hemisphere, the maximum tilting towards the Sun is at summer solstice in June and the minimum at winter solstice in December. At equinoxes in September and March, the axial tilt does not have an effect on the observed Sun track.

As viewed from Iceland (65° N.)                                   As viewed from Rome (42° N.)

Figure 2:  Sun’s path in the sky, with points of sunrise and sunset, at June solstice (red), equinox (black), and December solstice (blue)

Figure 2 shows the paths of the Sun in the sky at the solstices and equinoxes as viewed from Iceland (65° N.) and Rome (42° N.). The horizontal circle denotes the horizon. The further north the location is situated, the greater is the inclination of the Sun’s path with respect to the horizon. At the June solstice, the Sun’s path as viewed in Iceland begins and ends further north, and the visible path of the Sun is therefore longer than it is when viewed in Rome on this same date. In contrast, the visible path of the Sun at the December solstice is shorter when viewed in Iceland than it is when viewed in Rome.

Click here for a simulation showing how the declination of the Sun varies over the course of a year
(from the Astronomy Education at the University of Nebraska-Lincoln website).

(Note: Animation requires the Flash plug-in, included for instance in the Google Chrome browser.)

We will now model the observed Sun track at 65° North, the latitude of the farm of Thorsteinn Surtur, and for comparison at 42° North, the latitude of Rome. The Mediterranean area was considered the middle of the Earth in medieval times, and Rome was the center of the Christian Church that decided upon the official calendar of the Western World. In Figure 3, the altitudes of the Sun at 65°N and 42°N through 24 hours at equinoxes in March and September are approximated by the functions:

${\rm{Iceland}}(x) = - (90 - 65) \cos\left (\frac{2\pi x}{360} \right )$

${\rm{Rome}}(x) = - (90 - 42)\cos\left (\frac{2\pi x}{360} \right )$

The 0°–360° scale on the horizontal axis denotes the placement of the Sun on the 360° horizon (the azimuth) during 24 hours, while the scale on the vertical axis denotes the altitude in degrees.

Figure 3: Simulation of the Sun track seen from Iceland at 65°N and from Rome at 42°N at equinoxes in March and September;  points A and B show location of sunsets for each location.

Figures 4 and 5 show simulations of the Sun track at the two solstices.

Figure 4: Simulated tracks of the Sun, seen from Iceland at 65°N and Rome at 42°N, at winter solstice when the declination, –23.44°, is subtracted from formulas for altitude of the Sun at equinoxes.

Figure 5: Simulated tracks of the Sun, seen from 65°N and from Rome at 42°N,  at summer solstice when the declination, +23.44°, is added to formulas for altitude of the Sun at equinoxes.

We notice in all three models that the track of the Sun is flatter at northern latitudes than it is closer to the equator. At winter solstice the sunset in Iceland at 65°N is at 200° on the horizon, while at summer solstice the sunset is at 340° (see Figures 4 and 5). The range of the sunset in Iceland during half a year is therefore about 140° out of the 180° possible range. For comparison, the approximated sunset in Rome at 42°N at winter solstice is at 241° on the horizon, and at summer-solstice it is at 300°. The range in Rome is thus only about 60°.

Notice also that the altitude of the Sun at 65°N at noon is 90°– 65° = 25° at the equinoxes (Figure 3). At summer solstice (Figure 5), the Sun’s altitude in Iceland is therefore 25°+ 23.4° = 48.4° at its highest position at noon, and the Sun is only –25°+ 23.4° = –1.6° below the horizon at its lowest position in the night. Since the Sun is so close to the horizon at that time, the night is bright. The official calendar for Iceland does not record darkness at 65°N from May 21 until July 23 [8]. For the same reason, in the wintertime when the Sun is only seen for a few hours a day, dawn breaks well before sunrise, and darkness does not fall right after sunset, thus extending the daylight time.

As the track of the Sun is flatter at northern latitudes than at those closer to the equator, the sunsets move faster along the horizon there than at places closer to the equator. The difference in the sunset location over the course of a week or two is much larger in Iceland at 65° than in Rome at 42°. The difference in June for a one degree difference in declination is about 1.4° in Rome (see the coordinates of points B and D in Figure 6), while at 65°N latitude the difference is nearly 6° (see coordinates A and C in Figure 6).

Figure 6. Placement of sunsets at 22.4° declination, and at maximum declination 23.4°.

This may easily be seen from many places on the west coast of Iceland, e.g. from Öskjuhlíð Hill in Reykjavík, as shown in Figure 7.

Figure 7. Photographs of sunset on June 11 2008 at 23:55 (left) and on June 19 2008 at 24:04 (right), taken from Öskjuhlíð Hill in Reykjavík, Iceland, at 64° North. (Photographer: The author)

The sunset moved 2.2° clockwise along the horizon in 8 days in June 2008. The situation in nature cannot be compared computationally to the models drawn above, as refractions in the air have to be taken into account. However, the pictures in Figure 7 illustrate how Thorsteinn Surtur may have discovered (in the year 955) the error in the calendar first adopted in the year 930.

Click here for a simulation showing how the direction in which the position of the Sun at sunrise or sunset changes over the course of the year
(from the Astronomy Education at the University of Nebraska-Lincoln website).

(Note: Animation requires the Flash plug-in, included for instance in the Google Chrome browser.)

[8] Almanak fyrir Ísland 2015. Reykjavík: University of Iceland.

Kristín Bjarnadóttir (University of Iceland), "Misseri-calendar - Simulation of the Sun Track," Convergence (July 2016)