# THE SUN AND ITS DAILY AND ANNUAL PATHS

### Abstract

After some observations : shadow of the gnomon, salad bowl, slides from C.L.E.A. about the different places where the Sun rises or sets along the year (author : Daniel Toussaint), the participants will make a cardboard model of a sundial (author : Pierre Causeret), they will build a little sphere to understand the daily path of the Sun, the length of the day, the height of the Sun, its annual path through the constellations (authors : Béatrice Sandré and Cécile Schulmann)

First, we will have observed the results of one day registration of the positions of the Sun on Roland Szostak's salad bowl and the shadow of a vertical stick (gnomon) at different moments (for instance, every fifteen minutes). From that, we'll be able to determine the South direction and the instant of midday, and to see that on the floor, the progression of the Sun is not regular, whereas it is on the salad bowl.

So, let's study the problem of the daily path and understand how a sundial works.

### The apparent motion of the Sun at our latitudes

 At the moment of equinoxe (march, 20th and september, 23rd), the Sun rises at East and sets at West. Except those two dates, the Sun rises between South-East and North-East, and sets between South-West and North-West.The meridian plane is the vertical plane North-South. When the Sun crosses that plane, it is the highest in the day, and in the South direction. It is midday from the Sun (midday = middle of the day). At our latitudes, the Sun is never at the zenith (vertical).

The axis of the apparent motion of the Sun is parallel to the Earth axis, and so is pointed towards what is called the celestial North Pole.
That axis is vertical at the Earth poles and horizontal at the Earth Equator. For a particular place on Earth, the angle between that axis and the North horizontal direction an angle h equal to the latitude j .

### Solar time

First, the solar midday is defined as the instant when the Sun crosses the local meridian (see illustration beside). Then, one day is divided in 24 hours.

On the illustration, the line "15 h" represents the different positions of the Sun at 15.00 (solar time) during the year.

### Vertical gnomon or inclined style

Generally, a gnomon is a rudimentary sundial using the shadow of a vertical stick. It only allows to get midday according to the Sun.

On the other hand, if we direct the style (stick giving the shadow), towards a parallel to Earth axis, the shadow of the Sun at the same hour will keep the same direction.

### Equatorial sundial

 The principle of the equatorial sundial is easier to understand at the North Pole : a vertical style and an horizontal table graduated every 15° (15° x 24 = 360°). The table being parallel to the Equator, that type of sundial is called "equatorial sundial". It can be carried to any place : at the Equator, it will be an vertical table with an horizontal style, at a place (latitude j ) in the Northern hemisphere, the style makes with the North horizon an angle equal to j. The table is perpendicular to the style and so, makes with the vertical of the place an angle equal to j.

A disadvantage of that type of sundial is that it has to be graduated on the two sides. The shadow is projected on the Northern side during Spring and Summer, when the Sun is at the Northern part in relation to the Equator plane, and the other side will be used in Autumn and Winter.

### Horizontal or vertical sundials

 When we want to build a sundial, it is easier to have an horizontal or a vertical one. But the lines of the hours won't be any more drawn every 15°. They can be drawn from those of an equatorial sundial (see beside) or using trigonometry formulas.For an horizontal sundial (in a place at latitude j, the angle a between the hour line and the North-South direction is given according to the formula :tan a = sin j x tan h where h is the difference between the time and midday, with 15° each hour (h = 15° at 11.00 and 13.00, 30° at 10.00 and 14.00...)

 The style must always be parallel to the axis of the Earth and so make with the North horizon an angle equal to the latitude of the place.

### Legal time and solar time

#### LH = SH + longitude lag + time equation + 1 or 2 hours (Winter or Summer time)

with LH = legal time
SH = real local solar time (counting 12.00 for midday according to the Sun).
Longitude lag in minutes : it can be obtained by multiplying the longitude (in degrees) by 4 (the lag is of 24 hours for 360° that is 1 hour for 15° or 4 min for 1°). The longitude is negative when East of Greenwich and positive when West.

### We'll build a cardboard equatorial sundial.

Then, we can see the slides : where does the Sun rises ? sets ?
And why the paths of the Sun on the salad bowl are different according to the different days in the year ?

To understand all that, the best is to use a celestial sphere as people used to do it a few centuries ago. They are generally beautiful objects, but rather cumbersome… So we'll build little celestial spheres with : cardboard, a 12cm polystyrene sphere (for the sphere of the fixed stars), pins, paste and sellotape.

With thin cardboard (0,8 to 1 mm thick) :
First plane : horizontal plane, with a circular hole (radius : 7.5 cm), graduated every 10° for the azimuth (0° at the South, increasing in retrograde sense)
Vertical plane : half a ring (radius between 7.8 cm and 8.8 cm), graduated every 10° from 0° (on horizontal plane) to 90° (zenith). It will be stuck on a ring (radius between 7.8 cm and 8.8 cm) that will keep it vertical when set down on the horizontal plane.
Meridian plane : a ring (radius between 6.25 cm and 7.75 cm), on which will be stuck two half rings (radius between 6.25 cm and 7.25 cm) graduated from 0° to + 90° (North Pole) and from 0° to - 90° (South Pole) every 10° for the declination.
Equatorial plane : two half rings (radius between 6.25 cm and 7.25 cm), graduated in the retrograde sense, one from 0 h (South) to 12 h (North), the other from 12 h to 24 h for the hour angle. They will be stuck perpendicularly to the local meridian (the graduated side towards the North) after having positioned the sphere of the fixed stars.

In the thickness of the cardboard of the local meridian, we'll stick the two pins : one at + 90° (North Pole), the other at - 90° (South Pole), then push them into the polystyrene ball in such a way that it turns correctly in the ring. The graduations of right ascension have to be done on the sphere (for example every 15° for one hour) : one of these graduations will be chosen as g (0 h), and then increasing in the direct sense (the polar axis orientated from South towards North).
We can then draw on the sphere some remarkable stars (from their co-ordinates), twelve positions of the Sun along the year (so, we get the ecliptic) and perhaps the limits of the zodiacal constellations.

From that, we can understand, at any latitude, the path of the Sun in the sky at a special date (make a comparison with the path on the salad bowl), see where the Sun rises and sets at that moment, guess how long day and night will be, and notice the differences between the different moments in the year.