The apparent path of the sun across the sky varies with time and place: it depends on the season of the year and on the latitude of the point on the earth’s surface from which the sun is observed. The actual shape of the sunpath is the result of two separate rotations of the earth: the orbit around the sun, and the 24-hour rotation around the axis of the north and south poles.
Because the polar axis is tilted in relation to the plane of the solar orbit, the sunlight received at any point on the earth’s surface varies during the year: this causes the seasons.
The actual angle of tilt, 23.45°, defines the limits of the tropics (23.45° north and south of the equator) and of the arctic and antarctic circles (23.45° from the poles). This is shown in Figure 5.1.
Figure 5.1 Rotations of the earth.
On two occasions—the spring equinox (about 21 March) and the autumn equinox (about 23 September)—day and night are equally long, and the midday sun is directly overhead at the equator; it rises exactly east and sets exactly west. During the northern summer, when the sun at noon is overhead somewhere between the equator and the tropic of Cancer, the direction of sunrise is somewhere between east and north, and the direction of the setting sun is between west and north. The greater the latitude, the more northerly the direction of sunrise and sunset. During the southern summer the sun is overhead between the equator and the tropic of Capricorn; sunrise occurs between east and south, sunset between west and south.
Figure 5.2 Apparent movement of the sun seen from London.
The latitude at which the sun is overhead at midday is the solar declination, . It is +23.45° for northern midsummer, 0° at the equinoxes, and !23.45° for northern midwinter. The value of is used to describe the time of year in formulae to find the sun’s apparent position.
At the north pole the path of the sun through the year is a gentle spiral. The sun becomes visible above the horizon in spring and remains continuously in the sky until autumn, reaching a maximum elevation of 23.45° on midsummer day. But seen from the equator the sunpath rises vertically; at the equinoxes it makes a semi-circle across the sky, passing through the zenith, the highest point. On the northern hemisphere’s midsummer and midwinter days, the sun seen from the equator reaches a height of 23.45° below the zenith.
36 The design of lighting
Figure 5.3 Apparent movement of the sun seen from the equator and the north pole.
At intermediate points on the earth’s surface the sunpaths slope across the sky dome, becoming more nearly horizontal as latitude increases. Figure 5.2 shows the daily paths of the sun as they appear in London, in comparison with Figure 5.3, which illustrates the paths at the equator and the north pole.
The maximum solar elevation (the angle of the sun above the horizon at midday) is (5.1) where ø is the latitude of the site (positive north of the equator, negative south of the equator).
The time of day indicated by the sun—the time that can be read from a sundial—is known as solar time. This is not necessarily the same as clock time, for three reasons:
• Solar time depends on longitude—sunrise occurs an hour later every 15° west—but all clocks in the same time zone are synchronized. In Britain, for example, clocks are based on the average solar time at Greenwich, 0° longitude: so at Falmouth, 5° west, solar time appears to lag by 20 minutes.
• Summer time or daylight saving may be adopted. This is alteration of clock time to give longer daylight in the evening at the cost of a later sunrise in the morning.
• And because the earth’s orbit around the sun is elliptical, solar time seems irregular by comparison with a perfect clock. In February, solar time is about 14 minutes ahead of a regular clock time; in November it is about 16 minutes late. This effect is called the equation of time.
Figure 5.4 Stereographic sunpath diagram for London, latitude 51° north.
Graphs showing the solar angles are called sunpath diagrams. The most common form is a stereographic projection in which the concentric circles give the angle of elevation above the horizon, with 90° in the centre, and the radial lines are compass bearings, or azimuth angles. Figure 5.4 illustrates the sunpaths for London in midsummer, midwinter and at the equinoxes. On 22 December the sun rises at about 8.15 am at 130° from north, rises to a maximum elevation of 15°, due south, then sets before 4 pm in the south-west. Sunpath diagrams are usually given in solar time, so they are valid for all sites along a line of latitude.
38 The design of lighting
Figure 5.5 Outline of building as seen from a point on the site, plotted on sunpath diagram.
The outlines of window openings or of buildings surrounding a site can be plotted on a sunpath diagram to discover when the sun would be obscured by obstructions. This is done by finding the angles of elevation and azimuth from a point on the site to the corners of the obstructions.
Various graphical aids are available to simplify the procedure. Figure 5.5 depicts the outline of a building plotted on the sunpaths for London. It shows that in mid-December sunlight would reach the point from sunrise until 11.15 am but be cut off for the remainder of the day.
The three-dimensional forms of buildings and their shadows can be cumbersome to plot graphically. An alternative is to use scale models with a light source to view or photograph the shadow patterns directly. A heliodon is a device that supports a model building, rotating it and tilting it as required, with an associated light source. The equipment is graduated so that the direction of the solar beam onto the building can be simulated for any time at any latitude. But no special equipment is necessary: a model of a building and its site can be illuminated with any suitable source, provided that the beam direction onto the model can be set up correctly. This can be done with a sundial — effectively a type of sunpath diagram used in reverse—adjacent to the model. Various publications on daylighting contain patterns for sundials; one especially convenient
version fits into the form of a matchbox. The light source must be small, such as an incandescent lamp, and must be located sufficiently far from the model for the divergence of rays to be minimal. The real sun, with the model outdoors, can be used.